Hooke anisotropy

All crystals are anisotropic many other structures also have elastic anisotropy. The propagation of elastic waves in anisotropic media is described by the Christoffel equation. This still depends on Newton s law and Hooke s law, but it is expressed in tensor form so that elastic anisotropy may be included. The tensor description of elastic stiffness was summarized in 6.2, especially eqns (6.23)—(6.29). The Christoffel equation is... 

Observe that the Reuss laminate is identical to the Voigt laminate, except for a rotation with respect to the direction of load. Therefore, the stiffness of the laminate is anisotropic, that is, dependent on direction [Lekhnitskii, 1963 Nye, 1976 Agarwal and Broutman, 1980]. Anisotropy is characteristic of composite materials. The relationship between stress cty and strain in anisotropic materials is given by the tensorial form of Hooke s law as follows ... 

Another important special case of a homogeneous deformation (i.e. y is positionally constant) in which now, however, pressure anisotropies are effective, is the uniaxially stressed cubic crystal. Let us assume this time that there is tensile stress in the x-direction. There it holds for small effects (i.e. Hooke s law fulfilled) that s = P. dcxx/dsxx const = xx/ xx nnd also dsyy/dsxx — ds z/dsxx yy/sxx zz/ xx const. 

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