Abstract

This paper is concerned with the two simple numerical implementation methods for a damage-coupled Chaboche-type viscoplastic constitutive model. By considering the damage variable as a constant in each incremental step, the return-mapping procedure is reduced to the solution of only one nonlinear scalar equation. Depending on the use of damage value in the current or prior incremental state, the two methods are named the backward difference implicit integration scheme and the two-step explicit integration scheme respectively. These two numerical algorithms are implemented into the ansys software by developing the usermat subroutine and verified by comparing them with available experimental data. Several numerical examples on the Gauss point level are studied in terms of stability, accuracy, computational efficiency, and applicability for further numerical observation. In addition to higher computational efficiency and lower memory requirements, the two methods can be easily extended to other damage models due to their simplicity.

References

1.
Benaarbia
,
A.
,
Rae
,
Y.
, and
Sun
,
W.
,
2018
, “
Unified Viscoplasticity Modelling and Its Application to Fatigue-Creep Behaviour of Gas Turbine Rotor
,”
Int. J. Mech. Sci.
,
136
, pp.
36
49
.
2.
Hong
,
H.
,
Wang
,
W.
, and
Liu
,
Y.
,
2019
, “
High-Temperature Fatigue Behavior of a Steam Turbine Rotor Under Flexible Operating Conditions With Variable Loading Amplitudes
,”
Int. J. Mech. Sci.
,
163
, p.
105121
.
3.
Zhu
,
X.
,
Chen
,
H.
,
Xuan
,
F.
, and
Chen
,
X.
,
2019
, “
On the Creep Fatigue and Creep Rupture Behaviours of 9–12% Cr Steam Turbine Rotor
,”
Eur. J. Mech. A/Solids
,
76
, pp.
263
278
.
4.
Azeez
,
A.
,
Eriksson
,
R.
,
Leidermark
,
D.
, and
Calmunger
,
M.
,
2020
, “
Low Cycle Fatigue Life Modelling Using Finite Element Strain Range Partitioning for a Steam Turbine Rotor Steel
,”
Theor. Appl. Fract. Mech.
,
107
, p.
102510
.
5.
Li
,
B.
,
Zheng
,
Y.
,
Shi
,
S.
,
Zhang
,
Z.
, and
Chen
,
X.
,
2020
, “
Cyclic Deformation and Cracking Behavior of 316LN Stainless Steel Under Thermomechanical and Isothermal Fatigue Loadings
,”
Mater. Sci. Eng. A
,
773
, p.
138866
.
6.
Bartošák
,
M.
,
Horváth
,
J.
, and
Španiel
,
M.
,
2020
, “
Isothermal Low-Cycle Fatigue and Fatigue-Creep of a 42CrMo4 Steel
,”
Int. J. Fatigue
,
135
, p.
105538
.
7.
Fournier
,
B.
,
Sauzay
,
M.
, and
Pineau
,
A.
,
2011
, “
Micromechanical Model of the High Temperature Cyclic Behavior of 9–12%Cr Martensitic Steels
,”
Int. J. Plast.
,
27
(
11
), pp.
1803
1816
.
8.
Li
,
D.
,
Golden
,
B. J.
, and
O’Dowd
,
N. P.
,
2014
, “
Multiscale Modelling of Mechanical Response in a Martensitic Steel: A Micromechanical and Length-Scale-Dependent Framework for Precipitate Hardening
,”
Acta Mater.
,
80
, pp.
445
456
.
9.
Yu
,
C.
,
Kang
,
G.
, and
Kan
,
Q.
,
2014
, “
Crystal Plasticity Based Constitutive Model of NiTi Shape Memory Alloy Considering Different Mechanisms of Inelastic Deformation
,”
Int. J. Plast.
,
54
, pp.
132
162
.
10.
Guan
,
Y.
,
Chen
,
B.
,
Zou
,
J.
,
Britton
,
T. B.
,
Jiang
,
J.
, and
Dunne
,
F. P. E.
,
2017
, “
Crystal Plasticity Modelling and HR-DIC Measurement of Slip Activation and Strain Localization in Single and Oligo-Crystal Ni Alloys Under Fatigue
,”
Int. J. Plast.
