A nonassociative plasticity model of Drucker–Prager yield surface coupled with a generalized nonlinear kinematic hardening is considered. Conforming to the plasticity model, two exponential-based methods, called fully explicit and semi-implicit, are recommended for integrating its constitutive equations. These techniques are proposed for the first time to solve nonlinear hardening materials. The integrations are thoroughly investigated by utilizing stress and strain-updating tests along with a boundary value problem in diverse grounds of accuracy, convergence rate, and efficiency. The results indicate that the fully explicit scheme is more accurate and efficient than the Euler's, but the same convergence rate as the classical integrations is also perceived. Having a quadratic convergence, the semi-implicit is noticeably the most accurate and efficient procedure to use for this plasticity model among the algorithms in question. Since the plasticity model is in a great consistency with discontinuously reinforced aluminum (DRA) composites, the suggested formulations can be utilized pragmatically. The tangent moduli of the proposed and Euler's strategies are derived and examined, as well, due to their vital role in achieving the asymptotic quadratic convergence rate of the Newton–Raphson solution in nonlinear finite-element analyses.

References

References
1.
Bridgman
,
P. W.
,
1947
, “
The Effect of Hydrostatic Pressure on the Fracture of Brittle Substances
,”
J. Appl. Phys.
,
18
, pp.
246
258
.10.1063/1.1697610
2.
Bridgman
,
P. W.
,
1952
,
Studies in Large Plastic Flow and Fracture With Special Emphasis on the Effect of Hydrostatic Pressure
,
McGraw-Hill
,
New York
.
3.
Spitzig
,
W. A.
,
Sober
,
R. J.
, and
Richmond
,
O.
,
1975
, “
Pressure Dependence of Yielding and Associated Volume Expansion in Tempered Martensite
,”
Acta Metall.
,
23
, pp.
885
893
.10.1016/0001-6160(75)90205-9
4.
Spitzig
,
W. A.
,
Sober
,
R. J.
, and
Richmond
,
O.
,
1976
, “
The Effect of Hydrostatic Pressure on the Deformation Behavior of Maraging and HY-80 and Its Implication for Plasticity
,”
Metall. Trans. A
,
A7
, pp.
457
463
.
5.
Spitzig
,
W. A.
, and
Richmond
,
O.
,
1984
, “
The Effect of Pressure on the Flow Stress of Metals
,”
Acta Metall.
,
32
, pp.
457
463
.10.1016/0001-6160(84)90119-6
6.
Wilson
,
C. D.
,
2002
, “
A Critical Reexamination of Classical Metal Plasticity
,”
ASME J. Appl. Mech.
,
69
, pp.
63
68
.10.1115/1.1412239
7.
Drucker
,
D. C.
, and
Prager
,
W.
,
1952
, “
Soil Mechanics and Plastic Analysis or Limit Design
,”
Q. Appl. Math.
,
10
, pp.
157
165
.
8.
Singh
,
A. P.
,
Padmanabhan
,
K. A.
,
Pandy
,
G. N.
,
Murty
,
G. M. D.
, and
Jha
,
S.
,
2000
, “
Strength Differential Effect in Four Commercial Steels
,”
J. Mater. Sci.
,
35
, pp.
1379
1388
.10.1023/A:1004738326505
9.
Altenbach
,
H.
,
Stoychev
,
G. B.
, and
Tushtev
,
K. N.
,
2001
, “
On Elastoplastic Deformation of Grey Cast Iron
,”
Int. J. Plast.
,
17
, pp.
719
736
.10.1016/S0749-6419(00)00026-7
10.
Chait
,
R.
,
1973
, “
The Strength Differential of Steel and Ti Alloys as Influenced by Test Temperature and Microstructure
,”
Scr. Metall.
,
7
, pp.
351
363
.10.1016/0036-9748(73)90054-9
11.
Gil
,
C. M.
,
Lissenden
,
C. J.
, and
Lerch
,
B. A.
,
1999
, “
Yield of Inconel 718 by Axial-Torsional Loading at Temperatures Up to 649 °C
,”
J. Test. Eval.
,
27
, pp.
327
336
.10.1520/JTE12233J
12.
Iyer
,
S. K.
, and
Lissenden
,
C. J.
,
2000
, “
Initial Anisotropy of Inconel 718: Experiments and Mathematical Representation
.”
J. Eng. Mater. Technol.
,
122
, pp.
321
326
.10.1115/1.482804
13.
Lewandowski
,
J. J.
,
Wesseling
,
P.
,
Prabhu
,
N. S.
,
Larose
,
J.
, and
Lerch
,
B. A.
,
2003
, “
Strength Differential Measurements in IN-718: Effects of Superimposed Pressure
,”
Metall. Mater. Trans. A
,
34A
, pp.
