CRYSTAL PLASTICITY FINITE ELEMENT ANALYSIS OF
DEFORMATION BEHAVIOUR IN
SAC305 SOLDER JOINT
By
Payam Darbandi
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mechanical Engineering-Doctor of Philosophy
2014
ABSTRACT
CRYSTAL PLASTICITY FINITE ELEMENT ANALYSIS OF
DEFORMATION BEHAVIOUR IN
SAC305 SOLDER JOINT
By
Payam Darbandi
Due to the awareness of the potential health hazards associated with the toxicity of lead (Pb),
actions have been taken to eliminate or reduce the use of Pb in consumer products. Among
those, tin (Sn) solders have been used for the assembly of electronic systems. Anisotropy is of
significant importance in all structural metals, but this characteristic is unusually strong in Sn,
making Sn based solder joints one of the best examples of the influence of anisotropy.
The effect of anisotropy arising from the crystal structure of tin and large grain microstructure on
the microstructure and the evolution of constitutive responses of microscale SAC305 solder
joints is investigated.
Insights into the effects of key microstructural features and dominant plastic deformation
mechanisms influencing the measured relative activity of slip systems in SAC305 are obtained
from a combination of optical microscopy, orientation imaging microscopy (OIM), slip plane
trace analysis and crystal plasticity finite element (CPFE) modeling.
Package level SAC305 specimens were subjected to shear deformation in sequential steps and
characterized using optical microscopy and OIM to identify the activity of slip systems.
X-ray micro Laue diffraction and high energy monochromatic X-ray beam were employed to
characterize the joint scale tensile samples to provide necessary information to be able to
compare and validate the CPFE model.
A CPFE model was developed that can account for relative ease of activating slip systems in
SAC305 solder based upon the statistical estimation based on correlation between the critical
resolved shear stress and the probability of activating various slip systems.
The results from simulations show that the CPFE model developed using the statistical analysis
of activity of slip system not only can satisfy the requirements associated with kinematic of
plastic deformation in crystal coordinate systems (activity of slip systems) and global coordinate
system (shape changes) but also this model is able to predict the evolution of stress in joint level
SAC305 sample.
ACKNOWLEDGMENTS
I would like to express my deepest appreciation to my advisors, Dr. Farhang Pourboghrat and Dr.
Thomas Bieler, for all their supports, guidance, and encouragement during my PhD study. I
would also like to thank the members of my PhD committee; Dr. Alejandro Diaz, and Dr.
Thomas Pence for their constructive criticism and suggestions during the course of this work.
I would also like to acknowledge Dr. Carl Boehlert and Dr Tae-kyu Lee for their valuable
assistance and support during this work.
My colleagues; Dr. Bite Zhou and Quan Zhou are greatly appreciated for their help and
suggestions on this research.
Most importantly, I would like to express my greatest appreciation to my parents and Farnaz for
their encouragement through the years.
iv
TABLE OF CONTENTS
LIST OF TABLES ............................................................................................................ vii
LIST OF FIGURES ......................................................................................................... viii
CHAPTER 1
INTRODUCTION .............................................................................................................. 1
CHAPTER 2
LITERATURE REVIEW ................................................................................................... 5
2.1.1 Crystal structure and physical properties of tin ................................................. 7
2.1.2 Activity of slip systems in tin: ........................................................................... 8
2.1.3 Microstructural considerations......................................................................... 10
2.1.4 Importance of slip in SAC 305 alloys .............................................................. 17
2.1.5 Effect of aging on the mechanical properties of SAC 305 solders .................. 20
2-2- Characterization techniques ...................................................................................... 24
2.2.1 Polarized Light Microscopy (PLM) ................................................................. 24
2.2.2 - EBSD and polarized light microscopy .......................................................... 28
2.2.3 Differential Aperture X-ray Microscopy (DAXM) ......................................... 29
2.2.4 2-D Radiography............................................................................................. 32
2.3 Modeling approaches used in solder joints ................................................................. 34
2.3.1 Comparison of the phenomenological basis for different models ................... 38
2.3.2 -Explicit integration algorithms: ...................................................................... 39
2.3.1 -Implicit integration algorithms: ...................................................................... 40
2.3.3 -Semi-implicit integration algorithms: ............................................................. 40
2.4. Summary of the literature review ...........................................................................................48
CHAPTER 3
CRYSTAL PLASTICITY MODEL DESCRIPTION ...................................................... 49
3.1.Orientation matrices in crystal structures: .................................................................. 57
CHAPTER 4
EXPERIMENTAL PROCEDURES ................................................................................. 59
4.1.Single joint tensile samples ......................................................................................... 59
4.1.1. Sample preparation for joint level tensile test samples:.................................. 59
4.1.2.Tensile test set up ............................................................................................. 60
Figure 4.4 Set up employed for tensile testing, (a) RSA III equipment. .................. 64
4.2.1 Shear samples................................................................................................... 65
4.3 Procedure for OIM characterizations: ......................................................................... 65
4.4 X-ray diffraction ........................................................................................................ 67
4.4.1- Beamline 34-ID-E........................................................................................... 67
4.4.2- Beamline 6-ID-E............................................................................................. 68
v
CHAPTER 5
CRYSTAL PLASTICITY FINITE ELEMENT ANALYSIS OF DEFORMATION
BEHAVIOR IN MULTIPLE-GRAINED LEAD-FREE SOLDER JOINTS ................... 71
5.1 Introduction ................................................................................................................. 71
5.2 Single shear lap simulations........................................................................................ 76
5.3. Tri-crystal simulations ............................................................................................... 81
5.4 Summary ..................................................................................................................... 93
CHAPTER 6
CRYSTAL PLASTICITY FINITE ELEMENT STUDY OF DEFORMATION
BEHAVIOR IN COMMONLY OBSERVED MICROSTRUCTURES IN LEAD FREE
SOLDER JOINTS ............................................................................................................. 94
6.1. Modeling of single crystal solder balls using CPFE .................................................. 99
6.2. Modeling of heterogeneous plastic deformation in Beach-ball microstructure of
solder joints .................................................................................................................. 104
6.3. Summary .................................................................................................................. 109
CHAPTER 7
SLIP SYSTEM ACTIVITIES OF SAC305 SOLDER BALLS UNDERGOING SIMPLE
SHEAR DEFORMATION ............................................................................................. 110
7.1. Experimental procedures ......................................................................................... 112
7.2. Slip-trace analysis and Schmid factor calculation ................................................... 114
7.4. Summary .................................................................................................................. 141
CHAPTER 8
EXPERIMENTAL AND NUMERICAL ASSESSMENTS OF TENSILE TEST ON
JOINT SCALE SAC 305 SAMPLES ............................................................................. 143
8.1 Introduction ............................................................................................................... 143
8.2 Statistical analysis on the activity of slip systems .................................................... 148
8.3 Experimental results.................................................................................................. 154
8.4 Simulation results: .................................................................................................... 163
8.5 Case study- CPFE modeling of sample A12 (in Figure 8.4) .................................... 167
8.6 Prediction capability of CPFE model base on new parameters based on the calibration
of tensile test ................................................................................................................ 170
8.7 Summary and conclusion .......................................................................................... 170
CHAPTER 9
CONCLUDING REMARKS AND FUTURE WORKS ................................................ 177
9.1 Concluding Remarks ................................................................................................. 177
9.2 Recommendation for future works ........................................................................... 178
BIBLIOGRAPHY ........................................................................................................... 181
vi
LIST OF TABLES
Table 2.1 Commonly observed Sn slip systems and schematic of slip systems ............... 14
Table 2.2 Summary of the main phenomenological equations used in solder joints ........ 43
Table 2.3 Summary of different modeling approaches that employed to model the
mechanical behavior of lead free solders .......................................................................... 47
Table 5.1 Elastic Constants (GPa) of Tin used in numerical analysis .............................. 77
Table 5.2 Elastic constants (GPa) of Cu. .......................................................................... 77
Table 5.3 Hardening parameters of tin for different slip systems used in numerical
analysis .............................................................................................................................. 77
Table 5.4 Element Types and number of elements used in CPFE .................................... 77
Table 5.5. Bunge Euler Angle values for different tri-crystal sets ................................... 82
Table 7.1 Test conditions for performing the simple shear tests .................................... 113
Table-8.1 Euler angles value for samples which are characterized using beamline 34
(labeled by ―A‖) and beamline 6 labeled with ―B‖......................................................... 158
Table 8.2 Element Types and number of elements used in CPFE. ................................. 165
Table 8.3 Elastic Constants (GPa.) of Tin used in numerical analysis. .......................... 165
Table 8.4 Hardening parameters of tin for different slip systems used in numerical
analysis. ........................................................................................................................... 165
Table 8.5 Critical resolved shear stress calculated based on statistical analysis ............ 166
vii
LIST OF FIGURES
Figure 2.1. Variation of Young‘s modulus on the (100) plane (solid line) and (110) plane
(dashed line), compared with CTE (units are in legend). ................................................. 11
Figure 2. 2. Variation of CTE along the (100) direction and plane and (001) direction
(Bieler et al. (2012)). ......................................................................................................... 11
Figure 2.3. Tin unit cell (c/a = 0.5456) (Bieler et al. (2012)). .......................................... 12
Figure 2.4. Schematic representation of difference between CTE in the interface plane for
(a) c-axis parallel to the interface (b) a-axis is parallel to the interface ............................ 13
Figure 2.5 (a) Optical micrograph showing the slip trace for a sample that was subjected
to creep at 48 MPa (Zhang(2012)) and,(b) BE image of slip traces after 1 × 103/s
constant-rate creep test at 50°C to shear strain equals to 0.38 (Hertmeyer et al. (2009) )
(c) Magnified micrograph showing slip traces were developed after 112Thermal cycling))
(Zhou et al.(2012)) ............................................................................................................ 16
Figure 2.6 Polarized light image and corresponding c-axis orientation maps and pole
figures showing a thermo-mechanically cycled single crystal (left), and a tri-crystal
showing the 60 about [100] axis twin relationship in SAC305 solder joint (right) . In the
single crystal, the white lines delineate low-angle boundaries, and orientation gradients
are apparent in the spread peaks on pole figures (Bieler et al. (2008)).. .......................... 18
Figure 2.7 Schematic diagrams showing the experimental set up for EBSD observation.
........................................................................................................................................... 26
Figure 2.8 Polarized light image showing the application of this method in
characterization of grains morphology in lead free solder balls (adapted from Lehman et
al. (2010)). ......................................................................................................................... 27
Figure 2.9 Schematic diagram showing the experimental set up for DAXM observation.
The incident X-ray beam can have either a polychromatic or monochromatic spectra. A
50 m m-diameter platinum wire is translated near the sample surface to decode the origin
of the overlapping Laue patterns....................................................................................... 31
Figure 2.10: Basic 2-D x-ray system configuration (Benard (2003)) ............................... 33
Figure 4.1 Schematic diagrams and dimensions of the jig used for polishing and
fabrication of joint level tensile samples. ......................................................................... 61
viii
Figure 4.2 Schematic configurations of tensile samples inside the miniature jig (a)
locations of polishing and cutting of copper wires are illustrated, (b) configuration of
solder inside the jig and on silicon plates. ........................................................................ 62
Figure 4.3 Temperature profiles used for fabricating solder joints .................................. 63
Figure 4.5 Schematic diagrams of a 4 × 4 solder joint array sectioned from a PBGA
package. The solder ball assembly geometry and shear test apparatus is also shown. .... 66
Figure 4.6 General procedures for conducting the EBSD analysis. ................................. 69
Figure 4.7 Experimental setup at synchrotron beamline 34-ID-E for conducting the
characterization of joint level tensile samples (a) platinum wire used as a differential
aperture. (b) Experimental station for conducting the synchrotron X-ray microdiffraction
(c) sample stage 45º inclined with respect to beam direction. .......................................... 70
Figure 5.1 Simplified model of a of tri-crystal microstructure in half-joint configuration,
with coordinate system used for Euler angles................................................................... 74
Figure 5.2 Geometry and dimensions of (a) lap-shear samples, and(b) solder balls used in
this study. The sense of shear is to the left on the upper surface of the joint. ................. 75
Figure 5.3 The orientation 1--2 = 75°-45°-0° was used to estimate crystal plasticity
model parameters using experimental data of Darveaux et al. (2005). Using these model
parameters, other orientations deformed differently in simulated single shear lap
deformation. The distribution of von Mises stress in the joint are shown to illustrate
effects of orientation on stress distribution. ...................................................................... 78
Figure 5.4 Shear strain in one element in the middle of the joint in the single shear lap
geometry. Some slip systems have conjugate partners that are 90 apart, and have the
same Schmid factors; these two distinct systems are illustrated with lines and overlaid
dots. ................................................................................................................................... 80
Figure 5.5 Average shear strain on the most active slip systems are compared in different
orientation sets A,B,C, to illustrate the effects of location and interaction between
different orientations. Slip system facility is based upon Table 2.1 Each row has the
same crystal orientation (Table 5.4), but in a given row, the upper right grain (position 1,
on the diagonal), exhibits different strain evolution than when the same orientation is in
the other two positions. ..................................................................................................... 85
Figure 5.6 Average shear strain on the most active slip systems are compared in different
orientation sets A,B,C, to illustrate the effects of location and interaction between
different orientations. Slip system facility is based upon Table 5.3, but slip system 9 is
suppressed. Each row has the same crystal orientation (Table 5.5), but in a given row, the
upper right grain (position 1, on the diagonal), exhibits different strain evolution than
when the same orientation is in the other two positions. .................................................. 86
ix
Figure 5.7. When slip system 9 <101]{10-1) is suppressed, to be more consistent with
experimental observations ( Zhou et al (2009)), activation on slip systems 4<111>{110)
and 10 <211]{-101) increases. .......................................................................................... 87
Figure 5.8 Distribution of von-Mises stress in different tri-crystals deformed to a simple
shear strain of 0.84 with orientations defined in table 5.5 is rather similar. The grain
orientation 120-90-0 in the upper left position resists shear more effectively than other
orientations. ....................................................................................................................... 88
Figure 5.9 Distribution of von-Mises stress at a strain of 0.5 in different tri-crystals
with orientations defined in table 5.5 differs when slip system 9 is suppressed. The grain
orientation 120-90-0 in the upper left position in orientation sets B and C resists shear
more effectively than other orientations. .......................................................................... 89
Figure 5.10 .Shear deformation to a simple shear strain of 0.42 where slip systems 9
<101]{10-1) and 10 <211]{-101) are suppressed to be more consistent with experimental
observations (Zhou et.al (2009). Differential strains at the grain boundary on the right are
evident in orientation sets A and C (compare with Figure 2.6) ....................................... 91
Figure 5.11. The generalized Schimd factor (computed from the stress tensor in each
element) varies substantially with the crystal orientation in the elements along the line in
the upper left grain orientation in orientation set C in Figure 5.10. Slip system
4<111>{110) has the greatest resolved shear stress and is greater than on suppressed slip
systems 9, <101]{10-1), and 10 <-101]{121). .................................................................. 92
Figure.6.1 Schematic diagrams of a 3-D visualization of the CPFE mesh located in a the
4 × 4 solder joint array sectioned from a PBGA package. The solder ball assembly
geometry and shear test apparatus is also shown. ............................................................. 97
Figure 6.2. Optical micrographs of 2 different solder balls with corresponding c-axis
orientation maps with respect to the substrate normal direction, before (a-d) and after (eh) 0.65 mm shear displacement. After shear, strains are concentrated in upper right
regions. The c-axis orientation was retained (no color change) as the crystal rotated
about the c-axis (note overlaid unit cell prisms) it during shear. .................................... 100
Figure.6.3. Comparison of simulation and experimental results for single crystal
deformed solder joints illustrated in Figure 6.2. 565 C3D10M elements were used in (a)
and 569 C3D10M elements were used in (b).................................................................. 101
Figure 6.4. (a) Optical image of joint b in Figures 6.2 and 6.3, showing local unit cell
orientations with shaded slip planes and corresponding plane traces, and slip vectors
(blue lines) with high Schmid factors (there is some evidence for (121) slip in the highly
sheared upper right part of the joint). Red and yellow-green edges of the unit cells
represent the crystal x and y axes, respectively. (b) Comparison of the average activity in
the simulation of the four most active slip systems at each integration point in the single
crystal joint...................................................................................................................... 103
x
Figure 6.5. Pole figures obtained from OIM for two solder joints with beach ball
microstructure (a,b), FE meshes colored according to the c-axis color scale in the vertical
direction before deformation (c,d) and after deformation, (e,f) showing shape after
indicated displacements were imposed and released. 2550 C3D10M elements were used
in (c,e) and 2600 C3D10M elements were used in (d,f). ............................................... 105
Figure.6.6 (a) X direction c-axis orientation maps for beach ball microstructure before
deformation and, (b) after 0.1mm shear, backscattered SEM image tilted -45 about X
axis showing a ledge in the lower left area (c), and a path along which misorientation and
topography are traced in the experiment (b). A similar trace in the CPFE simulation (d),
is plotted with experimental data in (e). The CPFE model is geometrically simpler and
does not include the material indicated by the dotted line in (d), but it is able to semiquantitatively capture the localized deformation observed experimentally in the lower left
corner. ............................................................................................................................. 106
Figure.6.7. Polarized light micrograph (a) of beach ball microstructure that experienced a
large shear displacement of 0.4 mm (red line), deformed solder ball predicted by CPFE
illustrating distribution of shear stress in the 1-2 plane indicating a higher stress due to
more shear localization in the upper area. ...................................................................... 107
Figure 7.1 Sn grain orientation color code used with OIM c-axis maps to identify the caxis inclination from the interface. ................................................................................. 117
Figure 7.2 Load displacement diagrams for step 2 of shear deformations at (a) room
temperature (samples 1 and 3 are aged and samples 5 and 7 are non-aged) for (b) aged
samples 100ºC (samples 2 and 4 are aged and samples 6 and 8 are non-aged). ............. 118
Figure 7.3 Load displacement diagrams for step 3 of shear deformation at (a) room
temperature (samples 1 and 3 are aged and samples 5 and 7 are non-aged), and for (b)
aged samples 100ºC (samples 2 and 4 are aged and samples 6 and 8 are non-aged). .... 119
Figure 7.4.Optical micrographs of 4 cross sectioned samples in in the pre-aged condition,
showing the deformation in three steps at room temperature (a,c) and at 100ºC (b,d)... 120
Figure 7.5 Optical micrographs of 4 rows of cross sectioned samples in the unaged
condition, showing the effects of deformation in three steps at room temperature (a,c) and
at 100ºC (b,d). ................................................................................................................. 121
Figure 7.6 PLM micrographs of 4 rows of cross sectioned samples in the pre-aged
conditions, showing the effects of deformation in three steps at room temperature (a,c)
and at 100ºC (b,d). .......................................................................................................... 122
Figure 7.7. Cross-polarized light micrographs of 4 cross sectioned samples in non-aged
samples, showing the effects of deformation in three steps at room temperature (a,c) and
at 100ºC (b,d). ................................................................................................................. 123
xi
Figure 7.8. C-axis EBSD maps of 4 rows of cross sectioned samples in the pre-aged
condition, showing the effect is of deformation in three steps at room temperature (a,c)
and at 100ºC (b,d). .......................................................................................................... 125
Figure 7.9 C-axis EBSD maps of 4 cross sectioned samples in the unaged condition,
showing effects of deformation in three steps at room temperature (a,c) and at 100ºC
(b,d). ................................................................................................................................ 126
Figure 7.10 Optical micrograph of a solder ball showing slip planes in the right and left
areas, (b) c-axis EBSD map corresponding to this solder ball. ...................................... 127
Figure 7.11 (a) inset showing magnified BSE images of an area shown in optical
micrograph (b) of sample depicted in (c) to show how the crystal orientation is related to
the x-axis slip vector in the ―orange‖ area, (d) c-axis EBSD map. ................................. 129
Figure 7.12 (a) c-axis EBSD map for deformed sample after step-1,(b) c-axis EBSD map
for deformed sample after step-2, low angle boundaries are illustrated with white lines
and high angle grain boundaries are shown with black lines.......................................... 130
Figure 7.13. Vertical direction c-axis orientation map in large deformation regions of
solder joints for three samples with different c-axis orientations and different steps of
deformation showing different deformation behavior (a-c). Difference in distribution of
misorientation for three orientations in (a-c) showing an increase in the amount of (1525º) grain boundaries after the large deformation step for blue and red orientations and a
decrease in yellow-green orientation (d)......................................................................... 132
Figure 7.14. Difference in distribution of misorientation for three orientations in (a-c)
showing increase in amount of (15-25º) grain boundaries after large deformation for
blue and red orientations and decrease in green orientation. .......................................... 133
Figure 7.15 Fine-step EBSD c-axis orientation map and image quality maps for step1(a,b), step-2 (c,d), step-3(e,f ) for an unaged sample with blue orientation deformed at
100 ºC, where low angle boundaries are illustrated with white lines and high angle grain
boundaries are shown with black lines. Rotation of crystal orientation at different
locations (as illustrated with red arrows) due to shear deformation in step-3 (g),
evolutions of c-axis pole figures showing the spread in crystal orientations (h). ........... 134
Figure 7.16 Fine-step EBSD c-axis orientation map and image quality maps for step1(a,b), step-2 (c,d), step-3 (e,f) for an unaged sample with a red orientation deformed at
100 ºC, low angle boundaries are illustrated with white lines and high angle grain
boundaries are shown with black lines, crystal rotation at different locations (as
illustrated with red arrows) due to shear deformation in step-3 (g), evolutions of c-axis
pole figures showing the evolutions of crystal orientations (h). ..................................... 136
Figure 7.17 Fine-step EBSD c-axis orientation map and image quality maps for step1(a,b), step-2 (c,d), step-3(e,f) for a bi-crystal unaged sample deformed at 100 ºC, low
xii
angle boundaries are illustrated with white lines and high angle grain boundaries are
shown with black lines. ................................................................................................... 137
Figure 7.18 Fine-step EBSD c-axis orientation map and image quality maps for step1(a,b), step-2 (c,d), step-3(e,f) for an aged bi-crystal sample deformed at room
tempereture. Low angle boundaries are illustrated with white lines and high angle grain
boundaries are shown with black lines. .......................................................................... 139
Figure 7-19. Number of observations of slip on 32 polished half-joints after room
temperature shear deformation, normalized by the number of slip systems in the family,
and separated by the c-axis orientation. Black bars are the sum (divided by 2 to be on a
similar scale) of all observed slip activities on joints with c-axis orientations indicated by
the colors. ........................................................................................................................ 140
Figure 8.1 Flowchart showing the methodology of CPFE modeling of SAC 305 tensile
test of SAC 305 joint scale samples using the statistical analysis of slip system activities.