,
88
, pp.
70
88
.
11.
Barrett
,
R. A.
,
O'Donoghue
,
P. E.
, and
Leen
,
S. B.
,
2017
, “
A Physically-Based Constitutive Model for High Temperature Microstructural Degradation Under Cyclic Deformation
,”
Int. J. Fatigue
,
100
, pp.
388
406
.
12.
Estrada Rodas
,
E. A.
, and
Neu
,
R. W.
,
2018
, “
Crystal Viscoplasticity Model for the Creep-Fatigue Interactions in Single-Crystal Ni-Base Superalloy CMSX-8
,”
Int. J. Plast.
,
100
, pp.
14
33
.
13.
Sun
,
F.
,
Meade
,
E. D.
, and
O'Dowd
,
N. P.
,
2019
, “
Strain Gradient Crystal Plasticity Modelling of Size Effects in a Hierarchical Martensitic Steel Using the Voronoi Tessellation Method
,”
Int. J. Plast.
,
119
, pp.
215
229
.
14.
Zhao
,
P.
, and
Xuan
,
F.
,
2011
, “
Ratchetting Behavior of Advanced 9–12% Chromium Ferrite Steel Under Creep-Fatigue Loadings
,”
Mech. Mater.
,
43
(
6
), pp.
299
312
.
15.
Zhang
,
S.
, and
Xuan
,
F.
,
2017
, “
Interaction of Cyclic Softening and Stress Relaxation of 9–12% Cr Steel Under Strain-Controlled Fatigue-Creep Condition: Experimental and Modeling
,”
Int. J. Plast.
,
98
, pp.
45
64
.
16.
Zhou
,
J.
,
Sun
,
Z.
,
Kanouté
,
P.
, and
Retraint
,
D.
,
2018
, “
Experimental Analysis and Constitutive Modelling of Cyclic Behaviour of 316L Steels Including Hardening/Softening and Strain Range Memory Effect in LCF Regime
,”
Int. J. Plast.
,
107
, pp.
54
78
.
17.
Hormozi
,
R.
,
Biglari
,
F.
, and
Nikbin
,
K.
,
2015
, “
Experimental Study of Type 316 Stainless Steel Failure Under LCF/TMF Loading Conditions
,”
Int. J. Fatigue
,
75
, pp.
153
169
.
18.
Tong
,
J.
,
2004
, “
Modelling of Cyclic Plasticity and Viscoplasticity of a Nickel-Based Alloy Using Chaboche Constitutive Equations
,”
Int. J. Fatigue
,
26
(
8
), pp.
829
837
.
19.
Chaboche
,
J. L.
,
2008
, “
A Review of Some Plasticity and Viscoplasticity Constitutive Theories
,”
Int. J. Plast.
,
24
(
10
), pp.
1642
1693
.
20.
Cailletaud
,
G.
, and
Saï
,
K.
,
1995
, “
Study of Plastic/Viscoplastic Models With Various Inelastic Mechanisms
,”
Int. J. Plast.
,
11
(
8
), pp.
991
1005
.
21.
Velay
,
V.
,
Bernhart
,
G.
, and
Penazzi
,
L.
,
2006
, “
Cyclic Behavior Modeling of a Tempered Martensitic Hot Work Tool Steel
,”
Int. J. Plast.
,
22
(
3
), pp.
459
496
.
22.
Chaboche
,
J. L.
,
1989
, “
Constitutive Equations for Cyclic Plasticity and Cyclic Viscoplasticity
,”
Int. J. Plast.
,
5
(
3
), pp.
247
302
.
23.
Barrett
,
R. A.
,
O’Donoghue
,
P. E.
, and
Leen
,
S. B.
,
2013
, “
An Improved Unified Viscoplastic Constitutive Model for Strain-Rate Sensitivity in High Temperature Fatigue
,”
Int. J. Fatigue
,
48
, pp.
192
204
.
24.
Bodner
,
S. R.
, and
Partom
,
Y.
,
1975
, “
Constitutive Equations for Elastic-Viscoplastic Strain-Hardening Materials
,”
ASME J. Appl. Mech.
