Positive Definiteness of Anisotropic Elastic Constants

Abstract

When the elastic constants of an anisotropic material are written as a 6 x 6 symmetric matrix C, the elastic energy of the material is positive if the matrix C is positive definite. There are two criteria that one can use to see if C is positive definite. We present each criterion and discuss its merits and drawbacks. For a two-dimensional deformation, it suffices to consider a 5 x 5 symmetric matrix C'. In the Stroh formalism for two-dimensional deformations, the matrix C⁰ is replaced by three 3 x 3 matrices N₁, N₂, N₃. Deleting the elements that are either zero or unity, the N₃ is reduced to a 2 x 2 matrix &3, and the N₁ is reduced to a 3 x 2 matrix N1. We show that C' is positive definite if and only if N₂ and - N₃ are positive as well. The matrix N₁ can be arbitrary. In particular, a new relation 1C⁰ I = I -N₃ j IN2 I - is obtained. Generalized to three-dimensional deformations, it is shown that positive definiteness of the 6 x 6 matrix C is equivalent to positive definiteness of two 3 x 3 matrices. In the special case of monoclinic materials with the symmetry plane at xl = O,x2 = 0, or x₃ = 0, positive definiteness of C is equivalent to positive definiteness of three 2 x 2 matrices.