Editing Linear elasticity (section)

=== Elastodynamics in terms of displacements ===
{{Expand section|more principles, a brief explanation to each type of wave|discuss=Talk:Linear elasticity#New section needed|date=September 2010}}
Elastodynamics is the study of '''elastic waves''' and involves linear elasticity with variation in time.  An '''elastic wave''' is a type of [[mechanical wave]] that propagates in elastic or [[viscoelastic]] materials.  The elasticity of the material provides the restoring [[force]] of the wave.  When they occur in the [[Earth]] as the result of an [[earthquake]] or other disturbance, elastic waves are usually called [[seismic wave]]s.

The linear momentum equation is simply the equilibrium equation with an additional inertial term:
<math display="block"> \sigma_{ji,j}+ F_i = \rho\,\ddot{u}_i = \rho \, \partial_{tt} u_i.</math>

If the material is governed by anisotropic Hooke's law (with the stiffness tensor homogeneous throughout the material), one obtains the '''displacement equation of elastodynamics''':
<math display="block">\left( C_{ijkl} u_{(k},_{l)}\right) ,_{j}+F_{i}=\rho \ddot{u}_{i}.</math>

If the material is isotropic and homogeneous, one obtains the (general, or transient) '''Navier–Cauchy equation''':
<math display="block">
\mu u_{i,jj} + (\mu+\lambda)u_{j,ij}+F_i=\rho\partial_{tt}u_i
\quad \text{or} \quad
\mu \nabla^2\mathbf{u} + (\mu+\lambda)\nabla(\nabla\cdot\mathbf{u}) + \mathbf{F}=\rho\frac{\partial^2\mathbf{u}}{\partial t^2}.</math>

The elastodynamic wave equation can also be expressed as
<math display="block"> \left(\delta_{kl} \partial_{tt} - A_{kl}[\nabla]\right) u_l = \frac{1}{\rho} F_k</math>
where
<math display="block"> 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]].

In [[Hooke's law#Isotropic materials|isotropic]] media, the stiffness tensor has the form
<math display="block"> C_{ijkl}
= K \, \delta_{ij}\, \delta_{kl}
+ \mu\, (\delta_{ik}\delta_{jl} + \delta_{il} \delta_{jk} - \frac{2}{3}\, \delta_{ij}\, \delta_{kl})</math>
where
<math>K</math> is the [[bulk modulus]] (or incompressibility), and <math>\mu</math> is the [[shear modulus]] (or rigidity), two [[elastic moduli]]. If the material is homogeneous (i.e. the stiffness tensor is constant throughout the material), the acoustic operator becomes:
<math display="block">A_{ij}[\nabla] = \alpha^2 \partial_i \partial_j + \beta^2 (\partial_m \partial_m \delta_{ij} - \partial_i \partial_j)</math>

For [[plane waves]], the above differential operator becomes the ''acoustic algebraic operator'': 
<math display="block">A_{ij}[\mathbf{k}] = \alpha^2 k_i k_j + \beta^2(k_m k_m \delta_{ij}-k_i k_j)</math>
where
<math display="block"> \alpha^2 = \left(K+\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. The associated waves are called ''longitudinal'' and ''shear'' elastic waves. In the seismological literature, the corresponding plane waves are called P-waves and S-waves (see [[Seismic wave]]).