A finite element solution for plasticity with strain-hardening

Three basic boundary value problems areformulated in terms of stresses and hardening parameters Piecewise hnear simphcial éléments are employed and some error estimâtes derived, provided certain regulanty of the exact solution holds Ifthe solution is not regular, the convergence is proven Resumé — Trois problèmes aux limites fondamentaux sont exprime s en font tion des contraintes et de patamettes deaouissage On utilise des éléments finis simplutain (ineaues paf moueaux et on établit des majorations d'erreur, moyennant une certaine régulante de la solution exacte Si la solution n'est pas reguliere, on démontre la convergence The flow theory of plasticity with stram-hardenmg matenal (cf. [5]) has been studiedrecentlybyC. Johnson [8],Groger [3]andNeöas [6] from a new point of view, pioneered by Nguyen Quoc Son [14] and Halphen-Nguyen Quoc Son [4] The common idea of their existence proofs is to formulate the problem by means of vanational mequahty of évolution and to use a penalty method. Vanous incrémental fmite element solutions have been published in the engineering literature. To the author's knowledge, ho wever, the only theoretical convergence analysis have been presented by C. Johnson [9] In the present paper, we propose another variant of the incrémental fini te element method, startmg from the formulation of the quasi-staticj^roblem in terms of stresses and^ ïïârdenmg parameters only. Whereas m the mixed method of [9] the stresses and hardemng parameters are approximated by piecewise constant functions and the displacements by piecewise hnear functions, we employ piecewise hnear functions for both the stresses and the hardemng parameters. The stress approximations consist of equihbnated tnangular or tetrahedral blockelements respectively (cf [15, 7, 11, 12]) (*) Reçu novembre 1979 (*) Mathematical Institute of the Czechoslovak Academy of Sciences, Zitna 25, Praha 1, Tchécoslovaquie R A I R O Analyse numenque/Numencal Analysis, 0399-0516/1980/347/$ 5 00 © Bordas-Dunod