Generalized Cowin–Mehrabadi theorems and a direct proof that the number of linear elastic symmetries is eight

The Cowin–Mehrabadi theorem is generalized to allow less restrictive and more flexible conditions for locating a symmetry plane in an anisotropic elastic material. The generalized theorems are then employed to prove that the number of linear elastic symmetries is eight. The proof starts by imposing a symmetry plane to a triclinic material and, after new elastic symmetries are found, another symmetry plane is imposed. This process exhausts all possibility of elastic symmetries, and shows that there are only eight elastic symmetries. At each stage when a new symmetry plane is added, explicit results are obtained for the locations of the new symmetry plane that lead to a new elastic symmetry. It takes as few as three, and at most five, symmetry planes to reduce a triclinic material (which has no symmetry plane) to an isotropic material for which any plane is a symmetry plane.