,
42
(
2
), pp.
385
389
.
25.
Prager
,
W.
,
1955
, “
The Theory of Plasticity: A Survey of Recent Achievements
,”
Proc. Inst. Mech. Eng.
,
169
(
1
), pp.
41
57
.
26.
Ziegler
,
H.
,
1959
, “
A Modification of Prager's Hardening Rule
,”
Q. Appl. Math.
,
17
(
1
), pp.
55
65
.
27.
Armstrong
,
P. J.
, and
Frederick
,
C. O.
,
1966
, “A Mathematical Representation of the Multiaxial Bauschinger Effect,”
Central Electricity Generating Board
,
Berkeley, UK
, Report No. RD/B/N731.
28.
Chaboche
,
J. L.
,
Van
,
K. D.
, and
Cordier
,
G.
,
1979
, “Modelization of the Strain Memory Effect on the Cyclic Hardening of 316 Stainless Steel,” Smirt5 Div L.
29.
Chaboche
,
J. L.
,
1991
, “
On Some Modifications of Kinematic Hardening to Improve the Description of Ratchetting Effects
,”
Int. J. Plast.
,
7
(
7
), pp.
661
678
.
30.
Ohno
,
N.
, and
Wang
,
J. D.
,
1993
, “
Kinematic Hardening Rules With Critical State of Dynamic Recovery, Part I: Formulation and Basic Features for Ratchetting Behavior
,”
Int. J. Plast.
,
9
(
3
), pp.
375
390
.
31.
McDowell
,
D. L.
,
1995
, “
Stress State Dependence of Cyclic Ratchetting Behavior of Two Rail Steels
,”
Int. J. Plast.
,
11
(
4
), pp.
397
421
.
32.
Abdel-Karim
,
M.
, and
Ohno
,
N.
,
2000
, “
Kinematic Hardening Model Suitable for Ratchetting With Steady-State
,”
Int. J. Plast.
,
16
(
3
), pp.
225
240
.
33.
Bari
,
S.
, and
Hassan
,
T.
,
2002
, “
An Advancement in Cyclic Plasticity Modeling for Multiaxial Ratcheting Simulation
,”
Int. J. Plast.
,
18
(
7
), pp.
873
894
.
34.
Abdel-Karim
,
M.
,
2009
, “
Modified Kinematic Hardening Rules for Simulations of Ratchetting
,”
Int. J. Plast.
,
25
(
8
), pp.
1560
1587
.
35.
Kang
,
G.
,
Gao
,
Q.
, and
Yang
,
X.
,
2002
, “
A Visco-Plastic Constitutive Model Incorporated With Cyclic Hardening for Uniaxial/Multiaxial Ratcheting of SS304 Stainless Steel at Room Temperature
,”
Mech. Mater.
,
34
(
9
), pp.
521
531
.
36.
Chen
,
X.
, and
Jiao
,
R.
,
2004
, “
Modified Kinematic Hardening Rule for Multiaxial Ratcheting Prediction
,”
Int. J. Plast.
,
20
(
4–5
), pp.
871
898
.
37.
Guo
,
S.
,
Kang
,
G.
, and
Zhang
,
J.
,
2013
, “
A Cyclic Visco-Plastic Constitutive Model for Time-Dependent Ratchetting of Particle-Reinforced Metal Matrix Composites
,”
Int. J. Plast.
,
40
, pp.
101
125
.
38.
Chaboche
,
J. L.
,
Kanouté
,
P.
, and
Azzouz
,
F.
,
2012
, “
Cyclic Inelastic Constitutive Equations and Their Impact on the Fatigue Life Predictions
,”
Int. J. Plast.
,
35
, pp.
44
66
.
39.
Malinin
,
N. N.
, and
Khadjinsky
,
G. M.
, “
Theory of Creep With Anisotropic Hardening
,”
Int. J. Mech. Sci.
,
14
(
4
), pp.
235
246
.
40.
Yaguchi
,
M.
, and
Takahashi
,
Y.
