The constitutive model for glassy polymers proposed by Arruda and Boyce (BPA model) is reviewed and compared to experimental data for long-term loading. The BPA model has previously been shown to capture monotonic loading accurately, but for unloading and long-term behavior, the response of the BPA model is found to deviate from experimental data. In the present paper, we suggest an efficient extension that significantly improves the predictive capability of the BPA model during unloading and long-term recovery. The new, extended BPA model (EBPA model) is calibrated to experimental data of polycarbonate (PC) in various loading–unloading situations and deformation states. The numerical treatment of the BPA model associated with the finite element analysis is also discussed. As a consequence of the anisotropic hardening, the plastic spin enters the model. In order to handle the plastic spin in a finite element formulation, an algorithmic plastic spin is introduced. In conjunction with the backward Euler integration scheme use of the algorithmic plastic spin leads to a set of algebraic equations that provides the updated state. Numerical examples reveal that the proposed numerical algorithm is robust and well suited for finite element simulations.

References

References
1.
Haward
,
R. N.
, and
Thackray
,
G.
,
1968
, “
The Use of a Mathematical Model to Describe Isothermal Stress-Strain Curves in Glassy Thermoplastics
,”
Proc. R. Soc. London, Ser. A
,
302
, pp.
453
472
.10.1098/rspa.1968.0029
2.
Gibson
,
A. G.
,
Hope
,
P. S.
, and
Ward
,
I. M.
,
1980
, “
The Hydrostatic Extrusion of Polymethylmethacrylate
,”
J. Mater. Sci.
,
15
, pp.
2207
2220
.10.1007/BF00552111
3.
Stokes
,
V. K.
, and
Nied
,
H. F.
,
1986
, “
Solid Phase Sheet Forming of Thermoplastics, Parts I and II
,”
J. Eng. Mater. Technol.
,
108
, pp.
107
118
.10.1115/1.3225846
4.
Bowden
,
P. B.
, and
Raha
,
S.
,
1970
, “
The Formation of Micro Shear Bands in Polystyrene and Polymethylmethacrylate
,”
Philos. Mag.
,
22
, pp.
463
482
.10.1080/14786437008225837
5.
G'Sell
,
C.
, and
Jonas
,
J. J.
,
1981
, “
Yield and Transient Effects During the Plastic Deformation of Solid Polymers
,”
J. Mater. Sci.
,
16
, pp.
1956
1974
.10.1007/BF00540644
6.
Arruda
,
E. M.
,
Boyce
,
M. C.
, and
Quintus-Bosz
,
H.
,
1993
, “
Effects of Initial Anisotropy on the Finite Strain Deformation Behavior of Glassy Polymers
,”
Int. J. Plast.
,
9
, pp.
783
811
.10.1016/0749-6419(93)90052-R
7.
Arruda
,
E. M.
, and
Boyce
,
M. C.
,
1993
, “
Evolution of Plastic Anisotropy in Amorphous Polymers During Finite Straining
,”
Int. J. Plast.
,
9
, pp.
697
720
.10.1016/0749-6419(93)90034-N
8.
Krempl
,
E.
, and
Khan
,
F.
,
2003
, “
Rate(Time)-Dependent Deformation Behavior: An Overview of Some Properties of Metals and Solid Polymers
,”
Int. J. Plast.
,
19
, pp.
1069
1095
.10.1016/S0749-6419(03)00002-0
9.
Zari
,
F.
,
Naït-Abdelaziz
,
M.
,
Woznica
,
K.
, and
Gloaguen
,
J.-M.
,
2007
, “
Elasto-Viscoplastic Constitutive Equations for the Description of Glassy Polymers Behavior at Constant Strain Rate
,”
J. Eng. Mater. Technol.
,
129
, pp.
564
571
.10.1115/1.2400256
10.
Dreistadt
,
C.
,
Bonnet
,
A. S.
,
Chevrier
,
P.
, and
Lipinski
,
P.
,
2009
, “
Experimental Study of the Polycarbonate Behaviour During Complex Loadings and Comparison with the Boyce, Parks and Argon Model Predictions
,”
Mater. Des.
