Abstract
The numerical solution of non-associated elastoplasticity is still a key aspect of research and development in computational plasticity. Approximate solution procedures are based, in the context of a displacement method, on a weak form of the equilibrium and reply upon two main ingredients: the numerical integration of the rate constitutive relations over a generic time step (local stage) and the iterative algorithm exploited to solve the nonlinear equilibrium equations (global stage). The fully discrete problem is the obtained by performing a spatial discretization of the field equations and a time-integration of the evolution rule. The interest is here given to the discretization errors, which are caused by the numerical discretization of the continuous mathematical model in order to define an adaptive strategy.
The aim of this paper is to extend the concept of error in the constitutive equations to non-associated plasticity Drucker-Prager model to handle non-associative rate-independent plasticity problems solved by employing the incremental displacement conforming finite element method.
Numerical examples by PLSAER2D (a Matlab program) for both the associated and the non-associated cases for Drucker-Prager model with hardening are also presented.
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References
Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. John Wiley & Son and B G Teubner, Chichester (2000)
Halphen, B., Nguyen, Q.S.: Sur les matériaux standards généralisés. J. Méc. 14, 39–63 (1975)
Babuska, I., Rheinboldt, W.C.: A posteriori error estimates for the finite element method. Int. J. Numer. Methods Eng. 12, 1597–1615 (1978)
Drucker, D.C.: On the postulate of stability of material in the mechanics of continua. J. Méc. 3(2), 235–249 (1964)
Lam, D.D.: A posteriori error estimation for non-associated plasticity problems, Ph. D. thesis, INSA de Rennes (2009)
Lam, D.D.: A bi-potential update algorithm for the non-associated plasticity model with hardening. In: Proceedings of CIGOS-2015: Innovations in Construction, Paris, France (2015)
de Saxcé, G.: Une généralisation de l’inégalité de Fenchel et ses applications aux lois constitutives. Comptes Rendus de l’Académie des Sciences de Paris (1992)
Hjiaj, M.: Sur la classe des matériaux standard implicites: Concept, Aspects discrétisés et Estimation de l’erreur a posteriori. Ph. D. thesis, Polytechnic Faculty of Mons. (1999)
Hjiaj, M., Lam, D.D., de Saxcé, G.: A family of bi-potentials describing the non-associated flow rule of pressure-dependant plastic models. Acta Mech. 220, 237–246 (2011)
Ladevèze, P., Maunder, E.A.W.: A general procedure for recovering equilibrating element tractions. Comput. Methods Appl. Mech. Eng. 137, 111–151 (1996)
Ladevèze, P., Moës, N., Douchin, B.: Constitutive relation error estimators for (visco) plastic finite element analysis with softening. Comput. Methods Appl. Mech. Eng. 176, 247–264 (1999)
Ladevèze, P., Moës, N.: A new a posteriori error estimation for nonlinear time dependent finite element analysis. Comput. Methods Appl. Mech. Eng. 157, 45–68 (1997)
Ladevèze, P., Pelle, J.P.: Mastering Calculations in Linear and Nonlinear Mechanics. Springer, New York (2004)
Ladevèze, P., Passieux, J.-C., Néron, D.: The LATIN multiscale computational method and the proper generalized decomposition. Comput. Methods Appl. Mech. Eng. 199, 1287–1296 (2009)
Maunder, E.A.W., Moitinho de Almeida, J.P.: The stability of stars of triangular equilibrium plate elements. Int. J. Numer. Methods Eng. 77, 922–968 (2009)
Moreau, J.: Evolution problem associated with a moving convex set in a hilbert space. J. Diff. Equ. 26, 347–374 (1977)
Orlando, A., Peri´c, D.: Analysis of transfer procedures in elastoplasticity based on the error in the constitutive equations: Theory and numerical illustration. Int. J. Numer. Methods Eng. 60, 1595–1631 (2004)
Simo, J.C., Hughes, T.J.: Computational Inelasticity. Springer, New York (2000)
Zienkiewicz, O.C., Liu, Y.C., Huang, G.C.: Error estimation and adaptivity in flow formulation for forming problems. Int. J. Numer. Methods Eng. 25, 23–42 (1988)
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Lam, D.D. (2018). A Posteriori Error Estimation for the Non-associated Plasticity Drucker-Prager Model with Hardening. In: Shehata, H., Rashed, Y. (eds) Numerical Analysis of Nonlinear Coupled Problems. GeoMEast 2017. Sustainable Civil Infrastructures. Springer, Cham. https://doi.org/10.1007/978-3-319-61905-7_9
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DOI: https://doi.org/10.1007/978-3-319-61905-7_9
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