A discontinuous Galerkin formulation for classical and gradient plasticity. Part 2: Algorithms and numerical analysis

Computer Methods in Applied Mechanics and Engineering

Volumn 197, Issue 1-4, 2007, Pages 1-21

  Djoko, J K ^a   Ebobisse, F ^a   McBride, A T ^a   Reddy, B D ^a  

^a UNIVERSITY OF CAPE TOWN   (South Africa)

Author keywords

Algorithms; Consistent tangent modulus; Discontinuous Galerkin; Elastoplasticity; Gradient plasticity

Indexed keywords

ALGORITHMS; CONVERGENCE OF NUMERICAL METHODS; ELASTIC MODULI; ELASTOPLASTICITY;

CONSISTENT TANGENT MODULUS; DISCONTINUOUS GALERKIN METHOD; GRADIENT PLASTICITY; PREDICTOR-CORRECTOR ALGORITHM;

GALERKIN METHODS;

ALGORITHMS; CONVERGENCE OF NUMERICAL METHODS; ELASTIC MODULI; ELASTOPLASTICITY; GALERKIN METHODS;

EID: 35348897873     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2007.06.027     Document Type: Article

References (54)

1. Djoko J.K., Ebobisse F., McBride A.T., and Reddy B.D. A discontinuous Galerkin formulation for classical and gradient plasticity, Part 1: Formulation and analysis. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3881-3897
2. Lasry D., and Belytschko T. Localization limiters in transient problems. Int. J. Solids Struct. 24 24 (1998) 581-597
3. de Borst R., and Mülhaus H.-S. Gradient-dependent plasticity: formulation and algorithmic aspects. Int. J. Numer. Methods Engrg. 35 (1992) 521-539
4. Mülhaus H.B., and Aifantis E.C. A variational principle for gradient plasticity. Int. J. Solids Struct. 28 7 (1991) 845-857
5. J. Pamin, Gradient-Dependent Plasticity in Numerical Simulation of Localization Phenomena, Ph.D. Thesis, Delft University of Technology, 1994.
6. de Borst R., and Pamin J. Some novel developments in finite element procedures for gradient-dependent plasticity. Int. J. Numer. Methods Engrg. 39 (1996) 2477-2505
7. Liebe T., and Steinmann P. Theory and numerics of a thermodynamically consistent framework for geometrically linear gradient plasticity. Int. J. Numer. Methods Engrg. 51 (2001) 1437-1467
8. de Borst R., Pamin J., and Geers M.G.D. On coupled gradient-dependent plasticity and damage theories with a view to localization analysis. Eur. J. Mech. Solids 18 (1999) 939-962
9. Wells G.N., Garikipati K., and Molari L. A discontinuous Galerkin formulation for a strain gradient-dependent damage model. Comput. Methods Appl. Mech. Engrg. 193 (2004) 3633-3645
10. Garikipati K. Variational multiscale methods to embed the macromechanical continuum formulation with fine-scale gradient theories. Int. J. Numer. Methods Engrg. 57 (2003) 1283-1298
11. Fleck N.A., and Hutchinson J.W. Strain gradient plasticity. Adv. Appl. Mech. 33 (1997) 295-361
12. Li X., and Cescotto S. Finite element method for gradient plasticity at large strains. Int. J. Numer. Methods Engrg. 39 (1996) 619-633
13. Mikkelsen L.P. Post-necking behaviour modelled by a gradient dependent plasticity theory. Int. J. Solids Struct. 34 35-36 (1997) 4531-4546
14. Ramaswamya S., and Aravas N. Finite element implementation of gradient plasticity models Part I: Gradient-dependent yield functions. Comput. Methods Appl. Mech. Engrg. 163 (1998) 11-32
15. Comi C., and Perego U. A generalized variable formulation for gradient dependent softening plasticity. Int. J. Numer. Methods Engrg. 39 (1996) 3731-3755
16. Comi C., and Corigliano A. On uniqueness of the dynamic finite-step problem in gradient-dependent softening plasticity. Int. J. Solids Struct. 33 26 (1996) 3881-3902
17. Li X., and Cescotto S. A mixed element method in gradient plasticity for pressure dependent materials and modelling of strain localization. Comput. Methods Appl. Mech. Engrg. 144 (1997) 287-305
18. Svedberg T., and Runesson K. An algorithm for gradient-regularized plasticity coupled to damage based on a dual mixed FE-formulation. Comput. Methods Appl. Mech. Engrg. 161 (1998) 49-65
19. D.J. Bammann, D. Mosher, D.A. Hughes, N.R. Moody, P.R. Dawson, Using spatial gradients to model localization phenomena, Technical report, Sandia National Laboratories, 1999.
20. Zervos A., Papanastasiou P., and Vardoulakis I. Modeling of localisation and scale effect in thick-walled cylinders with gradient elastoplasticity. Int. J. Solids Struct. 38 (2001) 5081-5095
21. Liebe T., Menzel A., and Steinmann P. Theory and numerics of geometrically non-linear gradient plasticity. Int. J. Engrg. Sci. 41 (2003) 1603-1629
22. Al-Rub R.K.A., and Voyiadjis G.Z. A direct finite element implementation of the gradient dependent theory. Int. J. Numer. Methods Engrg. 63 (2005) 603-629
23. Bird W.W., and Martin J.B. A secant approximation for holonomic elastic-plastic incremental analysis with a von mises yield condition. Engrg. Comput. 3 (1986) 192-201
