Problèmes capacitaires en viscoplasticité avec effets de torsion

Abstract

We study the homogenization of elasticity problems like (1) when f, g are strictly convex functions satisfying a growth condition of order p∈(1,+∞), g is positively homogeneous of degree p, kε→+∞, and Trε consists of an ε-periodic distribution of parallel fibers of cross sections of size rε≪ε. The problem (1) corresponds to a simplified model of small deformation nonlinear elasticity describing, for instance, the small deformations of an Ogdenʼs material (Ogden, 1972 [8]). When 1<p<2, it may also characterize the viscoplastic creep experienced, at high temperatures, by a metallic composite governed by the Norton–Hoff model (Friaâ, 1979 [7]). In this case, uε represents the velocity vector field. We show that if p⩽2, a concentration of strain energy appears in a small region of space surrounding the fibers. This extra contribution is characterized by a local density of the sections of the fibers with respect to some appropriate capacity depending, if p<2, on the angles of rotation of the fibers with respect to their principal axis. This rotating behavior generates, in parallel, the emergence of torsional strain energy within the fibers.