Inversions related to the stress-strain-fabric relationship

The stress-strain-fabric relationship is an extension of the anisotropic form of Hooke's law to include a dependence of the elastic coefficients upon a second-rank tensor called the fabric tensor. The fabric tensor represents features of the material microstructure associated with the type and the degree of the anisotropy. The inversion considered first in this work is that in which the stress-strain-fabric relation is constructed from the strain-stress-fabric relation and vice versa. Next, a semi-inversion of the relationship between the fourth-rank tensor of elastic coefficients and the fabric tensor is developed. This latter inversion permits the determination of the fabric tensor from a fourth-rank tensor of elastic constants. Explicit, approximate forms of these results, including a numerical example, are given for the case when the fabric tensor is normalized and terms of order three and higher in the fabric tensor are neglected.