Some materials, however, are not invariant to coordinate inversions and this type of anisotropy can be expected to result in qualitatively different behavior in comparison with isotropic solids. Structural noncentrosymmetry is characteristic of bone, as well as synthetic composites containing twisted fibers. In this paper the behavior of a noncentrosymmetric micropolar elastic solid is examined. In classical elasticity, inversion symmetry has no effect on the elastic modulus tensor as shown in the following. C ijkl .
Inversion symmetry has no effect on the elastic modulus tensor or on any tensor of even rank.
Tensor properties of odd rank are zero if there is inversion symmetry, and can only be nonzero if there is handedness. Examples include piezoelectricity, governed by a third rank tensor, and strain gradient elastic theories which are governed by a fifth rank tensor. Consider the constitutive eq'ns for an isotropic centrosymmetric Cosserat solid, for review. The usual Einstein summation convention for repeated indices is used and the comma denotes differentiation with respect to spatial coordinates. In three dimensions there are six independent elastic constants required to describe an isotropic centrosymmetric Cosserat elastic solid, a , b , g , k , l , and m .