......................................................................................................................................... 147
Figure 8.2 (a) Probability of observation of each slip system in data set that was studied
in chapter 7. (b) Critical resolved shear stress that is estimated using methodology that is
described in Figure 8.1 .................................................................................................... 149
Figure 8.3 (a) Dislocation velocity versus resolved shear stress for different single
crystals (Meyers (1984)), (b) Stress dependence of the passage rate of glide dislocations
through obstacles in tin (Fujiwara (1987). ...................................................................... 151
Figure 8.4 Radiographic micrographs of tensile samples which are characterized using
beamline 34 before (a) and after (b) deformation. Voids are shown as bright spots. ..... 155
Figure 8.5 Radiographic micrographs of tensile samples which are characterized using
beamline 6 before (a) and after (b) deformation. Voids are shown using bright spots. .. 156
Figure 8.6 Orientation image for samples A in Figure 8.4 which are characterized using
beamline 34 before deformation. .................................................................................... 159
Figure 8.7 Force –displacement curves for tensile sample set A, which were partially
characterized using beamline 34. The crystal orientation is overlaid close to each
mechanical response to show the correlation between orientations and mechanical
responses. ........................................................................................................................ 160
Figure 8.8 Force–displacement curves for different tensile samples which are partially
characterized using beamline 6. The crystal structure is overlaid close to each
mechanical response show the correlation between orientations and mechanical
responses. ........................................................................................................................ 162
xiii
Figure 8.9 Simplified geometry and dimensions of of sample A 12 in Figure 8-5 , with
coordinate system used for Euler angles. ........................................................................ 168
Figure 8.10 Distribution of von-Mises stress in sample-12 obtained using the CPFE
modeling. Localization of the stress and rotation of the sample is predicted. ................ 172
Figure 8.11 Comparison of the average activity in sample 12 predicts three active slip
systems, with one dominant, during tensile deformation. ............................................. 173
Figure 8.12 Force displacement for two tensile deformed samples 6 and 12 ( solid lines)
dashed lines shows the simulation results. Curve fitting was conducted based on the
assessments on the relative activity of slip systems....................................................... 174
Figure 8.13 SEM image of sample 12 shows the slip plane traces on the surface, sample
rotation and shear localization after shear deformation. Plane traces (magenta line)
shows that slip system mode 2 is the most active slip systems (a) CPFE simulation of
sample 12 predicts that slip system mode 2 is the most active slip system (b). Crystal
structure of tin shows the joint orientation and slip trace (c). ......................................... 175
Figure 8.14 Deformed solder ball predicted by CPFE utilizing the material parameters
tabulated in Table 8.5 indicating more realistic values of stress compared to Figure 6.7 (a)
Comparison of the outline of the model illustrated in (a) and Figure 6.7 (b) indicating
the similarity in kinematic of deformation predicted by both models. Polarized light
micrograph of beach ball microstructure that was modeled in (a) and (b). ................... 176
xiv
CHAPTER 1
INTRODUCTION
Due to the awareness of the potential health hazards associated with the toxicity of lead, actions
have been taken to eliminate or reduce the use of Pb in consumer products. Among the resulting
changes, tin solders are now used for the assembly of electronic systems.
Anisotropy is of significant importance in all structural metals, but this characteristic is
unusually strong in Sn, making Sn based solder joints one of the best examples of the influence
of anisotropy. The mechanical properties and damage evolution in SAC 305 alloys strongly
depend on the underlying microstructures. The existence of single or multi-grain microstructure
in this alloy causes each joint to exhibit unique mechanical response and stress and strain
history. An outcome of the inhomogeneous behavior of microscale SAC joints is reflected in
their thermal cycling fatigue behavior. In general, it is expected that the first joint to fail will be
located at the die or package corner where extrinsic shear strains arising from coefficients of
thermal expansion (CTE) mismatch are the largest.
But in reality, failed joints are not
necessarily at the above stated regions. This issue is especially important for package designers.
While design engineers look for simple isotropic models to use for package design, this
approach is by no means effective for examining damage nucleation processes arising from the
large grain microstructure and the anisotropic properties of Sn.
Given the wide range of initial microstructures that SAC 305 alloys can have, in order to conduct
an experimental analysis across the entire space of feasible microstructures one needs to have
unlimited resources in terms of time and cost. Therefore, developing a material model which is
1
sensitive to the microstructure and the initial grain orientation is particularly desirable for
electronic assembly industry.
In addition to the complexity of the microstructure, the stress state that electronic components
can experience is quite complicated.
In service conditions, mechanical loads on the packages
can vary from high temperature, low strain-rate loads such as thermal cycling, to high strain-rate
loads at relatively low temperatures, such as in vibration, shock, drop and mechanical cycling at
ambient temperatures [Abtew and Selvaduray (2000)].
This dissertation aims to provide an improved understanding of the modeling of anisotropy in the
elastic-plastic behavior of SAC solder at sub-mm length scales, using a combination of
microstructural and mechanical characterization and crystal plasticity finite element (CPFE)
modeling approaches. The results of this study are applicable to the mechanical response of
solders in low temperature, high strain-rate loading conditions where plastic deformation
dominates over creep deformation mechanisms. This is particularly important for modeling of
accelerated thermal cycling tests in which deformation based on slip mechanisms are dominant
and are in common within the industry.
Specific objectives of this dissertation, regarding the unresolved research issues, are to:
1-Propose a mechanistic framework based upon the crystal plasticity finite element modeling of
idealized geometries to provide insights into the effect of initial coarse-grained Sn
microstructures and orientations on joint dependent deformation behavior of SAC305 solders.
2- Investigate the activities of slip systems in SAC 305 alloys utilizing the orientation imaging
microscopy, SEM, and plane trace analysis on prior polished shear deformed samples to assess
the effect of initial crystal orientation and evolved crystal orientations on anisotropic plastic
deformation arising from the activity of different slip systems in different locations.
2
3- Examine the predictive capabilities of the CPFE model to capture the kinematics of
inhomogeneous plastic deformation in microstructures that are commonly observed in real
SAC305 solders, and to assess the relative activity of different slip systems and the evolution of
microstructural features.
4- Correlate the mechanical properties, initial microstructure, and crystal orientation based on the
CPFE modeling of tensile deformation of joint level samples characterized using 3D x-ray
method.
A detailed literature review on the microstructural features and deformation mechanisms related
to the lead free solder alloys is given in chapter 2. This chapter also describes the most
significant material models that are used to model the mechanical behavior of lead free solders.
The numerical framework regarding the crystal plasticity finite element method is detailed in
chapter 3. Chapter 4 describes the experimental procedures that are used in this study
Chapter 5 focuses on the applicability of the CPFE model to study the impact of external and
internal constraints associated with the tri-crystal microstructure of lead free solder balls on the
activity of slip systems. The correlation between the crystal grain orientation and evolution of
microstructural features and activity of slip systems using CPFE, OIM, and comparative studies
is given in chapter 6.
A statistical analysis on the relative activity of slip systems based on the orientation imaging
microscopy method to characterize 32 shear deformed samples is detailed in chapter 7. An
experimental and CPFE study on the plastic deformation of microscale tensile samples and
variability in the plastic deformation response under identical loading histories due to the crystal
orientation is reported in chapter 7.
3
Chapter 9 summarizes the results, provides conclusions, and describes needed future work.
Chapter 5 is extracted from a paper that is published in the Journal of Electronic Materials.
Chapter 6 also is extracted from a paper that is submitted to the Journal of Computational
Materials Science.
The results of chapter 7 will be submitted to Materials Science and
Engineering A.
4
CHAPTER 2
LITERATURE REVIEW
Due to actions that have been taken to eliminate or reduce the use of lead (because of health risk
associated with the toxicity of this metal ) there has been a great amount of interest in using lead
free solder alloys within the electronic assembly industry in last several years.
Reliability is one of the most important concerns regarding the elimination or reduction of Pb
based solders. The most important issue with Pb-free soldering is to replace the Sn-Pb solders
with alloys that have equivalent mechanical properties, satisfy the standards developed for Sn-Pb
soldering, and to insure the reliability of these alloys. Parameters that are important in this
regard are the coefficient of thermal expansion, elastic modulus, yield strength, shear strength,
fatigue and creep behavior.
Among different choices, the tin-silver-copper (Sn-Ag-Cu or SAC) family of alloys has earned a
great deal of positive response from the electronic assembly industry in recent years. Although
this family of alloys has relatively low melting points, good reliability, and reasonable cost, there
are significant differences between these alloys and Sn-Pb solder alloys. In lead-tin based solder
alloys the prediction of failure is straightforward, but due to the anisotropy of tin and the more
complicated crystal structure of this metal, it is very difficult to predict its damage initiation and
evolution. This issue is clearly reflected in thermal cycling fatigue behavior of Sn based solder
alloys. The highest stressed solder ball, which according to the simple mechanical calculations
could be the one at the package or die corner, is not necessarily the first joint to fail during the
thermal cycling of Plastic Ball Grid Arrays (PBGA).
5
This unexpected failure location of Pb-free solder joints creates special challenges for package
designer who in the past successfully used the isotropic material models for Sn-Pb solders.
In this regard, it is important to assess the differences between the Pb- free solders and Sn-Pb
alloys in the light of metallurgical considerations. The main difference between Sn-Pb and Sn
based alloys is that soft and isotropic Pb in Sn-Pb alloys can dramatically reduce the anisotropic
effects of Sn.
Furthermore, the anisotropy in the coefficient of thermal expansion (CTE) plays an important
role in the failure process. Figure 1.2 illustrates the anisotropy in the CTE. The crystal structure
of tin is BCT (a squashed diamond cubic structure), as shown in Figure 1.3. The coefficient of
thermal expansion is greatest in the c-axis, and the stiffness is also highest in this direction.
Because of the different values of the CTE and stiffness in different directions, tin naturally is
conflicted at grain boundaries. So, the morphology of grains and grain boundary orientation can
have a strong effect on the mechanical properties of solders.
The stress state which electronic components can experience is also complicated. The main
source of this stress, however, comes from the fact that electronic components and their
supporting board have different coefficients of thermal expansion. Therefore, as the temperature
rises, the board expands quite differently than electronic components mounted on it. This gives
rise to a state of shear deformation, in turn developing shear stresses and strains in connecting
joints. Especially, as electronic circuits pass current on and off, the solder ball becomes
subjected to thermal cycling loads, which also lead to cyclic shear stresses.
Furthermore,
bending that may occur due to mechanical boundary conditions imposed on the electronic board
can generate a state of tensile or compressive stress in the connecting solder ball. In the
6
automotive industry, vibration is of significant importance and in these applications; fatigue
failure has to be considered. Vibrational failure is one of the most significant failure modes in
electronic packages during their service life. In summary, in service, mechanical loads on
packages can vary from high temperature, low strain-rate loads such as thermal cycling, power
cycling, quasi-static isothermal mechanical cycling at elevated temperatures, to high strain-rate
loads at relatively low temperatures, such as vibration, shock, drop and mechanical cycling at
ambient temperatures [Abtew and Selvaduray (2000)].
There are two levels of packaging that solders are utilized in electronic assembly industry. At
the first level, solder can assist in bonding of a die to substrate to provide the electrical
connection. In the next level of assembly, the electronic component is mounted on a printed
wiring board (PWB) using solders. One of the more common methods of surface mounting is to
use solder balls, which are arranged in ball grid arrays (BGA) between the package and the
board.
2.1.1 Crystal structure and physical properties of tin
The bct unit cell is shown in Figure. 2.3. For tin crystal structure a = 0.58194 nm and c =
0.31753 nm (c/a = 0.54564). As it is apparent, there are 8 atoms at the corners, one atom at the
center and four atoms on the four faces (4 atoms per unit cell). The lattice is a distorted diamond
cubic lattice (when c/a =
it is equivalent to the diamond cubic lattice Yang and Li (2007)).
The coefficient of linear thermal expansion at room temperature is 15.4 ×10–6 along the a-axis
and 30.5 ×10–6 along the c-axis. As is shown in Figure2.2, with increasing temperature, these
coefficients increase. The role of anisotropy in CTE in developing inhomogeneous shear
deformation is schematically illustrated in Figure 2.4. Since the coefficient of thermal expansion
7
in copper is about 15.4 ×10–6 , when the c-axis is parallel to the interface there is a maximum
difference in the coefficient of thermal expansion between solder and the board. Subsequent
thermal cycling can lead to oscillation in internal strains and cause grain boundary sliding,
dislocation creep, and plastic deformation at values of stress that exceed the yield strength
(Bieler et al (2012)).
2.1.2 Activity of slip systems in tin:
A review about the plastic properties of tin by Yang and Li (2007) indicates the need for better
understanding of the deformation mechanisms of tin. There is incomplete understanding of slip
in Sn, though single crystals have been investigated by several researchers such as Fujiwara and
Hirokawa ( 1986, 1987) , M. Fujiwara (1997), Nagasaka (1999), Ekinci et al. (2003) , Düzgün et
al. (1999) , Kouhashi and Koenronbunshu (2000) , Honda (1978,1979), Obinata and Schmid
(1933), Polanyi and Schmid (1925) , Lorentz (1968), Fiedler and Lang (1972), Fiedler and Lang
(1975) , Vook (1964) , Chu and Li (1979) , Obinata and Schmid (1933), Zhou at al. (2009)
Matin et al. (2006) , Sidhu (2008) Chu and Li (1979), Weertman and Breen (1956) , Kirichenko
and Soldatov( 1982) , Ojima and Hirokawa (1983), Nagasaka (1989), Düzgün and Aytas
(1993), Mark and Polanyi (1923), Telang and Bieler (2007, 2009) and Kinoshita et al.(2012) .
There is especially uncertainty about the critical resolved shear stress for different slip systems.
Evidence presented by Düzgün et al. (1999, 2003), suggests that the initial critical resolved shear
stress is similar on different slip systems, but they have different rates of hardening. In these
studies, a limited set of crystal orientations have been investigated using tensile tests, and also
the effect of alloying and strain rate is unknown. Table 2.1 presents an estimate of the relative
ease of activating the 10 relevant slip systems based upon an assessment from the literature data.
8
The schematic of deformation in these slip systems in the tin crystal structure is illustrated in this
table. This table which was developed by Fujiwara and Hirokawa (1987) using the etch-pit /
hillock study is not extensive. Other studies by Fiedler and Vagera (1975) to find the relative
activity of slip systems are based on the minimum dislocation energy criteria which shows that
slip systems #1, #2, and #4 in Table1 are the most active slip systems.
In addition to these two methods, other researchers (Matin et al. (2006) and Sidhu and Chawla
(2008)) defined a parameter called ―effective yield strength‖ which is the value of the critical
resolved shear stress over the Schmid factor for different slip systems. In this method, the
authors used values of critical resolved shear stress for pure tin, which can be different from
SAC305 due to the effect of alloying elements. Friedel (1964) and Labusch (1970) showed that
the critical resolved shear stress increases with alloying elements. Especially, for some slip
systems, there are some assumptions made about the values of the critical resolved shear stress
that are based on the linear atomic density, which are not experimentally evaluated.
Kinoshita et al. (2012) recently used the first-principles density functional theory to study the
activity of slip systems in pure tin. They used the ideal shear resistance as the values of critical
resolved shear stress. They employed a computational tensile test study to assess the activity of
different slip systems and investigated the effect of crystal orientation on relative activity of slip
systems. Zhou et al. (2009) used a methodology based on the Orientation Imaging Microscopy
(OIM), and calculated the Schmid factor together with partial observation of slip traces to assess
the activity of slip systems on shear deformed samples. Their study implies that slip in [001] and
[111] directions is likely, and slip on (010)[101] may contribute significantly.
9
2.1.3 Microstructural considerations
Crystal features such as grain size and morphology have significant effects on mechanical
properties and the reliability of sub-mm scale joints. Studies about the microstructure and Sn
grain morphologies in lead free solder joints show peculiar microstructures, such as the beachball morphology, in which 60 cyclic twins are formed during the solidification (Lehman et al.
2004, 2010). Figure 2.6 illustrates two solder joints after thermal cycling (Bieler et al. 2008)).
A shear band of different orientation is developed in the middle of the left sample, and a bulge is
developed in the right sample along the upper edge of the grain boundary, which illustrates
heterogeneous deformation in two types of joint morphologies.
In a series of recent studies on the most commonly used lead-free solder joint, SAC305 (Sn3.0Ag-0.5Cu, wt%), microstructural issues were examined using polarized light image
microscopy, and orientation imaging microscopy (OIM), which showed that most joints are
single crystals or multicrystals with no more than a few Sn grain orientations (Bieler et al. 2009,
2012, 2011). Using transmission synchrotron x-ray diffraction patterns, this idea was proven by
observing that there are usually one or three orientations present in diffraction patterns (Bieler et
al. 2009). Alloying affects nucleation of Sn, leading to formation of single crystal or tri-crystals
depending on the concentration of Cu and Ag at the point of solidification (Matin et al. 2005,
Lehman et al. 2004). In multicrystals, there is usually a solidification twin relationship with
about 55 to 65 degree rotations about a common [100] axis (Figure 2.6).
10
Figure 2.1 Variation of Young‘s modulus on the (100) plane (solid line) and
(110) plane (dashed line), compared with CTE (units are in legend).
Figure 2. 2 Variation of CTE along the (100) direction and plane and (001)
direction (Bieler et al. (2012)).
11
Figure 2.3 Tin unit cell (c/a = 0.5456) (Bieler et al. (2012)).
12
CTE: ~15ppm/K
CTE: ~30.5 ppm /K
CTE: ~15.4ppm/K
Figure 2.4 Schematic representation of difference between CTE in the interface
plane for (a) c-axis parallel to the interface (b) a-axis is parallel to the interface
13
Slip system
Mode 1
{100)<001]
Table 2.1 Commonly observed Sn slip systems and schematic of slip systems
Characterization
Crystal structure References
method
Etch hillock, Energy
Barret, Fujiwara, Obinata and Schmid, Fiedler and Lang,
calculations, Plane
Fiedler and Vagera, Honda
trace
Mode 2
{110)<001]
TEM, energy
calculations
Mode 3
{100)<010]
Plane trace, Energy
calculations
Barret, Fujiwara, Chu and Li, , Tyte,Mark and polany
Sidhu and Chawla, Matin, Obinata and Schmid , Fiedler
and Lang Fiedler and Vagera
Honda, Vook
Fujiwara, Duzgun ,Ojima, Bausch, Chu and Li, Weertman
and Breen, Ojima and Hirokawa, Düzgün et al, Düzgün
and Aytaş , Nagasaka , Honda,
Mode 4
Energy calculation
{110)<1-11]/2 Etch hillock
Plane trace
Barret, Fujiwara, Chu and Li Weertman and Breen ,
Honda et al., Matin et al., Nagasaka et al., Fiedler and
Lang, Fiedler and Vagera, Honda, Zhou et al.
Mode 5
{110)<1-10]
Weertman and Breen
Energy calculation
14
Slip system
Mode 6
{100)<011]
Characterization
method
TEM, Plane trace
Table 2.1 (cont‘d)
Crystal structure References
Vook, Düzgün et al. , Düzgün and Aytaş ,
Fujiwara and Hirokawa, Nagasaka et al.
Fiedler and Vagera, Zhou et al.
Mode 7
{001)<010]
Energy calculations,
Plane trace
Schmid, Telang et al. , Fiedler and Vagera, Honda
Mode 8
{001)<110]
Plane trace
Schmid, Telang et al. ,Aytas
Mode 9
{011)<01-1]
TEM, Energy
calculations
Barret, , Vook, Sidhu and Chawla, Obinata and
Schmid, Fiedler and Vagera , Düzgün and Aytaş
Mode 10
{211)<01-1]
Plane trace
Düzgün et al. (1999), Sidhu and Chawla , Matin et al.,
Fiedler and Vagera , Düzgün and Aytaş , Zhou
15
a
b
c
Figure 2.5 (a) Optical micrograph showing the slip trace for a sample that was subjected
to creep at 48 MPa (Zhang (2012)) and,(b) BE image of slip traces after 1 × 103/s
constant-rate creep test at 50°C to shear strain equals to 0.38 (Hertmeyer et al. (2009) )
(c) Magnified micrograph showing slip traces were developed after 112Thermal
cycling)) (Zhou et al.(2012))
16
2.1.4 Importance of slip in SAC 305 alloys
Due to the complexity of the problem arising from the high homologous temperature of tin,
anisotropy, and the complexity of loading conditions in real electronic assembly products, it is
very difficult to develop a reliable model that takes into account many considerations. Below is a
list of important considerations that affect the complex interaction between material properties
and loading conditions in real electronic products.
1- Mismatch in the coefficient of thermal expansion along with the high homologous
temperature of tin causes the room temperature to be a ‗hot‘ deformation temperature.
Residual stresses caused by this mismatch can produce creep phenomena.
2- Change in the ambient temperature, shutting down, and turning the circuit on and off can
produce thermo-mechanical fatigue and recrystallization.
3- The anisotropy in the coefficient of thermal expansion, and elastic properties of tin (that
has body centered tetragonal crystal structure), cause the deformation to be quite
inhomogeneous.
