Certain plane contact problems of elasticity theory for strongly anisotropic media

Abstract

The asymptotic behavior is studied of the solutions of the second boundary value (displacements are given on the boundary) and two contact problems of elasticity theory in a rectangular domain for a curvilinear orthotropic medium with one quite high elasticity coefficient. The sides of the rectangle are parallel to the orthotropy directions. It is shown that for a curvature different everywhere from zero for families of very rigid fibers, we always obtain a medium with inextensible fibers in the limit (the model of the medium with inextensible fibers is introduced in a number of papers, /1/, say). The presence of a large parameter in the generalized Hooke's law results in a singular perturbation of the boundary value problems. Such singularly perturbed problems occur in studying structures from composite materials reinforced by high-modulus fibers /1,2/. It was studied the question of regularity of the degeneration of pre-limiting boundary value problems in limit problems /3/. The uniform asymptotic of the solution in a closed domain contains functions of the angular boundary layer.