definition of elastic symmetry?

Answer 1

Elastic properties of materials can be represented by a matrix or tensor containing values of stiffness or modulus of elasticity. The different coefficients of the matrix give stress vs strain along different axes. The diagonal terms (e.g., the x,x term) of the matrix represent stress/strain for a given axis (x), and the off-diagonal terms represent the coupling of stress in one axis to strain in another axis (e.g., the x,y coefficient might specify strain in the y axis resulting from stress along the x axis). In a symmetric matrix, corresponding coefficients to either side of the diagonal are equal; that is, C(x,y) = C(y,x), etc.
Unfortunately I couldn't find a simple explanation on the web of the physical significance of such symmetry other than the obvious conclusions you can draw from those equalities, that stress-strain coupling between any two axes has the same value in either direction. Ref. 1 is a rather advanced article on mathematical manipulation of arbitrary tensors to find a set of axes in which the tensor is close to symmetrical. It contains examples of the symmetric elastic matrices mentioned above.
Since you asked me for a simple description (in another question on this subject) I'll add some references to wikipedia that may get you going in the right direction (refs. 2 and up). (Ref. 2 is the "index" to several useful pages, including refs. 3 and up.)