Considerations on incompressibility in linear elasticity

The classical way to treat incompressible linear elastic materials is to use the inverse constitutive relationship (strain as a function of the stress), based on the compliance tensor, in place of the direct constitutive equation (stress as a function of strain), based on the elasticity (stiffness) tensor. This is because the elasticity tensor is affected by a diverging bulk modulus, required in order to allow the material to sustain any hydrostatic load, and is therefore not defined. In this work we show that there is a part of the elasticity tensor that can be "saved" also for incompressible materials, by "filtering" the components that deal with hydrostatic loads. The procedure is based on the treatment of incompressibility by means of the constraint of isochoric motion, i.e. of conservation of volume, and fourth-order tensor algebra.