Linear elasticity: Difference between revisions

Revision as of 15:07, 11 May 2006

Revision as of 15:11, 11 May 2006

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

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

Taking the divergence of both sides of this static equation and assuming a conservative force, (<math>\partial_i f_i=0</math>) we have

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:<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>

Noting that summed indices need not match, and that the partial derivatives commute, the two differential terms are seen to be the same and we have:

:<math>\partial_i A_{ij}u_j = \alpha^2\partial_i\partial_i\partial_ju_j = 0</math>