Abstract

The Anand model is a unified viscoplastic model, which is widely employed to describe the solder material deformation. The parameters in the Anand model for a certain material are usually identified by using the classical method based on two algebraic equations derived from the original differential equation of the Anand model. However, the second algebraic equation describing the relationship between the stress and inelastic strain is obtained with some terms about the unsteady value of internal variable neglected. But the effects induced by the omission of some unsteady terms on the effectiveness of classical method are not researched comprehensively. Therefore, in this paper, the effects of the omitted terms on the accuracy of the classical method are discussed. The inelastic deformation for the material which the second algebraic equation cannot describe due to the omission of unsteady terms is presented. The precondition for obtaining accurate results from the second algebraic equation is given out. Two criteria used to judge the effectiveness of the second algebraic equation are derived. To reduce the error related to the second algebraic equation of the classical method for some materials, two alternative identification methods are proposed. By combining the step of solving differential equation and genetic algorithm, the parameters in the Anand model originally identified by the second algebraic equation are determined in the processes of the two proposed methods. The effectiveness of the two alternative methods is presented by identifying the material Anand parameters where the classical method cannot be applied.

References

1.
Anand
,
L.
,
1985
, β€œ
Constitutive Equations for Hot Working of Metals
,”
Int. J. Plast.
,
1
(
3
), pp.
213
–
231
.10.1016/0749-6419(85)90004-X
2.
Brown
,
S. B.
,
Kim
,
K. H.
, and
Anand
,
L.
,
1989
, β€œ
An Internal Variable Constitutive Model for Hot Working of Metals
,”
Int. J. Plast.
,
5
(
2
), pp.
95
–
130
.10.1016/0749-6419(89)90025-9
3.
Wilde
,
J.
,
Becker
,
K.
,
Thoben
,
M.
,
Blum
,
W.
,
Jupitz
,
T.
,
Wang
,
G. Z.
, and
Cheng
,
Z. N.
,
2000
, β€œ
Rate Dependent Constitutive Relations Based on Anand Model for 92.5Sn2.5Ag2.5Ag Solder
,”
IEEE Trans. Adv. Packag.
,
23
, pp.
408
–
414
.10.1109/6040.861554
4.
Darveaux
,
R.
, and
Banerji
,
K.
,
1992
, β€œ
Constitutive Relations for Tin-Based Solder Joints
,”
IEEE Trans. CHMT
,
15
(
6
), pp.
1013
–
1024
.10.1109/33.206925
5.
Wang
,
G. Z.
,
Cheng
,
Z. N.
,
Becker
,
K.
, and
Wilde
,
J.
,
2001
, β€œ
Applying Anand Model to Represent the Viscoplastic Deformation Behavior of Solder Alloys
,”
ASME J. Electron. Packag.
,
123
(
3
), pp.
247
–
253
.10.1115/1.1371781
6.
Zhang
,
L.
,
Xue
,
S. B.
,
Gao
,
L. L.
,
Zeng
,
G.
,
Sheng
,
Z.
,
Chen
,
Y.
, and
Yu
,
S. L.
,
2009
, β€œ
Determination of Anand Parameters for SnAgCuCe Solder
,”
Modell. Simul. Mater. Sci. Eng.
,
17
(
7
), p.
075014
.10.1088/0965-0393/17/7/075014
7.
Zhang
,
L.
,
Han
,
J.
,
Guo
,
Y.
, and
He
,
C.
,
2014
, β€œ
Anand Model and FEM Analysis of SnAgCuZn Lead-Free Solder Joints in Wafer Level Chip Scale Packaging Devices
,”
Microelectron. Reliab.
,
54
(
1
), pp.
281
–
286
.10.1016/j.microrel.2013.07.100
8.
Liu
,
J. C.
,
Yu
,
H. J.
,
Zhang
,
G.
,
Wang
,
Z. H.
, and
Ma
,
J. S.
,
2014
, β€œ
Constitutive Behavior and Anand Model of Novel Lead-Free Solder Sn-Zn-Bi-in-P
,”
International Conference on Electronics Packaging
, Toyama, Japan, Apr. 23–25, pp.
156
–
161
.10.1109/ICEP.2014.6826681
9.
Ahmed
,
S.
,
Suhling
,
J. C.
, and
Lall
,
P.
,
2017
, β€œ
The Anand Parameters of Aging Resistant Doped Solder Alloys
,”
16th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems
,
IEEE
,
Orlando, FL
, May 30–June 2, pp.
1416
–
1424
.10.1109/ITHERM.2017.7992647