,
2005
, “
Ratchetting of Viscoplastic Material With Cyclic Softening, Part 2: Application of Constitutive Models
,”
Int. J. Plast.
,
21
(
4
), pp.
835
860
.
41.
Kullig
,
E.
, and
Wippler
,
S.
,
2006
, “
Numerical Integration and FEM-Implementation of a Viscoplastic Chaboche-Model With Static Recovery
,”
Comput. Mech.
,
38
(
6
), pp.
1
13
.
42.
Chen
,
W.
,
Kitamura
,
T.
, and
Feng
,
M.
,
2018
, “
Creep and Fatigue Behavior of 316L Stainless Steel at Room Temperature: Experiments and a Revisit of a Unified Viscoplasticity Model
,”
Int. J. Fatigue
,
112
, pp.
70
77
.
43.
Zhang
,
Z.
,
Delagnes
,
D.
, and
Bernhart
,
G.
,
2002
, “
Anisothermal Cyclic Plasticity Modelling of Martensitic Steels
,”
Int. J. Fatigue
,
24
(
6
), pp.
635
648
.
44.
Rae
,
Y.
,
Benaarbia
,
A.
,
Hughes
,
J.
, and
Sun
,
W.
,
2019
, “
Experimental Characterisation and Computational Modelling of Cyclic Viscoplastic Behaviour of Turbine Steel
,”
Int. J. Fatigue
,
124
, pp.
581
594
.
45.
Chaboche
,
J. L.
,
1986
, “
Time-Independent Constitutive Theories for Cyclic Plasticity
,”
Int. J. Plast.
,
2
(
2
), pp.
149
188
.
46.
Ohno
,
N.
,
1982
, “
A Constitutive Model of Cyclic Plasticity With a Nonhardening Strain Region
,”
ASME J. Appl. Mech.
,
49
(
4
), pp.
721
727
.
47.
Morin
,
C.
,
Moumni
,
Z.
, and
Zaki
,
W.
,
2011
, “
A Constitutive Model for Shape Memory Alloys Accounting for Thermomechanical Coupling
,”
Int. J. Plast.
,
27
(
5
), pp.
748
767
.
48.
Yu
,
C.
,
Kang
,
G.
,
Kan
,
Q.
, and
Zhu
,
Y.
,
2015
, “
Rate-Dependent Cyclic Deformation of Super-Elastic NiTi Shape Memory Alloy: Thermo-Mechanical Coupled and Physical Mechanism-Based Constitutive Model
,”
Int. J. Plast.
,
72
, pp.
60
90
.
49.
Wang
,
J.
,
Moumni
,
Z.
, and
Zhang
,
W.
,
2017
, “
A Thermomechanically Coupled Finite-Strain Constitutive Model for Cyclic Pseudoelasticity of Polycrystalline Shape Memory Alloys
,”
Int. J. Plast.
,
97
, pp.
194
221
.
50.
Krishna
,
S.
,
Hassan
,
T.
,
Ben Naceur
,
I.
,
Saï
,
K.
, and
Cailletaud
,
G.
,
2009
, “
Macro Versus Micro-Scale Constitutive Models in Simulating Proportional and Nonproportional Cyclic and Ratcheting Responses of Stainless Steel 304
,”
Int. J. Plast.
,
25
(
10
), pp.
1910
1949
.
51.
Nouailhas
,
D.
,
Cailletaud
,
G.
,
Policella
,
H.
,
Marquis
,
D.
,
Dufailly
,
J.
,
Lieurade
,
H. P.
,
Ribes
,
A.
, and
Bollinger
,
E.
,
1985
, “
On the Description of Cyclic Hardening and Initial Cold Working
,”
Eng. Fract. Mech.
,
21
(
4
), pp.
887
895
.
52.
Xu
,
L.
,
Nie
,
X.
,
Fan
,
J.
,
Tao
,
M.
, and
Ding
,
R.
,
2016
, “
Cyclic Hardening and Softening Behavior of the Low Yield Point Steel BLY160: Experimental Response and Constitutive Modeling
,”
Int. J. Plast.
,
78
, pp.
44
63
.
53.
Ahmed
,
R.
,
Barrett
,
P. R.
, and
Hassan
,
T.
