Effective biaxial modulus of ideally (hkl)-fiber-textured hexagonal, tetragonal, and orthorhombic films

Abstract

A detailed study on the effective biaxial modulus of ideally (hkl)-fiber-textured films is presented for the cases where the constituent grains are of hexagonal, tetragonal, and orthorhombic crystal symmetries. The analytical form of the Voigt [Lehrbuch der kristallphysik (Teubner, Leipzig, 1910)] and Reuss [Z. Angew. Math. Mech. 9, 49 (1929)] averages of M in these noncubic cases is first given, coupled with the numerical values of four averages for selected films and textures. The coincidences of the Voigt and Reuss averages of M are then explained in terms of the symmetry of the constituent grain about the fiber axis, followed by several useful implications. For other textures with noncoincident Voigt and Reuss averages, our results indicate that the Voigt-Reuss-Hill [R. Hill, Proc. Phy. Soc., London, Sect. A 65, 349 (1952); D. H. Chung, Philos. Mag. 8, 833 (1963)] average is a good theoretical estimation.