24. Bird W.W., and Martin J.B. Consistent predictors and the solution of the piecewise holonomic incremental problem in elasto-plasticity. Engrg. Struct. 12 (1990) 9-14
25. Eve R.A., Gültop T., and Reddy B.D. An internal variable finite strain theory of plasticity. Quart. Appl. Math. 48 (1990) 625-644
26. Reddy B.D., and Martin J.B. Algorithms for the solution of the internal variable problem in plasticity. Comput. Methods Appl. Mech. Engrg. 93 (1991) 253-273
27. Martin J.B., and Nappi A. An internal variable formulation for perfectly plastic and linear hardening relations in plasticity. Eur. J. Mech. Solids 9 2 (1990) 107-131
28. Eve R.A., Reddy B.D., and Rockafellar R.T. An internal variable theory of elasto-plasticity based on the maximum plastic work inequality. Quart. Appl. Math. 48 (1990) 59-83
29. Han W., and Reddy B.D. Plasticity: Mathematical Theory and Numerical Analysis vol. 9 (1999), Springer
30. Simo J.C., and Hughes T.J.R. Computational Inelasticity (1998), Springer-Verlag, New York
31. Hansbo P., and Larson M.G. Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. M2AN Math. Model. Numer. Anal. 37 (2003) 63-72
32. Rivière B., Shaw S., Wheeler M.F., and Whiteman J.R. Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity. Numer. Math. 95 2 (2003) 347-376
33. Lew A., Neff P., Sulsky D., and Ortiz M. Optimal BV estimates for a discontinuous Galerkin method for linear elasticity. Appl. Math. Res. Exp. 2004 3 (2004) 73-106
34. Martin J.B., and Caddemi S. Sufficient conditions for convergence of the Newton-Raphson iterative algorithm in incremental elastic-plastic analysis. Eur. J. Mech. Solids 13 3 (1994) 351-365
35. Simo J.C. Topics on the numerical analysis and simulation of plasticity. In: Ciarlet P.G., and Lions J.-L. (Eds). Handbook of Numerical Analysis vol. VI (1998), North-Holland, Amsterdam 183-499
36. Simo J.C., and Taylor R.L. Consistent tangent operators for rate independent elasto-plasticity. Comput. Methods Appl. Mech. Engrg. 48 (1985) 101-118
37. Alfano G., and Rosati L. A general approach to the evaluation of consistent tangent operators for rate-independent elastoplasticity. Comput. Methods Appl. Mech. Engrg. 167 (1998) 75-89
38. Simo J.C. A framework for finite strain elasto-plasticity based on maximum plastic dissipation and their multiplicative decompositions: Part iii computation aspects. Comput. Methods Appl. Mech. Engrg. 68 (1988) 1-31
39. R.A. Eve, Theoretical and Numerical Aspects of Problems In Finite Strain Plasticity, Ph.D. Thesis, University of Cape Town, 1992.
40. Alberty J., Carstensen C., and Funken S.A. Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algorithms 20 (1999) 117-137
41. Carstensen C., and Klose R. Elastoviscoplastic finite element analysis in 100 lines of Matlab. J. Numer. Math. 10 3 (2002) 157-192
42. Engelen R.A.B., Geers M.G.D., and Baaijens F.P.T. Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour. Int. J. Plast. 19 (2003) 403-433
43. R.K.A. Al-Rub, Material Length Scales in Gradient-Dependent Plasticity/Damage and Size Effects: Theory and Computation, Ph.D. Thesis, Louisiana State University and Agricultural and Mechanical College, 2004.
44. Khoei A.R., Tabarraie A.R., and Gharehbaghi S.A. H-adaptive mesh refinement for shear band localization in elasto-plasticity cosserat continuum. Commun. Nonlinear Sci. Numer. Simulat. 10 (2005) 253-286
45. Begley M.R., and Hutchinson J.W. The mechanics of size-dependent indentation. J. Mech. Phys. Solids 46 10 (1998) 2049-2068
46. Nix W.D., and Gao H. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46 3 (1998) 411-425
47. Shu J.Y., and Fleck N.A. The prediction of size effect in micro-indentation. Int. J. Solids Struct. 35 13 (1998) 1363-1383
48. Yuan H., and Chen J. Analysis of size effects based on a symmetric lower-order gradient plasticity model. Comput. Mater. Sci. 19 (2000) 143-157
49. Yuang H., and Chen J. Identification of the intrinsic material length in gradient plasticity theory from micro-indentation tests. Int. J. Solids Struct. 38 (2001) 8171-8187
50. Thoa K.K., Swaddiwudhipong S., Hua J., and Liu Z.S. Numerical simulation of indentation with size effect. Mater. Sci. Engrg. A 421 (2006) 268-275
51. W.W. Bird, Formulations For The Incremental Elastic-Plastic Analysis and the Consolidation of Porous Media, Ph.D. Thesis, University of Cape Town, March 1987.
52. Gurtin M.E., and Anand L. A theory of strain gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. Int. J. Plast. 53 (2005) 1624-1649
53. B.D. Reddy, F. Ebobisse, A.T. McBride, Well-posedness of a model of strain gradient plasticity for plastically irrotational materials, in press.
54. Fleck N.A., and Hutchinson J.W. A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49 (2001) 2245-2271