Visualization of Symmetries in Fourth-Order Stiffness Tensors

Abstract

Many materials like wood, biological tissue, composites or rock have anisotropic mechanical properties. They become increasingly important in modern material, earth, and life sciences. The stress-strain response of such materials can be characterized (to first-order) by the three-dimensional fourth-order stiffness tensor. There are different anisotropy classes, i.e. material symmetries, that differ in the number and orientation of symmetry planes characteristic of the material. A three-dimensional fourth-order stiffness tensor of a hyperelastic material has up to 21 independent coefficients representing both moduli and orientation information which challenges any visualization method. Therefore, we use a fourth-order tensor decomposition to compute the anisotropy classes and the position of the corresponding symmetry planes. To facilitate judgment of the significance of the amount of anisotropy, we construct an isotropic material. Based on these computations, we design a glyph that represents the stiffness tensor. We demonstrate our method in a finite deformation setting of an initially isotropic hyperelastic material of Ogden class which is often modeling biological tissue. Upon deformation, the stiffness tensor can evolve along with its symmetry creating an inhomogeneous, unsteady fourth-order tensor field in three dimensions.