NOTATION FOR ANISOTROPIC ELASTIC MEDIA

In tensor notation, the relationship between components of stress $\sigma_{ij}$and strain u_k,l is given by

_ij = c_ijklu_k,l,   where c_ijkl is the adiabatic stiffness tensor. Repeated indices on the right hand side of (ssgen) are summed. In (ssgen), u_k is the kth Cartesian component of the displacement vector ${\bf u}$, and $u_{k,l} = {\partial u_k}/{\partial x_l}$.Whereas for an isotropic elastic medium the stiffness tensor has the form

c_ijkl = _ij_kl + (_ik_jl + _il_jk),   depending on only two parameters (the Lamé constants, $\lambda$ and $\mu$). This tensor can have up to 21 independent constants for general anisotropic elastic media. The stiffness tensor has pairwise symmetry in its indices such that c_ijkl = c_jikl and c_ijkl = c_ijlk, which will be used later to simplify the resulting equations. The requirement that straining the medium must produce a positive change in its internal energy provides the additional constraint that c_ijkl = c_klij.

The general equation of motion for wave propagation through an anisotropic elastic medium is given by

ü_i = _ij,j = c_ijklu_k,lj,   where $\ddot{u}_i$ is the second time derivative of the ith Cartesian component of the displacement vector ${\bf u}$ and $\rho$ is the density (assumed constant). Equation (anisowaves) is a statement that the product of mass times acceleration of a particle is determined by the internal stress force $\sigma_{ij,j}$.

A commonly used simplification of the notation for elastic analysis is given by introducing the strain tensor, where

e_ij = 12(u_i,j + u_j,i) = 12(u_ix_j + u_jx_i).   Then, using one version of the Voigt convention, in which the pairwise symmetries of the stiffness tensor indices are used to reduce the number of indices from 4 to 2 using the rules $11 \to 1$, $22 \to 2$, $33 \to 3$, $23\hbox{ or }32 \to 4$,$13\hbox{ or }31 \to 5$, and $12\hbox{ or }21 \to 6$,we have

_11 _22 _33 _23 _31 _12 = c_11 & c_12 & c_13 & & & c_12 & c_22 & c_23 & & & c_13 & c_23 & c_33 & & & & & & 2c_44 & & & & & & 2c_55 & & & & & & 2c_66 e_11 e_22 e_33 e_23 e_31 e_12 .   Although the Voigt convention introduces no restrictions on the stiffness tensor, we have chosen to limit discussion to the form in (sscij), which is not completely general. Of the 36 coefficients (of which 21 are generally independent), we choose to treat only those cases for which the 12 coefficients shown (of which nine are generally independent) are nonzero. This form includes all orthorhombic, cubic, hexagonal, and isotropic systems, while excluding triclinic, monoclinic, trigonal, and some tetragonal systems, since each of the latter contains additional off-diagonal constants that may be nonzero. Nevertheless, we will restrict the discussion to (sscij) or to the still simpler case of transversely isotropic (TI) materials.

For TI materials, $c_{11} = c_{22} \equiv a$, $c_{12} \equiv b$,$c_{13} = c_{23} \equiv f$,$c_{33} \equiv c$, $c_{44} = c_{55} \equiv l$, and $c_{66} \equiv m$.There is also one further constraint on the constants that a = b + 2m, following from rotational symmetry in the x₁x₂-plane. In such materials, (sscij) may be replaced by

_11 _22 _33 _23 _31 _12 = a & b & f & & & b & a & f & & & f & f & c & & & & & & 2l & & & & & & 2l & & & & & & 2m e_11 e_22 e_33 e_23 e_31 e_12 ,   in which the matrix has the same symmetry as hexagonal systems and of which isotropic symmetry is a special case (having $a=c=\lambda+ 2\mu$, $b=f=\lambda$, and $l=m=\mu$).