On the Existence of Rotationally Symmetric Solution of a Constrained Minimization Problem of Elasticity

Abstract

We consider the equilibrium problem, with no body force, of a cylindrically orthotropic disk subject to a prescribed displacement along its boundary. In classical linear elasticity, the solution of this problem predicts material overlapping, which is not physically realistic. One way to prevent this anomalous behavior is to consider the minimization of the total potential energy of classical linear elasticity subject to the local injectivity constraint. In the context of this constrained minimization theory, bifurcation occurs from a radially symmetric solution to a secondary solution. In this work we present analytical and computational results indicating that this secondary solution is rotationally symmetric.

Initially, we assume that the solution is rotationally symmetric, solve the Euler-Lagrange equations of the corresponding minimization problem in the region where the local injectivity constraint is not active, and obtain a solution that depends on constants of integration that are determined numerically. In the region where the constraint is active, we determine a nonlinear relation between the radial and tangential displacements, which contains a constant of integration that is also determined numerically.

Still assuming rotational symmetry, we use an interior penalty formulation together with a standard finite element method to obtain sequences of numerical solutions that converge to a limit function that is in very good agreement with our analytical results in the non active region. To confirm these findings, we also search for an asymmetric solution numerically. In this case, there is no a priori assumption on symmetry and we only obtain either the radially or the rotationally symmetric solution. In all the cases investigated numerically, we have to introduce a small perturbation to obtain the latter solution. The rotationally symmetric solution presents a novel behavior that is not reported in the literature. The tangential displacement is linear near the center of the disk and the corresponding angle of rotation has the value $\pi$ at this center and decreases as we move away from it.

Finally, we investigate numerically the influence of both the shear modulus and the boundary condition on the existence of the rotationally symmetric solution. For a given mesh, there is a maximum value of the shear modulus above which and a minimum value of the boundary condition below which this solution is not possible. This research is of interest in the investigation of solids having stiffer response in the radial direction than in the tangential direction, such as in the case of carbon fibers with radial microstructure and certain types of wood.