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Linear elasticity

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Linear elasticity models the macroscopic mechanical properties of solids assuming "small" deformations.

Contents

[edit] Basic equations

Linear elastodynamics is based on three tensor equations:

  • dynamic equation
<math>

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

<math>

\sigma_{ij} = C_{ijkl} \, \varepsilon_{kl} </math>

  • kinematic equation
<math>

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

where:

  • <math> \sigma_{ij}=\sigma_{ji} \,</math> is the stress
  • <math> f_i \,</math> is the body force
  • <math> \rho \,</math> is the mass density
  • <math> u_i \,</math> is the displacement
  • <math> C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk} \,</math> is the elasticity tensor
  • <math> \varepsilon_{ij}=\varepsilon_{ji} \,</math> is the strain
  • <math>\partial_i</math> is the partial derivative <math>\partial/\partial x_i</math> and <math>\partial_t</math> is <math>\partial/\partial t</math>.

The elastostatic equations are given by setting <math>\partial_t</math> to zero in the dynamic equation. The elastostatic equations are shown in their full form on the 3-D Elasticity entry.

[edit] 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.

[edit] 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.

[edit] Isotropic homogeneous 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. 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>

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>

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).

[edit] The biharmonic equation

For a static situation (<math>\partial_t=0</math>) in isotropic materials, the wave equation becomes the elastostatic equation :

<math>A_{ij}u_j=(\alpha^2-\beta^2)\partial_i\partial_ju_j+

\beta^2\partial_m\partial_mu_i=-f_i/\rho</math>

Taking the divergence of both sides of the elastostatic equation and assuming a conservative force, (<math>\partial_i f_i=0</math>) we have

<math>\partial_i A_{ij}u_j = (\alpha^2-\beta^2)\partial_i\partial_i\partial_ju_j+\beta^2\partial_i\partial_m\partial_mu_i = 0</math>

Noting that summed indices need not match, and that the partial derivatives commute, the two differential terms are seen to be the same and we have:

<math>\partial_i A_{ij}u_j = \alpha^2\partial_i\partial_i\partial_ju_j = 0</math>

from which we conclude that:

<math>\partial_i\partial_i\partial_ju_j = 0</math>

Taking the Laplacian of both sides of the elastostatic equation, a conservative force will give <math>\partial_k\partial_kf_i=0</math> and we have

<math>\partial_k\partial_kA_{ij}u_j = (\alpha^2-\beta^2)\partial_k\partial_k\partial_i\partial_ju_j+\beta^2\partial_k\partial_k\partial_m\partial_mu_i=0</math>

From the divergence equation, the first term on the right is zero (Note: again, the summed indices need not match) and we have:

<math>\partial_k\partial_kA_{ij}u_j = \beta^2\partial_k\partial_k\partial_m\partial_mu_i=0</math>

from which we conclude that:

<math>\partial_k\partial_k\partial_m\partial_mu_i=0</math>

or, in coordinate free notation <math>\nabla^4 \mathbf{u}=0</math> which is just the biharmonic equation in <math>\mathbf{u}</math>.

[edit] See also

[edit] References

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