1736
1739
.10.1007/s11661-003-0319-2
14.
Lei
,
X.
, and
Lissenden
,
C. J.
,
2007
, “
Pressure Sensitive Nonassociative Plasticity Model for DRA Composites
,”
ASME J. Eng. Mater. Technol.
,
129
, pp.
255
264
.10.1115/1.2400273
15.
Prager
,
W.
,
1956
, “
A New Method of Analyzing Stresses and Strains in Work Hardening Plastic Solids
.”
ASME J. Appl. Mech.
,
23
, pp.
493
496
.
16.
Chakrabarty
,
J.
,
2000
,
Theory of Plasticity
,
3rd ed.
Elsevier Butterworth-Heinemann
,
Oxford, UK
.
17.
Bari
,
S.
, and
Hassan
,
T.
,
2000
, “
Anatomy of Coupled Constitutive Models for Ratcheting Simulation
,”
Int. J. Plast.
,
16
, pp.
381
409
.10.1016/S0749-6419(99)00059-5
18.
Armstrong
,
P. J.
, and
Frederick
,
C. O.
,
1966
, “
A Mathematical Representation of the Multiaxial Bauschinger Effect
,” CEGB Report No. RD/B/N 731.
19.
Chaboche
,
J. L.
,
1986
, “
Time-Independent Constitutive Theories for Cyclic Plasticity
,”
Int. J. Plast.
,
2
, pp.
149
188
.10.1016/0749-6419(86)90010-0
20.
Chaboche
,
J. L.
,
1991
, “
On Some Modifications of Kinematic Hardening to Improve the Description of Ratcheting Effects
,”
Int. J. Plast.
,
7
, pp.
661
678
.10.1016/0749-6419(91)90050-9
21.
Ohno
,
N.
, and
Wang
,
J. D.
,
1993
, “
Kinematic Hardening Rules With Critical State of Dynamic Recovery—Part I: Formulations and Basic Features for Ratcheting Behavior
,”
Int. J. Plast.
,
9
, pp.
375
390
.10.1016/0749-6419(93)90042-O
22.
Abdel-Karim
,
M.
, and
Ohno
,
N.
,
2000
, “
Kinematic Hardening Model Suitable for Ratcheting With Steady-State
,”
Int. J. Plast.
,
16
, pp.
225
240
.10.1016/S0749-6419(99)00052-2
23.
Kang
,
G.
,
2004
, “
A Visco-Plastic Constitutive Model for Ratcheting of Cyclically Stable Materials and Its Finite Element Implementation
,”
Mech. Mater.
,
36
, pp.
299
312
.10.1016/S0167-6636(03)00024-3
24.
Chaboche
,
J. L.
,
2008
, “
A Review of Some Plasticity and Viscoplasticity Constitutive Theories
,”
Int. J. Plast.
,
24
, pp.
1642
1693
.10.1016/j.ijplas.2008.03.009
25.
Abdel-Karim
,
M.
,
2009
, “
Modified Kinematic Hardening Rules for Simulations of Ratcheting
,”
Int. J. Plast.
,
25
, pp.
1560
1587
.10.1016/j.ijplas.2008.10.004
26.
Rezaiee-Pajand
,
M.
, and
Sinaie
,
S.
,
2009
, “
On the Calibration of the Chaboche Hardening Model and a Modified Hardening Rule for Uniaxial Ratcheting Prediction
,”
Int. J. Solids Struct.
,
46
, pp.
3009
3017
.10.1016/j.ijsolstr.2009.04.002
27.
Mroz
,
Z.
,
1967
, “
On the Description of Anisotropic Work Hardening
,”
J. Mech. Phys. Solids
,
15
, pp.
163
175
.10.1016/0022-5096(67)90030-0
28.
Dafalias
,
Y. F.
, and
Popov
,
E. P.
,
1976
, “
Plastic Internal Variables Formalism of Cyclic Plasticity
,”
ASME J. Appl. Mech.
,
43
, pp.
645
650
.10.1115/1.3423948
29.
Tseng
,
N. T.
, and
Lee
,
G. C.
,
1983
, “
Simple Plasticity Model of the Two-Surface Type
,”
ASCE J. Eng. Mech.
,
109
, pp.
795
810
.10.1061/(ASCE)0733-9399(1983)109:3(795)
30.
Krieg
,
R. D.
, and
Krieg
,
D. B.
,
1977
, “
Accuracies of Numerical Solution Methods for the Elastic-Perfectly Plastic Model
,”
ASME J. Pressure Vessel Technol.
,
99
, pp.
510
515
.10.1115/1.3454568
31.