,
30
, pp.
3126
3140
.10.1016/j.matdes.2008.11.028
11.
James
,
H. M.
, and
Guth
,
E.
,
1943
, “
Theory of the Elastic Properties of Rubber
,”
J. Chem. Phys.
,
11
, pp.
455
481
.10.1063/1.1723785
12.
Treloar
,
L. R. G.
,
1946
, “
The Elasticity of a Network of Long-Chain Molecules: Part III
,”
Trans. Faraday Soc.
,
42
, pp.
83
94
.10.1039/tf9464200083
13.
Boyce
,
M. C.
,
Parks
,
D. M.
, and
Argon
,
A. S.
,
1988
, “
Large Inelastic Deformation of Glassy Polymers, Part I: Rate-Dependent Constitutive Model
,”
Mech. Mater.
,
7
, pp.
15
33
.10.1016/0167-6636(88)90003-8
14.
Arruda
,
E. M.
, and
Boyce
,
M. C.
,
1991
, “
Anisotropy and Localization of Plastic Deformation
,”
Proceedings of Plasticity
,
J. P.
Boehler
and
A. S.
Khan
, eds.,
Elsevier Applied Science
,
London
,
483
pp.
15.
Anand
,
L.
, and
Ames
,
N. M.
,
2006
, “
On Modeling the Micro-Indentation Response of an Amorphous Polymer
,”
Int. J. Plast.
,
22
, pp.
1123
1170
.10.1016/j.ijplas.2005.07.006
16.
Silberstein
,
M. N.
, and
Boyce
,
M. C.
,
2010
, “
Constitutive Modeling of the Rate, Temperature, and Hydration Dependent Deformation Response of Nafion to Monotonic and Cycling Loading
,”
J. Power Sources
,
195
, pp.
5692
5706
.10.1016/j.jpowsour.2010.03.047
17.
Treloar
,
L. R. G.
and
Riding
,
G.
,
1979
, “
A Non-Gaussian Theory for Rubber in Biaxial Strain. I Mechanical Properties
,”
Proc. R. Soc. London, Ser. A.
,
369
, pp.
261
280
.10.1098/rspa.1979.0163
18.
Wu
,
P. D.
, and
van der
Giessen
,
E.
,
1993
, “
On Improved Network Models for Rubber Elasticity and Their Applications to Orientation Hardening in Glassy Polymers
,”
J. Mech. Phys. Solids
,
41
(
3
), pp.
427
456
.10.1016/0022-5096(93)90043-F
19.
Beatty
,
M. F.
,
2003
, “
An Average-Stretch Full-Network Model for Rubber Elasticity
,”
J. Elast.
,
70
, pp.
65
86
.10.1023/B:ELAS.0000005553.38563.91
20.
Arruda
,
E. M.
and
Boyce
,
M. C.
,
1993
, “
A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials
,”
J. Mech. Phys. Solids
,
41
, pp.
389
412
.10.1016/0022-5096(93)90013-6
21.
Govaert
,
L. E.
,
Timmermans
,
P. H. M.
, and
Brekelmans
,
W. A. M.
,
2000
, “
The Influence of Intrinsic Strain Softening and Strain Localization in Polycarbonate: Modeling and Experimental Validation
,”
J. Eng. Mater. Technol.
,
122
, pp.
177
185
.10.1115/1.482784
22.
Dupaix
,
R. B.
,
Palm
,
G.
, and
Castro
,
J.
,
2006
, “
Large Strain Mechanical Behavior of Polymethylmethacrylate (PMMA) Near the Glass Transition Temperature
,”
J. Eng. Mater. Technol.
,
128
, pp.
559
563
.10.1115/1.1924564
23.
Negahban
,
M.
,
Goel
,
A.
,
Delabarre
,
P.
,
Feng
,
R.
, and
Dimick
,
A.
,
2006
, “
Experimentally Evaluating the Equilibrium Stress in Shear of Glassy Polycarbonate
,”
J. Eng. Mater. Technol.
,
128
, pp.
