A common approach to generating fourth-order elastic-moduli tensors for linear, anisotropic elastic media is to start with the most complex case — the least symmetric case — of triclinic symmetry, in which all 81 elastic constants are non zero and 21 of them are independent. Then the relationships among the 21 independent elastic constants are determined for the higher-symmetry (less anisotropic) cases by using those coordinate transformations which, for each symmetry class, leave the material invariant. The most symmetric (least anisotropic) case is the isotropic case for which there are only two independent moduli and only 15 non zero components for the elastic-modulus tensor. Here, we take an alternate approach and start with the isotropic case which requires only two independent fourth-order tensors. We add fourth-order tensors which satisfy the lower-symmetry classes by inspection, continually increasing the number of independent con-stants as the symmetry decreases (or anisotropy increases). The selection of the fourth-order tensors which satisfy the more-anisotropic cases are shown to be easily determined and they are not restricted; that is, there are many additional fourth-order tensors that can be used to satisfy the higher-order anisotropy, but they are all related to each other and the fourth-order tensors previously used for the more-symmetric cases. For example, for the cubic-symmetry case, only one new independent elastic constant is needed and only one new independent fourth-order tensor is required compared to the isotropic case. All other possible fourth-order tensors that satisfy cubic symmetry can be composed of linear combinations of the new one and the two original ones for the isotropic case. In addition, we show that we can generate elastic-moduli tensors which are orthogonal under the double-dot product; these sets of tensors simplify the inversion of stiffness tensors into compliance tensors, and vice versa.
Figures & Tables
Anisotropy 2000: Fractures, Converted Waves, and Case Studies
“This volume contains 25 papers that represent most of the best work in seismic anisotropy in 1998 and 1999. Fracture characterizations and processing of converted waves are the two main topics covered in this volume. They are addressed from both theoretical and practical viewpoints. Also included are papers describing the historical roots of seismic anisotropy.”