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A complete-damage problem at small strains. (English) Zbl 1238.74005
Summary: Damage of a linearly-responding material that can completely disintegrate is addressed at small strains. Using time-varying Dirichlet boundary conditions, we set up a rate-independent evolution problem in multidimensional situations. The stored energy involves the gradient of the damage variable. This variable as well as stress and energies are shown to be well-defined even under complete damage, in contrast to displacement and strain. Existence of an energetic solution is proved, in particular, by a detailed investigation of the $$\Gamma$$-limit of the stored energy and its dependence on boundary conditions. Eventually, the theory is illustrated on a one-dimensional example.

##### MSC:
 74A45 Theories of fracture and damage 74R20 Anelastic fracture and damage 74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) 74G65 Energy minimization in equilibrium problems in solid mechanics 35Q74 PDEs in connection with mechanics of deformable solids
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