Linear elasticity: Difference between revisions

Revision as of 14:57, 11 May 2006

Revision as of 15:07, 11 May 2006

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Taking the divergence of both sides of this static 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 = \alpha^2\partial_i\partial_i\partial_ju_j = 0</math>

Noting that summed indices need not match, the two differential terms are the same and we have:

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

from which we conclude that:

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:<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>

From the divergence equation, the first term on the right is zero (Note: again, the summed indices need not match) and we conclude thathave:

:<math>\partial_k\partial_kA_{ij}u_j = \beta^2\partial_k\partial_k\partial_m\partial_mu_i=0</math>

from which we conclude that:

:<math>\partial_k\partial_k\partial_m\partial_mu_i=0</math>

or, in coordinate free notation <math>\nabla^4 \mathbf{u}=0</math> which is just the [[biharmonic equation]] in <math>\mathbf{u}</math>.