Limit analysis is one of the most fundamental methods of plasticity. For the nonstandard model, the concept of the bipotential, representing the dissipated plastic power, allowed us to extend limit analysis theorems to the nonassociated flow rules. In this work, the kinematic approach is used to find the limit load and its corresponding collapse mechanism. Because the bipotential contains in its expression the stress field of the limit state, the kinematic approach is coupled with the static one. For this reason, a solution of kinematic problem is obtained in two steps. In the first one, the stress field is assumed to be constant and a velocity field is computed by the use of the kinematic theorem. Then, the second step consists to compute the stress field by means of constitutive relations keeping the velocity field constant and equal to that of the previous step. A regularization method is used to overcome problems related to the nondifferentiability of the dissipation function. A successive approximation algorithm is used to treat the coupling question. A simple compression-traction of a nonassociated rigid perfectly plastic material and an application of punching by finite element method are presented in the end of the paper.

1.
Anderheggen
,
E.
, and
Knôpfel
,
H.
, 1972, “
Finite Element Limit Analysis Using Linear Programming
,”
Int. J. Solids Struct.
0020-7683,
8
, pp.
1413
1431
.
2.
Maier
,
G.
,
Grierson
,
D. E.
, and
Best
,
M. J.
, 1977, “
Mathematical Programming Methods for Deformation Analysis at Plastic Collapse
,”
Comput. Struct.
0045-7949,
7
, pp.
599
612
.
3.
Dang Hung
,
N.
, 1976, “
Direct Limit Analysis via Rigid-Plastic Finite Elements
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
8
, pp.
81
116
.
4.
Guennouni
,
A. T.
,
Letallec
,
P.
, 1982, “
Calcul à la rupture: Régularisation de Norton-Hoff et lagrangien augmenté
,”
J. Mec. Theor. Appl.
0750-7240,
2
(
1
), pp.
75
99
.
5.
Jiang
,
G. L.
, 1995, “
Non Linear Finite Element Formulation of Kinematic Limit Analysis
,”
Int. J. Numer. Methods Eng.
0029-5981,
38
, pp.
2775
2807
.
6.
Pontes
,
I. D. S.
,
Borges
,
L. A.
,
Zouain
,
N.
, and
Lopes
,
F. R.
, 1997, “
An Approach to Limit Analysis With Cone-Shaped Yield Surfaces
,”
Int. J. Numer. Methods Eng.
0029-5981,
40
, pp.
4011
4032
.
7.
Christiansen
,
E.
, 1980, “
Limit Analysis in Plasticity as a Mathematical Programming Problem
,”
Calcolo
0008-0624,
17
, pp.
41
65
.
8.
Christiansen
,
E.
, 1996, “
Limit Analysis of Collapse States
,”
Handbook of Numerical Analysis
,
P. G.
Ciarlet
and
J. L.
Lions
, eds.,
Elsevier
,
New York
, Vol.
4
.
9.
Chaaba
,
A.
,
Bousshine
,
L.
, and
De Saxcé
,
G.
, 2003, “
Kinematic Limit Analysis Modelling by Regularization Approach and Finite Element Method
,”
Int. J. Numer. Methods Eng.
0029-5981,
57
, pp.
1899
1922
.
10.
Huh
,
H.
,
Lee
,
C. H.
, and
Yang
,
W. H.
, 1999, “
A General Algorithm for Plastic Flow Simulation by Finite Element Analysis
,”
Int. J. Solids Struct.
0020-7683,
36
, pp.
1193
1207
.
11.
Hwan
,
C. L.
, 1997, “
Plane Strain Extrusion by Sequential Limit Analysis
,”
Int. J. Mech. Sci.
0020-7403,
39
(
7
), pp.
807
817
.
12.
Maier
,
G.
, 1969, “
Shakedown Theory in Perfect Elastoplasticity With Associated and Non Associated Flow Laws: A Finite Element, Linear Programming Approach
,”
Meccanica
0025-6455,
4
, pp.
250
260
.
13.
Fenchel
,
W.
, 1949, “
On Conjugate Convex Functions
,”
Can. J. Math.
0008-414X,
1
, pp.
73
77
.
14.
Moreau
,
J. J.
, 1968, “
La notion de sur-potential et les liaisons unilatérales en élastoplasticité
,”
C.R. Seances Acad. Sci., Ser. A
0366-6034,
267
, pp.
954
957
.
15.
De Saxcé
,
G.
, and
Feng
,
Z. Q.
, 1991, “
New Inequation and Functional for Contact With Friction: The Implicit Standard Material Approach
,”
Mech. Struct. Mach.
0890-5452,
19
(
3
), pp.
301
325
.
16.
De Saxcé
,
G.
, and
Feng
,
Z. Q.
, 1998, “
The Bipotential Method: A Constructive Approach to Design the Complete Contact Law With Friction and Improved Numerical Algorithms
,”
Math. Comput. Modell.
0895-7177,
28
(
4–8
), pp.
225
245
.
17.
Fihri
,
F. H.
, 2002, “
Simulation numérique des procédés de formage à froid des métaux en présence d'une loi non associée du contact avec frottement
,” Doctorat thesis., Faculty of Sciences Agdal, Rabat, Morroco.
18.
De Saxcé
,
G.
