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.
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