Show that any second rank Cartesian tensor in can be written as a sum of a symmetric tensor and an antisymmetric tensor. Further, show that can be decomposed into the following terms
where is symmetric and traceless. Give expressions for and explicitly in terms of .
For an isotropic material, the stress can be related to the strain through the stress-strain relation, , where the elasticity tensor is given by
and and are scalars. As in , the strain can be decomposed into its trace , a symmetric traceless tensor and a vector . Use the stress-strain relation to express each of and in terms of and .
Hence, or otherwise, show that if is symmetric then so is . Show also that the stress-strain relation can be written in the form
where and are scalars.