,
Berga
,
A.
, and
Bousshine
,
L.
, 1992, “
The Implicit Standards Materials for Non Associated Plasticity in Soil Mechanics
,”
Proceedings of the International Congress on Numerical Methods in Engineering and Applied Sciences
, Concepcion, Chile, Vo1.
1
, pp.
585
594
.
19.
Bousshine
,
L.
, 1994, “
Processus de mise en forme des métaux et des matériaux granulaires
,” Ph.D. thesis, Polytechnic Faculty of Mons, Mons, Belgium.
20.
Chaaba
,
A.
,
Bousshine
,
L.
,
De Saxcé
,
G.
, and
Guerlement
,
G.
, 2000, “
Granular Material Limit Analysis and Application for Slope Stability
,
Fourth EUROMECH Conference
, Metz, France, Jun. 26–30.
21.
De Saxcé
,
G.
, and
Bousshine
,
L.
, 1998, “
Limit Analysis Theorems for Implicit Standard Materials: Application to the Unilateral Contact With Dry Friction and Non-Associated Flow Rules in Soils and Rocks
,”
Int. J. Mech. Sci.
0020-7403,
40
(
4
), pp.
387
398
.
22.
Bousshine
,
L
,
Chaaba
,
A.
, and
De Saxcé
,
G.
, 2002, “
Plastic Limit Load of Frictional Contact Supports Plane Frames
,”
Int. J. Mech. Sci.
0020-7403,
44
(
11
), pp.
2189
2216
.
23.
Chaaba
,
A.
,
Bousshine
,
L.
, and
De Saxcé
,
G.
, “
Analyse limite des comportements mécaniques non associés
,”
Plasticité et contact avec frottement, CIMASI ‘2002
. Casablanca, Morroco, Oct. 14–16.
24.
Van Langen
,
H.
, and
Vermer
,
P. A.
, 1990, “
Automatic Step Size Correction for Non Associated Plasticity Problems
,”
Int. J. Numer. Methods Eng.
0029-5981,
29
, pp.
579
598
.
25.
Rudnicki
,
J.
, and
Rice
,
J. R.
, 1975, “
Conditions for the Localization of Deformation in Pressure-Sensitive Dilatant Materials
,”
J. Mech. Phys. Solids
0022-5096,
23
, pp.
371
394
.
26.
Bousshine
,
L.
,
Chaaba
,
A.
, and
De Saxcé
,
G.
, 2001, “
Softening in Stress-Strain Curve for Drucker-Prager Non-Associated Plasticity
,”
Int. J. Plast
,
17
(
1
), pp.
21
46
.
27.
Bousshine
,
L.
,
Chaaba
,
A.
, and
De Saxcé
,
G.
, 2003, “
A New Approach to Shakedown Analysis for Non Standard Elastoplastic Material by the Bipotential
,”
Jour. Plasticity
,
19
(
5
), pp.
583
598
.
28.
Chaaba
,
A.
,
Bousshine
,
L.
,
Elharif
,
A.
, and
De Saxcé
,
G.
, 1997, “
Une nouvelle approche des lois non associées et application aux matériaux non standards sous chargement cyclique
,”
Third Congress of Mechanics
, Faculty of Sciences, Tétouan, April, pp.
22
25
.
29.
De Saxcé
,
G.
,
Tritsh
,
J. B.
and
Hjiaj
,
M.
1998, “
Shakedown of Elastic-Plastic Materials With Non Linear Kinematic Hardening Rule by the Bipotential Approach
,”
Euromech 385 Colloquium
, Aachen, Germany, Sept. 8–11.
30.
Berga
,
A.
, and
De Saxcé
,
G.
, 1994, “
Elastoplastic Finite Elements Analysis of Soil Problems With Implicit Standard Materials Constitutive Laws
,”
Revue européenne des éléments finis (European Journal of Computational Mechanics)
,
3
, pp.
411
456
.
31.
Bouby
,
C.
,
De Saxcé
,
G.
, and
Tritsch
,
J.-B.
, 2006, “
A comparison Between Analytical Calculations of the Shakedown Load by the Bipotential Approach and Step-By-Step Computations for Elastoplastic Materials With Nonlinear Kinematic Hardening
,”
Int. J. Solids Struct.
0020-7683,
43
(
9
), pp.
2670
2692
.
32.
Murtagh
,
R. A.
, and
Saunders
,
M. A.
, 1987, Minos 5.1 User’s Guide, Stanford University.
33.
Drucker
,
D. C.
, 1956, “
On Uniqueness in the Theory of Plasticity
,”
Q. J. Mech. Appl. Math.
0033-5614,
14
, pp.
35
42
.
34.
Bleich
H. H.
, 1972, “
On Uniqueness in Ideally Elastoplastic Problems in Case of Nonassociated Flow Rules
,” JAM, Trans., ASME,
983
987
.
35.
De Saxcé
,
G.
, and
Bousshine
,
L.
, 1993, “
On the Extension of Limit Analysis Theorems to the Non Associated Flow Rules in Soils and to the Contact With Coulomb’s Friction
,”
Proceedings of the XI Polish Conference on Computer Methods in Mechanics
, Kielce, Poland, May 11–14,
2
, pp.
815
822
.
You do not currently have access to this content.