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NUMERICAL STUDIES ON THE STRESS-WAVE
PROPAGATION IN SOLID AND FOAM MATERIALS

presented by

Zhen Zheng

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NUMERICAL STUDIES ON THE STRESS—WAVE PROPAGATION IN SOLID
AND FOAM MATERIALS

By

Zhen Zheng

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Civil Engineering

2010

ABSTRACT

NUMERICAL STUDIES ON THE STRESS-WAVE PROPAGATION IN SOLID
AND FOAM MATERIALS

By
Zhen Zheng

Improved understanding on the propagation of stress waves and means of
controlling them could lead to techniques that minimize the damage resulting from
blast and impact loads. This research was aimed at investigating the propagation of
stress waves in homogeneous and non—homogeneous solid and open-cell foam
materials in one and two dimensions through numerical simulations. One-dimensional
wave propagation studies considered cantilever bar models to verify the theory related
to stress wave propagation and energy dissipation. Two-dimensional wave
propagation used simply supported sandwich plates, composed from aluminum face
sheets and specially designed foam core, to examine stress wave management. Results
from the bar simulations were consistent with theoretical solutions. Numerical
simulations in the sandwich plates showed that stress waves can be attenuated and
redirected, and that energy dissipation is observable, especially when using tailored, or
functionally graded, foam cores. It is concluded that “engineered” cellular solids can
not only absorb energy to protect against blast and impact loads but that they may be
used in strategic, or tailored, forms to manage the propagation of damaging stress
waves, the energy dissipation and the kinetic energy that imparted to a protected

structure of the structure’s occupants.

Copyfightby
Zhen Zheng

2010

Dedicated to my father, mother and younger sister

ACKNOWLEDGEMENTS

I wish to express my sincere appreciation to my advisor Dr. Rigoberto Burguefio,
although I know these words cannot state all my gratitude to him. I am grateful for his
invaluable guidance during the whole course of this investigation, and his excellent
advice on my future career. Many thanks are given on his patience and efforts on this
thesis as well. He is a remarkable researcher and has a talent on explaining concepts
and ideas in a clear and simple yet thoughtful way based on his broad knowledge. I
have always learnt a lot after each meeting with him, for example his creative
thinking, and this has helped me with a new insight into many problems. I believe all
the skills that I have learned from him will benefit my future life.

Sincere thanks are also given to my other committee members, Dr. Jung-Wuk
Hong and Dr. Nizar Lajnef, for being part of my committee and providing feedback on
the work.

It is also a pleasure to thank those graduate students who discussed with me and
gave me suggestions during my study. Here I want to thank one of my best friends
Jian Rui. He is smart, knowledgeable and always shows up if I need help. He helped
me to revise my thesis draft even when he was on vacation in China and rehearsed my
presentation online when in Buffalo. His suggestions and encouragement helped me to
overcome difficulty I faced and finally made it. This thesis has consolidated the

friendship between us.

Finally and most importantly, I would like to thank my family for their love,
patience, endless support and understanding, which has encouraged me to accomplish

my research. They are always there for me.

vi

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. ix
LIST OF FIGURES ............................................................................................................ x
CHAPTER 1.

INTRODUCTION ............................................................................................................... l
1.1 Motivation ............................................................................................................. 1
1.2 Concept of Stress Wave Propagation .................................................................... 2
1.3 Hypothesis ............................................................................................................ 5
1.4 Scope of the Study ................................................................................................ 5
1.5 Approach ............................................................................................................... 6
1.6 Organization .......................................................................................................... 6

CHAPTER 2.

THEORETICAL BACKGROUND AND LITERATURE REVIEW ................................ 8
2.1 General .................................................................................................................. 8
2.2 Stress Wave Propagation in Solid Material .......................................................... 8

2.2.1 Elastic Stress Wave Propagation in Solid Material ....................................... 9
2.2.2 Plastic Stress Wave Propagation in Solid Material ..................................... 17
2.3 Introduction on Foam Materials ......................................................................... 18
2.3.1 Properties of Open-cell Foams .................................................................... 19
2.3.2 Energy Dissipation in Foam Materials ........................................................ 27
2.4 Developments of Research on Stress Wave Propagation ................................... 29
2.4.1 Stress Wave Propagation in Homogeneous Solid Material ......................... 29
2.4.2 Stress Wave Propagation in Non-Homogeneous Solid Material ................ 31
2.4.3 Stress Wave Propagation in Homogeneous Foam Material ........................ 33
2.4.4 Stress Wave Propagation in Non-Homogeneous Foam Material ................ 35
2.5 Summary ............................................................................................................. 37

CHAPTER 3.

ONE-DIMENSIONAL STRESS WAVE PROPAGATION IN SOLID MATERIALS ..39
3.1 General ................................................................................................................ 39
3.2 Model Definition ................................................................................................. 40

3.2.1 Model Description in 1D Cases ................................................................... 40
3.2.2 Mesh Sensitivity Study ................................................................................ 41
3.3 Material Modeling .............................................................................................. 46
3.4 Stress Wave Propagation in Homogeneous Solid Materials .............................. 48
3.4.1 Case Studies ................................................................................................. 48
3.4.2 Results ......................................................................................................... 52
3.4.3 Discussion .................................................................................................... 61
3.5 Stress Wave Propagation in Non-Homogeneous Solid Materials ...................... 62
3.5.1 Model Particulars ......................................................................................... 62

vii

3.5.2 Results ......................................................................................................... 63

3.5.3 Discussion .................................................................................................... 68
3.6 Summary ............................................................................................................. 68

CHAPTER 4.

ONE-DIMENSIONAL STRESS WAVE PROPAGATION IN FOAM MATERIALS...7O
4.1 General ................................................................................................................ 70
4.2 Model Definition ................................................................................................. 71
4.3 Material Modeling .............................................................................................. 72
4.4 Stress Wave Propagation in Homogeneous Foam Materials .............................. 74

4.4.1 Case Study ................................................................................................... 74
4.4.2 Results ......................................................................................................... 75
4.4.3 Discussion .................................................................................................... 76
4.5 Stress Wave Propagation in Non-Homogeneous Foam Materials ...................... 78
4.5.1 Case Study ................................................................................................... 78
4.5.2 Results ......................................................................................................... 78
4.5.3 Discussion .................................................................................................... 81
4.6 Summary ............................................................................................................. 83

CHAPTER 5.

TWO-DIMENSIONAL STRESS WAVE PROPAGATION IN FOAM MATERIALS .. 84
5.1 General ................................................................................................................ 84
5.2 Model Definition ................................................................................................. 85
5.3 Material Modeling .............................................................................................. 88
5.4 Case Studies ........................................................................................................ 89

5.4.1 Case 1 .......................................................................................................... 90
5.4.2 Case 2 .......................................................................................................... 90
5.4.3 Case 3 .......................................................................................................... 91
5.5 Results ................................................................................................................. 92
5.5.1 Case 1 .......................................................................................................... 92
5.5.2 Case 2 ........................................................................................................ 100
5.5.3 Case 3 ........................................................................................................ 108
5.6 Discussion ......................................................................................................... 114

CHAPTER 6.

CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK ................... l 17
6.1 Conclusions ....................................................................................................... 117
6.2 Recommendations for Future Work ................................................................. 119

6.2.] Experiments on Sandwich Foam Beams ................................................... 119
6.2.2 Improvements on Numerical Simulation ................................................... 120
6.2.3 Design Optimization .................................................................................. 121
BIBLIOGRAPHY ........................................................................................................... 122

viii

LIST OF TABLES

Table 1. Mechanical Properties of Open—cell Foam Material [From (Gibson and Ashby
1988)] ........................................................................................................................... 21
Table 2. Mechanical Properties of Open-cell Metal Foam [From (Ashby et al. 2000)]
...................................................................................................................................... 23
Table 3. Parameters of Theoretical Stress-Strain Curve of Open-Cell Foam .............. 25
Table 4. Properties of Aluminum Plate in 1D Foam Bar ............................................. 72
Table 5. Material Properties of Foams in 1D Foam Bar .............................................. 73
Table 6. Results from Homogeneous Foam Model in ID Case ................................... 77
Table 7. Results from Non-homogeneous Foam Model in ID Case ............................ 82
Table 8. Properties of Aluminum Plate in-2D_ Sandwich Beams ................................. 89
Table 9. Material Properties of Foams Used in 2D Sandwich Beams ......................... 89
Table 10. Foam Materials Used in Case 1 in 2D .......................................................... 90
Table 11. Foam Materials Used in Case 2 in 2D .......................................................... 91
Table 12. Foam Materials Used in Case 3 in 2D .......................................................... 92
Table 13. Summary of Results for 2D Studies ........................................................... 115

LIST OF FIGURES

Figure 1. Schematic of Stress Wave Propagation in 2D ................................................ 3
Figure 2. Schematic of stress wave propagation in tailored foam structure ................... 5
Figure 3. Constant Velocity at Front End of Bar Due to Compressive Stress ............. 10
Figure 4. Wave Propagation in St Time ....................................................................... 12
Figure 5. (a) Transmission of a Compressive Wave Front; (B) Transmission of a
Tensile Wave Front ...................................................................................................... 13
Figure 6. History of Wave Velocities ........................................................................... 15

Figure 7. Reflection and Transmission at an Interface between Dissimilar Materials. 16
Figure 8. Cubic Model for Open-cell Solid [From (Gibson and Ashby 1988)] ........... 20
Figure 9. Normalized Theoretical Stress Strain Curves of Rigid Plastic Foams .......... 26

Figure 10. Plastic Foam Theoretical Stress-Strain Curves with Different Relative
Density .......................................................................................................................... 27

Figure 11. Schematic of Compression Stress Strain Curve of Plastic Foam ................ 28

Figure 12. Ring Packing Arrangements in Experiments: (3) Square-packed, (b)
Transverse Close-packed and (c) Vertical Close-packed Arrangement[From Shim et a]

(2008)] .......................................................................................................................... 35
Figure 13. Baseline Configuration and Different Cases of Interlayer Configurations
Used in the Finite Element Study. [ From Tasdemirci and Hall (2009) ] .................... 36
Figure 14. Homogeneous Slender Bar Model for Solid Material in 1D Cases ............ 40
Figure 15. Right-triangle Shape Shell Element ............................................................ 41
Figure 16. The Locations of Chosen Elements in 1D Models ..................................... 41
Figure 17. Stress History of T3 Element with Length 3 mm ................ ‘ ....................... 43
Figum 18. Stress History of T3 Element with Length 6 mm ....................................... 44
Figure 19. Square Element with Length 3 mm ............................................................. 45
Figure 20. Stress Strain Curve for Steel ....................................................................... 47

Figure 21. Stress Strain Curve for Aluminum .............................................................. 47

Figure 22. Unit Tension Pulse Force Loading History ................................................. 49
Figure 23. Unit Compression Pulse Force Loading History ........................................ 50
Figure 24. Compression Loading Pulse Force with lkN Force ................................... 51
Figure 25. Compression Pressure Pulse with Magnitude 100 MPa ............................. 51
Figure 26. Stress History of Homogeneous Bar in Case 1 in 1D ................................. 52

Figure 27. Stress History for Homogeneous Bar (2D Simulation) Subjected to
Transient Loading and Its Comparison with 1D Analytical Solution [ From Paulino

and Zhang (2007) ] ....................................................................................................... 53
Figure 28. Particle Velocity in Elastic Steel Model under 1N Compression Force ..... 54
Figure 29. Stress History of Homogeneous Bar in Case 2 in ID ................................. 55
Figure 30. Energy Distribution in Case 2 in 1D ........................................................... 56
Figure 31. Stress History in Case 3 in 1D .................................................................... 57
Figure 32. Energy Distribution in Case3 in 1D ............................................................ 57

Figure 33. Stress History at Chosen Locations in Steel Elastic Model under 100Mpa
within 0.6ms ................................................................................................................. 59

Figure 34. Stress History at Chosen Locations in Steel Elastic Model under 100Mpa
within 5 ms ................................................................................................................... 59

Figure 35. Stress History at Chosen Locations in Steel Plastic Model under 100Mpa

within 0.6ms ................................................................................................................. 60
Figure 36. Stress History at Chosen Locations in Steel Plastic Model under 100Mpa
within 5ms .................................................................................................................... 60
Figure 37. Bi-Material Bar Model in 1D ...................................................................... 63
Figure 38. lMPa Compression Pressure Load ............................................................. 63
Figure 39. Points Locations in Bi-material Model ....................................................... 64
Figure 40. Stress History at Given Locations in Bi-Material Model ........................... 64
Figure 41. Stress History of Two End Points in Bi-Material Model ............................ 65

xi

Figure 42.
Figure 43.
Figure 44.
Figure 45.
Figure 46.
Figure 47.
Figure 48.
Figure 49.
Figure 50.
Figure 51.
Figure 52.
Figure 53.
Figure 54.
Figure 55.
Figure 56.
Figure 57.
Figure 58.
Figure 59.
Figure 60.
Figure 61.
Figure 62.
Figure 63.
Figure 64.

Figure 65.

Internal Energy Distribution in Bi-Material Model .................................... 66
Kinetic Energy Distribution in Bi-Material Model ..................................... 67
Energy distribution in Elastic Bi-material Model ....................................... 67
Cantilever Bar Model for Foam Material in 1D Cases ............................... 72
Stress Strain Curves of Foam Materials in 1D Cases ................................. 73
Specified Locations on the Homogeneous Foam Model in ID Case ......... 74
Internal Energy History for Homogeneous Foam Model in 1D ................. 75
Kinetic Energy History for Homogeneous Foam Model in 1D .................. 76
Internal Energy History for Non-homogeneous Foam Model 1 in 1D ....... 79
Kinetic Energy History of Non-homogeneous Foam Model 1 in 1D ......... 79
Internal Energy History of Non-homogeneous Foam Model 2 in 1D ........ 80
Kinetic Energy History of Non-homogeneous Foam Model 2 in 1D ......... 81
Model 1 with Homogeneous Foam Core .................................................... 86
Model 2 with Irregularly Reinforced Foam Core ....................................... 87
Model 3 with Increasing Layered Foam Core ............................................ 87
Model 4 with Decreasing Layered Foam Core ........................................... 88
lOMPa Compression Pressure Load ........................................................... 91
Kinetic Energy of the Top Facesheets in Case 1 ........................................ 93
Displacements of the Top Facesheets in Case 1 ......................................... 94
Displacements of the Bottom Facesheets in Case 1 .................................... 94
Stresses in the Middle of the Bottom Facesheet in Case 1 ......................... 95
Redirected Stresses in the Bottom Facesheet in Case 1 .............................. 96
Internal Energy of Foam from Model 1 in Case 1 ...................................... 97
Internal Energy of Foams from Model 2 in Case 1 ..................................... 97

xii

Figure 66.
Figure 67.
Figure 68.
Figure 69.
Figure 70.
Figure 71.
Figure 72.
Figure 73.
Figure 74.
Figure 75.
Figure 76.
Figure 77.
Figure 78.
Figure 79.
Figure 80.
Figure 81.
Figure 82.
Figure 83.
Figure 84.

Figure 85.

Internal Energy of Foams from Model 3 in Case 1 ..................................... 98
lntemal Energy of Foams from Model 4 in Case 1 ..................................... 99
Kinetic Energy of the Top Facesheets in Case 2 ...................................... 100
Displacements of the Top Facesheets in Case 2 ....................................... 101
Displacements of the Bottom Facesheets in Case 2 .................................. 102
Stresses in the Middle of the Bottom Facesheet in Case 2 ....................... 103
Redirected Stresses in the Bottom Facesheet in Case 2 ............................ 104
Internal Energy of Foam from Model 1 in Case 2 .................................... 105
Internal Energy of Foams from Model 2 in Case 2 ................................... 106
Internal Energy of Foams from Model 3 in Case 2 ................................... 107
Internal Energy of Foams from Model 4 in Case 2 ................................... 107
Kinetic Energy of the Top Facesheets in Case 3 ...................................... 108
Displacements of the Top Facesheets in Case 3 ....................................... 109
Displacements of the Bottom Facesheets in Case 3 .................................. 109
Stresses in the Middle of the Bottom Facesheet in Case 3 ....................... 111
Redirected Stresses in the Bottom Facesheet in Case 3 ............................ 111
Internal Energy of Foam from Model 1 in Case 3 .................................... 112
Internal Energy of Foams from Model 2 in Case 3 ................................... 112
Internal Energy of Foams from Model 3 in Case 3 ................................... 113
Internal Energy of Foams from Model 4 in Case 3 ................................... 1 l4

xiii

CHAPTER 1. INTRODUCTION

1.1 Motivation

Wave propagation is an important phenomenon in engineering. All structures that
are subjected to an impact or dynamic loading, such as blast loading due to an
explosion, or impact loading from a moving object at high speed, experience the
propagation of stress waves within the material. If the magnitude of stress wave varies
within the elastic limit, the structures will have an elastic deformation and recover
their original configuration (neglecting second-order effects). However, if the
magnitude of the stress wave exceeds the material elastic limit, then the structure will
suffer permanent plastic deformation, and could potentially fail. It is thus important to
study the mechanism of stress wave propagation and the way in which they may be
controlled so as to minimize damage.

Stress wave propagation management is an important area in engineering
practice. It includes military applications, automotive crashworthiness design and
aerospace research. In the military, research in stress wave propagation is conducted to
design protective devices such as warheads and body armor with material that can
maintain its integrity and attenuate the magnitude of stress wave under a high speed
impact or blast loading. In civil engineering, more and more significant structures
need to be protected to mitigate strong stress waves from accidental or malicious
impact or blast events. In the automotive industry, crashworthiness is an essential
design aspect of the automotive product. Research in impact stress wave propagation

helps researchers understand crash behavior of structures and materials, and develop

energy-absorbing systems that can handle the potential dynamic impact loads. In
aerospace applications, engineers investigate the propagation of the dynamic stress in

aircraft frames by using supercomputers to simulate non-linear numerical models.

1.2 Concept of Stress Wave Propagation

Stress wave propagation is a common but complicated phenomenon in the world.
In an isotropic elastic medium, stresses propagating inside can be divided into waves
of dilatation and waves of distortion(Kolsky 1963). In a plane dilatational wave the
direction of particle velocity is parallel to the direction of wave propagation; in a plane
distortional wave those two directions are perpendicular to each other. Only these two
waves propagate in an extended solid if it is unbounded. In addition, Rayleigh surface
waves propagate in a free surface of the solid or in a surface boundary between two
solids.

