Linear elasticity: Difference between revisions

Revision as of 15:14, 11 May 2006

Revision as of 15:19, 11 May 2006

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== The biharmonic equation ==

For a static situation (<math>\partial_t=0</math>) the wave equation becomes athe staticelastostatic equation :

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

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

Taking the [[Laplacian]] of both sides of the staticelastostatic 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>