4- In fact, slip in different slip systems, dislocation climb, sub-grain boundary formation,
grain boundary sliding, and grain boundary migration and recrystallization may happen
as the result of creep, fatigue, and inhomogeneous deformation.
17
Figure 2.6 Polarized light image and corresponding c-axis orientation maps and pole figures
showing a thermo-mechanically cycled single crystal (left), and a tri-crystal showing the 60
about [100] axis twin relationship in SAC305 solder joint (right) . In the single crystal, the white
lines delineate low-angle boundaries, and orientation gradients are apparent in the spread peaks
on pole figures (Bieler et al. (2008))..
18
Hertmeyer et al. (2009) employed an in-situ optical creep observation of joint-scale SAC 305
alloy shear samples, and observed that dislocation slip can have a large role in the deformation
mechanism in SAC 305 alloy at various stress values and temperatures. Zhou et al. (2012)
conducted a study under different
stress
states
and showed that the slip phenomena is
extremely important in inhomogeneous deformation of particular grains, as well as in developing
new grains through gradual lattice rotation during the continuous recrystallization process.
Zhang (2012) utilized in-situ tensile creep and electron backscatter diffraction to understand the
mechanism of deformation and grain rotation in Sn–4Ag/Cu solder joints during the creep
deformation. He observed slip traces on the deformed surface during the primary creep process.
Figure 1.5 shows the slip traces that can be observed in different loading condition and
temperetures.These studies clearly show that dislocation slip can have a crucial role in thermomechanical cycling and creep, and developing a model based on slip mechanism that can define
a desirable framework to define a path for further improvement in modeling is necessary.
In reality, due to the large variation in CTE, internal strains arising from both extrinsic (package–
board) and intrinsic (anisotropic) CTE mismatches lead to strains on the order of 0.1% or more
during the temperature change arising from thermal cycling. However, during the accelerated
thermal cycling condition (
, which is common in the industry, plastic deformation
can occur and knowledge about the mechanisms of plastic deformation is important in this
regard.
19
2.1.5 Effect of aging on the mechanical properties of SAC 305 solders
One important aspect of Sn-based solders that has important impact on mechanical properties
and stability of the microstructure during service is the presence of intermetallic compounds
(IMC). Depending on the composition of minor elements, lead free Sn based solder can contain
different IMCs. Generally SAC305 contains Ag3Sn and Cu6Sn5. There are many studies about
the role of intermetallics in strengthening the solder matrix. Lee and Subramanian (2005)
investigated the impact of Cu6Sn5 on the microstructural evolution of Sn based solders. They
pointed out that this IMC can pin the Sn grain boundaries.
Kerr and Chawla (2004) examined the influence of micron size Ag3Sn on the mechanical
behavior of Sn based solders and concluded that these IMCs can act as obstacles to reduce the
dislocation motion. One important practical aspect of IMCs is that the coarsening of these
compounds can degrade mechanical properties of the solder during service conditions.
Due to the anisotropy present in lead free solder balls, which arises from inhomogeneity
associated with the crystal structure of tin and large grain microstructures of tin, the macroscopic
mechanical response is highly sensitive to the microstructure and geometrical constraints. It
should be emphasized that realistic models that can assist in the prediction of the failure location
in electronic packages have to consider the unique nature of the stress and strain evolution. This
unique stress and strain history is due to the irregular deformation arising from both large grain
microstructure of tin and presence of 32 slip systems in the crystal structure of this metal.
Due to the fact that the dimension of a solder BGA ball in the real application is on the order of
grain sizes, it is important to examine the mechanical behavior and correlate mechanical
20
properties and the microstructure at the solder ball scale. Therefore information that can be
obtained from the bulk material is not necessarily helpful for these applications.
In these size scales, less constrained surface grains deform more easily, and their behavior could
be totally different from grains in the bulk materials. Furthermore, a higher cooling rate in the
microscale samples leads to different dendritic structure, and creation of different IMC
morphologies. Hence, the mechanical response of solder joints to external loading can be
different from the bulk solders due to fine microstructure, free surfaces, grain orientation (single
grain/ multigrain), and the presence of intermetallic compounds at joint boundaries.
On the other hand, using the assembly level BGA packages as test specimens is not helpful in
this regard. The mechanical response which can be obtained using these samples represents the
average behavior of an array of many, and is very difficult to correlate the microstructure and
mechanical properties in this condition. However, one can still obtain some ideas about the
kinematic of deformation and activity of different slip systems using this test configuration.
In addition to the anisotropy, isothermal aging is another parameter that has significant impact on
mechanical properties of lead free solder joints during service conditions. This mechanism can
reduce the strength and degrade the reliability of solder joints. This issue was examined by
several researchers:
Coyle, et al. (2000) reported 20% reduction in shear strength of BGA solder joints after 240
hours of aging at room temperature.
21
Ding, et al. (2007) studied the effect of aging on fracture behavior of Sn-Ag solder using tensile
tests. They showed that the tensile strength is reduces dramatically after aging at 180 ºC for 120
hours.
Ma, et al. (2006) investigated the effect of different aging conditions on Young‘s modulus, yield
strength, and ultimate tensile strength of SAC305 and SAC405 solder alloys. They showed that
the strength decreased dramatically in the first 20 days for both room temperature and elevated
temperature aging. After 20 days of aging, the properties changed slowly.
Zhang, et al. (2008) also studied the aging effects on tensile properties of SACN05 (SAC105,
SAC205, SAC305 and SAC405) series solders for different amounts of aging at temperatures
between 25-125 ºC.
They demonstrated that the mechanical properties degraded more
dramatically when the aging temperature was increased.
The data also showed that the
degradation becomes linear with longer aging time.
Cai, et al. (2010) also indicated that aging effects are significant for lead free solders for room
temperature aging as well as elevated temperature aging. They have also shown that the aging
effects can be reduced by using certain dopants (e.g. Bi, In, Ni, La, Mg, Mn, Ce, Co, Ti, Zn, etc.)
to SAC solder alloys to enhance the reliability of lead free solders.
Isothermal aging effects have also been reported to lower the strength and to reduce the
reliability of solder joints. Li, et al. (2002) studied the elevated temperature aging effects on flipchip packages with SAC solders. They suggested that the shear strength of solder bumps
subjected to aging at 80ºC can decrease gradually with aging. Also, for aging temperatures of
150 ºC and 175 ºC, the degradation of shear strength of the bumps were much faster.
22
Darveaux (2005) indicated that after 24 hours of aging at 125 °C, all alloys showed a 10% to
30% reduction in solder joint strength. All solder joints failed within the bulk solder and
exhibited high ductility. In addition, the ductility of all Pb-free solder joints decreased with
increased aging.
Chen, et al. (2006) studied the effects of aging on the solder bump shear strength for both Sn-Pb
and Sn-3.5Ag solders. They reported that the shear strength for both solder materials decreases
after aging at 150 ºC for 1500 hours, 8.9% for Sn-Pb solder bumps, and 5.3% for Sn-3.5Ag.
Kim, et al. (2004) obtained similar results in which they reported 5% decrease in the strength of
joints in stud bump samples for aging at 150 ºC for 300 hours.
Lee et al. (2010), indicated that aging degrades the Thermal Cycling Reliability (TCR) of lead
free PBGA assemblies subjected to Accelerated Life Testing (ALT) for SAC305 samples, which
were subjected to aging at elevated temperature aging (100-150 ºC). Lee, et al. (2012) showed
that the lifetime of wafer-level chip scale packages with SAC305 solder interconnects was
reduced by 29% for 500 hours of aging at 150 ºC.
Zhang, et al. (2012) studied the effects of aging on the reliability of PBGA components. They
showed that for 6 months aging at 125 ºC the reliability of SAC105 components dropped by
53%.
In summary, the main reason for softening effect after aging is that when the particles are small
they can pin the dislocation movement and can act as strengthening mechanisms, however when
they grow coarser their strengthening ability is reduced dramatically.
23
2-2- Characterization techniques
To assess the evolution of Sn crystal orientation during monotonic loading (shear and tension),
different characterization techniques were employed in this study.
2.2.1 Polarized Light Microscopy (PLM)
Polarized light microscopy is an optical contrast-enhancing technique that uses birefringent
(or doubly-refracting) property of non-cubic materials. A polarization filter is introduced into
the beam of light before it passes the sample, and a second polarization filter, the analyzer, is put
into the beam of light after it passes. In order to be able to conduct the characterization, polarizer
and analyzer have to be oriented perpendicular to each other. Anisotropy in the birefringent (or
doubly-refracting) property of the sample causes bending in some of the plane polarized light, so
that it can pass through the analyzer and can be imaged. The amount of the light, which passes
the analyzer is dependent on the orientation of the anisotropic structures in the sample in relation
to the rotation angle (azimuth) of the polars. By rotating the stage, one can change the relative
orientation of near cross-polarized light with respect to the sample and take advantage of
birefringent property of tin and distinguish the contrast (Lee et al. (2009)).
Application of this method for characterization of tin has been proved by several prior studies
[Bieler et al. (2008), Seo et al. (2009), Seo et al. (2010), Telang et al. (2004), Elmer et al. (2010),
Lehman et al. (2004, 2010), Lee et al. (2009), Mattila et al. (2010), Yin et al. (2012), Sundelin et
al. (2008), Henderson et al. (2004), Chen et al. (2011), Bieler et al. (2011), and Mattila and
Paulasto-Krckel (2011)].
One of the most important applications of PLM in lead free solder joints is discovering the
cyclic-twin grain structure (also known as Kara‘s beachball) in a SAC alloy solder which is
24
shown in Figure 1.8). Although this method is a useful qualitative method to identify the crystal
structure and different grain structures, it does not provide any quantitative information about
crystal structures. Therefore, other techniques such as electron backscatter diffraction (EBSD) or
X-ray diffraction (XRD) are required to be able to quantitatively identify grain orientations.
These methods will be described later.
25
Figure 2.7 Schematic diagrams showing the experimental set up for EBSD observation.
26
Figure 2.8 Polarized light image showing the application of this method in characterization of
grains morphology in lead free solder balls (adapted from Lehman et al. (2010)).
27
2.2.2 - EBSD and polarized light microscopy
Electron backscatter diffraction (EBSD) is a technique used to examine the crystallographic
orientation of crystalline materials. This technique is conducted using a Scanning Electron
Microscope (SEM) equipped with an EBSD detector containing a phosphor screen, compact lens
and low light CCD camera chip.
For an EBSD measurement, a polished crystalline sample is mounted inside the SEM chamber at
a highly tilted angle (~60-70° from horizontal) towards the diffraction camera to increase the
contrast in the electron backscatter diffraction pattern. The phosphor screen is located inside the
chamber of the SEM at an angle off by approximately 90° from the pole piece, and is coupled to
a compact lens which focuses the image from the phosphor screen onto the CCD camera. This
configuration is schematically shown in Figure 2.7.
In this configuration, some of the electrons which enter the sample will backscatter and exit at
the Bragg condition (For a crystalline solid, the waves are scattered from lattice planes separated
by the distance d). The Bragg condition describes the constructive interference from successive
crystallographic planes as:
where n is an integer determined by the order given, λ is the wavelength, d is lattice plane
distance, and θ is the scattering angle. Some of these diffracted electrons collide and excite the
phosphor causing it to fluoresce. Different crystalline planes diffract different electrons and
form a pattern consisting of pairs of white and dark parallel lines known as Kikuchi bands. Each
band can be indexed individually, but most commercial systems use look up tables with crystal
28
data bases to perform indexing. An automatic indexing of these patterns known as Orientation
Image Microscopy (OIM) is utilized to obtain a complete description of the crystallographic
orientations in polycrystalline materials.
2.2.3 Differential Aperture X-ray Microscopy (DAXM)
Although the EBSD is very helpful and popular in probing the grain orientation, obtaining 3D
information using this method is only possible when this tool is combined with the destructive
procedures such as serial sectioning or milling with a focused ion beam (FIB). 3D-xray is one of
few tools that can probe local grain structure nondestructively.
In the DAXM methods, a pair of elliptical total external reflection Kirkpatrick-Baez (K-B)
mirrors focuses a polychromatic x-ray beam to a 0.5 µm×0.5 µm spot on the sample. The
sample is mounted on a stage, with the sample normal making an angle of 45° with the incoming
x-ray beam. A schematic of the experimental setup is shown in Figure 2.9. A charged-coupled
device (CCD) detector located above the sample records Laue diffraction patterns generated by
volume elements along the x-ray beams. A differential aperture made of a 50 m platinum wire
scans across the surface of the sample and decode the overlapping beams by subtraction of CCD
image pairs, which differ only by a small (differential) motion.
A computer program developed at Oak Ridge National Lab determines the positions of Laue
diffraction spots in the CCD camera images, indexs the patterns, and calculates the orientation
matrix for each 1 m voxel from which a discernible diffraction pattern could be decoded.
The program first makes a list of possible indices (hkl) for each reflection from the known band
pass of the radiation, the known unstrained unit cell parameters, and the angle of reflection. The
29
measured angles between reflections are compared to the angles calculated for an unstrained
grain with all possible indices. If the angles are within the expected experimental and strain
uncertainty, the indices are assigned to the same grain. These computational procedures are
performed using powerful computer servers equipped with multiple processers.
30
Figure 2.9 Schematic diagram showing the experimental set up for DAXM
observation. The incident X-ray beam can have either a polychromatic or
monochromatic spectra. A 50 m m-diameter platinum wire is translated near the
sample surface to decode the origin of the overlapping Laue patterns.
31
2.2.4 2-D Radiography
The 2-D radiography which is based on the x-rays shadow microscopy was employed in this
study to assess the effect of voids in solder balls on mechanical properties.
This method
schematically is illustrated in Figure 2.10. This technique is based on the idea that different
materials within a sample absorb the x-ray diffraction differently depending on their density and
atomic numbers. This phenomenon produces a shadow on the detector such that when the
material is denser, the shadow becomes darker. Hence, big voids clearly are characterized as
brighter areas.
The detector creates optical images from the incident x-rays and allows
identifiacation of defects in the sample that are larger than the detector pixel size.
32
Figure 2.10: Basic 2-D x-ray system configuration (Benard (2003))
33
2.3 Modeling approaches used in solder joints
There are numerous computational methods that have been utilized to model deformation
behavior and damage nucleation in solder alloys. Dong et al. (2005) Gao and et. al (2010) used
molecular dynamics simulations to find the atomic scale properties of solders required for multiscale simulation of damage nucleation and evolution. The disadvantage of this approach is that it
is computationally expensive and cannot be used for realistic sample sizes, and these calculations
require long periods of time. Due to the fact that dislocation motion is the most significant
mechanism in creep (time-dependent) and plasticity (time-independent) deformation, one can
develop the same constitutive equation for both creep and plasticity. These types of studies
based upon unified creep plasticity (UCP) modeling have been conducted by several researchers.
In the following, some of these models are briefly described. Hart et al. (1976) introduced a
model based upon the idea that dislocation pile up occurs in two states, strong (macro plastic)
and weak (micro plastic), and one can define a state variable for each case. This model was later
employed for solder alloys by Wilcox et al. (1990) ignoring the viscous effect. Although this
model was fairly promising in the prediction of several deformation behaviors such as creep,
load relaxation, and tensile test, it was quite unsuccessful in predicting the transient behavior
such as the Bauschinger effect. Yao and Krempl (1986) proposed a viscoplasticity theory based
on overstress (VBO), which was later developed by Tachibana and Krempl (1995, 1997, 1998).
This model is essentially based on a viscoplastic flow potential and defines three state variables
as overstress (the difference between the actual stress and back stress), kinematic stress for work
hardening, and isotropic stress.
Maciucescu et al. (1999) employed this model for creep
behavior of solder alloys.
34
Anand (1985) and Brown et al. (1989) developed a model for hot working of metals which was
employed for solder joints by Adams (1986), Wilde et al. (2000) , Wang et al. (2001), Chen et
al. (2005), and Qing et al. (2007). This model is very popular due to using a single scalar state
variable to define the deformation resistance. Furthermore, this model is quite successful in
describing deformation behaviors such as strain hardening, constant strain rate behavior at
different temperatures, steady-state creep, and thermal cycling hysteresis loops. The
disadvantages of this model is that the oversimplification arising from defining one state variable
makes it very difficult for applications that need more physics based understanding. Busso
(1992) introduced a model in which the Bauschinger effect was captured using a back stress.
This back stress can be described as an internal stress arising from dislocation pile-ups at
barriers. This model was employed for solder deformation by Busso (1992, 1994). McDowell et
al. (1994) introduced a thermo-viscoelastic model to simulate the deformation behavior of solder
alloys. In their model, the back stress can be written as a summation of short range and long
range effects.
A modified, unified creep and plasticity model was developed by Wen et al. (2001, 2002) using a
similar framework. Employing different rules of evolution for short-range back stress, in this
framework, Bai and Chen (2009) could model uniaxial tensile tests, strain rate jump tests, short
term creep tests with stress jumps, and uniaxial ratcheting tests. The governing equations of the
last four models are reported in Tables 2.2.
35
Bodner and Partom (1975) presented a set of constitutive equations to describe the elasticviscoplastic strain-hardening behavior for large deformation and arbitrary loading. They used
the additive decomposition to divide the total plastic strain into elastic and inelastic strains. The
inelastic strains follow the Prandtl-Reuss flow law, which is independent of any yield criteria or
loading and unloading conditions. There are two state variables in this model; one corresponds
to isotropic hardening and the other is related to directional hardening. Skipor (1996), Whitelaw
et al. (1999), and He et al. (2006) used this model to explain the deformation behavior of
different solder alloys.
Johnson and Cook (1983) presented a phenomenological model to
describe the strain rate- and temperature-dependent response of metals. Fei et al. (2008) used
this model to consider the effect of strain rate for lead-free solder joints to simulate the board
level drop impact test deformation and damage. Xuming et al. (2011) presented a modified
Johnson Cook constitutive equation to model the damage evolution in these alloys.
There are other unified constitutive models that have been introduced for solders. A temperature
dependent yield function was developed by Desai et al. (1997). Ju et al. (1996) proposed a scalar
function to obtain the inelastic strain rate. Qian and Liu (1997) used the back stress to represent
the transient stage of a stress–strain curve in a unified constitutive model for tin–lead solder. In
an attempt to measure the mechanical properties for SAC305, a parametric study was conducted
by Chawla et al. (2005) to simulate the lap-shear test using a simple isotropic constitutive
equation. Recently, Lederer et al. (2012) proposed a constitutive model for plasticity and
damage of solder joints. This model shows better agreement with experimental results for
ultimate tensile strength of solder joints with various thicknesses.
36
The models discussed above generally lack anisotropic material behavior, so the sophistication in
these formulations are based upon the assumption that there are sufficient numbers of grains that
anisotropic behavior can be homogenized. The next set of models are simpler but contain the
important anisotropic property considerations.
Matin et.al (2005) used a simple linear elastic constitutive equation to study the role of intrinsic
anisotropy of Sn on the thermal fatigue damage in real microstructures and grain orientations.
Similar study was conducted by Park et al. (2008). Although Matin‘s and Park‘s studies were
very important for understanding the effect of real microstructure and grain boundaries in strain
localization, they assumed very simple elastic models.
In order to examine the solder
deformation, one has to understand the evolutions of plastic stress and strain as well as
microstructural changes arising from the plastic deformation.
Gong et.al (2007) used the crystal plasticity finite element developed by Huang (1991) and
incorporated a visco-plasticity constitutive equation in that model.
Although they could
realistically simulate the response of a solder joint under thermal-cycling loading condition, they
did not incorporate the elastic and thermal expansion anisotropy in this model.
Bieler and Telang (2005, 2009) performed modeling of a shear test using the Visco-Plastic SelfConsistent (VPSC) polycrystal plasticity model developed by Lebensohn and Tomé (1993).
Using some assumptions about the activity of slip systems, they were able to achieve excellent
agreement with measured texture evolution from an OIM data set.
Zamiri et al. (2009) used an anisotropic elastic-plastic finite element model with
phenomenological flow models for slip systems based upon semi quantitative information
37
available in the literature. Using a set of simulations, they were able to correlate changes in the
crystal orientation to the internal elastic energy of the system.
The main limitation of crystal plasticity models that are developed so far (models which were
used by Gong, Bieler, and Zamiri) is that they did not validate their model by comparing the
stress response with experimental data. Table 2.3 shows a comparison between these models and
phenomenological models.
2.3.1 Comparison of the phenomenological basis for different models
The phenomenological basis for most of the proposed models can be explained by Kocks‘
discussion of the thermodynamics of slip (Kocks 1987). Hart proposed a unified viscoplasticity
model based on these principles. He started his derivations based on the existence of barriers to
dislocations and suggested the existence of two stresses: one due to the pile-up of these
dislocations at barriers and the other due to glide resistance to dislocation movement within
barriers. Anand‘s model also was based on the same physics. He defined averaged isotropic
resistance that can represent strengthening mechanisms such as dislocation density, solid solution
strengthening, and sub-grains. In the hardening formulation, the effect of static recovery was
taken into account. Busso, used a single state variable that is derived from the physical basis of
dislocation pile-ups at barriers that create a back stress. The Krempl and McDowell models
clearly differ from the other three models.
The main difference is that they define the
viscoplastic flow potentials and a lack of thermodynamic activation energy associated with the
plastic strain rate. Bower and Wininger (2004) proposed a two dimensional FE model that could
simulate the constitutive response and microstructure evolution of polycrystals during high
temperature plastic deformation. Their model takes into account the effects of grain boundary
38
sliding, grain boundary diffusion, grain boundary migration, surface diffusion, as well as
thermally activated dislocation creep within the grains.
The ability to relate the microstructure to the macroscopic mechanical properties is important
from the materials design point of view, and crystal plasticity assists in conducting the
microstructure sensitive design for crystalline materials. The mechanical modeling of single
crystal behavior is well established (Taylor and Elam, (1923), Asaro (1983), Hill and Rice
(1972), Mandel, (1965)). However, the efficiency of numerical modeling is still a controversial
subject especially when these algorithms are used as tools to design industrial products using
FEM codes. In this context, development of numerical integration algorithms with improved
efficiency, speed and accuracy is of great importance.
