Numerical investigation of bifurcation instability in constrained minimization problem of elasticity

Resumo

There are problems in classical linear elasticity whose closed form solutions, while satisfying the
governing equations of equilibrium and well-posed boundary conditions, predict material overlapping, which is
not physically realistic. One possible way to prevent this anomalous behavior is to consider the minimization of the
total potential energy of classical linear elasticity subjected to the local injectivity constraint. In two dimensions,
the corresponding constrained minimization problem has a solution, which may not be unique. To investigate this
class of problems, we consider the equilibrium problem of a cylindrically anisotropic disk subjected to a prescribed
displacement along its boundary. We search numerically for either radially or rotationally symmetric solutions
defined in a one-dimensional domain. Our discretization strategy uses linear finite elements and yields convergent
sequences of solutions at a very low computational cost when compared to results reported in the literature. It
is clear from our investigation that a small perturbation must be introduced to obtain the rotationally symmetric
solution, for otherwise the solution obtained is radially symmetric. The total potential energy from the rotationally
symmetric solution is lower than the corresponding energy evaluated from the radially symmetric solution.