,
2016
, “
Unified Viscoplasticity Modeling for Isothermal Low-Cycle Fatigue and Fatigue-Creep Stress-Strain Responses of Haynes 230
,”
Int. J. Solids Struct.
,
88–89
, pp.
131
145
.
54.
Kachanov
,
L. M.
,
1958
, “
Time of Rupture Process Under Creep Conditions
,”
Isv. Akad. Nauk. SSR. Otd Tekh. Nauk
,
23
, pp.
26
31
.
55.
Rabotnov
,
Y. N.
,
1968
,
Creep Rupture
,
Springer, Stanford University
,
Stanford, CA
.
56.
Voyiadjis
,
G. Z.
, and
Dorgan
,
R. J.
,
2007
, “
Framework Using Functional Forms of Hardening Internal State Variables in Modeling Elasto-Plastic-Damage Behavior
,”
Int. J. Plast.
,
23
(
10–11
), pp.
1826
1859
.
57.
Zhu
,
H.
, and
Sun
,
L.
,
2013
, “
A Viscoelastic-Viscoplastic Damage Constitutive Model for Asphalt Mixtures Based on Thermodynamics
,”
Int. J. Plast.
,
40
, pp.
81
100
.
58.
Xu
,
L.
,
Zhao
,
L.
,
Gao
,
Z.
, and
Han
,
Y.
,
2017
, “
A Novel Creep-Fatigue Interaction Damage Model With the Stress Effect to Simulate the Creep-Fatigue Crack Growth Behavior
,”
Int. J. Mech. Sci.
,
130
, pp.
143
153
.
59.
Devaney
,
R. J.
,
O'Donoghue
,
P. E.
, and
Leen
,
S. B.
,
2018
, “
Experimental Characterisation and Computational Modelling for Cyclic Elastic-Plastic Constitutive Behaviour and Fatigue Damage of X100Q for Steel Catenary Risers
,”
Int. J. Fatigue
,
116
, pp.
366
378
.
60.
Lemaitre
,
J.
, and
Desmorat
,
R.
,
2005
,
Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures
,
Springer
,
Berlin
, Chap. II.
61.
Kang
,
G.
,
Liu
,
Y.
,
Ding
,
J.
, and
Gao
,
Q.
,
2009
, “
Uniaxial Ratcheting and Fatigue Failure of Tempered 42CrMo Steel: Damage Evolution and Damage-Coupled Visco-Plastic Constitutive Model
,”
Int. J. Plast.
,
25
(
5
), pp.
838
860
.
62.
Zhang
,
G.
,
Zhao
,
Y.
,
Xue
,
F.
,
Mei
,
J.
,
Wang
,
Z.
,
Zhou
,
C.
, and
Zhang
,
L.
,
2011
, “
Creep-Fatigue Interaction Damage Model and Its Application in Modified 9Cr-1Mo Steel
,”
Nucl. Eng. Des.
,
241
(
12
), pp.
4856
4861
.
63.
Lu
,
J.
,
Sun
,
W.
,
Becker
,
A.
, and
Saad
,
A. A.
,
2015
, “
Simulation of the Fatigue Behaviour of a Power Plant Steel With a Damage Variable
,”
Int. J. Mech. Sci.
,
100
, pp.
145
157
.
64.
Wang
,
W. Z.
,
Buhl
,
P.
,
Klenk
,
A.
, and
Liu
,
Y. Z.
,
2016
, “
Study of Creep-Fatigue Behavior in a 1000 MW Rotor Using a Unified Viscoplastic Constitutive Model With Damage
,”
Int. J. Damage Mech.
,
25
(
2
), pp.
178
202
.
65.
Wang
,
W.
,
Buhl
,
P.
,
Klenk
,
A.
, and
Liu
,
Y.
,
2016
, “
Influence of High-Temperature Dwell Time on Creep-Fatigue Behavior in a 1000 MW Steam Turbine Rotor
,”
Eng. Fract. Mech.
,
166
, pp.
1
22
.
66.
Kyaw
,
S. T.
,
Rouse
,
J. P.
,
Lu
,
J.
