MECHANICAL CHARACTERIZATION OF HYBRID AND FUNCTIONALLYGRADED ALUMINUM OPEN-CELL FOAMS WITH NANOCRYSTALLINE-COPPER
COATINGS
By
Yi Sun

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Civil Engineering—Doctor of Philosophy
2015

ABSTRACT
MECHANICAL CHARACTERIZATION OF HYBRID AND FUNCTIONALLYGRADED ALUMINUM OPEN-CELL FOAMS WITH NANOCRYSTALLINE-COPPER
COATINGS
By
Yi Sun
Cellular/foam materials found in nature such as bone, wood, and bamboo are usually
functionally graded by having a non-uniform density distribution and inhomogenous
composition that optimizes their global mechanical performance. Inspired by such naturally
engineered products, the current study was conducted towards the development of functionally
graded hybrid metal foams (FGHMF) with electrodeposited (ED) nanocrystalline coatings. First,
the deformation and failure mechanisms of aluminum/copper (Al/Cu) hybrid foams were
investigated using finite element analyses at different scales. The micro-scale behavior was
studied based on single ligament models discretized using continuum elements and the macroscale behavior was investigated using beam-element based finite element models of
representative unit volumes consisting of multiple foam cells. With a detailed constitutive
material behavior and material failure considered for both the aluminum ligament and the nanocopper coating, the numerical models were able to capture the unique behavior of Al/Cu hybrid
foams, such as the typically observed sudden load drop after yielding. The numerical models
indicate that such load drop is caused by the fracture of foam ligaments initiated from the rupture
of the ED nano-copper coating due to its low ductility. This failure mode jeopardizes the global
energy absorption capacity of hybrid foams, especially when a thick coating is applied. With the
purpose of enhancing the performance of Al/Cu hybrid foams, an annealing process, which
increased the ductility of the nanocrystalline copper coating by causing recovery,

recrystallination and grain growth, was introduced in the manufacturing of Al/Cu hybrid foams.
Quasi-static experimental results indicate that when a proper amount of annealing is applied, the
ductility of the ED copper can be effectively improved and the compressive and tensile behavior
of Al/Cu hybrid foams can be significantly enhanced, including better energy absorption
capacity. The behavior of Al/Cu hybrid foams under high-strain-rate condition was then
investigated using experiments on a split Hopkinson pressure bar. It was found that the ED nanocopper coating can also effectively enhance the energy absorption capacities of aluminum opencell foams under high strain rate. Similar to the quasi-static behavior, a large stress drop was
observed in the compressive response of Al/Cu hybrid foams under high strain rate, which was
accompanied by dramatic shattering of material. It is shown that a more ductile behavior and
better energy absorption performance under high strain rate condition can be also obtained by
introducing an annealing process. Finally, the manufacturing process of Al/Cu hybrid foams was
customized to fabricate FGHMF systems with two dimensional property gradients. The
performance of these FGHMFs at both quasi-static and dynamic conditions was evaluated.
Under quasi-static condition, two flexural type loading conditions were considered, namely, a
three point bending condition and a cantilever beam condition. The dynamic behavior of
FGHMFs was investigated by conducting drop weight tower tests on a three point bending setup.
It was found that the failure mechanism of hybrid metal foams can be modified and the
mechanical properties, such as stiffness and strength, and energy absorption capacities of hybrid
metal foams can be optimized under both quasi-static and dynamic conditions by introducing
strategically designed coating patterns. The presented novel approach and findings in this study
provide valuable information on the development of high performance hybrid and functionallygraded cellular materials.

Copyright by
YI SUN
2015

This is dedicated to my parents and my wife Danna,
for their understanding and support all along;
and to my daughter Olivia,
for all the happiness she brought to my life.

v

ACKNOWLEDGMENTS

Firstly, I would like to dedicate my gratitude to my advisor, Dr. Rigoberto Burgueño.
With his vision, enthusiasm and rigorous style of work, he showed me the key characteristics of
an excellent researcher. His encouragement and sound advices helped me find the right
directions during the difficult times. He has been inspired and enlightened me throughout my
PhD studies and helped to make it a great and interesting journey in my life.
I would like to thank the members of my PhD committee, Dr. Alejandro Diaz, Dr. Nizar
Lajnef and Dr. Srinivasan Arjun Tekalur for their valuable time and comments. Special thanks
are due to Dr. Srinivasan Arjun Tekalur for his guidance and help in the dynamic testing of
materials. I would like to thank Dr. Ilsoon Lee and Dr. Wei Wang for sharing with me their
expertise in the manufacturing of hybrid foam with nanocrystalline coatings. Also, I would like
to thank Dr. David Grummon for his guidance and help in the material processing.
I would like to thank Mr. Siavosh Ravanbakhsh and Mr. Joseph Nguyen,for their help at
Civil Engineering Laboratories. I would also like to thank Mr. Michael Rich, Mr. Per Ashkeland,
and Mr. Edward Drown at Composite Materials and Structures Center for their help in material
characterization and testing, I would also like to thank Dr. Stanley Flegler and Ms. Abby
Vandera at MSU’s Center of Advanced Microscopy for their help in electron microscopy
imaging.
I would like to thank Wassim Borchani, Andy Venderklok, David Gonzalez, Aishu
Venkat for their help in material testing. And I would like to thank Scott Sutton and Andrew

vi

Maximiuk for their help in material processing. I would also like to thank my colleagues for their
support during my stay at MSU.
Lastly, and most importantly, I would like to thank my parents and my wife Danna, for
their understanding and support. I would also like to thank my daughter Olivia for all the
happiness she brought to my life. To them, I dedicate this dissertation.
This study was mainly funded by the National Science Foundation under grant NSF grant
CMMI-0928835. Additional financial supports were obtained from the Department of Civil and
Environmental Engineering, and the Graduate School at Michigan State University. All
contributions are gratefully acknowledged.

vii

TABLE OF CONTENTS

LIST OF TABLES………………………………………………………………………………xi
LIST OF FIGURES…………………………………………………………………………….xii
CHAPTER 1 INTRODUCTION............................................................................................. 1
1.1
Motivation and Vision ..................................................................................................... 1
1.2
Background ..................................................................................................................... 2
1.2.1
Impact and Blast Effects on Structures ........................................................... 2
1.2.2
Energy Dissipation in Materials and Structures.............................................. 3
1.2.3
Nature’s Functionally Graded Materials ......................................................... 4
1.2.4
Tailoring Energy Dissipation Mechanisms ..................................................... 5
1.2.5
Electrodeposition of Metal Coatings .............................................................. 6
1.2.6
Hybrid Metal Foams ....................................................................................... 6
1.2.7
Annealing of Metals ........................................................................................ 7
1.3
Research Hypothesis and Objectives .............................................................................. 7
1.3.1
Hypothesis....................................................................................................... 7
1.3.2
Objectives ....................................................................................................... 8
1.4
Scope ............................................................................................................................. 10
1.5
Organization ................................................................................................................. 11
CHAPTER 2 BACKGROUND ............................................................................................. 12
2.1
General ......................................................................................................................... 12
2.2
Behavior of Open Cell Metallic Foams ........................................................................ 12
2.3
Electrodeposition of Metal Coatings ............................................................................ 18
2.3.1
Electrodeposition Process [37, 55] ............................................................... 19
2.3.2
Structures and Properties of Electrodeposited Metals .................................. 21
2.3.3
Properties of Electrodeposited Metals .......................................................... 22
2.4
Hybrid Metal Foams through Nanocrystalline Coatings ............................................. 24
2.4.1
State-of-the-art Review ................................................................................. 24
2.4.2
Manufacture of Al/Cu Hybrid Foam............................................................. 25
2.4.3
Mechanical Behavior of Al/Cu Hybrid Foam with Nano-coating ................ 28
2.5
Annealing of Deformed Metals ..................................................................................... 31
2.6
Modeling and Simulation of Open-cell Foams ............................................................. 34
2.7
Concepts of Stress-wave Propagation in Solids ........................................................... 38
2.8
Functionally Graded foam materials ............................................................................ 42
CHAPTER 3 MODELING AND SIMULATION OF QUASI-STATIC COMPRESSIVE
BEHAVIOR OF AL/CU HYBRID FOAMS ............................................................................ 47
3.1
Overview ....................................................................................................................... 47
3.2
Material Model ............................................................................................................. 48
3.3
Section flexural response .............................................................................................. 50
3.4
Single Ligament Modeling ............................................................................................ 53
viii

3.4.1
Inelastic Buckling of Single Ligaments ........................................................ 56
3.4.2
Effect of Coating Ductility............................................................................ 60
3.5
Multi-cell Modeling ...................................................................................................... 65
3.5.1
Foam Geometry ............................................................................................ 66
3.5.2
Ligament Geometry ...................................................................................... 67
3.5.3
Boundary Conditions .................................................................................... 68
3.5.4
Contact Simulation........................................................................................ 69
3.5.5
Size Effect in Multi-cell Models ................................................................... 71
3.5.6
Quasi-static Compressive Response ............................................................. 73
3.5.7
Effect of Coating Ductility............................................................................ 76
3.6
Discussion ..................................................................................................................... 79
3.7
Conclusions ................................................................................................................... 81
CHAPTER 4 EFFECT OF COATING DUCTILITY ENHANCEMENT ....................... 83
4.1
Overview ....................................................................................................................... 83
4.2
Annealing Procedure .................................................................................................... 84
4.3
Mechanical Testing and Characterization ................................................................... 86
4.4
Results ........................................................................................................................... 88
4.4.1
Crystallite Size .............................................................................................. 88
4.4.2
Compressive Response ................................................................................. 90
4.4.2.1 Effect of Annealing Temperature ......................................................................... 94
4.4.2.2 Effect of Annealing Time on Compressive Behavior ........................................... 95
4.4.3
Tensile Response .......................................................................................... 99
4.5
Discussion ................................................................................................................... 102
4.6
Conclusions ................................................................................................................. 104
CHAPTER 5 ANALYTICAL SOLUTION FOR HYBRID FOAM ................................ 106
5.1
Modulus of Hybrid Foam ............................................................................................ 111
5.2
Analytical Solution for Hybrid Foams with Ductile Coating ..................................... 113
5.2.1
Plastic Stress ............................................................................................... 113
5.2.2
Results ......................................................................................................... 116
5.3
Analytical Solution for Hybrid Foams with Low Ductility Coating ........................... 119
5.3.1
Stress at Fracture ......................................................................................... 120
5.3.2
Post-peak Stress .......................................................................................... 122
5.3.3
Results ......................................................................................................... 126
5.4
Conclusions ................................................................................................................. 128
CHAPTER 6 CHARACTERIZATION OF DYNAMIC BEHAVIOR OF AL/CU
HYBRID FOAMS ..................................................................................................................... 129
6.1
Overview ..................................................................................................................... 129
6.2
High Strain Rate Experiments .................................................................................... 130
6.2.1
SHPB Tests ................................................................................................. 130
6.2.2
Quasi-static Experiments ............................................................................ 133
6.2.3
Compressive Behavior of Hybrid Foams .................................................... 133
6.2.4
Effect of Electrodeposited Copper Coating ................................................ 136
6.2.5
Effect of Annealing ..................................................................................... 141
6.2.6
Results for Effect of Annealing .................................................................. 142
ix

6.3
6.4

Discussion ................................................................................................................... 146
Conclusions ................................................................................................................. 149

CHAPTER 7 STRUCTURAL CHARACTERIZATION AND DESIGN OF
FUNCTIONALLY GRADED AL/CU HYBRID FOAMS .................................................... 151
7.1
Overview ..................................................................................................................... 151
7.2
Behavior of Al/Cu Hybrid Foam ................................................................................. 152
7.3
Selective Electrodeposition of Nc-Cu ......................................................................... 153
7.4
Quasi-static Behavior of Functionally Graded Al/Cu Hybrid Foams ........................ 157
7.4.1
Design of Functionally Graded Hybrid Foam ............................................ 157
7.4.2
Test Setup (3-point bending and cantilever beam) ..................................... 161
7.4.3
Results ......................................................................................................... 163
7.4.3.1 Three-point Bending Tests .................................................................................. 163
7.4.3.2 Cantilever Beam Tests ........................................................................................ 167
7.5
Dynamic Behavior of Functionally Graded Al/Cu Hybrid Foams ............................. 170
7.5.1
Design of Coating Pattern ........................................................................... 170
7.5.2
Test Setup (Drop Tower) ............................................................................ 174
7.5.3
Results and Discussion ............................................................................... 176
7.6
Discussion ................................................................................................................... 181
7.7
Conclusions ................................................................................................................. 181
CHAPTER 8 CONCLUSIONS ........................................................................................... 183
8.1
Research Significance ................................................................................................. 183
8.2
Conclusions ................................................................................................................. 183
8.3
Future Research .......................................................................................................... 185
8.3.1
Optimized Material System ........................................................................ 185
8.3.2
Multi-scale Modeling .................................................................................. 185
8.3.3
Ductility Enhancement................................................................................ 186
8.3.4
Wave Propagation through Functionally Grade Metal Foams ................... 186
REFERENCES.. ........................................................................................................................ 188

x

LIST OF TABLES

Table 2-1 Equations for the Mechanical Properties of Open-cell Metal Foams........................... 18
Table 3-1 Material model parameters for aluminum (Al) and copper (Cu). ................................ 49
Table 3-2 Energy absorption of single ligament models from 40 PPI foam with 60 µm Cu
coating (Δ = 0.6L). ........................................................................................................................ 65
Table 3-3 Non-linear stiffness of spring elements for ligament contact simulation. .................... 70
Table 3-4 Comparison of stress values between results from experiment and FE simulations. ... 76
Table 3-5 Energy absorption of multi-cell models of 40 PPI Al/Cu foams up to 50% strain ...... 79
Table 4-1 Test Matrix ................................................................................................................... 86
Table 4-2 Information for Al and Al/Cu foam samples ................................................................ 87
Table 4-3 Summary of results from XRD measurements on Al/Cu hybrid foam specimens....... 89
Table 4-4 Width of intermetallic compound with different annealing time ................................. 98
Table 5-1 Input parameters for analytical solution ..................................................................... 117
Table 5-2 Contributions for moment capacity of the fractured ligament cross-section ............. 125
Table 5-3 Material properties used for analytical model of Al/Cu hybrid foams ....................... 126
Table 5-4 Initial stress peak and post-peak stresses for Al/Cu hybrid foams with different coating
thicknesses .................................................................................................................................. 127
Table 6-1 Summary of results from all experiments .................................................................. 136
Table 7-1 Information of Al/Cu hybrid foams for three-point bending and cantilever beam tests
..................................................................................................................................................... 159
Table 7-2 Results summary of tests ............................................................................................ 167
Table 7-3 Sample design for drop tower tests ............................................................................ 174

xi

LIST OF FIGURES

Figure 1-1 Compressive behavior of a typical metal foam ............................................................. 4
Figure 1-2 Schematic of the research motivation and the integration across scales for functionally
graded hybrid foam systems ........................................................................................................... 8
Figure 2-1 Metallic open-cell foam .............................................................................................. 12
Figure 2-2 Kelvin’s tetrakaidecahedron cell structure .................................................................. 13
Figure 2-3 Compressive behavior of open-cell metal foams (Adapted from Gibson and Ashby
[13])............................................................................................................................................... 15
Figure 2-4 Tensile behavior of open-cell metal foams (Adapted from Gibson and Ashby [13]) . 15
Figure 2-5 Cubic unit cell (Adapted from Gibson and Ashby [13]) ............................................. 17
Figure 2-6 Electrolytic cell for electrodeposition of metal ........................................................... 20
Figure 2-7 Hall-Petch relationship ................................................................................................ 23
Figure 2-8 Schematic of electrodeposition setup .......................................................................... 26
Figure 2-9 SEM images of hybrid Al/Cu foams: a) SEM image of micro-structure of hybrid
Al/Cu foams with ligament cross sections exposed; b) SEM image of cross-section of a single
ligament of hybrid Al/Cu foam ..................................................................................................... 26
Figure 2-10 Comparison of coating thickness estimates to experimental measurements ............ 28
Figure 2-11 Typical stress-strain curves of uncoated foam samples and with nickel coating
(Adapted from Bouwhuis et al. [6]) ............................................................................................. 30
Figure 2-12 Compressive stress-strain response of 40 ppi plain Al foam and copper coated Al/Cu
hybrid foams [8]............................................................................................................................ 31
Figure 2-13 Recrystallization temperature for various copper materials (Adapted from Sherlin
and Bjelland [69]) ......................................................................................................................... 33
Figure 2-14 Relative ultimate strength loss and relative total elongation gain at 23 ˚C for
electrodeposited copper foil (Adapted from Merchant [45]) ........................................................ 34
Figure 2-15 Transmission of disturbance from atom to atom....................................................... 38
Figure 2-16 Wave propagation in a bar ........................................................................................ 39

xii

Figure 2-17 Reflection and refraction of longitudinal elastic wave encounters different medium
....................................................................................................................................................... 41
Figure 2-18 Longitudinal wave encountering boundary between medium A and medium B ...... 41
Figure 3-1 Stress-strain curve and pictures of 40 PPI plain Al foam and Al/Cu hybrid foam with
60 μm coating under compressive loading [8]. ............................................................................. 47
Figure 3-2 Behavior of aluminum (Al) 6101-T6 and electrodeposited copper (Cu) from literature
[66, 80] and corresponding material models used in the finite element simulations .................... 50
Figure 3-3 Results from sectional analyses on the mid-span sections of 40 PPI ligaments. (a)
Characteristic moment-curvature response of an Al/Cu hybrid foam. (b) Moment-curvature
response of mid-sections of Al/Cu hybrid foam ligaments with different coating thickness. ...... 52
Figure 3-4 Schematic of the bending strain and stress profiles in the cross-section of a ligament
in a Al/Cu hybrid foam ................................................................................................................. 53
Figure 3-5 Single ligament models for 40 PPI open-cell foam. (a) Uncoated ligament model; (b)
Coated ligament model ................................................................................................................. 54
Figure 3-6 Two boundary conditions used for single ligament models. (a) Free rotation boundary
condition; (b) Fixed-rotation boundary condition......................................................................... 55
Figure 3-7 Maximum principal strain profiles on the core surface of the free-rotation single
ligament model at Δ = 0.5 L .......................................................................................................... 58
Figure 3-8 Maximum principal strain profiles on the core surface of the fixed-rotation single
ligament model at Δ = 0.5 L .......................................................................................................... 59
Figure 3-9 Maximum principal strain profiles of 40 PPI single ligament models coated with 60
μm Cu with different ductility capacity at Δ = 0.5 L .................................................................... 60
Figure 3-10 Von-Misses stress contours of 40 PPI ligament models coated with 60 μm Cu with
different ductility capacities at different deformation states......................................................... 61
Figure 3-11 Load-displacement response of single 40 PPI ligaments with different ductility Cu
coatings under compression .......................................................................................................... 63
Figure 3-12 Internal Energy-displacement response of single 40 PPI ligaments with different
ductility Cu coatings under compression ...................................................................................... 64
Figure 3-13 Multi-cell model with rendered cross-section (4×4×4 cells). (a) Front view; (b)
Isotropic ........................................................................................................................................ 67
Figure 3-14 Single ligament in multi-cell models with rendered section: (a) uncoated ligament;
(b) coated ligament ....................................................................................................................... 68

xiii

Figure 3-15 Quasi-static compressive stress-strain response from beam-element based finite
element models with different sizes and boundary conditions ..................................................... 71
Figure 3-16 Simulated and experimental quasi-static compressive stress-strain response for 40
PPI plain Al and Al/Cu hybrid foams ........................................................................................... 73
Figure 3-17 Simulated collapse progression of coated (120 μm) foam under uniaxial quasistatic
compression; (a) Low ductility coating; (b) High-ductility coating. ............................................ 74
Figure 3-18 Simulated quasi-static compressive stress-strain response of Al/Cu hybrid foams
with low and high coating ductility .............................................................................................. 77
Figure 3-19 Simulated and experimental quasi-static compressive stress-strain response for 20
PPI plain Al and Al/Ni hybrid foams ............................................................................................ 80
Figure 4-1 Naming convention of test samples ............................................................................ 85
Figure 4-2 Images of hybrid Al/Cu foams samples with uniform coating in a) Uniaxial
compressive test; and b) Uniaxial tensile test ............................................................................... 87
Figure 4-3 First two peaks of XRD patterns of copper (Cu) coated aluminum (Al) foam
specimens with and without annealing ......................................................................................... 89
Figure 4-4 Compressive stress-strain responses of Al/Cu hybrid foams with different coating
thicknesses .................................................................................................................................... 91
Figure 4-5 Sequential images from the compression testing of Al/Cu hybrid foam samples of (a)
C90 and (b) C90-HT-A1 ............................................................................................................... 92
Figure 4-6 SEM images of ligament deformation in 40 PPI Al/Cu hybrid foam samples (50 μm
copper coating) compressed after the onset of plastification to about 6 % strain: (a) without
annealing; (b) with 1 hour annealing at 400 °C ............................................................................ 94
Figure 4-7 Compressive response of Al/Cu hybrid foams annealed at different temperature...... 95
Figure 4-8 Images from the compressive testing of sample C60-LT-A1 at 60 % and 80 % strain
....................................................................................................................................................... 95
Figure 4-9 Compressive behavior of Al/Cu hybrid foams with different annealing time ............ 96
Figure 4-10 SEM images of Al-Cu interface in Al/Cu hybrid foams after annealing time of (a) 5
minutes; (b) 1 hour; and (c-d) 2 hours .......................................................................................... 97
Figure 4-11 Images from the tensile testing of foam sample C30-HT-A1 ................................... 99
Figure 4-12 Tensile stress-strain response of Al/Cu hybrid foams............................................. 101
Figure 4-13 Comparison of energy absorption capacities for different foam samples: (a) per unit
volume; (b) per unit mass. The error bars represent the standard deviations ............................. 102
xiv

Figure 5-1 Geometry of Unit Cell [71] ....................................................................................... 107
Figure 5-2 Simplification of unit cell model............................................................................... 108
Figure 5-3 Cross section of composite ligament in hybrid foam ................................................ 109
Figure 5-4 Plastic hinge location ................................................................................................ 113
Figure 5-5 Comparison between experimental data and analytical solution .............................. 117
Figure 5-6 Stress strain curve of Al/Cu hybrid foam (RD=6%, RD’=10.7%) with different
anisotropy.................................................................................................................................... 118
Figure 5-7 Typical behavior of Al/Cu hybrid foams .................................................................. 120
Figure 5-8 Strain and stress across ligament cross section ......................................................... 120
Figure 5-9 Schematic of fractured ligament cross-section of Al/Cu hybrid foam ...................... 123
Figure 5-10 Schematic of a circular segment ............................................................................. 123
Figure 5-11 Comparison of analytical solution and experimental data on the compressive
behavior of brittle Al/Cu hybrid foams ....................................................................................... 127
Figure 6-1 Schematic of split Hopkinson pressure bar (SHPB) setup ........................................ 131
Figure 6-2 Cross section pictures of (a) uncoated circular-shaped (5 mm thick with a diameter of
15.9 mm) specimen for dynamic experiments and (b) Cu-coated square-shaped specimen (25.4
mm × 25.4 mm × 5 mm) for quasi-static experiments................................................................ 131
Figure 6-3 Typical oscilloscope records from SHPB experiments ............................................. 132
Figure 6-4 Schematic of typical compressive behavior of Al/Cu hybrid foams with peak stress
(  P ), lowest stress in collapse region (  L ), densification stress (  D ) and densification strain

(  D ) ............................................................................................................................................ 134
Figure 6-5 Quasi-static compressive behavior of Al/Cu hybrid foams (3.3×10-3 s-1) ................ 138
Figure 6-6 High strain rate compressive behavior of Al/Cu hybrid foams (2.8×103 s-1) ........... 138
Figure 6-7 High strain rate compressive behavior of Al/Cu hybrid foams (4.9×103 s-1) ........... 139
Figure 6-8 Initial peak stress and lowest stress level in the compressive behavior of Al/Cu hybrid
foams under different strain rates ............................................................................................... 141
Figure 6-9 Compressive behavior of annealed specimen and un-annealed specimen with nominal
copper coating thicknesses of 30 μm under high strain rate loading (4.9×103 s-1) ..................... 143

xv

Figure 6-10 Compressive behavior of annealed specimen and un-annealed specimen with
nominal copper coating thicknesses of 60 μm under high strain rate loading (4.9×103 s-1) ....... 144
Figure 6-11 Compressive behavior of annealed specimen and un-annealed specimen with
nominal copper coating thicknesses of 120 μm under high strain rate loading (4.9×103 s-1) ..... 144
Figure 6-12 Images of (a) un-annealed and (b) annealed Al/Cu hybrid foam specimens with 120
μm nominal copper coating thickness during high-rate compressive loading ............................ 145
Figure 6-13 Comparison of energy absorption capacity for quasi-static and high strain rate
experiments ................................................................................................................................. 147
Figure 6-14 Comparison of stress drop ratio for quasi-static and high strain rate experiments . 147
Figure 6-15 Comparison of energy absorption efficiency for quasi-static and high strain rate
experiments ................................................................................................................................. 148
Figure 7-1 Compressive and tensile behavior of plain Al and hybrid Al/Cu foams with uniform
coating ......................................................................................................................................... 154
Figure 7-2 Manufacturing of graded hybrid foam using a) Partial immersion method and b)
Masking -Washing method ......................................................................................................... 155
Figure 7-3 Hybrid foams with two dimensional graded coating pattern .................................... 156
Figure 7-4 Foam samples with graded reinforcement: a) bottom reinforced sample for three-point
bending test; b) sample for cantilever beam test with three stepped reinforcement ................... 158
Figure 7-5 Topology optimized reinforcement design: a) For three-point bending test; b) for
cantilever beam test..................................................................................................................... 161
Figure 7-6 Test setup of: a) Three-point bending test; and b) Cantilever beam test. ................. 162
Figure 7-7 Images of foam samples in three point bending tests ............................................... 163
Figure 7-8 Bottom displacement vs. time from three point bending tests .................................. 164
Figure 7-9 Force vs. loading tip displacement from three point bending tests ........................... 164
Figure 7-10 Force vs. displacement at sample bottom center from three point bending tests .... 165
Figure 7-11 Images of foam samples in cantilever beam tests ................................................... 168
Figure 7-12 Results from cantilever beam tests .......................................................................... 169
Figure 7-13 Optimized design of structural panel with Al/Cu hybrid foam core [110] ............. 172
Figure 7-14 Design of Al/Cu hybrid foams for drop tower tests ................................................ 172

xvi

Figure 7-15 Test setup for drop tower test .................................................................................. 175
Figure 7-16 Sample loading setup for drop tower test ................................................................ 176
Figure 7-17 Force-displacement curves for samples in drop tower tests .................................... 177
Figure 7-18 Energy versus time curves for samples in drop tower tests .................................... 178
Figure 7-19 Comparison of energy absorption capacity of foams with similar solid volumn ... 179
Figure 7-20 Comparison of energy absorption capacity of foams with similar weight .............. 180

xvii

CHAPTER 1 INTRODUCTION
1.1

Motivation and Vision
The effect of impact and blast loads on structures can lead to devastating damage to

structures and their occupants alike. Thus it is of great interest to protect people and the
enclosures that they use (vehicles, buildings) from the natural, accidental or purposeful effects of
explosions and impacts. An approach towards this goal is to design materials and protective
devices that mitigate the effect of impact and blast loads by through energy dissipation and
energy management (i.e., reduction in magnitude and direction.) In this effort, metallic foams
have attracted great interest due to the large amount of energy dissipated during their plastic
deformation under compressive loading. Numerous studies have been conducted on traditional
metallic foams (e.g., aluminum foams) and their mechanical behavior and performance are
understood to a great extent. However, while the use of metal foams as a resisting
material/structure has been recognized, the design of the material itself as well as structural
components based on metal foams has been challenging. Lately, increased attention has been
paid to the development of strategically engineered functionally-graded metal foam systems with
enhanced energy absorption capacity or the ability to tailor the propagation of stress waves
through the foam [1-3]. While methods to fabricate functionally-graded foams by changing the
pore size and density distribution in the foam have been reported [4, 5], an alternative way with
great potential is to create hybrid foam systems by reinforcing the conventional open-cell foam
with nano-coatings through electrodeposition [6-8]. The effectiveness of manipulating the
properties of open-cell metallic foams using electrodeposited nano-crystalline coating has been
demonstrated and reported [6-8]. However, the unique behavior of such hybrid foams has not
been fully understood. In addition, few studies have been conducted to evaluate the performance
1

of functionally graded hybrid foam systems manufactured using such methods, and the
efficiency of these new hybrid material systems in the management of stress-waves has not been
addressed.
1.2

Background

1.2.1 Impact and Blast Effects on Structures
Attention to the design of buildings and vehicles again blast and impact loads from events
such as explosions has seen a recent increase due to the growing threat of terrorist activities [911]. The blast wind, debris and fires caused by the explosion of a bomb may cause catastrophic
damage on the structural components of a building and can lead to injuries and even death of the
occupants or people nearby. When an explosion occurs, a large amount of energy is rapidly
released and usually generates very hot and dense gases under high pressure. The expansion of
the hot gas creates a layer of compressed air called a blast wave.
A pressure profile for a blast wave after arrival to a resisting structure consists of a
positive duration phase, during which the pressure is higher than ambient level, followed by a
negative duration, in which the pressure is below ambient level [9, 12]. When the shock front
from an explosion encounters a structural component, such as a wall, it is instantly reflected;
which may significantly amplify the incident pressure level, depending on the orientation of the
structure with respect to the incident wave. The impulse from the blast will cause stresses to
propagate through the structural component in the form of an compressive stress wave. This
compression wave is reflected as a tensile wave when it reaches the rear end of the structural
member on which it is acting. When the compressive wave or tensile wave exceeds the material

2

capacity, the structural element may fail due to fracture. Continuation and accumulation of this
phenomena may cause partial collapse in a building.
1.2.2 Energy Dissipation in Materials and Structures
Protective mechanisms can be provided to structural components in order to absorb or
dissipate the imported kinetic energy from impact or blast effect, and thus minimize or mitigate
damage due to impact or blast effects. An ideal energy absorber should have a yielding stress
that is lower than the failure stress of the material being protected and a long and flat postyielding stress-strain curve. This will lead to the maximum energy absorption while retaining the
safety and integrity of the protected object. Metal foams, such as aluminum foam, have such
desired response features and have thus been used as energy absorbers for impact resistance
(usually in the form of core material in a sandwich panel). The compressive behavior of a metal
foam (e.g., Al open-cell foam) can be described by three stages [13-18]: an elastic stage, a
collapse stage (or stress plateau stage), and a densification stage. Due to the plastic deformation
of cell ligaments, or cell walls, in metal foams after yielding, a relative constant stress level can
be maintained up to a larger strain (densification strain), after which the metal foam goes into the
densification region where the stress increases rapidly due to contact of cell ligaments or walls
[13]. The area underneath the stress-strain curve within the plateau state is usually considered as
the energy dissipation capacity of a metal foam. The mechanical properties (e.g., stiffness,
plateau stress, densification strain, etc.) of a metal foam such as aluminum foam is mainly
controlled by its relative density, which can be calculated as the ratio of density of the foam to
the density of the solid material [13]. As a result, the mitigated pressure can be adjusted to the
desired level by varying the density of the metal foam material used.

