In this paper, the orientation dependence of the ‘effective modulus’, M_eff , in statics relating the applied stress to the applied strain for a single crystal subjected to full transverse constraint is specified as a general problem for which up to now analytical solutions have only existed for special cases of symmetry.

For isotropic materials, the result is a simple one in terms of the isothermal Young’s modulus, E, and the isothermal Poisson ratio, v:

and the strain state is one of uniaxial strain. Similar states of uniaxial strain can occur along axes of symmetry in all crystalline materials, such as <100>, <110> or <111> directions in cubic single crystals which have which have three independent elastic stiffnesses c₁₁, c₁₂ and c₄₄. For such axes of symmetry, M_eff can easily be shown to be the elastic stiffness tensor c_ij resolved along the direction under consideration: c'₁₁.

For other orientations of applied stress with full transverse constraint for cubic single crystals, conditions of either zero shear strains or zero shear stresses are the two limiting cases. If zero shear strains are taken to be the determining boundary conditions for a general direction of applied stress [uvw], it is evident that the measured stiffness will be c'₁₁. If, instead, zero shear stresses are taken to be the determining boundary conditions, M_eff can be shown both analytically and by computation to be less than c'₁₁. For some cubic materials such as In–27 at% Tl at 290 K just before it undergoes a martensitic transformation on cooling to a tetragonal phase with rubber-like behaviour, the differences between M_eff and c'₁₁ are very noticeable around <001> orientations, as shown in the accompanying figure.

Figure caption: (a) contour map of M_eff plotted within the standard 001 – 011 – 111  stereographic triangle for In–27 at% Tl at 290 K, and (b) contour map of c'₁₁ plotted within the standard 001 – 011 – 111  stereographic triangle for In–27 at% Tl at 290 K.