Among the waves produced by shock loading in two dimensions, the dilatational
wave is of prime interest since it induces large deformations and damages in the solid.
When a load reaches the structure, a large amount of kinetic energy will be imparted.
From a microstructural perspective, particles will start to move quickly once they gain
kinetic energy. They hit other conjoint particles thus stress magnitude starts to rise and
energy is transmitted to other particles. After the energy transmission, stress
magnitude begins to decay and the particles will slow down. This transit stress travels
through the material in the form of a compressive stress wave(Smith and Hetherington
1994). As shown in Figure 1, a simply-supported beam is under a pressure loading in

the middle of top side. Once the pressure is applied on the top surface of the beam, the

stress starts to propagate inside of the beam in all the directions (denoted in dashed

an‘ows) in the form of a compressive stress wave.

 

 

 

 

 

Figure 1. Schematic of Stress Wave Propagation in 2D

Stress wave propagation occurs in different mediums. If the stress wave travels in
a model with single material, it can be specified as stress wave propagation in a
homogeneous material. If it travels in a medium consisting of two or more materials, it
may be referred to as stress wave propagation in a non-homogeneous material. Two
types of materials are widely used in analysis and experiments: solid materials such as
steel and aluminum, and foam materials which are porous solids.

Stress wave propagation has been investigated extensively during the past
decades. Researchers studied stress wave propagation in homogeneous solid materials
such as steel and aluminum, in homogeneous foam materials such as aluminum foam,
in non-homogeneous solid materials such as graded metal-ceramic composites, and in

non-homogeneous foam materials that can be layered or functional graded foam. It has

been found that the efficiency of energy dissipation and stress mitigation in foam
materials is much higher than in solid materials.

However, the concept of guiding or controlling the propagation of stress waves in
two dimensions has not been explored. To the author’s knowledge there are no
experimental, theoretical or numerical studies available to date on this concept.
Therefore, in the studies carried out in this thesis, a tailored foam structure was
employed to demonstrate that the path of the stress wave propagation could be
controlled. In the structure, stress waves may not propagate in a straight or linear way
due to modified foam material. This idea is illustrated in Figure 2, which shows a
simply—supported foam beam with reinforced arch foam (zone in grey color)
surrounded by a weaker foam and subjected to a pressure load at the middle of the top
surface. This study is aimed at demonstrating that after the impact load is applied on
the structure, most of the stress wave (denoted in solid lines) will propagate very fast
in the reinforced zone and will be directed to the supported region, and that stress
waves of reduced intensity (shown in dashed lines) will travel in the weaker foam
regions. As a result, the magnitude of the stress in the middle of the bottom facesheet
decreases while the stresses in the simply-supported region increase when compared
with a model with a homogeneous foam core. During the process of the stress wave
propagation, the kinetic energy imparted by the exterior force is dissipated and stored

as internal energy in the form of plastic deformation in the strong and weak foam.

 

 

 

 

Figure 2. Schematic of stress wave propagation in tailored foam structure

1.3 Hypothesis

The hypothesis behind this study is that the stress waves created from blast and
impact loads can be managed, or directed, by using open-cell foam materials with
engineered constitutive behavior. Such behavior could lead to the development of new
materials and structures with enhanced resiliency to impact and blast loads for use in

stand-alone structures or as protection devices.

1.4 Scope of the Study
This study is to provide first-level validation of the hypothesis that the stress—
waves from blast and impact loads can be managed (i.e., directed, attenuated, and

controlled) through design of high performance cellular materials. The results will be

used to establish the design criteria for the fabrication of the nano-reinforced cellular
structures. The study was undertaken to numerically evaluate the propagation of the
stress waves in homogeneous and non-homogeneous materials in one and two
dimensions. One—dimensional (1D) behavior was studied by means of a cantilever bar,

and two-dimensional (2D) behavior was studied by means of sandwich beams.

1.5 Approach

The approach to this study is the numerical simulation of stress-wave propagation
using the nonlinear finite element program LS-DYNA (LS-DYNA). First, theoretical
solutions for one-dimension, which have been extensively studied during past decades,
were used to validate the numerical results from 1D models. Upon validation of ID
models with theoretical solutions, the model was used to evaluate other one—
dimensional problems. The validated approach in one-dimension was then assumed to
be appropriate to study two-dimensional response. Further, the features of stress wave
propagation gained from the 1D studies were used as criteria to evaluate results from
the 2D analyses. Results for the 2D models are considered to provide a level-l
validation of the mentioned research hypothesis and provide guidance for future

experimental studies.

1.6 Organization

The remainder of this thesis is organized as follows. Chapter 2 provides a
summary on the theory of elastic and plastic wave propagation in homogeneous and
non-homogeneous solid material. Elastic and plastic wave energy distribution in bi-

material is presented. Details on the modification of foam materials are presented as

well. In addition, a review is given on the studies and works done in the past decades
on stress wave propagation in homogeneous and non-homogeneous solid and foam
materials. Chapter 3 deals with the study of stress wave propagation in solid material
in one dimension. A cantilever bar model was employed. Steel was used for the
homogeneous model, while a steel/aluminum hybrid bar was used for the non-
homogenous model. Results from numerical simulations were discussed and compared
with theories. Chapter 4 deals with the stress wave propagation in foam material in
one dimension. Two kinds of foam materials were used, and results from
homogeneous and non—homogeneous models highlight the features of stress wave
propagation in foam materials. In Chapter 5, studies on the stress wave propagation in
foam material in two dimension are presented. Analyses were conducted on four
aluminum foam sandwich plates with varied configurations. Results derived from the
numerical studies in this thesis confirm the postulated research hypothesis. Finally in

Chapter 6, conclusions are provided and future research needs are identified.

CHAPTER 2. THEORETICAL BACKGROUND AND LITERATURE REVIEW

2.1 General

This chapter presents the concept of stress wave propagation, theoretical
background on stress wave propagation in solid materials, introduction to foam
materials, and developments of research on stress wave propagation.

Sections 2.2 and 2.3 provide well-established and widely accepted knowledge on
the mechanism of stress wave propagation in solid materials, and the properties of
foam materials with a concept of energy dissipation to help comprehend the complex
behavior of stress waves from sophisticated studies and experiments. Meanwhile, it is
important to know that all the theories provided here will be used to examine and
validate the accuracy of numerical models conducted in this study, as no experimental
data or theoretical solutions are available at present for models employed in this study.

Section 2.4 shows research that has been done before on four fields based on the

classification of material.

2.2 Stress Wave Propagation in Solid Material

More than one century ago, researchers started to study elastic stress wave
propagation in solid materials (Lamb 1904), where deformations occurred in the
material while it was still in its elastic region. In 1930 Donnell extended the work for
materials behaving in under plastic deformations (Donnell 1930). Stress wave
propagation in solid material such as steel has been studied extensively and

documented very well (Smith and Hetherington 1994). Therefore, it constitutes

important and fundamental knowledge to understand more complex phenomena of

stress wave propagation in homogeneous and non-homogeneous materials.

2.2.1 Elastic Stress Wave Propagation in Solid Material

This section deals with two important concepts of velocity related to the
phenomenon of stress wave propagation, stress wave reflection and transmission in
homogeneous and non-homogeneous models.

At the beginning of this section, the elastic wave speed relation in homogeneous
material is presented. The concept of particle velocity is then introduced. After
introducing the two velocities, mechanism of stress wave reflection and transmission
in homogeneous material explains how those two velocities determine if a propagating
stress wave is compressive or tensile based on a certain boundary condition. One
example on homogeneous cantilever bar is added to illustrate how to apply this
mechanism in different models. In addition, energy transmission of elastic stress wave

in non-homogeneous (bi-material) model is explained.

2.2.1.1 Elastic Wave Speed

All materials propagates the effects of external loads by means of elastic wave
propagation. In a one-dimensional homogeneous structure, the elastic wave speed
depends on the homogeneous material’s mass density and Young’s modulus

(Achenbach 1973), that is:

This is a widely used equation and has been validated by many experimental
data. It indicates that elastic stress wave propagates at a constant speed in a
homogeneous material. A theoretical proof of Equation (1) is available (Davis 2000).
Figure 3 shows a homogeneous bar with forces applied on the left surface. During the
propagation time t, the distance of a propagation of particle velocity is vt in the
positive x direction. Since the area of the cross-section is A, a volume (zone in black)
formed by the particle velocity is calculated asV 2 AW . At the same time, stress wave
has propagated from 0 to a distance ct in the positive x direction, the volume formed
by the propagation of wave velocity (denoted in black box and white box) is

calculated as V = Act, where v is the particle velocity, cis the wave velocity, andA

IS the area of cross-section. According to Fr gure 3, strain IS computed as E = —. Given
c

Hooke’s law and Newton’s equation of motion F = m(v/t), the following equation is

derived:

F=A0=AE8=AE(v/c)=m(v/t)=pctA(v/t) (2)

From the Equation (2) above, c = .lE/p is obtained.

vol=Avt V01=ACt Front of Stress
Wave

 

 

 

0 vt ct x

Figure 3. Constant Velocity at Front End of Bar Due to Compressive Stress

10

When propagating in two and three dimensions, an elastic wave speed depends

on a mass density, Young's modulus and Poisson's ratio as shown in Equation (3),

c . . . .
where l is speed of longitudinal wave, E is Young’s Modulus, '0 IS densrty, V rs
Poisson’s ratio. This equation also shows that wave velocity is a constant value when

it propagates in a homogeneous material.

E
: —— 3
Cl 113,0(1— 2v) ( )

Another equation to calculate wave velocity at a specific location was proposed

by Paulino and Zhang (Paulino and Zhang 2005):

E (x)(wl - mm
C ,- :
d (x) \[(l+v(x))(l—20(x))p(x) (4)

 

 

where Cd(x) is speed of longitudinal wave, E(x) is Young’s Modulus, p(x) is

density, v(x) is Poisson’s ratio, xindicates the specified location.

The three equations presented above give similar results under the same
conditions. Equation (3) and (4) were derived from different experimental data.
Equation (1) is widely accepted and offers reasonable results, thus it would also be

used in following studies.

2.2.1.2 Particle Velocity

When an exterior load is applied on a structure, the imparted kinetic energy
makes particles move. In Figure 4, a pressure load with magnitude 0' is applied on the

free surface of a bar. The black box represents an area formed by the propagation of

11

stress waves at a time internal (5! .Because of the conservation of momentum. particle
velocity v can be calculated through Equation (5):

0'

 

S

v =

a
‘Q

where 0 is stress applied on the structure, E is Young’s Modulus, p is density.
Equation (5) indicates that particle velocity is constant if the material and loading are
determined.

5.5
p t

——>
——->

Figure 4. Wave Propagation in St Time

 

 

2.2.1.3 Elastic Wave Reflection and Transmission in Homogeneous Material

In general, two kinds of velocities appear in the defining the phenomena of wave
propagation: particle velocity and wave velocity. Particle velocity is used to describe
the velocity of a particle (real or imaginary) in a medium as it transmits a wave. Wave
velocity represents how fast the message of movement is transmitted from one group
0f particles to others. They are different in that particle velocity is much slower than
the wave velocity in a single material.

Figure 5 offers a model to describe mechanism of elastic stress wave propagation
in homogeneous material(Smith and Hetherington 1994). Particle velocity and stress

Wave velocity, as discussed above play important roles in this model. The mechanism

derived from this model is significant for studies conducted in the following chapter,
as it is one important criterion to evaluate and comprehend results from numerical

models.

 

’//////////////////////////////////////////////////////////////////

(b)

Figure 5. (a) Transmission of a Compressive Wave Front; (B) Transmission of a
Tensile Wave Front

In Figure 5, each small truck represents a particle while the springs between each
truck represent the intermolecular force. The truck velocity is a particle velocity, and
the velocity that the movement transmits from one truck to another is the wave
velocity. When the direction of the particle velocity is parallel to the direction of the
wave velocity, this wave will be a compressive wave. When the particle velocity and
the wave velocity move in opposite directions, this wave will be a tensile wave. For
example, Figure 5(a) shows that particle velocity is directed to the right due to a force
in the same direction. The wave velocity will also be directed to the right. Therefore,
this wave is a compressive wave. Figure 5(b) shows that the particle velocity is
directed to the left due to a force applied to the left, but the movement is transmitting
from left to right, meaning the wave velocity is in the right direction. Hence, this wave
is a tensile wave.

The mechanism from Figure 5 can be applied to other elastic homogeneous

models that have dissimilar configurations and boundary conditions. Figure 6 presents

13

an analysis on elastic wave propagation in a homogeneous cantilever bar modeled by
using the concepts state above. A pressure load is applied at the free end of the
cantilever bar. Particle and stress wave velocities are generated simultaneously. In
Figure 6, the big arrow represents wave velocity and the small arrow represents
particle velocity.

In Figure 6(a), the wave and particle velocities are in the same direction. Thus
this wave is a compressive wave.

In Figure 6 (b), because no displacement and velocity is allowed at the fixed end,
a compressive wave reflects back with the same velocity magnitude but in the opposite
direction. According to the conservation of momentum, a particle also rebounds back
with the opposite velocity. Thus, again, the wave and particle velocities are in the
same direction. This wave is also a compressive wave. However, the stress intensity at
the fixed end will double because of the overlap of two compressive waves.

In Figure 6 (c), displacement and velocity is allowed at the free end. Thus, the
wave reflects back at the free end, but particles continue to move because of their
inertia. At that time the bar experiences stretching. Wave velocity and particle velocity
act in the opposite directions, so this wave is a tension wave. Stress intensity is zero at
the free end because of an overlap of compressive and tensile waves with the same
magnitude.

In Figure 6 ((1), because no displacement and velocity is allowed at the fixed end,
the tension wave reflects back with the same velocity magnitude but in an opposite

direction. Because of the intermolecular force, the particles are also being pulled back

14

with the opposite velocity. As a result, the wave and particle velocities propagate in

opposite directions. This wave is a tensile wave.

<::3 {——

 

A

 

 

.\\\\\\\\\\\\\

 

(a) Velocity directions when the wave travel from the right side to the left side

::> ——->

 

 

 

\\\\\\\\\\\\\'

 

(b) Velocity directions when the wave rebounds at left fixed end and travels from the
left side to the right side

C: ——-)

 

 

 

.\\\\\K\\\\\&

 

(c) Velocity directions when the wave rebounds at the right free end and travels from
the right side to the left side

l:> {—

 

 

 

.\\\\\\\\\\\\\

 

(d) Velocity directions when the wave rebounds at the left fixed end and travels from
the left side to the right side

Figure 6. History of Wave Velocities

2.2.1.4 Elastic Stress Wave Reflection and Transmission in Non-homogeneous
Material

Figure 7 below shows a non-homogeneous model (bi-material) to illustrate the
concept of stress wave reflection and transmission at the interface of two dissimilar
materials(Smith and Hetherington 1994). When a stress wave travels from material 1
to material 2, it breaks into two parts: in one part the wave reflects back into material 1
and in the other part wave transmits into material 2. The magnitude of reflected and
transmitted waves can be calculated with Equations (6) and (7). From these equations,
stresses can be calculated once two materials’ mass densities, Young’s modulus and

original stress are known.

45%

0b :2 \/E1’01+\/E2’02 Ga (6)

a _ JEzpz'xlErpi a

c— x/E1”1+\/E2"’2 a (7)

 

 

 

 

C

 

 

Figure 7. Reflection and Transmission at an Interface between Dissimilar
Materials

16

2.2.2 Plastic Stress Wave Propagation in Solid Material

Plastic stress wave refers to a stress wave propagating inside a plasticized
material. It is much more complicated than elastic wave since the material is in a non-
linear situation. Once a plastic wave is generated within a material, the speed of wave
propagation slows down and the kinetic energy is dissipated (explained in section

2.42) due to permanent plastic deformations in the material.

2.2.2.1 Plastic Wave Speed

Equation (8) below is used to compute the velocity of plastic wave propagation

C
(Donnell 1930). where p is velocity of plastic stress wave, pis mass density, 0' is

stress and E is strain. This equation indicates that the speed of plastic wave varies if
the stress-strain curve of a material is non-linear beyond elastic limits. In general, the
Speed of plastic wave is smaller than the speed of elastic wave, since the tangent

modulus of the non—linear region in the stress-strain curve is smaller than elastic

C : liq. (8)
P pds

Because plastic waves shows different features in dissimilar materials,

modulus.

researchers developed other models to deal with such situations. Equation (9) offers

another model to explain the speed of plastic stress waves (G.Vaughn ct al. 2005),

17

do . . . . . .
where Et 2 17—,prs mass densrty, 0' rs stress, e rs mathematical constant and 5 IS
8

strain. However, this equation has three limitations:
l)If yielding happens with small strains and Er <<|0'|, c: lEt/p that is

identical to the equation proposed by Donnell.

2) If a plastic compression wave propagates along an elastic-perfectly plastic rod
(E! =0), the speed of plastic wave c: l0), /,0 where 0'), rs the yield

SII‘CSS.

3) If Er = 0', the wave speed of a tensile wave is close to zero.

(9)

 

2.3 Introduction on Foam Materials

A foam material is a porous material commonly existing in nature and people’s
life. It is widely used in architectural, industrial, medical and consumer applications
primarily due to its light weight compared to its strength/stiffness and the many
functions that can be given to the open cavities of the material..

Foam materials are generally divided into two groups: open-cell and closed-cell
foams. The mechanical behavior of open-cell foam materials is easier to predict since
the gas inside closed-cell foams increases stiffness under compressive loading.

Foam materials are available in a variety of forms. They can be made of metal,

metal alloys or ceramic materials. Foam materials made of aluminum have been often

18

used in research related to understanding the mechanical properties of foams as well as
their design features.

Studies in this thesis are based on open-cell aluminum foams due to the large
amount of research has been done on this material, and the availability of well-
developed models to simulate their constitutive behavior. Further, the research being
considered as an extension to the presented study is to be conducted on open-cell
aluminum foams. It was thus meaningful to simulate this material in the numerical
simulation in the current research.

Since one objective of these studies is to demonstrate that a modified, or
“engineered,” foam material may allow control over the propagation of stress waves, it
is helpful to know the properties of foam material and the means of modifying them.
Considerable research evidence exists to prove that foam materials are good at
protecting structures by means of mitigating stress and dissipating kinetic energy. It is
then also important to learn the mechanisms of stress mitigation and energy

dissipation in foam materials.

