Linear elasticity: Difference between revisions

Revision as of 01:34, 23 October 2007

Revision as of 19:35, 18 November 2007

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

:<math> C_{ijkl}

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

+\mu\, (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\frac{2}{3}\, \delta_{ij}\,\delta_{kl})</math>

where

<math>\kappaK</math> is the [[Bulkbulk modulus|incompressibility]] (or incompressibility), and

<math>\mu</math> is the [[Shearshear modulus|rigidity]] (or rigidity), two [[elastic moduli]]. If the material is homogeneous (i.e. the elasticity tensor is constant throughout the material), the acoustic operator becomes:

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

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

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