Shear moduli under uniaxial loading

This paper presents stress/strain dependence of shear moduli under uniaxial homogeneous loading in direction 3 of an elastic solid, which has initially an isotropic or cubic symmetry. The stress/strain dependence of the Young’s modulus and Poisson’s ratio under uniaxial loading was previously treated in detail by this author and Sachse for a similar case. Stress/strain dependence of shear moduli is expressed in terms of thermodynamic elastic coefficients and uniaxial stress ?3 = ?33. The thermodynamic elastic coefficients are then expressed in terms of the second-order elastic constants (SOEC) and the third-order elastic constants (TOEC). The relation between the Young’s modulus E(a), Poisson’s ratio ?(a) , and shear modulus G(a) at stress-free isotropic state is given by G(a) = E(a)/2[1+ ? (a) ], where vector a represents a stress-free natural state. This relation no longer holds valid for a stressed media. However, an isotropic solid at stress-free state becomes transversely isotropic under uniaxial loading around the loading direction 3 and a similar relation holds at a uniaxially stressed state X. Letting shear modulus G66(X) denote a shear modulus defined by infinitesimal Cauchy shear stress ?? 6(X) divided by infinitesimal shear strain ??6(X), a relation G66(X) = E3(X)/2[1+ ?(a)] holds at a stressed state X. A similar relation fails to hold for other shear moduli G44 (X) and G55 (X) defined similar to G66(X).