On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy

This paper discusses the multiplicative decomposition of the deformation gradient into its volumetric and isochoric parts and its implications in the case of anisotropy. An analysis is carried out showing that the volumetric-isochoric split of the stored energy function can be justified and systematically derived on the basis of the physical assumption that the spherical part of the stress depends on the determinant of the deformation gradient without ad hoc introduction of the multiplicative split. The analysis shows that care must be exercised in the case of anisotropic material description in order not to violate certain physical requirements. Additive splits of the energy can be justified on the basis of certain physical observations and independent of the multiplicative decomposition of the deformation gradient. Specifically, it is shown that a spherical state of stress will cause even in the incompressible case, a change of shape. In fibre reinforced materials, the split of the stored energy function into a part related to the matrix and a part related to the fibre is considered, showing that the volumetric-isochoric split should be applied to the matrix part only.