Elastic Solids

In a linear, isotropic (but not necessarily homogeneous) elastic medium we have

• Stress-strain relation:

  $$\begin{bmatrix}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{zz}\\ \sigma_{xy}\\ \sigma_{xz}\\ \sigma_{yz}\\ \end{bmatrix}= \begin{bmatrix}\lambda+2\mu&\lambda&\lambda&0&0&0\\ \lambda&\lambda+2\mu&\lambda&0&0&0\\ \lambda&\lambda&\lambda+2\mu&0&0&0\\ 0&0&0&2\mu&0&0\\ 0&0&0&0&2\mu&0\\ 0&0&0&0&0&2\mu\\ \end{bmatrix}\begin{bmatrix}e_{xx}\\ e_{yy}\\ e_{zz}\\ e_{xy}\\ e_{xz}\\ e_{yz}\\ \end{bmatrix}$$

  
  

  with

  $$e_{ij} = \frac{1}{2}\left(u_{i,j}+ u_{j,i}\right),\qquad i,j=x,y,z.$$

  
  
  $\lambda$ ,$\mu$ are the Lame constants. $(u_{x}, u_{y}, u_{z})$ is the displacement vector at any point in the medium. Also, $\sigma_{ij} = \sigma_{ji}$ (Cons. of angular momentum)

• Force balance

  
  

  $$\rho \frac{\partial^{2}u_{x}}{\partial t^{2}} = \frac{\partial\sigma_{xx}}{\partial x}+ \frac{\partial\sigma_{xy}}{\partial y}+\frac{\partial\sigma_{xz}}{\partial z}$$

  $$\rho \frac{\partial^{2}u_{y}}{\partial t^{2}} = \frac{\partial\sigma_{yx}}{\partial x}+ \frac{\partial\sigma_{yy}}{\partial y}+\frac{\partial\sigma_{yz}}{\partial z}$$

  $$\rho \frac{\partial^{2}u_{z}}{\partial t^{2}} = \frac{\partial\sigma_{zx}}{\partial x}+ \frac{\partial\sigma_{zy}}{\partial y}+\frac{\partial\sigma_{zz}}{\partial z}$$

  
  

  (neglecting body forces)