WAVE PROPAGATION IN 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, and 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 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, I 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, I 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 $l=m=\mu$).

Recall that the equation of motion may be written as

ü_i = c_ijklu_k,lj.   After Fourier transforming in both space and time [i.e., ${\bf u}(\vecx, t) = {\bf u}\exp{i({\bf k}\cdot\vecx - \omega t)}$,where ${\bf k}$ is the wavevector and $\omega$ is the angular frequency], I find

(^2 _ik - c_ijklk_jk_l)u_k = 0,   which provides three equations for the components of displacement u₁, u₂, u₃. These equations can be solved if and only if the determinant of the coefficients vanishes, which implies

det(^2 _il - c_ijklk_jk_l) = 0.   Equation (dispgen) is known as the dispersion relation, giving $\omega$ as a function of ${\bf k}$.The magnitude of the phase velocity of a mode of propagation is given by $v = \omega/k$, where $k = \vert{\bf k}\vert$.Then, if the stiffness tensor is assumed to have the symmetry of a transversely isotropic material, the equation of motion can be written in the following form

^2 u_1 u_2 u_3 = k_1 & & & k_2 & & & k_3 a & b & f b & a & f f & f & c k_1 & & & k_2 & & & k_3 u_1 u_2 u_3      + 0 & k_3 & k_2 k_3 & 0 & k_1 k_2 & k_1 & 0 l & & & l & & & m 0 & k_3 & k_2 k_3 & 0 & k_1 k_2 & k_1 & 0 u_1 u_2 u_3 .  

It is not difficult to solve (wavesinTI) in a completely general fashion, but for simplicity I will consider the equations under the assumption that the direction of propagation lies in the x₁x₃-plane. Then, k₂ = 0 and the equations become

(ak_1^2 + lk_3^2 - ^2)u_1 + (f+l)k_1 k_3 u_3 = 0,  

(f+l)k_1 k_3 u_1 + (ck_3^2 + lk_1^2 - ^2)u_3 = 0,   and

(mk_1^2 + lk_3^2 - ^2)u_2 = 0.   The first two equations are coupled and can be solved for the dispersion relation of the propagation modes given by

_^2 = 12{ (a+l)k_1^2 + (c+l)k_3^2 [(a-l)k_1^2 - (c-l)k_3^2]^2 + 4(f+l)^2k_1^2 k_3^2}.   These modes of propagation are called quasi-P (or qP) and quasi-SV (or qSV) because the particle motion does not generally satisfy the requirements for pure compressional waves (${\bf k}\times{\bf u} = 0$) or pure shear waves (${\bf k}\cdot{\bf u} = 0$), although they may satisfy these conditions at a few special angles of propagation. Since compressional waves generally have higher wave speeds than shear waves, it is easy to see that $\omega_+$ is the expression for the quasi-P-wave dispersion and that $\omega_-$ is the expression for the quasi-SV-wave dispersion. The third equation gives the dispersion relation for shear waves polarized in the horizontal direction (i.e., SH waves) given by

_sh^2 = mk_1^2 + lk_3^2.  

The argument of the radical in (PSVdisp) can be rewritten (uniquely) into the form

[(a-l)k_1^2 - (c-l)k_3^2]^2 + 4(f+l)^2k_1^2 k_3^2 =                  [(a-l)k_1^2 + (c-l)k_3^2]^2 + 4[(f+l)^2 - (a-l)(c-l)]k_1^2k_3^2.   The left hand side of this equality would be a perfect square for all values of k₁² and k₃² if $f+l \equiv 0$, while the right hand side would be a perfect square if the quantity defined as $A \equiv (a-l)(c-l) - (f+l)^2 = 0$. The first case with f+l = 0 will virtually never happen because both f and l are normally positive quantities. The second case with A = 0 can occur for some types of anisotropic media, but I show later that this cannot occur for finely layered media. Nevertheless, if A = 0, then the dispersion relations of (PSVdisp) reduce to

_p^2 = ak_1^2 + ck_3^2   for $\omega_p = \omega_+$ and

_sv^2 = l(k_1^2 + k_3^2) = lk^2   for $\omega_{sv} = \omega_-$,showing that the P-wave surface for velocity squared is elliptical if $a \ne c$, while the SV-wave surface for velocity squared is circular and therefore isotropic. The dispersion relation (SHdispersion) shows that the SH-wave surface for velocity squared is always an ellipse as long as $l \ne m$.I call A the anellipticity parameter because, if $A \ne 0$, then the dispersion relations for qP- and qSV-waves are anelliptical (i.e., something other than elliptical) in shape.

Phase velocities are obtained as a function of angle from these expressions by first defining the wavevector angle $\theta$ such that

k = k(x_1 + x_3).   Then, the phase velocity vector for each type of wave is given in general by

v_ph = v_ph(x_1 + x_3),   where $v_{ph} \equiv \omega/k$.

The group velocity is defined by

v_gr = k_1 x_1 + k_3 x_3 = v_gr(x_1 + x_3),   where the group angle $\phi$ is determined by

= /k_11pt/k_3.  

One other angle is particularly important, since it is the one that is most easily measured, and that is the angle of particle motion $\psi$ for a wave passing a particular point in space. The particle motion is given by the displacement vector ${\bf u}$, so

u = u(x_1 + x_3),   where

= u_1/u_3.