General compliance transformation relation and applications for anisotropic hexagonal metals

In anisotropic crystals, the compliance ( s i j ) and the stiffness ( c i j ) matrices are usually specified in the orthogonal coordinate systems ( X 1 , Y 1 , Z 1 ) , which do not coincide with the crystal axes ( X , Y , Z ) used commonly, excepting cubic and orthorhombic crystal systems, and must be transformed to an arbitrary orthogonal coordinate system chosen to be convenient for the question. Such a transformation has been done in this paper for hexagonal crystals and a general compliance transformation relation is given. Accordingly, the useful expressions of the Young’s modulus E ( h k l ) , Poisson’s ratio υ ( h k l ) and X-ray elastic constants (XREC) s 1 ( h k l ) = − υ ( h k l ) E ( h k l ) and 1 2 s 2 ( h k l ) = 1 + υ ( h k l ) E ( h k l ) are also given in terms of the Miller indices of the lattice plane ( h k l ) in the crystal axes ( X , Y , Z ) used commonly.