3

Stress

Elastic
Region

Plateau
Region

Densification
Region

Energy
Dissipated
Strain

Figure 1-1 Compressive behavior of a typical metal foam

1.2.3 Nature’s Functionally Graded Materials
For a structural component subjected to a given loading under a set of boundary
conditions, the demand in material performance varies with location within the component. In a
functionally graded material, the microstructure and composition has a gradual variation over the
structure. This results in an optimized overall performance of the components while meeting the
requirement of a specific function and application [19, 20]. Cellular materials found in nature
such as bone, wood, and bamboo usually have a space varying density distribution that allows
them to optimize intrinsic properties of the constituents that form them [21, 22].
The human bone, for example, is a naturally engineered functionally graded material.
Generally, the strength of bone is highest at the surface and gradually lower towards the inside
by changing its porosity. In addition, the bone structure has a sponge-like appearance at the ends
(cancellous bone) while being very dense and with low porosity (cortical bone) in the middle.
The gradient in mechanical properties resulting from the non-uniform structure can be used to
4

optimize the global mechanical performance of the structural element subjected to unevenly
distributed external loads and thus lead to higher specific strength and more efficient use of the
material.
1.2.4 Tailoring Energy Dissipation Mechanisms
Foam materials are widely used as energy absorbers and for impact resistance due to the
large energy dissipated during the plateau region of their response [13, 23]. Many studies have
been conducted to investigate the potential benefit in tailoring energy dissipation mechanisms
using functionally graded foam materials. Functionally graded foam materials can be fabricated
by creating a density gradient directly in the metallic porous materials through chemical or
electrochemical processing [24, 25], by replication process based on graded precursors or
performs [4, 5, 26, 27], or by combining layers of cellular materials with different densities to
create a step-wise property gradient [28, 29].
The compressive behavior of functionally graded foams has been studied using
experimental, analytical and numerical approaches under both quasi-static [2, 4, 5, 28] and
dynamic conditions [1, 29-34]. These studies have shown that functionally graded foam
materials can be used to control the overall stress-strain behavior of the foam and that enhanced
performance in energy dissipation can be achieved. However, the experimental studies
conducted thus far on functionally graded foams have been limited to those with a onedimensional property gradient (usually the loading direction). And only few studies, those based
on numerical methods, on functionally graded foams with two-dimensional property gradients
have been reported [35, 36]. This may be may be partly due to the difficulty in creating
properties gradients in higher dimensions using the manufacturing process of traditional cellular

5

materials. Thus, development in the fabrication and investigation of cellular/foam structures with
two-dimensional property gradients is of great interest.
1.2.5 Electrodeposition of Metal Coatings
Electrodeposition is a electroplating process in which a metal coating is applied to a
metallic or other conducting surface by an electrochemical process [37]. Electrodeposition is
usually used to modify or improve the surface properties of an object, such as enhancing wear
resistance, provide corrosion protection, or for aesthetic qualities [37]. During an
electrodeposition process, the electrons on the surface of the cathode (part to be coated)
neutralize the positive charges on the metal ions in the solution and lead to the deposition of the
metal on the surface of cathode. On the other hand, electrons are removed from the anode
(usually the coating material) and thus metal ions are released into the solution [37]. The
electrodeposited metals, which usually have a nanocrystalline structure and high concentration of
dislocations, have much higher yielding strength and are much harder compared to their bulk
counterparts [37, 38]. However, this is achieved at the expense of ductility and toughness [39].
1.2.6 Hybrid Metal Foams
Lately, increased attention has been paid to the development of hybrid metal foams by
coating or reinforcing conventional open-cell foams through electrodeposition to enhance or
tailor their mechanical performance [6-8, 40, 41]. Different hybrid foam systems have been
reported, namely, aluminum (Al) open-cell foam with nickel (Ni) based coatings [6, 7, 40] and
aluminum/copper (Al/Cu) hybrid foam systems [8]. The deposited coating material (e.g., Ni or
Cu), which usually has a higher modulus than the substrate, (e.g., Al) works in synergy with the
substrate and leads to optimized composite behavior of the foam ligaments. It has been shown

6

that the stiffness and strength of open-cell hybrid metallic foams with nano-crystalline coating
can be controlled by varying the coating thickness [6-8]. Thus, it is considered that such method
can be adopted to create functionally graded cellular structures by introducing strategically
designed coating patterns. However, the unique mechanical response features of as-deposited
hybrid foams, such as their brittle failure mechanism due to its nanocrystalline coating, have not
been understood.
1.2.7 Annealing of Metals
Similar to metal materials after large plastic deformations, electroplated metals in the asdeposited state usually have fine grain size and a large concentration of structural defects [37,
42-44]. These microstructure features of electrodeposited metals lead to their high strength and
hardness but low ductility [37, 42-44]. The large amount of energy stored in electrodeposited or
highly deformed metals due to the high concentration of structural defects may provide a great
driving force for recovery and recrystallization when subjected to heat (i.e., annealing) [45-47].
When enough annealing is provided, the defects in electrodeposited or deformed metals can be
removed or rearranged, and also a new microstructure with larger grain sizes can be obtained
[47]. Such changes in microstructure usually result in a decrease in strength and increase in
ductility of the metal.
1.3

Research Hypothesis and Objectives

1.3.1 Hypothesis
The hypothesis behind this research is that open-cell metal foams can be strategically
reinforced with nanostructured coatings through electrodeposition to achieve functionally graded

7

hybrid foam systems with optimized performance for different loading conditions and with the
ability to mitigate or redirect damaging stress waves from blast and impact loads (see Figure 1-2).

Figure 1-2 Schematic of the research motivation and the integration across scales for functionally
graded hybrid foam systems

1.3.2 Objectives
The first objective of this research is to gain a comprehensive understanding of the microand macro-scale deformation and failure mechanisms of hybrid metal foams fabricated through
electrodeposition, which have a distinct behavior compared to conventional metal foams due to
their nanocrystalline coating. The second objective is to study the behavior of hybrid foams
under high strain rate loading and also show that the mechanical performance of hybrid foams

8

can be efficiently enhanced by the nano-structured reinforcement. The third objective is to show
that the mechanical performance of hybrid foam structural elements can be optimized according
to the loading condition at different loading rates by means of strategically designed coating
patterns. The objectives in this research were met by completing the following tasks:
1) Characterization of hybrid foam material. Finite element models at the micro- and macroscales were established and used to investigate the unique mechanical behavior of Al/Cu hybrid
foams and the effect of coating material properties. The micro-scale behavior of hybrid foams
were investigated using sectional analyses combined with finite element models of single hybrid
foam ligaments discretized using continuum elements. On the other hand, beam-element-based
finite element models consisting of multiple foam cells were used to study the macro-scale
behavior of hybrid metal foams.
2) Fabrication of material and structure. The low ductility of the nanocrystalline coating in
hybrid metal foams can have a negative effect in their performance, such as localized damage
and a reduction in energy dissipation capacity. Such detrimental effect can be improved by
introducing annealing after the electrodeposition process in the manufacturing of hybrid metal
foams. In addition, hybrid foam systems with strategically designed two dimensional (2D)
coating patterns were successfully fabricated using a selective electrodeposition approach.
3) High strain rate characterization. The behavior of Al/Cu hybrid foams under high-strainrate condition was investigated using Split Hopkinson Pressure Bar (SHPB) tests. In order to
investigate the influence of the nanocrystalline coating, hybrid foams with different coating
thicknesses were tested while two different strain rates were considered. The effect of coating
ductility was also studied by testing annealed hybrid foams samples.

9

4) Structural characterization Al/Cu hybrid metal foams with strategically designed coating
patterns were fabricated and tested. Two flexural type loading conditions were considered under
quasi-static conditions, namely, a three point bending beam and a cantilever beam. The dynamic
performance of functionally graded Al/Cu hybrid foams was investigated using drop-weight
tower tests with a three point bending setup. In both conditions, different coating designs were
considered to demonstrate the capability of modifying the deformation and failure mechanisms
and optimizing mechanical performance by adopting functionally graded designs.
1.4

Scope
This study aims to develop a functionally graded hybrid foam systems with optimized

performance responding to different loading conditions and the ability to mitigate or redirect
damaging stress waves from blast and impact loads by introducing strategically designed
nanocrystalline coatings onto open-cell metal foams through electrodeposition. The aim is
achieved by meeting the objectives and completing the tasks discussed in Section 1.3.
While metal foams have many applications due to their unique properties (e.g., heat
exchangers and sound absorbers), the current study focuses on the mechanical behavior and
energy absorption capacity of hybrid and functionally-graded hybrid metal foams. The study and
investigation in this research is mainly based on Al/Cu hybrid foams, which were obtained by
depositing nanocrystalline copper coatings onto open-cell aluminum foams through an
electrodeposition process. Copper was used as the coating material due since its electrodepostion
is well-established and because of its relatively low cost. However, other types of hybrid foams
have been reported (e.g., aluminum/nickel hybrid foams) [6, 7, 40]. In addition, electrodeposition
processes for other metals or alloys have been established. While it is of great interest to
investigate the optimized combinations of core and coating material for hybrid metal foam

10

systems, it is not within the scope of this study. However, as it will be shown in later chapters,
some of the findings and conclusions are applicable to other types of hybrid metal foams.
1.5

Organization
This dissertation is organized into seven chapters. The remainder of this dissertation is

organized as follows. Chapter 2 provides a background and literature review for open cell metal
foams, electrodeposited metal coatings, hybrid metal foams, the annealing of metals, and
functionally graded materials. Chapter 3 documents the investigation on micro- and macro-scale
behavior on hybrid Al/Cu hybrid foams using the finite element method. Chapter 4 describes the
study on the effect of ductility of electrodeposited coatings in hybrid metal foams, in which the
influence of annealing was investigated. Chapter 5 reports the experimental characterization of
Al/Cu hybrid foams under high strain rate loading using a Split Hopkinson Pressure Bar (SHPB).
Chapter 6 describes the experimental characterization of Al/Cu hybrid foam structure elements
with strategically designed coating patterns under quasi-static and dynamic loading. Finally,
Chapter 7 summarizes the research by noting its significance, main conclusions and future
research needs.

11

CHAPTER 2 BACKGROUND
2.1

General
This chapter presents the background and fundamental knowledge based on which the

current work is established. The contents presented in this chapter will be used and referenced in
the following chapters.
2.2

Behavior of Open Cell Metallic Foams
Metallic foams are a cellular material made of metal with polyhedral cells packed in three

dimensions to fill space [13]. An open-cell foam only contains the cell edges and forms an
interconnected network of solid struts (see Figure 2-1). There are many methods currently
employed in the manufacturing of metallic foams, including liquid metals, powder metallurgy,
electrochemical deposition, or vapor deposition techniques. [13-18]. Metallic foams excel in
different kinds of mechanical properties, such as great impact energy absorption, high specific
strength, and good stiffness [48]. As a result, metallic foams can have numerous applications in
different fields such as energy absorption systems for impact protection and lightweight
structural components [13, 23].

Figure 2-1 Metallic open-cell foam

12

The unique physical properties of open-cell foam materials can be attributed to its cell
structure at the micro scale. A proper representation of the foam structure is thus usually the key
to study its behavior using analytical or numerical approaches. A well-known method to
resemble the foam structure is to fill the space with Kelvin’s tetrakaidecahedron cell (see Figure
2-2), which has 14 faces consisting of 6 square faces and 8 hexagonal faces [49]. Such structure
was believed to minimize surface area per unit volume for over a century. With the help of
computer software, Weaire and Phelan identified a better structure solution consisting of six
tetrakaidecahedron cells (with 2 hexagonal and 12 pentagonal faces) and two irregular
dodecahedron cells (with pentagonal faces), that lead to even lower surface area per unit volume
[50]. In addition, computer software has been developed to generate random soap froth micro
structures, based on which an open-cell foam with random cell structures [51].

Figure 2-2 Kelvin’s tetrakaidecahedron cell structure

Numerous studies have been conducted on metallic foams due to their excellent
properties and wide range of applications, and their mechanical behavior has been studied
extensively using experimental, analytical and numerical approaches. Deshpande and Fleck

13

studied the yield behavior of aluminum alloy foams by conducting tests considering a range of
axisymmetric compressive stress states and proposed phenomenological isotropic constitutive
models [52]. Zhu et al. [53] established elastic constants for open-cell foams based on
tetrakaidecahedral cells (Figure 2-2 (a)). Finally, in what is perhaps the most well-known and
followed work on the subject, Gibson and Ashby [13] derived semi-empirical equations to
predict the performance of foam materials under different failure mechanisms based on
simplified unit cell models.
The compressive behavior of a metal foam (e.g., Al open-cell foam) can be described by
three stages [13-18]: an elastic stage, a collapse stage (or stress plateau stage), and a
densification stage. Figure 2-3 shows a schematic of the compressive stress-strain curves for a
typical metal (elastic-plastic) foam. Within elastic stage, the deformation mechanism of opencell metal foam is controlled by cell wall bending and the stress-strain relationship is basically
linear and can be described using a Young’s modulus E*. The plateau stage in the compressive
behavior of open-cell metal foam is associated with collapse of the cells during which material
yielding and the creation of plastic hinges occurs. A relatively constant stress level can be
maintained within the plateau stage up to a large strain. The area underneath the stress-strain
curve within the plateau stage is usually considered as the energy dissipation capacity of a metal
foam. When the foam cells have completely collapsed, the deformation enters the densification
stage during which the stress increases rapidly due to contact between the foam ligaments.

14

Stress

Densification

 pl
Plateau (Plastic Yielding)
Linear Elasticity (Bending)
Strain

D

Figure 2-3 Compressive behavior of open-cell metal foams (Adapted from Gibson and Ashby

Stress

[13])

 pl

Cell
Wall Alignment

Plastic Yield
E*
Linear Elasticity (Bending)
Strain

Figure 2-4 Tensile behavior of open-cell metal foams (Adapted from Gibson and Ashby [13])

The tensile behavior of open-cell metal foam is shown in Figure 2-4. Under low stress,
the behavior is linear elastic, which is caused by cell wall bending. At larger strains, plastic
bending occurs and the cell walls rotate towards the tensile axis, thus leading to a yield point

15

followed by a rising (stiffening) stress-strain curve [13]. Fracture occurs in open-cell metal
foams upon reaching the material’s ultimate tensile strain.
One of the most important structural properties of metallic foams is their relative density,
which is usually calculated using the density of the foam  foam , divided by that of the solid of
which it is made  solid [13]. However, the following relationship can be easily obtained
considering single material foam:

 foam

V
 solid
 solid V foam

Equation 2-1

One of the pioneering studies on foam materials is that by Gibson and Ashby [13], in
which they studied the mechanical behavior of open-cell foams based on a simplified cubic
models (Figure 2-5). The unit cell model used by Gibson and Ashby for three-dimensional opencell foams consists of a cubic array of members of length l and square cross-section with an edge
thickness t. The mechanical properties of open-cell metal foams were studied by considering
different deformations and failure mechanisms. The semi-empirical equations obtained by
Gibson and Ashby [13] for the mechanical properties of open-cell foams are summarized in

Table 2-1. It can be seen that the mechanical properties of open-cell metal foams are
related to the base material properties and the relative density of the foam. According to Gibson
and Ashby, the modulus and yield stress under compress and tension are approximately the same
for open-cell metal foams.

16

Figure 2-5 Cubic unit cell (Adapted from Gibson and Ashby [13])

In the work by Gibson and Ashby [13], the post-yield behavior of open-cell metal foam is
described using Equation 2-2 and Equation 5-37, in which constants D and m are determined by
fitting the curves with experimental data. Gibson and Ashby suggested m=1  0.4 and D = 2.3
for elastic-plastic open-cell foams based on experimental data for polymethacrylimid foams.


1

 1 when    D 1  
 pl
 D

Equation 2-2

m


1

1 

  D  when    D 1  
 pl D   D   
 D

17

Equation 2-3

Table 2-1 Equations for the Mechanical Properties of Open-cell Metal Foams
Mechanical Properties

Formula
RD 

Relative density RD

*
s

Young’s Modulus

Shear Modulus

3   *
G  Es 

8  s 

Poisson’s Ratio

* 

*

 *pl

 *pl

2

1
3
3/2

  *
 0.3 ys 

 s 

Plastic Collapse Strength
  *
 0.23 ys 

 s 

2

1/2 



1    * 
  s 






  *

 s 

 D  1  1.4 

Densification Strain

2.3

2

  *
E  Es 

 s 
*

Electrodeposition of Metal Coatings
Electrodeposition is a electroplating process in which a metal coating is applied to a

metallic or other conducting surface by an electrochemical process [37]. Hybrid metal foams can
be manufactured by reinforcing a conventional single-material foam with a different coating
material through electrodeposition [6, 8, 40, 54]. This section discusses the electrodeposition
process, and the physical characteristics and mechanical properties of electrodeposited metals.

18

2.3.1 Electrodeposition Process [37, 55]
The electrolysis cell for an electrodeposition process basically consists of four parts,
namely, the cathode, the anode, the electrolyte, and a power supply (i.e., rectifier). The cathode
is the negative electrode which is the part being plated (coated). The anode is the positive
electrode which is the source of the metal ions for the electrolyte and provides electrons for the
circuit. The electrolyte, which is called “the bath”, is mostly aqueous and contains metal ions.
During an electrodeposition process, the electrons on the surface of the cathode neutralize the
positive charges on the metal ions in the solution and lead to the deposition of the metal on the
surface of cathode. On the other hand, electrons are removed from the anode and thus metal ions
are formed and released into the solution. A power supply is needed to create an electric field
that moves the metal ions in the solution. Equation 2-4 and Equation 2-5 show the reactions
taking place at the cathode and anode.
Cathode reaction:

M z  (aq)  ze  M ( s)

Equation 2-4

Anode reaction:

M (s)  M z  (aq)  ze

Equation 2-5

19

Anode
(+)

Cathode
(-)

+

+
+

Electrolyte
+

+

Metal Cations

Figure 2-6 Electrolytic cell for electrodeposition of metal

The amount of metal deposited on the cathode (coating) can be estimated using Faraday’s
laws of electrolysis [56], which can be summarized by Equation 2-6. In the equation, m is the
mass of the substance liberated at an electrode, Q is the total electric charge passed through the
substance, F is the Faraday constant (96485 C mol-1 ), M is the molar mass of the substance, and
z is the valence number of ions of the substance (electrons transferred per ion). In the case of
constant-current electrolysis, Q  It and it leads to Equation 2-7.
 Q  M 
m    
 F  z 

Equation 2-6

 It  M 
m    
 F  z 

Equation 2-7

20

2.3.2 Structures and Properties of Electrodeposited Metals
The physical, chemical and mechanical properties of electrodeposited metals usually
differ from those of their bulk counterparts. This can be mainly attributed to two factors: the
microstructure of electrodeposted materials and the presence of impurities [37]. During the
electrodeposition process, the diffusing atoms at the surface may not be able to achieve an
equilibrium position and end up be in a metastable configuration due to the deposition rates. This
phenomenon leads to a structure containing high concentration of vacancies, dislocations, twins,
etc. The refinement mechanisms based on these structural defects leads to a small grain structure
(usually nanocrystalline structure [8, 57]). Due to the high concentration of these structural
defects and small-grained structure, the properties of the deposits are quite different from those
of bulk materials.
The movement of dislocations is the cause of plastic deformation. On the other hand, due
to the change in orientation, grain boundaries, twin boundaries, other dislocations act as
obstacles that dislocation motion. Such resistance to dislocation motion, or plastic deformation,
defines the yield strength and hardness of a metal material [58]. As a result, the electrodeposited
metals, which usually have a small grain size and high concentration of dislocations, have higher
yielding strength and are harder compared to their bulk counterparts [37, 38]. An
electrodeposited metal can be two or three times as strong as the corresponding bulk metal [38].
Electrodeposited metals can have various impurity contents originated from additives,
metal or nonmetal particles, metal complexes, hydroxides, and gas bubbles (e.g., hydrogen) [59].
The amount of impurities can noticeably influence the strength of electrodeposited metals [60,
61].

21

2.3.3 Properties of Electrodeposited Metals
Similar to plastically deformed (cold worked) metals, the high dislocation densities, fine
grain structure found in electrodeposited metals are due to the existence of a strained condition.
However, the dislocation density in electrodeposited metal is usually much higher (in the order
of 1015 -1016 m-2 ) compared with heavily deformed or quenched metals [37]. As discussed in
Section 2.3.2, the dislocation densities as well as the grain size affect the mechanical properties
of metals, such as strength, ductility and hardness.
A deposited metal with a small grain size can have a high strength. Petch and Hall [62, 63]
proposed an expression to describe relationship between the grain size in a metal and its
mechanical properties, as shown in Equation 2-8 and Figure 2-7. In this equation, H can
represent the hardness or yield strength of the metal and d is the grain size. H0 and K H are
determined through experiment for a specific metal. More specifically, H0 is determined by
dislocation blocking and K H represents the penetrability of the moving dislocation boundary
[62, 63]. This empirical expression is applicable to many electrodeposited metals and provides
useful information on the relationship between mechanical properties and microstructure of
deposits. It needs to be noticed that the strength of deposited metal films also depends on film
thickness. Generally, the strength of a thin film is higher. However, the change in strength is only
for films thinner than 10 µm, with respect to the value measured in thick films [37].

K
H=H0 + H
d

22

Equation 2-8

H

d 1/2
Figure 2-7 Hall-Petch relationship

An electrodeposited metal usually has a nanocrystalline structure, which leads to a high
tensile strength. However, the increased strength is achieved at the expense of ductility and
toughness [39]. The ductility of a metal material can be defined as the degree of plastic
deformation that can occur before fracture [38]. Since plastic deformations are the result of
dislocation motions, ductility is affected by the dislocation structure. As a result, the ability for
dislocation motion usually leads to ductile behavior. On the other hand, fracture occurs if the
higher stresses at the structural defects cannot be relieved by plastic deformation [43]. Thus, due
to the small grain size and the existence of obstacles to dislocation motion, electrodeposited
metals usually show low ductility and relatively brittle behavior compared to their bulk
counterparts. The low ductility of electrodeposited metal films can also be caused by the
precipitation of impurities at grain boundaries [37].
It also needs to be noted that the structure formation of electrodeposited metals results
from the nucleation and growth of metal crystals during the electrochemical process and is

23

related to the particular electrodeposition condition [37]. As a result, the mechanical properties
of electrodeposited metals, such as tensile strength, ductility can be affected by the variables in
the electrodeposition process, including metal ion concentration, additives, current density,
temperature, agitation and polarization [64].
2.4

Hybrid Metal Foams through Nanocrystalline Coatings

2.4.1 State-of-the-art Review
Increased attention has been recently paid to the development of hybrid metal foams by
coating or reinforcing conventional open-cell foams through electrodeposition to enhance, or
tailor, their mechanical performance [6-8, 40, 41]. Different hybrid foam systems have been
reported, namely, aluminum (Al) open-cell foam with nickel (Ni) based coatings [6, 7, 40] and
aluminum/copper (Al/Cu) hybrid foam systems [8]. The deposited coating material (e.g., Ni or
Cu), usually has a higher modulus than the substrate, (e.g., Al) and works interactively with the
substrate to create improved composite behavior of the foam ligaments. Boonyongmaneerat et al.
[40], Bouwhuis et al. [6] and Jung et al. [7] investigated the quasi-static and high-strain-rate
performance of open-cell aluminum (Al) foam reinforced with nanocrystalline nickel-tungsten
(Ni-W) or nanocrystalline nickel (Ni) coatings and found significantly enhanced modulus,
strength and energy absorption capacity. Wang et al. [8] found similar enhancements under
quasi-static loading for aluminum open-cell foams reinforced with nanocrystalline copper (Cu).
The investigation of this study focuses on Al/Cu hybrid foams. One of the reasons is the
well-established electrodeposition process of copper. In addition, copper has a high plating
efficiency and a low cost. However, as it is shown in later chapters, the conclusions and findings

24

of this study are also applicable to other types of hybrid metal foam systems (e.g., Al.Ni hybrid
foam).
2.4.2 Manufacture of Al/Cu Hybrid Foam
The Al/Cu hybrid foams were manufactured by applying a nanocrystalline copper
coatings on an aluminum open-cell foams using electrodeposition (see Figure 2-8) [8]. The
electrodeposition process was conducted in a glass container filled with an electrolyte and placed
on top of a stirrer/hot plate. The temperature was monitored and controlled by a thermocouple
placed into the electrolyte. A rectangular titanium basket with copper strips tied on the sides was
used as the anode and the foam specimen was placed in the center of the basket without any
contact with it. The copper strips were necessary to maintain the copper concentration and pH
value within the electrolyte.
Copper can be electrodeposited from acid, alkaline, and neutral bath formulations [55]. In
this study, copper pyrophosphate plating bath was used. A commercial electrolyte (Uyemura
International Co., Ontario, CA, US), which mainly contained copper pyrophosphate as the
copper source was used at 65 °C and a PH of 7.5 A stirring bar was applied throughout the entire
deposition process at 180 rpm to minimize the gradient of electrolyte concentration. The
electrodeposition was carried out at a constant current density of about 13 mA/cm2. Pretreatment according to ASTM Standard B 253 was applied to the foam specimens prior to
electrodeposition to yield a better coating result. SEM images of the hybrid foam fabricated are
shown in Figure 2-9. It can be seen that a uniform coating can be obtained using the setup
discussed.