, and
Sun
,
W.
,
2016
, “
Determination of Material Parameters for a Unified Viscoplasticity-Damage Model for a P91 Power Plant Steel
,”
Int. J. Mech. Sci.
,
115–116
, pp.
168
179
.
67.
Xie
,
X.
,
Jiang
,
W.
,
Chen
,
J.
,
Zhang
,
X.
, and
Tu
,
S.
,
2019
, “
Cyclic Hardening/Softening Behavior of 316L Stainless Steel at Elevated Temperature Including Strain-Rate and Strain-Range Dependence: Experimental and Damage-Coupled Constitutive Modeling
,”
Int. J. Plast.
,
114
, pp.
196
214
.
68.
Guo
,
Z.
,
Huang
,
D.
,
Yan
,
X.
,
Zhang
,
X.
,
Qi
,
M.
, and
Fan
,
J.
,
2020
, “
A Damage Coupled Elastic-Plastic Constitutive Model and Its Application on Low Cycle Fatigue Life Prediction of Turbine Blade
,”
Int. J. Fatigue
,
131
, p.
105298
.
69.
Li
,
D.
,
Li
,
M.
,
Shang
,
D.
,
Gupta
,
A.
, and
Sun
,
W.
,
2021
, “
Physically-Based Modeling of Cyclic Softening and Damage Behaviors for a Martensitic Turbine Rotor Material at Elevated Temperature
,”
Int. J. Fatigue
,
142
, p.
105956
.
70.
Benallal
,
A.
,
Billardon
,
R.
, and
Doghri
,
I.
,
1988
, “
An Integration Algorithm and the Corresponding Consistent Tangent Operator for Fully Coupled Elastoplastic and Damage Equations
,”
Commun. Appl. Numer. Meth.
,
4
(
6
), pp.
731
740
.
71.
Büttner
,
J.
, and
Simeon
,
B.
,
2002
, “
Runge-Kutta Methods in Elastoplasticity
,”
Appl. Numer. Math.
,
41
(
4
), pp.
443
458
.
72.
Eckert
,
S.
,
Baaser
,
H.
,
Gross
,
D.
, and
Scherf
,
O.
, “
A BDF2 Integration Method With Step Size Control for Elasto-Plasticity
,”
Comput. Mech.
,
34
(
5
), pp.
377
386
.
73.
Simo
,
J. C.
, and
Taylor
,
R. L.
,
1985
, “
Consistent Tangent Operators for Rate-Independent Elastoplasticity
,”
Comput. Meth. Appl. Mech. Eng.
,
48
(
1
), pp.
101
118
.
74.
Kobayashi
,
M.
, and
Ohno
,
N.
,
2002
, “
Implementation of Cyclic Plasticity Models Based on a General Form of Kinematic Hardening
,”
Int. J. Numer. Meth. Eng.
,
53
(
9
), pp.
2217
2238
.
75.
Kang
,
G.
,
2006
, “
Finite Element Implementation of Visco-Plastic Constitutive Model With Strain-Range-Dependent Cyclic Hardening
,”
Commun. Numer. Meth. Eng.
,
22
(
2
), pp.
137
153
.
76.
Kang
,
G.
,
2004
, “
A Visco-Plastic Constitutive Model for Ratcheting of Cyclically Stable Materials and Its Finite Element Implementation
,”
Mech. Mater.
,
36
(
4
), pp.
299
312
.
77.
Kan
,
Q. H.
,
Kang
,
G. Z.
, and
Zhang
,
J.
,
2007
, “
Uniaxial Time-Dependent Ratchetting: Visco-Plastic Model and Finite Element Application
,”
Theor. Appl. Fract. Mech.
,
47
(
2
), pp.
133
144
.
78.
Han
,
J.
,
Marimuthu
,
K. P.
,
Koo
,
S.
, and
Lee
,
H.
,
2020
, “
Numerical Implementation of Modified Chaboche Kinematic Hardening Model for Multiaxial Ratcheting
,”
Comput. Struct.
,
231
, p.
106222
.
79.
Kobayashi
,
M.
,
Mukai
,
M.