Regarding the numerical algorithms behind the crystal plasticity, two issues are important. The
first one is how to integrate the set of highly non-linear constitutive equations and the second one
is how to resolve the ambiguity arising from non-uniqueness in the activity of slip systems.
In order to solve this set of non-linear equations, several incrementally based numerical
algorithms have been proposed. These numerical schemes can be classified into three main
families: explicit algorithms based on a forward Euler scheme, implicit algorithms based on a
backward Euler scheme, and semi-implicit algorithms.
2.3.2 -Explicit integration algorithms:
Developing explicit algorithms is straightforward, but it generally requires very small times or
loading increments to avoid numerical instabilities. This numerical scheme was employed by
Nemat-Nasser and Amirkhizi,(2007), Anand and Kothari (1996), Ben Bettaieb (2006),
39
Knockaert et al. (2000), Kuchnicki et al. (2006), Maniatty et al. (1992), Mathur and Dawson
(1989), Peirce et al.(1982), Zamiri et al. (2007), Zamiri and Pourboghrat (2010), and Rossiter et
al. (2010). The critical resolved shear stresses and the crystal rotation are assumed to be constant
over the time increment and chosen to be equal to their values at the beginning of the time step.
2.3.1 -Implicit integration algorithms:
The implementation of implicit schemes is more complicated than for explicit ones. In the
implicit family, critical shear stresses and the rotation of the crystal lattice are unknown,
producing a set of nonlinear equations. This set has to be solved by traditional iterative methods
(generally those such as the Newton–Raphson procedure or the fixed point method). Kalidindi et
al. (1992) used this method of solving crystal plasticity finite element simultions.
2.3.3 -Semi-implicit integration algorithms:
In this class of numerical schemes, the crystal lattice rotation is evaluated at the beginning of the
time increment, whereas, the computation of critical resolved shear stresses is based on an
implicit scheme. McGinty and McDowell (2006) and Watanabe et al. (2010) employed the
iterative resolution of the consistency conditions to solve the non-linear constitutive equations.
Débordes et al. (2005) incorporated the linearization of the hardening law over each time step
into semi-implicit scheme.
The second difficulty in numerical algorithms is related to the non-uniqueness of active slip
systems. Several researchers tried to solve this problem using two distinct procedures. The first
method that is based on the thermodynamic motivation was first proposed by Taylor (1923).
One of the fundamental assumptions of the Taylor theory is that among the many possible slip
40
combinations that can achieve the prescribed crystallographic strain, the combination that
produces the minimum amount of internal work is the set that is activated (Taylor and Elam
(1923)).
Renouard and Wintenberger (1981), Van Houtte (1988), and recently Hamelin et al. (2011)
developed a first order selection criterion, based on the minimization of the change in the internal
work with respect to von-Mises strain. Driver et al. (1984), Fortunier and Driver (1987) and
Skalli et al. (1985) examined this method for aluminum single crystals.
Franciosi and Zaoui (1991) used a criterion based upon the minimization of the hardening in
order to select the set of active slip systems.
Raabe et. al (2001) implemented a local Taylor factor schemes using the local stress tensor to
determine the local Taylor factor, to better track heterogeneous deformation evolution, and this
matched experiments better than using a global Taylor factor approach.
On the other hand, several researchers tried to use a rate-dependent formulation based on powertype laws without differentiation of slip systems into active and inactive sets (Asaro and
Needleman, (1985), Mathur and Dawson, (1989), Peirce et al. (1982, 1983)). This method has
attracted attention since the empirical equations can be easily incorporated in this framework
using formulations that define the relationship between the dislocation density, velocity, and
shear strain rate.
Anand and Kothari (1996) and Knockaert et al. (2000), employed the well-known singular value
decomposition (SDV) method (which projects the deformation rate on the set of positive slip
rates. Schröder and Miehe (1997) and Miehe and Schröder (2001) proposed an alternative
general inverse method where the reduced space is obtained by dropping columns of the local
41
Jacobian associated with zero diagonal elements within a standard factorization procedure. All
these models attempt to use the Schmid law and define multi-surface yield functions.
Many other authors (Gambin (1991, 1992), Montheillet et al. (1985), Toth et al. (1991), Van
Houtte (1987), Guan et al.(2007), Zamiri et al. (2007)) proposed robust schemes based upon
defining smooth yield surfaces to solve the ambiguity in defining the active slip systems. The
derivative of this yield function with respect to the shear stress can define the strain rate.
Recently, Zamiri and Pourboghrat (2010) developed an effective methodology to define a
smooth yield function based on the logarithms of the Schmid law for all slip systems and
incorporating a regulation parameter which will be described later.
42
Table 2.2 Summary of the main phenomenological equations used in solder joints
Original Model
--Application to Solder
1. Hart (1976)
-- Wilcox et. al. (1990)
Forms of Important Constitutive Definition of variables
Relations
ln(
s1
a1
)(
1
)
1
1 , f1 , Q, , , C : Materials parameters
1
R : Boltzmann‘s constant
s
Q
( 1 ) m1 f1 exp(
)
G
RT
d ln( s1 ) / dt h( a1 , s1 ) r (s1 , T )
*
1
a1
) ( )
s1
s1
C a 2
s2 a 2 e
h(
2. Anand (1982)
--Adams (1986),
--Wilde et al. (2000),
--Wang et al. (2001)
S1
1
r : re cov ery var iable
1 , s1 : macro strain and hardening
2 , s 2 : micro strain and hardening
cS
c
s1 , s1 , h : hardening var iable
, Q, A, m : Materials parameters
sinh 1 [
p
exp(
Q m
) ]
RT
A
1
Q
p A exp(
) sinh( ) m
RT
s
s h( , s, T ) p r(s, T )
a
s
s
s h0 1 * sign(1 * ) p
s
s
p
Q n
s * s[ exp(
)]
A
RT
43
R : Boltzmann‘s constant
s : hardening var iable
r : re cov ery var iable
s * : saturatedh ardening var iable
Table 2.2 (cont‘d)
Original Model
--Application to Solder
3. Krempl(1985)
--Maciucescu et al. (1999)
Forms of Important Constitutive Definition of variables
Relations
x B
E , Et , m, , , : Materials parameters
m
g x x
m 1 x
g : Equilibriu m stress
B : Backstress
x (3 / 2)tr ( x.x)
f : Kinematic stress
3 x x
g ( ) m
2 x
p
A : Isotropic stress
x B f
B Ep (
)
x
A
2
x
f Et p Et p
3
x
A
A
2
A (
)
A2
p (2 / 3)tr ( p . p )
44
A2 : Initial isotropic stress
: dimensionl ess shape constant
Table 2.2 (cont‘d)
Original Model
--Application to Solder
4. McDowell(1994)
--Bai,Chen (2009)
Forms of Important Constitutive Relations
A, Q, G, B, , Y0 :
Sv n
S v n1
3
A(
) exp(( B
) )N
2
D
D
S v (3 / 2)1 / 2 s Y
in
Materials parameter
S v : over stress
T
Q
) for T m
RT
2
Tm
T
2Q
exp(
[ln( ) 1]) for T m
R Tm
2T
2
s
N
s
exp(
C (T ) : ( D n D in k T ) C 1 : :
G in N
Y Yin Y0
45
Definition of variables
C
T
T
Y : Yield stress
: Backstress
R : Boltzmann‘s constant
D : Darg stress
Table 2.2 (cont‘d)
Original Model
--Application to Solder
5. Huang(1991)
--Gong(2007)
Forms of Important Constitutive Definition of variables
Relations
C1 sign(C 2 ) n exp(
e c
C : e
Q
)
RT
C1 , C 2 , Q, A, m, C :
Materials parameter
R : Boltzmann‘s constant
s : hardening var iable
r : re cov ery var iable
N
c .P
i 1
s * : saturatedh ardening var iable
1
(m n n m )
2
Q
A sign( ) exp( )
RT
:P
P
C1 sign(C 2 ) n exp(
Q
)
RT
46
n : Plane normal
m : Slip direction
: Re solved shear stress
Table 2.3 Summary of different modeling approaches that employed to model the mechanical behavior of lead
free solders
Original
Model
Application
to Solder
#S.V.*
#M.P.s*
Prediction
Capability
Limitations
Hart (1976)
Wilcox (1990)
2
18
Cyclic tests, transient effect
primary creep, Strain rate
jump
Anand (1990)
Maleki (2011)
1
9
Strain rate jump,
Steady state creep
primary creep
Busso (1992)
Busso (1994)
1
10
Isothermal tests
validated test
Krempl (1986)
Maciucescu
(1999)
3
9
Baushinger effect,
Fatigue life
validated test
McDowell
(1994)
Bai (2009)
2
10
Strain rate jump,
Ratcheting test
primary creep
Tome(1993)
Bieler (2009)
1
7
Texture evolution
stress-strain validation
Zamiri (2009)
Zamiri (2009)
1
7
Damage evolution
stress-strain validation
Huang (1991)
Gong (2008)
1
7
Possible multi-scale
modeling
* #S.V.: numbers of state variables, * #M.P.s: number of material parameters
47
validated test
2.4. Summary of the literature review
Anisotropy is of significant importance in all structural metals, but this characteristic is unusually
strong in Sn, making Sn based solder joints one of the best examples for studying the influence
of anisotropy. This anisotropy is due to the crystal structure of tin and the existence of coarse
grain microstructures that are observed in sub-mm solder balls.
Studies about the microstructure and Sn grain morphologies in lead free solder joints show
peculiar microstructures, such as the beach-ball morphology, in which 60 cyclic twins can be
formed during solidification.
Understanding the slip activity in tin provides valuable insights into the understanding of plastic
anisotropy. There is incomplete understanding of slip in Sn, though single crystals have been
investigated. There is especially uncertainty about the critical resolved shear stress for different
slip systems in SAC alloys.
The majority of previous studies are based on the isotropic plasticity models, which may be
useful for polycrystals but for lead-free solder joints that are inherently either single crystal or
multi-crystals, using isotropic plasticity models is not physically sufficient.
The purpose of this study is to develop a model to correlate the microstructure to the mechanical
properties. Due to significant effect of anisotropy arising from the slip activity combination of
CPFE and materials characterization methods was employed to propose a microstructure
sensitive material model.
48
CHAPTER 3
CRYSTAL PLASTICITY MODEL DESCRIPTION
A velocity gradient in plastic deformation in the material coordinate system can be decomposed
into a rate of deformation and a spin tensor as :
(5.1)
An elasto-plastic problem is usually defined as a constrained optimization problem aimed at
finding the optimum stress tensor and internal variables for a given strain increment. In such a
problem, the objective function is defined based on the principle of maximum dissipation and is
made of terms describing the incremental release of elastic strain energy and the dissipation due
to the incremental plastic work and the constraint is the yield function
(5.2)
where
is the design variable (stress tensor) to be found, q is a vector containing the internal
variables such as strain hardening and kinematic hardening variables to be found, C is the
material stiffness matrix, f( , q) is the yield function, and E is the so-called matrix of generalized
hardening moduli. One of the solutions to the above problem is the following equation for the
plastic rate of deformation
(5.3)
49
In a crystal plasticity problem, the deformation is defined by several yield surfaces, and the
number of yield functions depends on the number of slip systems in a crystal. Assuming the
validity of the Schmid law for the plastic deformation of a single crystal, then for any slip system
a yield function can be defined as:
(5.4)
The constraints of problem (2) can be combined and replaced by an equivalent single constraint
defined as:
(5.5)
Where
tensor,
is the critical shear stress on slip plane , and
is the symmetric part of the Schmid
, describing the orientation of a slip system, defined as:
(5.6)
where
is a unit normal to the slip plane, and
is a unit vector denoting the slip direction.
The plastic deformation matrix can be expressed as
(5.7)
where
are the slip rates.
50
And spin tensor, which represents the material axis rotation, can be expressed as:
(5.8)
where
matrix is the anti-symmetric part of
,defined as:
(5.9)
It can be shown that during the plastic deformation of a single crystal the slip rate on any slip
system could be expressed by:
(5.10)
where m and
are material parameters that control the shape of the single-crystal yield surface,
and been shown to have a direct relationship with the stacking fault energy (SFE) of the material
as following:
(5.11)
Where
is the SFE of the material, G is the shear modulus, and b is the magnitude of Burgers
vector. For most materials m = 1 Parameter is a Lagrange multiplier which has been shown to
be a measure of the rate of plastic work in a crystal,
51
The generalized Schmid factor can be found for each slip system by finding the scalar product
of the normalized stress tensor (using the Frobenious norm) and the Schmid matrix.
In order to define the resistance against the shear, Hill (1966) and Asaro and Needleman (1985)
proposed the following formulation:
(5.12)
where
is the plastic slip rate on the active slip system , and
hardening matrix
are known as the self-hardening moduli while
are the components of the
for
are known as
the latent-hardening moduli. Hutchison (1976) proposed the following model for evolution of the
components of hardening matrix
(5.13)
Here q is the so-called latent-hardening ratio, which is the hardening on a secondary slip system
caused by slip on a primary slip system (Kapoor and Nemat-Nasser (1999)) and can be measured
by the ratio of the latent-hardening rate to the self-hardening rate of a slip system with typical
values in the range of 1 < q < 1.4. The parameter q can be considered as 1 for coplanar slip
systems and 1.4 for non-coplanar slip systems. There are different kinds of hardening models
presented by researchers for the evolution of
(so-called self-hardening). One of the most
well-known formulations is:
(5.14)
52
where
, a, and
for all slip systems.
are slip system hardening parameters, which are considered to be identical
denotes the initial hardening rate,
the saturation value of the slip
resistance, and a the exponent describing the shape of the stress-strain yield function. These
parameters can be obtained by fitting the model to experimental data.
Finite element software ABAQUS is utilized as a tool to build the geometric shape of grains (single
crystal or cyclic twining morphologies) and to conduct the FEM solution. ABAQUS provides a
special interface, called user subroutine VUMAT, which allows the user to define the mechanical
behavior of a material and to interface with any externally defined programs.
53
Algorithm for combined constrains single crystal plasticity model
If timestep=0 then
Set
1.1-Build Q using Equation 5-38
1.2. Calculate
and
using
and
(5.15)
1.3. Initialize the critical resolve shear stress:
(5.16)
For
EndIf
Else
Rotate ,
equations:
from the global coordinate system to crystal coordinate system using the following
Q
(5.17)
Q
(5.18)
A-Calculate the trial stress, initial resolved shear stress, Initial Lagrange multiplier and shear
strain as following:
(5.19)
54
(5.20)
(5.21)
(5.22)
Find the initial incremental rotation of the grain using:
(5.23)
Find the stress tensor for current crystal axes:
(5.24)
Calculate
If
go to A
Calculate the incremental spin tensor using the following equation:
(5.25)
Compute the rotation of grain at current iteration:
(5.26)
55
Rotate the stress tensor to the crystal coordinate system
Find the increment of plastic consistency parameter:
(5.27)
Compute the stress, shear strain slip systems hardening increment and shear resistance using
plastic corrector
(5.28)
(5.29)
(5.30)
(5.31)
Update the increment of incremental slip rate:
(5.32)
Set i←i+1
(5.33)
56
Knowing the orientation matrix
as:
at the beginning of increment, update the orientation matrix
(5.34)
(5.35)
(
(5.36)
Rotate the updated stress tensor back to the reference coordinate system
(5.37)
3.1. Orientation matrices in crystal structures:
Orientation of a crystal can be defined either by three Euler angels (
) or for calculation
purposes using, different orientation matrices. According to Kocks et al (2000) three kinds of
conventions to define the crystal orientations are: Bunge, kocks, and Roe.
Bunge defined the orientation matrix as:
(5.38)
Kocks defined the crystal orientation by three Euler angels
57
as:
(5.39)
and finally Roe presented the crystal orientation as:
(5.40)
58
CHAPTER 4
EXPERIMENTAL PROCEDURES
4.1. Single joint tensile samples
4.1.1. Sample preparation for joint level tensile test samples:
The specimen preparation procedures are intended to duplicate the structure (grain size and
orientation) which is observed in the actual solder joints. For fabrication of 6 tensile samples a
piece of 0.55 mm diameter copper wire with a thin lacquer insulation layer was cut into 25mm
pieces. A jig was designed and fabricated to hold the samples during the polishing and solder
fabrication processes. A schematic diagram of this jig is illustrated in Figure 4.1 where copper
wires are placed in the jig and fixed. Polishing of the ends of the copper wires that were
intended to be in contact with solder balls was conducted until a sufficient surface finish was
obtained. In the next step, the copper wires were cut in the middle, and then flipped around so
the polished sides faced each other in the jig. By leaving a particular distance between polished
sides of the copper wire, it was possible to place a solder ball in the gap, to complete the
fabrication of the tensile testing sample. Small pieces of Si sheets were utilized to prevent
contact between the solder and the copper jig. Flux was applied on the ends of copper wires to
allow the activation of solder.
Lead-free SAC 305 (Sn-3.0Ag-0.5Cu (wt.%)) solder balls
approximately 550 micron diameters were placed on the copper ends as shown in Figure 4.2 The
assembly was heated on a hot plate with a thermocouple monitoring the temperature, to make the
solder joint, and then cooled on an aluminum plate. The heating and cooling profile is shown in
Figure 4.3. After the fabrication of the solders, the samples were aged for 500 hours at 150 ºC.
59
X-ray characterization of the joints were performed at the Argonne National Laboratory using
beamlines 34 and 6 to obtain the crystallographic orientation before and after tensile deformation
to just after maximum load. Radiographic evaluations were also made to assess the effects from
porosity and voids in solder joints, on the mechanical properties of the samples.
4.1.2.Tensile test set up
Mechanical characterization of single solder balls were carried out using a Rheometric Solid
Analyzer (RSA-III). Figure 4.4.a shows the setup which was utilized to conduct the mechanical
testing. Single solder ball samples were subjected to tensile deformation at 25 ºC with the
displacement
rate of 0.003mm/s. Force and
displacements were recorded using the data
acquisition software, and after the test stress relaxation data were also recorded.
Optical microscopy, radiography, and orientation x-ray characterization were conducted before
and after the tensile test to identify changes in geometry and identify fraction locations.
60
Figure 4.1 Schematic diagrams and dimensions of the jig used for polishing and fabrication of
joint level tensile samples.
61
Figure 4.2 Schematic configurations of tensile samples inside the miniature jig (a) locations of
polishing and cutting of copper wires are illustrated, (b) configuration of solder inside the jig and
on silicon plates.
62
350
300
Temperature(C)
250
200
150
Set1
100
Set2
Set 3
50
0
0
200
400
Time(seconds)
600
800
Figure 4.3 Temperature profiles used for fabricating solder joints
63
Figure 4.4 Set up employed for tensile testing, (a) RSA III equipment.
64
4.2.1 Shear samples
Eight (8) specially prepared samples of a 4×4 ball grid array (BGA) were cut from a 15 mm × 15
mm body-size plastic ball grid array (PBGA) package with Sn-3.0Ag-0.5Cu (wt.%) solder joints.
Four (4) samples were aged at 100ºC for 500 hours. The samples were sectioned and carefully
polished so that the plane of shear at the center of the joint could be observed on one set of four
balls before and after shear displacements. Due to the path dependency in plasticity, the total
displacement was applied in three steps in order to obtain the information about the shape change
and crystal rotation during the shear deformation.
A fixture was used to impose a two-
dimensional shear strain, so that small out of the plane variations of shear resulting from local
strain effects could be detected topographically on the previously polished surface. Figure 4.5
shows the schematic of the apparatus in this experiment.
To permit accurate orientation
measurements, samples were lightly re-polished to improve surface cleanliness after optical
observations, and before orientation imaging microscopy (OIM) measurements. Optical
microscopy, analysis of polarized light microscopy, OIM, and SEM were conducted initially, and
after each deformation step.
4.3 Procedure for OIM characterizations:
The resin surrounding the balls and the package was carefully painted over with carbon paint to
cover the nonconducting surfaces. Also, the polished samples were covered by copper tape away
from the solder joints to provide a conductive path, and to prevent the charging effect which
causes image distortion or beam drift.
OIM data were obtained in a CamScan 44FE electron microscope. The digital image resolution
of this microscope is 640×480 pixel size, and the accelerating voltage was set to be 20 kV and
65
Package
Cu pad
Solder joint
Figure 4.5 Schematic diagrams of a 4 × 4 solder joint array sectioned from a PBGA package.
The solder ball assembly geometry and shear test apparatus is also shown.
66
the working distance was 33 mm. A TSL EDAX system with a Digiview SEM detector was
utilized. An acquisition speed of 15-20 points per second was obtained using 2×2 binning
setting. EDAX TSL OIM Analysis software was used to generate the orientation maps.
Maps were cleaned up using the neighbor confidence index (CI) correlation to replace pixels that
had low CI with their neighbors having a high CI. This step is required to remove the minority
data points coming from un-indexed or wrongly indexed patterns. In the neighbor-confidenceindex-correlation method, a datum point having a low CI is replaced with the neighboring point
that has the highest CI value
A user defined orientation map was employed to correlate the orientation of Sn c-axis [001] with
respect to solder/package interface. For this orientation map, a graded 5 color scale (Purpleblue-green-yellow-red) was used to represent the orientation of the c-axis with respect to the
solder/package interface. When the c-axis is on the interface plane, the orientation is expressed
as ―red‖ while when the c-axis is perpendicular to the surface the orientation is ―purple‖. Other
colors (blue, green and yellow) fill in the rest of the orientation space. Figure 4.6 shows the
general procedure for OIM analysis.