25

Figure 2-8 Schematic of electrodeposition setup

Al

Cu
Al

Cu

500 µm

100 µm

Figure 2-9 SEM images of hybrid Al/Cu foams: a) SEM image of micro-structure of hybrid
Al/Cu foams with ligament cross sections exposed; b) SEM image of cross-section of a single
ligament of hybrid Al/Cu foam

26

The nominal copper coating thicknesses on Al/Cu hybrid foam samples can be calculated
based on the gained weight. Assuming that the copper coating is uniformly applied on the foam
ligaments with circular cross sections of radius r, the weight of copper coating is proportional to
the cross section area of the coating layer as shown in Equation 2; where W is the copper weight
density (g/cm3), C is a constant and tc is the coating thickness (μm). The radius r was defined to
be 97 μm based on measurements of ligaments on 40 PPI foams by Wang et al. [8]. The constant
in Equation 2 was determined by fitting the equation with the measurements reported by Wang et
al. [8] in which coated foam struts of representative samples were imaged with a scanning
electron microscopy and measured coating thickness were averaged based on at least 30
measurements at different locations for each sample. Figure 2-10 shows the comparison between
the coating thicknesses estimated using Equation 2 and the measured thicknesses with different
weight gain.
W C

r  t 
c

2

 



 r 2  C tc 2  2rtc , C  2.76 105

27

Equation 2-9

3

Weight Gain Per Volume (g/cm )

0.8

Calculation
Measurement [15]

0.6
0.4
0.2
0.0
0

20

40

60

80

100

Coating Thickness ( m)
Figure 2-10 Comparison of coating thickness estimates to experimental measurements
2.4.3 Mechanical Behavior of Al/Cu Hybrid Foam with Nano-coating
The works by Boonyongmaneerat et al. [40], Bouwhuis et al. [6], Jung et al. [7] and
Wang et al. [8] demonstrate the effectiveness in manipulating the properties of open-cell Al
foams by using electrodeposited nano-crystalline metal coatings. However, it can be observed
from their reported work that the hybrid foams had a behavior that differs considerably from that
of conventional (uncoated) metal foams. The typical compressive response of a metal foam (e.g.,
Al open-cell foam) can be described by three stages [13-18]: an elastic stage, followed by a
collapse stage (or stress plateau stage) where a relatively constant stress level can be maintained

28

up to a large strain level, and then the densification stage where the stress level rises rapidly due
to the contact between cell walls (closed cell foam) or ligaments (open-cell foam), as shown in
Figure 2-3. However, a large stress drop after the initial peak, especially for specimens with
thicker coatings, was observed in the behavior of the Al/Ni-W and Al/Ni hybrid foams reported
by Boonyongmaneerat et al. [40], Bouwhuis et al. [6] and Jung et al. [7] and the Al/Cu hybrid
foams reported by Wang et al. [8], as shown in Figure 2-11 and Figure 2-12. No detailed
discussion on this behavior was made in the noted studies except for the observation noted by
Bouwhuis et al. [6] of cracks and wrinkling of the Ni sleeve (coating) after the peak stress was
reached. Thus, it is clear that the large stress drop greatly decreases the energy absorption
capacity and efficiency of hybrid foams and that the failure mechanism needs to be investigated
in detail in order to get a better understanding of their performance and for possible improvement
of their manufacturing process.
The Ni-W, Ni or Cu coatings in the studies previously mentioned were applied using
direct current or pulsed electrodeposition and had a nanocrystalline structure. According to the
Hall-Petch relationship [62, 63] (see Equation 2-10), where C is a constant and d is the mean
grain size), the strength of a material increases with smaller grain size. However, the higher
strength of electrodeposited nanocrystalline material is usually accompanied by a significantly
reduced ductility capacity [65-68]. The increased brittleness may cause local coating failure in
the hybrid foam material and lead to concentrated deformations in weakened critical sections at
low strain levels. As most of the energy absorbed by metal foams is in the form of large plastic
deformations, the consequences of a brittle coating in hybrid foams may overtake the
enhancement gained from the additional strength and inhibit the improvement of energy
absorption capacity.

29

σ y =σ0 +

C
d

Equation 2-10

While the macroscopic performance of hybrid foams with nano-coating has been
evaluated experimentally. An understanding of the detailed failure mechanisms and influence
parameters such as the coating properties is still lacking; and thus the unique behavior of such
material cannot be clearly explained. In addition, methods to numerically study and simulate the
behavior of metal hybrid foams are not well established.

Al

Stress (MPa)

8

Ni/Al

4

0.3
0.6
Strain (mm/mm)
Figure 2-11 Typical stress-strain curves of uncoated foam samples and with nickel coating
(Adapted from Bouwhuis et al. [6])

30

Figure 2-12 Compressive stress-strain response of 40 ppi plain Al foam and copper coated Al/Cu
hybrid foams [8]
2.5

Annealing of Deformed Metals
As discussed in Section 2.3, electroplated films in the as-deposited state usually have

rather low ductility due to the fine grain size and large concentration of structural defects. These
features are responsible for the mechanical properties of electrodeposited metals, such as high
strength and hardness, and low ductility. The microstructure of electrodeposited metal is similar
to that of metal material after large plastic deformation [37].
A large amount of energy can be stored in an electrodeposited, or highly deformed
polycrystalline material (e.g., metals), due to the structural defects such as dislocations, twins,
grain boundaries. This leads to a material that is thermodynamically unstable. The unstable
defects are retained in the material as the atomistic mechanisms needed to remove the defects are

31

usually very slow at normal temperature. However, when the material is subjected to high
temperature, or heat treatment such as annealing, the large amount of stored energy may provide
the driving force for recovery and recrystallization [37]. By this way, the defects can be removed
or rearranged to a lower energy state due to thermally activated processes such as solid state
diffusion [47].
Usually cold worked metals undergo three stages when subjected to annealing, namely,
recovery, recrystallization and grain growth [47]. In the recovery stage, annihilation and
rearrangement of the dislocations occurs and the microstructure of the metal can be partially
restored. The changes in microstructure during recovery are usually within the deformed grains
and do not affect the grain boundaries. The dislocations are not completely removed during
recovery but rather entering a metastable state. New grains without any dislocation are formed in
the next stage, called recrystallization, which grow and consume the old grains. This leads to a
new grain structure with low dislocation density. With further annealing, the smaller grains are
eliminated while the larger grains grow (grain grow stage). A lower energy configuration is then
achieved with the reduction of grain boundaries.

32

Pyrophosphate
Electrodeposited
Copper

Cold Work (%)

100
75
50

25

Wrought
Copper

200

400
800
600
Recrystallization temp (˚F)

Figure 2-13 Recrystallization temperature for various copper materials (Adapted from Sherlin
and Bjelland [69])
Figure 2-13 shows a plot of recrystallization temperature for electrodeposited
pyrophosphate copper and wrought copper with different levels of cold work [69]. It indicates
that the electrodeposited copper has behavior similar to that of 100% cold worked material. In
addition, it can be seen from the figure that the higher the percentage of cold work, the quicker
the recrystallization behavior is upon heating.
The mechanical properties of metals heavily depend on the dislocation densities and grain
sizes. The dislocation density of an electrodeposited metal, or a heavily deformed metal, can go
up to 1016 m-2 , which is much higher compared with a dislocation density of about 1011 m-2
for a metal in an annealed state. As discussed in Section 2.3, this leads to a much higher yield
strength and lower ductility for electrodeposited or heavily deformed metals. On the other hand,

33

dislocation loss and rearrangement can occur once the electrodeposited or deformed metals are
subjected to annealing, which results in a decrease in strength and increase in ductility, as shown

1

1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1

Relative Strength

0.98
0.96
0.94
0.92
Relative Strength
Relative Elongation

0.9
0.88
0.01

0.1
1
10
Anneal Time, min

Relative Elongation

for example in Figure 2-14 for electrodeposited copper.

100

Figure 2-14 Relative ultimate strength loss and relative total elongation gain at 23 ˚C for
electrodeposited copper foil (Adapted from Merchant [45])
2.6

Modeling and Simulation of Open-cell Foams
The macroscopic behavior of open-cell foams can be studied by investigating the

behavior and mechanism of foam cell at the micro-scale. As shown in Section 2.2, Gibson and
Ashby [13] derived semi-empirical solution for open-cell foams by studying a simple cubic cell
model. The effect of the cell geometry was accounted for by introducing constants, which were

34

calibrated based on experimental data, into the analytical solutions. Meanwhile, numerical
approaches are often used to study of open-cell foam materials as it allows a more sophisticated
way to analyze both micro- and macro-scale behavior.
In many studies on the simulation of open-cell foam materials, numerical models are
established with a foam microstructure idealized with Kelvin’s cell structure, while the ligaments
are usually modeled using beam-type elements. Such method has been shown to be successful in
simulating the behavior of open-cell materials in their ability of reproducing most of its
mechanical properties.
Laroussi et al.[70] investigated the plateau stress of an elastic open-cell foam material
under compression based on a periodic lattice structure consists of tetrakaidecahedral cells. Gong
et al. [14, 15] also established numerical models based on a Kelvin cell assuming linear elastic
material behavior to study the response of a polyester urethane foam. The ligaments were
modeled as shear-deformable extensional beams with non-uniform cross section along ligament
length, based on a cell morphology study. Equation 2-11 shows the relationship between the
ligament cross-sectional area along their length and the distance from the mid-span of the
ligament [15], in which A is the cross sectional area at location of interest, A0 is the crosssectional area at mid-span of ligament, ξ is the ratio of the distance from the mid-span to the
location of interest and the ligament length. The constants were determined by fitting the
experimental data and it was found that a = 86 and b = 1 for the polyester urethane foam tested
by Gong et al [15]. The model established by Gong et al. was able to adequately predict the
modulus of the elastic foam studied and was used to investigate the crushing of elastic foam due
to ligament buckling.

35

Jang and Kyriakides [18, 71] conducted thorough microscopic measurements on the
geometry of aluminum (Al) open-cell foam ligaments and established beam-element-based finite
element models using unit or multiple characteristic cells to predict the macroscopic compressive
behavior of aluminum open-cell foams. Equation 2-11 was also used to define the cross-sectional
area of the aluminum foam ligament along its length. The constants used in the equation were a
= 36 and b = 1 for the aluminum foam studied by Jang and Kyriakides [18, 71]. Through this
work Jang and Kyriakides showed that that proper definition of the ligament geometry is
essential in the numerical simulation of metallic foams [18, 71].
Another factor that needs to be considered in the geometry of open-cell foam models is
the randomness of the cell structure. It has been shown that the introduction of small
perturbations to the basic Kelvin cell microstructure has advantages over the use of a perfect
geometry, by breaking up the symmetry of the banded collapse localization of the perfect case
and leading to a more random distributed buckling and collapse [71]. Furthermore, Gaitanaros et
al. [72] established foam models using realistic random soap froth generated using the Surface
Evolver software [51, 73] and found crushing patterns that were more similar to observations in
corresponding experiments. As a result, it is necessary to consider certain randomness in the
modeling of open-cell foams.





A  ξ   A0 f  ξ   A0 aξ 4  bξ 2  1 , where   x / l

Equation
2-11

In addition to a proper representation of the foam cell structure, there are several other
important factors in the modeling and simulation of open-cell foams when large deformation are
considered, such as the contact between ligaments and material nonlinearity. Cell collapse occurs
during the plateau regime in the compressive behavior of open-cell foams. When a cell collapses

36

completely, contact between ligaments occurs, which stiffens the local region and causes
sequential collapse of neighboring cells. For numerical models based on Kelvin cell structures,
spring elements can be introduced between collinear nodes in the loading direction to simulate
ligament contacts [14, 15, 17, 71]. It needs to be noted that such method is usually only
applicable to simulate uniaxial compressive behavior. On the other hand, beam to beam contact
has been adopted to simulate the ligament contact in foam with random foam structures [72]. In
this case, the cross-section size needs to be considered so that contact occurs on the outer surface
of the ligament. Material nonlinearity needs to be included in the modeling of open-cell metal
foams when large deformation is considered. Previous researchers have demonstrated that the
compressive response of plain metal alloy foams is governed by localization at the ligament/cell
level and thus simple material constitutive models (elastic-perfect plastic) have been shown to be
sufficient to appropriately simulate the behavior. [13, 71].
Compared to the conventional open-cell metal foams, hybrid open-cell metal foams have
more than one material and more complicated failure mechanisms. Thus, existing numerical
models for are inadequate to capture the behavior of hybrid foams. First, existing models assume
a homogeneous material, while a coating material with constitutive characteristics different from
the base material need to be considered for simulating hybrid foams. Another factor needs to be
addressed in numerical simulations of a composite material system is the differences in stiffness,
strength and post-yield behavior (hardening and/or softening) of the constituents. This implies
the use of constitutive models that are more realistic. Zhou et al. (Zhou et al., 2005) investigated
the tensile behavior of an individual strut (ligament) extracted from an Al open-cell foam using
micro-tests and found it to behave differently than the corresponding bulk alloy. In addition, as
discussed in Section 2.3.3, the coating materials in hybrid foams are usually produced using

37

electrodeposition, leading to a nanocrystalline structure with higher yield stress but lower
ductility compared to its bulk counterpart [46, 66, 74]. The noted material features can have great
influence on the performance of hybrid foams and need to be considered. Thus, the use of
elastic-perfect plastic material behavior obtained from bulk material testing, which has been
typically used in most reported studies on open-cell metal foams, is inappropriate for the
simulation of hybrid foams.
2.7

Concepts of Stress-wave Propagation in Solids
The behavior of objects when subjected to rapid changing loads can be quite different

from its behavior under static or quasi-static conditions. A static equilibrium is maintained in
quasi-static deformation in which the summation of forces acting on any element in the body is
close to zero. However, when the deformation occurs at a very high rate, the associated stress
and strain is not instantaneously transmitted from the deformed region to other regions of the
body. Rather, they travel through the body at specific velocities and thus form waves [75].
The wave propagation in solids can be understood by imaging atoms connected by
interatomic forces as spheres connected by springs, as shown in Figure 2-15. When force is
applied, the resulted momentum is transferred from sphere to sphere (atom to atom) at a rate
which provides the velocity of wave propagation.

Figure 2-15 Transmission of disturbance from atom to atom

38

P

  P/ A

dx






dx
x

dx

Figure 2-16 Wave propagation in a bar
The simplest case is wave propagation in a uniform, homogeneous bar, as shown in the
schematic in Figure 2-16. By considering a segment in the bar and using Newton’s second law,
one can get the following equation [75]:

 2u

x
t 2

Equation 2-12

If assuming the deformation is elastic and Hookes’s law holds:

E




Equation 2-13



u
x

Equation 2-14

The following equation is then obtained:
 2u
t

2



E  2u

 x 2

And the velocity of the wave is:
39

Equation 2-15

C0 

E



Equation 2-16

It can be seen that the speed of elastic wave in solids is related to the modulus and density
of the material. It can be shown that the general solution to Equation 2-15 is in the form of
Equation 2-17 [75], in which F and G are functions that describe the shape of pulses propagating
in the positive and negative directions along the x axis, at a velocity of C0 . The shapes of these
waves are unchanged with time.

u(x,t)= F(x - C0t)+G(x+C0t)

Equation 2-17

Another important concept is the sonic impedance of the material, which is defined as the
product of the medium density and its elastic wave velocity (note C  E  ). When an elastic
wave encounters a medium with different sonic impedance, it will be reflected and refracted, as
shown in Figure 2-17. It needs to be noted that the figure only shows the reflected and refracted
longitudinal waves while actually there will be reflected and refracted transverse waves (shear
waves) [75]. The refraction and reflection angles are related to each other as stated in Equation
2-18 [75].

sin 1 sin  2 sin 3


C1
C1
C2

40

Equation 2-18

Refracted

Reflected
θ2

θ3

θ1
Incident

C1

C2

Figure 2-17 Reflection and refraction of longitudinal elastic wave encounters different medium
It is much simpler when the incident wave is normal to the interface, in which case a
longitudinal wave only reflects and refracts longitudinal waves [75]. Considering a longitudinal
wave travels from medium A to medium B, while the medium A has a larger sonic impedance
(see Figure 2-18), one can calculate the amplitudes of the transmitted and reflected waves based
on the densities and wave velocities of the two media, as given by Equation 2-19 and Equation
2-20 [75].

A σR
ρA, EA

B
σT

σI

ρB, EB

Figure 2-18 Longitudinal wave encountering boundary between medium A and medium B

41

T
2  B CB

 I  BCB   AC A

Equation 2-19

 R  BCB   AC A

 I  BCB   AC A

Equation 2-20

Equation 2-19 and Equation 2-20 show that the amplitude of the transmitted and reflected
pulses are determined by the impedances of the materials. If  BCB   AC A , the reflected pulse
has the same sign as the incident pulse. On the other hand, if  BCB   AC A , the reflected pulse
would have an opposite sign to that of the incident pulse.
In a ductile material, plastic deformation occurs when the stress in a material exceeds the
elastic limit. When the amplitude of a pulse transmitted to a material has an amplitude exceeding
the elastic limit of the material, the pulse will be decomposed in to two components, an elastic
wave and a plastic wave. The velocity of the plastic wave is given by Equation 2-21, in which

d / d is the slope of the plastic region of the stress-strain curve.
1/2

 d / d  
Vp  

  

Equation 2-21

It can be noted that within elastic limit, d / d is the modulus E for a material, which is
usually higher in the elastic regime than the plastic one. As a result, the plastic wave travels at a
lower speed than an elastic wave.
2.8

Functionally Graded foam materials
When a structural component is subjected to certain loading and boundary conditions, the

demand in material performance may vary with location within the component. In a functionally
graded material the microstructure and composition has a gradual variation over the structure,
42

which results in an overall optimized performance of the components while meeting the
requirement of a specific function and application [19, 20].
As discussed in Section 1.2.3, many cellular materials found in nature such as bone,
wood and bamboo are functionally graded. Inspired by these naturally engineered products,
many studies have been conducted over the past two decades in the development of so called
functionally graded cellular/foam materials. The fabrication of graded cellular materials can be
basically grouped into three categories. First, cellular materials with graded properties can be
obtained by creating a density gradient directly in metallic porous materials through chemical or
electrochemical processing. Neubrand [24] fabricated functionally graded porous materials based
on the evolution of porosity gradients in porous electrodes that undergo dissolution or deposition
due to electrochemical gradation. Alternatively, Matsumoto et al. [25] used chemical dissolution
to create a density gradient in Al open-cell foams. However, the strength and stiffness of the
fabricated materials were found to decrease more rapidly with respect to density than that of
typical metallic foams.
Metallic cellular materials with density gradient can also be fabricated through
replication process based on graded precursors or performs. Pollien et al. [26] produced
sandwich beams with graded foam core using replication processing with a preform of
compacted salt layers of different density. It was found that the graded foam beams had little
advantage in stiffness-limited design but may be beneficial in load-limited design. Brothers and
Dunand [2, 4] fabricated density-graded Al foams using replication based on non-uniformly
compressed polymer foam precursors and observed a smoothly rising plateau (plastic) stress.
Hangai et al. [5] produced functionally graded closed-cell Al foam by bonding precursors that
have various amounts of blowing agent using friction stir processing and the obtained material

43

had two different deformation stages and two plateau regions. Hassani et al. [76] fabricated
graded foams using a powder-space holder technique and found that the cell-size graded foams
has better energy absorbing performance compared to the non-graded foams.
In another approach to obtain graded designs, layers of cellular materials with different
densities can be combined to create step-wise property gradients. Gupta [28] fabricated graded
syntactic foams filled with hollow particle (microballoons) with a wall thicknesses gradient and
found it possible to control the compressive modulus, strength and total energy absorption. Zeng
et al. [29] studied the impact behavior of functionally graded cellular materials made of graded
polymeric hollow sphere agglomerates and found that by placing the hardest layer at the impact
end and the weakest layer at the remote end leads to maximized energy absorption and
minimized transmitted forces. In addition, studies on functionally graded foams fabricated by
bonding foam layers with different densities have also been reported [30, 77, 78].
Since foams structures are usually used as energy absorbers and impact resistant layers,
many studies have been conducted to investigate the performance of functionally graded foam
structures under dynamic loading conditions. Cui et al. [1, 33] investigated the behavior of
functionally graded foam materials under impact using finite element analyses and found that
functionally graded foams can reduce the duration of the high acceleration during an impact and
have superior energy absorption performance over equivalent uniform foams under low energy
impacts. They also found that convex gradients perform better than concave gradients and that a
better performance can be achieved by increasing the density range. Zeng et al. [29, 79] found
that placing the hardest layer as the first impacted layer and the weakest layer on the remote side
had some benefit to maximize energy absorption with a minimum force level transmitted to the
protected structures. However, placing the weakest layer at the impacted end is preferable to

44

avoid the breakage of the face sheet for sandwich panels subjected to localized impact. Ajdari et
al. [34] studied the in-plane crushing of regular, irregular and functionally graded honeycombs
using numerical methods and found that a density gradient could significantly change the
deformation mode and energy absorption of cellular structures under both low and high crushing
velocities. Wang et al. [30] and Gardner et al. [31, 32] conducted shock tube experiments to
study the dynamic response of sandwich panels with graded styrene foam cores. They found that
the overall blast performance and structural integrity can be improved by increasing the number
of monotonically graded foam core layers and by applying polyurea between the foam core and
the back face sheet. Zhou et al. [78] studied low velocity impact response on sandwich structures
with foam core that has graded densities using experimental and numerical method and found a
better performance of graded core structures over the monolithic cores.
The experimental studies conducted on functionally graded cellular/foam materials so far
have been limited to one-dimensional property gradient (usually the loading direction). And to
the author’s knowledge, there is no reported experimental effort for the fabrication and
investigation of two-dimensional functionally graded foams in public literature. This can be
partly attributed to the difficulty in creating gradients with higher dimensions. However, the
higher dimensional spatial variations in the properties of cellular/foams in analogy to cortical
bone [22] offer greater opportunity to optimize mechanical performance when foam materials are
used as structural elements. Wang et al. [35] considered graded cellular structure in conceptual
design of an acetabular component which changes from solid metal at the liner section to high
porosity at the implant-bone interface to encourage bone growth upon implantation. Daxner et al.
[36] conducted a numerical study and found that the distribution of foam density can be
optimized with respect to structural strength and stiffness for a given loading condition. However,

45

none of the complex designs can be realized with manufacturing processes for traditional cellular
materials. Thus, development of fabrication methods and the investigation of cellular/foam
structures with two-dimensional property gradient are of great interest.

46

CHAPTER 3 MODELING AND SIMULATION OF QUASI-STATIC COMPRESSIVE
BEHAVIOR OF AL/CU HYBRID FOAMS
3.1

Overview
In this chapter, finite element method approaches considering both coating and base

material damage are used to investigate the failure mechanisms and mechanical performance of
nanocrystalline hybrid open cell foams at the micro- and macro-scales. First, models of single
foam ligaments combined with sectional analyses were used to obtain further understanding of
micro-scale failure mechanisms. Experimentally validated multi-cell models are then presented

Nominal Stress (MPa)

to study the macroscopic compressive behavior.
4
3

Al/Cu Hybrid Foam
(60 m coating)

2

Plain (After)

1
0
0.0

Plain Al
Foam
0.2

0.4

0.6

Nominal Strain

0.8

Spalling
Plain (Before)

60 μm (After)

60 μm (Before)

Figure 3-1 Stress-strain curve and pictures of 40 PPI plain Al foam and Al/Cu hybrid foam with
60 μm coating under compressive loading [8].
Due to the multi-material characteristics and the more complicated failure mechanisms in
hybrid foams, existing numerical models for traditional open cell metal foams are inadequate to
capture the behavior of hybrid foams. First, existing models assume a homogeneous material,
while a coating material with constitutive characteristics different from the base material need to

47

be considered for simulating hybrid foams. Another factor needs to be addressed in numerical
simulations of a composite material system is the differences in stiffness, strength and post-yield
behavior (hardening and/or softening) of the constituents. This implies the use of constitutive
models that are more realistic. Zhou et al. (Zhou et al., 2005) investigated the tensile behavior of
an individual strut (ligament) extracted from an Al open-cell foam using micro-tests and found it
to behave differently than the corresponding bulk alloy. In addition, the coating materials in
hybrid foams are usually produced using electrodeposition, leading to a nanocrystalline structure
with higher yield stress but lower ductility compared to its bulk counterpart [46, 66, 74]. The
noted material features can have great influence on the performance of hybrid foams and need to
be considered. Thus, the use of elastic-perfect plastic material behavior obtained from bulk
material testing, which has been typically used in most reported studies on open-cell metal foams,
is inappropriate for the simulation of hybrid foams.
3.2

Material Model
As mentioned in Section 3.1, material properties based on macro-scale tests are not

appropriate in the simulation of Al/Cu hybrid foams, especially for the nanocrystalline coating.
In this study, the material models defined for the Al core and Cu coating were based on the
results from micro-scale tensile experiments on single Al foam ligaments conducted by Zhou et
al. [80] and tensile experiments on thin electrodeposited Cu films by Zhang et al. [66]. Both of
these studies have shown that the behavior of Al and Cu at the micro-scale is quite different from
their bulk response. Figure 3-2 shows the material behavior from experiments [66, 80] and the
material models used in this study. The true stress σ in the hardening region ( σ  σ y ) was
defined using Equation 3-1, where σ 0 and n are the hardening parameters.

48

σ  σ0 ε n

Equation 3-1

A ductile damage model [81] was used for both Al and Cu in the numerical study such
that damage would initiate once the defined fracture strain was reached and elements would
degrade and fail. A summary of the key parameters used in the material model definitions is
given in Table 3-1, where E is the Young’s modulus, υ is the Poisson’s ratio, σ y is the yield
strength, ε ft is the fracture strain in tension and δ ft is the effective plastic displacement at
failure.

Table 3-1 Material model parameters for aluminum (Al) and copper (Cu).
E
Material

ν

(GPa)

σy

σ0

n

(MPa) (MPa)

ε ft

δ ft

(mm/mm)

(mm)

Aluminum

70

0.33

193

511

0.17

0.5

0.005

Copper

102

0.33

150

329

0.12

0.08

0.0008

49

Figure 3-2 Behavior of aluminum (Al) 6101-T6 and electrodeposited copper (Cu) from literature
[66, 80] and corresponding material models used in the finite element simulations
3.3

Section flexural response
The collapse of cellular foams in compression is dominated by inelastic flexural behavior.

Thus, the first step taken to understand the behavior of Al/Cu hybrid foam was to evaluate their
sectional flexural response. Sectional, or moment-curvature, analyses were performed on
ligament cross-sections of a 40 PPI Al/Cu hybrid foam. The analyses were performed using a
fiber-based approach for unidirectional bending and with uniaxial constitutive material
definitions as presented in Section 3.2. The characteristic moment curvature response a hybrid
foam ligament section is shown in Figure 3-3(a) while the effect of coating thickness is shown in

50

Figure 3-3(b). Examining Figure 3-3(a) it can be seen that the section flexural response of a
hybrid foam ligament features a pronounced peak before a large capacity drop, followed by a
plateau region and later further gradual degradation at larger deformations. This behavior can be
explained using the bending strain and stress schematics shown in Figure 3-4. Point “a”
corresponds to a state when the extreme tensile fiber in the Cu coating reaches failure. At this
point, the maximum moment capacity of the section has been reached and further deformation
causes progressive fracture of the coating. Point “b” refers to the state when cracking has
propagated through the coating and the core is exposed, at which point the moment capacity of
the section has been reduced due to the loss of coating. It can be seen in Figure 3-3(b) that such
capacity loss is more significant in ligaments with thicker coating as a larger percentage of the
load is carried by the coating before failure. During section deformations from point “b” to “c”
there is a combination of plastification and hardening of the core and coating materials as well as
further fracture of the coating material on the tensile region. Combination of these two effects
leads to a relatively constant moment capacity. The onset of core fracture (point “d”) takes place
at large section curvature and is accompanied by rapid capacity loss. It can also be seen in Fig.
8(b) that point “c” occurs at smaller curvatures as the coating thicknesses increases. This is
because fracture of core starts earlier in hybrid foams with thicker or stronger coating since both
of these conditions lead to a larger compressive region in the ligament’s section. The sectional
analyses indicate that failure of the coating material in hybrid foam ligaments can lead to a
significant loss in load carrying capacity and early ligament fracture, both of which are more
severe with thicker coating.

51

Moment (N*mm)

1.2

a

60 m

1.0
0.8

b

0.6

c
d

0.4
0.2
0.0
0

2

4

6

8

Curvature (1/mm)

(a)

Moment (N*mm)

3.0
Plain
30 m
60 m
120 m

2.5
2.0

c120

1.5

c60

1.0
0.5

c30

cp

0.0
0
(b)

2

4

6

8

10

Curvature (1/mm)

Figure 3-3 Results from sectional analyses on the mid-span sections of 40 PPI ligaments. (a)
Characteristic moment-curvature response of an Al/Cu hybrid foam. (b) Moment-curvature
response of mid-sections of Al/Cu hybrid foam ligaments with different coating thickness.

52

a
Al Core

b

-

Cross-Section

d

et-Al

et-Cu
+

+ e
t-Cu

Cu
Coating

c

et-Al
+

+

-

-

Bending Strains
+

+

-

-

+

-

+

-

Bending Stresses
Figure 3-4 Schematic of the bending strain and stress profiles in the cross-section of a ligament
in a Al/Cu hybrid foam
3.4

Single Ligament Modeling
Studying the behavior of individual ligaments can offer great information on the

macroscopic performance of open-cell foams [13]. A numerical study was thus conducted on
single ligament models from an Al 6101-T6 foam[81][81][81][81] to achieve a better
understanding of the microscopic behavior and failure mechanisms of Al/Cu hybrid foams. The
numerical analyses in this study were carried out using the commercial finite element program
ABAQUS/Explicit [81] with considerations for geometric and material nonlinearities. At the
micro-level the ligaments of loaded open-cell foams are subjected to complicated loading and
boundary conditions and usually undergo large inelastic deformations. However, only two types
of boundary conditions that represent two typical deformation modes were considered in this
53

study. Both the Al core and the Cu coating were discretized using solid elements (C3D8R: eightnode linear bricks with reduced integration and hourglass control). A circular shape cross-section
was assumed and it was defined to be non-uniform along the length (L) of the ligament based on
the model proposed by Jang and Kyriakides [71]. In the noted model the cross-sectional area A at
different locations depends on the distance from the mid-span (x) and the cross-sectional area at
the mid-span (A0) of the ligament (see Figure 3-5(a) and Equation 3-2). The diameter (d0) of the
cross-section at mid-span of a base 40 PPI Al foam ligament was 0.192 mm based on the
measurements by Wang [8] and the average ligament length for a 40 PPI Al-6101-T6 foam (1.04
mm) measured by Jang and Kyriakides [18] was used as the length (L) of the ligament.





