Spatial representation of evolving anisotropy at large strains

A phenomenological model for evolving anisotropy at large strains is presented. The model is formulated using spatial quantities and the anisotropic properties of the material is modeled by including structural variables. Evolution of anisotropy is accounted for by introducing substructural deformation gradients which are linear maps similar to the usual deformation gradient. The evolution of the substructural deformation gradients is governed by the substructural plastic velocity gradients in a manner similar to that for the continuum. Certain topics related to the numerical implementation are discussed and a simple integration scheme for the local constitutive equations is developed. To demonstrate the capabilities of the model it is implemented into a finite element code. Two numerical examples are considered: deformation of uniform plate with circular hole and the drawing of a cup. In the two examples it is assumed that initial cubic material symmetry applies to both the elastic and plastic behavior. To be specific, a polyconvex Helmholtz free energy function together with a yield function of quadratic type is adopted.