# # Isotropic Material

The material properties are

– elastic modulus

– Poisson’s ratio

– shear modulus

The shear modulus is related to the Elastic modulus and Poisson’s ratio through

The general elasticity matrix is

If the default value of is overridden the material matrix need to be constructed differently to preserve symmetry in this case.

The Cauchy stress tensor is

But it is convenient to think instead of the stress tensor made up of two parts

• a mean hydrostatic stress tensor – tending to change the volume
• a stress deviator tensor – tending to distort The mean stress is gives by

And the Cauchy stress tensor can then be expressed as the sum of the mean hydrostatic and deviatoric stresses

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

and is as defined.

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