POSITIVE DEFINITENESS OF ANISOTROPIC ELASTIC-CONSTANTS

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 posit ive 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 criterio n and discuss its merits and drawbacks. For a two-dimensional deformat ion, it suffices to consider a 5 x 5 symmetric matrix C-0 In the Stroh formalism for two-dimensional deformations, the matrix C-0 is replace d by three 3 x 3 matrices N-1,N-2,N-3. Deleting the elements that are either zero or unity, the N-3 is reduced to a 2 x 2 matrix (N) over ca p(3), and the N-1 is reduced to a 3 x 2 matrix (N) over cap(1). We sho w that C-0 is positive definite if and only if N-2 and -(N) over cap(3 ) are positive as well. The matrix (N) over cap(1) can be arbitrary. I n particular, a new relation \C-0\ = \-(N) over cap(3)\.\N-2\(-1) is o btained. Generalized to three-dimensional deformations, it is shown th at positive definiteness of the 6 x 6 matrix C is equivalent to positi ve definiteness of two 3 x 3 matrices. In the special case of monoclin ic materials with the symmetry plane at x(1) = 0,x(2) = 0, or x(3) = 0 , positive definiteness of C is equivalent to positive definiteness of three 2 x 2 matrices.