2.3.1 Properties of Open-cell Foams

Gibson and Ashby have summarized an extensive treatise on the behavior and
modeling of foam materials (Gibson and Ashby 1988). Their work has shown that
open-cell foams may be adequately modeled based on the behavior of simple unit
cells. One of the simplest three-dimensional cells that can be used to simulate the
behavior of open-cell foams is shown in Figure 8 (Gibson and Ashby 1988).. This

cellular solid is a cubic cell whose edges are made up of solids struts at a micro-

l9

structural level. At each face, two half-length struts are assumed to connect with other
cells with identical configuration. The collection of many cells forms a foam block

commonly used in experiments.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Cell Edge
L in I
7 -> + t
L E \
LU
L“ Open Cell Face
K———->l
L

Figure 8. Cubic Model for Open-cell Solid [From (Gibson and Ashby 1988)]

Generally, the tensile modulus E, of a metal foam is greater than its compressive
one, Ec' It has been observed that open-cell foams experience three stages while

loaded in compression. The first stage consists of deformations of the cell edges while
the solid material is in its linear elastic range. Once the compressive load goes beyond
the yield stress of the solid material, the foam enters the plastic collapse region and its
macroscopic response features a plateau region in its stress strain response. As the
cells collapse completely, the edges and faces of cells touch each other and the foam
material enters the densification region at the macro scale, which leads to a rapid

increase in the resisted stresses while the strains do not change significantly.
* .
Results from experiments show that the relative density,,0 /,0g (i.e., the mass
density of the foam divided by the mass density of solid material) and the properties of

20

the material of which the foam is made from determine the crushing behavior of open-
ccll foams. Table 1 lists the equations proposed by Gibson and Ashby (Gibson and
Ashby 1988) to calculate the mechanical properties of open-cell foams based on the

simple unit cell in Figure 8.

In Table 1, p*is the mass density of the foam material; ps refers to the mass
density of the solid material from which the foam is made of; ES is the Young’s
modulus of solid material; ays refers to the yield strength of the solid material; afsis

the fracture strength of the solid material.

Table 1. Mechanical Properties of Open-cell Foam Material [From (Gibson and
Ashby 1988)]

 

 

 

 

 

 

 

Property Open-cell foam material
:1:
. . p
Relative Densrty —
10S
Modulus of * 2
Elasticity, E 13* z [L]
2 .'
[GN/m ] ‘ p,
Shear Modulus, * 2
G 2 0* .. 0.3755g[p—]
[GN/m 1 ‘ P,
Porsson 3 Ratio U = g — 1
20"
- 2
Compressrve ,, 2 [/2
Elastic Strezngth 0* : O‘O3Eg[p—] 1+ [L]
[MN/m 1 e! ‘ p... 9..

 

 

 

 

21

Table 1(cont’d)

 

 

 

 

 

 

 

 

 

* 3/2
0* =0.30W[—)—-]
)
Compressive PI '1 5
Plastic Strezngth * 3/ 2 * 1/2
[MN/m] 0* 20230 ‘ P_ 1, P_
P1 'H ps ’05
. (including density correction)
Compressrve 3 / ,,
Crushing * pi“
Strength 0’ - 0 650']- It?
2
[MN/m ]
Densification 8 =1-1.4 i
strain D p
S
Tensile Plastic *
Stiffneszs E: = E5[L]
[ON/m l ”s
Tensile Plastic =1:
Collapsze 0* : 0vs[E—]
[MN/m] a " ”s
Tensile Brittle * 3/2
Strenggjlz ch 20.650fS,/7z1 9—
[MN/m ] ”s

 

 

In addition, Table 2 lists formulas to predict the behavior of open-cell metal

foams derived from experimental data (Ashby et al. 2000).

Table 2. Mechanical Properties of Open-cell Metal Foam [From (Ashby et a1.

 

 

2000)]
Mechanical Open-cell metal foam
properties
Young’s 2
modulus (GPa), E = (0_] _ 4)E‘ _/)_
E 5 pg

 

 

 

 

Shear modulus 3

(GPa), G G ” g E
Bulk modulus

(GPa) K ~ 1.1E
Flexural
E z E

modulus (GPa) f
Poisson’s ratio 0.32-0.34

 

Compressive
strength (MPa)

0' = (0.1 - 1.0)0'6,

312
i
c S :05

 

Tensile strength

at z(1.1—1.4)ac

 

 

(MPa)
Endurance limit 0- 2 (0,5 —0,75)0'
(MPa) 6 C
3
Densrfrcatron e : (0.9—I.0)x 1_1.4 L +0.4 _p_
strain D ps ps

 

Loss coefficient

27 z (0.95-1.05)xt;;7;—)
S

 

 

 

 

Hardness (MPa) H = 0" 1 + 2L
C p s
Initiation P
toughness * 3 L
2 JIC flatt,s'l[p
(J/m ) S

 

 

Comparing Table l and Table 2 it can be seen that the formulae are similar. The

parameters in Table 2 were chosen based on experimental data and empirical

experience. Formulas in Table l are more specific and can be plotted easily. Thus the
stress-strain curves used for the analyses in this thesis are based on these formulas
presented in Table 1.

As previously noted. the compressive stress-strain curve for open-cell foam has
three regions: linear elasticity. plastic collapse and densification. In experiments, the
dramatic increase of the stress-strain curve happens after the foam has been
compressed to a relative density of about 0.5.

Based on the formulas offered in Table 1, a new set that describes the macro-
scale compression behavior of open-cell metal foams is deduced and provided in
Table 3. The equations can be used to define the complete compressive stress-strain
model for an open cell foam of given relative density and known base solid material.
Two parameters D and m that are derived from experimental data are used to calculate
strain in plastic and densified regions, and stresses in densification region. In the

current thesis. D223 and m=l.

24

Table 3. Parameters of Theoretical Stress-Strain Curve of Open-Cell Foam

 

 

 

 

 

 

 

Strain Stress
1 2
Elastic region 0- “" ‘2‘ * ,0
0<8<Ee Ee =O.3—Eyi[£ = 535 ,0—
3 ps 5
=I< >l<
0 =0
. 8 =8 [l—i) P1
Plateau region d D D 3
8 <8<8 -
e d 8021—14—10— 0 =0 x0.3x 40- 2
ps I S ps
8 m
l 0*=o* i D
Densification Ed :80 1‘3 PI D 80 —-8
region 3
8 <8<8 g =1_14_E_ _
d D D ' .0 p 2
V 0' [=0 Sx0.3>< —
’05

 

 

 

By using relative density 0.3 and ays /ES 2 1/ 30 the compressive stress-strain

response for 'an open-cell foam was calculated and the complete trace is shown in
Figure 9. The values and plot are in agreement with the results provided by Gibson
and Ashby (Gibson and Ashby 1988). Normalized stress-strain curves with different

relative densities are also plotted.

25

r

 

10

 

I
Relative Density 0.30 :
-- -- Relative Dcnsityl_).20 I
I
I

J

— — Relative Density 0.10

 

 

 

I
I

I

— ----- Relative Density 0.03 ' II

I

I

I

I

 

 

C

X

El:

99

e 5—

?. I

g I

V? I

C3

5: l

E ,’ /
z 2‘ x /

 

 

 

 

1.0

Compressive Strain 8

Figure 9. Normalized Theoretical Stress Strain Curves of Rigid Plastic Foams

Since these formulas can predict the identical curves from (Gibson and Ashby
1988), they can also offer a stress—strain curve of plastic foam by using real material
properties. By using a relative density of 0.3, a yield stress for aluminum of 80MPa,
and a Young’s modulus 80 GPa, a normalized compressive stress-strain curve of open-
cell aluminum foam is shown in Figure 10, in which the onset of plastic strain is
0.55x10'4, the onset of densification strain is 0.327 and the limit densification strain is
0.580. These values are identical to data from (Gibson and Ashby 1988). The plastic

stress is calculated to be 3.9 MPa can be obtained as well from Figure 10.

 

25

 

 

 

 

 

 

I I

— RD 0.30 I I

20 ~ -— - RD 0.24 I l

-- RD 0.10 I |

; ---- RD 0.03 I l

c. 15-

E ' i

.52 I I
g 10~ I

/ I

5 — // I

L— -- -— -- -_ --' l/ I

._———-""’ /

 

 

 

0.0 0.2 0.4 0.6 0.8 1.0
Compressive Strain 8

Figure 10. Plastic Foam Theoretical Stress-Strain Curves with Different Relative
Density

By decreasing the relative density of the foam the plastic plateau region extends,
the plastic, or yield, stress drops, and the onset of densification strain moves toward
higher values. For example, for a foam with a relative density of 0.1, the stress-strain
curve is shown in Figure 10. The onset of plastic strain is 0.0316, the onset of

densification strain is 0.486, the limit end of densification strain is 0.860, and the

plastic stress is 0.75 MPa.

2.3.2 Energy Dissipation in Foam Materials
The macro-scale compressive stress-strain response of a typical metal foam is
composed of three regions: linear elasticity region, plastic collapse plateau region and

densification region. The kinetic energy produced by impact and blast loading is

27

mainly dissipated under the plateau region. In this region, the kinetic energy is
transferred into strain energy and fracture energy due to permanent deformation. A

schematic of this compressive stress strain behavior is shown in Figure 1 1.

{(3 'Young sé/Iodulus, Densification
g l
q I Yield Plateau

Energy
dissipation

 

Strain, a

Figure 11. Schematic of Compression Stress Strain Curve of Plastic Foam

Tests on aluminum-based foams show that the dependence of the plastification
stress, or plateau stress, on strain rate is not strong. Nonetheless, it is important to
separate the effect of strain rate and impact velocity on the dynamic response of
metallic foam. The negligible effect of stain rate is associated with the fact that
aluminum displays only minor strain rate sensitivity. In contrast, material inertia leads
to enhanced stresses at high impact velocities.

Homogeneous foam is most efficient for absorbing energy when the demand

strain just reaches to onset of densification. This is because most energy is absorbed

28

under the plastic plateau region (denoted in shadow area in Figure 11) without a
significant increase in stress.

In order to maximize the energy dissipation area shown in Figure 11, one
approach is to modify the relative density of the foam, and/or the Young’s modulus
and yield strength of solid material. As these three parameters increase, the yield

stresses of the foam material increase but the densification strain decreases.

2.4 Developments of Research on Stress Wave Propagation

The following sections provide an overview of the research that has been done in
the past decades related to stress wave propagation, which may be categoraized the
medium of propagation: stress wave propagation in homogeneous solid materials, in
non-homogeneous materials, in homogeneous foam materials and in non-

homogeneous foam materials.

2.4.1 Stress Wave Propagation in Homogeneous Solid Material

The problem of stress wave propagation has been studied extensively over the
past century. Analytical solutions to elastic wave propagation in homogcnous material
were among the first and most significant contributions to the topic. An analytical
solution for one dimensional (1D) stress wave propagation in a homogeneous material

(Meirovitch 1967) was given as follows:

_ _ 0 -"' *0 x
0'(.t.I)—0' f(t+E:—d_)+0 flf‘q) (10)

where Cd rs the speed of waves. 0 IS initial stress, O'Ors dynamic stress. I IS time

and x is a location.

Equation (9) was validated by (Paulino and Zhang 2007) through numerical
studies on a slender bar model subjected to a half-sine impulsive tension load. In a 1D
homogeneous bar, the half-sine stress wave travels in the slender bar at constant speed
and maintains its half-sine shape in spite of experiencing reflection at the boundaries.
Comparisons between the numerical and analytical solutions showed that these two
results agreed with each other very well.

However, no widely accepted theoretical solutions are available for stress wave
propagation in 2 dimensions (2D) so far. Some researchers have implemented
theoretical and numerical studies(Li et al. 2001; Berezovski et al. 2003). But the
solutions were not considered suitable for the models studied in this thesis. Although
(Dizaji and Jafari 2009) provided a three-dimensional (3D) fundamental solution of
wave propagation, it is still not widely accepted and too complicated for direct
application.

Therefore, validating numerical models in one dimension with analytical
solutions and theoretical principles will provide confidence on the framework of
numerical simulations on two-dimensional models, under the condition that no
experimental data and analytical solution is available for the validation of the

numerical simulations.

2.4.2 Stress Wave Propagation in Non-Homogeneous Solid Material

Research on stress wave propagation in non-homogeneous solid material includes
studies on layered solid material and functionally graded metal (FGM) materials.

The transmission and reflection of stress wave and energy at the interface
between dissimilar materials in layered solid material is a unique feature when
compared with homogeneous materials. Thus, studies in this field originated when the
attention shifted from homogeneous models to layered solid materials.

Much of the work on understanding wave propagation in layered non-
homogeneous solid materials was focused on the geometric dispersion of elastic waves.
As the interest on the influence of scattering effects induced by internal interfaces on
shock wave propagation in a layered non-homogeneous model emerged, a series of
experimental studies were conducted on this topic(Zhuang et al. 2003). Results
showed that the interface scattering in periodically layered composite affected bulk
response of the composites to shock compression loading. The effect was to slow
down the propagating speed of shock wave.

Further, shock energy was dissipated and dispersed by the impendence mismatch
at the interface during propagation. This is because the planar interfaces interacted
with the incident shock wave, and the transmitted and reflected waves, whose
amplitudes were determined by impedance factors (Chen et al. 2004). The
discontinuities are not only due to the dissimilarity of materials, but also the non—linear
reactions from these materials. An application of nonlinear stress—strain relation for
materials in numerical simulations of the plate impact problem of a layered non-

hOmogeneous medium was developed to capture the effects form material’s non-

31

linearity (Berezovski et al. 2006). This introduced a modified nonlinear stress—strain

relationazpc28(l+A8), and a modified sound velocity5=c\/1+2A8 , in which
,0 is the density, c is conventional longitudinal wave speed, A is a parameter of

nonlinearity.

As stated above, dispersion and attenuation were observed in experiments on
layered solid models. This indicates that this kind of structures can be one option for
energy dissipation, and scattering effects occurring at the interface, and which need to
be considered during numerical analyses and experiments.

As a new kind of material, functionally graded materials (FGM), attracted
researchers’ attentions since its properties can be changed according to a linear or non-
linear function as specified by designers (Erdogan 1995). This feature of high
customization minimizes the limitation on the material and gives researchers more
opportunities to optimize a structure to meet its demands.

From experiments and numerical simulations, results have shown that layered
materials and FGM material with different functions of gradation have have an
influence not only on the dynamic behavior of these structures but also on the
probability and locations of failures (Li et al. 2001.). A metal-ceramic composite
material was used, and the volume fractions of the reinforcement were varied for
different cases. Different volume fractions of reinforcement were assigned for the
individual layer in a layered. model, and smooth gradations of ceramic volume
fractions across the thickness of the plate were assigned to the functionally graded

model. These numerical models were verified by experiments. Results shows the

dispersion of waves as a result of the gradation of the material and the gradation of
material properties, can affect the dissipate energy and strain energy in the model.
Meanwhile, in order to improve the accuracy of the numerical simulation on FGM
materials, new kinds of finite elements with graded properties to simulate stress wave
propagation were developed (Santare et al. 2003). Researchers have not developed an
analytical solution to the two-dimensional problem, so they compared results between
graded and ordinary elements. Results show that conventional elements offer a
discontinuous stress field in the direction perpendicular to the material property
gradient while the graded elements give a continuous distribution. An eight-noded
isoparametric formulation was used to solve one-dimensional examples. Results were

compared with the analytical solution provided by (Chiu and Erdogan 1999).

2.4.3 Stress Wave Propagation in Homogeneous Foam Material

Porous materials or foams have unique characteristics such as light weight, low
yield stress, easy to reach large deformation due to its microstructure. There exists a
large amount of research that has showed that the way of absorbing energy or
attenuating stress is through plastic deformation (see 2.3.2). Thus, foam materials have
been the subject of study in many research projects related to energy dissipation and
stress mitigation.

Considerable research on stress wave propagation in homogeneous cellular
material was conducted during recent years. The compressive properties and energy
dissipation of aluminum alloy foam and aluminum foam under dynamic loading was

recently studied by (Wang et al. 2001). Results show that energy absorption efficiency

33

peaks at a compressive strain of 0.2-0.3, and it decreases when porosity increases at a
fixed value of strain. The efficiency of energy absorption in aluminum alloy foam was
found to be greater than the one in pure aluminum foam.

Meanwhile, validation studies of the material constitutive foam model in LS-
DYNA were conducted (Hanssen et al. 2001). Results show that constitutive model 63
is closed to the deformed shape of an experimentally-crushed cube. Therefore, this
constitutive model was employed in the current studies.

Research on the dynamic testing and modeling of soft structural material made of
metals such as aluminum honeycombs and foams (Zhao 2001) found an enhancement
of the crushing strength under high speed impact loading in foam materials. The
inertia effects associated with the localization of crushing contributed to this
phenomenon. The experimental data also showed the existence of a shock front in
cellular structure under impact load, especially at low critical velocities around 50 m/s
(Zhao et al. 2007). A high speed camera was used to capture the strain field image in
the cellular material, and this image was used to measure the shock front velocity and
analyze shock enhancement (Pattofatto et al. 2009). The image analysis showed that
the shock wave propagates at a constant velocity through the foam.

Research on elastic waves propagating through an ideal cellular structure with a
closely packed array of thin-walled rings was carried out by Shim et al. (Shim et al.
2008). Results showed that the dominant mode of elastic energy propagation in a thin-
walled ring arrays is flexural instead of extensional and shear waves. Three kinds of
ring packing arrangements, as shown in Figure 12, were employed in experiments:

square-packed, transverse close-packed and vertical close-packed arrangement. An

34

impact force was applied on the ring arrays. Results showed that a higher bar velocity

c and a thicker ring wall It lead to higher wave propagation speeds through the ring

0

array.

 

Figure 12. Ring Packing Arrangements in Experiments: (3) Square-packed, (b)
Transverse Close-packed and (c) Vertical Close-packed Arrangement[From
Shim et a1 (2008)]

2.4.4 Stress Wave Propagation in Non-Homogeneous Foam Material

Research has shown that foam materials have a better performance than solid
materials in terms of energy dissipation and stress mitigation. In addition, scattering
effects at the interface of two dissimilar materials reduces the stress wave energy and
disperses the stress wave. Therefore, it is natural to combine these two advantages to
maximize the capacity of energy dissipation and stress mitigation of foam material.