A  ξ   A0 f  ξ   A0 36ξ 4  ξ 2  1

Equation 3-2

, where ξ  x / L .

Ax

Core

x

A0
d0
L

(a)

(b)

Coating

Figure 3-5 Single ligament models for 40 PPI open-cell foam. (a) Uncoated ligament model; (b)
Coated ligament model

54

Δ

Δ
L

L
(a)

(b)

Figure 3-6 Two boundary conditions used for single ligament models. (a) Free rotation boundary
condition; (b) Fixed-rotation boundary condition

The Cu coating in the model was defined by adding a sleeve with corresponding
thickness around the Al core part along the ligament length (Figure 3-5(b)). A perfect bond was
prescribed between the coating and core model parts. Three thicknesses were considered, namely,
30 μm, 60 μm and 120 μm. The cross-sectional surface on each end was coupled with a
controlling node on which the boundary conditions were applied. Two different types of
boundary conditions (see Figure 3-6) were defined for the single ligament models, namely a freerotation boundary condition and a fixed-rotation boundary condition. The free-rotation boundary
condition was used to represent ligaments with flexible ends where the deformation would be
dominated by inelastic buckling and bending. For this model the ends of the ligament were
allowed to rotate while only one end could move along the ligament’s longitudinal axis. The
loading was defined by applying a displacement on the free translational end in the axial
direction. To invoke inelastic buckling behavior, an initial imperfection was seeded to the
ligament geometry based on the first buckling mode shape, which was obtained from a separate
eigenvalue buckling analysis. The model with fixed-rotation boundary conditions was used to

55

investigate the behavior of ligaments dominated by shear demands. In this case, the rotational
degrees of freedom of both ends were constrained. Lateral and axial translational displacements
of equal magnitude were applied on one end of the ligament while the other end was fixed.
As previously noted, ligaments in a real loaded foam specimen have more complicated
boundary conditions and loading demands than what was considered in these single-ligament
simulations. However, the models in this study were used to gain a better understanding of the
effect of coating ductility on the failure mechanisms and performance of Al/Cu hybrid foams
rather than to accurately simulate actual micro-scale behavior. Thus, the simplified models
presented here are considered adequate and necessary for the purpose of this study since the
major mechanism involved in the macro-level deformation of metal foams is the inelastic
buckling response of ligaments [13].
3.4.1 Inelastic Buckling of Single Ligaments
Figure 3-7 and Figure 3-8 show representative results of the micro-scale evaluation of
single ligament response. Specifically, the figures show the deformed shapes of the ligaments
during their inelastic response and profiles of the maximum principal strain along one the surface
of the Al core. The information shown in both figures corresponds to the stage at which the
vertical displacement

at the center of the top surface of the ligament model reaches half of it

its original length L (i.e.,

= 0.5L). Elements with excessive deformation were removed from

the deformed shape views. It can be observed that the inelastic deformations were mainly
concentrated at the center region in the free-rotation ligament models and at two critical regions
in the fixed-rotation models. The noted critical regions are those with higher curvature (

)

demands, which are qualitatively shown by considering the moment (M) gradient and the section
stiffness (EI) along the ligament (Note:

= M/EI). However, the response of coated and
56

uncoated ligaments was very different. Coating rupture is predicted for all hybrid models at the
locations of high curvature. In addition, rupture of the Al core can be observed in the coated
ligaments and is more severe in ligament with thicker coatings. This can be seen by observing
the curvature profiles and the threshold line that defines the fracture strain limit for the Al core.
Thus, Fig. 10 shows how at the noted deformation level the ligament with a 120 μm Cu coating
has almost ruptured completely while no apparent rupture can be seen in the uncoated ligament.
This can be explained by the shift of neutral axis that occurs after coating rupture as shown in
Figure 3-4 and discussed in Section 3.3. The single ligament model results also show that
damage tends to concentrate in the ruptured section as the coating thickness increases. That is,
the regions of high curvature are smaller in models with thicker coatings. This follows from the
larger capacity drop experienced by sections with thicker coatings as shown in Figure 3-3(b).
The same behavior features can be seen in the fixed-rotation ligament modeling results shown in
Figure 3-8. From these results it can be seen that while a thicker coating reduces the strain in the
compression side of the core at critical sections, while concentrated peak strains can be seen on
the tensile side. Such strain concentrations suggest possible ligament fracture at these locations.
The observations and findings from the single ligament models imply that Al/Cu hybrid foams
with thicker coatings will display a more brittle response during their inelastic collapse.

57

1.0

 ft  Al

Plain

tc = 30 μm

Position (%L)

0.8
0.6

M

∅

EI

0.4

Plain
30 m

0.2

Path

60 m
120 m

tc = 60 μm tc = 120 μm

0.0
0

1

2

3

Strain (mm/mm)

Figure 3-7 Maximum principal strain profiles on the core surface of the free-rotation single
ligament model at Δ = 0.5 L

58

1.0

Plain

tc = 60 μm

tc = 30 μm

tc = 120 μm

Position (%L)

0.8

Path

0.6
0.4

Plain
30 m

0.2
0.0
0.0

M

EI

60 m

 ft  Al

∅

0.2

0.4

120 m
0.6

0.8

1.0

1.2

Strain (mm/mm)

Figure 3-8 Maximum principal strain profiles on the core surface of the fixed-rotation single
ligament model at Δ = 0.5 L

59

3.4.2 Effect of Coating Ductility

1.0

Position (%L)

0.8

40-60

0.6

40-60-2D 40-60-4D

0.4

Path
60 m
60 m (2D)
60 m (4D)

0.2
0.0

 ft  Al
0.0

0.5

1.0

1.5

2.0

2.5

Strain (mm/mm)

Figure 3-9 Maximum principal strain profiles of 40 PPI single ligament models coated with 60
μm Cu with different ductility capacity at Δ = 0.5 L

60

Δ = 0.1L

Δ = 0.3L

Δ

Δ = 0.5L

L
60 μm

60 μm
(2D)

60 μm
(4D)

Figure 3-10 Von-Misses stress contours of 40 PPI ligament models coated with 60 μm Cu with
different ductility capacities at different deformation states
The previous discussions and results have shown that rupture of the coating and ligament
core can occur in Al/Cu hybrid foams at large deformations and thus lead to brittle macro-scale
behavior for foams with thicker coatings. As noted earlier, this is due to the low ductility of the
nanocrystalline coating material. As a result, it is of interest to investigate the effect of coating
ductility on the performance of Al/Cu hybrid foams to provide guidance on possible
improvements in the processing methods. Figure 3-9 shows deformed shapes and maximum
principal strain profiles on the Al core for single ligament models of 40 PPI Al foam with 60 μm
61

nanocrystalline Cu coating and free-rotation boundary conditions. Three cases with different
assumed ductility capacities for the coating material were considered. Cases with two times
( ε ft Cu  0.16 ) and four times ( ε ft Cu  0.32 ) the original ductility capacities are identified with
“2D” and “4D”, respectively. It can be seen that rupture of both coating and core is predicted for
the model with original coating ductility. However, with increased coating ductility capacity the
localized failure is reduced and no ligament fracture is predicted when the ductility capacity of
the coating is increased four times. The maximum principal strain profiles show that the strain
concentrations are significantly reduced and that the region experiencing large deformations
becomes larger with increased coating ductility.
Figure 3-10 shows Von-Mises stress contours along a longitudinal section of the single
ligament models with different coating ductility levels. It can be seen that at small axial
deformations (  < 0.1L), the stress distributions are similar in all cases. However, the decrease
and redistribution of stresses is obvious as the coating and core rupture and the ligament loses
load carrying capacity. It can also be seen that increased coating ductility delays fracture of the
ligament, thus allowing it to resist additional load even at large deformations. Figure 3-12 shows
plots of axial force and total internal strain energy versus the axial shortening displacements for
ligament models with different coating ductility. The total strain energy reported by ABAQUS,
which is the sum of the recoverable strain energy, the energy dissipated by plastic deformations,
and the energy dissipated by damage, can be considered an indication of the energy absorption
capacity. It can be seen that in all cases the axial forces peak at small deformation levels and
drops rapidly during the inelastic buckling response of the ligament. From Figure 3-11 it can be
seen that the peak load carrying capacity of the coated ligaments is considerably higher
compared to the uncoated ligament. However, the coated ligament model with original coating

62

ductility loses its entire load carrying capacity at large deformations (  > 0.45L). Conversely,
increased coating ductility notably improves the ligament’s residual load carrying capacity.
Figure 3-12 and Table 3-2 show that the energy absorption capacity of the ligament model with
original coating ductility reaches a plateau at about 0.45 strain while coatings with increased
ductility can significantly improve the ligament’s energy absorption capacity.

14

Plain
60 m
60 m (2D)
60 m (4D)

Axial Force (N)

12
10
8
6
4
2
0

0.0 0.1 0.2 0.3 0.4 0.5 0.6
Displacement (L)
Figure 3-11 Load-displacement response of single 40 PPI ligaments with different ductility Cu
coatings under compression

63

Internal Energy (N*mm)

5
4
3

Plain
60 m
60 m (2D)
60 m (4D)

2
1
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Displacement (L)

Figure 3-12 Internal Energy-displacement response of single 40 PPI ligaments with different
ductility Cu coatings under compression

64

Table 3-2 Energy absorption of single ligament models from 40 PPI foam with 60 µm Cu
coating (Δ = 0.6L).
Energy Absorption Energy absorption
Coating Thickness
(N·mm)

enhancement

60 µm

1.85

-

60 µm (2D)

2.93

58%

60 µm (4D)

3.22

74%

In reality the deformation mechanism of a single ligament involves axial, bi-directional
bending, shear and torsion effects, which lead to more complicated loading and boundary
conditions than those used in this study. However, the presented single ligament simulations
have shown that coatings with low ductility will cause concentrated failure at critical sections of
the foam ligaments and reduce the ligament’s energy dissipation capacity. Such effect is also
expected to occur under the more complex conditions in an actual loaded foam ligament.
3.5

Multi-cell Modeling
Numerical models with multiple characteristic cells were established to investigate

macroscopic behavior. Jang and Kyriakides [71] have shown that a single characteristic cell
model based on the Kelvin cell geometry with periodic boundary conditions can be used to
adequately predict the compressive behavior of Al foam, but that obtaining a more realistic
crushing response (the initiation and expansion of collapsing cell bands) requires using models
with multiple characteristic cells. It needs to be noted that the numerical models used by Jang
and Kyriakides [71] assumed an elastic-perfect plastic material response and did not consider

65

material damage or failure, which is a valid assumption in the simulation of homogeneous Al
foams. However, the definition of material failure is essential for the simulation of Al/Cu hybrid
foams since their failure mechanism has been observed to involve rupture of the coating, even
ligaments, and to have an increasingly brittle response with thicker coatings [8]. As will be
shown in Section 3.5.5, a unit-cell model with periodic boundary conditions is not suitable for
the simulation of hybrid foams if the low ductility capacity of the coating is considered. As a
result, numerical models with foam geometry of finite dimensions (multiple cells) were used for
the investigation.
3.5.1 Foam Geometry
The models used in this study were established based on the Kelvin cell model (or
tetrakaidecahedron) by packing characteristic cells as done by Jang and Kyriakides [71]. Kelvin
cells have 6 squares and 8 hexagons for a total of 14 faces per cell and are highly optimized
partitions of space to minimize surface energy. Such structure have been widely used to study
cellular materials [14, 15, 71, 82-85] as it is deemed to naturally appear in foams due to the
growth and interaction of gas bubbles during the foaming process [73]. The multi-cell models
consisted of 64 characteristic cells with four cells in each dimension. Figure 3-13 shows front
and isotropic views of a multi-cell model. It has been shown that the introduction of small
perturbations to the basic Kelvin cell microstructure has advantages over the use of a perfect
geometry, by breaking up the symmetry of the banded collapse localization of the perfect case,
and leading to a more random distributed buckling and collapse [71]. The model’s nodal
coordinates were thus perturbed according to Equation 3-3 [71], where x is the perturbed position
vector, x0 is the initial position vector, p is a normally distributed random number between -0.5
and 0.5, φ is the amplitude of the disturbance and L is the average length of the ligament. Since
66

the effect of randomness is not the main interest of this study, a relatively small value was
assumed for the disturbance in this study ( φ = 0.1).

x  x0  pφL , 0  φ  0.5

Equation 3-3

Anisotropy in the foam cellular network was introduced to the Kelvin microstructure by
using the same approach as Gong and Kyriakides [14]. The anisotropy λ of the foam geometry
is defined by Equation 3-4, where h1 and h2 are the heights in rise and lateral direction of the
generated structure, respectively; and was assumed to be equal to 1.2 in this study based on the
measurement on similar foams in literature[71].

h1 / h2  λ

(a)

Equation 3-4

(b)

Figure 3-13 Multi-cell model with rendered cross-section (4×4×4 cells). (a) Front view; (b)
Isotropic
3.5.2 Ligament Geometry
The ligaments in the multi-cell models were discretized using eight beam elements (B32:
three-node quadratic shear deformable beam elements). As for the single ligament models, non67

uniform sections were defined for the ligament in the multi-cell model based on Equation 3-2. A
circular section was used for the Al core. The diameter of the core section at mid-span was
defined by calibrating the measured plateau stress from plain foam experiments with the results
from numerical models. The calibrated mid-span diameter for the Al core was 0.18 mm, which is
in the range of measured diameters of 0.192 ± 0.014 mm by Wang [8]. The coatings were
modeled by duplicating the original foam structure with a pipe section with wall thickness equal
to that of the corresponding coating. A "tie" constraint was used to connect the base foam part
and the coating part, which lead to the assumption of a perfect bond between the coating and the
base material. Representative plain and coated ligaments from the multi-cell models are shown in
Figure 3-14 in rendered views to show the different section assignments.

(a)

(b)

Figure 3-14 Single ligament in multi-cell models with rendered section: (a) uncoated ligament;
(b) coated ligament

3.5.3 Boundary Conditions
Jang and Kyriakides [71] used periodic boundary conditions for their unit-cell model and
relaxed the periodic boundary conditions (with free surfaces in one or both lateral directions) for

68

the multiple cell models. Periodic boundary conditions are applied by prescribing relations
between the displacements of corresponding nodes on opposite surfaces of the representative
volume element (RVE). Such boundary conditions work well for traditional metal foams
composed of ductile ligaments. However, as shown in later sections, the deformation of Al/Cu
hybrid foams (especially those with thicker coatings) under compression generally involves the
fracture of ligaments. If fracture occurs in the ligaments connected to the nodes with prescribed
boundary conditions, the constraint definitions for periodic boundary conditions as well as the
applied displacement will not be appropriate. Thus, a continuous loading surface is preferred for
the simulation of Al/Cu hybrid foams. Two rigid surfaces (normal to the loading direction) were
defined at the top and bottom surfaces of the foam model. The bottom surface was fixed while a
displacement boundary condition was defined on the top surface. Coulomb friction with a
coefficient of 0.5 was used in the contact definition between the rigid surfaces and the foam
model.
3.5.4 Contact Simulation
The densification feature in the compressive response of open cell foams follows from
the contact that develops between ligaments as they collapse. Cell collapse starts locally and then
propagates to other regions after densification of the initially collapse region is reached. This
pattern continues and propagates throughout the foam domain and densification at a macro-scale
is observed when most of the cells have collapsed [13]. In this study, the contact between
ligaments was simulated based on the approach proposed by Jang and Kyriakides [71]. Nonlinear spring elements were defined between collinear nodes in the cell vertices in the loading
direction (see Figure 3-13(a)). The stiffness of the nonlinear springs defined in this study follow
the recommendation of the work by Jang and Kyriakides [71] and are shown in Table 3-3, where
69

y is the gap between adjacent nodes in the rise (i.e., loading) direction as defined by Equation 3-5,
where dn is the core diameter of the ligament cross section evaluated using Eq. 1 at x = 0.5L plus
two times the coating thicknesses, if any.

ψ  2 λL  dn

Equation 3-5

Table 3-3 Non-linear stiffness of spring elements for ligament contact simulation.
K

Displacement

(N/mm)

δ (mm)

0

δ < 0.2 ψ

5

0.2 ψ < δ < 0.7 ψ

110

0.7 ψ < δ

It can be seen from Table 3-3 and Equation 3-5 that the stiffness of the nonlinear springs
depends on the ligament size, which increases with coating thickness. However, the deformation
of Al/Cu hybrid foams under compression usually involves fracture and separation of the coating
and sometimes complete ligament fracture. This leads to a complicated contact phenomenon at
large deformation (densification region). Thus the densification strain of Al/Cu hybrid foams
(with non-ductile coatings) is not solely related to the original ligament size. Consequently, the
focus of this study was to understand the failure mechanisms and the prediction of peak and
plateau stresses in Al/Cu hybrid foams, and not intended to characterize the densification region.

70

3.5.5 Size Effect in Multi-cell Models

3.0

Stress (MPa)

2.5
2.0
1.5
1.0
Unit Cell & PBC
3x3x3 Cells
4x4x4 Cells
5x5x5 Cells

0.5
0.0
0.0

0.1

0.2

0.3

0.4

Strain (mm/mm)

Figure 3-15 Quasi-static compressive stress-strain response from beam-element based finite
element models with different sizes and boundary conditions

Multi-cell models were established to investigate the macroscopic compressive behavior
of Al/Cu hybrid foams. Different size models were evaluated to determine a proper size. Figure
3-15 shows the result of three different model sizes for 40 PPI Al/Cu hybrid foams with 60 μm
Cu coating. Also shown in the figure is the response obtained by using a single cell with periodic
boundary conditions. It can be seen that even though all models predict almost the same initial

71

stiffness and similar peak stress, the post-peak behavior varies significantly. The unit cell model
with periodic boundary condition shows a negative slope in the stress plateau region. This is
because the model assumes uniform collapse of all the foam material that it represents. Such
assumption is valid if the foam behavior is ductile and has no localized softening region (e.g., Al
foam). Localized damage of the coating material in Al/Cu hybrid foams causes softening in
concentrated regions and leads to fluctuations on the stress plateau in its macroscopic behavior,
which can only be captured using models with multiple cells. Such stress fluctuation on the stress
plateau is shown to be captured by the multi-cell models. However, the 18-cell model (3x3x3)
had an irregular behavior due to the instability of the spring elements at large deformation. On
the other hand, predictions from 64-cell (4x4x4) and 125-cell models (5x5x5) were very close.
The 4x4x4 multi-cell models were thus chosen for this study.

72

3.5.6 Quasi-static Compressive Response

Plain FE
30 m FE
60 m FE
120 m FE
Plain EXP
30 m EXP
60 m EXP

Stress (MPa)

6

4

2

0
0.0

0.1

0.2

0.3

0.4

Strain (mm/mm)

Figure 3-16 Simulated and experimental quasi-static compressive stress-strain response for 40
PPI plain Al and Al/Cu hybrid foams

73

(a)

(b)

Figure 3-17 Simulated collapse progression of coated (120 μm) foam under uniaxial quasistatic
compression; (a) Low ductility coating; (b) High-ductility coating.

Figure 3-16 presents results for the quasi-static response of hybrid foams obtained from
the 4x4x4 multi-cell model. Responses for plain Al foam as well as hybrid foams with 30, 60,
for

results compare well with the experimental data, being able to capture the peak stress, the
follow-up stress drop (which increases with coating thickness), and the fluctuating response, with
a slight hardening global trend, in the plateau region. Of all these features, the ability to capture
the large stress drop after the initial peak stress in the hybrid foams (and its increase with coating
thickness) is the most notable characteristic of the simulation results. Values for the elastic
modulus E foam , maximum stress σ max (i.e., initial peak stress), post-peak minimum stress σ min
and plateau stress σ plateau (defined as the average stress within 15% to 40% strain range) are
74

shown in Table 3-4. The models led to a good prediction of the elastic modulus for plain Al foam
but overestimated the elastic modulus of Al/Cu hybrid foams, especially with thick coatings.
This may be due to differences between the actual stiffness of the electrodeposited copper and
the values used in the numerical models, which were adopted from the literature [66]. However,
the deviations in predicted vs. experimental elastic modulus do not affect the main findings of
this study. The post-peak stress drop in the behavior of Al/Cu hybrid foams is caused by the
localized collapse of cells in an inclined shear band (Figure 3-16(a)). The fluctuation observed
after the peak is primarily caused by the successive collapse and densification of cells at different
regions during loading. As presented in Section 3.3, section loss from the failure of thicker
coatings has a bigger impact on the overall load carrying capacity of foam ligaments. As a result,
the macro-scale initial stress drop, as well as the magnitude of the fluctuations in the plateau
region, is larger in Al/Cu hybrid foams with thicker coatings; which is captured by the simulation
results. It should be noted that the simulation for a hybrid foam with 120 μm coating showed a
very brittle response (see Figure 3-16), as shown by the significant initial stress drop and the
large fluctuations in the stress plateau region. The solution also suffered from convergence
problems at large deformation. This indicates that low ductility nanocrystalline depositions can
be detrimental to the stable compressive response and energy absorption capacity of foams with
thick coatings.

75

Table 3-4 Comparison of stress values between results from experiment and FE simulations.
Plain

E foam

Exp

LD

80

76

30 μm
HD Exp

60 μm

120 μm

LD

HD

Exp

LD

HD

Exp

LD

HD

174

135

322

819

-

122

162

322

-

819

0.77 0.79

-

1.6

1.39 1.53 2.34 2.38 2.55

-

5.64 5.93

0.72 0.72

-

0.97 1.25 1.53 1.34 1.70 2.35

-

2.46 4.35

0.98 1.00

-

1.40 1.44 1.64 2.05 2.05 2.62

-

2.44 5.80

(MPa)

σ max
(MPa)

σ min
(MPa)
σ plateau

(MPa)
Note: Exp = Experimental data; LD = Numerical simulations with low ductility coating; HD =
Numerical simulations with high ductility coating.

3.5.7 Effect of Coating Ductility
This section presents simulation results to highlight the effect of increased ductility of the
coating material on the macro-scale compressive behavior of Al/Cu hybrid foams. Multi-cell
model simulations employing coating material definitions with large ductility were conducted.
The tensile fracture strain of Cu coating was assumed to be the same as the Al core
( ε ft Cu  ε ft  Al  0.5 ).

76

12
30 m FE
60 m FE
120 m FE
30 m FE Ductile
60 m FE Ductile
120 m FE Ductile

Stress (MPa)

10
8
6
4
2
0
0.0

0.1

0.2

0.3

0.4

0.5

Strain (mm/mm)
Figure 3-18 Simulated quasi-static compressive stress-strain response of Al/Cu hybrid foams
with low and high coating ductility

Figure 3-17 shows the simulated collapsing response of a Al/Cu hybrid foam (120 μm Cu
thickness) with low and high ductile coatings. In both cases failure is caused by the formation of
a shear band of cells that starts collapsing under the compressive load. This state corresponds to
the peak stress response and occurs at low global deformations. The collapse sequence of for the
low-ductility coating model (Figure 3-17(a)) shows that as global deformations increase damage
concentrates in the collapse region and fracture of the ligaments occurs. Conversely, the
deformation sequence for the high-ductility coating model (Figure 3-17(b)) shows that other
77

regions in the foam start collapsing after the first collapsing band forms, which leads to a more
uniform densification of the foam domain. Thus, a hybrid foam with high-ductility coating will
have a failure mode similar to that exhibited by homogeneous metal foams. Otherwise, hybrid
foams with low-ductility coatings will exhibit a more brittle failure and non-uniform damage
regions; especially highly reinforced foams through thicker or stronger coatings.
Figure 3-18 compares the simulated stress-strain compressive response of Al/Cu hybrid
foams with low and high coating ductility. Values for the maximum stress σ max , post-peak
minimum stress σ min and plateau stress σ plateau are given in Table 3-4. It can be seen that coating
ductility does not affect the peak stress level. However, the subsequent stress drop that is so
prominent in hybrid foams with low-ductility coatings is mostly eliminated by the use of a highductility coating, and the stress fluctuation in the plateau region is reduced significantly. This is
because the high-ductility coating permits the ligaments to remain integral and thus capable of
sustaining load at large deformations, even if they have reached their plastic limit at critical
sections along their length. The energy absorbed by the Al/Cu hybrid foam models up to 50%
strain were calculated by determining the area under a stress-strain curve and listed in Table 3-5.
The results show the significant enhancement in energy absorption capacity that can be obtained
with improved coating ductility, especially for foams with thicker coatings.

78

Table 3-5 Energy absorption of multi-cell models of 40 PPI Al/Cu foams up to 50%
strain
Low coating ductility High coating ductility
Coating Thickness

3.6

Enhancement
(N·mm)

(N·mm)

30 μm

0.71

0.79

11%

60 μm

1.05

1.26

20%

120 μm

1.43

2.76

93%

Discussion
The quasi-static behavior of Al/Cu hybrid foams under compression has been

investigated using nonlinear finite element simulations and it has been shown that their overall
performance can be significantly enhanced if the ductility of the coating material is improved.
The ductility of metals can be improved by annealing due to the recrystallization and grain
growth that occurs at high temperature [46, 58]. The effect of coating ductility on the
performance of Al/Cu hybrid foams will be presented in CHAPTER 4. It needs to be noted that
the discussion on the effect of coating ductility also reflects the consequences of lacking reliable
material properties at the scale of the base metal foam ligaments and the electrodeposited
coatings. It is clear that if the bulk material properties for Cu and Al, which usually have a high
ductility level, were used in the simulations predicted capacities and the deformation
mechanisms of the hybrid foams would have been incorrect.
The presented simulations and discussions were based on Al/Cu hybrid foams; however,
the methods and findings are considered to apply similarly to the compressive behavior Al/Ni
hybrid foams manufactured using similar processes due to the reported low ductility of
79

electrodeposited nickel [86]. Actually, since electrodeposited nickel usually has much higher
strength compared to electrodeposited Cu, the effect of embrittlement and reduction in energy
absorption efficiency due to the low-ductility coating will be more significant for Al/Ni hybrid
foams. Figure 3-19 shows the compressive stress-strain curves of Al/Ni hybrid foams with
various coating thicknesses reported by Bouwhuis et al. [6] and simulated responses using
numerical methods presented in this study. It can be seen that both the initial stress peak and the
post-peak stress drop are more significant for Al/Ni hybrid foams compared to Al/Cu hybrid
foams with similar coating thicknesses.

Plain FE
25.6 m FE
39 m FE
72.3 m FE
Plain EXP
25.6 m EXP
39 m EXP
72.3 m EXP

Stress (MPa)

8

6

4

2

0
0.00

0.05

0.10

0.15

0.20

Strain (mm/mm)

Figure 3-19 Simulated and experimental quasi-static compressive stress-strain response for 20
PPI plain Al and Al/Ni hybrid foams
80

3.7

Conclusions
The quasi-static compressive behavior of Al/Cu hybrid open-cell foams manufactured by

the electrodeposition of nanocrystalline Cu onto open cell Al foams was studied through
nonlinear finite element simulations at different scales. Detailed failure mechanisms at the
micro-scale level were studied conducting fiber-based sectional analyses and simulations on
single ligament models discretized using continuum-type finite elements. The stress-strain
compressive response at the macro-scale was studied through multi-cell models, based on
repeating Kelvin cells in three dimensions, established using composite beam elements. The
following conclusions are drawn from this study:
1)

The numerical simulations have shown that rupture of the electrodeposited Cu

coating is a prevailing failure mechanism and fracture of the ligaments can occur in Al/Cu hybrid
foams loaded in compression. Such behavior is due to the low ductility of the electrodeposited
nanocrystalline coating and needs to be considered in the simulation of hybrid foams.
2)

Sectional analyses and micro-scale simulations of single ligaments have shown

that coating rupture occurs at locations under large flexural curvature demands, with an
increasing detrimental effect proportional to the coating thickness; an effect that can lead to
fracture of the cell ligaments. As a result, ligaments in Al/Cu hybrid open cell foams with thicker
coating have a brittle response as they collapse under compressive loads.
3)

Macro-scale simulations using multi-cell models were shown to adequately

capture the unique quasi-static behavior of Al/Cu hybrid foams as validated by experimental data.
The models were able to capture the enhancement of strength and stiffness with coating
thickness as well as the increased stress drop upon the initiation of cell collapse, and the large
response fluctuation in the stress plateau region. These features were shown to increase in

81

proportion to the Cu coating thickness and the stress drops and fluctuations are due to the
reduced ductility capacity of the electrodeposited Cu. Thus, while increased coating thickness
improve strength and stiffness, the lack of coating ductility limits and jeopardizes the overall
energy absorption capacity of the system, especially for foams with thick coatings. While such
behavior had been previously observed in experimental results no explanation or ways to
numerically simulate it had been reported before this study.
4)

The micro- and macro-level simulations have shown that the performance of

Al/Cu hybrid foams under compression can be enhanced by improving the ductility of the
coating material. Coatings with higher ductility can reduce fracture of the ligaments at the micro
level and avoid the localization of collapsed regions at macroscopic level. Such improvement can
lead to a sustained load carrying capacity under compression and improved energy absorption
characteristics, an enhancement that would be more significant for Al/Cu hybrid foams with
thicker coatings.

