Numerical study of a two-dimensional problem in a constrained minimization theory of elasticity

Resumo

We consider the equilibrium of a cylindrically orthotropic disk subject to a prescribed displacement on

its boundary. In the context of classical linear elasticity, this problem has a solution that predicts material inter-
penetration. To prevent this unphysical behavior, we minimize the energy functional of classical linear elasticity

subject to the local injectivity constraint. In previous work, we have obtained computational results showing that

this problem has a rotationally symmetric solution that bifurcates from a radially symmetric solution. This sec-
ondary solution differs from a secondary solution reported in the literature, which was obtained by making no

assumptions about the symmetry of the solution. Here, we extend the investigation of our previous work by mak-
ing no a priori assumption on the symmetry of the displacement field. Using two different formulations, we obtain

sequences of numerical solutions that converge to the rotationally symmetric solution mentioned above as the mesh
is refined. In addition, there is no evidence of the existence of a third solution, which indicates that the rotationally
symmetric solution is the only possible secondary solution. This research is of interest in the investigation of solids
with radial microstructure, such as carbon fibers and certain types of wood.