Linear elasticity: Difference between revisions

Revision as of 04:33, 19 December 2007

Revision as of 01:04, 20 December 2007

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* <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 basic 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]] entryarticle.

Just as a spring which is compressed or expanded holds potential energy, so a strained material will possess an energy density due to the deformation. The energy density due to deformation is given by:

:<math>F=\frac{1}{2}\sigma_{ij}\,\varepsilon_{ij}</math>

== Isotropic homogeneous media ==

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:<math>

\sigma_{ij}=\lambdafrac{2\nu}{1-2\nu}\mu\epsilon_{kk}\delta_{ij}+2\mu\epsilon_{ij}\,

</math>

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</math>

It can be seen that there is a component of the displacement in the direction of the force, which diminishes, as is the case for the potential in electrostatics, as 1/r. There is also an additional &rho;-directed component which will be zero when &nu;=1.

=== Boussinesq's solution - point force at the origin of an infinite isotropic half-space===