Yoder
,
P. J.
, and
Whirley
,
R. G.
,
1984
, “
On the Numerical Implementation of Elastoplastic Models
,”
ASME J. Appl. Mech.
,
51
, pp.
283
288
.10.1115/1.3167613
32.
Loret
,
B.
, and
Prevost
,
J. H.
,
1986
, “
Accurate Numerical Solutions for Drucker-Prager Elastic-Plastic Models
,”
Comput. Methods Appl. Mech. Eng.
,
54
, pp.
259
277
.10.1016/0045-7825(86)90106-4
33.
Ristinmaa
,
M.
, and
Tryding
,
J.
,
1993
, “
Exact Integration of Constitutive Equations in Elastoplasticity
,”
Int. J. Numer. Methods Eng.
,
36
, pp.
2525
2544
.10.1002/nme.1620361503
34.
Wei
,
Z.
,
Perić
,
D.
, and
Owen
,
D. R. J.
,
1996
, “
Consistent Linearization for the Exact Stress Update of Prandtl–Reuss Non-Hardening Elastoplastic Models
,”
Int. J. Numer. Methods Eng.
,
39
, pp.
1219
1235
.10.1002/(SICI)1097-0207(19960415)39:7<1219::AID-NME901>3.0.CO;2-7
35.
Szabó
,
L.
,
2009
, “
A Semi-Analytical Integration Method for J2 Flow Theory of Plasticity With Linear Isotropic Hardening
,”
Comput. Methods Appl. Mech. Eng.
,
198
, pp.
2151
2166
.10.1016/j.cma.2009.02.007
36.
Rezaiee-Pajand
,
M.
,
Sharifian
,
M.
, and
Sharifian
,
M.
,
2011
, “
Accurate and Approximate Integrations of Drucker-Parger Plasticity With Linear Isotropic and Kinematic Hardening
,”
Eur. J. Mech. A/Solids
,
30
, pp.
345
361
.10.1016/j.euromechsol.2010.12.001
37.
Rezaiee-Pajand
,
M.
, and
Sharifian
,
M.
,
2012
, “
A Novel Formulation for Integrating Nonlinear Kinematic Hardening Drucker-Prager's Yield Condition
,”
Eur. J. Mech. A/Solids
,
31
, pp.
163
178
.10.1016/j.euromechsol.2011.08.004
38.
Wilkins
,
M. L.
,
1964
, “
Calculation of Elastic-Plastic Flow
,”
Methods in Computational Physics
, Vol.
3
, Academic Press, New York.
39.
Rice
,
J. R.
, and
Tracey
,
D. M.
,
1973
, “
Computational Fracture Mechanics
,”
Numerical and Computer Methods in Structural Mechanics
, S. J. Fenves, ed., Academic Press, New York, pp. 585–623.
40.
Ortiz
,
M.
, and
Popov
,
E. P.
,
1985
, “
Accuracy and Stability of Integration Algorithms for Elastoplastic Constitutive Relations
,”
Int. J. Numer. Methods Eng.
,
21
, pp.
1561
1576
.10.1002/nme.1620210902
41.
Kobayashi
,
M.
, and
Ohno
,
N.
,
2002
, “
Implementation of Cyclic Plasticity Models Based on a General Form of Kinematic Hardening
,”
Int. J. Numer. Methods Eng.
,
53
, pp.
2217
2238
.10.1002/nme.384
42.
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. Methods Eng.
,
58
, pp.
1523
1543
.10.1002/nme.825
43.
Kang
,
G.
,
2006
, “
Finite Element Implementation of Viscoplastic Constitutive Model With Strain-Range Dependent Cyclic Hardening
,”
Commun. Numer. Methods Eng.
,
22
(
2
), pp.
137
153
.10.1002/cnm.803
44.
Kan
,
Q. H.
,
Kang
,
G. Z.
, and
Zhang
,
J.
,
2007
, “
A Unified Visco-Plastic Constitutive Model for Uniaxial Time-Dependent Ratchetting and Its Finite Element Implementation
,”
Theor. Appl. Fract. Mech.
,
47
, pp.
133
144
.10.1016/j.tafmec.2006.11.005
45.
Coombs
,
W. M.
,
Crouch
,
R. S.
, and
Augarde
,
C. E.
,
2010
, “
Reuleaux Plasticity: Analytical Backward Euler Stress Integration and Consistent Tangent
,”
Comput. Methods Appl. Mech. Eng.
,
199
, pp.
1733
1743
.10.1016/j.cma.2010.01.017
46.
Hong
,
H. K.
, and
Liu
,
C. S.
,
1999
, “
Internal Symmetry in Bilinear Elastoplasticity
,”
Int. J. Non-Linear Mech.
,
34
, pp.
