Elastic constants of cubic crystals

The simplest material symmetry beyond isotropy is cubic symmetry, a property of crystals that possess three fourfold axes of rotational symmetry, the cube axes, and four threefold axes of rotational symmetry, the cube diagonals. Alternatively, cubic symmetry may be described as invariance of material structure under a translation of a certain distance in any of three mutually orthogonal directions these directions are usually identified as the cube axes. Consider a cubic material for which the [100], [010] and [001] cube axes are parallel to the axes of an underlying rectangular a i,0 2,0 3—coordinate system. For this case, it is evident that 

All the other elastic constants vanish because of the fourfold rotational symmetry of the reference axes. This point can be demonstrated by reasoning that the remaining constants must vanish so as to avoid introducing a change in the sign of the shear strain for a given shear stress, merely to account for rotation. It follows from (3.5) that the elastic response of any cubic crystal is characterized by three independent elastic constants, and that the 

Unique relationships among the compliance constants and stiffness constants for cubic crystals can be derived by considering simple states of stress and deformation of the crystal. For example, consider a uniaxial tensile stress Ti applied in the direction of the xi—axis, which is ahgned with one of the cube axes. From (3.3), (3.4) and (3.6), it follows that 

If these relations are to be applied for arbitrary values of ci, then elimination of the strain components implies the condition that 

On the other hand, imposition of a hydrostatic tension of magnitude a on the material results in an extensional strain e in each of the coordinate directions, so that 

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Elastic constants of cubic crystals complete stiffness matrix is... 

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