4.4 X-ray diffraction
4.4.1- Beamline 34-ID-E
Synchrotron X-ray microdiffraction experiments were conducted on beamline 34-ID-E at the
Advanced Photon Source (APS) at Argonne National Laboratory. The experimental set-up is
illustrated in Figure 4.7 and described in chapter 2. A white beam with an energy range of 7-30
keV was used in the microdiffraction experiment with a beam size of approximately
67
5μm×0.5μm. The scanning step-size was set to be 1μm in a non-depth-resolved surface areascan (in the horizontal X and vertical H directions as defined in their coordinate system). Fifteen
(15) joint level tensile samples were characterized using beamline 34-ID-E before deformation.
Two (2) sides of each sample were scanned by the X-ray beam. For each step along the scan, the
diffraction pattern provided information from about 100 μm in depth along the incident beam
direction.
4.4.2- Beamline 6-ID-E
Characterization of Sn crystal orientation in joint level tensile samples using a high energy
monochromatic X-ray beam was accomplished at beamline 6-ID-D at the Advanced Photon
Source (APS) at the Argonne National Laboratory. A 2048 × 2048 pixel PerkinElmer XRD
1621 AN/CN area detector was employed to collect the diffraction patterns. To extract accurate
intensities from the detector, dark current images without the beam were used as the background.
Subtraction of background from the diffraction patterns were conducted automatically using the
acquisition software. To calibrate the beam diffraction, patterns from a silicon powder standard
was used, which resulted in calibration wavelength of 0.12492 A˚ and ‗‗distance to detector‘‘ of
1112.75 mm. Fit2D software was utilized to integrate and analyze the diffraction peaks from
selected portions of the pattern.
68
Figure 4.6 General procedures for conducting the EBSD analysis.
69
Figure 4.7 Experimental setup at synchrotron beamline 34-ID-E for conducting the
characterization of joint level tensile samples (a) platinum wire used as a differential aperture.
(b) Experimental station for conducting the synchrotron X-ray microdiffraction (c) sample stage
45º inclined with respect to beam direction.
70
CHAPTER 5
CRYSTAL PLASTICITY FINITE ELEMENT ANALYSIS OF
DEFORMATION BEHAVIOR IN MULTIPLE-GRAINED LEAD-FREE
SOLDER JOINTS
5.1 Introduction
Anisotropy is of significant importance in all structural metals, but this characteristic is unusually
strong in Sn, making Sn based solder joints one of the best examples of the influence of
anisotropy. The fracture probability in lead-based solder alloys is quite different from lead-free
solder alloys, mostly because of anisotropy present in the tin phase [House( 1960)]. While in
lead-tin based solder alloys, the location in the solder area that is most prone to the failure can be
predicted, the crystal structure of tin in lead-free solder alloys can highly affect the damage
initiation and evolution. This is because the coefficient of thermal expansion (CTE) plays an
important role in thermo-mechanical loading of solder joints, leading to significant axial as well
as shear strains acting on joints [ Bieler et.al (2008)]. Also, the residual stresses that are
generated during the initial cooling and thermal-cycling can cause dislocation motion.
Crystal features such as grain size and morphology have significant effects on the mechanical
properties and reliability of sub-mm scale joints. Some studies about the microstructure and Sn
grain morphologies in lead free solders joints show peculiar microstructures, such as the beachball morphology, in which 60 cyclic twins can be formed during the solidification.
There are a few papers about using the crystal plasticity to model the slip behavior in lead free
solder joints. The majority of previous studies are based on the isotropic plasticity models which
may be useful for polycrystals but for the lead-free solder joints that are inherently either single
crystal or multi-crystals, using just the anisotropic plasticity models is not physically sufficient.
71
In this type of analysis it is very important to take into account both spatial (effect of particular
morphologies) and orientation space (effect of grain orientation and its consistency with
morphology).
Incompatibilities along grain boundaries arise in lead-free solder joints from the anisotropy in the
CTE in the specific configuration of grains in the microstructure, which increases the complexity
of the problem and makes using the CPFE necessary for understanding of the deformation
behavior. One of the advantages of crystal plasticity modeling is its ability to solve mechanical
problems under the complicated external and internal constraints imposed by inter and intragrain slip mechanisms. Using microstructural scale CPFE is of great importance because at
these scales experimental boundary conditions are very difficult to control and monitor. In such
cases experimental results will be very difficult to interpret without the assistance of a model
with similar complexity. Miniaturization is a very important issue in solder joints because the
dimension in the joint is in the range of the grain scale. The design of solder joints will
increasingly require consideration of grain scale anisotropy.
Since there is a lack of consistent data available to establish models for dislocation slip activity,
this work will examine the effects of grain geometry and orientation using overly simple models.
Thus, this work will establish a basis for comparison that can be used with future refinements in
the models that can be developed once sufficiently detailed experimental data sets are established
for comparison. This will provide a basis for assessing the ability of better models to make
better predictions.
Furthermore, establishing this simple model will also assist in the
interpretation of experimental characterization. As there is little experimental work that can be
72
easily compared with a CPFE simulation in detail, this work will consider partial agreement with
experimental observations as a success to be improved upon. Thus, this work provides a basis
for an integrated incremental model development strategy based upon experiments, modeling
and comparative analysis.
A crystal plasticity model developed by Zamiri and Pourboghrat (2010) which is implemented
into ABAQUS finite element software was used to analyze the elastic–plastic deformation in the
tin phase.
The finite element analysis was performed using commercial finite element code ABAQUS. The
crystal plasticity material model was implemented using a user material subroutine in
FORTRAN (VUMAT). Solid works was used to make the model of a tri-crystal shown in
Figure 5.1 and ABAQUS was used to create the finite element mesh with the assignment of the
boundary conditions shown in Figure 5.2, and building the input parameters for the analysis.
Element types and number of elements used in CPFE are shown in Table 5.4. The elements are
8-node linear bricks with reduced integration and hourglass control. Hourglassing can be a
problem with first-order, reduced-integration elements (CPS4R, CAX4R, C3D8R, etc.)
in stress/displacement analyses. Since the elements have only one integration point, it is possible
for them to distort in such a way that the strains calculated at the integration points are all zero,
which, in turn, leads to uncontrolled distortion of the mesh. First-order, reduced-integration
elements in ABAQUS include hourglass control, but they should be used with reasonably fine
meshes.
Hourglassing can also be minimized by distributing point loads and boundary
conditions over a number of adjacent nodes. The elastic constants for tin and copper are known
73
Figure 5.1 Simplified model of a of tri-crystal microstructure in half-joint configuration, with
coordinate system used for Euler angles.
74
Figure 5.2 Geometry and dimensions of (a) lap-shear samples, and (b) solder balls used in this
study. The sense of shear is to the left on the upper surface of the joint.
75
and tabulated in Tables 5.1, and 5.2, respectively. The parameters used for hardening in Table
5.4 were chosen so that they generated trends that are commonly observed in polycrystal or
multi-joint experiments (Darveaux et. al (2005)) and various single crystal experiments (Zhou et.
al (2009), Bieler et al (2012)).
5.2 Single shear lap simulations
Single lap-shear samples with different crystal orientations were computationally deformed to
identify how the shear stress-shear strain behavior depends on crystal orientation. The behavior
of one element in the center of the joint is illustrated in Figure 5.3 for five different crystal
orientations.
In each computation, all elements in the joint started with the initial crystal
orientation indicated by the Bunge Euler angles 1--2. Deformation of the orientation 75°45°-0° showed a smooth response similar to measured data from shear-lap samples consisting of
a group of joints (Darveaux et. al (2005)). Hardening parameters were adjusted to the values in
Table 5.4 so that this orientation matched the Darveaux data, as illustrated in Figure 5.3. These
hardening parameters for the slip system flow curves were used for all subsequent simulations in
this chapter. It is interesting that based upon these hardening parameters, the response of the
joint varied with different initial orientations; some provided higher stress-strain behavior, others
lower. A couple of the orientations had changes in the flow behavior at different places along
the stress-strain curve (Figure 5.3). The parameters for these phenomenological flow curves
(Table 5.4) are considered estimates that serve as a starting point for iterative development of
improved model parameters, as comparisons between experimental data and simulations of these
experiments progress.
76
Table 5.1 Elastic C onstants (GPa) of Tin used in numerical analysis
Table 5.2 Elastic constants (GPa) of Cu.
C11
C12
C44
178
76
45
Table 5.3 Hardening parameters of tin for different slip systems used in numerical
analysis
Table 5.4 Element Types and number of elements used in CPFE.
Material
No. of elements
Type of element
Tin in lap-shear
360
C3D8R
Cu in lap-shear
2780
C3D8R
Tin in solder ball
494
C3D8R
Tin in Tri-Crystal
1065
C3D8R
77
Figure 5.3 The orientation 1--2 = 75°-45°-0° was used to estimate crystal plasticity model
parameters using experimental data of Darveaux et al. (2005). Using these model parameters,
other orientations deformed differently in simulated single shear lap deformation. The
distributions of von Mises stress in the joint are shown to illustrate effects of orientation on stress
distribution.
78
The shear strain on the most active slip systems that contribute to the shear stress-shear strain
behavior in Figure 5.3 are plotted in Figure 5.4. The effect of conjugate slip systems is evident
in Figure 5.4, as there are four slip systems where the plane normal and slip directions are
perpendicular to each other, so they have the same Schmid factors. For example, the slip system
(001)[100] and (100)[001] have the same Schmid factor, but the (100)[001] is expected to be
more facile and is frequently observed, due to a smaller Burgers vector. Slip on the (001)[100]
system has been observed (Zhou et. al (2009)), but less frequently. Thus, as the hardening
parameters for Sn are refined, it is expected the (001)[100] should have greater slip resistance
than (100)[001], but at this point they are made equal (there are similar conjugate systems on
(010) planes and [010] directions, too). The other two pairs of conjugate slip systems, e.g.
and
should truly be equally balanced, as they are members of the same
family (#9 in Table 1.1, and similarly for (011) planes and [011] directions).
As can be seen for orientation 75-45-0 in Figure 5.4, there are more slip activity on
than any other slip system throughout the deformation. There is some shear strain developing
on
and
slip systems, but their magnitudes are small and therefore have a
small impact on the evolution of the shear stress. The combined effect of slip activities on these
systems leads to a smooth and moderate hardening behavior that is expected for predominantly
single slip orientation with some activity on secondary slip systems. In contrast, the 45-60-0
orientation had four slip systems with similar activity at the early stage of strain, and the least
active of these top four became the most active at larger strains. With four systems initially
favored, the yield stress was low, but the initial hardening rate was high. The stress-strain plot
shows several changes in the slope that reflects changes in the dominant contribution of different
79
Figure 5.4 Shear strain in one element in the middle of the joint in the single shear lap geometry.
Some slip systems have conjugate partners that are 90 apart, and have the same Schmid factors;
these two distinct systems are illustrated with lines and overlaid dots.
80
slip systems and jumps in flow stress, where some latent systems became hardened when they
were favored at larger strains.
For the 90-75-0 orientation, two pairs of conjugate slip systems contributed to the hardening
process, and changed the slope, but one pair dominated in the beginning, (100)[001], and later,
the second pair,
, gradually became favored. As this second system became active,
the flow stress and hardening rate started to increase at a strain of about 13%. In contrast, for the
60-90-0 orientation, the same two pairs of slip systems were both active in the beginning, and
(100) [001] became favored with increasing strain. The operation of all four at the beginning led
to the highest initial flow stress, and as the
pair became less active, the hardening
rate decreased. In the 75-90-0 orientation, the
pair was not very active until much
later, resulting in a low initial hardening rate that led to softening, presumably due to rotation to
a more favorable orientation with a lower Schmid factor, but eventually, the
pair
became more active, and led to hardening at large strain.
5.3. Tri-crystal simulations
Tri-crystals having a triple junction in the center of the joint were computationally deformed,
using orientations defined in Figure 5.1 and Table 5.5. These orientations are all related by 60
rotations about the crystal [100] axis to provide misorientations across grain boundaries that are
similar to beach ball morphology, but geometrically simpler; similar geometries are sometimes
observed. The computational model was deformed to a simple shear strain of 0.84. The average
shears on active slip systems for all elements in each initial orientation in Samples A, B, and C
are laid out for comparison in Figure 5.5, where each column represents a different orientation
set permutation, and each row represents the same crystal orientation. The colors of lines are
81
Table 5.5. Bunge Euler Angle values for different tri-crystal sets
1 (φ1 Φ φ2) °
2 (φ1 Φ φ2) °
3 (φ1 Φ φ2) °
A
120,90,0
180,90,0
240,90,0
B
240,90,0
120,90,0
180,90,0
C
180,90,0
120,90,0
240,90,0
D
150,90,0
30,90,0
90,90,0
82
coded to the slip systems used in Table 1.1. Activity of multiple members of a slip system
family are given the same color, but have different line thicknesses and dashes for specific
systems as indicated in the legends.
Study of the shear response in each row shows that the shear deformation behavior in position 1
(along the diagonal) differs from the activated systems when the crystal is in one of the other two
positions. This shows that the activation of slip systems depends on both the geometrical
position and the orientation of neighboring orientations. For example, in the first row of Figure
5.5, for orientation 120-90-0, the activity of the
exceeds the other slip systems
substantially in position 1, whereas in the same orientation in positions 2 and 3 still show this
system as most active, but there is more balanced activity on several other similar slip systems.
In the third row, for orientation 180-90-0, (100)[001] is dominant when this orientation is in
position 1, but not when it is in position 2 or 3. These observations indicate that the deformation
kinematics of position 1 differs significantly from positions 2 and 3.
The activated slip systems in Figure 5.5 commonly show activity on slip system 9,
,
but this system is generally thought to be less easily activated than other slip systems. Thus the
yield stress for activation of this slip system was increased by a factor of 10, and the same
deformation path was imposed, and the difference in activity of slip systems is shown in Figure
5.6.
By suppression of the slip on system #9, slip on systems 4 <111>{110) and 10
were more highly activated. Figure 5.7 illustrates a different permutation (D) of the
crystal orientations where the first Euler angles were all smaller by 90 from the (B) orientation
83
set for the two conditions described above. Suppression of slip system 9 clearly requires greater
activity on slip systems 4 and 10, and this figure makes it easier to see this comparison.
Figure5.8 shows the distribution of von-Mises stress in different tri-crystals at a shear strain of
0.84. The distribution of von-Mises stress at upper left corner of the solder balls is strongly
affected by the crystal orientation; orientation sets B and C have the same orientation in the
upper left, and both show greater resistance to shear deformation. Elsewhere, the von-Mises
stress distribution is not much different for different orientation sets, though there are
discontinuous stress jumps across the upper grain boundary for cases A and D. The effective
stress value averages out heterogeneous deformation conditions if the magnitudes of strains are
similar on different systems, and from the perspective of understanding slip behavior, it does not
reveal much insight.
However, following the experimental observations by (Bieler et.al (2009)), it is likely that slip on
both systems 9 and 10 are less easily activated, so von Mises stress plots for deformation with
slip system 9 suppressed in Figure 5.9 at a lesser shear strain of 0.5 shows stresses that are a bit
more heterogeneous, as less favorable slip systems need to be activated. With the additional
suppression of slip system 10, the von Mises stress in Figure 5.9 at a shear strain of 0.5 is
distributed much more heterogeneously, and the stresses are very large. There are bands of high
stress in orientation sets B and C, and large stress discontinuities are evident at the triple point
and along the lower right grain boundary in orientation sets C and D.
Also, differential
deformation in grains with different orientations is much more apparent (compare the right side
of orientations A and C in Figure 5.10 with the right side of the tricrystal in Figure 5.3). As a
84
A
B
C
Figure 5.5 Average shear strain on the most active slip systems are compared in different
orientation sets A, B, C, to illustrate the effects of location and interaction between different
orientations. Slip system facility is based upon Table 2.1 Each row has the same crystal
orientation (Table 5.4), but in a given row, the upper right grain (position 1, on the diagonal),
exhibits different strain evolution than when the same orientation is in the other two positions.
85
A
B
C
Figure 5.6 Average shear strain on the most active slip systems are compared in different
orientation sets A, B, C, to illustrate the effects of location and interaction between different
orientations. Slip system facility is based upon Table 5.3, but slip system 9 is suppressed. Each
row has the same crystal orientation (Table 5.5), but in a given row, the upper right grain
(position 1, on the diagonal), exhibits different strain evolution than when the same orientation is
in the other two positions.
86
D
Figure 5.7 When slip system 9 <101]{10-1) is suppressed, to be more consistent with
experimental observations ( Zhou et al (2009)), activation on slip systems 4<111>{110) and 10
<211]{-101) increases.
87
B
A
C
D
Figure 5.8 Distribution of von-Mises stress in different tri-crystals deformed to a simple shear
strain of 0.84 with orientations defined in table 5.5 is rather similar. The grain orientation 12090-0 in the upper left position resists shear more effectively than other orientations.
88
B
A
C
D
Figure 5.9 Distribution of von-Mises stress at a strain of 0.5 in different tri-crystals
with orientations defined in table 5.5 differs when slip system 9 is suppressed. The grain
orientation 120-90-0 in the upper left position in orientation sets B and C resists shear more
effectively than other orientations.
89
higher stress is the outcome of more hardening, this implies higher total shear strain activity in
the band of elements with higher stress. Such shear banding is occasionally observed in joints,
especially in the form of orientation gradients surrounding a shear band.
One way to illustrate the presence of an orientation gradient is to plot the generalized Schmid
factor based upon the local stress tensor for the most active slip systems in elements that go
across the shear band. Figure 5.11 shows how the generalized Schmid factor varies spatially; an
increase in the Schmid factor in one slip system and a corresponding decrease in another slip
system shows that a substantial orientation gradient developed in regions of high stress. Under
such conditions, it is not as meaningful to plot average shear for all elements in each grain
orientation (as was done in Figures 7-9) due to the heterogeneous deformation. Indeed, shear
activity plots from the dozen elements in the upper left corner in the simulation shown in Figure
5.8 differs substantially from the average (not shown). The degree of heterogeneous strain in
Figure 5.10 is more dramatic than is commonly observed in solder joints, such as Figure 2.6 or
shear deformation samples described in references (Bieler et.al (2009)), so complete suppression
of slip systems 9 and 10 is probably too extreme, as reference (Bieler et.al (2009)) shows
evidence for some activity on systems 9 and 10. Further optimization of model parameters using
comparisons with experiments will be needed to identify more suitable rules for stress evolution
among the 10 slip systems in Sn, and this will be addressed in future chapters. However, it is
clear that the crystal plasticity finite element approach will be able to capture realistic
heterogeneous deformation phenomena, and provide a means to predict slip behavior in SAC
solder joint in a form that can be used with computational models.
90
y
B
A
D
C
Figure 5.10 .Shear deformation to a simple shear strain of 0.42 where slip systems 9 <101]{10-1)
and 10 <211]{-101) are suppressed to be more consistent with experimental observations (Zhou
et.al (2009). Differential strains at the grain boundary on the right are evident in orientation sets
A and C (compare with Figure 2.6)
91
Figure 5.11 The generalized Schimd factor (computed from the stress tensor in each element)
varies substantially with the crystal orientation in the elements along the line in the upper left
grain orientation in orientation set C in Figure 5.10. Slip system 4<111>{110) has the greatest
resolved shear stress and is greater than on suppressed slip systems 9, <101]{10-1), and 10 <101]{121).
92
5.4 Summary
Crystal features such as special morphology as well as anisotropy present in lead free tin based
solder joints have significant effects on the mechanical properties of sub-mm solder joints.
CPFE models can solve the mechanical problems under complicated external and internal
constraints imposed by grain morphology. In this study the effects of both anisotropy and grain
morphology are examined using a very simple phenomenological constitutive model. Hardening
behavior based upon information in the literature was used to simulate several different
geometrical morphologies with different crystal orientations (Table 1.1).
Comparison of tri-
crystals with different permuted sets of orientations shows that the activity of slip systems
depends on both position and orientations of neighboring grains. The simulations showed
deformation characteristics similar to experimental observations when less facile slip systems
were suppressed. For example, shear bands and orientation gradients are observed in von-Mises
stress contours and Schmid factor calculations. While the qualitative behavior of the model is
realistic, further improvements to the model require optimization based upon direct comparison
with well characterized specific experiments.
93
CHAPTER 6
CRYSTAL PLASTICITY FINITE ELEMENT STUDY OF DEFORMATION
BEHAVIOR IN COMMONLY OBSERVED MICROSTRUCTURES IN
LEAD FREE SOLDER JOINTS
As it was discussed in chapter 2, the strongly anisotropic elastic modulus and coefficient of
thermal expansion (CTE) play an important role in thermo-mechanical loading of solder joints,
leading to complex heterogeneous stress states acting on joints [Bieler et. al (2008)]. Due to this
anisotropy, crystal features such as grain size and grain morphology have significant effects on
the mechanical properties and reliability of sub-mm scale joints. There are very few prior studies
about modeling the real microstructure in lead free solders. Park et al. (2008) used an anisotropic
linear elastic constitutive model in an FEM simulation to simulate three-dimensional elastic
strains measured experimentally in lead-free solder balls.
Although they could predict the
location of damage near the grain boundary, the observation of plastic deformation by Bieler et
al. (2012) in the samples that experience thermo-mechanical loading suggests that application of
more sophisticated plastic constitutive models are required for prediction of damage evolution in
lead free solders that fail by ductile fracture caused by plastic deformation.
More recently, Maleki et al. (2013) used a J2 plasticity constitutive model to investigate the
effect of aging conditions on deformation behavior of the eutectic micro-constituent in SnAgCu
lead-free solder.
Although their study accurately modeled the geometry of Sn grains and
intermetallic, the isotropic plasticity associated with J2 model is unrealistic and oversimplified.
Darbandi et al. (2013) used a crystal plasticity model to investigate the effect of grain orientation
and its relationship with morphology (grain position within a simplified geometry of solder ball
microstructure).
Specifically, this study investigated the complex interaction between the
94
activity of slip systems, location of a particular grain within the solder ball and orientation of
different grains.
Miniaturization is a very important issue in solder joints because the dimension of a joint is
similar to the size of the Sn grains.
