Linear elasticity

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Linear elasticity

The linear theory of elasticity models the macroscopic mechanical properties of solids assuming "small" deformations.

Basic equations

Linear elastodynamics is based on three tensor equations:

  • dynamic equation

<math> \partial_j T_{ij} + f_i =\rho \, \partial_{tt} u_i <math>

<math> T_{ij} = C_{ijkl} \, E_{kl} <math>

  • kinematic equation

<math> E_{ij} =\frac{1}{2} (\partial_i u_j+\partial_j u_i) <math>

where:

  • <math> T_{ij}=T_{ji} <math> is stress
  • <math> f_i <math> is body force
  • <math> \rho <math> is density
  • <math> u_i <math> is displacement
  • <math> C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk} <math> is the stiffness tensor
  • <math> E_{ij}=E_{ji} <math> is strain

Wave equation

From the basic equations one gets the wave equation

<math> (\delta_{kl} \partial_{tt}-A_{kl}[\nabla]) \, u_l

= \frac{1}{\rho} f_k <math> where

<math> A_{kl}[\nabla]=\frac{1}{\rho} \, \partial_i \, C_{iklj} \, \partial_j <math>

is the acoustic differential operator, and <math> \delta_{kl}<math> is Kronecker delta.

Plane waves

A plane wave has the form

<math> \mathbf{u}[\mathbf{x}, \, t] = U[\mathbf{k} \cdot \mathbf{x} - \omega \, t] \, \hat{\mathbf{u}} <math>

with <math>\hat{\mathbf{u}}<math> of unit length. It is a solution of the wave equation with zero forcing, if and only if <math> \omega^2 <math> and <math>\hat{\mathbf{u}}<math> constitute an eigenvalue/eigenvector pair of the acoustic algebraic operator

<math> A_{kl}[\mathbf{k}]=\frac{1}{\rho} \, k_i \, C_{iklj} \, k_j <math>

This propagation condition may be written as

<math>A[\hat{\mathbf{k}}] \, \hat{\mathbf{u}}=c^2 \, \hat{\mathbf{u}}<math>

where <math>\hat{\mathbf{k}} = \mathbf{k} / \sqrt{\mathbf{k}\cdot\mathbf{k}}<math> denotes propagation direction and <math>c=\omega/\sqrt{\mathbf{k}\cdot\mathbf{k}}<math> is phase velocity.

Isotropic media

In isotropic media, the elasticity tensor has the form

<math> C_{ijkl}

= \kappa \, \delta_{ij}\, \delta_{kl} +\mu\, (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\frac{2}{3}\, \delta_{ij}\,\delta_{kl})<math> where <math>\kappa<math> is incompressibility, and <math>\mu<math> is rigidity. Hence the acoustic algebraic operator becomes

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

\alpha^2 \,\hat{\mathbf{k}}\otimes\hat{\mathbf{k}} +\beta^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 the seismological literature, the corresponding plane waves are called P-waves and S-waves (see Seismic wave).

References

  • Gurtin M. E., Introduction to Continuum Mechanics, Academic Press 1981
  • L. D. Landau & E. M. Lifschitz, Theory of Elasticity, Butterworth 1986
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