Identification of symmetry type of linear elastic stiffness tensor in an arbitrarily orientated coordinate system

We develop a method through the mirror plane (MP) to identify the symmetry type of linear elastic stiffness tensor whose components are given with respect to an arbitrarily oriented coordinate system. The method is based on the irreducible decomposition of high-order tensor into a set of deviators and the multipole representation of a deviator into a scalar and a unit-vector set. Since a unit-vector depends on two Euler angles, we can illustrate the MP normals of the elastic tensor as zeros of a characteristic function on a unit disk and identify its symmetry immediately, which is clearer and simpler than the methods proposed before. Furthermore, by finding the common MPs of three unit-vector sets using Fortran recipes, we can also analytically recognize the symmetry type first and then recover the natural coordinate system associated with the linear elastic tensor. The structures of linear elastic stiffness tensors of real materials with all possible anisotropies are investigated in detail.