The design of solder joints will increasingly require
considering the grain scale anisotropy. Since the damage evolution highly depends on stress,
strain, and the activity of slip systems, providing a model that can predict the local stress and
strain state will be helpful from both a scientific and industrial point of view. Furthermore, the
use of a microstructure scale FE mesh in a crystal plasticity finite element (CPFE) model is also
important for interpreting experimental measurements, because at these scales experimental
boundary conditions are very difficult to measure or monitor.
In this study, a CPFE model is used to account for deformation mechanisms of, and anisotropy
associated with the slip phenomena in Sn to compare with corresponding experimental
characterization of shear deformation, to assess the capability of the CPFE model to predict the
kinematics of plastic deformation and evolution of microstructural features in different solder
joints. Firstly, two different single crystal orientations were investigated to evaluate the
capability of CPFE to predict the kinematic plastic deformation. Next, the beach-ball
microstructure which is commonly observed in lead free solder was investigated, in two different
samples to evaluate the reliability of CPFE model to predict the plastic deformation in lead free
solders. Thus, this work provides a basis for an integrated incremental model development
strategy based upon experiments, modeling and comparative analysis.
Four specially prepared samples of a 4×4 ball grid array (BGA) were cut from a 15 mm x 15
mm body-size plastic ball grid array (PBGA) package with Sn-3.0Ag-0.5Cu (wt.%) solder joints.
The solder joints were approximately 0.5 mm in diameter. The samples were aged at 100°C for
95
500 hours, sectioned, and carefully polished (without metallurgical mounting) so that the plane
of shear at the center of the joint could be observed on one set of four balls. A fixture shown in
Figure 6.1 was used to impose a shear strain, so that evidence of local strain effects could be
detected topographically by small variations in out-of-plane shear on the previously polished
surface.
Optical microscopy, analysis of polarized light microscopy, and orientation imaging
microscopyTM (OIM), were conducted initially, and after the deformation. To permit accurate
orientation measurements, samples were lightly re-polished after the deformation to improve
surface smoothness. The resin in the package and board were carefully painted with carbon paint
to cover non-conducting surfaces. Also, copper tape was used to provide a conductive path to
prevent the charging that causes image distortion.
OIM data were obtained using a CamScan 44FE electron microscope. The accelerating voltage
was 20 kV and the working distance was 33 mm. A TSL EDAX OIM system with a digiview
SEM detector was utilized to generate the orientation maps. Maps were cleaned up using the
neighbor confidence index (CI) correlation to replace pixels that had low CI with their neighbors
having a high CI. This step is required to remove the minority data points coming from unindexed or wrongly indexed pixels, typically arising from intermetallic phases, which were not
considered in this work. A user defined crystal orientation map was employed to correlate the
orientation of the Sn c-axis with respect to solder/package interface. For this orientation map 5
colors (Purple-blue-green-yellow-red) were used to represent the orientation of the c-axis with
respect to the solder/package interface. When the c-axis is on the interface plane, the orientation
is expressed as ―red‖, and when the c-axis is perpendicular to the surface the orientation is
―purple‖. Other colors (blue, green and yellow) fill the rest of the c-axis orientation space.
96
Figure.6.1 Schematic diagrams of a 3-D visualization of the CPFE mesh located in a 4 × 4 solder
joint array sectioned from a PBGA package. The solder ball assembly geometry and shear test
apparatus is also shown.
97
The finite element analysis was performed on four samples using the commercial finite element
code ABAQUS.
The crystal plasticity material model was implemented using a user material
subroutine in FORTRAN (VUMAT). Details of the model can be found in Darbandi et al.
(2013).
The joints were modeled as asymmetrically truncated spheres to correspond with
measured cross sections. Element types and number of elements used are described in captions
of relevant figures. The material parameters used are shown in Tables 5.1 and 5.3.
A 3-D visualization of the CPFE mesh is presented in Figure 6.1, showing the position of one of
the joints analyzed with respect to the rest of the package.
Two different single crystal joints (Figures 6.2-6.4) and two different beach ball morphology
joints (Figures 6.5-6.7) were studied to investigate the capability of CPFE to model the
kinematics of deformation and microstructural features. The two single crystals were subjected
to 0.65 mm displacement. One of the beach-ball solder balls was subjected to 0.1mm shear
displacement, and the second one was displaced by 0.4 mm.
In order to reconstruct the real microstructure of a joint with beach ball morphology from OIM
and polarized light microscopy measurements, one can assume that the beach ball microstructure
is part of a sphere that is equally divided by three planes into six parts with the same volume.
Due to the fact that the beach ball microstructure can be produced by 60° rotations about an
axis, the cutting plane can be assumed to be a plane with its normal parallel to a common <100}
axis. This modeling method allows the morphology to be correlated appropriately with
crystal orientations. (The morphology obtained using the polarized light microscopy is not
sufficient to reconstruct the microstructure because it does not give the orientation of the
common axis required to determine the geometrical inclination of the grain boundary
through the depth of the solder balls).
98
6.1. Modeling of single crystal solder balls using CPFE
Figure 6.2(a, b) shows two different undeformed single crystal joints. The crystal orientations
overlaid on the figures in (c, d) show that the c-axis is nearly parallel to the interface according
to the c-axis color code. While both of these joints have a ‗red‘ orientation, the c-axis direction
differs with respect to the plane of shear. (The top of the map in both joints is exaggerated in the
vertical direction due to the topographic effects of the polish and the 70 tilt of the sample during
the measurement).
The optical images in Figure 6.2(e, f) show that deformation was
concentrated in the upper part of the joint, where the cross sectional area was the smallest. The
OIM maps for the samples after deformation in Figure 6.2 (g, h) show that the orientation of the
c-axis with respect to the interface did not change much (same color), but rotation is observed in
both samples about the c-axis, especially in the highly deformed area near the top.
In both joint orientations, the crystal rotated about the c-axis in the direction of shear about an
axis near the [001] direction. In Figure 6.2(g), the c-axis is inclined to the plane of shear, and the
rotation axis is close to the c-axis.
In Figure 6.2(h), the c-axis rotation axis is nearly
perpendicular to the plane of shear. In both cases, modest rotations about an axis near (001)
imply that slip on the (110) plane is facile and stable.
For the simulation, optical micrographs (Figures 6.2(a, b)) were used to define the geometry for
CPFE analysis. The initial orientations were used to establish the basis for the kinematics of
plastic deformation (Figure 6.2 (c, d)).
The simulations also show that large amounts of
deformation took place in the upper right area indicating the localization of plastic deformation
in these regions in Figure 6.3. The deformation simulated by the CPFE model compared closely
to the deformed shapes observed in the optical microscopy micrographs. This shows that the
99
a
b
c
d
f
e
g
hh
Figure 6.2 Optical micrographs of 2 different solder balls with corresponding c-axis orientation
maps with respect to the substrate normal direction, before (a-d) and after (e-h) 0.65 mm shear
displacement. After shear, strains are concentrated in upper right regions. The c-axis orientation
was retained (no color change) as the crystal rotated about the c-axis (note overlaid unit cell
prisms) it during shear.
100
b
a
Figure.6.3. Comparison of simulation and experimental results for single crystal deformed solder
joints illustrated in Figure 6.2. 565 C3D10M elements were used in (a) and 569 C3D10M
elements were used in (b).
101
crystal plasticity model can accurately predict the kinematics of plastic deformation due to the
shear loading.
Figure 6.4 shows the surface features on the previously polished surface of the joint in Figure
6.3(b). The micrograph shows different plane traces on the sample surface in different locations.
Using plane trace analysis based upon local orientations measured after deformation (details in
Zhou et al.(2009)), slip on {110)* planes is dominant, The complimentary shear strains required
for shear deformation are apparent in the perpendicular {110) slip traces. The highest Schmid
factors (m, based upon a pure shear stress assumption) are for slip in <110] directions, but they
are also high for slip in <111> directions. The most active slip systems in the simulation in
Figure 6. 4(b) also indicates that the most active slip systems are those with high Schmid factors.
To assess the most active slip systems from experimental measurements, rotation axes can
indicate which of the four slip directions on a {110) plane was the most active using the
orientation map data in Figure 6.2(d,h). The rotation axes were determined between the initial
and final orientations in nine locations evenly distributed in a 3x3 grid (after correcting for the
sample alignment in the measurements). In each of the nine locations, the rotation angle was
5.4, about an average rotation axis of <-154.5 -6.53.7 100]. This is an average of 5.44
from the <-15 -15 100] rotation axis that is 90 from the {1-10)<111] slip plane normal and slip
directions. As the <-15 -15 100 axis is 21 from the [001] axis, the measured rotation axes are
much closer to the expected rotation axes for <111> slip than the [001] rotation axis for <110]
slip (these vectors are illustrated in Figure 6.4(b)). This indicates that the <111> slip system is
much more active than the <110] slip direction. Furthermore this indicates that one of the <111>
directions is strongly favored, because equal amounts of slip in the two <111> directions in the
* Mixed { ) and < ] for planes and directions recognize the symmetry of the tetragonal crystal structure.
102
Figure 6.4 (a) Optical image of joint b in Figures 6.2 and 6.3, showing local unit cell
orientations with shaded slip planes and corresponding plane traces, and slip vectors (blue lines)
with high Schmid factors (there is some evidence for (121) slip in the highly sheared upper right
part of the joint). Red and yellow-green edges of the unit cells represent the crystal x and y axes,
respectively. (b) Comparison of the average activity in the simulation of the four most active slip
systems at each integration point in the single crystal joint.
103
same {110) plane will give the equivalent of <110] slip. Figure 6.4(b) shows the average amount
of shear on the four most active slip systems obtained from CPFE model (which is based on the
average initial orientation).
This simulation shows that (1-10)[110] slip is dominant, which is
not what the experiment shows, but this is a consequence of no significant difference in the
initial slip resistance on the various slip systems used in the simple material model used . This
experiment and others like it will lead to future refinements in the model. Nevertheless, even
this overly simple material model is able to effectively capture the kinematics of deformation.
6.2. Modeling of heterogeneous plastic deformation in Beach-ball microstructure of solder joints
In order to simulate the behavior of solder balls in different locations and at different stages of
thermo-cycling in a real ball grid array package, the behavior of solder balls subjected to both
small and large shear displacements have been investigated. From examination of incremental
strains resulting from small to larger shear displacements, it is apparent that orientations near the
center of the ball do not change very much, so the final orientation in the central area can be used
as a good approximation for the initial crystal orientation. This allows analytical comparison
with prior characterization studies where grain orientations were characterized only after the
deformation. Two cases are presented: the first, where the orientation was measured before and
after a small shear displacement (Figures 6.5 and 6.6), and second, a sample characterized in
prior work where solid modeling of the beach ball and sectioning based upon orientations in the
center of the deformed sample were used to identify a highly probable initial microstructure
condition (Figures 6.5 and 6.7).
Figure 6.5 shows the measured tricrystal orientations in pole figures, and the microstructure,
mesh, and deformed mesh for the two joint samples studied. The c-axis OIM map for a beach
ball sample before and after 0.1 mm shear displacement is shown in Figure 6.6 (a,b). A
104
Figure 6.5 Pole figures obtained from OIM for two solder joints with beach ball microstructure
(a, b), FE meshes colored according to the c-axis color scale in the vertical direction before
deformation (c, d) and after deformation, (e, f) showing shape after indicated displacements were
imposed and released. 2550 C3D10M elements were used in (c, e) and 2600 C3D10M elements
were used in (d, f).
105
Figure.6.6 (a) X direction c-axis orientation maps for beach ball microstructure before
deformation and, (b) after 0.1mm shear, backscattered SEM image tilted -45 about X axis
showing a ledge in the lower left area (c), and a path along which misorientation and topography
are traced in the experiment (b). A similar trace in the CPFE simulation (d), is plotted with
experimental data in (e). The CPFE model is geometrically simpler and does not include the
material indicated by the dotted line in (d), but it is able to semi-quantitatively capture the
localized deformation observed experimentally in the lower left corner.
106
Figure.6.7. Polarized light micrograph (a) of beach ball microstructure that experienced a large
shear displacement of 0.4 mm (red line), deformed solder ball predicted by CPFE illustrating
distribution of shear stress in the 1-2 plane indicating a higher stress due to more shear
localization in the upper area.
comparison of the c-axis map before and after deformation reveals the most noteworthy change
to take place in the crystal orientation due to localized plastic deformation in the lower left
corner of the c-axis map. In this area, the orientation color changed from green to blue, as made
evident by overlaid prisms. A corresponding backscattered electron micrograph in Figure 6.6(c)
indicates the formation of a ledge in the lower left corner area, where significant local orientation
changes were measured with OIM. The image of the surface in Figure 6.6(c) is tilted so that it
makes the ledge easier to see. The tilted image was one of a stereo pair from which it was
possible to prove that the material to the right of the ledge was at a lower elevation than the
material to the left.
Figure 6.6(e) illustrates the change in surface topography obtained using the CPFE analysis. It is
apparent that the CPFE model was able to predict a negative displacement from strain
localization similar to what happened in the real experiment.
Figure 6.6(e) compares the
orientation and topography along a diagonal path that crosses the region where the topographic
107
features were observed (diagonal lines in Figures 6.5(d, c)). As shown in Figure 6.6 (e), the
crystal orientation changed along this path by about 25 from the outer edge of the sample, and
the surface elevation dropped by 1 micron at the ledge, and then came back up to the original
height at the copper interface. The path in the CPFE simulation also shows a similar trend,
though the large size of the elements makes for a rough representation of the topography. Also,
because the CPFE model was geometrically simpler, without modeling the shape of the Cu pad
around which the solder flowed, the experimental geometry has more solder in the perimeter
region. However, when the trace covers similar paths with respect to the corner of the copper
pad, the surface topography from the CPFE matches the experimental trends reasonably closely
(noting that a ledge cannot be predicted by the model).
Figure 6.7 shows a simulation of the microstructure of a different deformed sample for which the
initial condition was not measured. With computational deformation, the distribution of
strain is quite similar to the observed strain in the real microstructure. The differences in
shape made evident by overlaying the shape of the CPFE model on the experimentally deformed
sample may arise from the fact that initial solder joints are not always perfect truncated spheres
(that were modeled), so it is not possible to assess the differences between the model and the
experiment quantitatively. In particular, the localization of shear strain in the upper grain,
which is a ―red‖ orientation (where the orientation of the c-axis is parallel to the interface)
shows consistency with the real microstructure where the greatest amount of shear occurs in
the ―red‖ orientation. This is an outcome of slip along the c-axis, which is known to be a facile
slip system (Bieler et. al (2012)).
108
6.3. Summary
In this chapter, the capability of CPFE modeling was examined using shear deformation applied
on real microstructures of SAC305 solder balls.
The simulation results clearly show the
capability of crystal plasticity model to predict the heterogeneous strain that is observed in
correlated physical samples.
The models used estimates of material properties that are
considered to be reasonable based upon limited prior study of slip behavior in Sn. In addition to
the kinematics of plastic deformation, more details can be predicted using this model, such as
accurate prediction of some of the microstructural features such as formation of ledges as well as
activity of slip systems. Therefore, this model can be used as a useful tool to predict damage, as
the damage phenomena is controlled by the plastic deformation, and it can be further refined
with constraints obtained by detailed comparisons with experimental data.
109
CHAPTER 7
SLIP SYSTEM ACTIVITIES OF SAC305 SOLDER BALLS UNDERGOING
SIMPLE SHEAR DEFORMATION
As discussed in chapter 2, since the anisotropy in plastic deformation emerges naturally from the
activity of slip systems, and due to the fact that plastic deformation is the main mechanism
responsible for fracture in ductile metals in different strain rates and temperatures, understanding
of the relative activity of different slip provides valuable insights into the inhomogeneous
deformation of interconnections in electronic industry. It also can help us to develop more
reliable modeling tools based on crystal plasticity finite element modeling.
A review about the plastic properties of tin by Yang and Li (2007) indicates the need for better
understanding of the deformation mechanisms of tin. There is incomplete understanding of slip
in Sn, though single crystals have been investigated by several researchers. There is especially
uncertainty about the critical resolved shear stress for different slip systems. Evidence presented
by Düzgün et al. (1999, 2003) suggests that the initial critical shear stress is similar on different
slip systems, but they have different rates of hardening. In these studies, a limited set of crystal
orientations have been investigated using tensile tests, and also the effect of alloying and strain
rate were not considered. Table 2-1 presents an estimate of the relative ease of activating the 10
relevant slip systems based upon an assessment from literature data. This table which was
developed by Fujiwara and Hirokawa (1987) based upon the etch-pit / hillock study is not
extensive. In addition to this method, other researchers (Matin et al. (2006) and Sidhu and
Chalwa (2008)) defined a parameter called ―effective yield strength‖ which is the value of
critical resolved shear stress over the Schmid factor for different slip systems. In this study, the
authors used values of the critical resolved shear stress of pure tin, which can be different from
110
SAC305 due to the effect of alloying elements. Friedel (1964) and Labusch (1970) showed that
the critical resolved shear stress increases with alloying elements. However, for certain slip
systems some assumptions based on the linear atomic density were made to calculate the critical
resolved shear stress, which are not experimentally validated.
Kinoshita et al. (2012) recently used the first-principles density functional theory to study the
activity of slip systems in pure tin. They used the ideal shear resistance for calculation of the
critical resolved shear stress. They employed a tensile test study to assess the activity of
different slip systems and investigated the effect of crystal orientation on relative activity of slip
systems. Zhou et al. (2009) used a methodology based on the OIM and calculating the Schmid
factor together with partially observing slip traces to assess the activity of slip systems on shear
deformed samples. Their study implied that slip in [001] and [111] directions is likely, and slip
on (010)[101] may contribute significantly.
In this chapter the activity of slip systems has been discussed using the shear test studies on lead
free solder joints in different orientations by applying shear deformation in three separate steps.
OIM and SEM were utilized to understand how the crystal rotates during the plastic deformation
after each deformation step.
In addition, using optical microscopy and polarized light microscopy, plane traces were analyzed
to understand the activity of slip systems. Thirty two solder balls were analyzed to provide
enough information to sort out the slip systems in terms of activity. Also, the activities of slip
systems were categorized based on the crystal orientation, in order to show the effect of the
crystal orientation on facilitating the operation in different slip systems.
111
7.1. Experimental procedures
Eight specially prepared samples of a 4×4 ball grid array (BGA) were cut from a 15 mm x 15
mm body-size plastic ball grid array (PBGA) package with Sn-3.0Al-0.5Cu (wt.%) solder joints.
Four (4) samples were aged at 100ºC for 500 hours. The samples were sectioned and carefully
polished so that the plane of shear at the center of the joint could be observed on one set of four
balls before and after shear displacements.
Due to the path dependency of plasticity, the total displacements were applied in three separate
steps in order to obtain the information about crystal rotation during the shear deformation. A
fixture was used to impose a two-dimensional shear strain, so that small variations out of the
plane of shear resulting from local strain effects could be detected topographically on the
previously polished surface. The loading conditions and pre-aging history of each sample is
reported in Table 7.1. Figure 4.4 shows the schematic of the apparatus which was utilized in this
experiment. To permit accurate orientation measurements, samples were lightly repolished to
improve surface cleanliness.
Optical microscopy, analysis of polarized light microscopy,
orientation imaging microscopy (OIM) and SEM were conducted initially and before and after
each deformation step. The resin surrounding the balls and the package was carefully painted
with carbon paint to cover the non-conducting surfaces. Also samples were covered by the
copper tape to provide a conductive path and to prevent the charging effect which causes image
distortion.
OIM data were obtained in a CamScan 44FE electron microscope. The accelerating voltage was
set to be 20 Kv and the working distance was 33 mm. A TSL EDAX system with digiview SEM
detector was utilized.
Acquisition speed of 15- 20 points per second was obtained using 2×2
binning setting. In order to generate the orientation maps, EDAX TSL OIM Analysis software
112
Table 7.1 Test conditions for performing the simple shear tests.
was employed. Maps were cleaned up using the neighbor confidence index (CI) correlation to
replace pixels that had low CI with their neighbors having a high CI. This step is required to
remove the minority data points coming from un-indexed or wrongly indexed patterns. In the
neighbor-confidence-index-correlation method, a datum point having low CI is replaced with
neighboring points that have the highest CI value. In the presentation of results, Sn unit cells
were overlaid on OIM maps.
A user defined orientation map was employed to correlate the orientation of Sn c-axis [001] with
respect to solder/package interface.
For this orientation map, 5 colors (Purple-blue-green-
yellow-red) were used to represent the orientation of the c-axis with respect to the solder/package
113
interface. When the c-axis is on the interface plane the orientation is expressed as ―red‖, while
when the c-axis is perpendicular to the surface the orientation is ―purple‖. Other colors (blue,
green and yellow) fill the rest of the orientation space.
7.2. Slip-trace analysis and Schmid factor calculation
Activity of slip systems can be correlated with microstructure and grain orientation using
calculation of Schmid factor and plane trace analysis. One can represent the crystal orientation
using three (Bunge) Euler angles
, ,
.
Using the concept of the rotation matrix (g) in linear algebra, three basic rotation matrices rotate
vectors about the x, y, or z axis to be able to relate the crystal coordinate system to the global
coordinate system using the definition of Euler angles defined by Bunge (1982) as follows:
(7.1)
(7.2)
(7.3)
And general rotation matrix can be expresses as:
114
(7.4)
The slip plane normal and slip direction can be expressed by Miller indices as (h k l) and [u v w]
respectively. Since slip traces are observed using the optical or electron microscopes, the slip
plane and direction have to be represented in global coordinate system using the general rotation
matrix. Due to the fact that the plane trace is the intersection between a slip plane and the plane
of a sample that the trace is observed, plane traces can be calculated as a cross product of slip
plane normal (in the global coordinate system) and normal to the sample surface.