82

CHAPTER 4 EFFECT OF COATING DUCTILITY ENHANCEMENT
4.1

Overview
It has been demonstrated that hybrid foams can have an enhanced and controllable

modulus, strength and energy absorption capacity. The numerical investigation in CHAPTER 3
has shown that the low ductility of the nanocrystalline structured coating materials in hybrid
foams can cause rupture and separation of the coating material and significantly reduce their
energy absorption capacity. In this chapter, Al/Cu hybrid foams manufactured through
electrodeposition were subjected to an annealing process with the aim of increasing the ductility
of the coating and thus achieve improved performance and the effect of annealing temperature
and time on their macro- and micro-level behavior under compression and tension were
investigated using experimental methods.
Figure 3-1 shows the compressive stress-strain response and test pictures of Al and Al/Cu
hybrid (with 60 μm Cu coating) open cell foams. Both samples had a cell size of 40 pores per
inch (PPI) and a relative density of 6 %. It can be observed that Al/Cu hybrid foam shows a large
stress drop after the initial peak (onset of cell plastification) and a plateau region with
considerably more fluctuations compared to the plain Al foam. The post-peak stress drop
observed in the Al/Cu hybrid foam was accompanied by ligament breakage and material spalling,
which is characteristic of Al/Cu hybrid foams and seen to increase with coating thickness [8].
Similar behavior has been reported for Al/Ni hybrid foams [6, 7, 40]. Bouwhuis et al. [6]
observed cracking and wrinkling of the nickel coating in Al/Ni hybrid foams and thought it that
could be caused by the low ductility of the coating.

83

The electrodeposited coatings in hybrid foams have a nanocrystalline structure [7, 8] that
usually leads to higher yielding strength but significantly less ductility compared to the bulk
material counterparts [46, 62, 63, 66, 74, 87]. Numerical simulations in CHAPTER 3 indicate
that the resulting local coating failures lead to concentrated deformations at weakened critical
sections at low strain levels and thus inhibit the energy absorption performance of hybrid foam
materials [88]. It is thus hypothesized that the energy absorption performance of metallic hybrid
foams can be effectively enhanced by improving the ductility of the electrodeposited coating.
4.2

Annealing Procedure
Annealing is a well-established heat treatment process for metals to release internal

stresses and enhance ductility through recrystallization and grain growth, which also usually
results in a lower modulus and yield strength [46, 58, 89]. All annealing processes were
performed in a Centorr M60 vacuum annealing furnace. The specimen chamber was evacuated
by a Varian VHS-4 diffusion pump. Annealing was conducted at temperatures of 180 °C and
400 °C at a base vacuum pressure of 3×10-5 torr. The annealing temperature was measured using
an Omega Type-K thermocouple inside the furnace, which was positioned close to the samples.
The noted temperatures were about one third or one half, respectively, of the melting temperature
for copper (absolute temperature scale), between which the atomic mobility is sufficient to affect
mechanical properties [58].
The annealing process was carried out by first evacuating the specimen chamber. The
heating process was not initiated until the desired pressure within the chamber was achieved. The
temperature was controlled to achieve a ramp-up rate of about 20 °C/min. Upon reaching the
target annealing time the heating element of the furnace was turned off and the specimen was
allowed to cool down through heat dissipation of the system. The annealed specimen was taken
84

out of the furnace after the chamber temperature was below 100 °C. The time at which heating
element was turned off and specimen was removed were recorded and the cool down rate was
estimated to be about 6 °C/min based on the records. Thus, the actual exposure time was longer
than the nominal annealing time, especially when annealing was carried out at 400 °C.
Table 4-1 shows the test matrix for the reported study on the effect of annealing on the
performance of Al/Cu hybrid foams. The samples were identified based on the testing method,
copper coating thickness, nominal annealing temperature and annealing time considered, and the
naming convention is shown in Figure 4-1. The nominal annealing time is defined as the time
during which the sample temperature was kept at the target value. Also listed in Table 4-1 is the
nominal relative density of the hybrid foam samples (RD’), defined as the ratio of the volume of
the solid (volume of Al and Copper, VAl + VCu) to the volume of the sample (Vs), was calculated
based on the relative density of the base aluminum foam (RD) and the weight gain from the
copper coating (mCu) after the electrodeposition process using Equation 4-1.

RD' 

“C” - Compression
“T” - Tension

VAl  VCu
m
 RD  Cu
Vs
ρCuVs

C30-HT-A1

Equation 4-1

Nominal annealing
time in hours
“LT” - 180 °C
“HT” - 400 °C

Nominal copper coating
thickness (μm)

Figure 4-1 Naming convention of test samples

85

Table 4-1 Test Matrix
Sample ID

Nominal Coating

Annealing

Annealing Weight needed Nominal Relative

Thickness (µm) Temperature (°C) Time (h)

(g)

Density (%)

C30/T30

30

-

-

6.1

8.1

C60/T60

60

-

-

13.8

10.7

C90

90

-

-

22.5

13.7

C30-LT-A1

30

180

1

6.1

8.1

C60-LT-A1

60

180

1

13.8

10.7

30

400

1

6.1

8.1

60

400

0.0833

13.8

10.7

60

400

1

13.8

10.7

C60-HT-A2

60

400

2

13.8

10.7

C90-HT-A1

90

400

1

22.5

13.7

C30-HT-A1
T30-HT-A1
C60-HT-A0*
C60-HT-A1
T60-HT-A1

*Note: Samples were annealed at the target temperature for 5 minutes.
4.3

Mechanical Testing and Characterization
Quasi-static compression and tension tests were conducted on the plain Al and hybrid

Al/Cu foams samples (size: 25.4 mm x 25.4 mm x 50.8 mm) using a universal loading frame as
shown in Figure 4-2. The samples were loaded along their long dimension. For compression
testing aluminum plates (3 mm thick) were bonded using epoxy to the ends of the foam samples
before testing to provide the sample with uniform loading surfaces. The tension tests were
conducted by mounting the foam samples ends using epoxy to two loading fixtures that were
then connected to the loading frame. In order to include more cells in the section area
perpendicular to the loading direction, no section reduction was applied to the prism specimen.
The rate of loading was 2.54 mm/s for the compression tests and 0.12 mm/s for the tension tests.

86

The annealed and unannealed Al/Cu hybrid foam samples were evaluated with a JEOL
JSM 6610 LV field emission scanning electron microscope (SEM). The effect of annealing on
the micro-scale deformation mechanism of Al/Cu hybrid foams were investigated using
secondary electrons (SE) with a 15 kV acceleration voltage. In addition, backscattered electrons
(BSE) with an acceleration voltage of 10 kV were used to evaluate the formation of intermetallic
compound at the Al/Cu interface.
Table 4-2 Information for Al and Al/Cu foam samples
Cu Coating Thickness Cu Coating Weight Relative Density
(μm)

(g)

(%)

0

0

6

30

6.1

8.1

60

13.8

10.7

a)

b)

Figure 4-2 Images of hybrid Al/Cu foams samples with uniform coating in a) Uniaxial
compressive test; and b) Uniaxial tensile test

87

X-ray diffraction (XRD) was used to evaluate the effect of annealing on the crystallite
size of the copper coating in the Al/Cu hybrid foams. XRD patterns were obtained on a Bruker
D8 DaVinci diffractometer equipped with Cu X-ray radiation operating at 40 kV and 40 mA.
Peak intensities were obtained by counting the Lynxeye detector every 0.01o at sweep rates of
0.5o 2/min. The crystallite size of the deposited Cu was determined using the Williamson-Hall
method [90] as detailed in the results section.
4.4

Results

4.4.1 Crystallite Size
Figure 4-3 shows the first two peaks of the XRD diagram obtained for two Al/Cu hybrid
foam specimens, one of which was annealed after electrodeposition while the other one was not.
The peak positions of the observed pattern are summarized in Table 4-3 together with the peak
positions for copper reported in the literature [91]. It can be seen that the peak positions in the
pattern of the Al/Cu foams agree well with those for copper since the aluminum was covered by
the copper coating.
The crystallite size of nano-copper coating was determined using the averaged estimation
of the first three peaks and the Williamson-Hall method [90]. In this method the true peak
broadening is assumed to be composed of crystallite size broadening and strain broadening,
given by Equation 4-2 where d is the mean crystallite size, K is shape factor, λ is the X-ray
wavelength (0.15406 nm), β is the line broadening at half the maximum intensity (FWHM), and
θ is the Bragg angle and ɛ is the microstrain. The microstrain was determined to be 0.4 percent
for Al/Cu hybrid foam and 0.06 percent for annealed Al/Cu hybrid foam. It can be seen in Table

88

4-3 that the annealed specimen had a significantly increased crystallite size, which indicates the
occurrence of grain growth from annealing.

 cos  

K
  sin 
d

Equation 4-2

2000

Intensity/Counts

Al/Cu Foam
Al/Cu Foam (Annealed)
1500

1000

500

0
42

44

46

48

50

52

2 (degrees), WL=0.15406 nm

Figure 4-3 First two peaks of XRD patterns of copper (Cu) coated aluminum (Al) foam
specimens with and without annealing

Table 4-3 Summary of results from XRD measurements on Al/Cu hybrid foam specimens.
Estimated
Material

1st Peak

2nd Peak

3rd Peak

Average

Position

Position

Position

Crystallite Size
(nm)

Al/Cu Hybrid Foam
Al/Cu Hybrid Foam
(Annealed)
Copper [91]

43.23°

50.32°

73.99°

38

43.15°

50.26°

73.97°

56

43.30°

50.43°

74.13°

-

89

4.4.2 Compressive Response
Figure 4-4 shows the compressive response for plain Al foam and Al/Cu hybrid foams
with and without an annealing process (400 °C for 1 h). It can be seen in Figure 4-4a that the
annealed plain Al foam had a reduced plateau stress and thus lower energy absorption capacity.
This is because the original plain Al foam has been heat treated to increase its strength and the
annealing process causes relaxation due to reorganization of the crystal structure.
The compressive response for the unannealed Al/Cu hybrid foams shown in Figure 4-4b
to Figure 4-4d feature an initial peak stress at the onset of plastification (defined as the onset of
plastic collapse of the foam cells) followed by a drop in stress and then a stress plateaus with
considerably more fluctuations than for the annealed samples. It can be seen that the noted stress
drop and the fluctuations become more pronounced as the coating thickness increases.
The response features just noted are caused by the concentrated local damage and brittle
behavior of the unannealed samples as shown in Figure 4-5a. It can be seen in Figure 4-5a that
an inclined collapsing band forms at a small strain level and that the subsequent deformation is
caused by crushing of foam material within that region. On the other hand, no significant stress
drop and a smoother stress plateau, with a magnitude similar to or higher than the stress level at
onset of plastification, were observed in the stress-strain response of the annealed Al/Cu hybrid
foam samples.
It is of interest to point out that the plastification stress of the annealed samples is close to
or higher than the corresponding stress level in the unannealed samples, even though the yielding
and plateau stress decreased in the plain Al foam samples after annealing. The enhanced
response of the annealed Al/Cu hybrid foam samples compared to the unannealed samples was
more significant in samples with thicker coatings. This can be expected as the influence of the
90

low ductility coating on foam performance is more severe as the ratio of coating thickness to the
original strut cross-section dimensions increases [92].

20
C0
C0-HT-A1

15

a)
Stress (MPa)

Stress (MPa)

20

10
5
0
0.0

0.2

0.4

0.6

10
5
0
0.0

0.8

0.4

0.6

0.8

20

20
C60
C60-HT-A1

15

c)
Stress (MPa)

Stress (MPa)

0.2

Strain (mm/mm)

Strain (mm/mm)

10
5
0
0.0

b)

C30
C30-HT-A1

15

0.2

0.4

0.6

15

d)

10
5
0
0.0

0.8

C90
C90-HT-A1

0.2

0.4

0.6

0.8

Strain (mm/mm)

Strain (mm/mm)

Figure 4-4 Compressive stress-strain responses of Al/Cu hybrid foams with different coating
thicknesses

The improved ductile macroscopic behavior of the annealed samples can be attributed to
the fact that there is no localized damage and the samples stay intact throughout the deformation,
as shown in Figure 4-5b. It can also be seen from Figure 4-4 that the densification region starts

91

earlier in the annealed samples. However, it needs to be noted that the densification strain (the
strain at which densification region starts, which can be defined as the strain level at the
intersection of the best fitting lines (using a least square fit) of the collapse region and the
densification region, respectively) is overestimated in the unannealed samples due to the severe
separation and breakage of material during the deformation (see Figure 4-5a).

C90
0%

18%

35%

53%

70%

a)

C90-HT-A1
0%

18%

35%

53%

70%

b)

Figure 4-5 Sequential images from the compression testing of Al/Cu hybrid foam samples of (a)
C90 and (b) C90-HT-A1

To provide further understanding of the effect of annealing on Al/Cu hybrid foams an
investigation on the micro-scale failure mechanisms was conducted. Two Al/Cu hybrid foams
samples (with and without annealing) with a nominal coating thickness of 50 μm were
92

compressed to about 6% strain and SEM (scanning electron microscope) images were taken at a
region of collapsed cells (see Figure 4-6).
The noted strain level was chosen since it is shortly after the stress drop after the onset of
plastification was observed in the compressive behavior of the unannealed sample. Rupture of
the copper coating (see arrows in Figure 4-6a) was observed in the unannealed sample. A similar
observation was reported by Bouwhuis et al. [6] for Al/Ni hybrid foams. The coating rupture can
be attributed to the low ductility of the electrodeposited metal which usually has a
nanocrystalline structure [66, 74, 92].
The resulting loss in load carrying capacity of the ligaments as a result of coating failure
is considered to be the main cause for the observed stress drop and the fluctuating plateau trace
in the compressive behavior of unannalead hybrid foams. Conversely, Figure 4-6b shows some
largely deformed ligaments in the annealed sample at the same deformation level, where no
obvious damage or rupture of the coating can be seen. This enhanced response of the ligaments
led to the more ductile macroscopic behavior observed in the prism samples. The effect of
annealing is more significant in Al/Cu hybrid foam samples with thicker coating as the rupture of
coating has larger influence on the load carrying capacity of the ligaments in those samples.

93

a)

b)

500μm

500μm

Figure 4-6 SEM images of ligament deformation in 40 PPI Al/Cu hybrid foam samples (50 μm
copper coating) compressed after the onset of plastification to about 6 % strain: (a) without
annealing; (b) with 1 hour annealing at 400 °C
4.4.2.1 Effect of Annealing Temperature
Figure 4-7 shows the compressive behavior of Al/Cu hybrid foams annealed at different
temperatures. Two temperature levels were considered, namely, 180 °C and 400 °C. All samples
in this discussion were annealed for one hour. It can be seen that the samples with 30 μm coating
had basically the same performance after been annealed at different temperatures (see Fig. 9a).
However, at a large strain level, fracture was observed in the sample with 60 μm coating
annealed at 180 °C (C60-LT-A1) as shown in Figure 4-8, which led to a delayed densification
region in the stress-strain response shown in Fig. 9b.
The observed fracture in sample C60-LT-A1 was not as severe as that observed in
unannealed samples and it is considered to be caused by insufficient annealing. The observed
behavior of Al/Cu hybrid foams annealed at different temperature indicates that the enhancement
on the mechanical performance from annealing can be achieved at lower temperature (180 °C).
However, in order to achieve a ductile behavior at large strain level, additional time may be

94

needed when annealing is conducted at lower temperature, especially for Al/Cu hybrid foams
with thicker coating.

15

20

a)

C30-LT-A1
C30-HT-A1

Stress (MPa)

Stress (MPa)

20

10
5
0
0.0

0.2

0.4

0.6

15

b)

10
5
0
0.0

0.8

C60-LT-A1
C60-HT-A1

0.2

Strain (mm/mm)

0.4

0.6

0.8

Strain (mm/mm)

Figure 4-7 Compressive response of Al/Cu hybrid foams annealed at different temperature

a)

60%

b)

80%

Figure 4-8 Images from the compressive testing of sample C60-LT-A1 at 60 % and 80 % strain
4.4.2.2 Effect of Annealing Time on Compressive Behavior
Figure 4-9 shows a comparison of the compressive behavior of Al/Cu hybrid foam
samples (60 μm nominal coating thickness) annealed for 5 minutes (A0), 1 hour (A1) and 2
hours (A2), respectively at 400 °C. It can be observed that the responses of the C60-HT-A0 and
C60-HT-A1 samples were almost identical. The two samples showed a very ductile behavior

95

(see Fig. 11b) and no fluctuations were observed on the stress-strain traces. Thus, a short
annealing period at 400 °C was sufficient for Al/Cu hybrid foams with 60 μm nominal copper
coating thickness. However, brittle ligament failures were observed during the deformation of
the sample annealed for 2 hours (C60-HT-A2). This is seen in the post-test debris in Figure 4-9c
as well as the stress-strain trace for this sample in Figure 4-9a, which shows increased
fluctuations in the plastification region and a larger onset for the densification region due to the
loss (i.e., breakage) of material.

Stress (MPa)

20
C60-HT-A0
C60-HT-A1
C60-HT-A2

15

b)

10

C60-HT-A1
5

a)
0
0.0

0.2

0.4

0.6

Strain (mm/mm)

0.8

c)
C60-HT-A2

Figure 4-9 Compressive behavior of Al/Cu hybrid foams with different annealing time

The observed behavior of sample C60-HT-A2 is similar to that of an unannealed Al/Cu
hybrid foam. However, the low coating ductility is not considered to be the cause in this case, as
it has been shown that the ductility of copper coating can be sufficiently enhanced with
annealing for an even shorter period. Rather, it is known that intermetallic compounds (IMC)
such as Al2Cu, Al4Cu9, AlCu, Al3Cu4, and AlCu3 can form at the Al/Cu interface during cold roll

96

welding, hot pressing, flash welding, friction stir welding, and annealing processes [93-100]. The
intermetallic compounds are usually brittle phases and the bond strength can reduce sharply and
promote crack propagation as this region increases in size [95, 98]. As the IMC thickness
increases with annealing time [94, 96, 97, 100], it is considered as a possible reason for the
observed brittle behavior of hybrid foam samples with longer annealing time, and thus an
investigation into this effect was carried out.

5 mins

a)

Al

Al
Cu

1 hr

b)

IMC

10 μm

2 hrs

c)

IMC

Cu

10 μm

d)

*
*

Al

Cu

IMC

Al
Cu

10 μm

*100 μm

Figure 4-10 SEM images of Al-Cu interface in Al/Cu hybrid foams after annealing time of (a) 5
minutes; (b) 1 hour; and (c-d) 2 hours

97

Table 4-4 Width of intermetallic compound with different annealing time
Annealing Time Thickness (μm)
5 minutes

1.6

1 hour

4.9

2 hours

10.5

Fig. 12 shows SEM images (using backscattered electrons) of the Al/Cu interface of
hybrid foam samples after different annealing times. The IMC region can be observed at the
Al/Cu interface in all annealed samples, which can be seen as an intermediate grey color region
separating aluminum and copper. The estimated IMC width in the different samples is shown in
Table 4-4. The reported values are the averages of at least five measurements taken for each
sample. It can be seen that the size of the IMC increases with annealing time and that the width
of the IMC in samples after 2 hours of annealing reached 10.5 μm.
Due to the different material thermal properties of Al and Cu (the thermal expansion
coefficient is 23 × 10-6/°C for aluminum and 17 × 10-6/°C for copper), stress may be accumulate
at the Al/Cu interface caused by the mismatch of deformation during the annealing and cooling
stages. Fracture of the IMC may thus occur once such buildup stress exceeds the strength of the
IMC, which, as previously mentioned, reduces in proportion to the thickness of the region.
Figure 4-10d shows fracture of the IMC, which led to separation between the Al core and
the Cu coating, indicated by arrows with an asterisk (“*”) in sample C60-HT-A2. Failure of the
IMC caused by reduction in bond strength is considered to be the main cause for the brittle
response observed in the C60-HT-A2 sample. Thus, a proper amount of annealing time needs to

98

be chosen to provide sufficiently improved ductility of the electrodeposited coating while
limiting the size of the IMC created.
4.4.3 Tensile Response
Foams are usually used in compression due to their unique energy absorption capacity,.
However, foams may be subjected to tension in certain cases, e.g., in a sandwich panel or when
used as structural components. Thus it is considered important to study the behavior of Al/Cu
hybrid foams and the effect of annealing under tensile loading conditions.

a)

b)

c)

d)

Figure 4-11 Images from the tensile testing of foam sample C30-HT-A1

Images from the tensile testing of sample C30-HT-A1 is shown in Figure 4-11. It can be
seen that failure was localized at the center of the sample and not at the bonding surfaces with
the loading fixtures. Figure 4-12 shows the tensile stress-strain curves for plain Al foam and
Al/Cu hybrid foam samples with and without annealing (400 °C for 1 h) treatment. In the figure,
99

plots (a) and (c) show complete traces for 30 m and 60 m Cu coatings, respectively, while
plots (b) and (d) show corresponding closer views of the initial region of the stress-strain
response. The yielding point was determined by finding the intersection of the first and second
linear portion of the stress strain curves.
It can be seen from Figure 4-12c that the plain Al foam yields at about 1 % strain
followed by some hardening until fracture happens at about 5 % strain. The post-yielding
hardening is attributed to the hardening of the material and the re-alignment of foam ligaments to
the tension loading axis [13]. A stiffer behavior was observed for the Al/Cu hybrid foams at low
strain levels since the foam ligaments are reinforced by the copper coating. The hybrid foam
specimens also show a lower strain for the onset of yielding compared to the plain unannealed Al
foam. This is due to the relatively low yielding strength of annealed of copper (the yield strength
of unnanealed Al is 193 MPa while that of annealed Cu is 33.3 MPa [101]).
Yielding of the copper coating leads to a stiffness change that is similar to that of the
plain Al foam during the post-yielding stage. A much earlier fracture (at about 2 % strain)
happens in the unannealed hybrid foam samples and the load carrying capacity drops more
rapidly compared to the plain Al foam. This is a direct consequence of the low ductility of the
copper coating, with a larger brittle effect observed for unannealed hybrid foam samples with
thicker coatings.

100

2.0

Stress (MPa)

Stress (MPa)

T0
T30
T30-HT-A1

3

2

1

a)
0
0.00

0.05

0.10

0.15

1.5

T0
T30
T30-HT-A1

1.0

Yielding

0.5

b)
0.0
0.00

0.20

Strain (mm/mm)

0.01

0.02

Strain (mm/mm)

2.0

Stress (MPa)

Stress (MPa)

T0
T60
T60-HT-A1

3

2

1

0
0.00

c)
0.05

0.10

0.15

1.5
1.0

Yielding
0.5
0.0
0.00

0.20

Strain (mm/mm)

T0
T60
T60-HT-A1

d)
0.01

0.02

Strain (mm/mm)

Figure 4-12 Tensile stress-strain response of Al/Cu hybrid foams

On the other hand, the annealed hybrid foam samples showed a similar behavior at lower
strain level to the unannealed samples but the post-yielding response branch continues to larger
strain level, especially in hybrid foams with 60 μm coating. It needs to be noted that the sample
with 30 μm coating was expected to fracture at a higher strain level (close to that with 60 μm
coating) if a proper amount of annealing was provided but the performance of the samples tested
may have been degraded due to the formation of excessive IMC at the Al/Cu interface.

101

It can be seen that the tensile behavior of Al/Cu hybrid foams is distinct from their
compressive behavior. More specifically, the low fracture strain of the hybrid foams significantly
limited the performance in tension and such behavior may be detrimental in certain applications.
Conversely, the annealed hybrid foams showed a promising performance with enhancement in
both strength and ductility. It is believed that similar tensile behavior as that observed in Al/Cu
hybrid foams and their enhancement with annealing will be characteristic in other hybrid foam
systems (e.g., Al/Ni hybrid foams [6, 7, 40]).
4.5

Discussion
Figure 4-13 shows the comparison of the energy absorption capacity under compression

of different foams with respect to coating thicknesses. The data points for plain Al foams with
different coating thicknesses represent unannealed Al foams with the same relative density (RD)

3

Al
Al/Cu
Al/Cu HT-A1

3

Energy / Unit mass (J/g)

Energy / Unit Volume (J/cm )

as the Al/Cu hybrid foams with corresponding coating thicknesses.

2

1

a)

0
0

20

40

60

80

4
3
2
Al
Al/Cu
Al/Cu HT-A1

1
0

100

0

20

40

b)
60

80

100

Nominal Coating Thickness ( m)

Nominal Coating Thickness ( m)

Figure 4-13 Comparison of energy absorption capacities for different foam samples: (a) per unit
volume; (b) per unit mass. The error bars represent the standard deviations

102

For Al/Cu hybrid foams, the energy absorption capacity per unit volume was calculated
as the area under the compressive stress strain curve up to a strain level of 50% ( εmax = 50%)
using Equation 4-3. The energy absorption capacity per unit volume of plain Al foams was
calculated using Equation 4-4 assuming a perfectly plastic stress plateau after yielding [40],
where the yield stress (or plateau stress) σ y , yield strain ε y and modulus of elasticity E were
calculated using Equation 4-5Equation 4-7 [13], respectively. The product of Cσ s in Equation
4-5 was determined to be 56.9 (MPa) based on the plateau stress of the uncoated Al foam (6%
nominal relative density) tested in this study.
ε
U h   max σdε

Equation 4-3

0



U p  σ y εmax  0.5ε y
σ y  Cσ s  RD 

3/ 2

εy  σ y / E
E  Es  RD 



Equation 4-4
Equation 4-5
Equation 4-6

2

Equation 4-7

It can be seen from Figure 4-13a that the energy absorption capacity per unit volume for
Al/Cu hybrid foams increases with the Cu coating thickness (i.e., relative density) since more
materials are used. This is expected and has also been shown in other studies on hybrid foams [7,
8, 40].
Compared to plain Al foams with the same relative density, the unannealed Al/Cu hybrid
foams showed similar energy absorption capacity per unit volume at coating thicknesses less
than 90 μm but showed a lower capacity for higher coating thicknesses. The limited performance
for the unnanealed hybrid foams with thicker coatings was due to the brittle behavior caused by
103

the low ductility of copper coating. In contrast, the annealed hybrid foams showed significantly
higher energy absorption capacity per unit volume compared to the unannealed Al/Cu hybrid
foams and the plain Al foams for different coating thicknesses.
A different trend was observed when the energy absorption capacity per unit mass for the
different foams was compared, as shown in Figure 4-13b. As the density of copper is more than
three times that of aluminum, Al/Cu hybrid foams (with and without annealing) generally
showed a lower energy absorption capacity per unit mass compared to plain Al foams with equal
relative density. In fact, in such comparison the performance of unannealed Al/Cu hybrid foams
decreases with coating thickness. This indicates a poor performance of unannealed Al/Cu hybrid
foams for weigh/mass critical applications. Similar behavior for Al foams with Ni-W coatings
has been reported [40]. However, the energy absorption capacity per unit mass of annealed
Al/Cu hybrid foams shows an increasing trend with coating thickness, which is similar to the
behavior of plain Al foams. Such behavior indicates a more efficient performance of the
annealed Al/Cu hybrid foams with thicker coatings.
The difference in energy absorption performance of Al/Cu hybrid foams before and after
a proper annealing process indicates a significant enhancement due to annealing. It is believed
that a similar improvement can be obtained through annealing in other hybrid foams
manufactured using electrodeposition (e.g., Al/Ni hybrid foams).
4.6

Conclusions
1)

The energy absorption performance of Al/Cu hybrid foams under compression

can be effectively improved by proper annealing, which enhances the ductility of the
electrodeposited copper coating. Such effect is more significant in hybrid foams with thicker
coatings.
104

2)

Annealing of hybrid foams can be conducted at one third to one half of the

melting temperature of the coating material. However, additional time is needed when annealing
at a lower temperature.
3)

To efficiently enhance the energy absorption performance of Al/Cu hybrid foams,

the annealing time should be sufficient to cause a ductility increase in the coating material but
limited to avoid the formation of excessive intermetallic compounds (IMC) at the Al/Cu
interface, which may cause fracture within the IMC and consequently reduced performance.
4)

A low ductility coating in Al/Cu hybrid foams causes early fracture of in tension,

which limits the macroscopic strength and ductility of the foam. However, proper annealing can
significantly improve both strength and ductility of Al/Cu hybrid foams in tension.

