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Peano
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Peano
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The elastic wave equation describes the propagation of waves in elastic solids. They are commonly derived from the conservation of momentum and the constitutive relation, defined by Hookes's law. With the assumption that the relation between stress and strain can be linearised for relatively small deformations this results in the linear elastic wave equation.
A material is described by its density \( \rho \), shear modulus \( \mu \) and the Lamé constant \( \lambda \).
This tutorial implements the elastic wave equation in its first-order formulation:
\( \partial_t \begin{pmatrix} u\\v\\ \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{xy} \end{pmatrix} + \nabla \cdot F \begin{pmatrix} u\\v\\ \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{xy} \end{pmatrix} = 0 \)
With \( F=\left(F_1,F_2\right) \) being a tensor describing the linear flux:
\( F_1 = \begin{pmatrix} 0 & 0 & 1/\rho & 0 & 0 \\ 0 & 0 & 0 & 0 & 1/\rho \\ \left(\lambda + 2\mu \right) & 0 & 0 & 0 & 0 \\ \lambda&0&0&0&0 \\ 0&\mu&0&0&0 \end{pmatrix} \\ F_2 = \begin{pmatrix} 0 & 0 & 0 & 0 & 1/\rho \\ 0 & 0 & 0 & 1/\rho & 0\\ 0 & \lambda & 0 & 0 & 0 \\ 0 & \left(\lambda + 2\mu \right) & 0 & 0 & 0 \\ \mu & 0 & 0 & 0 & 0 \end{pmatrix} \)
In elastic materials, two types of waves can propagate through the domain: p-waves and s-waves. Their speeds are
\( c_s=\sqrt{\frac{\mu}{\rho}} \\ c_p=\sqrt{\frac{\lambda + 2 \mu}{\rho}} \)
These speeds also define the eigenvalues of the flux:
\( \left( -c_p, -c_s , 0 , c_s , c_p \right) \)
| t=0.0 s | t=1.0 s |
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The computed stress of the 2D elastic wave equation at times t=0.0 s and t=1.0 s.
1.9.1 