Calculation of the Schmid factor depends on the stress definition, but generally this parameter
has to be calculated in the local coordinate system. For a particular slip system with the
normalized plane normal and slip direction defined respectively as
and
, the Schmid factor
can be expressed as:
(7.5)
Evidence for operation of slip systems based on the calculation of Schmid factor can be assessed
using optical images and orientation maps.
In order to sort the activity of slip systems, observation of the plane trace and Schmid factor
calculation have to be accomplished to rank the slip systems in terms of slip activity. A Matlab
program based upon the global stress boundary condition of pure shear was utilized for this
purpose.
115
Figures 7.1 and 7.2 show the load displacement curves for step-2 and step-3 of deformation
respectively. It is important to point out that since these curves reflect the hardening of 16 (4×4)
solder balls it is not possible to correlate the deformation behavior of each solder to the loading
history. As it is apparent, general values of force for a particular displacement are higher for
unaged samples. Finer and more intemetallic particles were formed in SAC alloys, and they
strengthened the alloys by hindering the movement of dislocation. After aging by coarsening of
these particles their efficiency in pining and blocking the dislocations decreases and strength of
the solders decreases (Ding et al (2007)). Therefore, it is observed that for aged samples the
plastic deformation (non proportionality in the curves) starts at smaller values of displacements
in both step-2 and step-3.
On the other hand comparison between figures 7.2 (a), 7.2(b), 7.3(a) and 7.3 (b) reveals the
strong influence of deformation temperature on the hardening of SAC 305 alloys. This impact
corresponds to thermally activated phenomena which are more dominant at 100º C.
Figures 7.4 and 7.5 show optical micrographs of aged and unaged shear deformed samples at
various deformation steps. Considerable changes in morphology of grains as well as slip plane
traces are observed at large plastic deformations (step-2 and step-3). Cross-polarized light
images for different shear deformed samples are shown in 7.6 and 7.7. These micrographs
qualitatively show the existence of single crystals or multi-crystals in these sets of samples.
In order to quantify the extent of grain rotation during the plastic deformation one can use the
OIM characterization method. To conduct these investigations more effectively one can
116
Figure 7.1 Sn grain orientation color code used with OIM c-axis maps to identify the c-axis
inclination from the interface.
117
a
b
Figure 7.2 Load displacement diagrams for step 2 of shear deformations at (a) room temperature
(samples 1 and 3 are aged and samples 5 and 7 are non-aged) for (b) aged samples 100ºC
(samples 2 and 4 are aged and samples 6 and 8 are non-aged).
118
(a) Room T
(b) 100ºC
Figure 7.3 Load displacement diagrams for step 3 of shear deformation at (a) room temperature
(samples 1 and 3 are aged and samples 5 and 7 are non-aged), and for (b) aged samples 100ºC
(samples 2 and 4 are aged and samples 6 and 8 are non-aged).
119
Initial
Step-1
Step-2
Step-3
(a)
(b)
(c)
(d)
Initial
Step-1
Step-2
Step-3
500μm
Figure 7.4 Optical micrographs of 4 cross sectioned samples in in the pre-aged condition, showing the deformation in three steps at
room temperature (a,c) and at 100ºC (b,d).
120
Initial
Step-1
Step-2
Step-3
(a)
(b)
Initial
Step-1
Step-2
Step-3
(d)
(c)
Figure 7.5 Optical micrographs of 4 rows of cross sectioned samples in the unaged condition, showing the effects of deformation in
three steps at room temperature (a,c) and at 100ºC (b,d).
121
Initial
Step-1
Step-2
Step-3
(b)
(a)
Initial
Step-1
Step-2
Step-3
(d)
(c)
Figure 7.6 PLM micrographs of 4 rows of cross sectioned samples in the pre-aged conditions, showing the effects of
deformation in three steps at room temperature (a, c) and at 100ºC (b, d).
122
Initial
Step-1
Step-2
Step-3
(a)
(b)
Initial
Step-1
Step-2
Step-3
(d)
(c)
Figure 7.7 Cross-polarized light micrographs of 4 cross sectioned samples in non-aged samples, showing the effects of deformation in
three steps at room temperature (a, c) and at 100ºC (b, d).
123
categorize the grain orientation based upon the c-axis maps (as it is illustrated in Figure 7.1).
This approach was used to demonstrate evoloution of solders crystal orientations at different
steps of deformation for aged and unaged samples by mapping the c-axis based upon the color
code which was described in chapter 4 and demonstrated in Figure 7.1. Figures 7.8 and 7.9 show
the c-axis maps and evolution of crystal orientations and shape changes at various deformation
steps. Due to observation of different single and multi-crystals in terms of c-axis in these set of
32 solder balls, this data set can provide us enough information to conduct a reliable statistical
analysis on activity of slip systems.
In order to conduct more accurate examinations, microstructure and Sn grain orientation
evolution was observed using a fine scan EBSD map, particularly near the package-solder
interface which experienced a higher amount of deformation.
Figure 7.10 shows the optical microscopy, and the c-axis orientation map of a shear deformed
solder ball. The c-axis map shows tri-crystals with twinning relationships where two grains
intersected with the cutting plane. It is apparent in the optical micrograph that slip happens in
neighboring grains and due to the orientation of the grains, two different slip systems contribute
to the plastic deformation.
Comparison of calculated slip planes using the Schmid factor with observations of the plane
trace on the surfaces of solder balls show that in the left grain (010)[101] (6th mode in Table 2.1)
slip system is active. This result is consistent with VPSC simulations conducted by Bieler and
124
Initial
Step-1
Step-2
Step-3
(a)
(b)
Initial
Step-1
Step-2
(d)
Step-3
(c)
Figure 7.8 C-axis EBSD maps of 4 rows of cross sectioned samples in the pre-aged condition, showing the effect is of
deformation in three steps at room temperature (a, c) and at 100ºC (b, d).
125
Initial
Step-1
Step-2
Step-3
(a)
(b)
(c)
(d)
Initial
Step-1
Step-2
Step-3
Figure 7.9 C-axis EBSD maps of 4 cross sectioned samples in the unaged condition, showing effects of deformation in three
steps at room temperature (a, c) and at 100ºC (b, d).
126
Figure 7.10 Optical micrograph of a solder ball showing slip planes in the right and left areas, (b)
c-axis EBSD map corresponding to this solder ball.
127
Telang (2009). On the right side, (1-10)[-1-11] (4th mode in table 2.1) contributes significantly
to the deformation, which shows consistency with observations made by Fujiwara (1987).
Figure 7.11 shows optical microscopy, an SEM Image, and the c-axis orientation map of a shear
deformed solder ball. In regions that have ―red‖ orientations where the c-axis is parallel to the
interface, the sample undergoes larger deformation. As illustrated in the c-axis map, in the area
near the interface there is a grain with red orientation but according to the crystal structure
overlaid on this map, the c direction is nearly perpendicular to the shear direction (the shear
direction is shown using a red arrow in the top area). Therefore, the contribution of slip along
the c-axis is small, and plane trace analysis shows that (-1-21)[101] slip system (10th mode in
Table 2.1) was active in this solder ball. This may be due to the fact that increasing the
deformation temperature can lead to a decrease in the critical resolved shear stress. This result is
consistent with Düzgün et al. (1999) who found out that (121)[10-1] is facile at 90 and 100°C in
pure tin single crystals.
Figure 7.12 shows the SEM image and the magnified c-axis orientation map of a shear deformed
solder ball (an unaged sample which is deformed at 100 ºC) near the interface area. The c-axis
map is illustrated after 0.1mm (b), and 0.42mm (c) deformation. A significant contribution of
(1-10)[111] in plastic defo
128
(-1-21)[101]
Figure 7.11 (a) inset showing magnified BSE images of an area shown in optical micrograph (b)
of sample depicted in (c) to show how the crystal orientation is related to the x-axis slip vector in
the ―orange‖ area, (d) c-axis EBSD map.
129
(a)
0.45 mm
0.1 mm
(b)
Figure 7.12 (a) c-axis EBSD map for deformed sample after step-1, (b) c-axis EBSD map for
deformed sample after step-2, low angle boundaries are illustrated with white lines and high
angle grain boundaries are shown with black lines.
130
rmation along with deformation at 100°C can lead to recrystallization of new grains (orange area
in the left part of Figure 7.12(c), and also green and yellow regions in this figure). Observation
of this slip system shows consistency with the observation made by Fujiwara (1987).
Figure 7.13 (a-c) shows a comparison between different solder balls in terms of the c-axis
orientations (yellow, blue, and red orientations). The figure shows the evolution of the crystal
rotation at different steps in the large deformation regions of solder joints (near the interface). It
is apparent that in the red orientation c-axis, in which a close packed direction is parallel to the
interface, by applying the shear deformation parallel to the interface, a larger amount of
deformation and rotation occurs in this area. However, in the blue orientation where the c-axis is
perpendicular to the interface, the contribution to the deformation is minimal.
This is
demonstrated in Figure 7-14 shows the difference in the distribution of misorientation in steps 3
and 1 for three orientations in (a-c). It is evident that a relatively high amount of increase in the
misorientation (15-25º) occurs at grain boundaries for the red orientation while for the blue
orientation this increase is smaller.
For the yellow-green orientation, a decrease in
misorientation is observed in the fraction of boundaries misoriented by 15-25º. This indicates
that the red orientation has the maximum amount of energy and dislocation content that develops
during the large deformation and may represent the early stages of damage evolution.
Observation of more damage in the red orientation during the thermo mechanical loading can be
related development of more high energy boundaries (Bieler et al (2012)).
Fine-step EBSD c-axis orientation maps and image quality maps for an unaged sample with a
blue orientation deformed at 100 ºC is illustrated in Figure 7.15. The c-axis map clearly shows
131
(a)
Step-1
Step-2
Step-3
(b)
Step-1
Step-2
Step-3
(c)
Step-1
Step-2
Step-3
Figure 7.13 Vertical direction c-axis orientation map in large deformation regions of solder joints
for three samples with different c-axis orientations and different steps of deformation showing
different deformation behavior (a-c). Difference in distribution of misorientation for three
orientations in (a-c) showing an increase in the amount of (15-25º) grain boundaries after the
large deformation step for blue and red orientations and a decrease in yellow-green orientation
(d).
132
Figure 7.14 Difference in distribution of misorientation for three orientations in (a-c) showing
increase in amount of (15-25º) grain boundaries after large deformation for blue and red
orientations and decrease in green orientation.
133
(a)
(b)
(c)
(d)
(e)
(f)
23@ [4 -25 -3] ~ [010]
(g)
85.3@ [6 5 -1]
Orientation
in step-2
50.6 @ [-21 19 -10]
(h)
Step-1
Step-2
Step-3
Figure 7.15 Fine-step EBSD c-axis orientation map and image quality maps for step-1(a, b),
step-2 (c, d), step-3(e, f ) for an unaged sample with blue orientation deformed at 100 ºC, where
low angle boundaries are illustrated with white lines and high angle grain boundaries are shown
with black lines. Rotation of crystal orientation at different locations (as illustrated with red
arrows) due to shear deformation in step-3 (g), evolutions of c-axis pole figures showing the
spread in crystal orientations (h).
134
the change in the crystal orientation due to the large deformation especially at step 3 and at the
neck area which shows a shear band (in region that is highlighted with a white arrow). In order
to demonstrate inhomogeneous deformation at step-3 the crystal orientation in different locations
were compared with the orientation of a location that shows small rotation with respect to step-2.
As it is illustrated in 7.15 (g) the rotation generally does not happen around a low index rotation
axis in this grain. Development of low angle boundaries [Figure 7.15 (f)] shows the evolution of
crystal orientation and increasing the GND‘s due to the large deformation.
Fine-step EBSD c-axis orientation map and image quality maps for an unaged sample with a red
orientation deformed at 100 ºC is illustrated in Figure 7.16. In this case due to the fact that the
the c-axis is parallel to the interface global coordinate system and the crystal coordinate system
is coplanar (a plane which is consists of two a-axis in crystal coordinate system is identical to the
joint section plane). Therefore, deformation in this orientation occurs with rotation about the caxis. Similar to the blue orientation the rotation were calculated with respect to a location that its
rotation is negligible with respect to the step-2 to be able to represent the crystal rotation that
occurs in step-3.
Another important observation regarding these sets of samples is localization of shear
deformation, which depending on the crystal orientation can occur in the interior region of grains
or along the grain boundaries. Figure 7.17 shows the evolution of microstructure using fine-step
EBSD c-axis orientation maps and image quality maps for two steps of shear deformation of an
unaged multi-crystal sample deformed at 100 ºC, low angle boundaries are illustrated with white
lines and high angle grain boundaries are shown with black lines. As illustrated, a shear band is
135
(a)
(b)
(c)
(d)
(e)
(f)
11.4 @ [3 5 -7]
(g)
28@ (1 -2 21)~[001]
19@ [1 2 -21]~)~[001]
Step-1
Step-2
Step-3
(h)
Figure 7.16 Fine-step EBSD c-axis orientation map and image quality maps for step-1(a, b),
step-2 (c, d), step-3 (e, f) for an unaged sample with a red orientation deformed at 100 ºC, low
angle boundaries are illustrated with white lines and high angle grain boundaries are shown with
black lines, crystal rotation at different locations (as illustrated with red arrows) due to shear
deformation in step-3 (g), evolutions of c-axis pole figures showing the evolutions of crystal
orientations (h).
136
(a)
(b)
(c)
(d)
(e)
(f)
Figure 7.17 Fine-step EBSD c-axis orientation map and image quality maps for step-1(a, b),
step-2 (c, d), step-3(e, f) for a bi-crystal unaged sample deformed at 100 ºC, low angle
boundaries are illustrated with white lines and high angle grain boundaries are shown with black
lines.
137
developed in a grain with the blue orientation (7.17(f)) rather than along the grain boundaries.
The history of deformation at step-2 (7.17(d)) can assist interpretation of this observation.
Development of a shear band at step-3 occurred at regions where sub-grain boundaries were
formed at previous steps. Since the deformation happened at 100 ºC it is expected that plastic
deformation leads to the grain boundary movement but in this case the localization of plastic
deformation occurs within the grain. This clearly shows the importance of anisotropy associated
with the large grain microstructure and crystal structure of Sn which leads to heterogeneous
plastic deformation.
In contrast, a completely different phenomenon is observed in Figure 7.18. This figure shows
fine-step EBSD c-axis orientation maps and image quality maps for different steps of
deformation of an aged sample deformed at room temperature. As illustrated in figure 7.18 (c)
deformation at step 3 leads to formation of sub-grain boundaries in one grain but at step-3 the
dominant deformation mechanism is movement of grain boundaries. (The grain boundary
movement is shown with the red arrow).
Based on the observed activities of these 32 slip systems, OIM, calculated Schmid factor for
different slip systems, and observation of plane traces in optical and SEM micrographs, a
statistical analysis of the relative activity of slip systems was conducted. Figure 7.19 shows the
distribution of the number of observed slip activities on 32 polished half-joints after room
temperature shear deformation. These numbers are normalized by the number of slip activities
for each family of slip systems. This figure also shows the effect of crystal orientation on the
activity of slip systems. For example, for the blue orientation, slip system number 9 is the most
138
(a)
(b)
(c)
(d)
(e)
(f)
Figure 7.18 Fine-step EBSD c-axis orientation map and image quality maps for step-1(a, b),
step-2 (c, d), step-3(e, f) for an aged bi-crystal sample deformed at room tempereture. Low
angle boundaries are illustrated with white lines and high angle grain boundaries are shown with
black lines.
139
Figure 7-19. Number of observations of slip on 32 polished half-joints after room temperature
shear deformation, normalized by the number of slip systems in the family, and separated by the
c-axis orientation. Black bars are the sum (divided by 2 to be on a similar scale) of all observed
slip activities on joints with c-axis orientations indicated by the colors.
140
active, while for the red orientation the most active slip system is number 2. The number of
observations for slip system number 4 is highest in these set of data.
(note that in Figure 7.19 the number of observation of slip traces is divided by the number of slip
systems in each family of slip systems to represent the probability of slip activity and for slip
family # 4 which consists of 4 slip systems the number of observation is higher than slip family #
2).
This ranking is different from what Fujiwara found based on the etch-hillocks method for pure
tin samples (Table 1). Kinoshita et al. (2012) performed a numerical study using first-principles
density functional theory. Based on their orientation-dependence of active slip systems in pure
tin samples subjected to uniaxial tension when the c-axis is perpendicular to the loading, the slip
system number 9 is the most active slip system, and for the case that c-axis is parallel to the
loading direction the slip system number 4 is the most active. These results are similar to what
it‘s observed in this study in terms of the number of observation of each slip systems with
respect to c-axis orientation map.
7.4. Summary
Shear deformation was conducted in 3 steps to study the activity of slip systems in SAC305
alloys. Orientation imaging microscopy was performed to identify the orientation evolution of
solder balls during shear deformation.
Crystal orientation along with SEM and optical
micrographs of shear deformed samples, as well as plane trace analysis were conducted to find
the relative activity of slip systems. The orientation dependency of the facility in operation of
slip systems also was performed. The relative activity of slip systems using this dataset can be
141
utilized to develop a more reliable crystal plasticity model for SAC 305 alloy which accounts for
correlation between critical resolved shear stress and the probability of activating the slip
systems based upon the statistical analysis.
142
CHAPTER 8
EXPERIMENTAL AND NUMERICAL ASSESSMENTS OF TENSILE
TEST ON JOINT SCALE SAC 305 SAMPLES
8.1 Introduction
The length scale of the test specimen used to measure the deformation behavior of SAC
alloys has significant impact on the mechanical response of these alloys. As it was discussed in
chapters 5 to 7, due to the elastic and plastic anisotropy arising from the crystal structure of tin
and its large grain microstructure, the sub-mm scale SAC joint exhibits pronounced
inhomogeneity in mechanical response as well as damage evolution. Since the length scales of
current solder interconnects in the surface mount packages are less than 1000 µm, and this scale
is comparable to the characteristic length scale of SAC 305 solders microstructures, the length
scales of a reliable test specimen are expected to be comparable with those of functional solder
joints to represent the mechanical property and microstructure evolution of lead free solders.
It is important to point out that mechanical properties of sub-mm scale solder balls can be
very different from those of bulk materials. At these scales materials experience faster cooling
rates than bulk specimens under similar profiles owing to the volume of solder. On the other
hand, the larger scale in a bulk specimen results in a gradient in the cooling rate across the entire
specimen. Thus, the dendritic and IMC distributions on the edges of the specimen exposed to the
ambient are subject to a faster cooling rate compared to those in interior areas. Comparative
studies of the microstructures of bulk solder alloys with solder joints of the same alloys show
distinctive differences (Sidhu and Chawla (2008) and Sidhu et al. (2008)). In a study by Hegde
et al. (2009), the effect that size has on the stress–strain properties of Sn-3.8A g-0.7Cu solder
joints was investigated and compared with bulk solder of the same alloy. Their experimental
143
results show a strengthening effect of the constrained joints. The authors suggested that this
strengthening is a contribution of the solder joint aspect ratio, state of the stress, and the
microstructure.
Due to the complexities arising from the effect of miniaturization in solder balls, prior
studies attempted to divide these effects into single parameters in order to treat them
individually. Among those parameters, microstructural effects and constraining effects attracted
more attention (Khatibi et al (2012)). In order to develop a reliable model that could predict the
behavior of SAC alloys at the joint scale, an appropriate framework that accounts for these
variables in a single model needs to be developed. Since CPFE models can solve mechanical
deformation problems under complicated external and internal constraints imposed by grain
morphology (microstructure effects), their use is promising in this context.
Using CPFE is of great importance since at microstructural scales experimental boundary
conditions are very difficult to control and monitor. In such cases, experimental results will be
very difficult to interpret without the assistance of a model with similar complexity arising from
the microstructure and constraining effects.
In this work, a combination of experimental and numerical investigation is utilized to
understand the mechanical behavior of joint-scale lead free solders. Although the information in
chapter 7 provides valuable insights into the activity of slip systems in SAC 305, these results
cannot provide complete information required for developing a reliable model for sub-mm scale
solders. Firstly, the load-displacement responses of these shear deformed samples reflect the
behavior of 16 solder balls (shear samples have 4×4 configurations) which cannot be directly
correlated to the microstructure and crystal orientations of each solder ball.
144
Furthermore,
interaction between different solder balls in the specimen can have a strong impact on the stress
evolution at each joint.
The goal of this chapter is to develop a model that not only can satisfy the requirements
arising from the kinematic of plastic deformation in terms of both global (shape changes) and
crystal (slip activities) coordinate systems, but also accurately predict the magnitude of the stress
evolution. Hence, it is important to use the dataset that was developed in chapter 7 to insure that
a developed model can satisfy the constraints arising from slip activities.
Previous investigations based on the OIM characterizations and plane trace analysis will define a
framework for ranking the slip systems and subsequently defining the critical resolved shear
stress based on the probability of activating each slip system. Since the sample preparation for
OIM analysis is generally destructive, Synchrotron X-ray microdiffraction experiments were
conducted at beamlines 6-ID-D and 34-ID-E at the Advanced Photon Source (APS) to
characterize the orientation of each solder joint.
This methodology for developing a new CPFE model, using a combination of
experimental and numerical methods, is based on the following steps:
1- Ranking the slip system activities and estimating the probability of activating each slip
system based on the statistical data presented in chapter 7.
2- Modeling of the tensile test for a solder ball with a particular orientation which was
obtained using X-ray microdiffraction experiments.
3- Finding the value of critical resolved shear stress as a function of the probability of
activating each slip system and also fitting the force-displacement data.
Thus, this work provides the basis for developing an integrated, incremental model based on
experiment observations, modeling, and comparative analysis. This method is illustrated in
145
Figure 8.1, and discussed in more detail in section 8.2.
The results of non-destructive
characterization, as well as tensile test are described in section 8.3. Finally, section 8.4 describes
the simulation results obtained from our model using steps described above.
146
A
Calculate the probability of activating one slip system
based on chapter 7.