During recent studies, five novel layered material models for impact application
have been assessed by experiments and numerical simulation (Tasdemirci and Hall
2009). Results showed that geometrically tailoring a multilayered material system to

minimize stress propagating into the backing plate is possible for any specified impact

35

 

Véhk

COD?

um“

COP'

€\?L

 

 

 

 

Idt‘

sin

I {I ‘t

.-\f

 

velocity. In other words, stress waves or impact loading energy could be mitigated and
controlled in an appropriate design by using dissimilar solid materials. In the designs,
a rubber or Teflon foam was used as an interlayer to separate a ceramic faceplate and a
composite backing plate (Figure 13). Four FEM models were validated by
experiments and the results showed that use of multiple Teflon foam interlayers could

reduce average stresses in the layered material.

 

Figure 13. Baseline Configuration and Different Cases of Interlayer
Configurations Used in the Finite Element Study. [ From Tasdemirci and Hall
(2009) ]

With the development of modern materials engineering knowledge and
manufacture techniques, functionally graded foam material (FGFM) appear to be an
ideal candidate to maximize the energy absorbing capacity of a material or structure,
since its properties can be customized based on required functions.

During recent studies, functionally graded foam model perform perfectly under
low energy impact loads (Cui et al. 2009). One model established by

ABAQUS(ABAQUS) was used to develop the material’s characteristics of energy

36

dissipation. Besides the results that FGFM performed perfectly under low energy
impacts, the model showed that when the impact energy was high, the low-density
region of the graded foam absorbed only a small fraction of the total energy since it
densified very soon after plastic collapse. For foam materials with a given density, it
was determined that at a certain critical stress the efficiency of foam is maximized.
Meanwhile the duration of a high acceleration due to an impact can be reduced.

The propagation of stress waves through functionally graded foam material has
been studied as well. In the study by Kieman et al. (Kieman et al. 2009), a finite
element model of a Split Hopkinson Pressure Bar (SHPB) test was developed to carry
out simulations on uniform density foam and the results were validated against
laboratory SHPB tests. It was found that the magnitude of transmitted stress wave was
smaller when the stress wave propagated from a stiff material into a soft one. An
increasing density gradient would lead to an amplified transmitted stress. Once the
finite element model was validated by experimental data, the optimization work was
conducted by the combination of yield stress and density range of foam materials

through numerical models.

2.5 Summary

The foregoing review shows that the phenomenon of stress wave propagation has
been investigated extensively during the past decades. Researchers have identified the
fundamental mechanisms of stress wave propagation in solid materials, discovering
that plastic deformation causes energy dissipation. In addition, the scattering effect in

layered materials has been shown to play an important role in reducing induced kinetic

energy. An increasing or decreasing gradation of stiffness in a non-homogenous
material leads to an increasing or decreasing transmitted stress wave, respectively,
when compared with the propagation of a stress wave in a homogeneous material. As
researchers’ attention shifted from solid to foam materials, it has been found that the
efficiency of energy dissipation and stress mitigation in foam materials is much higher
than in solid materials. Similar conclusions were derived from research on non-
homogeneous materials as well. The dispersion of stress wave at the interface between
two dissimilar materials has been shown to reduce stress magnitude and propagated
kinetic energy. When stress waves travel from stiff material to soft material, the
magnitude of the stress reduces.

Based on an extensive review, the research done to date has focused on the
concept of energy dissipation and stress mitigation. However, another concept, the
direction of stress wave propagation in two dimensions remains undeveloped. By
conducting studies on the concept of guiding the stress wave propagation in this thesis,
the complete thought of stress wave management, which includes control on the
energy dissipation and control of the speed, magnitude and direction of the stress

wave, will be fully extended.

CHART ER 3. ONE-DIMENSIONAL STRESS WAVE PROPAGATION IN
SOLID MATERIALS

3.1 General

This chapter presents numerical studies on stress wave propagation in solid
materials in one dimension. The study is based on the response of a cantilever slender
bar subjected to a compressive impulsive load at its free end. Studies were performed
on a homogeneous model, consisting of a single material bar, and a non-homogeneous
model, consisting of a bi-material bar.

The speed of stress wave propagation and the energy transfer between kinetic
energy and internal energy were the responses of interest for the homogeneous model.
The energy transfer between distinct materials (i.e., different stiffness) was the
response of main interest for the non-homogeneous model.

The homogeneous model employed steel as the material, while the non-
homogeneous model featured steel and aluminum segments (half each). A mesh
sensitivity study was carried out to determine an appropriate size of mesh element.

Results showed that stress wave propagated in a homogeneous material at a
constant speed. Part of the kinetic energy is stored as internal energy due to permanent
plastic deformation. At the interface of distinct materials (different stiffnesses), one
part of the stress is reflected and the other part is transmitted to other material in
proportion to the mass density and Young’s modulus of the materials.

Although stress wave propagation in 1D has been extensively studied during past

decades and it is well documented in papers and books, numerical simulations of this

39

behavior were performed to validate the numerical approach. The results showed that
the numerical models in this chapter offered reasonable results and the conclusions

drawn from them was consistent with theoretical solutions.

3.2 Model Definition
In this section, information of homogeneous model would be offered. A study on
mesh sensitivity would be presented as well. And this study was used to determine an

appropriate size of a mesh element.

3.2.1 Model Description in 1D Cases

A cantilever bar with a length 0.9m and a height 0.045m (Figure 14) was used to
investigate elastic and plastic stress wave propagation in a homogeneous material. In
this model, the right surface, the top and the bottom of the bar are free of fractions and

the left surface is fixed to a rigid boundary.

 

45 mm

 

 

.s\ \\\\\\V
LJ
VI

900 mm J
7]

Figure 14. Homogeneous Slender Bar Model for Solid Material in 1D Cases

 

 

After a mesh sensitivity study, an appropriate mesh discretization was determined
to be 300 x 15 squares, each was divided into 2 triangular elements (Figure 15). The

3-node element type was a shell element with linear shape functions and a single

40

integration point. Since this plane strain element can capture the deformation of the

model, and it reflects the behavior of dilatational wave from the impacted load.

wtug

_
3mm

Figure 15. Right-triangle Shape Shell Element

3.2.2 Mesh Sensitivity Study

The mesh sensitivity study was performed for the homogeneous elastic model.
The material was steel (Figure 20) and a unit impulsive tension force was applied
(Figure 22). Three different meshes were evaluated. To compare the performance of
the different models, three locations were chosen to extract stress histories, as shown

in Figure 16. All of 3 points were at mid-height of the bar.

L B

 

\\\\\\\S

 

 

A
% L/2 . L/Z ¢
a a %

 

Figure 16. The Locations of Chosen Elements in 1D Models

41

3.2.2.1 Mesh Case 1

The bar was discretized into 300 x 15 quads, each divided into two T3 elements
(3-node element with linear shape functions and has a right-triangle shape as shown in
Figure 15). This led to a mesh with 4,816 nodes and 9,000 T3 elements.

Figure 17 shows the stress history of mesh case 1. A solid line shows the stress
history at point A; the dash-double-dot line represents stress from point B, and the
dashed line indicates the stress at point C. As shown in the graph, the triangular stress
wave travels in the slender bar at a constant speed and maintains its triangular shape
wave in spite of reflections at the boundaries. The stress magnitude at point A is 0.323
MPa. Stress wave reaches point C at 0.15 ms and the magnitude of stress wave
doubles to 0.667 MPa. When the rebound stress wave comes back to point A at 0.30
ms, the magnitude of stress wave is zero. This agrees with the theory provided in the

Chapter two.

 

 

 

 

 

 

 

 

 

 

 

 

 

0.8
0‘6 I [[1 Point A
0.4- I l ----- :05an
. .. - - — ornt
A _ "1 I, I I \
is 0.2 .I \ \ .. ..
E I :91, L -’ \. _ A ‘ _
g; 0.0 '4 — __ __ — fit .r \ 7. \ I.
E -0.2 — \. .I \\ I, °\ :
. C B A V \ I ..I
-04 — 0 §—> \
L/2 L/2 F(t) \ I
-0.6 — i 'I' 'l J
-0.8 I I I I I
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time (ms)

Figure 17. Stress History of T3 Element with Length 3 mm

3.2.2.2 Mesh Case 2

The bar was discretized into 150 x 8 quads, each divided into two T3 elements
(3-node element with linear shape function and has a right-triangle shape). This led to
a mesh with 1359 nodes and 2,400 T3 elements.

Figure 18 shows the stress history of mesh case 2. Stress histories from the three
points follow the same plotting convention as presented in mesh case 1. In this case,
the stress magnitude at point A is 0.188 MPa and it doubles to 0.372 MPa at point C.

The difference in the stress magnitude is only due to the different mesh size.

43

 

 

 

 

 

 

 

 

 

 

 

 

 

 

06
Point A
0'4 I -~-- Point B
— - - Point C
’E?
r; 0.2 ~ . ,
2‘ /\ 0‘. o‘
I / \ .I .\ .I .‘
g 0 O __L J. ' v - 2 J 4w__.~‘ ' L _ -
\. I \ -’
-02 - C B A V "
D o §—>
U2 U2 F(t)
'0.4 l I I I I
0.0 0.1 0.2 0.3 0.4 0.5 0.6

Time (ms)

Figure 18. Stress History of T3 Element with Length 6 mm

3.2.2.3 Mesh Case 3

The bar was discretized into 300 x 15 quads. This led to a mesh with 4,816 nodes
and 4,500 elements. Figure 19 shows the stress history in mesh case 3. Stress histories
from the three points follow the same plotting convention as presented in mesh case 1.
At 0.025 ms the stress magnitude at point A is 0.327 MPa and it doubles to 0.670 MPa
at point C (at 0.175 ms). Comparing Figure 19 with Figure 17, it can be seen that

stress histories from both cases are very close.

44

 

0.8

 

 

 

 

 

 

 

 

 

 

 

 

 

0‘6 d I“ Point A
04 - I \\ -°°-° Point B
’ a I .- — — -- Point c
”a 0.2 - I °° I \\ .’ \.
C- o \ I ' '
E." j 1 l \ I \.
a 0'0 ’ T' " ' WET: 71 TA I.
5. -02 — ‘1 .I \ [I °'\ I:
C B A V \ I °.
.04 — P O H \ I
L/2 L/2 F(t) \ I
-0.6 — I I i I v
”0.8 F T I I I
0.0 0.1 0.2 0.3 0.4 0.5 0.6

Time (ms)

Figure 19. Square Element with Length 3 mm

3.2.2.4 Conclusion

In mesh case 1, the maximum stress is around 0.64 MPa and the minimum stress
is around —0.63 MPa when the element is a T3 with 3mm length. In mesh case 2, the
maximum stress is around 0.37 MPa and the minimum stress is around —0.37 MPa
when the element is a T3 with 6 mm length. In mesh case 3, the maximum stress is
around 0.65 MPa and the minimum stress is around -O.67 MPa when element is a
square with length 3mm.

From these results, it can be seen that minimum element size should be set in
order to capture accurate wave propagation behavior. Elastic wave velocity is the same
in each case since wave velocity is a material’s property that is determined by

Young’s modulus and mass density. However, stresses vary with element size. It was

45

also seen that stresses did not change much when different element types were

employed. Mesh case I was thus chosen for the numerical models.

3.3 Material Modeling
Two different materials were used in the numerical simulations: steel and
aluminum.

The materials properties for Steel were: Young’s modulus: 200 GPa, mass

3
density: 7800kg/m , Poisson’s ratio: 0.3, yield stress: 200MPa; plastic tangent

modulus: 50MPa, and ultimate tension strength: 400MPa. A stress-strain curve for the
steel material is shown in Figure 20. The MAT_001 material model in LS-DYNA was
used as the elastic steel constitutive model. The MAT_024 material model in LS-
DYNA was used as the elastic-plastic steel constitutive model.

The properties of Aluminum were: Young’s modulus: 80 GPa, mass density:

3
2800kg/m , Poisson’s ratio: 0.33, yield stress: 80MPa, plastic tangent modulus:

lOMPa, and ultimate tension strength: IIOMPa. A stress-strain curve for aluminum is
shown in Figure 21. The MAT_OOI material model in LS-DYNA was used as the
elastic aluminum constitutive model. The MAT_024 material model in LS-DYNA was

used as the elastic-plastic steel constitutive model.

46

 

 

 

 

TotN/mmz)
400 ----------
I
I
200 —— E
l
-5 -1 l I _>
I I 1 5 3
8(10— )
i ' — -200
l
' —————————— -400
Figure 20. Stress Strain Curve for Steel
A0(N/mm2)
110 """"""" .
I
80 :
| I
I I
' l
I
_ _ l
.4 .1 i a 3*
i i 8(10_ )
l l
: I
: "—7 -80
--------- -110

Figure 21. Stress Strain Curve for Aluminum

47

3.4 Stress Wave Prepagation in Homogeneous Solid Materials

In this section, results from the simulation of homogeneous solid models are
presented and discussed. The aim of these studies is to develop and validate the
framework for the finite element models within which the remainder of the analyses
will be conducted. Numerical models, corresponding loading histories and materials
are presented. Simulation results, namely particle velocities, stresses, and energy
distributions are compared with values from elastic and plastic wave propagation
theory. The locations chosen to extract the response measures are those shown in

Figure 16.

3.4.1 Case Studies

This section presents three case studies on the stress wave propagation in
homogeneous materials. Case 1 deals with the elastic wave propagation in a
homogeneous material under a unit tensile force. Results from this case are compared
with results from (Paulino and Zhang 2007) and comparison with close-form
solutions validated the accuracy of the results from the numerical model. Case 2 deals
with the elastic wave propagation in a homogeneous material under a unit
compression force. Case 3 studies the phenomena of plastic wave propagation in a
homogeneous material under a lkN compressive force. Results were be verified by
comparing it to existing knowledge on 1D stress wave propagation in homogeneous

materials.

48

3.4.1.1 Case 1

Case study 1 was defined for the study on elastic wave propagation in
homogeneous material under a unit tension force. The model used material model
MATOOI in LS-DYNA with steel properties. The loading condition was a 1N tensile

pulse load as shown in Figure 22.

A
F (N)

._____——_-_————

 

’

0 2 5 113 5011s 1000us T(s)

 

Figure 22. Unit Tension Pulse Force Loading History
3.4.1.2 Case2
Case study 2 was defined to study elastic stress wave propagation in a
homogeneous material under a unit compression force. The model used material
model MAT001 in LS-DYNA with steel properties. The loading condition was a 1N

compressive pulse load as shown in Figure 23.

49

F(N)

._—___.___.-_————

 

P

0 2505 5005 100005 T(s)

 

Figure 23. Unit Compression Pulse Force Loading History

3.4.1.3 Case 3

Case study 3 was defined to study the propagation of plastic stress waves in a
homogeneous material. The loading condition was a compression force of lkN at
cantilever end (Figure 24). The material model used was MAT024 in LS-DYNA with
elastic-plastic steel properties. A study using the MAT024 material model in LS-
DYNA was also carried out. The purpose of this study was to determine whether
MAT024 was an appropriate constitutive model to simulate elastic-plastic behavior.

The load used in this case study was 100 MPa (see Figure 25).

50

 

 

 

P(N)
-1000 -------- l
i >
0 25 us 5003 1000118 T(s)
Figure 24. Compression Loading Pulse Force with lkN Force
A
P(MPa)
-100 -------- .
1 b
0 25113 501.18 _ 1000115 T(S)

 

Figure 25. Compression Pressure Pulse with Magnitude 100 MPa

51

3.4.2 Results

3.4.2.1 Case 1: Validation of 2D Homogeneous Model under Tension Loading

Figure 26 shows results from the homogeneous numerical model. As shown in
the graph, the triangular stress wave travels in the slender bar constantly (in this case,
the speed of stress wave propagation is 5064 m/s) and maintains its triangular shape
despite of reflection at the boundaries. The stress magnitude at point A is 0.323 MPa.
The stress wave reaches point C at 0.15 ms and the magnitude of the stress wave
doubles to 0.667 MPa. When the rebounded stress wave comes back to point A at 0.30
ms, the magnitude of stress wave is zero. This is response is consistent with the theory

of stress wave propagation in solids as presented in Chapter two.

 

 

 

 

 

 

 

 

 

 

 

 

 

2 71
II
I \ Point A
1, I \ —«-- PointB
[1‘ I I i‘. ——-- PointC
i '. I l : \
0y : \. I I I ’.
b 0.. I '—1__ \:-° \ .
r. I, : \ I ° 7
b '\ I I I I. :
. . j
‘- l’ l I \ :
-1- c B A \.- l I v
I o H \ I
L/2 L/2 F(t) III
I
'2 I I I \I
o 1 2 3 4

IXCd /L

Figure 26. Stress History of Homogeneous Bar in Case 1 in 1D

Figure 27 shows the result from (Paulino and Zhang 2007) who provided a

numerical solution to the same problem. In Figure 27, numerical results obtained at

location x = L are denoted by a dark solid line, those at location x 2 0.5L are shown by

the dashed line and those at location x = O are shown by the dash-dot line. Analytical

results are shown by a thin solid line. It can be seen magnitude of stress wave doubles

at the fixed boundary and becomes zero at the free surface end. Thus, the results

obtained by (Paulino and Zhang 2007) (Figure 27) agree with the ones obtained from

the finite element simulation shown in Figure 26.

