Numerical investigation of orthotropic finite elasticity problem with discontinuous deformation gradient

Resumo

We consider the problem of an elastic annular disk 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 is stiffer in the radial direction than in the tangential direction. Such material properties are found in certain types of wood and carbon fibers with radial microstructure. We consider that the disk is made of a cylindrically orthotropic St Venant-Kirchhoff material, which is a natural constitutive extension from the linear to the nonlinear elasticity theory. The solution of this problem predicts material overlapping,
which is unphysical if either the pressure is large enough or the inner radius is small enough. A way to prevent this anomalous behavior consists of imposing the local injectivity constraint through a constrained minimization problem of the energy functional. We use both a penalty and an augmented Lagrangian formulation to obtain convergent sequences of finite element approximations. Our results indicate that, to impose the local injectivity constraint accurately, it is preferable to increase the degree of the finite element approximation than to increase the number of finite elements in the mesh.