Linear elasticity: Difference between revisions

Revision as of 09:31, 22 April 2006

Revision as of 02:36, 6 May 2006

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Linear elastodynamics is based on three [[tensor]] equations:

* dynamic equation

:<math>

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

</math>

* [[constitutive equation]] (''anisotropic [[Hooke's law]]'')

:<math>

T_\sigma_{ij} = C_{ijkl} \, E_\varepsilon_{kl}

</math>

* kinematic equation

:<math>

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

</math>

where:

* <math> T_\sigma_{ij}=T_\sigma_{ji} \,</math> is stressthe [[stress]]

* <math> f_i \,</math> is isthe body force

* <math> \rho \,</math> is the ismass density

* <math> u_i \,</math> is isthe displacement

* <math> C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk} \,</math> is the [[elasticity tensor]]

* <math> E_\varepsilon_{ij}=E_\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>.

TheseThe 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.

== Wave equation ==