105

CHAPTER 5 ANALYTICAL SOLUTION FOR HYBRID FOAM
The enhanced performance of Al/Cu hybrid foam with nano-copper coating through
electrodeposition has been demonstrated. In addition, it has been shown both numerically and
experimentally that the performance of Al/Cu hybrid foams can be further improved with a
proper amount of annealing after the electrodeposition process. By varying the coating thickness,
the properties of Al/Cu hybrid foams can be modified. The behavior of Al/Cu hybrid foam with
different levels of coating thicknesses can be predicted using the beam-element based finite
element models developed in CHAPTER 3. However, a simple but reliable analytical model is
usually more appropriate and convenient for the purpose of engineering design. As a result,
analytical solution for the prediction of compressive behavior of Al/Cu hybrid foams was
derived and is presented in this section.
Among many theoretical studies on the behavior of cellular materials [13, 52, 102, 103],
the work by Gibson and Ashby [13] is widely recognized due to its simplicity as well as
accuracy. The semi-empirical solution derived by Gibson and Ashby [13] can be used to predict
properties of open-cell foam materials by simply using the relative density of the foam and the
properties of base material. The theoretical solution for open-cell foam material derived by
Gibson and Ashby is based on a cubic unit cell model with square cross section (see Figure 2-5).
When combined with several experimentally calibrated constants, solution with a good accuracy
can be obtained using such unit cell model. However, such unit cell is not a good representation
of the actual foam structure. In the current work, the analytical solution was based on the
characteristic cell geometry used by Jang and Kyriakides [71], which was based on a Kelvin cell
model.

106

A Kelvin cell has 24 ligaments and 18 nodes (see Figure 5-1). It is assumed that
ligaments in the vertical plane have a length L and an angle of θ with respect to the horizontal
plane. According to the study by Jang and Kyriakides [18], an anisotropy λ can be observed in
open-cell foam structures, which can be defined using Equation 5-1. As a result, the space
occupied by the unit cell has a width and depth of 4Lcosθ and a height of 4Lsinθ.

  tan 

Equation 5-1


4L sin 

L

4L cos 
4L cos 
Figure 5-1 Geometry of Unit Cell [71]
The size of the characteristic unit cell used is much smaller compared to the size of the
component made with foam material. Thus it is assumed that the unit cell is within an infinite
domain and thus any boundary effect is neglected. When the unit cell is subjected to uniaxial
stress (e.g., stress in vertical direction in Figure 5-2), it is assumed the unit cell can expand freely
in the direction perpendicular to the loading direction (e.g., horizontal direction in Figure 5-2).
As a result, the 8 ligaments within horizontal plane will not be loaded when the unit cell is
loaded in vertical direction.

107

Figure 5-2 Simplification of unit cell model

The loaded 16 ligaments have identical geometry and are subjected to the same loading
condition due to symmetry (see Figure 5-2). In other words, the problem of a unit cell subjected
to uniaxial stress loading condition can be simplified to the problem of one loaded ligament with
boundary conditions and loading as shown in Figure 5-2, in which the moment at the ends of
ligament can be calculated by:

M   P cos  L  / 2

108

Equation 5-2

Figure 5-3 Cross section of composite ligament in hybrid foam

It has been demonstrated that relative density is the most important parameter that
affecting the properties of open-cell foam materials [13]. The relative density of single-material
open-cell foams can be calculated as the ratio of the density of the foam material with respect to
the base material density. However, there are more than one base material in a hybrid foam and
thus it is difficult to calculate the relative density in terms of density. Instead, the relative density
is considered to be the ratio of the volume of solids with respect to the volume/space occupied by
foam material. As a result, the relative density of plain foam RD can be calculated using
Equation 5-3 while the relative density for a hybrid foam can be calculated using Equation 5-4.

RD 

RD ' 

Vcore  Vcoating
V foam

Vcore
V foam
 RD 

Equation 5-3
mcoating

coatingV foam

Equation 5-4

The relative density of hybrid foam can thus be calculated based on the relative density of
base foam and the mass of the coating material, which is convenient as the mass of coating is
usually easy to know in a hybrid foam. Similar to the numerical model discussed in CHAPTER 3,
it is assumed that the ligament in base foam and hybrid foam has a circular cross section (see
109

Figure 5-3). For the purposes of simplification, it is assumed that the ligament has a uniform
cross-section along the ligament length. With the cross section and the unit cell geometry, the
relationship between relative density and ligament size can be calculated as follows.
For unit cell in a plain foam, the volume of solid is:
Equation 5-5

Vcore  24   d 2 / 4  L  6 d 2 L
The space occupied by the unit cell is:

V foam  (4 Lcos  )2  4 Lsin   64sin  cos 2  L3

Equation 5-6

Then the relative density RD of plain form can be calculated as:

V
6 d 2 L
d 
RD  core 
 k 
2
3
V foam 64sin  cos  L
L

2

Equation 5-7

where

3


k 
2 
 32sin  cos  

Equation 5-8

Equation 5-7 can also be written in the following form:

d
RD

L
k

Equation 5-9

Similar to plain foam, the relationship between relative density of hybrid foam (RD’) and
ligament geometry can be obtained as shown in Equation 5-10 and Equation 5-11:
 d  2t 
RD '  k 

 L 

or,

d  2t

L

2

RD '
.
k

Equation 5-10

Equation 5-11

The assumptions in ligament cross-section and unit cell geometry discussed were used for
all analytical solutions presented in this Chapter. First, the analytical solution to predict the

110

modulus of hybrid foam consisting two different materials are presented in Section 5.1. It has
been shown that the coating ductility has great influence on the macro behavior of hybrid foams.
Thus the analytical solutions for post-yield behavior for hybrid foams were derived considering
two schemes. More specifically, the analytical solutions in Section 5.2 are based on hybrid foam
with ductile coating material (e.g., annealed Al/Cu hybrid foam). And Section 5.3 presents the
analytical solutions for the post-yield behavior of hybrid foams with brittle coating material (e.g.,
as-built Al/Cu hybrid foam).
5.1

Modulus of Hybrid Foam
The modulus of a foam can be calculated based on the load applied to the unit cell and

the resulting deformation. The force P applied to one ligament can be calculated based on the
stress σ applied to the unit cell (note that there are four loaded ligaments in a horizontal plane).

P

  (4 L cos  )2

or,

4



 4( L cos  )2 

P
.
4( L cos  ) 2

Equation 5-12

Equation 5-13

Based on classic beam theory, the displacement Δ at the top of the ligament can be
calculated according to Equation 5-14:



P cos  L3
12 EI

Equation 5-14

The displacement component δ and the corresponding strain ε in the loading direction can
then be calculated with Equation 5-15 and Equation 5-16. The modulus of the foam material can
be calculated using σ and ε Equation 5-17.

111

   cos  




L sin 

E



P cos 2  L3
12 EI

Equation 5-15

P cos 2  L2
12 EI sin 

Equation 5-16

 3EI sin 

 (Lcos  )4

Equation 5-17

The cross section of a ligament in a hybrid foam is a composite section. Thus, the section
flexural stiffness, EI, can be calculated using Equation 5-18.
EI  E1I1  E2 I 2


E d
64

=

 
( E1  E2 )d 4  E2 (d  2 t) 4 



1

4



 E2 (d  2 t) 4  d 4 

64 

Equation 5-18



Assuming that E2/E1=a, then:
EI 


64

E1 (1  a)d 4  a(d  2 t)4 

Equation 5-19

Then modulus of a hybrid foam can then be calculated with Equation 5-20 and Equation
5-21, in which the relationship between foam relative density and ligament geometry derived in
Equation 5-9 and Equation 5-11 were used.

E

4

3 sin 
d 2 t 4 
d 
E
(1

a
)

a
(
) 

1


L
64 cos 4  
L


E
sin  
4
 0.147
(1  a)  RD   a( RD ')4 
4
2 

E1
cos  k

112

Equation 5-20

Equation 5-21

In Equation 5-21, θ and k are constants based depending on the anisotropy of the foam.
Thus, the modulus of a hybrid foam can be calculated simply using the relative density of the
plain foam and the hybrid foam and the properties of two base materials.
5.2

Analytical Solution for Hybrid Foams with Ductile Coating

5.2.1 Plastic Stress
The stress plateau observed in the stress-strain curve of conventional metal foams is
caused by plastic collapse of the foam ligaments when loaded beyond the linear-elastic range
[13]. Plastic collapse occurs when the localized bending moment demands exceeds the plastic
moment capacity of the ligament section and creating plastic hinges, as shown in Figure 5-4.
Based on the boundary conditions of a single ligament, it can be seen plastic hinges will form at
the ends of the ligament, where the maximum moment is.

P ΔP
Le

Lp
2

M

φ
θ

M

Plastic hinge

Lp
2

P
Figure 5-4 Plastic hinge location
An idealized plastic hinge has zero length and all plastic deformation occurs at the plastic
hinge. However, in reality plastic deformations occurs within a region of finite length (plastic
hinge length). Thus, it is assumed that the location of the plastic hinge is at a distance of Lp/2

113

from both ends, where Lp is plastic hinge length. The plateau stress can be obtained by equating
the external work done by the force during plastic rotation ϕ of the two plastic hinges in each
ligament to the internal plastic work done at the hinges:
2M p  P p cos 

Equation 5-22

P  4  L cos   

Equation 5-23

where,
2



 p   Le   L  Lp



Equation 5-24

A simple estimate of the plastic hinge length is [104]:
d
2

Equation 5-25

Le  xL

Equation 5-26

Lp 

Then,

Where, x  1 

1D
RD
 1
2L
k

Equation 5-27

Then from Equation 5-22, the yield stress (plastic stress) can be derived:

   pl 

Mp

Equation 5-28

2 x(Lcos  )

3

The same result can be obtained by assuming that the moment at the plastic hinge
location is equal to the plastic moment Mp:
M x   P cos   xL   / 2  2 x  L cos     M p
3

Equation 5-29

And then,



Mp
2 x(Lcos  )

114

Equation 5-30
3

Assuming an elastic-perfect plastic constitutive model for both core and coating material,
the moment for full section plastification, Mp, can be calculated using Equation 5-31, in which
Zp=d3/6 is the plastic section modulus for circular section.
M p   yZ p
  y1Z p1   y 2 Z p 2
1
1
  y1d 3   y 2 (d  2 t)3  d 3 
6
6
1
 ( y1   y 2 )d 3   y 2 (d  2 t)3 
6

Equation 5-31

Assuming

 y 2  b y1

Equation 5-32

1
M p   y1 (1  b)d 3  b(d  2 t)3  .
6

Equation 5-33

Then:

The ratio between plastic stress and yielding stress of core material is given in Equation
5-34, where the relations between relative density and ligament geometry from Equation 5-9 and
Equation 5-11 were used.
3
3

 pl
1
d 
 d 2 t  

(1

b
)

b

 

 
 y1 12 x(cos  )3 
L
 L  



Equation 5-34

1
(1  b)  RD 3/2  b  RD '3/2 
3 3/2 

12 x(cos  ) k

Similar to the solution obtained by Gibson and Ashby [13], Equation 5-34 does not
require information about the ligament geometry, Instead, it shows that the plateau stress of a
hybrid foam made with two metal materials can be calculated based on the relative density of the
foam before and after electrodeposition.

115

The post-yield behavior was modeled based on the method used by Gibson and Ashby
[13] using Equation 5-35Equation 5-37. The values for constants D and m were determined by
fitting the stress-strain curve to the equations. A value of 1.2 for D and 1 for m were found to
yield a good prediction of the post-yield behavior of Al/Cu hybrid foam.

 D  1  1.4RD

Equation 5-35


 1
 1 when    D 1  
 pl
 D

Equation 5-36

m



1 
 1
  D  when    D 1  
 pl D   D   
 D

Equation 5-37

5.2.2 Results
The derived analytical solution was used to predict the compressive response of Al/Cu
hybrid foams. Only samples subjected to an annealing process were considered, since the
analytical solution was based on an elastic-perfect plastic constitutive material model for the core
and coating. The input information is for the analytical solution is shown in Table 5-1. It needs to
be noted that annealed copper has a very low yielding stress (about 70 MPa) and a long
hardening branch in its constitutive model [105]. As a result, the ultimate stress is much higher
compared to its yield stress.
Beyond the elastic regime, the deformation of metal foams involves localized large
deformations (i.e., localized collapse of foam cell structures) and thus large rotations at the
plastic hinge location. Therefore, it is considered that significant hardening of the copper coating
will happen at the plastic hinge location and thus the ultimate stress was used. A value of 1.2 was

116

used for the anisotropy factor λ as suggested by Jang and Kyriakides [71] based on
measurements of ligament geometry on open-cell Al foams.

Table 5-1 Input parameters for analytical solution
E

y

(GPa)

(MPa)

Core (Al6101, annealed) [80]

70

90

Coating (Copper, annealed) [105]

110

210 (σu)

Material

λ

D

m

1.2

1.2

1

14
Plain Al
Al/Cu-1 (RD'=8.1%)
Al/Cu-1 (RD'=8.1%)
Al/Cu-1 (RD'=8.1%)
Plain-Al (Model)
Al/Cu-1 (Model)
Al/Cu-2 (Model)
Al/Cu-3 (Model)

12

Stress (MPa)

10
8
6
4
2
0
0.0

0.2

0.4

0.6

0.8

Strain (mm/mm)

Figure 5-5 Comparison between experimental data and analytical solution

Figure 5-5 shows the comparison between the presented analytical solution and
experimental data from Al/Cu hybrid foams with different levels of coating. The experimental
117

data was based on annealed samples (both plain Al and Al/Cu hybrid foams). It can be seen that
the plateau stress and post yield behavior predicted by the analytical solution agrees well with
the experimental data. However, deviation was observed between the analytical solution and the
experimental data within elastic region. This difference in results can be attributed to the fact that
the analytical solution was based on a simple elastic-perfect plastic material model while the
copper coating in Al/Cu hybrid foams actually has early yielding and a significant hardening.

10

Stress (MPa)

8
6
4
2
0
0.0

0.2

0.4

0.6

0.8

Strain (mm/mm)
Figure 5-6 Stress strain curve of Al/Cu hybrid foam (RD=6%, RD’=10.7%) with different
anisotropy

Figure 5-6 shows the stress strain curve for Al/Cu hybrid foams with a nominal coating
thickness of 60 m and different levels of anisotropy. The relative density for the base foam and
the hybrid foam is 6% and 10.7%, respectively. It can be seen that hybrid foam with higher

118

anisotropy has a higher yielding stress and plateau stress. This agrees with the observation by
Jang and Kyriakides [71] using multi-cell finite element models.
5.3

Analytical Solution for Hybrid Foams with Low Ductility Coating
The analytical solution derived in 0 provides a good prediction on the compressive

behavior of Al/Cu hybrid foams with annealing process. The solution assumed that both the
coating and core material have a ductile behavior and fracture of the material will not occur.
However, the analytical solution is not appropriate for as-built Al/Cu hybrid foams (i.e., no
annealing process introduced), which shows brittle behavior in compression due to the low
ductility of the electrodeposited copper coating. The compressive behavior of as-built Al/Cu
hybrid foams can be simplified as shown in Figure 5-7. Upon reaching the stress peak (  fr ),
fracture of the ligaments occurs and the stress drops to a lower level (  low ). The low stress level
continues through the plateau region until the densification allows the stress to increase at large
strains. It can be seen that a good prediction of the two stress level (  fr and  low ) is the key for
an analytical model. In this section, an analytical solution is derived for Al/Cu hybrid foams
without annealing process. The derivation is based on the single ligament model used for ductile
Al/Cu hybrid foams (i.e., with annealing process introduced after electrodeposition).

119

Stress

 fr

 low
Strain
Figure 5-7 Typical behavior of Al/Cu hybrid foams
5.3.1 Stress at Fracture
The strain and stress distribution across the cross section of a ligament subjected to pure
bending is shown in Figure 5-8. Based on classic beam theory the strain distribution across the
cross section is linear with zero strain at the neutral axis. The normal stress across the cross
section can be simply calculated using   E . It is assumed that coating fracture in AL/Cu
hybrid foams happens when the stress level at the extreme fiber of the copper coating exceeds
the fracture stress of electrodeposited copper,  fr Cu .

1

2

2
1

Figure 5-8 Strain and stress across ligament cross section

120

The moment capacity of the pre-fractured cross section, M fr ,can be calculated as the
sum of contributions from the core ( M1 ) and the coating ( M 2 ):
Equation 5-38

M fr  M1  M 2

It is assumed that the core and coating material remain elastic prior to fracture. Thus M1
and M 2 in Equation 5-38 can be calculated using Equation 5-39, Equation 5-40, Equation 5-41,
and Equation 5-42:

 I  d 3
M1  1 1  1
d /2
32
where, I1 
M2 

 2 I2

(d  2t ) / 2

where, I 2 



d4

Equation 5-39

Equation 5-40

64

 2 (d  2t )4  d 4
32

Equation 5-41

d  2t

1 
(d  2t )4  d 4 


64

Equation 5-42

Prior to ligament fracture, the stress level at the extreme fiber in the coating material
equals the fracture stress of the coating material (  2   fr Cu for Al/Cu hybrid foams). And the
stress level at the extreme fiber in the core can be calculated using Equation 5-43.

1 

 2 E1

d
E2 d  2t

Equation 5-43

It is assumed that strength degradation of the as-built (i.e., brittle) Al/Cu hybrid foam
initiates at the onset of coating fracture in the ligament. As a result, the initial peak stress in the
compressive behavior of as-built Al/Cu hybrid foams equals the strength prior to coating fracture

121

(  fr ), which can be calculated based on M fr using a relation derived for plastic stress, as
shown below in Equation 5-44.

 fr


M fr
2 x( L cos  )3

  d 3
 d  2t 3  d 3  d   



 1 
2 
   
 
 L   L   d  2t   
64 x(cos  )3   L 
3
3
3
 E
 fr
d  d   d  2t   d   d   
1





 

   
 
64 x(cos  )3  E2 d  2t  L   L   L   d  2t   




 E
1

3 E
64 x(cos  )  2

 fr

3/2

RD  RD 


RD '  k 

 RD ' 3/2  RD 3/2
 
 

 k 
 k 

Equation 5-44

RD  

RD '  


5.3.2 Post-peak Stress
Sectional behavior is of the coated ligament is more complex once coating fracture
occurs. Thus, several assumptions were made to ease the derivation of an analytical solution of
the post-fracture behavior of brittle Al/Cu hybrid foams. First, both compressive and tensile
behavior of the core material (i.e., Al for Al/Cu hybrid foams) is assumed to be elastic-perfect
plastic. Secondly, since the electrodeposited copper coating fails at a small strain, it is assumed
that the electrodeposited copper coating has a brittle behavior in tension. In other words, the
strength of copper coating in tension is neglected. Meanwhile, the compressive behavior of
copper coating is assumed to be elastic-perfect plastic. Thirdly, it is assumed that the full section
(except the coating area in tension) has plastified, or yielded, when equilibrium is achieved in the
fractured ligament section.

122

Figure 5-9 Schematic of fractured ligament cross-section of Al/Cu hybrid foam

The fractured cross-section of a brittle Al/Cu hybrid foam ligament can thus be separated
as three parts as shown in Figure 5-9. To maintain force equilibrium the resulting tension force
from the core area in tension region needs to balance the resulting compressive force from the
core and coating areas in the compression region.

Figure 5-10 Schematic of a circular segment
For a circular segment, the area and the distance of its centroid from the center of the
circle (see Figure 5-10) can be calculated using Equation 5-45 and Equation 5-46.
123

A

R2
  sin  
2

y

4 R sin 3

Equation 5-45



2
3   sin  

Equation 5-46

The relations in Equation 5-47 to Equation 5-50 can be found for the fractured ligament
cross section shown in Figure 5-10, in which  is the distance from the center of the original
ligament cross section to the current neutral axis (fractured section).

or,

or,

cos(1 / 2)   / R1

Equation 5-47

1  2arccos( / R1)

Equation 5-48

cos(2 / 2)   / R2

Equation 5-49

2  2arccos( / R2 )

Equation 5-50

Thus the areas of core in compression ( A1c ), core in tension ( A1t ) and coating in
compression ( A2 c ) can be calculated using Equation 5-51, Equation 5-52, and Equation 5-53.

R2
A1c  1 1  sin 1 
2

Equation 5-51

A1t   R12  A1c

Equation 5-52

R2
A2c  2 2  sin 2   A1c
2

Equation 5-53

To maintain force equilibrium, Equation 5-54 needs to be satisfied, from which  can be
solved.

A1c1  A2c 2  A1t1

124

Equation 5-54

The moment capacity of the fractured ligament cross-section can be calculated as the sum
of three moment contributions about the neutral axis, as shown in Table 5-2, in which y is the
distance from centroid of a circular segment to the center of the original ligament cross-section;
and d is the distance from the centroid of the circular segment to the neutral axis.

Table 5-2 Contributions for moment capacity of the fractured ligament cross-section
Part

Area

y

d

stress

A1t

(2  1)
2
ya 
3 (2  1)  sin(2  1) 

da    ya

 y1

A1c  A2c


4 R2 sin 3 2
2
yb 
3 2  sin 2 

db  yb  

 y2

A1c


4 R1 sin 3 1
2
yc 
3 1  sin 1 

dc  yc  

 y1   y 2

a

b

c

4 R1 sin 3

The moment capacity of fractured ligament cross-section can thus be calculated using
Equation 5-55.





M low   y1 A1t da   y 2  A1c  A2c  db   y1   y 2 A1c dc

Equation 5-55

The lower stress level  low after the initial peak stress  fr can be calculated from M low
using Equation 5-56.

 low 

M low

3

2 x  L cos  

125

Equation 5-56

5.3.3 Results
The expressions solutions derived in Section 5.3.1 and 5.3.2 were used to predict the
response of brittle Al/Cu hybrid foams with different levels of coating thicknesses. All
calculations were based on a 40 PPI foam and the ligament diameter and its length were taken as
0.192 mm and 1.04 mm, respectively. The material properties used for the calculations are
summarized in Table 5-3. The values for the initial stress peak  fr and post-peak stress  low
are summarized in Table 5-4.

Table 5-3 Material properties used for analytical model of Al/Cu hybrid foams
Modulus Yielding stress Fracture stress in tension
Material
(MPa)

(MPa)

(MPa)

6101-T6 Al (Core)

70000

330

-

Electrodeposited Cu (Coating)

102000

225

225

Figure 5-11 compares the analytical solution and experimental data for the compressive
behavior of brittle Al/Cu hybrid foams. The negative slope after the initial stress peak in the
analytical solution was assumed to 5% of the initial stiffness. The post-peak behavior was
modeled based on the method used by Gibson and Ashby with D = 1.2 and m = 0.8 (Equation 442 to Equation 4-44). It can be seen that the analytical model provides a good prediction on the
initial peak stresses and the plateau stresses for the compressive behavior of brittle Al/Cu hybrid
foams with different coating thicknesses.

126

Table 5-4 Initial stress peak and post-peak stresses for Al/Cu hybrid foams with different coating
thicknesses
Coating
( m )

 fr

 low

(MPa) (MPa)

30

1.72

1

60

2.37

1.36

90

3.28

1.89

8
40-30 EXP
7

40-60 EXP

6

40-90 EXP

Stress (MPa)

40-30 ANA
5

40-60 ANA

4

40-90 ANA

3
2
1
0
0

0.2

0.4
Strain (mm/mm)

0.6

0.8

Figure 5-11 Comparison of analytical solution and experimental data on the compressive
behavior of brittle Al/Cu hybrid foams

127

5.4

Conclusions
Analytical solutions for hybrid foams consisting two different materials were derived to

predict the uniaxial compressive behavior of hybrid foams. Different solutions were presented
considering different coating ductility level. The following conclusions are drawn from this
study:
1)

The modulus of hybrid foams consisting two materials can be predicted using the

obtained analytical solution. It has been shown that the modulus of hybrid foam is directly
related to the relative density of the base foam and hybrid foam.
2)

The analytical solution derived for hybrid foam with ductile coating provides a

good estimation on the plastic stress. The solutions were used to predict the compressive
behavior of annealed Al/Cu hybrid foams with different coating levels and the results agree well
with experimental data.
3)

By considering coating failure in tension, the analytical solutions for hybrid foam

with low ductility coating can predict the distinct behavior of as-built electrodeposited hybrid
foams with a reasonable accuracy.

128

CHAPTER 6 CHARACTERIZATION OF DYNAMIC BEHAVIOR OF AL/CU HYBRID
FOAMS
6.1

Overview
In this chapter, the mechanical properties of Al/Cu hybrid foams under high stain-rate

compression demands were investigated using a split Hopkinson pressure bar. It is hypothesized
that the energy absorption capacity of aluminum open-cell foams under high strain rate loading
can be effectively enhanced by electrodeposited metal coatings. However, the low ductility
capacity of nanocrystalline coatings is detrimental to the performance of the fabricated hybrid
foams, and thus the performance of such hybrid foams can be further enhanced by improving the
ductility of the coating material. Al/Cu hybrid foams were manufactured by electrodepositing
copper on aluminum open-cell foams to investigate the compressive response and failure
mechanism of the composite material under high strain rate loading. Compared to the current
literature the present work provides an extended experimental evaluation on the dynamic
behavior of hybrid foams with different coating thicknesses to provide guidance on the design of
functionally graded foam material systems under high-rate compressive strains. In order to study
the influence of the ductility capacity of the nano-copper coating, an annealing process was
performed on part of the specimens after electrodeposition and their behavior was compared with
that of specimens without annealing.

129

6.2

High Strain Rate Experiments

6.2.1 SHPB Tests
The compressive performance of Al/Cu hybrid foams under high strain rate loading was
investigated using a split Hopkinson pressure bar (SHPB). Figure 6-1 shows a schematic of the
SHPB set up, which consists of an incident bar, a transmission bar, and a striker. The incident
bar and the striker were two solid aluminum bars with a diameter of 19 mm and lengths of 1829
mm and 508 mm, respectively. Due to the low impedance of the foam material studied, a hollow
aluminum bar (I.D. = 15 mm, O.D. = 19 mm) with a length of 1829 mm was used as the
transmission bar. A 1 mm thick copper plate was used as the pulse shaper. Strain gages were
placed at the half length point of both the incident and transmission bars to measure the incident
strain  i (t ) , the transmitted strain  t (t ) and the reflected strain  r (t ) .
To reduce noise in the measured signal, four strain gages were placed on the transmission
bar and connected to a full bridge configuration while a half bridge configuration was used for
the strain gage on the incident bar. In order to achieve equilibrium during the SHPB experiments
the specimen length (LS) needs to be small, especially for materials with low impedance [106].
However, when evaluating foams it is necessary that the test sample contains several cells in the
loading direction. As a result, the SHPB specimens used in this study (see Figure 6-2a) had a
diameter of 15.9 mm and a length of 5 mm (about eight times the pore size), which was the same
length used in the study by Jung et al. [7].

130

Figure 6-1 Schematic of split Hopkinson pressure bar (SHPB) setup

(a)

(b)

Figure 6-2 Cross section pictures of (a) uncoated circular-shaped (5 mm thick with a diameter of
15.9 mm) specimen for dynamic experiments and (b) Cu-coated square-shaped specimen (25.4
mm × 25.4 mm × 5 mm) for quasi-static experiments

Figure 6-3 shows an oscilloscope record obtained during one of the SHPB experiments.
The flat plateau of the incidence and reflected pulse indicates a constant strain rate of about 180
microseconds for the noted setup. The stress, strain and strain rate of the specimen were
reconstructed from the strain-time records using Equation 6-1 to Equation 6-3 [107]:

A
σ s (t )  Et t εt (t )
As

131

Equation 6-1

εs (t ) 

2C0 t
 εi (t )  εr (t )  εt (t ) dt
Ls 0

Equation 6-2

2C0
 εi (t )  εr (t )  εt (t )
Ls

Equation 6-3

εs (t ) 

where C0 is the wave speed within the bar; At and AS are the cross-sectional areas of the
specimen and the transmission bar, respectively; and Et is the Young’s modulus of the
transmission bar.