537
542
.10.1115/1.2345445
24.
Wang
,
M. C.
, and
Guth
,
E. J.
,
1952
, “
Statistical Theory of Networks of Non-Gaussian Flexible Chains
,”
J. Chem. Phys.
,
20
(
7
), pp.
1144
1157
.10.1063/1.1700682
25.
Argon
,
A. S.
,
1973
, “
A Theory for the Low-Temperature Plastic Deformation of Glassy Polymers
,”
Philos. Mag.
,
28
(
4
), pp.
839
865
.10.1080/14786437308220987
26.
Cohen
,
A.
,
1991
, “
A Pade Approximant to the Inverse Langevin Function
,”
Rheol. Acta
,
30
(
3
), pp.
270
273
.10.1007/BF00366640
27.
Agah-Tehrani
,
A.
,
Lee
,
E. H.
,
Mallett
,
R. L.
, and
Onat
,
E. T.
,
1987
, “
The Theory of Elastic-Plastic Deformation at Finite Strain With Induced Anisotropy Modeled as Combined Isotropic-Kinematic Hardening
,”
J. Mech. Phys. Solids
,
35
(
5
), pp.
519
539
.10.1016/0022-5096(87)90015-9
28.
G'Sell
,
C.
, and
Gopez
,
A. J.
,
1985
, “
Plastic Banding in Glassy Polycarbonate Under Plane Simple Shear
,”
J. Mater. Sci.
,
20
, pp.
3462
3478
.10.1007/BF01113753
29.
Hasan
,
O. A.
, and
Boyce
,
M. C.
,
1995
, “
A Constitutive Model for the Nonlinear Viscoelastic Viscoplastic Behavior of Glassy Polymers
,”
Polym. Eng. Sci.
,
35
, pp.
331
344
.10.1002/pen.760350407
30.
Cao
,
K.
,
Wang
,
Y.
, and
Wang
,
Y.
,
2012
, “
Effects of Strain Rate and Temperature on the Tension Behaviour of Polycarbonate
,”
Mater. Des.
,
38
, pp.
53
58
.10.1016/j.matdes.2012.02.007
31.
Weber
,
G.
, and
Anand
,
L.
,
1990
, “
Finite Deformation Constitutive Equations and a Time Integration Procedure for Isotropic, Hyperelastic-Viscoplastic Solids
,”
Comput. Methods Appl. Mech. Eng.
,
79
, pp.
173
202
.10.1016/0045-7825(90)90131-5
32.
Steinmann
,
P.
, and
Stein
,
E.
,
1996
, “
On the Numerical Treatment and Analysis of Finite Deformation Ductile Single Crystal Plasticity
,”
Comput. Methods Appl. Mech. Eng.
,
129
, pp.
235
254
.10.1016/0045-7825(95)00913-2
33.
Miehe
,
C.
,
1998
, “
Comparison of Two Algorithms for the Computation of Fourth-Order Isotropic Tensor Functions
,”
Comput. Struct.
,
66
(
1
), pp.
37
43
.10.1016/S0045-7949(97)00073-4
34.
Ekh
,
M.
, and
Runesson
,
K.
,
2001
, “
Modeling and Numerical Issues in Hyperelasto-Plasticity With Anisotropy
,”
Int. J. Solids Struct.
,
38
, pp.
9461
9478
.10.1016/S0020-7683(01)00132-9
35.
Wallin
,
M.
, and
Ristinmaa
,
M.
,
2005
, “
Deformation Gradient Based Kinematic Hardening Model
,”
Int. J. Plast.
,
21
, pp.
2025
2050
.10.1016/j.ijplas.2005.01.007
36.
Shampine
,
L. F.
,
Reichelt
,
M. W.
, and
Kierzenka
,
J. A.
,
1999
, “
Solving Index-1 DAEs in MATLAB and Simulink
,”
SIAM Rev.
,
41
(
3
), pp.
538
552
.10.1137/S003614459933425X
You do not currently have access to this content.