Linear elasticity: Difference between revisions

Revision as of 15:19, 11 May 2006

Revision as of 15:22, 11 May 2006

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== 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 thisthe staticelastostatic 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>