Linear elasticity: Difference between revisions

Revision as of 14:51, 7 May 2006

Revision as of 23:05, 7 May 2006

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and <math>c=\omega/\sqrt{\mathbf{k}\cdot\mathbf{k}}</math> is phase velocity.

== Isotropic media ==

In [[Hooke's Law#Isotropic materials|isotropic]] media, the [[elasticity tensor]] has the form

:<math> C_{ijkl}

= \kappa \, \delta_{ij}\, \delta_{kl}

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<math>\kappa</math> is [[Bulk modulus|incompressibility]], and

<math>\mu</math> is [[Shear modulus|rigidity]].

Hence theThe acoustic algebraic operator becomes :

:<math>A[\hat{\mathbf{k}}]=

\alpha^2 \,\hat{\mathbf{k}}\otimes\hat{\mathbf{k}}

+\beta^2 \, (k^2\mathbf{I}-\hat{\mathbf{k}}\otimes\hat{\mathbf{k}} )

</math>

where

<math> \otimes </math> denotes the [[tensor product]],

<math> \mathbf{I} </math> is the identity matrix, and

:<math> \alpha^2=(\kappa+\frac{4}{3}\mu)/\rho

\qquad \beta^2=\mu/\rho </math>

are the eigenvalues of

<math>A[\hat{\mathbf{k}}]</math>

with eigenvectors <math>\hat{\mathbf{u}}</math> parallel and orthogonal to the propagation direction <math>\hat{\mathbf{k}}</math>, respectively. In index notation:

:<math>A_{ij}[\nabla]=\alpha^2 k_ik_j\partial_i\partial_j+\beta^2(k_mk_m\partial_m\partial_m\delta_{ij}-k_ik_j\partial_i\partial_j)\,</math>

and the acoustic algebraic operator becomes

:<math>A_{ij}[\mathbf{k}]=\alpha^2 k_ik_j+\beta^2(k_mk_m\delta_{ij}-k_ik_j)\,</math>

where

:<math> \alpha^2=\left(\kappa+\frac{4}{3}\mu\right)/\rho \qquad \beta^2=\mu/\rho </math>

In the seismological literature, the corresponding plane waves are called P-waves and S-waves (see [[Seismic wave]]).

are the [[eigenvalue]]s of <math>A[\hat{\mathbf{k}}]</math> with eigenvectors[[eigenvector]]s <math>\hat{\mathbf{u}}</math> parallel and orthogonal to the propagation direction <math>\hat{\mathbf{k}}</math>, respectively. In indexthe notation:seismological literature, the corresponding plane waves are called P-waves and S-waves (see [[Seismic wave]]).

== References ==