 

 

 

 

 

 

 

 

 

2 l l "(1.. l r I I I
0:" 1" x=0
1 .5 ’ X: ,'t n / . ‘
,r' f! ’3 numerical (2D)
x’ .‘I .1
1 f "I" l. l! {613‘ d
" 3% !| l. ‘0', '3' ‘
0': 9’ .H 1‘ "'1 ‘1‘
f l '_ .‘ l ‘1 ..
0.5 0., 4’ “I '1‘ .‘ ' "1‘
Q " ? I l 'l '\
>< ] a I ,1 '.( fl, ‘ ,s
\ O .4 "7: 4A {AV‘A‘A' ’ ‘1‘? “ 'l—‘vt' ' aliA’ Y+A r "V v
Q fiT I l7 V' . ‘ , : ' . , '
>< ° x=0 5L . .' .‘ . '~ ‘. '
-0 5 u analytical (1D) to ‘t ' '0 . l l l. ‘ f. I'—
. K. | . ' l g .1. ‘| '
l‘ I "1 9 | 1 . U.
.- F“) 2 . 1 1
x." 9 fi——"' l. |_ .‘ 1
—1.5 f». i | f ' #
X20 X2051. x: . 1.."
‘ l
_2 1 1 1 1 1 Lg” 1
O 0.5 1 1 .5 2 2.5 3 3.5
lXCd/L

Figure 27. Stress History for Homogeneous Bar (2D Simulation) Subjected to

Transient Loading and Its Comparison with 1D Analytical Solution [ From Paulino
and Zhang (2007) ]

'Jl
DJ

3.4.2.2 Case 2: Elastic Wave Propagation in Homogeneous Material

3.4.2.2.] Particle Velocity of 2D Homogeneous Model under Compression
Loading

Figure 28 shows the particle velocities in the elastic steel models under a lN
impulsive compressive load. The results in Figure 28 show how the wave traveled
from the right free end towards the fix support and then and back during the first
period. It can be seen that the velocity changes sign (i.e., from a compression wave to
a tension wave) at approximately 0.19ms (most easily seen in dashed line), as it
rebounds at the fixed end. It can be seen that particle velocity at the fixed end is
almost zero and that the velocity at the free end doubles from the initial particle

velocity upon completing a full period.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.020

0.015 4 Point A
A -°--- PointB
g 0010‘ --- PointC
>
C.
5; 0.005 .
a
'2 0.000«
E
Q, 0005 -
g 00104 C

-0.015 - .

L/2 L/2 P(t)
d——*—>{
‘0.020 I I r T l
0.0 0.1 0.2 0.3 0.4 0.5 0.6

Time (ms)

Figure 28. Particle Velocity in Elastic Steel Model under 1N Compression Force

54

3.4.2.2.2 Homogeneous 2D Model under Compression Loading

In this model, a compression impulse force was added on the free end of the bar.
Figure 29 shows the stress history extracted from the model in Case 2. It can be seen
that at the beginning the bar was subjected to a compression wave and later subjected
to a tension wave. The absolute values of magnitude and rising times are the same as

the ones in the tension model; the only difference is the signs.

 

 

 

 

 

 

 

 

 

 

 

 

 

0.8
a
0'6 _ C B A H
l C ”— I \
0-4 ‘ L/2 L/2 F(t) A I \
A - -' ° I \
g 0.2 .’ \. I l :1:
5 00 « r. - - _—fig' L1_ if, \.
é . ‘0 I. \\ F.\ I
b - ~ . . I '. -°
W 0.2 ‘11. \\ I ‘3’ Point A
‘04 ‘ \ ’ ---—- PointB
\ I —-- PointC
-O.6 - J
'0.8 l l l I l
0.0 0.1 0.2 0.3 04 0.5 0.6

Time (ms)

Figure 29. Stress History of Homogeneous Bar in Case 2 in 1D

The internal energy, kinetic energy and total energy distributions in the elastic
steel model are shown in Figure 30. Internal energy and kinetic energy transfer to each

other completely without any residual energy. Meanwhile, the total energy, which is

2.1x10_ 6 J, stays constant after the impulse load fades away.

55

0.0030

 

 

 

 

 

 

 

 

 

 

 

F(t) -— Internal Energy
0.0025 A 14— -'°-° Kinetic Energy
--— Total Energy
:_7 0.0020- r—--—- ————— T ————— --————
‘7; I I:
97 0.0015 - ll i l,
.33 : "
5 l I .
C‘ c
[J— 00010 _. l . . . oo-oo\ roe-00-004.)
I 3 l : i
l : ‘
0.0005 - I ‘ l l. l:
: i t,
l! :'
0.0000 . - . . . J .
0.0 0.1 0.2 0.3 0.4 0.5 0.6
'l‘ime (ms)

Figure 30. Energy Distribution in Case 2 in 1D

3.4.2.3 Case 3: Plastic Wave Propagation in Homogeneous Material

The aim of this case study was to evaluate the characteristics plastic stress wave
propagation. A 1 kN impulsive compression force was again applied at the free end of
the bar. Material model MAT 024 in LS-DYNA was used to simulate elastic-plastic
behavior of steel with properties as presented in Section 3.3.

Figure 31 shows the stress history in a slender bar with a plastic tangent modulus
of SOGPa. It can be seen that the slope of stress history curve changes due to decreased
tangent modulus after yielding at a time of approximately 0.19 ms. Yielding of the bar
causes a sudden drop in stress which is then recovered by the traveling stress wave.

Nonetheless, the speed of the loading stress wave, seem by the front end of the stress

history trace slope is reduced.

56

Stress (GPa)

y (10" J)

(r
C

liner

 

0.6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\ I
- I
\ ’ l I
. -° l I Point A
04 _ I ‘I\I —---- PointB
' l - — - Point C
'0.6 I l l l l
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time (ms)
Figure 31. Stress History in Case 3 in 1D
3.5
P(t) Internal Energy
3-0 ‘ K_ -°-—- Kinetic Energy
——- Total Energy
2.5 d
r- ____________________ _._
2.0 — I / i ,
I \f
1.5 a I
I o o o.
1.0 a I \
I i I‘-
05 - I .0 .- 00000000 d: \ 00000 —oo-oo\ roe-I
\ ' . :
0.0 I 3’ , I I v
0.0 0.1 0.2 0.3 0.4 0.5 0.6

Time (ms)

Figure 32. Energy Distribution in Case3 in 1D

57

Figure 32 shows the internal energy, kinetic energy and total energy distributions
in a steel material that suffers plastic deformations during the transmission of the
stress wave. The total energy is 2.3x10'3J; about 1.5x10’3J of energy are stored as
internal energy due to plastic deformation and about 0.8x10'3J of elastic strain energy
is transferred between kinetic energy and internal energy.

In this model, MAT 024 was used to simulate an elastic-plastic behavior for steel.
To verify the constitutive model the first step was to apply a pressure load below the
yield stress of material and thus obtain a stress history equal to that of the elastic
model. The magnitude of the compressive impulse pressure was lOOMPa. The load
history employed in this model is shown in Figure 25. Figure 33 and Figure 34 show
the stress histories in the elastic model. Figure 35 and Figure 36 show the stress
histories in the elastic—plastic model.

Comparing Figure 35 and Figure 36 with Figure 33 and Figure 34, it can be seen
that those stress histories in elastic-plastic model were almost the same as the ones
extracted from elastic materials, when the load pressure was less than yield stress in
elastic-plastic model. As a result, MAT 024 in LS-DYNA was considered appropriate

to simulate elastic-plastic behavior.

 

0.20

 

 

 

 

 

 

 

 

 

 

 

 

0.15 - C B A /‘
p o k— l \
’7
0.0 fl . L/._ L/2 F(t) I \
l
3 0.05 - l
5 I
E: 000 -1
i 005 ~ . ' I,
-0.10 . \’ i I PointA
l I --°-- PointB
-0.15 - \I ——- PointC
-020 T I I I I
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time (ms)

Figure 33. Stress History at Chosen Locations in Steel Elastic Model under 100Mpa
within 0.6ms

 

 

 

 

 

 

 

 

 

 

0.3
C B A PointA
l . “—— -oo-o '
0.2. ~*—L/g—*I‘—L/—2—>IF(t) .._— 13:22:?
{‘55 0'1“ I II I I I ll
V ~l I" O," II' I Q I
Ills. III. .1: I t 11' J I‘
5 0.01 .. . w H JV!" l

 

 

.6
N
a

Time (ms)

Figure 34. Stress History at Chosen Locations in Steel Elastic Model under 100Mpa
within 5 ms

 

 

 

L/2

 

Stress (GPa)

 

A l‘
J‘— ,\
L/2 F(t) I l
I
l
\

 

 

Point A
-°--- Point B

 

 

 

 

 

 

-0.10 'I
\ I
-045 ~ \I -.—— PointC.
’0.20 l I I I I
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time (ms)

Figure 35. Stress History at Chosen Locations in Steel Plastic Model under 100Mpa

within 0.6ms

 

 

 

 

 

 

 

 

 

 

 

 

0.3
'C A PointA
’ -00-. .
0-2 - w L/2 L/2 P(t) ___ EEIEIE
E 0.1- g ‘ i q l‘ ‘
0 3 I . I
v ~| 8'1 3' III I I
:g 00 ’Ils. I. . LI! I '1' . o'yI‘ I :' I 11“ «I... ,I
o k: I o o z . . ‘ ' f .0 J. 1 I :3"
I": Ii III I“? In I III, III; I II
l
.02 , , I I .
2 3 4 5

Time (ms)

Figure 36. Stress History at Chosen Locations in Steel Plastic Model under 100Mpa

within 5ms

60

3.4.3 Discussion

The presented results have shown that the simulation of elastic wave propagation
by means of finite element analysis in LS-DYNA led to results that were consistent
with theoretical solutions. The behavior of a slender bar under impulsive tensile load
was first modeled and compared with the results by (Paulino and Zhang 2007). The
models correctly capture the amplification (double) of the stress wave at the fixed
boundary and the dissipation of the wave at the free end. The model also correctly
computes zero particle velocity at the fixed end while that at the free end doubles. The
element type and the mesh geometry were selected based on a sensitivity study.

With such evaluation, the model was considered appropriate for use in the study
of stress wave propagation under compressive impulse loads and elastic-plastic
materials. As expected, the studies of stress wave propagation under compressive
loads showed that the behavior is the same as for the ease of a tensile load, except for
the obvious change in sign and the direction of the stress wave propagation.

With respect to the propagation of plastic stress waves in homogeneous materials,
the models seem to have correctly captured the reduction in transferred kinetic energy
due to the energy that is dissipated in the plastic deformations. The reduction of the
stress wave velocity was also apparent by observing the reduced slope of the leading

edge of the stress wave from the stress history traces.

61

3.5 Stress Wave Propagation in Non-Homogeneous Solid Materials

In this section, results from the simulation of non-homogeneous solid models are
presented and discussed. Non-homogeneity in the current context refers to the
presence of two materials with distinct elastic modulus (i.e., stiffness). The aim of the
analyses presented in this section was to evaluate the characteristics of stress-wave
propagation at the interfaces of dissimilar materials. Numerical models, corresponding
loading histories and materials are presented. Simulation results, namely particle
velocities, stresses, and energy distributions are compared with values from elastic and
plastic wave propagation theory. The locations chosen to extract the response

measures are those shown in Figure 16.

3.5.] Model Particulars

The model used to evaluate the characteristics of elastic and plastic wave
propagation in bi-material (or non-homogeneous) bars is shown in Figure 37. The
model consists of a slender cantilevered bar of 0.9m in length and a height of 0.045m.
The bar geometry was divided into two equal parts such that the material in Part 1 was
defined to be steel and the material for Part 2 was defined with the properties of

aluminum. The model was loaded with an impulsive compressive pressure load of

lMPa at the bar free end (Figure 38).

 

 

 

 

 

 

’/
¢ 2 1
é I __L 45 mm
é.
450 mm _L 450 mm _l
T 'I
Figure 37. Bi-Material Bar Model in 1D
A
P(MPa)
-1 ________ t
i n
0 25 “5 SOus IOOOHS T(S)

 

Figure 38. lMPa Compression Pressure Load

3.5.2 Results

In a non-homogeneous model it is important to notice that when stress wave
travels across two dissimilar materials (different stiffness) it will separate into two
parts at the material interface. One part wave would reflects back to the material in
which the wave was initially traveling while the other part is propagated or transmitted

to the neighboring material.

In this model the elastic wave travel from Part 1 (steel) to Part 2 (aluminum).
According to the equations (6) and (7), 55% of the original stress magnitude
propagates, or travels, into the aluminum part of the bar, and 45% of the stress reflects
back. Results from the model are shown next to match this theory very well. Two
points were chosen to obtain the stress history response as shown in Figure 39. The

stress history at points A and B are shown in Figure 40.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/
? 2 1
2 O I O I A: 45 mm
/
g 100 mm |.—.|H| 100 mm
B A
:4 >11: a
450 mm 450 mm
Figure 39. Points Locations in Bi-material Model
1
0.0004 « 2 P l l} 41:52
U9 W L/9
_ H——-*H-—-——*l
00°02 U2 U2 r.
E : \
c: . -. ’-
T; 0.0000 4 -—, , f - . / ,
° -‘ \ . \.
5;), '0 .‘I 2‘ / .\'.
0.0002 — V °
Point A
“0.0004 "‘ -oo—o Point B
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time (ms)

Figure 40. Stress History at Given Locations in Bi-Material Model

64

From Figure 40 it can be seen that the magnitude of the stress in location A at
0.08 ms is -0.33 MPa. According to the presented theory, the compressive stress that
would be transmitted into aluminum should be —0.1815 MPa. It can be observed in
Figure 40 that the numerically computed value at location B is approximately -0.l8
MPa at 0.12 ms. Similarly, the theoretical magnitude of the reflected tension stress

wave is 0.1485 MPa, while from Figure 40 the value is about 0.15 MPa at 0.12 ms.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.0006
B A
00004 - F k—
» F(t)
L/2 L/2
0.0002 —
E C
a '\ 0
T; 0.0000 ~ .. __fi... _-_;. / . .p
8’5 1 ,' -\ :' ' ' "
r}: \ -° '. .’
-0.0002 - .. I \ ,
\-’ -.'
-0.0004 « 1] Point A
-°°-- Point B
-00006 , , T I I
0.0 0.1 0.2 0.3 0.4 0.5 05
Time (ms)

Figure 41. Stress History of Two End Points in Bi-Material Model

Figure 41 shows the stress history at locations near the boundaries of the bar. It
can be seen that at the free end (Point A) the maximum stress was about 0.33 MPa at

0.025 ms, while the peak stress at the fixed end (Point B) was 0.38 MPa at 0.175 ms,

65

less than what can be observed from the results for the homogeneous model. This
results are consistent with the refraction of stresses that takes place at the material
interface and indicates that the stress transferred at a material interface is reduced
since part of it reflects back to the source material.

Figure 42, Figure 43 and Figure 44 show energy distributions for the non-
homogeneous model. It can be seen that total energy was 2.25 ><10'6 J. About 78% of
the total energy (1.75 ><10"6 J) was transmitted into the aluminum part while the other
22% (0.5 x10"6 J) reflected back into the steel part when the stress wave reaches the

material interface at 0.12 ms.

 

 

 

 

 

 

 

 

 

 

 

2.5e-6
2 l P(t) — l:Stecl
T J‘— —oo-o 2: A]
2.0e-6 .
*4————>l<—-———*l
A L/2 L/Z
E3 1.5e-6 ~ .I"
3.3 I .
E : i ,. A
E I °. \ °°'
1L3 1.08'6 “ .0 \ :l o .I \. ,00#)
E ooJ 00’ o o .
u—n ./ . .’ \ - o
5 08-7 "" I ..‘
o . ’
: oo—oo ’-ooj
0.0 ..:--_41 , , r I
0.0 0.1 0.2 0.3 0.4 0.5 0.6

'l‘ime (ms)

Figure 42. Internal Energy Distribution in Bi-Material Model

66

5.!

Kinetic Energy (.1)

w (. J)

ha

i
9‘

Total liner

2.59-6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
   

 

 

 

2 1 F(t) -— 1: Steel
1 1‘— _...... 2; Al
2.0e-6 ~
1.5e-6 -
1.06'6 " F001
00‘ ooq :
°. _I '- _
5.0e-7 ~ I 1, ° ‘. i ' I ..
-' . | - '. 1 \ :
.’ z! 2 I : °. I
00 “:“‘-J.T Lo 1 ‘ ..l T\oo-.‘p:o-:
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time (ms)
Figure 43. Kinetic Energy Distribution in Bi-Material Model
3.5e-6
7- ‘ P(t) 1- s
—— . . teel
3.0e-6 « I I"— _"_. 2:Al
2.5e-6 -
2.0e-6 - _A _
0-00-0.-. 0’
156-6 ‘ I .\ i
1.0e-6 - v .
I f '- i
5.0e-7 ~ -’ \z :
! I
00 AA:::="'T ‘ I I ..-..-..I-.
0.0 0.1 0.2 0.3 0.4 0.5

0.6

Time (ms)

Figure 44. Energy distribution in Elastic Bi-material Model

67

3.5.3 Discussion

The results provided above show that the propagation of stress waves across
materials with different stiffness brings about important features. As shown in Figure
40, stresses reflect and transmit at the interface of dissimilar materials in proportion to
their properties, namely, mass density and Young’s modulus. An important effect of
the wave refraction at the material’s interface is that the propagated stress is reduced
since part of it is reflected (see Figure 41). Finally, the way in which the energy is
shared by the two materials becomes more complex, but clearly complimentary. The
results from the finite element simulation were found to be in agreement with

theoretical models, thus providing confidence on the approach.

3.6 Summary

This chapter presented a series of numerical studies aimed at evaluating the
characteristics of stress wave propagations in homogeneous and non-homogeneous
(bi-material) solid materials. The goal of studying one—dimensional behavior was to
validate the framework of using finite element simulations to study the propagation of
stress waves in foams and two-dimensional domains (Chapters 4 and 5). Compliance
between the numerical (finite element) models and theoretical principles was
confirmed thus giving confidence to the use of the approach for further studies. The
following summarizes important features learned from the presented studies.

Stress waves in elastic homogeneous models propagate at a constant speed that is
controlled by the material’s mass density and elastic stiffness (Young’s modulus).For

the cantilever bar studied here, the magnitude of the stress doubles upon being fully

68

reflected at the fixed boundary and reduces to zero at the free boundary. When the
stress magnitude exceeds the proportionality limit of the material. the speed of the
stress wave reduces since it then depends on the instantaneous plastic elastic modulus
of the material. In addition, there is a transfer from kinetic energy to permanent strain
energy due to plastic deformations. Such behavior leads to the reduction of the
propagated stresses, which is consistent with the concept of using plastic deformation
as an effective means for stress mitigation of impulsive loads.

Stress wave propagation across dissimilar materials (non—homogeneous) leads to
the decomposition of the stress wave into transmitted and reflected components whose
magnitude depends on the relative material properties (mass density and elastic
stiffness) of the materials meeting at the interface. The interface of dissimilar materials
will clearly affect the magnitude of the stress waves that travel inside the structure,
leading to a more complex distribution of energy participation than what occurs in a

homogeneous material.

