Planes of isotropic shear moduli in anisotropic materials

Abstract

The elastic properties of crystalline solids are known to be anisotropic in general. The shear modulus G(n,m) depends upon the shear plane normal n and the shear direction m. It is known that G(n,m) is independent of m when n is normal to planes with three, four or sixfold symmetries. Additionally, G(n,m) is always independent of m when n is parallel to the <001> directions in cubic crystals, even in point groups 23 and m3‾ where they possess only a twofold symmetry. In the current study, an expression for ΔG(n), the difference between the extreme values of G(n,m) within a shear plane, is derived for a general anisotropic material. Novel 3D surfaces, with the radii along n related to ΔG(n), representing the anisotropy of the shear modulus within the shear plane, are presented. It is further shown that shear planes on which G(n,m) is independent of m can exist in all crystal systems. A general equation to determine n with isotropic G is derived. For all materials except those with monoclinic and triclinic crystal symmetries, explicit expressions for the isotropic shear planes are obtained. It is found that G(n,m) can be independent of m on planes without symmetry in addition to the ones with three, four and sixfold symmetries.