# Linear elasticity

 Contents

## 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

[itex] \partial_j T_{ij} + f_i =\rho \, \partial_{tt} u_i [itex]

[itex] T_{ij} = C_{ijkl} \, E_{kl} [itex]

• kinematic equation

[itex] E_{ij} =\frac{1}{2} (\partial_i u_j+\partial_j u_i) [itex]

where:

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

## Wave equation

From the basic equations one gets the wave equation

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

= \frac{1}{\rho} f_k [itex] where

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

is the acoustic differential operator, and [itex] \delta_{kl}[itex] is Kronecker delta.

## Plane waves

A plane wave has the form

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

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

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

This propagation condition may be written as

[itex]A[\hat{\mathbf{k}}] \, \hat{\mathbf{u}}=c^2 \, \hat{\mathbf{u}}[itex]

where [itex]\hat{\mathbf{k}} = \mathbf{k} / \sqrt{\mathbf{k}\cdot\mathbf{k}}[itex] denotes propagation direction and [itex]c=\omega/\sqrt{\mathbf{k}\cdot\mathbf{k}}[itex] is phase velocity.

## Isotropic media

In isotropic media, the elasticity tensor has the form

[itex] C_{ijkl}

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

[itex]A[\hat{\mathbf{k}}]=

\alpha^2 \,\hat{\mathbf{k}}\otimes\hat{\mathbf{k}} +\beta^2 \, (\mathbf{I}-\hat{\mathbf{k}}\otimes\hat{\mathbf{k}} ) [itex] where [itex] \otimes [itex] denotes the tensor product, [itex] \mathbf{I} [itex] is the identity matrix, and

[itex] \alpha^2=(\kappa+\frac{4}{3}\mu)/\rho

\qquad \beta^2=\mu/\rho [itex] are the eigenvalues of [itex]A[\hat{\mathbf{k}}][itex] with eigenvectors [itex]\hat{\mathbf{u}}[itex] parallel and orthogonal to the propagation direction [itex]\hat{\mathbf{k}}[itex], 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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