69

CHAPTER 4. ONE-DIMENSIONAL STRESS WAVE PROPAGATION IN
FOAM MATERIALS

4.1 General

This chapter presents numerical studies on stress wave propagation in foam
materials in one dimension. The studies were carried out on a cantilever foam bar
subjected to a compressive impulsive load at its free end. A homogeneous model
including a single foam material bar and two non-homogeneous models consisting of
two kinds of foam materials were used. The responses of interest were the capacities

of energy dissipation, stress mitigation and large deformation.

The homogeneous model used an aluminum foam with a relative density (p*/ps)

of 0.06 while the non-homogeneous models used considered a bar made of an
aluminum foam with two relative densities (half each) of 0.06 and 0.24. The only
difference between the two non-homogeneous models was the sequence of the foam
segments with different relative density. Non-homogeneous model 1 had the foam
with 0.06 relative density in the first half part of the bar (attached to the fixed
boundary condition) and the foam with 0.24 relative density in the second half part (at
the free end). In the non-homogeneous model two, the sequence was opposite.

Results indicate that stress magnitude decreases dramatically in the process of
plastification and that the dissipated kinetic energy is stored as internal energy in the
foam materials. This is due to the large deformations that occurred in a crushable foam.
Thus, foam materials are highly effective in mitigating stress and reducing kinetic

energy. The sequence of foam material in a non-homogeneous model has been shown

70

to effect stress wave propagation, where stiffer foam near the load application will
lead to smaller deformations. with the opposite design having an inverse consequence.

The findings from these studies can be used as criteria to understand the
behaviors of stress wave propagation in two dimensional foam structures such as those

presented in Chapter 5.

4.2 Model Definition

The previously used cantilever bar is used again in this chapter to evaluate the
performance of foam materials under impact loads. One difference to the models
presented in Chapter 3 is that the bar was modeled with two metal plates added on its
two ends (Figure 45). The reason is that foams are much weaker than a solid material,
thus the two thin plates were added to avoid local deformations and allow a more even
distribution of stress demands at the reaction and load application locations. The
height of foam and metal plate is 45mm. The length of the foam bar is 900 mm and the
thickness of metal plate is 3 mm. In this model, the foam part was divided into two
equal parts (Parts 1 and 2). This configuration was helpful to detect the locations of
large deformation and to compare results between homogeneous and non-
homogeneous designs, where in the later the foam materials for each part were
different. The interaction between two foam parts and between foam and metal plate

were assumed to be a perfect bond.

71

 

45 mm

 

 

 

 

2:3

 

l

 

r 4 4r—

m 450 mm 450 mm 3 mm

m
3

Figure 45. Cantilever Bar Model for Foam Material in 1D Cases

4.3 Material Modeling

The materials used in this chapter were aluminum and aluminum foam. The
properties of aluminum are listed in Table 4. The properties of aluminum foam are
presented in Table 5 as well. The constitutive model used to simulate aluminum was
the MAT_OOI material model from LS-DYNA. Considering that aluminum foam has
the capability of withstanding large deformations, MAT_O63 crushable foam material
model was used to simulate open-cell foam material. As mentioned in Chapter 2.
stress-strain curves of open-cell foam material can be defined based on the relative
density of the foam and the properties of the material of which the foam is made of.
The relative densities of aluminum foam used here are 0.06 and 0.24. The compressive

stress-strain curves are shown in Figure 46.

Table 4. Properties of Aluminum Plate in 1D Foam Bar

 

 

 

 

 

Al Plate
E (GPa) 80
v 0.3
3
p (kg/m ) 280°
0 (MPa) 80

 

 

 

 

Stress 0' (MPa)

25

20

_L
01

_x
C

Table 5. Material Properties of Foams in 1D Foam Bar

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Property Foam
Relative density 0.06 0.08
E (GPa) 0.29 0.51
v 0.33 0.33
3 7
p (kg/m ) 168 2-4
0 (MPa) 0.35 0.54
l
l
- RD 0.24 :
-- -- -- R1) 0.06 |
l
. l
l
l
l
‘ r
l
I
I
‘ I
I
0.0 0.2 0.4 0.6

CompressivL Strain a

Figure 46. Stress Strain Curves of Foam Materials in 1D Cases

1.0

4.4 Stress Wave Propagation in Homogeneous Foam Materials

This section presents results from the simulation of a homogeneous foam model.
The purpose of this study is to show features of stress wave propagation in a
homogeneous cantilever bar made of open-cell foam material. Meanwhile,
comparisons between homogeneous and non-homogeneous models can show the
influence of changing materials on stress wave propagation. The locations chosen to

extract the response measures are those shown in Figure 47.

E D C B

>

 

o o o H

 

350 mm 100 mmlOO mm 350 mm

l‘—+—>l+—+l‘———>l

Figure 47. Specified Locations on the Homogeneous Foam Model in ID Case

4.4.1 Case Study

This case was defined for the study on plastic wave propagation in a
homogeneous foam model under an impulse compressive pressure with a magnitude
of 100 MPa. The model used material model MAT_063 in LS-DYNA with the

properties of foam with 0.06 relative density. The loading condition is shown in Figure

25.

74

4.4.2 Results

Due to the low speed of stress wave propagation in a foam, all the information
shown here is within time domain from 0 to 5 ms. Figure 48 shows the internal energy
in the homogeneous model. Almost all the internal energy is stored in the Part 1 of
foam core. The energy in Part 1 increases dramatically before 1.0 ms, and then tends

to stabilize at 14.7 J at 5 ms. The internal energy in Part 2 has a very small value 0.252

 

 

 

 

 

 

 

J at 5 ms.
20
Part l
.00.. Part2
15 . ——- Total _
_____ ;—_-__-_-=——-t
If.
:50
5:)
E

 

 

 

 

 

 

Time (ms)

Figure 48. Internal Energy History for Homogeneous Foam Model in 1D

Figure 49 presents the kinetic energy history in the homogeneous foam bar model.
It shows that kinetic energy in Part 1 increases rapidly upon the load application,
peaks at 3.9 J at 0.24 ms and then goes down. The kinetic energy in Part 2 is 0.0293],

indicating that this part almost remained static during the first 5 ms. Finally, a curve of

75

the overall kinetic energy can be seen to overlap the one of Part 1 and is only

marginally higher after 1.4 ms.

 

 

 

 

 

 

 

 

 

 

 

 

 

5
2 ‘ "P(t) P
artl
4 _ l r -0.-. Part2
<————>l<———+l ——— Total
‘3
is 3 1
‘43
C‘.
Lu
.2
r2 2-
$2
1-
0 WW»

Time (ms)

Figure 49. Kinetic Energy History for Homogeneous Foam Model in 1D

4.4.3 Discussion

The results listed in Table 6 below show that the stress wave propagation in
homogeneous foam material has great features. Table 6 shows that the peak stress
decreased dramatically in the foam material during the process of propagation. In Part
1, the stress magnitude decreased from 42.6 MPa at point A to 0.35 MPa at point C

due to the large plastic deformations that took place in this part. Thus, most of the

76

foam in Part 1 crushed, which causes a great amount of kinetic energy to be dissipated
in the part as internal energy. The foam in Part 2 suffered a much smaller deformation
during the wave propagation because the magnitude of the stress wave in it was lower
than the yield stress of foam material.

In addition, the speed of stress wave propagation in the foam material was much
slower than that in a solid material. It took 0.15 ms for the stress wave to propagate
from one end of the bar to the other in the solid bar. However, the same stress

propagation in the foam took 1. .4 ms.

Table 6. Results from Homogeneous Foam Model in ID Case

 

 

 

 

 

 

 

 

 

 

 

 

Homogeneous Model
Stress at point A (MPa) 42.6
Stress at point B (MPa) 0.36
Stress at point C (MPa) 0.35
Stress at point D (MPa) 0.35
Stress at point E (MPa) 0.36
Kinetic energy of Part 1 foam (J) 3.9
Kinetic energy of Part 2 foam (J) 0.0293
Internal energy of Part 1 foam (J) 14.7
Intemal energy of Part 2 foam (J) 0.252
Deformation of Part 1 in x direction (mm) 238.699
Deformation of Part 2 in x direction (mm) 17.7953

 

 

 

 

77

4.5 Stress Wave Propagation in Non-Homogeneous Foam Materials

In this section, results from the simulation of non-homogeneous foam models are
presented. The aim of the analyses shown in this section was to observe the features of
stress-wave propagation in multiple foam materials. Numerical models, corresponding
loading histories and materials pr0perties are presented. Simulation results, namely
stress and energy history are shown as well. The locations chosen to extract the

response measures are shown in Figure 47.

4.5.1 Case Study

The bi-material (or non-homogeneous) model used to capture the characteristics
of plastic wave propagation is shown in Figure 45. The configuration of this model is
identical to the homogeneous model. However, dissimilar materials were assigned to
two parts. In Model 1, a foam material with a relative density 0.06 was assigned for
Part 1 and a foam material with a relative density 0.24 for Part 2. Model 2 followed an

opposite sequence. The models were loaded with an impulsive compressive pressure

load of lOOMPa at the free end of the bar.

4.5.2 Results

Figure 50 and Figure 51 show results from Model 1, where Part 1 is a foam with
a relative density of 0.06 and Part 2 is a foam with a relative density of 0.24. Figure 50
shows the internal energy (consisting of elastic and plastic strain energy) stored in the
foam parts. The energy in Part 1 increases rapidly before 1.0 ms, then tends to

stabilize at 14.9 J at 5 ms. The energy in Part 2 has a very small value 0.0182 J at 5 ms.

78

20

 

 

 

 

 

 

 

 

 

 

 

 

 

Part 1
-°--- Part2
15 .J —-- Total
I.
P“
2
m 10 -
":6
E
Q)
.2: , 1
- - Ht)
5 I 1 11+—
'<——-——-—‘>l‘———->l
L/2 L0
0 vv:C::“;33:32:“;‘3=3'::“:':“=“:“:“:C‘:;‘
O 1 2 3 4 5
Time(ms)

Figure 50. Internal Energy History for Non-homogeneous Foam Model 1 in 1D

 

 

 

 

 

 

5
2 I F t
l 1 II‘( ) Part 1
4 ‘ -oo—o Pan 2
L/2 1J2 “’ “‘8'

 

 

 

Kinetic Energy (J)

 

 

 

 

 

Time (ms)

Figure 51. Kinetic Energy History of Non-homogeneous Foam Model 1 in 1D

79

Figure 51 shows the kinetic energy history for Model 1. The kinetic energy in
Part 1 increases rapidly, it peaks at 3.9 J in 0.24 ms and then decays. The kinetic
energy in Part 2 is only 0.00294J, indicating that this part remained essentially static
during the first 5 ms of response.

Figure 53 is results from Model 2, where Part 1 is a foam with a relative density
of 0.24 and Part 2 is a foam with a relative density of 0.06, Figure 52 shows the
internal energy stored in the foam parts. The internal energy in Part 1 increases rapidly
before 0.8 ms and then tends to stabilize at 12.0 J at 5 ms. The energy in Part 2 has a

very small value 0.357 J at 5 ms.

 

 

 

 

 

 

20
Part 1 2 1 F(t)
-00-. Part2 l I II‘ ' ‘
1 _ ——- Total
5 U2 L/2

 

 

 

(J)

V-

 

lnternal Eneruv

 

 

 

 

0 AA—A-—-A_AA_AA—AA_IQ-OO- -.O-OO-OO-IO-OO-OO-O

 

Time (ms)

Figure 52. Internal Energy History of Non-homogeneous Foam Model 2 in 1D

Figure 53 shows the kinetic energy history in for Model 2. As the load is applied

on the bar the kinetic energy in Part I quickly increases. peaking at 3.44 J in 0.07 ms.

80

and then decreased rapidly after 0.07 ms. The kinetic energy in Part 2 was 0.0342 I,

which means that it almost remained static during the first 5 ms of response.

 

 

 

 

 

 

 

 

 

 

 

 

 

5
2 ] P(t) qut1
4 — l l J] .00.. Panz
A —--— Total
3 L/Z L/Z
a: 3 .
§
3
g 2-
l
1 a
0 q ———————— :

Time (ms)

Figure 53. Kinetic Energy History of Non-homogeneous Foam Model 2 in 1D

4.5.3 Discussion

The results listed in Table 7 show that the stress wave propagation in non-
homogeneous foam models has interesting characteristics. As shown in Figure 50, the
internal energy is mainly stored in the weaker material in Part 1. Given that large
deformations (see Table 7) happen in the same part, Model 1 confirms the conclusion
from the solid models in Chapter 3 that larger plastic deformation will contribute to
larger internal energy dissipation. When comparing deformations between the two

models. Model 2 had much smaller values for both parts, while the energy dissipation

81

in Model 2 was slightly lower than that of Model I. This indicates that adding stiffer
material to locations close to loads can prevent severe damage. Considering the trend
of stress from the free end (Point A) to the fixed end (Point E), the magnitude of stress
is affected by the stiffness of material. In a non-homogeneous model, an increasing
stiffness of the material in the direction of the stress wave propagation will lead to
higher transferred stresses, compared with a model with a deceasing material stiffness.

This is consistent with the results from (Cui et al. 2009).

Table 7. Results from Non-homogeneous Foam Model in ID Case

 

 

 

 

 

 

 

 

 

 

 

 

 

Non-Homogeneous Model
Model 1 Model 2
Stress at point A (MPa) 42.6 54.9
Stress at point B (MPa) 0.366 2.82
Stress at point C (MPa) 0.395 0.443
Stress at point D (MPa) 0.734 0.351
Stress at point B (MPa) 0.968 0.369
Kinetic energy of the part 1 foam (J) 3.9 3.44
Kinetic energy of the part 2 foam (J) 0.00294 0.0342
Internal energy of the part I foam (J) 14.9 12.0
Internal energy of the part 2 foam (J) 0.0182 0.357
Deformation of part 1 in x direction (mm) 255.377 36.0426
Deformation of part 2 in x direction (mm) 1.149 24.1994

 

 

 

 

 

4.6 Summary

This chapter showed numerical studies aimed at evaluating the characteristics of
stress wave propagation in homogeneous and non-homogeneous (bi-material) foam
materials. The goal of studying one-dimensional models was to understand the
behavior of stress wave propagation in foams and thus evaluate the performance of
two-dimensional foam models (Chapter 5). The results were reasonable and confirmed
findings by other researchers. The following summarizes important features learned
from the presented studies.

Stress waves in homogeneous foam models propagate at a lower speed compared
with homogeneous solid models. The stress magnitude deceases dramatically because
most of the kinetic energy transfers to internal energy by means of large plastic
deformations in the foam material. This is consistent with the conclusion from Chapter
3. It is clear that foams are suitable to be used as a buffer material for energy
dissipation and stress mitigation.

The propagation of stress waves across dissimilar foam materials (non-
homogeneous) highlighted that arranging material’s stiffness and strength sequence
can lead to a better performance on stress mitigation and energy dissipation. A stiffer
material near the load application will reduce the overall deformation in the model
dramatically without losing much kinetic energy dissipation. A layered non-
homogeneous with an increasing sequence of stiffness of material in the direction
propagation will lead to a higher stresses at the end of the bar, while an opposite will

lead to a lower stresses.

83

CHAPTER 5.TWO-DIMENSIONAL STRESS WAVE PROPAGATION IN
FOAM MATERIALS

5.1 General

This chapter presents studies on the two-dimensional stress wave propagation in
foam materials. Deformations, stress mitigation and energy dissipation are of prime
concern. The aim is to evaluate the performance of using homogeneous and non-
homogeneous open-cell foam cores in a sandwich beam subjected to an impulsive
pressure load.

Four types of sandwich aluminum foam beams, with a length of 72 mm and
height of 27 mm, were evaluated. The sandwich foam panels featured an open-cell
aluminum foam core and two outer face sheets assumed to be of solid aluminum
material. The model differences were in the open cell foam core. The foam core in the
first design consisted of a homogeneous foam material with a low relative density and
thus a relatively low strength. The second design features a foam core with a
strategically reinforced region in which the reinforcement effect was implemented by
considering a foam with a lower relative density. The third design features a core with
layers of foams made of different relative densities, with lower relative density at the
top and a higher relative density at the bottom. The fourth design also features a
layered foam core but in this case the relative stiffness decreases from top to bottom.
The panels were modeled as two-dimensional simply-supported beams and

constrained from horizontal deformations to represent testing of a circular (or disk)

84

panel inside a shock tube chamber. Three levels of impulse pressure loads, lMPa,
2MPa and 3MPa, were applied on the full span of the top facesheet.

Results from the numerical simulations indicate sandwich beams with
strategically reinforced non-homogeneous cores performed best in terms of
deformations, stress mitigation and energy dissipation. The results further show that
the findings from Chapter 4 on 1D stress wave propagation in foam materials can be
used to predict the behavior of foam sandwich plates in 2D. Finally, the studies in this
chapter indicate that strategically reinforced (or tailored) foam structures can be used

for stress wave management.

5.2 Model Definition

Four models were used in this study. Following conventional sandwich plate
designs, each model is made up of 3 parts: top facesheet, foam core and bottom
facesheet. Differences exist on the design of the foam cores. The top and bottom
facesheets have a length of 72 mm and height of 1.5 mm. The overall length of the
foam core is 72 mm and height is 24 mm. In order to detect the stress redirection,
simple supports are modeled as a vertical restraint along a 6 mm region at the ends of
the lower facesheet. The facesheets and foam core are assumed to be perfectly bonded
and so are the interfaces between foam regions with different relative densities. The
material in the facesheet is aluminum. The sandwich beam cores are assumed to be
made from open-cell aluminum foams with varied relative densities. The facesheet
was simulated with the elastic material model MAT001 in LS-DYNA, and the

crushable foam material model MAT063 was used to simulate the foam material.

85

Details for the mesh were the same as the ones used for the one—dimensional studies in

Chapter 4.

 

I:(t)

 

 

 

 

 

 

 

 

 

 

 

 

A
‘

.v: ........ ...... . .v.v.v.v ....... -_-_ . .. 15 mm

 

 

X“
n11 A

F acesheet 24 mm

 

 

 

\\Z\\\\
V
"I1
, O
‘ §
1 o
-1
(D
I (HI TH) Ill]
\ \\\\WY\\\

 

 

1.5 mm

uuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu

 

 

 

Figure 54. Model 1 with Homogeneous Foam Core

Figure 54 shows the sandwich plate geometry for Model 1. The facesheet
thickness is 1.5mm, 6.25% of 24mm the foam core thickness. The length of the plate
is 72 mm, and the free span is 60 mm due to the 6 mm support regions at each beam
end. The foam core in Model 1 is taken as a homogeneous aluminum open-cell foam.
The simply-supported boundary conditions for the beam were modeled as vertical
deformation supports at the ends of the bottom facesheet. In addition, lateral

deformations along the beam thickness were restrained.