6
Incidence
Transmission

Signal (V)

4
2
0
-2
-4
-6

0

200

400
Time (s)

600

800

Figure 6-3 Typical oscilloscope records from SHPB experiments

Four different types of specimens were considered: uncoated, and Cu coated with 30 μm,
60 μm, and 120 μm nominal coating thicknesses; and three experiments were conducted for each

132

specimen type. Two different strain rates were obtained in the SHPB experiments conducted in
this study, namely, 2.8×103 s-1 and 4.9×103 s-1.
6.2.2 Quasi-static Experiments
To provide a reference for the high strain rate experiments, quasi-static compressive
experiments were also performed with a universal testing frame. It is important to maintain the
same specimen length in the loading direction in both the high strain rate and quasi-static
experiments as the non-uniform collapse of cells in foam specimens of larger size may lead to a
different overall behavior [18]. Thus, the specimens used for the quasi-static experiments (Figure
6-2b) also had a thickness of 5 mm and a cross-section of 25.4 mm by 25.4 mm. The
experiments were conducted in a displacement control mode at a strain rate of 3.3×10-3 s-1. Three
experiments were conducted for each specimen type.
6.2.3 Compressive Behavior of Hybrid Foams
Figure 6-4 shows a typical compressive stress strain curve for the Al/Cu hybrid foams
reported in this study. Nominal stress (load divided by original cross-sectional area) and nominal
strain (deformation divided by original length) are used throughout the results presented here.
Similar to the behavior of a conventional single-material metal foam, the response features an
elastic region, a collapse region and a densification region. However, it was observed that the
stress level in the collapse region dropped significantly after the initial peak stress P for the
hybrid foams, while such difference is usually small in conventional single-material metal foams
[13, 18]. Ideally the stress level should remain constant at a nominal level close to the initial
peak stress throughout the collapse region to maximize the energy absorbed during plastic
deformations under a constant stress level. Thus, the significant drop after the initial peak stress

133

is considered an unfavorable behavior feature and needs to be considered when evaluating the
energy absorption performance of hybrid foams.

Nominal stress

Elastic
Region

Collapse
Region

σP

Densification
Region

Best Fitting
Lines

σD
σL
Energy Absorption
Nominal strain

eD

Figure 6-4 Schematic of typical compressive behavior of Al/Cu hybrid foams with peak stress
(  P ), lowest stress in collapse region (  L ), densification stress (  D ) and densification strain
(D )

In this study, the densification strain D was defined as the strain level at the intersection
of the best fitting lines (using a least square fit) of the ascending part in the collapse region and
the densification region, respectively, as shown in Figure 6-4. The energy absorption capacity
was defined as the area under the stress-strain curve up to the densification strain. In addition,
energy absorption efficiency was defined as the ratio of the absorbed energy to an ideal energy
absorption level, which was defined as the product of the densification strain D and the initial
peak stress P, or the stress at densification strain D, whichever is larger (see Equation 6-4).

134

The quantitative parameters of all experiments are summarized in Table 6-1, where L is
the lowest stress level in the collapse region and  is the ratio of the stress drop to the initial
peak stress calculated using Equation 6-5. The letters “S” and “D” in the sample identification
(ID) names represent quasi-static (with strain rate of 3.3×10-3 s-1) and dynamic experiments,
respectively. The number after letter “C” represents the nominal thickness of the nano-copper
coating in µm and the last number indicates the strain rate (“28” and “49” refer to strain rates of
2.8×103 s-1 and 4.9×103 s-1, respectively). Sample ID names with an “A” refer to annealed
samples. The experimental data summarized in Table 6-1 is based on the average of three
repeated experiments for each specimen type. It needs to be noted that at the strain rate of
2.8×103 s-1 the densification region in the dynamic response of Al/Cu hybrid foams could not be
fully captured with the current setup and thus the densification strain D and the energy
absorption capacity could not be determined at such strain rate.

Efficiency 

Energy Absorption
max(σ P , σ D )  εD

σ  σL
σ  P
 100%
σP

135

Equation 6-4

Equation 6-5

Table 6-1 Summary of results from all experiments

Sample ID

P

L



D

(MPa) (MPa) (%) (mm/mm)

Energy
Absorption
(N·mm)

Ideal Energy Efficiency
(N·mm)

(%)

S.C0

0.64

0.63

2

0.68

0.70

2.11

33

S.C30

2.66

1.35

49

0.68

1.28

2.65

48

S.C60

4.21

1.83

57

0.65

1.71

3.79

45

S.C120

10.59

4.32

59

0.62

3.68

6.57

56

D.C0.28

0.67

0.54

19

-

-

-

-

D.C30.28

1.96

0.74

62

-

-

-

-

D.C60.28

4.00

1.14

71

-

-

-

-

D.C120.28

10.82

1.68

85

-

-

-

-

D.C0.49

0.73

0.52

29

0.68

0.69

2.52

27

D.C0.49

2.97

0.73

75

0.65

1.11

2.80

40

D.C60.49

4.39

1.02

77

0.61

1.31

2.68

49

D.C120.49

12.68

1.91

85

0.61

2.79

7.73

36

D.C30.49.A

2.10

1.16

45

0.59

1.22

2.56

48

D.C60.49.A

4.43

2.29

48

0.55

1.93

3.70

52

D.C120.49.A

9.10

5.92

35

0.52

3.94

6.10

65

6.2.4 Effect of Electrodeposited Copper Coating
The stress-strain curves obtained from quasi-static and high-strain rate compression
experiments are shown in Figure 6-5 to Figure 6-7. The naming convention for the different
experiments can be found in Section 6.2.3. The behavior for each specimen type presented in this
paper is the average of three separate experiments. Specimen size was dictated by the high-rate
experiment requirements. As the foam response is affected by sample size, the behavior reported
here differs from that obtained by Wang et al. [8] with larger specimens. Yet, the objectives of
136

this work are not thought to be compromised by this fact since the results are used for mutual
comparison.
It can be seen from Figure 6-5 to Figure 6-7 that the performance of the aluminum foam
under both quasi-static and high strain rate compressive loading was effectively enhanced by the
electrodeposited copper coating. In general, the initial peak stress and the plateau stress of the
coated foam specimen increased with the coating thickness. This is expected as the moment of
inertia of a reinforced ligament section is larger and thus its load bearing capacity (controlled by
flexural yielding) is increased. Meanwhile, a slight decrease of the densification strain for
specimens with thicker coating was observed in both quasi-static and dynamic experiments,
which is attributed to the fact that the densification of a foam is basically a volumetric
phenomenon and that specimens with thicker coating have more material.
As mentioned in Section 6.2.3, a significant stress drop in the compressive stress-strain
curve was observed in all experiments on coated samples and such drop was more significant for
specimens with thicker coatings. A similar behavior was reported in previous studies on opencell Al foam with nanocrystalline Ni coatings by Bouwhuis et al. [6] and Jung et al. [7] and the
study on Al/Cu hybrid foams by Wang et al. [8].

137

Stress (MPa)

20
S.C0
S.C30
S.C60
S.C120

15
10
5
0
0.0

0.2

0.4

0.6

0.8

Strain (mm/mm)
Figure 6-5 Quasi-static compressive behavior of Al/Cu hybrid foams (3.3×10-3 s-1)

Stress (MPa)

20
D.C0.28
D.C30.28
D.C60.28
D.C120.28

15
10
5
0
0.0

0.2

0.4

0.6

0.8

Strain (mm/mm)
Figure 6-6 High strain rate compressive behavior of Al/Cu hybrid foams (2.8×103 s-1)

138

Stress (MPa)

20
D.C0.49
D.C30.49
D.C60.49
D.C120.49

15
10
5
0
0.0

0.2

0.4

0.6

0.8

Strain (mm/mm)
Figure 6-7 High strain rate compressive behavior of Al/Cu hybrid foams (4.9×103 s-1)

Considering the previously noted definition of energy absorption capacity, a high peak
stress followed by a large drop can significantly reduce the foam’s energy absorption efficiency.
As demonstrated in the annealing effect evaluation (CHAPTER 4) and numerical studies
(CHAPTER 3), the observed stress drop is mainly attributed to the rupture of the
electrodeposited coating material, as electrodeposition usually leads to a material with low
ductility [66].
The larger stress drop for specimens with thicker coatings is due to the increased capacity
loss of the ligament, and the foam, upon brittle failure of the thicker section reinforcement. It
was also observed that the stress drop was more significant under high strain rate loading (see
Figure 6-5 to Figure 6-7). This may be attributed to the fact that the shattering of foam ligaments
and the spalling of coating material was more severe under high strain rate loading. Such

139

phenomenon also led to a slightly higher densification strain in Al/Cu hybrid foams under high
strain rate loading conditions, especially with thicker coatings. It can be seen from Table 6-1 that
the lower plateau stress from high strain rate experiments lead to a relatively lower energy
absorption capacity and efficiency compared to the quasi-static experiments.
Figure 6-8 compares the performance of foam specimens with different coating thickness
under different loading rates. No significant difference on the initial peak stress (see Figure 6-8a)
with respect to loading rates was observed for plain Al foams and Al/Cu hybrid foams with
nominal copper coating thicknesses less than 120 μm. However, the hybrid foams with 120 μm
nominal coating thickness showed slightly higher initial peak stress at a strain rate of 4.9×10 3 s-1
compared to that at lower strain rates. Figure 6-8b shows that the lowest stress level in the
collapse region of the coated foams was lower in the dynamic experiments, which can again be
explained by the more severe specimen shattering and material spalling under high-rate loading.
No significant difference in the lowest stress values was observed under strain rates of 2.8×103 s1

and 4.9×103 s-1.

140

Initial Peak Stress (MPa)

20
15

C0
C30
C60
C120

10
5
0
0

2000

4000

6000

Strain Rate (s-1)

Lowest Stress (MPa)

10
8
6

C0
C30
C60
C120

4
2
0
0

2000

4000

6000

Strain Rate (s-1)

Figure 6-8 Initial peak stress and lowest stress level in the compressive behavior of Al/Cu hybrid
foams under different strain rates

6.2.5 Effect of Annealing
The low ductility capacity of the electrodeposited metal coating is considered to be
unfavorable for the overall energy absorption performance of hybrid foams as most of the energy

141

absorbed is due to large plastic deformation of the cell struts. Thus, it is necessary to investigate
the effect of ductility capacity of the coating material on the performance of the hybrid foam.
An annealing process was thus added to some of the specimens after the
electrodeposition process to investigate the effect of enhanced coating ductility in hybrid foams.
Annealing is a well-established heat treatment process for metals to release internal stresses and
enhance ductility through recrystallization and grain growth, which also usually results in a
lower strength due to the change of crystal structure [46, 58, 89]. Annealing was conducted at a
temperature of 400 °C in a vacuum furnace (3×10-5 torr) for one hour. The noted temperature
was chosen since it is about half of the melting temperature of copper (1084 °C) at which
recrystallization can occur [58] and at the same time it is not too close to the melting temperature
of aluminum (660 °C). The temperature was increased at a rate of about 10 °C/min and the
furnace was cooled down slowly after the annealing process.
6.2.6 Results for Effect of Annealing
The compressive behavior of annealed and un-annealed Cu/Al hybrid foams under high
strain rate loading (4.9×103 s-1) is shown in Figure 6-9 to Figure 6-11. For reference purposes,
the result of a plain aluminum foam specimen is included in all plots. First, it can be seen from
the plots in Figure 6-9 to Figure 6-11 that the annealed specimens generally show a similar or
lower peak stress compared to the un-annealed specimens. This is because the recrystallization
and grain growth that occurs during annealing changes the crystal structure and leads to a lower
yield strength [46, 58, 89].
Secondly, the results in Figure 6-9 to Figure 6-11 also show that the stress plateau after
the peak stress was higher for the annealed foam specimens. As a result, the annealing process

142

reduced the difference between the initial peak stress and the stress level in the collapse region,
which is beneficial for energy absorption efficiency. It can also be seen that the effect of
annealing was more significant for foam specimens with thicker coatings as the improvement in
ductility is mainly on the copper coating.

Stress (MPa)

20
D.C0.49
D.C30.49
D.C30.49.A

15
10
5
0
0.0

0.2

0.4

0.6

0.8

Strain (mm/mm)
Figure 6-9 Compressive behavior of annealed specimen and un-annealed specimen with nominal
copper coating thicknesses of 30 μm under high strain rate loading (4.9×103 s-1)

143

Stress (MPa)

20
D.C0.49
D.C60.49
D.C60.49.A

15
10
5
0
0.0

0.2

0.4

0.6

0.8

Strain (mm/mm)
Figure 6-10 Compressive behavior of annealed specimen and un-annealed specimen with
nominal copper coating thicknesses of 60 μm under high strain rate loading (4.9×103 s-1)

Stress (MPa)

20
D.C0.49
D.C120.49
D.C120.49.A

15
10
5
0
0.0

0.2

0.4

0.6

0.8

Strain (mm/mm)
Figure 6-11 Compressive behavior of annealed specimen and un-annealed specimen with
nominal copper coating thicknesses of 120 μm under high strain rate loading (4.9×103 s-1)
144

Thirdly, it can be observed that the densification onset starts earlier in the annealed
specimens, which can also be seen in Table 6-1. The lower densification strain may reduce the
improvement in energy absorption capacity. However, it was observed that the annealed
specimens stayed intact during the experiments even at high strain loading, while dramatic
shattering occurred in the un-annealed specimens (especially the ones with the thickest coating,
see Figure 6-12).

Spalling
(a)

(b)

Figure 6-12 Images of (a) un-annealed and (b) annealed Al/Cu hybrid foam specimens with 120
μm nominal copper coating thickness during high-rate compressive loading

Thus, considering the amount of material broken away from the loaded domain, the
densification strain obtained in the plots for the un-annealed specimens may be overestimated. In
addition, it is considered that the relatively ductile nature of the annealed hybrid foams under
compression is favorable as it may avoid local concentrated damage and thus further improve the
energy absorption efficiency of the material.

145

6.3

Discussion
Figure 6-13 to Figure 6-15 shows the comparison of energy absorption, stress drop ratio

and energy absorption efficiency of Al/Cu hybrid foams with different coating thicknesses. The
discussion herein is based on the experimental results from the two extreme strain rates in this
study, namely, 3.3×10-3 s-1 (quasi-static) and 4.9×103 s-1. It can be seen that the energy
absorption capacity of the Al/Cu hybrid foams increases with coating thickness (see Figure
6-15a).
The energy absorption capacity of the Al/Cu hybrid foams from high strain rate
experiments are lower than those from quasi-static experiments due to the specimen shattering
and the material breaking away as mentioned earlier. Even though the initial peak stress was
lower for the annealed specimens, the energy absorption capacity was significantly higher
compared to the specimens without an annealing process. This indicates that ductility is as
important as strength for the coating of hybrid metal foams. Compared to plain aluminum foams,
specimens with a nominal coating thickness of 120 μm had an energy absorption capacity
approximately four times larger under high strain rate loading and more than five times larger
under quasi-static loading.

146

Energy Absorption (N*mm)

4

S
D.49
D.49.A

3
2
1
0
0

20

40

60

80

100

120

140

Coating Thickness ( m)
Figure 6-13 Comparison of energy absorption capacity for quasi-static and high strain rate

Stress Drop Ratio (%)

experiments

80
60
40
S
D.49
D.49.A

20
0
0

20

40

60

80

100

120

140

Coating Thickness ( m)
Figure 6-14 Comparison of stress drop ratio for quasi-static and high strain rate experiments

147

Efficiency (%)

80
60
40
S
D.49
D.49.A

20
0
0

20

40

60

80

100

120

140

Coating Thickness ( m)
Figure 6-15 Comparison of energy absorption efficiency for quasi-static and high strain rate
experiments

The larger stress drop after the initial peak stress for un-annealed specimens with thicker
coatings can be seen in Figure 6-15b. Such stress drop significantly limits or even reduces the
energy absorption efficiency as shown in Figure 6-15. However, by applying an annealing
process the stress drop ratio was significantly reduced, especially for thicker coatings, which lead
to a relatively uniform stress level in the collapse region. The enhanced ductility of the copper
coating leads to a proportional increase in energy absorption efficiency with coating thickness as
shown in Figure 6-15c.
It has thus been shown that Al/Cu hybrid foams manufactured through electrodeposition
have a better energy absorption performance that can be effectively modified by controlling the
coating thicknesses and processing. Such controllable enhancement indicates the potential to

148

fabricate functionally graded foams through a designed electrodeposition procedure that provides
non-uniform reinforcing coatings.
6.4

Conclusions
The high strain rate compressive behavior of Al/Cu hybrid foams manufactured by

electrodepositing nanocrystalline copper onto open-cell aluminum foam was investigated using
experimental methods. The effect of ductility enhancement of the coating material on the
performance of such hybrid foams was evaluated by introducing an annealing process. It was
found that:
1)

The energy absorption capacity of aluminum open-cell foams under high strain

rate loading can be effectively enhanced by electrodeposited copper coatings and that such
enhancement increases with coating thickness.
2)

The compressive behavior of Al/Cu hybrid foams with a thick coating (120 μm)

had a slightly increased initial peak stress at higher strain rates, while no significant difference in
the initial peak stress was observed for hybrid foams with thinner coatings. However, in all cases
the stress level in the collapse region was relatively low under high strain rate loading due to
specimen shattering and the material breaking away, which lead to a lower energy absorption
capacity compared to that under quasi-static loading.
3)

A large obvious drop in the stress level following the initial peak stress was

observed in the compressive response of Al/Cu hybrid foams. This can be attributed to the
rupture of the coating material and the concentration of deformations at weakened critical
sections due to the low ductility of the electrodeposited nano-coating. Such phenomenon is more
significant in foams with thicker coating and is unfavorable for the energy absorption
performance of hybrid metal foams as it limits or even reduces their energy absorption efficiency.
149

4)

The ductility capacity of electrodeposited copper coatings has a great influence on

the overall energy absorption efficiency of Al/Cu hybrid foams. Coatings with better ductility
(e.g., obtained by adding an annealing process) lead to significant improvements in energy
absorption capacity and efficiency, and a more favorable inelastic deformation mechanism. It is
believed that such improvement can also be applied to other hybrid foams manufactured using a
similar procedure. The more ductile failure mechanism due to the reduction in coating brittleness
may be even more beneficial for hybrid foams with larger specimen size since brittle failure
mechanisms may cause localized damage and thus reduce their energy absorption efficiency.

150

CHAPTER 7 STRUCTURAL CHARACTERIZATION AND DESIGN OF
FUNCTIONALLY GRADED AL/CU HYBRID FOAMS
7.1

Overview
In this Chapter, functionally graded hybrid foam structures with property gradient in two-

dimensions were successfully fabricated by strategically reinforcing conventional open-cell
aluminum (Al) foams with nanocrystalline copper (Cu) through electrodeposition. The quasistatic mechanical performance of the fabricated FGHF structural elements were experimentally
evaluated under two different bending loading conditions and compared to Al/Cu hybrid foam
structure elements with uniform coating. The dynamic behavior of FGHF structural element was
investigated using drop weight tower tests considering three-point bending loading conditions.
Functionally graded cellular structures (FGCS) can be used to optimize the mechanical
performance of structural elements such as sandwich panels. However, while many studies have
been conducted in the development of FGCS, most of the studies are limited to FGCS with
property gradients in one-dimension only (usually in the loading direction). And to the author’s
knowledge, there is no reported experimental effort for the fabrication and investigation of twodimensional functionally graded foams in public literature. This can be partly attributed to the
difficulty in creating gradients with higher dimensions. However, higher dimensional spatial
variations in the properties of cellular/foams, in analogy to cortical bone [22], offers a greater
opportunity to optimize mechanical performance when foams are used as structural elements.
Wang et al. [35] considered a graded cellular structure in the conceptual design of an
acetabular component that changed from solid metal at the liner section to one with high porosity
at the implant-bone interface to encourage bone growth upon implantation. Daxner et al. [36]

151

conducted a numerical study and found that the distribution of foam density can be optimized
with respect to structural strength and stiffness for a given loading condition. However, complex
designs such as these ones cannot be realized with the manufacturing process of traditional
cellular materials. Thus, the development in fabrication and investigation in cellular/foam
structures with two-dimensional property gradient is of great interest.
It is hypothesized that the manufacturing process for hybrid foams can be customized to
modify and enhance the mechanical performance of open-cell hybrid cellular structural elements
by applying reinforcement in a strategic pattern or in critical regions compared to applying the
reinforcement uniformly across the domain.
7.2

Behavior of Al/Cu Hybrid Foam
The behavior of Al/Cu hybrid foams have been discussed in previous chapters. Figure 7-1

show the comparison of compressive and tensile behavior of Al plain foam and Al/Cu hybrid
foams with different coating thicknesses. It can be seen that hybrid foams have enhanced
performance in both compression and tension compared to plain Al foam. As expected, the
enhancement in performance is more significant for hybrid foams with thicker coatings. Such
controllable behavior serves as the basis for the design of functionally graded foam systems.
The compressive response of plain Al and Al/Cu hybrid foams has a stress plateau after
yielding and the compressive stress increases significantly at large deformations due to
densification. However, plain Al and hybrid Al/Cu foams have distinct behavior characteristics
when under tensile demands. When the foam samples are in tension fracture occurs at a
relatively low strain level (less than 10% strain). Al/Cu hybrid foam samples show a higher
failure strain (i.e., higher ductility) in tension than plain Al foam samples. The maximum tensile

152

strength of plain Al and hybrid Al/Cu foams is slightly higher than their respective critical plastic
stress in compression. For both plain and coated foams, tensile strength drops sharply after
rupture occurs and the samples fail in a brittle form. This constitutive response indicates that
when such foam materials are used as structural elements, the global structural level failure
mechanism may be governed by tension and material failure will occur in regions where high
tensile stress exists. Such tension controlled failure mechanism may be alleviated if the coating
material can be strategically distributed to provide more reinforcement on tensile stressed regions.
7.3

Selective Electrodeposition of Nc-Cu
The hybrid foams manufactured using the method discussed in Section 2.4.2 have a

uniform coating thickness throughout the sample [8]. While it has been demonstrated that the
hybrid foams have enhanced and controllable modulus, strength and energy absorption capacity
compared to the base material, it is expected that the deformation and failure mechanisms of the
hybrid foams can be modified and improved by reinforcing specific regions in a designed pattern.
In addition, the mechanical performance of the hybrid foams can also be further optimized by
strategically distributing the coating material according to the loading conditions.
Hybrid foam samples with graded coating thicknesses can be fabricated by controlling
the surface of the base foam sample in contact with electrolyte during electrodeposition, which
can be achieved using two approaches. The first approach is to immerse only part of the sample
into the electrolyte during the electrodeposition process, as shown in Figure 7-2a. In this manner,
the coating material is deposited on to the part of the sample immersed in the electrolyte, while
the rest of the sample is not coated since there is no direct contact between the surfaces of the
foam substrate to the electrolyte. This approach can thus be used to fabricate graded hybrid
foams with partially coated regions.
153

10
40-P
40-30 Annealed
40-60 Annealed

Stress (MPa)

8
6
4
2
0
0.0

0.2

0.4

0.6

0.8

0.06

0.08

Strain (mm/mm)

Stress (MPa)

3

40-P
40-60 Annealed

2

1

0
0.00

0.02

0.04
Strain (mm/mm)

Figure 7-1 Compressive and tensile behavior of plain Al and hybrid Al/Cu foams with uniform
coating

154

Sample

Sample

Electrolyte
Immersing (Stage 1)
Sample

Electrolyte
Immersing (Stage 2)

(a)

Uncoated
Level 2
Level 1
Final Product

Masked Area

Electrolyte
Electrodeposition

Washing Coated
Area
Uncoated

(b)

Final Product

Figure 7-2 Manufacturing of graded hybrid foam using a) Partial immersion method and b)
Masking -Washing method

Hybrid foams with graded coating thicknesses can also be fabricated using the noted
method of immersing the foam substrate into the electrolyte in several stages. The level of
coating thicknesses in different regions can be controlled by the time of electrodeposition for
different stages. This method is straight forward and easy to conduct for producing hybrid foams
with a simple coating thickness gradient. However, it is difficult to use this method for fabricate
hybrid foams with more complicated two–dimensional patterns such as that shown in Figure 7-3.

155

b)

a)

Figure 7-3 Hybrid foams with two dimensional graded coating pattern

Another method to fabricate hybrid foams with graded coating thicknesses is to mask a
region of the foam samples using a non-conductive mask agent during the electrodeposition
process, as shown in Figure 7-2b. A liquid acrylic laquer conformal coating (419B, M.G.
Chemicals, Surrey, BC, Canada) was used in this study as mask agent. The mask agent had
service temperature limit of 105 °C and was not reactive to the electrolyte used. In addition, the
mask agent can be removed by acetone after the electrodeposition process. The mask agent was
applied on to the desired region of the foam sample after surface pretreatment procedure. When
the mask agent fully cures after 24 hours, the entire foam sample was immersed into the
electrolyte and the electrodeposition procedure was initiated. Once the desired amount of coating
material was applied, the sample was washed and then immersed into acetone for about 10
minutes to remove the mask agent. The method just described is able to produce twodimensional coating patterns such as the one shown in Figure 7-3. It needs to be noted that a
small amount of coating was observed in the masked region after electrodeposition. This may be
due to the unevenness of the mask agent thickness and some coating material can be deposited
on to the foam substrate where the mask agent is too thin. The coating material within the

156

masked region can be further reduced by enhancing the masking procedure (i.e., by applying a
second layer of masking agent after curing of the first layer). However, the amount of coating
within the masked region is insignificant compared to the total amount of coating. Thus, it is
considered that the performance of the hybrid foam samples will not be affected.
7.4

Quasi-static Behavior of Functionally Graded Al/Cu Hybrid Foams
The performance of hybrid Al/Cu foams as structural elements with graded coating

patterns was evaluated under quasi-static loading conditions. Two flexure type loading
conditions with linear bending moment and constant shear were considered, namely, a threepoint bending beam and a cantilever beam. While both loading conditions have critical sections,
the cantilever beam condition has a lower shear to bending moment ratio compared to the three
point bending beam tests. For each loading condition, four samples with different coating
designs were tested. Foams with pore size of 40 PPI were used for three point bending tests
while 20 PPI foams were used for cantilever beam tests. It needs to be noted that all samples
tested in this investigation were subjected to an annealing process.
7.4.1 Design of Functionally Graded Hybrid Foam
The functionally graded hybrid foam (FGHF) structural element designs followed two
objectives. The first objective was to show that the failure mechanism of foam structures can be
modified and less deformation at critical locations can be achieved by applying reinforcement in
critical areas of a foam domain. The second objective was to show that the mechanical
performance of open-cell foam structural elements, such as stiffness and strength, can be
optimized by employing a strategically designed reinforcement pattern.

157

In order to evaluate the design’s performance, plain Al and hybrid Al/Cu foams with
different reinforcement patterns were manufactured and tested. The same amount of coating
material was applied to all hybrid foam samples. That is, for each test setup, the hybrid foam
samples have approximately the same final weight. Information on the samples is summarized in
Table 7-1.
It has been shown in Section 7.2 that the plain Al foam and Al/Cu hybrid foams have
ductile behavior in compression. In addition, the compressive strength of these foam materials
increases at larger strain due to densification. On the other hand, the tensile behavior of both
plain Al and hybrid Al/Cu foams are relatively brittle and rupture occurs at low deformation. As
a result, when these foam materials are used in a structural element, their tension capacity will
govern the element’s global response. Thus, a more balanced failure mechanism can be achieved
if the tensioned region of the structural element is reinforced. Based on this criterion, samples
with graded reinforcement were manufactured for the two bending conditions for study, which
are shown in Figure 7-4.

(a)

Root
25.4 mm

6.4 mm

25.4 mm

50.8 mm
Plain
Region

50.8 mm

Tip

Region Region Region
3
2
1

Reinforced Region
(b)

17 mm 17 mm 17 mm

Figure 7-4 Foam samples with graded reinforcement: a) bottom reinforced sample for three-point
bending test; b) sample for cantilever beam test with three stepped reinforcement

158

Table 7-1 Information of Al/Cu hybrid foams for three-point bending and cantilever beam tests

Loading

Sample

Condition

Name
T0

Three point

T1

bending
(40 PPI

T2

Foams)
T3
C0
Cantilever

C1

beam
(20 PPI

C2

Foams)
C3

Weight
Description
(g)
Plain Al foam
Al/Cu hybrid foam with uniform
reinforcement
Al/Cu hybrid foam with bottom region
reinforced
Al/Cu hybrid foam with reinforcement
pattern based on topology optimization
Plain Al foam
Al/Cu hybrid foam with uniform
reinforcement
Al/Cu hybrid foam with three stepped
graded reinforcement
Al/Cu hybrid foam with reinforcement
pattern based on topology optimization

Coating
Thicknesses
(μm)

5.0

0

11.0

30

11.8

93

11.0

-

5.7

0

15.9

50

16.1

28,50,70

16.3

-

As is well known, the highest tensile stresses occur at the extreme fiber in the tension
side of the section with maximum bending moment. For the three-point bending test, the
maximum tensile stress is at the bottom of the mid-section and failure at that location will govern
the element’s load-carrying capacity. A sample with reinforcement applied at the bottom of the
sample (T2) was manufactured with the aim of delaying such failure mechanism (see Figure

159

7-4a). On the other hand, a sample with a three step graded reinforcement (C2) was
manufactured for the cantilever beam test. Different reinforcement levels were applied to three
regions (see Figure 7-4b) by controlling the amount of coating material at each region. The
coating level increased incrementally from region 1 to 3 with the highest coating level at the
region (region 3) close to the fixed end where maximum bending occurs. For the 3 step sample
tested, the amount of coating material applied to region 1, 2 and 3 was 16.7%, 33.3% and 50% of
the total coating.
To achieve the second design objective and demonstrate the potential of optimizing the
mechanical performance of open-cell hybrid foam structural elements, samples with a
reinforcement design based on topology optimization were manufactured. The optimized
reinforcement design was developed through a simple power-law topology optimization
approach using the educational program developed by Sigmund [108]. The design optimization
objective was to minimize compliance for the two bending configurations in the study. Figure
7-5 shows the optimized design and manufactured samples with optimized coating pattern for
both the three-point bending test (T3) and cantilever test (C3).

