Numerical investigation of an orthotropic finite elasticity problem using a constrained minimization theory to prevent material overlapping

Resumo

We consider the problem of an elastic annular disk with uniform thickness in equilibrium in the absence of body force. The disk is fixed on its inner surface and compressed by a uniform pressure on its outer surface. The disk is made of a cylindrically orthotropic material that has a constitutive response that is stiffer in the radial direction than in the tangential direction. Material properties of this type are found in carbon fibers with radial microstructure and some kinds of wood.
In the context of the classical linear elasticity theory, the solution of this problem predicts material overlapping for a large enough pressure, which is not physically acceptable. An approach to prevent this anomalous behavior consists of minimizing the total potential energy functional subject to the constraint that the determinant of the deformation gradient, J, be positive. We have used this approach to eliminate material overlapping, yielding solutions with J not close to one, which contradicts the basic assumption of infinitesimal strains upon which the classical linear theory is founded.
In this work, we extend our investigation to the nonlinear elasticity theory. Necessary conditions for a deformation field to be a minimizer were found elsewhere. It includes a nonzero Lagrange multiplier that represents a reaction pressure to prevent material overlapping. We use a finite element formulation to find that, differently from its linearly elastic counterpart, the Lagrange multiplier field associated with the local injectivity constraint remains bounded. In addition, we obtain convergent sequences of approximate solutions of the disk problem formulated with both this constrained minimization theory and a compressible and orthotropic Mooney-Rivlin material. This material has certain growth conditions on the deformation field that prevents material overlapping in classical problems of mechanics. Both sequences yield convergent solutions that are in very good agreement with each other.