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Characterization of the limit load in the case of an unbounded elastic convex. (English) Zbl 1330.74070
Summary: We consider a solid body $$\Omega\subset{\mathbb R}^3$$ constituted by a nonhomogeneous elastoplastic material, submitted to a density of body forces $$\lambda f$$ and a density of forces $$\lambda g$$ acting on the boundary where the real $$\lambda$$ is the loading parameter. The problem is to determine, in the case of an unbounded convex of elasticity, the limit load denoted by $$\overline{\lambda}$$ beyond which there is a break of the structure. The case of a bounded convex of elasticity is treated in [H. El Fekih and T. Hadhri, RAIRO, Modélisation Math. Anal. Numér. 29, No. 4, 391–419 (1995; Zbl 0831.73016)]. Then assuming that the convex of elasticity at the point $$x$$ of $$\Omega$$, denoted by $$K(x)$$, is written in the form of $$K^{\mathrm{D}}(x)+{\mathbb R}I$$, $$I$$ is the identity of $$\mathbb R^9_{\mathrm{sym}}$$, and the deviatoric component $$K^{\mathrm{D}}$$ is bounded regardless of $$x\in\Omega$$, we show under the condition ‘$$\mathrm{Rot} f \not= 0$$ or $$g$$ is not colinear to the normal on a part of the boundary of $$\Omega$$’, that the limit load $$\overline{\lambda}$$ searched is equal to the inverse of the infimum of the gauge of the elastic convex translated by stress field equilibrating the unitary load corresponding to $$\lambda =1$$; moreover we show that this infimum is reached in a suitable function space.
##### MSC:
 74G65 Energy minimization in equilibrium problems in solid mechanics 35J56 Boundary value problems for first-order elliptic systems 35Q74 PDEs in connection with mechanics of deformable solids 49J45 Methods involving semicontinuity and convergence; relaxation 74C99 Plastic materials, materials of stress-rate and internal-variable type 74R99 Fracture and damage