 

 

F“) t
1:— 1.5 mm
Facesheet 12 mm
-11-
12
Reinforced Weak Foam JL mm
Foam ' T15 mm

6mm 24mm 12mm 24mm 6mm

86

Figure 55. Model 2 with Irregularly Reinforced Foam Core

Figure 55 shows Model 2, which features a strategically reinforced foam core in
which a region shaped in the form of an arch has a foam with lower relative density
while the surrounding foam is a weaker (higher relative density) foam. The contact
region between top facesheet and reinforced region lies in the center of top facesheet
and it is 12 mm long. The contact regions between the reinforced foam and the bottom
facesheet lie on the two support regions and their length is 6 mm each. The interaction
among the foam regions and between the facesheets and the foam core is assumed to

be one of perfect bond.

 

F(t) J— 15 mm

Layer 1 v 7.5 mm

F acesheet Layer 2 :r‘ 9 mm
fi_'l51nni

 

Layer 3

/

 

2% 1.5 mm
l<->l<—;-——————->l1—>l
6mm 60 mm 6mm

Figure 56. Model 3 with Increasing Layered Foam Core

Figure 56 shows the foam core for Model 3, which is composed by three stacked
homogeneous foam layers. Due to the size of mesh element used in this study, the
thickness of the first and the third layer is 7.5 mm and that of the middle layer is 9 mm.

The relative densities for the foam layers increase linearly from the top to the bottom.

87

F(t)

k_

1.5 mm
7.5 mm

  

_I

F acesheet 9 mm

_L
'l‘ '1‘ 'l

7.5 mm

 

I‘LL
F

1.5 mm

‘—>|

r<—>I<——————>I<—>I

6 mm 60 mm 6 mm

Figure 57. Model 4 with Decreasing Layered Foam Core

The foam core for Model 4 is shown Figure 57. This model also features a core
with layered homogeneous foam regions. As for the previous model, the thickness of
the first and the third layer is 7.5 mm and that of the middle layer is 9 mm. The
relative densities for the foam layers in this model decrease linearly from top to

bottom.

5.3 Material Modeling

The material used in this study is aluminum and the assumed material properties
are listed in Table 8. The elastic material model MATOOI in LS-DYNA was used to
model the facesheets. The properties for the open-cell foams used in this study are
shown in Table 9, where the main differences between the foams (columns) is the
relative density. They were computed based on the formulas presented in Section 2.3
The foam regions were modeled in LS-DYNA with the crushable foam material model

MAT063.

88

Table 8. Properties of Aluminum Plate in 2D Sandwich Beams

 

 

 

 

 

 

E (GPa) 80
v 0.3
3
p (k g/m ) 2800
0 (MPa) 80

 

Table 9. Material Properties of Foams Used in 2D Sandwich Beams

 

 

 

 

 

 

 

No.1 No. 2 No. 3 N0. 4 No. 5 No. 6 No. 7

Relative density 0.06 0.08 0.10 0.24 0.26 0.28 0.30
E (GPa) 0.29 0.51 0.80 4.61 5.41 6.27 7.20

v 0.33 0.33 0.33 0.33 0.33 0.33 0.33

p (kg/m3) 168 224 280 672 728 784 840

0' (MPa) 0.35 0.54 0.76 2.82 3.18 3.56 3.94

 

 

 

 

 

 

 

 

 

 

5.4 Case Studies

All four of the previously described models were evaluated under three loading
cases, namely, pressure loads of l, 10 and 100 MPa, which define Cases 1, 2, and 3,
respectively. Load intensity has a significant effect on the performance of the foam
material. Thus, the relative densities for the foam materials considered in this chapter
were selected based on pilot studies so that the simulations could be successfully
completed and so that the results between models were comparable. Consequently, the

foam properties (relative density) for Case 1 and Case 2 were identical. However, the

89

foams used for Case 3 were stiffer so that the load of 100MPa would yield meaningful

results and aid convergence of the numerical simulation.

5.4.1 Case 1
In this case a lMPa impulse pressure load was applied on the full span of the top
facesheet for all four models. The loading history is shown in Figure 38. The foam

materials used in Case 1 are listed in Table 11. Please refer to Table 9 for the foam

engineering properties.

Table 10. Foam Materials Used in Case 1 in 2D

 

 

 

 

Model Model 2 Model 3 Model 4
1
Part Foam Weak Reinforced Layer Layer Layer Layer Layer Layer
Core Foam Foam 1 2 3 1 2 3
Foam No.1 No.1 No.4 No.1 No.2 No.3 No.3 No.2 No.1
No.

 

 

 

 

 

 

 

 

 

 

 

5.4.2 Case 2
In this case, a lOMPa impulse pressure load was applied on the full span of the
top facesheet for all four models. The loading history is shown in Figure 58. The foam

materials used in Case 2 are listed in Table 11. Please refer to Table 9 for the foam

engineering properties.

90

 

 

 

 

 

P(MPa)
—10 -------- ,
I p
0 2511s sorts 1000115 T(s)
Figure 58. lOMPa Compression Pressure Load
Table 11. Foam Materials Used in Case 2 in 2D
Model Model 2 Model 3 Model 4
1
Part Foam Weak Reinforced Layer Layer Layer Layer Layer Layer
Core Foam Foam 1 2 3 1 2 3
Foam No.1 No.1 No.4 No.1 No.2 No.3 No.3 No.2 No.1
No.

 

 

 

 

 

 

 

 

 

 

 

5.4.3 Case 3

This case featured a lOMPa impulse pressure load applied on the top facesheet

for all models. The loading history is shown in Figure 25 in Section 4.4.1.3. The foam

materials and their relative densities used in this case study are listed in Table 12.

Table 9 provides the engineering properties for the foam materials.

91

 

 

Table 12. Foam Materials Used in Case 3 in 2D

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Model Model 2 Model 3 Model 4
1
Part Foam Weak Reinforced Layer Layer Layer Layer Layer Layer
Core Foam Foam 1 2 3 1 2 3
Foam No.4 No.4 No.7 No.4 No.5 No.6 No.6 No.5 No.4
No.
5.5 Results

This section presents the results from the mentioned study cases with emphasis
on induced deformations, stress magnitudes, transferred kinetic energy and energy
absorption. Because the perfect bond assumption between the foam and facesheets is
not realistic, the deformations obtained from these models may not be accurate after
the pressure loads are fully acting on the top facesheet of sandwich plate. The effect of
delamination of the foam core from the facesheets was included in a study from
Bahei-El—Din (2006). Therefore, the data up to 0.2 ms of response is considered to be

more realistic and representative for evaluating and comparing the models.

5.5.1 Case 11

In this case, a lMPa compressive pressure impulse load was applied on the full
span of the top facesheet. The four models show distinct response. Figure 59 shows
the kinetic energy in the top facesheet for all models. It can be seen that as the
pressure load was applied on the sandwich plate the kinetic energy of the top face
sheet increases gradually reaching a peak and then decaying equally fast. Model 1 and
Model 3 had similar behavior with peak kinetic energy values at around 0.042 ms of

2.47x10'6J for Model 1 and 2.45x10'6 J for Model 3. The response of Model 2 and

92

model 4 was similar to each other with peak kinetic energy values at approximately

0.039 ms of 1.59x10'6 J and 1.50x10'6 J, respectively.

 

 

 

 

 

 

 

 

 

0.0030
—— Modcll

.19 0.0020 - -"'- Model 4
Z
:30
55 0.0015 -‘
5
.2
g 0.0010 -
:4

0.0005 6

0.0000

0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 59. Kinetic Energy of the Top Facesheets in Case 1

Figure 60 shows the vertical displacement history of the top facesheet for all
models. It can be seen that Model 1 had the greatest vertical displacement: -0.326 mm
at a time 0.18 ms. Model 3 reached its maximum vertical displacement of -0.266 mm
at 0.15 ms, and Model 4 reached its maximum vertical displacement of -0.254 mm at
0.17 ms. Model 2, with a reinforced arch foam core had the least vertical displacement:

-0.198 at 0.16 ms.

93

 

Vertical Displacement (mm)

Vertical Displacement (mm)

 

0.1

 

 

 

 

 

 

 

 

 

 

Model 1
-- -° Model 3
—~--- Metre-l 4
-0.1 -
-0.2 9
~03 ‘
-04 r l T
0.00 0.05 0.10 0.15

Time (ms)

Figure 60. Displacements of the Top Facesheets in Case 1

0.20

 

 

 
 
 
 

 

 

 

 

 

 

 

 

 

 

0.05
Model 1
— — Model 2
0 00 _ -- -' Model 3
-'°-° Modcl4
-0.05 7
-0.‘lO -+
-0.15 r r r
0 00 0.05 0.10 0.15

Time (ms)

Figure 61. Displacements of the Bottom Facesheets in Case 1

94

0.20

Figure 61 shows the vertical deformation in the bottom face sheet for the four
models. It can be seen be seen that the displacement response at bottom facesheet in
the four models was similar, ranging from -0.0979 mm to 0.118 mm for Model 2 and
Model 1, respectively.

Figure 62 shows the stress histories extracted from the middle of the bottom
facesheet. It can be seen that the stress response for all models was very similar. The
lowest stress peak values was for Model 1: -1.78x10'2 GPa at 0.15 ms, while the higest

stress value was for Model 1: -2.34x10‘2 GPa at 0.14 ms.

 

0.010

 

Model 1

 

0.005 -

— — Model 2
0.000 — -- -' Model 3
—--—° Model 4

 

 

 

 

-0.005 a

-0.010 1

Stress (GPa)

-0.015 ~

-0.020 -

 

-0.025 ~

 

 

 

-0.030 t T r
0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 62. Stresses in the Middle of the Bottom Facesheet in Case 1

95

 

 

 

 

 

 

 

 

 

0.0010
—— Model]
00005 ‘ —- Mode12
-- -- Model3
----- Model4
”T I
5; 0.0000 '1' H . 3.
I \ \. mallk '1‘..f*m. ‘l mu
8’3 \ \' Q r i: )' ‘l‘“ l
5 000054 \l t: \. . \l
\ l~ A, 1' J
\qu‘l
00010 -
-0.0015 I i r
0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 63. Redirected Stresses in the Bottom Facesheet in Case 1

The stress histories extracted from the support regions of the bottom facesheet are
shown in Figure 63 It can be seen that the traces are similar in featuring an increase in
the reaction stresses and then oscillations due to the reflection of waves off the
supports. It is of interest to note that Model 2, which featured the arch reinforced foam
had the highest reaction stress values, followed by Model 3, which used an
increasingly denser foam in a layered fashion.

The history of i'ntemal strain energy stored in the foam core of Model 1 is shown
in Figure 64, whereas that for Model 2 is shown in Figure 65. The results for Model 2
provide traces for the intemal energy in the reinforced (arch shape) and weaker
surrounding foam. It can be seen that the most of the energy absorption takes place in

the weaker foam regions.

96

 

5

Internal Energy (10.3 .1)

Internal Energy (10-3.1)

0.005

 

 

 

Foam Core

 

 

 

I

0.004

0.003

1

0.002

1

 

 

0.001

1

 

 

 

 

 

 

 

 

 

 

0.000 I T I
0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 64. Internal Energy of Foam from Model 1 in Case 1

0.0035

 

 

 

 
   
 

 

F(t)

0.0030 ‘

 

Weak Foam
—---- Reinforced Foam

 

 

0.0025 -

 

 

 

 

 

 

1

0.0020

1

0.0015

0.0010

1

   

' ..‘OO’..-..-..-..-0
’
”°
..

0.0005

1

 

 

0.0000 1
0.00 0.05 0.10 0.15 0.20

 

Time (ms)

Figure 65. Internal Energy of Foams from Model 2 in Case 1

97

 

The strain energy in Model 1 stabilized at approximately 3.74 x 106] while the
total strain energy in Model 2 stabilized at a level of approximately 3.1 x 10'6 J. The
higher strain energy in Model 1 indicates that that panel suffered more damage even

though Model 2 had less total vertical deformation in the bottom and top facesheets.

 

 

0.0014

Layer 1
-"-- Layer 2
- — - Layer 3

I

 

0.0012

 

 

 

-3
".l)

1

0.0010
0.0008 —

0.0006 -
F(t)

Internal Energy (10

0.0004 ~

0.0002 ~

 

 

 

 

 

0.0000 - I 1
0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 66. Internal Energy of Foams from Model 3 in Case 1

Figure 66 shows the internal energy stored in the three layers of Model 3 (model
with increasing foam stiffness). It can be seen that there is a time phase lag on the
increase of strain energy for each layer. However, the strain energy for all layers
reaches similar magnitudes before stabilizing at a similar level. This seems to indicate
that the upper layers in the foam do not impede the transmission of stress waves and

the energy is transmitted to the layers below.

98

 

0.0025

 

 

 

 

 

 

Laycl‘l
00020 _ -0.-. l‘il}er 2
3. ——- Layer 3
r".
'A
10.0015— ...,-._.»-—--“
:11 ,v’
'— ’fi/
35 /
LL.- /
'5 0.0010 - /
E / go-..“
,1) fl..’00’
: / .’..’..—OO
; ”

 

0.0005

1

 

 

0.0000

1

 

 

0.00 0.05 0.10

Time (ms)

0.15 0.20

Figure 67. Internal Energy of Foams from Model 4 in Case 1

The strain energy history for Model 4 (decreasing foam

stiffness) is shown in

Figure 67. As for Model 3, the time lag shift on the increase of strain energy is also

seen for this model. However, it can be seen that the levels of energy do not reach the

same level for all layers. Rather, the inelastic deformation (and energy absorption) is

concentrated in the bottom layer while the upper stiffer foam layers experience a much

lower (about half) level of internal strain energy. However, the

total strain energy for

Model 4, which was 3.08 x 10'6 J at 0.2 ms is very similar to the total energy in Model

3 (3.52 x 10'6 J at 0.2 ms), which indicates that Model 4 was able to dissipate about

the same total amount of energy but with less damage within its structure.

99

5.5.2 Case 2

As previously noted, Case 2 consists on the application of a lOMPa impulsive
compressive pressure load applied on the full span of the top facesheet. This section
provides an overview of the response of the four different sandwich foam beam
models to this loading condition.

A plot of the kinetic energy history response of the top face sheet is shown in
Figure 68. Model 1 (homogeneous foam) and Model 3 (increasing foam stiffness)
showed a similar response transmitting the highest level of kinetic energy. Conversely,
Model 2 (reinforced foam) and Model 4 (decreasing foam stiffness) transmitted less

kinetic energy to the top face sheet. This indicates that Models 2 and 4 will experience

less deformation in their top face sheets compared to the other designs.

0.5

 

 

-—- Model]
0.4 ~ — -— Model 2
' Model 3

fi

Kinetic Energy (10'3 J)

Model 4

 

 

 

 

 

 

 

0.00 0.05 0.10 0.15

Time (ms)

Figure 68. Kinetic Energy of the Top Facesheets in Case 2

100

The vertical displacements of the top and bottom facesheets are shown in Figure
69 and Figure 70, respectively. From Figure 69 it can be seen that Models 2 and 4
have the least amount of deformation on the top facesheet, which is consistent with the
lower levels of kinetic energy for this part of the model. The response of Models 1 and
3 is essentially equal. Overall, Model 2 (reinforced foam) shows the least amount of
deformation. Deformations of the bottom facesheet are more similar for all models,
however, differences can be seen around the peak values. First, it can be seen that
Models 1, 2 and 4 experience a very similar deformation history, reaching a very
similar maximum value and then showing a rebound in the response. Model 3,

however, experienced larger deformations and a later rebound on the response.

 

 

 
  
 
  
 
   

 

 

 

 

 

 

0 _ —-— Model I
A --- Modcl2
g -- -- Model 3
E -2_ —---- Model4
’1)
E
Q)
U
E.
e -4-
Q
'3
.9.
_; -6~

'8 T l r

0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 69. Displacements of the Top F acesheets in Case 2

101

 

 

  
 

 

 

 

 

 

 

 

 

0.0 V.\
E
E -0.1 -
Q)
E
'J)
C)
.‘3
:2“ ~02 4
Q
‘8 Model l
g — - Model 2
g -0.3 - -- -- Model 3 ”
--°-- Mode|4
-0.4 r r I
0.00 O 05 0.10 015 0.20

Time (ms)

Figure 70. Displacements of the Bottom F acesheets in Case 2

Figure 71 shows that stress histories extracted from the middle of the bottom
facesheet. The stress demands for Models 1, 2 and 4 was very similar, while Model 3
experienced a 40% lager peak stress. Model 3 is the one with the layered foam core
with increasing stiffness. These results indicate that the use of a higher stiff material
near the bottom leads to increased stresses. Thus, this design option increases the

forces at the bottom of the beam rather than attenuating the loading effects.

 

0.01

 

  
 
 

 

  

 

 

 

 

 

 

0.00 _ —- Modell
— — Mode12
-0.01 _ -- -- Model 3
--—- Model4
’5
F: -0.02 ~
2’.
g -0.03 .
Cl)
-0.04 d ._
\ ”\u/
\ 4
“A e ‘0
\I
—0.06 I u v
0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 71. Stresses in the Middle of the Bottom F acesheet in Case 2

The concept of stress redirection was again explored by comparing the stress
history at the support regions of the beams. As shown in Figure 72, stresses at the
support region tend to increase rapidly upon arrival of the stress wave and then
stabilizes with only minor oscillations due to refraction in Model 3. With the
exception of Model 2, all models have a similar response, which indicates that while
their individual stress, deformation, and energy levels were different, the stress wave

propagated to the supports in a similar manner.

103

 

0.001

 

 

 

 

0.000
\\. r" I l
3 0001- \\\‘ -~ ‘1‘? ‘ 't""--‘ ”It -'“ ‘
2.. -. v! ‘0 'P‘», . \ ’ 4
‘3; \\ }¢\ ‘\ \Ill
:12: -0.002 . \v-“J *‘VI' A»,
-—- Modell

— — Model 2

 

 

 

 

 

 

'0‘003 d -- -- Model 3
—---- Model 4
-0.004 r r r
0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 72. Redirected Stresses in the Bottom F acesheet in Case 2

However, the stresses at the support region of Model 2 not only increase more
rapidly but have a higher magnitude (twice as high) and a more pronounced effect of
wave refraction indicating that the stress wave is going back and forth to this same
region. Such behavior indicates that, as intended, the strategic reinforced foam design
of Model 2 was effective in redirecting the path of the stress wave to the supports.