,
Takahashi
,
H.
,
Ohno
,
N.
,
Kawakami
,
T.
, and
Ishikawa
,
T.
,
2003
, “
Implicit Integration and Consistent Tangent Modulus of a Time-Dependent Non-Unified Constitutive Model
,”
Int. J. Numer. Meth. Eng.
,
58
(
10
), pp.
1523
1543
.
80.
Wang
,
W. Z.
,
Buhl
,
P.
, and
Klenk
,
A.
,
2015
, “
A Unified Viscoplastic Constitutive Model With Damage for Multi-axial Creep-Fatigue Loading
,”
Int. J. Damage Mech.
,
24
(
3
), pp.
363
382
.
81.
Wang
,
W.
,
Buhl
,
P.
,
Klenk
,
A.
, and
Liu
,
Y.
,
2016
, “
A Continuum Damage Mechanics-Based Viscoplastic Model of Adapted Complexity for High-Temperature Creep-Fatigue Loading
,”
ASME J. Eng. Gas Turbines Power
,
138
(
9
), p.
092501
.
82.
Murakami
,
S.
,
1988
, “
Mechanical Modeling of Material Damage
,”
ASME J. Appl. Mech.
,
55
(
2
), pp.
280
286
.
83.
Mitchell
,
G. P.
,
1990
, “
Topics in the Numerical Analysis of Inelastic Solids
,”
Ph.D. thesis
,
Department of Civil Engineering, University College of Swansea
,
Swansea, UK
.
84.
Betten
,
J.
,
1992
, “
Applications of Tensor Functions in Continuum Damage Mechanics
,”
Int. J. Damage Mech.
,
1
(
1
), pp.
47
59
.
85.
Lemaitre
,
J.
,
Desmorat
,
R.
, and
Sauzay
,
M.
,
2000
, “
Anisotropic Damage Law of Evolution
,”
Eur. J. Mech. A/Solids
,
19
(
2
), pp.
187
208
.
86.
Lemaitre
,
J.
,
1984
, “
A Three-Dimensional Ductile Damage Model Applied to Deep Drawing Forming Limits
,”
Mech. Behav. Mater.
, pp.
1047
1053
.
87.
Lemaitre
,
J.
,
1985
, “
Coupled Elasto-Plasticity and Damage Constitutive Equations
,”
Comput. Meth. Appl. Mech. Eng
,
51
(
1–3
), pp.
31
49
.
88.
Alfano
,
G.
,
Angelis
,
F. D.
, and
Rosati
,
L.
,
2001
, “
General Solution Procedures in Elasto/Viscoplasticity
,”
Comput. Meth. Appl. Mech. Eng.
,
190
(
39
), pp.
5123
5147
.
89.
Neto
,
E. S.
,
Peric
,
D.
, and
Owens
,
D.
,
2008
,
Computational Methods for Plasticity: Theory and Applications
,
Wiley
,
New York
, Chap. XI.
90.
Zhang
,
T.
,
McHugh
,
P. E.
, and
Leen
,
S. B.
,
2012
, “
Finite Element Implementation of Multiaxial Continuum Damage Mechanics for Plain and Fretting Fatigue
,”
Int. J. Fatigue
,
44
, pp.
260
272
.
91.
Khoei
,
A. R.
, and
Eghbalian
,
M.
,
2012
, “
Numerical Simulation of Cyclic Behavior of Ductile Metals With a Coupled Damage-Viscoplasticity Model
,”
Comput. Mater. Sci.
,
55
, pp.
376
389
.
92.
Hartmann
,
S.
, and
Haupt
,
P.
,
1993
, “
Stress Computation and Consistent Tangent Operator Using Non-Linear Kinematic Hardening Models
,”
Int. J. Numer. Meth. Eng.
,
36
(
22
), pp.
3801
3814
.
93.
Nesnas
,
K.
, and
Saanouni
,
K.
,
2002
, “
Integral Formulation of Coupled Damage and Viscoplastic Constitutive Equations: Formulation and Computational Issues
,”
Int. J. Damage Mech.
,
11
(
4
), pp.
367
398
.
You do not currently have access to this content.