Bifurcating solutions in a constrained minimization problem of elasticity

Abstract

We consider the problem of a cylindrically anisotropic disk subject to an imposed  displacement on its boundary. In the context of classical linear elasticity, the solution of this  problem is not locally injective. This characterizes material overlapping, which is not physically  admissible. To prevent this anomalous behavior, we minimize the energy functional of classical  linear elasticity subject to the local injectivity constraint. One possible solution of the associated  Euler-Lagrange equations, reported in the literature, is radially symmetric. In this work we search  for a secondary solution, which is rotationally symmetric. In the region where the constraint is  not active, we determine closed-form expressions for the displacement field. In the region where  the constraint is active, which is annular, we determine a relation between the components of the  displacement field to ensure the imposition of this constraint. The expressions obtained in both  regions depend on constants of integration that are determined numerically. In addition, we also  determine the inner and outer radii of the active annular region. This research is of interest in the  investigation of solids with radial microstructure, such as carbon fibers.