Linear elasticity: Difference between revisions

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Revision as of 14:57, 11 May 2006

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:<math> \alpha^2=\left(\kappa+\frac{4}{3}\mu\right)/\rho \qquad \beta^2=\mu/\rho </math>

are the [[eigenvalue]]s of <math>A[\hat{\mathbf{k}}]</math> with [[eigenvector]]s <math>\hat{\mathbf{u}}</math> parallel and orthogonal to the propagation direction <math>\hat{\mathbf{k}}</math>, respectively. In the seismological literature, the corresponding plane waves are called P-waves and S-waves (see [[Seismic wave]]).

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== The biharmonic equation ==

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

From the expression for the acoustic operator for isotropic homogeneous materials, it is seen that:

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

:<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>

from which we conclude that:

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

Taking the Laplacian of both sides of the static equation, a conservative force will give <math>\partial_k\partial_kf_i=0</math> and we have

:<math>\partial_ipartial_k\partial_jA_partial_kA_{ij}[\nabla]u_j = (\alpha^2 -\partial_ibeta^2)\partial_ipartial_k\partial_jpartial_k\partial_i\partial_jpartial_ju_j+\beta^2(\partial_mpartial_k\partial_k\partial_m\partial_j\partial_jpartial_mu_i=0</math>

-\partial_i\partial_i\partial_j\partial_j)=\alpha^2\partial_i\partial_i\partial_j\partial_j\,</math>

From the divergence equation, the first term on the right is zero and we conclude that:

which is just <math>\alpha^2</math> times the [[biharmonic operator]]. It follows that for the elastostatic case, where all variation in time is zero, each of the <math>u_k</math> obey a biharmonic equation:

:<math>\alphapartial_k\partial_kA_{ij}u_j = \beta^2\partial_ipartial_k\partial_ipartial_k\partial_jpartial_m\partial_ju_kpartial_mu_i=-f_k/\rho\,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>.

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== References ==