Linear elasticity

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1 Linear elasticity

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

$\partial_j T_{ij} + f_i =\rho \, \partial_{tt} u_i$

constitutive equation (anisotropic Hooke's law)

$T_{ij} = C_{ijkl} \, E_{kl}$

kinematic equation

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

where:

$T i j = T j i$ is stress
$f i$ is body force
$ρ$ is density
$u i$ is displacement
$C i j k l = C k l i j = C j i k l = C i j l k$ is the stiffness tensor
$E i j = E j i$ is strain

Wave equation

From the basic equations one gets the wave equation

$(\delta_{kl} \partial_{tt}-A_{kl}[\nabla]) \, u_l = \frac{1}{\rho} f_k$

where

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

is the acoustic differential operator, and $δ k l$ is Kronecker delta.

Plane waves

A plane wave has the form

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

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

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

This propagation condition may be written as

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

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

Isotropic media

In isotropic media, the elasticity tensor has the form

$C_{ijkl} = \kappa \, \delta_{ij}\, \delta_{kl} +\mu\, (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\frac{2}{3}\, \delta_{ij}\,\delta_{kl})$

where $κ$ is incompressibility, and $μ$ is rigidity. Hence the acoustic algebraic operator becomes

$A[\hat{\mathbf{k}}]= \alpha^2 \,\hat{\mathbf{k}}\otimes\hat{\mathbf{k}} +\beta^2 \, (\mathbf{I}-\hat{\mathbf{k}}\otimes\hat{\mathbf{k}} )$

where $\otimes$ denotes the tensor product, $\mathbf{I}$ is the identity matrix, and

$\alpha^2=(\kappa+\frac{4}{3}\mu)/\rho \qquad \beta^2=\mu/\rho$

are the eigenvalues of $A[\hat{\mathbf{k}}]$ with eigenvectors $\hat{\mathbf{u}}$ parallel and orthogonal to the propagation direction $\hat{\mathbf{k}}$ , 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

Categories: Continuum mechanics