Use the hardening parameters obtained in chapters 5
and 6 as initial guess
B
Calculate new hardening parameters for two samples
based on curve fitting
Do the parameters
satisfy the requirements
in A & B
No
Yes
Adjust the critical resolved shear stress and calculate
these values as a function of relative activity of slip
systems
Figure 8.1 Flowchart showing the methodology of CPFE modeling of SAC 305 tensile test of
SAC 305 joint scale samples using the statistical analysis of slip system activities.
147
8.2 Statistical analysis on the activity of slip systems
Since the goal is to develop a reliable material model for SAC305 alloys, an appropriate
framework is required to represent the relative activity of slip systems. As it was discussed in
chapter 2, anisotropy arising from both the complex crystal structure of tin and the large grain
microstructures that are observed in SAC 305 alloys have a significant effect on the
inhomogeneity in deformation and damage evolution. Anisotropy in the plastic deformation
emerges naturally from the activity of slip systems, and in complex crystal structures such as the
32 slip systems in BCT Sn. Understanding the relative activity of slip systems provides a
valuable insight into the inhomogeneous deformation in solder alloys. Hence, a model that is
developed for SAC 305 has to account for these complexities. Since the critical resolved shear
stress is a parameter that represents the relative activity of slip systems in CPFE models, one has
to define a framework in which the initial critical resolved shear stress can be correlated with the
relative activity of slip systems.
Figure 8.2(a) shows the probability of observation of slip systems based on the
assessments that are provided in chapter 7. Based on the methodology that was described above,
one can estimate the critical resolved shear stress as following:
(8.1)
where X is the probability of observation for each slip system. This probability is based on a
data set comprising 32 solder balls as illustrated in Figure 8.2 (a). These values are shown in
Figure 8.2 (b). In order to be able to propose a more comprehensive model, the chapter 7 dataset
was used, which includes both room temperature and 100 C shear deformation. Due to the fact
that in tin based solders room temperature is higher than half of its homologous temperature,
148
Figure 8.2 (a) Probability of observation of each slip system in data set that was studied in
chapter 7. (b) Critical resolved shear stress that is estimated using methodology that is described
in Figure 8.1
149
similar behavior in terms of activity of slip systems and critical resolved shear stress is expected,
and as a first generation of the model, the two data sets were merged.
Employing this framework, one can use statistical assessments based on the observation of plane
traces using optical microscopy and OIM to estimate the correlation between critical resolved
shear stress and the probability of activating slip systems. Based on the flowchart that is
illustrated in Figure 8.1, one can estimate the correlation between the critical resolved shear
stress and the probability of observation of each slip system based on CPFE modeling of
different samples.
Although equation 8.1 is obtained by the curve fitting process, the equations that
correlate the dislocation density, shear rate, and the critical resolved shear stress can assist us in
understanding the physics associated with that equation. These equations, which were developed
by Gilman and Johnson (1957), define a relationship between dislocation density and shear
stress.
Figure 8.3 (a) shows this correlation for different materials. The stress dependence of the
passage rate of glide dislocations through obstacles in tin is studied by Fujiwara (1987) (Figure
8.3 (b)) .
One can calculate the rate of shear strain based on Orowan‘s equation as following:
(8.2)
Where
,
,
are dislocation density, burger‘s vector, and dislocation velocity respectively
at each slip system enumerated by the superscript.
150
Figure 8.3 (a) Dislocation velocity versus resolved shear stress for different single crystals
(Meyers (1984)), (b) Stress dependence of the passage rate of glide dislocations through
obstacles in tin (Fujiwara (1987).
151
Johnston and Gilman (1957 and 1959), Hahn (1962), and Greenman (1967) found the following
relationship between the dislocation velocity and the resolved shear stress at a constant
temperature:
(8.3)
Using equations (8-2) and (8-3) one can define the shear strain rate and initial critical resolved
shear stress as following
(8.4)
Or:
(8.5)
Where
On the other hand, using Taylor‘s equation one can calculate the resolved shear stress as follows:
(8.6)
Where
are materials constant and
In this equation it is assumed that:
One can assume that m=2 in equation 8.5 and write this equation as following:
152
(8.7)
where X is the probability of observation for each slip system as it was defined in equation 8.1.
By conducting curve fitting on tensile test data one can find that
introduced in equation 8.1.
153
similar to what is
8.3 Experimental results
In this study, in order to develop and evaluate the CPFE model, 27 single joints were
prepared using a miniature jig. The tensile sample fabrication procedure is detailed in chapter 3.
It should be noted that fracture in ductile materials, such as Sn solder alloys, damage nucleation
and evolution is controlled by slip or void growth. To investigate the contributions of each
parameter, it is necessary to examine the quality of the samples. Radiographic investigations
were conducted to examine the quality of single joint samples. Figures 8.4 and Figure 8.5 show
the radiographic images of the joints.
Since two different characterization methods were
employed to extract the crystal orientation in these samples, these sample sets are labeled as A
and B.
These examinations assisted in selecting defect free samples with minimal voids and
porosity to conduct the CPFE analysis of tensile testing using a realistic geometry. On the other
hand, the presence of voids in the samples is a mechanism that has a significant impact on
damage evolution. The radiographic images for the tensile deformed samples are shown in
Figures 8.4b, and 8.5b. These micrographs clearly show the impact of voids on the nucleation
and evolution of crack in the tensile samples.
It is apparent (upper left corner and lower right corners of Figure 8.4-A1), that growth of
two neighboring voids and coalescence of these voids can lead to the nucleation of cracks. In
addition, a big void at the center of sample A3(a) and development of crack in this area is strong
evidence that shows the influence of voids in nucleation and evolution of cracks.
One parameter that is required for conducting the CPFE analysis is grain orientation.
Different methods have been utilized to capture the grain orientations of lead free solder balls
(Bieler at al. (2012)). In chapter 7, OIM was employed to find the grain orientation for sectioned
154
(a)
(b)
Figure 8.4 Radiographic micrographs of tensile samples which are characterized using
beamline 34 before (a) and after (b) deformation. Voids are shown as bright spots.
155
(a)
(b)
Figure 8.5 Radiographic micrographs of tensile samples which are
characterized using beamline 6 before (a) and after (b) deformation. Voids are
shown using bright spots.
156
shear samples. One of the disadvantages of EBSD is that in order to obtain the back
scatter images, generally, it is required to have a flat surface. In this study, the possibility of
conducting the OIM investigation without polishing too much of the sample was examined. Due
to the roughness of the samples, this approach was unsuccessful. Hence an X-ray diffraction
method was employed to extract the grain orientations in joint scale samples.
Synchrotron X-ray microdiffraction experiments were conducted on beamline 34-ID-E at
the Advanced Photon Source (APS) at Argonne National Laboratory to characterize the grain
orientations for the samples that are shown in Figure 8.. These orientations are obtained up to a
200 micron depth underneath the surface. The characterization of orientation is shown in
Figures 8.6 for the front and back of the samples, and the orientations of the samples are
tabulated in Table 8.1.
Characterization of crystal orientations for joint scale tensile sample set B are shown in
Figure 8.6, using a high energy monochromatic X-ray beam, accomplished at beamline 6-ID-D
at the Advanced Photon Source (APS) at the Argonne National Laboratory. These methods are
described in chapter 3. Table 8.1 shows the three Euler angle values for samples which were
characterized at beamline 6..
The results of tensile testing for sample set A that were characterized at beamline 34-IDE are shown in Figure 8.7. The crystal orientation is represented by oriented crystal structure
with colors identical to each curve. In these sets of samples, due to limitations in preparation of
tensile samples, there is inconsistency in geometries of samples. Therefore, correlating the
mechanical response (force-displacement curves) which is the function of both sample geometry
and crystal orientation with each of these parameters is difficult. However, a few samples (such
157
Table-8.1 Euler angles value for samples which are characterized using beamline 34 (labeled by
―A‖) and beamline 6 labeled with ―B‖.
158
Figure 8.6 Orientation image for samples A in Figure 8.4 which
are characterized using beamline 34 before deformation.
159
Figure 8.7 Force –displacement curves for tensile sample set A, which were partially
characterized using beamline 34. The crystal orientation is overlaid close to each mechanical
response to show the correlation between orientations and mechanical responses.
160
as A12 and A6) had more uniform shapes compared to other samples, and their mechanical
behavior is without instability, so that the deformation could be effectively correlated with the
crystal orientation. This correlation will be described later.
The results of tensile testing for samples characterized at beamline 6 are shown in Figure
8.8. The crystal orientation is shown with the color similar to each curve. A large amount of
plastic deformation (Figure 8.5) and relatively small hardening is observed in sample B3 in
which the c-axis has a small deviation from the tensile direction.
161
Figure 8.8 Force–displacement curves for different tensile samples which are
partially characterized using beamline 6. The crystal structure is overlaid close to
each mechanical response show the correlation between orientations and
mechanical responses.
162
8.4 Simulation results:
The crystal plasticity model developed by Zamiri and Pourboghrat (2010) for FCC metals
was used to study tin. Firstly, the existing crystal plasticity model was modified in order to
account for the more complicated crystal structure of tin. Tin has a body centered tetragonal
(BCT) crystal structure with 32 slip systems modeled. Since not much is known from the
literature about the relative ease of slip activity of tin slip systems, slip resistance or hardening
characteristics, in chapters 5 and 6 a modified crystal plasticity model was developed. This
model was used to simulate the deformation of solder balls under shear loading, in order to
identify the likely active slip systems and hardening properties that allow comparison with
experiments. Experimental verification in those studies were limited to kinematics of plastic
deformation, and the hardening parameters were obtained using experiments reported by
Darveaux et al. (2005). In this chapter, in light of a better understanding about the activity of
slip systems which were obtained in chapter 7, and experimental tensile tests that were described
in this chapter, a more sophisticated model was developed which is based on the assumption that
the critical resolved shear stress is different in different slip systems.
Due to its formulation, the crystal plasticity model calculates shear rate for each slip
system, allowing the user to identify the most active slip systems for a given increment of plastic
deformation. The incremental hardening of slip systems is also a function of the magnitude of
shear rates, and hardening parameters. Once these parameters were fitted to an experimental
dataset, they were no longer modified when the code was used to simulate the deformation of tin
solder balls under different loading conditions.
The finite element analysis was performed using the commercial finite element code
ABAQUS.
The crystal plasticity material model was implemented using a user material
163
subroutine in FORTRAN (VUMAT). ABAQUS CAE 6.11-1 was used to make the model for
the solder joint scale samples, create the finite element mesh, assign boundary conditions, and to
build input parameters for the analysis. Element types and number of elements used in CPFE are
shown in Table 8.2.
The elements are 8-node linear bricks with reduced integration and
hourglass control. Hourglassing can be a problem with first-order, reduced-integration elements
(CPS4R, CAX4R, C3D8R, etc.) in stress/displacement analyses. Since the elements have only
one integration point, it is possible for them to distort in such a way that strains calculated at the
integration points are all zero, which in turn leads to uncontrolled distortion of the mesh. Firstorder, reduced-integration elements in ABAQUS include hourglass control, but they should be
used with reasonably fine meshes. Hourglassing can also be minimized by distributing point
loads and boundary conditions over a number of adjacent nodes. The elastic constants for tin are
known and tabulated in Tables 8.3. The parameters used for hardening in Table 8.4 and Table
8.5 were chosen using the algorithm that is shown in Figure 8.1 so that these parameters not only
generated trends that are commonly observed in multi-crystals, but also can satisfy the hardening
requirements. Estimation of materials parameters was conducted using the trial and error.
164
Table 8.2 Element Types and number of elements used in CPFE.
Table 8.3 Elastic Constants (GPa.) of Tin used in numerical analysis.
Table 8.4 Hardening parameters of tin for different slip systems used in numerical analysis.
165
Table 8.5 Critical resolved shear stress calculated based on statistical analysis
166
8.5 Case study- CPFE modeling of sample A12 (in Figure 8.4)
In order to validate the CPFE model and the framework which is developed based upon
the microstructural investigations, modeling of one of the solder samples is described in this
section. Due to the fact that Sample 12 has a cylindrical geometry and no voids were observed
in this sample from radiographic investigations, this sample deformed uniformly, making , this
sample is an appropriate candidate for this study.
A simplified geometry and dimensions of sample A12 is shown in Figure 8.9. The
coordinate system used to define the Euler angles is illustrated in this Figure. The Euler angles
and the geometry are defined in the OIM coordinate system. Copper is modeled as an elastic
material with Young‘s modulus, E =115 GPa and Poisson‘s ratio of v =0.35. Element types and
number of elements used for Cu and Sn are tabulated in Table 8.2.
167
Y
X
Cu wire
Solder
560 μm
804 μm
Cu wire
Figure 8.9 Simplified geometry and dimensions of of sample A 12 in Figure
8-5 , with coordinate system used for Euler angles.
168
Contours of the von-Mises stress obtained from the CPFE modeling are shown in Figure
8.10. It is apparent in this figure that the localization of the stress and the rotation of the sample
take place in the diagonal direction. The predicted maximum value of stress (~ 33 MPa) is in the
same range as that reported in the literature for the effective stress in the large plastic
deformation of tin alloys.
The relative activity of slip systems are shown in Figure 8.11. As illustrated, the slip
system (010)[001] (mode #2 in Table 2.1) has the highest value of activity followed by slip
system (011)[01-1] (mode #9 in Table 2.1).
Figure 8.12 shows the force-displacement response of this sample as it undergoes tensile
deformation. Curve fitting was conducted to obtain the hardening parameters, and to assess the
relative activity of slip systems, as reported in Tables 8.4 and 8.5. Also, in order to examine the
reliability of the current hardening parameters, sample 6 was also examined. As shown in Figure
8.12, the mechanical response of this sample is quite close to the experimental forcedisplacement curve corresponding to this sample.
This observation shows that the stress
evolution using this framework can predict the mechanical response of different samples with a
different initial orientation.
Figure 8.13(a) shows a magnified SEM picture for Sample A12 in Figure 8.4. It is
evident that plane traces are observed in different locations. In addition, a sample rotation is
clearly illustrated in this figure. Using the orientation corresponding to this sample, slip plane
estimations associated with this crystal orientation, and slip traces observed in this sample, it is
shown that (010)[001] (mode #2 in Table 2.1) is the most active slip system. This slip system is
shown using a magenta line on this plot.
169
Figure 8.13(b) shows a magnified picture of the slip system activity associated with this
slip system. It should be pointed out that there is a mix of crystal rotations and rigid body
rotations, which can impact the activity of slip systems. As it is shown, the model can predict the
crystal rotation which is in excellent agreement with experimental results. In addition, the model
can predict the dominant slip system.
8.6 Prediction capability of CPFE model base on new parameters based on the calibration of
tensile test
In order to compare the capability of the new model, the differences in kinematic of
plastic deformation is compared in Figure 8.14(b) by overlaying the predicted shape of the
deformed solder using CPFE model based upon Table 8.5 and the computationally deformed
sample using CPFE utilizing the hardening values that are tabulated in Table 5.3. Although
these two results are quite similar but compared to real deformed microstructure (Figure 8.14(c))
the new model shows more similarities in terms of kinematics of plastic deformation. Also, the
values of shear stress predicted by hardening parameters that are obtained in this chapter (Figure
8.14(a)) are more realistic compared to the values that are discussed in chapter 6 (Figure 6.7(b)).
8.7 Summary and conclusion
A CPFE model was proposed in this chapter to account for the relative activity of slip
systems in SAC305. A statistical analysis was introduced to define a correlation of the critical
resolve shear stress and the probability of activating each slip systems. In order to verify the
model, joint scale solder joints were fabricated. To characterize these samples non-destructively,
Synchrotron X-ray microdiffraction experiments on beamline 34-ID-E as well as high energy
monochromatic X-ray beam experiments at the Advanced Photon Source (APS) at Argonne
National Laboratory were employed. In addition, radiographic investigation was conducted
170
before and after deformation to check the quality of samples in terms of voids and porosities and
to assess the mechanism that controls the damage evolution.
The results of simulations show that the CPFE model that is developed based on the
statistical analysis of the activity of slip systems not only can satisfy the requirements associated
with kinematic of plastic deformation in crystal (activity of slip systems) and global (shape
changes) coordinate systems, but also is able to predict the evolution of stress effectively in jointscale SAC305 samples.
171
Y
X
Figure 8.10 Distribution of von-Mises stress in sample-12 obtained using the CPFE
modeling. Localization of the stress and rotation of the sample is predicted.
172
Figure 8.11 Comparison of the average activity in sample 12 predicts three active slip systems,
with one dominant, during tensile deformation.
173
Figure 8.12 Force displacement for two tensile deformed samples 6 and 12 (solid lines)
dashed lines shows the simulation results. Curve fitting was conducted based on the
assessments on the relative activity of slip systems.
174
Figure 8.13 SEM image of sample 12 shows the slip plane traces on the surface, sample
rotation and shear localization after shear deformation. Plane traces (magenta line)
shows that slip system mode 2 is the most active slip systems (a) CPFE simulation of
sample 12 predicts that slip system mode 2 is the most active slip system (b). Crystal
structure of tin shows the joint orientation and slip trace (c).
175
(a)
(b)
(c)
(c)
Figure 8.14 Deformed solder ball predicted by CPFE utilizing the material parameters tabulated
in Table 8.5 indicating more realistic values of stress compared to Figure 6.7 (a) Comparison of
the outline of the model illustrated in (a) and Figure 6.7 (b) indicating the similarity in
kinematic of deformation predicted by both models. Polarized light micrograph of beach ball
microstructure that was modeled in (a) and (b).
176
CHAPTER 9
CONCLUDING REMARKS AND FUTURE WORKS
9.1 Concluding Remarks
Anisotropy is of significant importance in all structural metals, but this characteristic is unusually
strong in Sn, making Sn based solder joints one of the best examples of the influence of
anisotropy.
The mechanical properties and damage evolution in SAC 305 alloys strongly
depends on the underlying microstructures. The existence of single or multi-grain microstructure
in this alloy causes each joint to exhibit a unique mechanical response to stress and strain history.
This dissertation provides an improved understanding of the modeling of anisotropy in the elastic
and plastic behavior of SAC solder at sub-mm length scales, using a combination of
microstructural and mechanical characterization and crystal plasticity finite element (CPFE)
modeling approaches.
In developing a mechanistic framework based upon the crystal plasticity finite element modeling
of idealized geometries, insights are provided into the effects of initial coarse-grained Sn
microstructures and orientations on joint-dependent deformation behavior of SAC305 solders.
Furthermore, the capability of a CPFE model to capture the kinematics of in-homogeneous
plastic deformation in microstructures that are commonly observed in real SAC305 solders,
activity of slip systems and evolution of micro structural features is successfully examined using
simple assumptions about the hardening parameters.
The critical issue in modeling of plastic anisotropy is the lack of understanding about the relative
ease in activating various slip systems. There is especially uncertainty about the critical resolved
shear stress values and hardening characteristics in SAC 305 alloys.
177
To provide more
understanding regarding the activity of slip systems in SAC 305 alloys, experimental
investigations were conducted utilizing the orientation imaging microscopy, SEM and plane
trace analysis on shear deformed samples.
An improved CPFE model was developed based on a combination statistical analysis on
probability of observation of various slip systems in SAC 305 and systematic modeling of tensile
tests on joint scale single crystal solder tensile specimens. In this model a correlation is observed
between the critical resolved shear stress and the probability of activating different slip systems.
In order to validate this model, joint scale solder tensile samples were fabricated and
characterized non-destructively using Synchrotron X-ray micro diffraction on beamline 34-ID-E
as well as high energy monochromatic X-ray beam on beamline 6 at the Advanced Photon
Source (APS) at Argonne National Laboratory. In addition, radiographic investigations were
conducted before and after deformation to check the quality of samples in terms of voids and
porosity to understand the mechanism that controls the stress-strain and damage evolution. The
results of simulation show that the CPFE model that is developed based on the statistical analysis
of activity of slip system not only can satisfy the requirements associated with kinematic of
plastic deformation in crystal coordinate systems (activity of slip systems) and global coordinate
system (shape changes) but also this model is able to predict the evolution of stress in joint level
SAC 305 samples.
9.2 Recommendation for future works
The results of this study are applicable to the mechanical response of solders at low temperature,
high strain-rate loading conditions where plastic deformation dominates over creep. Regarding
178
the further improvement of this model and future development in this area several
recommendations are provided as follows:
1- All finite element simulations performed in this thesis were based on this assumption that
deformation occurs at room temperature. Development of temperature sensitive models
requires further investigation in terms of implementing a temperature dependent
activation energy that has been measured in existing experimental studies.
2- Since the experimental measurements and the microstructural evidence indicates that
dislocation climb is an important deformation mode [Ghoniem et. al (1990)] in creep
deformation, CPFE models based upon the dislocation density can be developed to model
the creep in these alloys.
3- Rate effects are another area of interest that can be especially important for modeling of
shock performance in SAC 305. This characteristic is very important regarding the
reliability of electronic products.
Correlating of the shock performance and
microstructure is a subject of increasing interest as electronic systems become more
portable.
4- Since the CPFE modeling is naturally time expensive, macro scale modeling approaches
(which are less time consuming) can be developed by defining an anisotropic yield
function based on the CPFE analysis. In this approach the coefficients in the yield
function can be defined utilizing the CPFE modeling. Further experimental data in
different directions (including joint level shear test and compression test) are required to
develop these models.
5- The size effects which are a very important feature of miniaturization cannot be captured
using this CPFE model. The above crystal plasticity models can be further improved by
179
implementing the idea of non-local continuum theory to take into account the size effect
relevant to microstructure features and deformation mechanisms to make a more accurate
model.
6- Another important future research topic is to consider damage prediction in material
models. Further work can be conducted to implement damage in the proposed crystal
plasticity models. Damage can be implemented in these crystal plasticity models by using
several micro-scale damage parameters which evolve with deformation or time. These
micro-scale parameters can be a function of temperature and microstructural parameters.
These crystal plasticity models can then be used to obtain macro-scale damage
parameters for the material.
180
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