10.
Lall
,
P.
,
Yadav
,
V.
,
Suhling
,
J.
, and
Locker
,
D.
,
2021
, β€œ
Evolution of Anand Parameters With Elevated Temperature Aging for SnAgCu Lead-Free Alloys
,”
ASME J. Electron. Packag.
,
143
(
2
), p.
021005
.10.1115/1.4048181
11.
Chang
,
R. W.
, and
McCluskey
,
F. P.
,
2009
, β€œ
Constitutive Relations of Indium in Extreme-Temperature Electronic Packaging Based on Anand Model
,”
J. Electron. Mater.
,
38
(
9
), pp.
1855
–
1859
.10.1007/s11664-009-0765-8
12.
Yu
,
D.
,
Chen
,
X.
,
Chen
,
G.
,
Lu
,
G.
, and
Wang
,
Z.
,
2009
, β€œ
Applying Anand Model to Low-Temperature Sintered Nanoscale Silver Paste Chip Attachment
,”
Mater Des.
,
30
(
10
), pp.
4574
–
4579
.10.1016/j.matdes.2009.04.006
13.
Hu
,
X.
, and
Ju
,
D.
,
2006
, β€œ
Application of Anand's Constitutive Model on Twin Roll Casting Process of AZ31 Magnesium Alloy
,”
Trans. Nonferrous Met. Soc. China
,
16
, pp.
s586
–
s590
.10.1016/S1003-6326(06)60261-6
14.
Liu
,
L.
,
Li
,
Q.
,
Liao
,
B.
,
Gao
,
Y.
,
Wang
,
Y.
,
Ren
,
X.
, and
Yang
,
Q.
,
2010
, β€œ
Stress-Strain Behaviors Simulation of High Chromium Steel at Elevated Temperatures
,”
J. Mater. Eng. Perform.
,
19
(
7
), pp.
921
–
927
.10.1007/s11665-009-9575-7
15.
Guo
,
J.
,
Liu
,
Y.
,
Liu
,
L.
,
Zhang
,
Y.
, and
Yang
,
Q.
,
2014
, β€œ
3D Stress Simulation and Parameter Design During Twin-Roll Casting of 304 Stainless Steel Based on the Anand Model
,”
Int. J. Miner. Metall. Mater.
,
21
(
7
), pp.
666
–
673
.10.1007/s12613-014-0956-z
16.
Wang
,
Y.
,
Zhao
,
J.
, and
Zhang
,
C.
,
2017
, β€œ
Viscoplastic Equations Incorporated Into a Finite Element Model to Predict Deformation Behavior of Irradiated Reduced Activation Ferritic/Martensitic Steel
,”
Fusion Eng. Des.
,
118
, pp.
129
–
134
.10.1016/j.fusengdes.2017.03.145
17.
Teixeira-Dias
,
F.
, and
Menezes
,
L. F.
,
2001
, β€œ
Numerical Determination of the Influence of the Cooling Rate and Reinforcement Volume Fraction on the Levels of Residual Stresses in Al–SiC Composites
,”
Comput. Mater. Sci.
,
21
(
1
), pp.
26
–
36
.10.1016/S0927-0256(00)00213-5
18.
Bai
,
N.
,
Chen
,
X.
, and
Gao
,
H.
,
2009
, β€œ
Simulation of Uniaxial Tensile Properties for Lead-Free Solders With Modified Anand Model
,”
Mater. Des.
,
30
(
1
), pp.
122
–
128
.10.1016/j.matdes.2008.04.032
19.
Chen
,
X.
,
Chen
,
G.
, and
Sakane
,
M.
,
2005
, β€œ
Prediction of Stress-Strain Relationship With an Improved Anand Constitutive Model for Lead-Free Solder Sn3.5Ag
,”
IEEE Trans. Compon. Packag. Technol.
,
28
, pp.
111
–
116
.10.1109/TCAPT.2004.843157
20.
Pei
,
M.
, and
Qu
,
J.
,
2005
, β€œ
Constitutive Modeling of Lead-Free Solders
,”
International Symposium on Advanced Packaging Materials: Processes, Properties and Interfaces
, San Francisco, CA, July 17–22, pp.
1307
–
1311
.10.1109/ISAPM.2005.1432043
21.
Chen
,
G.
,
Zhang
,
Z. S.
,
Mei
,
Y. H.
,
Li
,
X.
,
Lu
,
G. Q.
, and
Chen
,
X.
,
2013
, β€œ
Ratcheting Behavior of Sandwiched Assembly Joined by Sintered Nanosilver for Power Electronics Packaging
,”
Microelectron. Reliab.
,
53
(
4
), pp.
645
–
651
.10.1016/j.microrel.2012.11.011
22.
Tucker
,
J. P.
,
Chan
,
D. K.
,
Subbarayan
,
G.
, and
Handwerker
,
C. A.
,
2014
, β€œ
Maximum Entropy Fracture Model and Its Use for Predicting Cyclic Hysteresis in Sn3.8Ag0.7Cu and Sn3.0Ag0.5 Solder Alloys
,”
Microelectron. Reliab.
,
54
(
11
), pp.
2513
–
2522
.10.1016/j.microrel.2014.04.012
23.
Motalab
,
M.
,
Mustafa
,
M.
,
Suhling
,
J. C.
, and
Lall
,
P.
,
2016
, β€œ
Improved Predictions of Cyclic Stress-Strain Curves for Lead Free Solders Using the Anand Viscoplastic Constitutive Model
,”
15th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems
, Las Vegas, NV, May 31–June 3, pp.
471
–
480
.10.1109/ITHERM.2016.7517586
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