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The obstacle problem for an elastoplastic body. (English) Zbl 0709.73020

The difficulties encountered by researchers in plastic-work hardening materials were mainly caused by inability to select a proper function space for strains and displacements. This difficulty was by-passed in such classical treatises as that of G. Duvant and J. L. Lions [Inequalities in mechanics and physics (1976; Zbl 0331.35002)].
For perfect plasticity a number of authors lead by Suquet [e.g.: P. M. Suquet, J. Méc. 20, 3-39 (1981; Zbl 0474.73030); Res. Notes Math. 46, 184-197 (1981; Zbl 0453.73040)] abandoned the Sobolev spaces and proved existence of solutions for the displacement problem in the space BD(\(\Omega\)) of bounded functions with the norm \(\| u\| =\| u\|_{L^ 1(\Omega)}+\| \epsilon (u)\|_{{\mathcal M}}\), where \({\mathcal M}\) is the space of measures assigned to the \(3\times 3\) symmetric matrices of strain components (which must not be identified with stress components!). R. Temam and G. Strang [e.g.: J. Méc. 19, 493-527 (1980; Zbl 0465.73033)] avoided some of the difficulties by using duality arguments.
The present authors formulate the plastic-work hardening problems in product spaces with two models of plasticity considered. One is based on displacement-plastic strain, the other is the well-known Hencky model. Existence of solutions are proved by assuming compatibility and so called “safe load” condition.
Reviewer: V.Komkov

MSC:

74C99 Plastic materials, materials of stress-rate and internal-variable type
49J27 Existence theories for problems in abstract spaces
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
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