# Isotropic Materials

The most commonly use material model is elastic isotropic where the properties are independent of orientation. This is a good model for materials like steel in the elastic range. Two independent parameters are all that are required to define this material, the elastic modulus $E$ and the Poisson's ratio $\nu$. From these the shear modulus $G$ can be defined as

$G = \dfrac{E}{2(1+\nu)}$

These are used to define the compliance matrix -- the inverse of the constitutive matrix

$C^{-1} = \begin{bmatrix} \dfrac{1}{E} & -\dfrac{\nu}{E} & -\dfrac{\nu}{E} & 0 & 0 & 0 \\ -\dfrac{\nu}{E} & \dfrac{1}{E} & -\dfrac{\nu}{E} & 0 & 0 & 0 \\ -\dfrac{\nu}{E} & -\dfrac{\nu}{E} & \dfrac{1}{E} & 0 & 0 & 0 \\ 0 & 0 & 0 & \dfrac{1}{G} & 0 & 0 \\ 0 & 0 & 0 & 0 & \dfrac{1}{G} & 0 \\ 0 & 0 & 0 & 0 & 0 & \dfrac{1}{G} \\ \end{bmatrix}$

If the default value of $G$ is overridden the material matrix need to be constructed differently to preserve symmetry in this case. It is convenient to think instead of the Cauchy stress tensor made up of two parts

• a mean hydrostatic stress tensor –- tending to change the volume $\sigma_h = \dfrac{\sigma_{xx} +\sigma_{yy} +\sigma_{zz}}{3}$
• a deviatoric stress tensor –- tending to distort

The stress tensor can then be expressed as the sum of the mean hydrostatic stress and the deviatoric stress

$\sigma = \begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{zz} \\ \sigma_{xy} \\ \sigma_{yz} \\ \sigma_{zx} \\ \end{bmatrix} = \begin{bmatrix} \sigma_{h} \\ \sigma_{h} \\ \sigma_{h} \\ 0 \\ 0 \\ 0 \\ \end{bmatrix} + \begin{bmatrix} \sigma_{xx} - \sigma_h \\ \sigma_{yy} - \sigma_h \\ \sigma_{zz} - \sigma_h \\ \sigma_{xy} \\ \sigma_{yz} \\ \sigma_{zx} \\ \end{bmatrix}$

This gives a basis for handling the construction of the material matrix where we the values for $E$, $\nu$ and $G$ form an incompatible set. To avoid asymmetries in the stiffness matrix the material matrix is constructed from the bulk $K$ and shear $G$ moduli. Where the bulk modulus is calculated from $E$ and $\nu$

$K = \dfrac{E}{3(1 - 2\nu)}$

By using this approach the hydrostatic and deviatoric terms are separated, thus preserving the symmetry in the equations.