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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
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