279
288
.10.1016/S0020-7462(98)00029-8
47.
Hong
,
H. K.
, and
Liu
,
C. S.
,
2000
, “
Internal Symmetry in the Constitutive Model of Perfect Elastoplasticity
,”
Int. J. Non-Linear Mech.
,
35
, pp.
447
466
.10.1016/S0020-7462(99)00030-X
48.
Hong
,
H. K.
, and
Liu
,
C. S.
,
2000
, “
Lorentz Group on Minkowski Spacetime for Construction of the Two Basic Principles of Plasticity
.”
Int. J. Non-Linear Mech.
,
36
, pp.
679
686
.10.1016/S0020-7462(00)00033-0
49.
Liu
,
C. S.
,
2003
, “
Symmetry Groups and the Pseudo-Riemann Spacetimes for Mixed-Hardening Elastoplasticity
,”
Int. J. Solids Struct.
,
40
, pp.
251
269
.10.1016/S0020-7683(02)00552-8
50.
Liu
,
C. S.
,
2004
, “
A Consistent Numerical Scheme for the Von-Mises Mixed-Hardening Constitutive Equations
.”
Int. J. Plast.
,
20
, pp.
663
704
.10.1016/S0749-6419(03)00077-9
51.
Liu
,
C. S.
,
2004
, “
Internal Symmetry Groups for the Drucker-Prager Material Model of Plasticity and Numerical Integrating Methods
.”
Int. J. Solids Struct.
,
41
, pp.
3771
3791
.10.1016/j.ijsolstr.2004.02.035
52.
Auricchio
,
F.
, and
Beirão da Veiga
,
L.
,
2003
, “
On a New Integration Scheme for Von-Mises Plasticity With Linear Hardening
,”
Int. J. Numer. Methods Eng.
,
56
, pp.
1375
1396
.10.1002/nme.612
53.
Artioli
,
E.
,
Auricchio
,
F.
, and
Beirão da Veiga
,
L.
,
2005
, “
Integration Schemes for Von-Mises Plasticity Models Based on Exponential Maps: Numerical Investigations and Theoretical Considerations
,”
Int. J. Numer. Methods Eng.
,
64
, pp.
1133
1165
.10.1002/nme.1342
54.
Artioli
,
E.
,
Auricchio
,
F.
, and
Beirão da Veiga
,
L.
,
2006
, “
A Novel ‘Optimal’ Exponential-Based Integration Algorithm for Von-Mises Plasticity With Linear Hardening: Theoretical Analysis on Yield Consistency, Accuracy, Convergence and Numerical Investigations
,”
Int. J. Numer. Methods Eng.
,
4
, pp.
449
498
.10.1002/nme.1637
55.
Artioli
,
E.
,
Auricchio
,
F.
, and
Beirão da Veiga
,
L.
,
2007
, “
Second-Order Accurate Integration Algorithms for Von-Mises Plasticity With a Nonlinear Kinematic Hardening Mechanism
,”
Comput. Methods Appl. Mech. Eng.
,
196
, pp.
1827
1846
.10.1016/j.cma.2006.10.002
56.
Rezaiee-Pajand
,
M.
, and
Nasirai
,
C.
,
2007
, “
Accurate Integration Scheme for Von-Mises Plasticity With Mixed-Hardening Based on Exponential Maps
,”
Eng. Comput.
,
24
(
6
), pp.
608
635
.10.1108/02644400710774806
57.
Rezaiee-Pajand
,
M.
, and
Nasirai
,
C.
,
2008
, “
On the Integration Schemes for Drucker-Prager's Elastoplastic Models Based on Exponential Maps
.”
Int. J. Numer. Methods Eng.
,
74
, pp.
799
826
.10.1002/nme.2178
58.
Rezaiee-Pajand
,
M.
,
Nasirai
,
C.
, and
Sharifian
,
M.
,
2010
, “
Application of Exponential-Based Methods in Integrating the Constitutive Equations With Multicomponent Kinematic Hardening
,”
ASCE J. Eng. Mech.
,
136
(
12
), pp.
1502
1518
.10.1061/(ASCE)EM.1943-7889.0000192
59.
Rezaiee-Pajand
,
M.
,
Nasirai
,
C.
, and
Sharifian
,
M.
,
2011
, “
Integration of Nonlinear Mixed Hardening Models
,”
Multidiscip. Model. Mater. Struct.
7
(
3
), pp.
266
305
.10.1108/1536-540911178252
60.
de Souza Neto
,
E. A.
,
Perić
,
D.
, and
Owen
,
D. R. J.
,
2008
,
Computational Methods for Plasticity: Theory and Applications
,
John Wiley and Sons, Ltd
, New York.
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