General compliance transformation relation and applications for anisotropic hexagonal metals

In anisotropic crystals, the compliance (sij)(sij) and the stiffness (cij)(cij) matrices are usually specified in the orthogonal coordinate systems (X1,Y1,Z1)(X1,Y1,Z1), which do not coincide with the crystal axes (X,Y,Z)(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(hkl)E(hkl), Poisson’s ratio υ(hkl)υ(hkl) and X-ray elastic constants (XREC) s1(hkl)=−υ(hkl)E(hkl) and 12s2(hkl)=1+υ(hkl)E(hkl) are also given in terms of the Miller indices of the lattice plane (hkl)(hkl) in the crystal axes (X,Y,Z)(X,Y,Z) used commonly.