160

a) T3 Design

b) T3 Sample

c) C3 Design

d) C3 Sample

Figure 7-5 Topology optimized reinforcement design: a) For three-point bending test; b) for
cantilever beam test

7.4.2 Test Setup (3-point bending and cantilever beam)
The test setup for three-point bending and cantilever beam tests is shown in Figure 7-6.
For the three-point bending tests the samples were placed spanning along their long dimensions
with 2 mm wide supports on each side. Loading was applied by a universal loading frame
through a semi-cylindrical (radius of 5 mm) loading tip that created a line loading condition at
the center of the top surface of sample. Two aluminum plates with a width of 12.7 mm and
thickness of 0.4 mm were attached to the bottom of the sample at the supports to avoid localized
damage during testing. Pilot test trials showed that placing an aluminum plate on the top region
in contact with the loading tip reduced the bending type failure. In addition, it was of interest to
study the behavior of foam material itself. Thus, no aluminum plate was attached to the top

161

surface of the sample tested. The tests were carried out with a loading rate of 0.05 mm/second.
For the cantilever beam tests, samples were attached to a T-shaped fixture using epoxy, and the
fixture was then mounted using bolts to a rigid steel beam to create a fixed boundary condition.

Foam Sample

Semi-cylindrical
Loading tip
(5 mm radius)

Foam Sample

12.7 mm

2mm

Aluminum Plate
(0.4 mm thickness)
50.8 mm

25.4 mm

25.4 mm

Semi-cylindrical
Loading tip
(5 mm radius)

50.8 mm
2mm

Figure 7-6 Test setup of: a) Three-point bending test; and b) Cantilever beam test.

The cantilever beam tests were conducted using the same loading frame and loading tip
as those used for the three-point bending tests were used for the cantilever beam tests. The line
loading from the semi-cylindrical loading tip was aligned with the edge of the top surface of the
sample (see Figure 7-6b). The cantilever beam tests were also conducted with a loading rate of
0.05 mm/second. Tests were ended when fracture of the sample was observed. The applied force
and the displacement of the loading tip were recorded during the tests.

162

7.4.3 Results
7.4.3.1 Three-point Bending Tests
Figure 7-7 shows the deformation of the three-point bending units at the instance when
fracture was observed. In all cases, failure by tensile fracture occurred at the bottom of the midsection. The force-deformation response was, however, noticeably, different for each design. The
displacement history at the center of sample bottom was obtained by using the image processing
tool in MATLAB [109] based on images captured from a video recording and shown in Figure
7-7.

a) T0

Rupture

b) T1

c) T2

Rupture

d) T3

Rupture

Rupture

Figure 7-7 Images of foam samples in three point bending tests

163

10
T0

Stress (MPa)

8

T1
T2

6

T3

4
2
0
0

100

200

300

Strain (mm/mm)
Figure 7-8 Bottom displacement vs. time from three point bending tests

T0

600

T1

Force (N)

T2

400

T3

200

0
0

10

20

30

Displacement (mm)
Figure 7-9 Force vs. loading tip displacement from three point bending tests

164

T0

600

T1

Force (N)

T2

400

T3

200

0
0

10

20

30

Displacement (mm)
Figure 7-10 Force vs. displacement at sample bottom center from three point bending tests

From Figure 7-8 it can be observed that the bottom mid-span deformation of the uniform
coating sample (T1) is close to that of the plain Al foam (T0). Conversely, the bottom
deformation for the bottom reinforced (T2) and optimized samples (T3) was “delayed,” that is
the mid-span deformation did not grow as rapidly. The bottom reinforced sample (T2) had
lowest bottom deformation among all hybrid Al/Cu foam samples. This is because the tension
region in sample T2 is significantly enhanced in stiffness and strength and thus deformation on
the compression region is involved in the deformation mechanism. The sample with optimized
coating pattern (T3) maximized the stiffness of the foam structure, which lead to reduced bottom
mid-span deformation. It can also be seen that the samples with designed coating pattern (T2 and
T3) have a more ductile response as failure occured at larger flexural curvatures (and also bottom
deformation). This can be attributed to the enhanced ductility provided to the bottom region with
the thicker nc-Cu coating. However, even though the bottom coated sample T2 had a larger

165

reinforcement amount in tension region, rupture occurred earlier in this sample compared to the
optimized design sample T3. This is because the increased compressive deformation in the
bottom reinforced sample effectively reduced the size of the mid-span section (i.e., the section
modulus is reduced) and thus leads to higher tensile stresses and early failure at the sample
bottom. This indicates a compression controlled deformation mechanism.
Figure 7-9 and Figure 7-10 show the force-deformation response at the loading tip and
sample bottom center for three point bending tests. Tests results are summarized and shown in
Table 7-2. It can be seen that all the hybrid foam samples had enhanced stiffness and strength
compared to the plain Al foam sample (T0). The sample with optimized coating pattern (T3) had
the highest stiffness and highest strength. In addition, the optimized pattern also led to a higher
ductility compared to the sample with uniform reinforcement. It can also be seen that, due to the
excessive deformation in the compression zone, the performance of the bottom reinforced
sample (T2) was significantly lower compared to the other hybrid foams. This indicates that
while the failure and deformation mechanism of the beam element can be modified by
reinforcing the tension region, a more strategically distributed coating pattern can lead to
improved, and even optimized, structural performance.

166

Table 7-2 Results summary of tests

Sample

Initial

Max

Stiffness

Force

(N/mm)

(N)

Top

Bottom

Deformation of

Displacement

Displacement

Mid-section

at Fracture

at Fracture

at Fracture

(mm)

(mm)

(mm)

T0

75

206

17.0

9.5

7.5

Three-point

T1

169

574

9.3

3.5

5.8

Bending

T2

114

435

13.5

4.6

8.9

T3

300

592

15.9

8.3

7.6

C0

67

75

11.5

NA

NA

Cantilever

C1

115

309

15.9

NA

NA

Beam

C2

90

369

No Fracture

NA

NA

C3

108

349

20.8

NA

NA

7.4.3.2 Cantilever Beam Tests
Figure 7-11 shows the cantilever samples at maximum deformation response. Tests were
ended when rupture was observed in the sample or when a displacement of 24 mm at the loading
tip was reached. As for the cantilever beam tests, failure was defined by fracture in tensile stress
region as indicated by the arrows in Figure 7-11.

167

Rupture

Rupture

a) C0

b) C1
Rupture

c) C2

d) C3
Figure 7-11 Images of foam samples in cantilever beam tests

For the plain Al foam (C0) and Al/Cu hybrid foam with uniform coating (C1) fracture
occurred near the fixed end where the bending moment is maximum. On the other hand, no
failure was observed in the three step graded foam sample (C2) in which more coating material
was provided near the fixed end. In the sample with optimized coating pattern (C3) fracture
happened at relatively large tip deformation (see Table 7-2) and the location was shifted close to
the center of the sample. This demonstrates that by strategically applying the coating material,
the location of material failure in the foam structure can be shifted or even avoided.
Figure 7-12 shows the force deformation response at the cantilever tip for all samples and
relevant test data is summarized in Table 7-2. It can be seen that the hybrid foams had

168

significantly better performance (i.e., increased stiffness, strength and deformation capacity)
compared to the plain foam. Compared to the uniformly reinforced foam sample (C1), the three
stepped graded foam (C2) and the optimized coating pattern foam (C3) had a 19% and 13%
larger strength, respectively. The elastic stiffness of the optimized coating pattern foam (C3) was
close to the uniformly reinforced foam sample (C1) while the stiffness of the three stepped
graded foam (C2) was slightly lower.
400

Force (N)

300

200

C0
C1

100

C2
C3

0
0

5

10

15

20

25

30

Displacement (mm)

Figure 7-12 Results from cantilever beam tests
It needs to be noted that the stiffness of the optimized coating pattern foam (C3) may be
further enhanced by improving the coating pattern. (Notice there is some difference between the
design of the coating pattern and the actual pattern on cantilever beam samples shown in Figure
7-5.) It can also be seen that higher ductility was achieved in the samples with strategically
distributed coatings compared to the one with uniform reinforcement. The efficiency in
enhancing structural performance by adopting a strategically distributed coating design is thus
demonstrated.

169

7.5

Dynamic Behavior of Functionally Graded Al/Cu Hybrid Foams
When metal foams are used for impact resistance or energy absorption, they are usually

subjected to high-strain-rate loading. Due to its unique feature, the short-duration incident pulse
can be delayed and lead to a relatively long-duration transmitted pulse. It has been shown in
Section 6.4 that by adopting a strategically designed coating pattern, the failure mechanisms of
Al/Cu hybrid foam can be modified and the mechanical properties can be optimized under quasistatic conditions. On the other hand, it is of interest to investigate the dynamic behavior of
functionally graded hybrid foams with designed reinforcements to control the stress wave
propagation through the material. The dynamic performance of Al/Cu hybrid foam structural
elements with graded coating patterns were evaluated using drop weight tower tests under a
three-point bending loading condition.
7.5.1 Design of Coating Pattern
Many studies have been conducted to investigate the performance of functionally graded
foam structures under dynamic loading conditions. Cui et al. [1, 33] investigated the behavior of
functionally graded foam materials under impact loading using finite element analyses and found
functionally graded foams can reduce the duration of the high acceleration during an impact and
had superior energy absorption performance over equivalent uniform foams under low energy
impacts. They also found that convex gradients perform better than concave gradients and a
better performance can be achieved by increasing the density range. Zeng et al. [29, 79] found
that placing the hardest layer as the first impacted layer and the weakest layer on the remote side
had some benefits to maximize energy absorption with a minimum force level transmitted to the
protected structures. However, placing the weakest layer at the impacted end is preferable to
avoid the breakage of the face sheet for sandwich panels subjected to localized impact.
170

Ajdari et al. [34] studied the in-plane crushing of regular, irregular and functionally
graded honeycombs using numerical methods and found that a density gradient could
significantly change the deformation mode and energy absorption of cellular structures under
both low and high crushing velocities. Wang et al. [30] and Gardner et al. [31, 32] conducted
shock tube experiments to study the dynamic response of sandwich panels with graded styrene
foam cores. They found that the overall blast performance and structural integrity can be
improved by increasing the number of monotonically graded foam core layers and applying
polyurea between foam core and the back face sheet. Zhou et al. [78] studied low velocity impact
response on sandwich structures with foam core that has graded densities using experimental and
numerical method and found a better performance of graded core structures over the monolithic
cores. In a parallel study with the current research,
Ohlsson [110] investigated and optimized the performance of a simplify supported
structural panel with Al/Cu hybrid foams core subjected to blast loading. Figure 7-13 shows the
optimized design found by Ohlsson with the object to minimize the displacement of the center of
the bottom plate. In this figure, the darker elements have more reinforcement (thicker coating). It
can be seen that the optimized design basically indicates an arch-shaped strengthened region that
“guides” the loading from the impacted surface to the supports. It needs to be noted that the
loading condition considered by Ohlsson was a uniform compressive loading on the top surface
of the panel. In addition, an isotropic material definition was assumed for the Al/Cu hybrid foam.
However, it has been shown in Section 6.1 that the tensile performance of Al/Cu hybrid foam is
quite different compared to its compressive performance, especially in terms of ductility.

171

Figure 7-13 Optimized design of structural panel with Al/Cu hybrid foam core [110]

The design of coating patterns for the drop tower tests were based the criteria discussed
in Section 6.4.1. The dynamic performance of Al/Cu hybrid foam was evaluated using drop
weight tower tests with a three-point bending condition. Similar to the quasi-static testing, plain
Al foam and Al/Cu hybrid foam samples with uniform coating thicknesses were also included in
the test matrix for comparison purposes. Besides these samples, Al/Cu hybrid foam samples with
two different coating patterns were fabricated and tested, as shown in Figure 7-14.

(a)

Level 2
Level 1
Plain

(b)

Figure 7-14 Design of Al/Cu hybrid foams for drop tower tests

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The first sample (see Figure 7-14a) had a three-step graded coating pattern. The coating
thicknesses increased in the upward direction. In other words, the top layer has the thickest
coating while the bottom layer is not coated at all. Such design is based on the conclusion from
the work by Zeng et al. [29], which indicates that such design is beneficial for energy absorption
and also leads to lower magnitude for transmitted stress.
The second design (see Figure 7-14b) is a modified version of the optimized design of
Al/Cu hybrid foam core in the study by Ohlsson [110]. Compared to the design found by
Ohlsson, the current design does not have reinforced region for the two top corner regions. This
is because the loading is only applied to the center region of the top surface of the sample in the
drop tower tests. Meanwhile, the current design still provides a “guide” from the loading to the
supports. The second difference is that coating was introduced to the bottom of the sample with
the purpose to reinforce the tension region in the bottom. The final coating pattern for the second
design in the dynamic test samples is the same as optimized coating pattern in the quasi-static
tests.
Table 7-3 shows the test matrix for the drop tower impact tests. Six types of specimens
were considered. Three plain Al foams with different relative densities were considered. The first
plain Al foam had the same relative density (6%) as the base foam for all Al/Cu hybrid foam
specimens. The second plain Al foam had a similar relative density (8%) as the Al/Cu hybrid
foam with uniform coating thickness. And the third plain foam specimen had a similar weight to
the hybrid foam tested. For the Al/Cu hybrid foam specimens, the first one had a uniform copper
coating throughout the sample. The second and third Al/Cu hybrid foam samples are the two
designs shown in Figure 7-14. All Al/Cu hybrid foams were designed to have a similar weight. It
also needs to be noted that all hybrid foam samples were annealed (400 oC for 5 min.) after the

173

eletrodeposition process as it has been shown that annealed hybrid foams have improved
performance. As a result, all plain Al foams were treated under the same annealing conditions to
eliminate the influence of material properties due to different processing.

Table 7-3 Sample design for drop tower tests
Loading

Sample

Condition

Name

Description

D-T0

5.6

Plain Al foam (Nominal RD=8%)

6.8

Plain Al foam (Nominal RD=12%)

10.5

RD8
Three-point
bending
(40 PPI

RD12

Al/Cu hybrid foam with uniform

D-T1

Foams)

(g)

Plain Al foam (Nominal RD=6%)

D-T0D-T0-

Weight

reinforcement

Coating
Thicknesses
(µm)
0

11.7

30

11.1

N/A

11.8

N/A

Al/Cu hybrid foam with graded coating
D-T2

thickness (thicker coating on impact
side)

D-T3

Al/Cu hybrid foam with reinforcement
pattern based on topology optimization

7.5.2 Test Setup (Drop Tower)
An INSTRON Dynatup 9250HV drop tower impact tester was used for the dynamic
investigation. In a drop tower impact test, a drop weight is raised to a specific height above the
specimen and then dropped onto the specimen. Figure 7-15 shows the setup for the drop tower
tests conducted in this study. The setup basically consists of two components: the drop tower

174

impact tester system and high speed camera system. In the current study, the drop weight and
specified impact velocity was same for all tests. Before each test, the tower system drives the
drop weight to an automatically calculated height to achieve the specified impact velocity. The
data acquisition system connected to the drop tower impact tester captures the load and
displacement data at a high frequency as the specimen deforms and fails under impact.

Figure 7-15 Test setup for drop tower test

The specimens used for the drop tower tests had the same size as those for the quasi-static
tests (25.4 mm by 25.4 mm by 50.8 mm) and the setup was similar to the three point bending
tests in the quasi-static tests conducted in previous section (see Figure 7-16). The specimen was
supported by two blocks with 5 mm sitting on the block on each side. Two small plates were
attached to the bottom of the specimen at the supports to avoid localized damage during testing.

175

An indenter with semi-cylindrical loading tip (similar to the one used for quasi-static testing) was
aligned with the center of the specimen. As a result, a line loading is applied to the top surface of
specimen at the impact. The speed at impact used was 10 m/s and the drop weight was 6.56 kg
for all tests.

Figure 7-16 Sample loading setup for drop tower test
7.5.3 Results and Discussion
The results from drop weight tower tests are discussed in this section. The comparisons
are based on force-displacement curves and energy input. Figure 7-17 shows the forcedisplacement results from drop tower tests for plain Al foam with a relative density of 6% and all
Al/Cu hybrid foams fabricated on similar base plain Al foams.

176

1400
Plain Al (RD=6%)
Al/Cu Uniform
Al/Cu Stepped
Al/Cu Optimized

1200

Force (N)

1000
800
600
400
200
0
0

2

4

6

8

10

Displacement (mm)
Figure 7-17 Force-displacement curves for samples in drop tower tests

The fluctuations observed in the force-deformation response for all tests (Figure 7-17)
can be attributed to the wave propagation through the sample while the loading is applied. In
spite of this effect, it can be seen that generally the Al/Cu hybrid foam had enhanced stiffness as
well as strength compared with the plain Al foam due to the additional copper coating. The
specimen with graded coating showed stiffer behavior at low deformations compared to other
Al/Cu hybrid foam samples. This is because the hardest layer is on the impact side of the sample.
Once the loading tip hits the sample, the loaded region in the top layer collapses first and
distributes the load more evenly to the remaining material, thus leading to a stiffer response. A
similar behavior was reported by Zeng et al [29] by putting stiffer material on the impact face.
However, at larger deformations the strength of the graded foam was lower compared to the
uniformly coated hybrid foam and the one with topology optimized coating pattern. Compared to
177

the uniformly coated hybrid foam specimen, the optimized coating pattern led to a higher
stiffness and higher strength. This may be attributed to that in the optimized design the coating
material is more efficiently distributed to the highly stressed regions, which leads to a higher
strength.

Plain Al (RD=6%)
Al/Cu Uniform
Al/Cu Stepped
Al/Cu Optimized

Energy (N-m)

6

4

2

0
0

2

4

6

8

10

Displacement (mm)
Figure 7-18 Energy versus time curves for samples in drop tower tests

Figure 7-18 shows the plot of input energy with respect to the loading tip displacement. It
can be seen that the Al/Cu hybrid foams had a significantly higher level of energy absorption
compared to the plain Al foam. Among the Al/Cu hybrid foams, the specimen with stepped
density gradient had higher energy absorption at low deformations compared to other hybrid
foams. This can be attributed to the high initial stiffness and strength of specimen D-T2 at low
deformations. However, the energy absorbed by foams D-T1 and D-T3 surpassed that of
specimen D-T2 as these two specimens has higher strength at larger deformation. Among all
178

hybrid foams, the specimen with a topology optimized design pattern had largest amount of
energy absorption after 5 mm of tip displacement.

Plain Al (RD=8%)
8%)
(RD=6%)
8
Al/Cu Uniform
Uniform
Al/Cu Optimized
Stepped

Energy
Energy (N-m)
(N-m)

6

4

2
2

0
0

0
0

2
2

44

66

88

10
10

Displacement
Displacement (mm)
(mm)
Figure 7-19 Comparison of energy absorption capacity of foams with similar solid volumn

Figure 7-19 compares the energy absorption capacity of the hybrid foams with uniform
and optimized coating pattern compared to a plain Al foam with a relative density of 8%. As
mentioned earlier, the uniformly coated sample in the dynamic tests had a relative density of
8.1%. In other words, the hybrid foam specimens tested had approximately the same solid
volume as the plain Al foam with relative density of 8%. From this comparison, it can be seen
that with the same amount of solid volume, the hybrid foam had much higher energy absorption
compared to the plain Al foam with 8% relative density. The enhancement in energy absorption
capacity seen in Figure 7-19 for the Al/Cu hybrid foam with optimized designs comes from two

179

sources: i) the coating material (annealed copper) is stronger compared to the base material
(annealed aluminum); and 2) the optimized pattern further enhanced the energy absorption
capacity by more efficiently redistributing the coating material.

Plain Al (RD=8%)
12%)
Al/Cu Uniform
Al/Cu Optimized

Energy (N-m)

6

4

2

0
0

2

4

6

8

10

Displacement (mm)
Figure 7-20 Comparison of energy absorption capacity of foams with similar weight

Figure 7-20 shows a comparison of the energy absorption of the hybrid foams with
uniform and optimized coating patterns compared to a plain Al foam with a relative density of
12%. The plain Al foam with 12% relative density had similar specimen weight compared to the
hybrid foams. As the density of copper is much higher (more than three times higher) compared
to aluminum, the hybrid foam with uniform coating had a much lower energy absorption
compared to the plain Al foam with similar weight. Meanwhile, due to the enhancement of the
designed coating pattern, specimen D-T3 had a similar energy absorption capacity compared to

180

the plain Al foam with similar weight. This comparison indicates that, compared to a plain Al
foam, the use of Al/Cu hybrid foams may not be efficient in weigh-sensitive applications, even
with a strategically designed coating pattern. However, certain measures can be adopted to
enhance the performance of hybrid foams. For example, the amount of coating can be increased
in which case the advantage of adopting a functionally graded foam may be more significant.
Nonetheless, the choice of coating material can be optimized by choosing a stronger material
(e.g., Nickel).
7.6

Discussion
The Al/Cu hybrid foam samples tested in this study had approximately the same weight

as a 40 PPI Al foam sample with a uniform 30 µm coating and a 20 PPI foam sample with
uniform 50 µm Cu coating for the three-point bending and cantilever beam tests, respectively.
The advantage of using strategically distributed reinforcement on the open-cell foam structural
elements was demonstrated. The samples with designed coating pattern were shown to have
superior performance, namely reduced deformations and damage, enhanced stiffness, higher
strength and larger ductility. The enhancements and advantages of using functionally graded
coating patterns will likely be more significant compared to uniform reinforcement for cases
with thicker coatings. The proposed manufacturing approach of open-cell foam structures with
two dimensional property gradient can also be applied to other material systems of recent
research interest such as Al/Ni hybrid foams [6, 7, 40, 111].
7.7

Conclusions
Functionally graded open-cell hybrid Al/Cu foam structural elements manufactured

through electrodeposition of nano crystalline Cu onto an open-cell Al foam were manufactured

181

and their performance was evaluated under different types of flexure loading conditions and
loading rate. The performance of functionally graded Al/Cu hybrid foams were compared with
plain foam with different relative density and Al/Cu hybrid foams with uniform coating
thickness. The following conclusions were reached:
1)

An approach to manufacture open-cell cellular structures with two-dimensional

property gradients was presented and demonstrated. The approach allows for strategic
reinforcement across the domain of an open-cell metal foam, in the form of nanocrystalline metal
coatings, for the provision of functionally graded properties.
2)

The tension-controlled deformation and failure mechanism of open-cell foam

structural elements under flexural demands can be controlled by reinforcing the tension region.
However, when the tension region is highly reinforced, the increased demands and possible large
in the compression region need to be considered, as the reinforcement strategy can reduce the
sectional flexural performance.
3)

The location of failure and damage in open-cell foam structural elements can be

shifted or eliminated by employing functionally graded coating patterns. This can be used to
avoid damage at critical regions within the foam structural elements.
4)

Under flexure-type loading conditions at different loading rates, the stiffness,

strength and ductility of open-cell foam structural elements can be optimized by implementing
strategically designed coating patterns.
5)

The energy absorption capacity of hybrid foam material under high strain rate

loading conditions can be optimized by implementing strategic coating pattern.

182

CHAPTER 8 CONCLUSIONS
8.1

Research Significance
This research provided new understanding on the micro- and macro-scale behavior of

hybrid metal foams created through electrodeposition. In addition, the manufacturing process for
such material was improved by introducing an annealing process that enhanced the ductility of
the nanocrystalline structured electrodeposited coating. The experimental characterization under
high strain rate condition provided useful information on the dynamic behavior of hybrid metal
foams. Finally, functionally graded hybrid metal foam systems with strategically designed
nanostrured coating through electrodeposition, which can be designed to achieve optimized
mechanical performances according to different loading conditions, were also developed. The
findings from this research provide essential information on the development of structural
components incorporating functionally graded cellular materials under both quasi-static and high
strain rate loadings.
8.2

Conclusions
Based on the discussions in previous chapters, the following general conclusions can be

provided:
1)

Rupture of the electrodeposited Cu coating is a prevailing failure mechanism in

Al/Cu hybrid foams and fracture of the ligaments can occur under quasi-static as well as
high-strain rate loading conditions. Such behavior is due to the low ductility of the
electrodeposited nanocrystalline coating and is proportional to the coating thickness. This
detrimental effect can jeopardize the the overall performance of hybrid foam systems
fabricated through electrodeposition, such as their energy absorption capacity.

183

2) Beam-element based macro-scale simulations created using multiple Kelvin cells were
established and were shown to adequately capture the unique quasi-static behavior of
Al/Cu hybrid foams. It was shown damage and fracture of the coating and core materials
need to be considered in the simulation of hybrid foams.
3) ,The ductility of the electrodeposited copper coating in Al/Cu hybrid foams can be
enhanced with proper amount of annealing. This improvement in coating material of
hybrid foams leads to an enhanced compressive and tensile behavior and a better overall
energy absorption performance under quasi-static and high-strain-rate conditions. Such
effect is more significant in hybrid foams with thick coatings.
4) Analytical solutions to predict the compressive and tensile response of annealed Al/Cu
hybrid foams were derived. With input on basic properties for base and coating materials
and the relative densities of base and hybrid foams, the analytical models were shown to
have an adequate predictive ability.
5) Hybrid open-cell foams with two-dimensional property gradient were successfully
fabricated and it was shown that the deformation, failure mechanism, and region of
damage of open-cell foam structural elements can be modified or controlled by
reinforcing specific regions. It was also demonstrated that the mechanical properties of
hybrid open-cell foam structural element, such as stiffness, strength and ductility, under
quasi-static and dynamic loading conditions can be optimized by implementing
strategically designed coating patterns.

184

8.3

Future Research

8.3.1 Optimized Material System
This study and investigation was based on Al/Cu hybrid foams systems manufactured by
introducing nanocrystalline copper coating onto plain aluminum foam through electrodeposition.
Copper was chosen as the coating material mainly due to the well-established knowledge in its
electrodeposition process, ease in availability and relatively low cost. However, many other
metals and alloys can also be electrodeposited. Furthermore, with other electroplating techniques
(i.e., electroless deposition), the choice of core and coating materials can be extended to
nonmetal materials such as polymers and ceramics. Using similar techniques and concept as
those presented in this study, it can be of great interest to identify other material systems that
lead to hybrid foams with optimized performance.
8.3.2 Multi-scale Modeling
The evaluation of functionally graded hybrid foam systems at the structural scale using
numerical methods can offer insight and guidance in the design of structural components
consisting of functionally graded foams. This requires proper definition of the macroscopic
behavior of hybrid foams. However, the behavior of hybrid foams is highly anisotropic and thus
defining a representative constitutive response for them through analytical or experimental
approaches can be complicated. On the other hand, to capture the unique behavior of hybrid
foams using a numerical approach requires detailed modeling of the microstructure and to
account for failure mechanisms in the ligaments, such as inelastic buckling, fracture and contact.
Such complicated mechanisms make it almost impossible to establish a macroscopic analysis
that explicitly models the microstructure due to the large computational expense. An alternative

185

way is to establish a micro-to-macro scale analysis, i.e., a multi-scale analysis). Multi-scale
analysis is based on the homogenization approach and can be implemented into the framework of
the finite element method in which a local periodicity is assumed and the macroscopic behavior
at certain integration point is determined through detailed modeling of the microstructure
assigned to that point [112-117]. As a result, there is no need to specify the homogenized
constitutive behavior at the macroscopic integration points. This approach is thus considered to
be preferable in the simulation of hybrid foams in order to capture the micro-scale mechanisms
while maintaining realistic computational demands.
8.3.3 Ductility Enhancement
It has been shown that rupture of the electrodeposited nanocrystalline coating can
jeopardize the overall performance of hybrid metal foams. This is due to the low ductility of the
electrodeposited metal material. In this study an annealing process was applied to hybrid metal
foams as an approach to enhance the coating ductility by modifying its grain structure and
eventually improving the performance of hybrid metal foams. However, the grain size also
depends on the electrochemical condition during the electrodeposition process and thus a better
understanding of the relationship between the parameters in the electrodeposition process and the
performance of hybrid metal foams may help improve the manufacturing of hybrid metal foams..
8.3.4 Wave Propagation through Functionally Grade Metal Foams
The behavior of Al/Cu hybrid foams under high-strain-rate conditions was characterized
in this study. However, one-dimensional propagation of stress wave through functionally graded
hybrid foams has not been fully investigated. A detailed investigation on stress wave propagation
through functionally graded foam structural elements with two dimensional properties gradient

186

also requires further understanding to better design functionally graded metal foams for use in
high-strain-rate condition.

187

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