The strain energy accumulated in the foam core is shown in Figure 73 to Figure
76. The internal energy in Model 1 (homogeneous foam) is shown in Figure 73. The
internal energy rises rapidly and uniformly throughout the loading time span. The

internal energy at 0.2 ms reaches 2.73 x 10'4 J.

104

 

 

 

 

 

 

 

 

 

F(I)
03 q — Foam Core
(Oi—5
'2
3’. 0.2 «
E.”
a.»
[:1
"F5
25
§ 0.1 4
0.0 . 1 I
0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 73. Internal Energy of Foam from Model 1 in Case 2

The strain energy for Model 2 (arch reinforced foam) is shown in Figure 74. It
can be seen that the accumulated internal energy for the weak and stiff foam regions is
similar, although slightly larger for the weak foam regions. The total internal energy
for this model reached 1.79 x 10'4 J at 0.2 ms. This is 34% lower than that for Model 1,

indicating the ability of this model to sustain less internal damage.

105

0.20

 

 

 

 

 

 

 

 

 

 

Weak Foam
--°—~ Reinforced Foam
.flod)

: 0.15 -
”P

O

:3

g 0.10 -+

LL]

E3

r:

2

5 0.05 ~

0.00 r r r
0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 74. Internal Energy of Foams from Model 2 in Case 2

The internal energy accumulated in the foam core of Models 3 and 4 are shown in
Figure 75 and Figure 76, respectively. In Model 3, most of the energy was stored in
the weaker foam at the top (layer 1). In Model 4, most of the energy was again at the
top layer but the bottom layers also contributed to dissipate energy, even though at a

lower level. Thus damage in Model 4 was not as concentrated as that in Model 3.

106

Internal Energy (10'3J)

v4

Internal Energv (10-3.!)

0.40

 

0.35 ~

0.30 ~

0.25 -

0.20 -

0.15 ~

 

 

 

 

 

 

 

 

0.05 0.10 0.15 0.20

Time (ms)

Figure 75. Internal Energy of Foams from Model 3 in Case 2

0.30

 

0.25

l

0.20

  

0.15

0.10

 

F(t)

 

 

 

 

Layer l
—---- Layer 2
— — - Layer 3

 

 

 

 

 

 

’00-°’-..’K-’:’
’II/
0.15 020

Time (ms)

Figure 76. Internal Energy of Foams from Model 4 in Case 2

107

5.5.3 Case 3

The loading condition for Case 3 is a 100MPa compressive impulse pressure load
applied on the full span of the sandwich beam top facesheet. As in the previous
sections, results for the different beam models to this load case are presented next.

Figure 77 shows the history of kinetic energy on the top facesheet of the beam.
The response for all models is similar, with slightly higher values for Models 1 and 3.
Displacements for the top and bottom facesheets are shown in Figure 78 and Figure 79,
respectively. In both cases, the displacement profiles are very similar, bounded by the
responses of Model 1 (larger deformations) and Model 2 (smaller deformations). The
difference in response is not as obvious as in previous load cases, which is thought to

be due to the increased damage caused by larger load in this case.

 

 

 

 

 

 

 

 

 

10 3 Model 1
—— Mode12
c; 8 _ -- -- Model3
‘1‘ VI; --°-- Model4
:3. "=39
w 6* ~
35 . ‘\
LL} 0
2 4 , \
E \{
2:4
kg
2 4 \\-‘Il-.
\
O r r
0.00 0.05 0.10 0.15 0.20
Time (ms)

Figure 77. Kinetic Energy of the Top Facesheets in Case 3

108

Vertical Displacement (mm)

Vertical Displacement (mm)

 

 
 
  
    

 

 

 

 

 

 

 

 

0
—— Modell
— — Model 2
-5 - -- -- Model 3
-"°- Model4
-10 _
-15 —1
~ _ _fl".
~~m$..-"—
__/
-20 r r r
0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 78. Displacements of the Top Facesheets in Case 3

 

 

 

 

 

 

     

 

 

 

 

0 ¥
-1 a
..2 d
-3 a
.4 a
Model I A

‘5 ‘ —— Model2

-- -. Model 3
-6 “ _.._. Model 4
'7 fiT I T

0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 79. Displacements of the Bottom Facesheets in Case 3

109

Figure 80 shows the stress histories extracted from the middle of the bottom
facesheet. The response for all models was very similar; indicating that the demand
transferred to this bottom element was essentially the same in all cases.

The stress histories extracted from the support regions along the bottom facesheet
are shown in Figure 81. The increased stress values for Model 2 indicate again that the
strategically reinforced foam core is able to direct the stress waves to the support
regions. However, the support stresses for the other models are now larger and closer
to those of Model 2. This is due to the increased load magnitude, which indicates that
as the level of damage increases the stress wave has no alternative but to increase the
demands on the support regions. Thus, the differences with tailored design of Model 2
are reduced. This brings to light that the efficiency of a foam core for stress redirection
and management will depend on the magnitude of the applied load.

The levels of internal stored energy in the foam cores of the beam models are
shown in Figure 82 to Figure 85. The internal energy for Model 1 is shown in Figure
82, which shows a similar trend as that seen in Cases 1 and 2. The fact that the strain
energy trace seems to stabilize indicates that most of the energy from the applied load
has been dissipated in the form of permanent internal deformations.

The internal energy for Model 2 is shown in Figure 83, which shows that the
weak and strong foams shared in the dissipation of energy almost equally. The total
amount of internal energy for Model 2 was 10.9 x 10‘3 J, which is about 43% of the
energy absorbed by Model 1. Thus, Model 2 was able to sustain the load with less

damage and with slightly less deformations to the top and bottom facesheets.

110

Stress (G Pa)

Stress (GPa)

0.2

 

 

   

 

 

 

 

 

 

 

 

rfl
0 0 L
-0.2 -
-0 4 a
'0'6 ‘ Model I J
— — Model 2 \
'0-8 ‘ -- -° Model 3
---—’ Model 4
-1.0 -
0.00 0.05 0.10 0.15 0.20
Time (ms)

Figure 80. Stresses in the Middle of the Bottom Facesheet in Case 3

 

 

 

 

 

 

 

 

 

 

 

0.00
-0.05 a
-0.10 - "
l
-0.15 - 1
020 - | '
MOdCl 1 I '

-O.25 — — — Model 2

-- -- Model 3 V
'0 30 ‘ ----' Model4
-0.35 l l r

0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 81. Redirected Stresses in the Bottom F acesheet in Case 3

111

Internal Energy (10'J J)

35

 

30

l

25‘

I

20

15a

10-

 

 

— Foam Core

 

 

 

 

 

 

I l l

0.00 0 05 0.10 0.15 0.20

Time (ms)

Figure 82. Internal Energy of Foam from Model 1 in Case 3

 

 

 

 

 

 

 

 

 

 

20
Weak Foam
----° Reinforced Foam
.1: 15 -
'7'

:51;
§ 10 ~
:3:
33
E. 5 a

0 — I l l

0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 83. Internal Energy of Foams from Model 2 in Case 3

112

The internal strain energy for Models 3 and 4 are shown in Figure 84 and Figure
85, respectively. The results for both models indicate that the internal energy
accumulated in the core was shared by the three layers, with the largest intemal energy
being taken by the top layer. In Model 3 the top two layers had about the same level of
internal energy, indicating that the damage was extensive up to this level. In Model 4,
the damage was decreasingly smaller through the layers, which signifies more
distributed damage through the beam foam core. The total internal energy for the

models was very similar, with Model 4 being only 10% lower than that in Model 3.

 

 

 

 

 

 

 

 

 

 

 

 

20
I:(t)
1 Layer 1
a 2 ' E -°--- I.ayer2
15 5 " ' ‘ 3*" ii ""' ——- Layer3
10 JA

 

..-..-..—..’..-..-..ql

00-00-00,

lntemal Energy (10'3J)

 

 

 

 

5 _~._..——-/‘_——"'"‘
/
/
/
/
/
0 F F
0.00 0.05 0.10 0.15 0.20
Time (ms)

Figure 84. Internal Energy of Foams from Model 3 in Case 3

113

18
16 — F(t)

 

 

Layer 1
—..-. La)’cr 2
14 ‘ -—- Layer3

 

 

-3
' J)

 

 

12‘

(10

U

 

10

l

   

09-00—00—00-00—00-00

’oo—oo-oo’

Internal Energy

 

 

0.00 0.05 0.10 0.15 0.20

Time (ms)

Figure 85. Internal Energy of Foams from Model 4 in Case 3

5.6 Discussion

This section provides a discussion on the results obtained from the two-
dimensional stress wave simulations in sandwich foam beams. The key numerical
results from the responses discussed in the previous sections are summarized in Table
13, which is used for comparisons later.

From Table 13, it can be seen that Model 2, with a reinforced foam arch had a
superior performance in terms of stress mitigation, stress redirection, kinetic energy

dissipation, and reduction structural deformation.

114

Table 13. Summary of Results for 2D Studies

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Model 1 Model 2 Model 3 Model 4
Kinetic E., top facesheet (x1061) 2.47 1.59 2.45 1.50
Kinetic E.,bott. facesheet (x lO'GJ) 0.286 0.159 0.352 0.245
lntemal E., foam (x10'6 J) 3.74 3.10 3.52 3.08
Case 1 Disp., top facesheet (mm) -0.326 -0.198 -0.266 -0.254
Disp., bottom facesheet (mm) -0.1 18 -0.098 -0.1 16 -0.1 11
Stress, mid-bot. facesheet (MPa) -23.4 -l7.8 -21.5 —20.1
Redirected stress (MPa) -0.574 -0.962 -0.876 -0.617
Kinetic E., top facesheet (x10'4J ) 4.0 3.2 4.0 3.1
Kinetic E.,bott. facesheet (x 1071) 6.53 6.34 8.05 7.09
lntemal E., foam (x104 J) 2.73 3.41 3.28 2.94
Case 2 Disp., top facesheet (mm) -7.37 —4.80 -7.17 -5.69
Disp., bottom facesheet (mm) —0.228 -0.231 -0.310 -0.243
Stress, mid-bot. facesheet (MPa) —41.4 -4I.2 -59.1 -44.4
Redirected stress (MPa) -0.809 -2.04 -I .47 -0.884
Kinetic 13., top facesheet (x1031) 10.0 8.60 9.98 8.44
Kinetic E.,bott. facesheet (x 1031) 1.12 0.94 0.95 1.02
lntemal 13., foam (110'3 J) 26.1 25.7 26.7 24.0
Case 3 Disp., top facesheet (mm) -18.4 -16.5 -l7.0 -l7.l
Disp., bottom facesheet (mm) -6. 12 -5.68 —6.04 -5.84
Stress, mid-bot. facesheet (MPa) -963 -875 -973 -948
Redirected stress (MPa) -l3.4 -28.3 -18.2 -16.0

 

 

 

 

 

 

115

 

For all load cases, Model 2 had the lowest kinetic energy at the top and bottom
facesheets, and had the smallest deformation at the top facesheet. These two features
make Model 2 a highly efficient buffer structure to prevent objects from severe
damage due to impact or blast loading. This agrees with the conclusions from Bahei-
El-Din (2006). The modification of the sandwich plate in Model 2 increases stiffness
of the whole structure; in addition, this increment reduces vibrations of the facesheet
and overall deflection of the plate and avoids severe deformations of the foam core.

The response of Model 2 also indicates that stress waves could be redirected by
design. For the three load cases studied, Model 2 effectively redirected the stresses to
the supported regions on the bottom facesheet. In Case I, the support region stresses
for Model 2 were 1.7 times greater than those of Model 1. For Cases 2 and 3, this ratio
was 2.5 and 2.1, respectively. These results indicate that some regions in the structure
would be protected since part of stresses was redirected to other regions, in this case
the stiff (and typically much stronger) support regions. This explains why Model 2 had
the lowest stress at the middle of the bottom facesheet for all 3 load cases.

Another finding from these studies is that the stresses in the middle of the bottom
face in Model 3 were greater than those for Model 4. This also agrees with the
findings from other researchers. The reason is that in Model 3, the stiffness is
increasing from the top to the bottom while in Model 4 it is the opposite. Increased
stiffness leads to a larger stress in the end while decreased stiffness reduces it, thus

leading to a more efficient energy absorbing design.

116

CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE
WORK

6.] Conclusions

In this thesis, numerical studies on the stress wave propagation in solid and foam
material were carried out. Theoretical principles of stress wave propagation in one
dimension and the work from (Zhang and Paulino 2007) confirmed results from
numerical simulations of cantilever bar in one dimension. Analyses on stress wave
propagation on foam bars in one dimension and sandwich foam beams in two
dimensions were conducted based on the validated numerical model. Results from
these studies were in agreement with documented analytical and experimental research
findings. The results show that a foam material performs very well in dissipating
energy and thus mitigating transmitted of stresses. In addition, studies on sandwich
beam models with reinforced foam presented characteristic of guiding stress wave
propagation, a promising notion that has not been studied.

As a result, the concept of stress wave management has been fully evaluated
through the presented study. The concept not only includes stress wave speed control,
stress mitigation and energy dissipation, but also addresses the redirection of paths of
stress wave propagation. For now, some practical principles of designing a sandwich
beam model with desired capacities of stress wave management can be derived from
the four sandwich beam models studied in this thesis.

The findings from this study lead to the conclusions stated below:

117

By modifying the configurations of foam structures, stress waves can be
directed. Among four sandwich aluminum foam beam models with
dissimilar foam core configurations, the model consisting of a reinforced
arch foam surrounded by relatively weaker foam showed an effective way
of guiding the stress wave propagation from the middle region of the
beam to its simply—supported regions. This implies the possibility that
stress waves can be directed to desired regions in a specified path by
using tailored foam structures.

Adding a stiffer foam material near the location of load application can
reduce deformations of the whole structures and the kinetic energy
distributed to the structure. Foam bars and sandwich foam beam models
with such configurations demonstrated smaller deformations and kinetic
energy. This offers the possibility of optimizing foam structures to
maximize their ability of escaping from severe damage as stand-alone or
protect structures.

The energy dissipation efficiency of a given configuration of tailored
foam structure is dependent on the load magnitude. For instance, Model 2
of the sandwich beam studies dissipated the most energy under a lOMPa
compressive impulse pressure btlt it dissipated the least amount of energy
when the pressure load was lMPa. This is an important consideration for

the optimization of energy dissipating structures in future studies.

118

6.2 Recommendations for Future Work

Recommendations for future work are offered on three directions: experiments on
sandwich foam beams, improved numerical simulations, and design optimization. The
first and most important one is to perform experiments on sandwich foam beam model
with reinforced foam regions to validate the findings from in this study. In addition,
improving the numerical simulation on sandwich beam models can provide more
precise results. Finally, it is necessary to develop an approach of optimizing the
disposition of the reinforced foam regions for future study to offer guidance for the

manufacture and design of experiments.

6.2.1 Experiments on Sandwich Foam Beams

In this study, cross section designs for the four sandwich foam beam models were
as follows: a homogeneous sandwich foam beam, an arch reinforced foam sandwich
beam, a decreasing layered sandwich foam beam, and an increasing layered sandwich
foam beam. Of the three study cases corresponding to impulsive pressure loads of
lMPa, lOMPa and 100MPa, numerical simulations indicated that the sandwich beam
with a reinforced arch foam showed the features of directing the path of the stress
wave propagation. However, no experimental data or analytical solution is available
for this innovative design. Therefore, experiments on reinforced sandwich foam beams
are the most direct and important way of validating the conclusions derived from the
numerical simulations of this thesis. In addition, experiments on homogeneous
sandwich foam beams should be conducted, to obtain baseline data for comparison

purposes.

119

Dimension for experimental samples of the homogeneous sandwich beam can be:
72 mm in length, 24 mm in width, a foam core thickness of 24 mm, and 1.5 mm thick
face sheets. The width of the beam should be in the order of 24 mm. Reinforced foam
designs should adopt the similar cross-section dimensions. The experimental program
may include the following. Impact pressure test may be performed by drop tests. The
relation between the generated pressure loads in the drop test should be well
established. In addition, the sandwich beam sample should be simply-supported and
constrained by a mold frame to prevent lateral displacements. Finally, measurement
equipment should be provided to obtain the information of the stress, reaction forces,

displacements, and kinetic and internal energy distribution.

6.2.2 Improvements on Numerical Simulation

Studies on sandwich beam models need to be modified to more realistically
simulate the interaction properties between dissimilar foam materials and between
foam materials and aluminum plates close. By modifying the interaction definition
between at these interfaces the numerical models will be able to better assess
delamination that may occur at these locations. Similar work has been done by (Bahei-
El-Din et al. 2006). An additional consideration is to provide elasto-plastic constitutive
properties to the face sheets in sandwich beam models in order to better capture
localized failure and input load to the foam core. These improvements will increase
the fidelity of the numerical simulations and offer better guidance to further

experiment studies.

6.2.3 Design Optimization

The studies conducted in this thesis assumed materials properties based on
existing analytical and experimental information, and the configurations for the
sandwich foam beam models were arbitrarily designed based on engineering
judgments. Therefore, in order to maximize the potential of stress wave management,
further research on design optimization should be conducted in three directions: the
material’s properties, combinations of dissimilar materials and configurations of of
reinforced foam in sandwich structures.

In the presented work, the material used to simulate foam behavior was
aluminum. Other materials such as steel and ceramic can also be used to build
reinforced foam because they have relatively high yield stress and could be stiffer. In
addition, other foam materials such as functionally graded metal materials (FGM) and
functionally graded foam material (FGF) can be introduced in the future because FGM
and FGF can be designed and manufactured to meet the desired demands. Variations
in material properties are possible based on specified functions (linear or non-linear)
on mass density, Young’s modulus, and relative density for FGF. The vast variation
on material properties offers many possibilities for optimization.

Optimization can also be conducted on combinations between dissimilar
materials. The combination of solid materials, foam materials, FGM and FGF, offer
many combination possibilities that can be obviously optimized for improved behavior.
Promising combinations should be investigated through optimization algorithms and

experiments.

121

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