Prev Subsection 2.4.2: Isotropic Elasticity in Principal Directions Up Section 2.4: Hyperelasticity Subsection 2.4.4: Transversely Isotropic Hyperelasticity Next
2.4.3 Nearly-Incompressible Hyperelasticity
A material is considered incompressible if it shows no change in volume during deformation, or otherwise stated, if holds throughout the entire body. It can be shown  that if the material is incompressible the hyperelastic constitutive equation becomes where is the deviatoric strain energy function and is the hydrostatic pressure. The presence of may seem unnecessary, but retaining has the advantage that equation (2.4.3-1) remains valid in the nearly incompressible case. Further, in practical terms, a finite element analysis rarely enforces in a pointwise manner, and hence its retention may be important for the evaluation of stresses.
The process of defining constitutive equations in the case of nearly incompressible hyperelasticity is simplified by adding a volumetric energy component to the distortional component : The second Piola-Kirchhoff tensor for a material defined by (2.4.3-2) is obtained in the standard manner with the help of equation (2.4.1-3). where the pressure is defined as An example for that will be used later in the definition of the constitutive models is The parameter will be used later as a penalty factor that will enforce the (nearly-) incompressible constraint. However, can represent a true material coefficient, namely the bulk modulus, for a compressible material that happens to have a hyperelastic strain energy function in the form of (2.4.3-2). In the case where the dilatational energy is given by (2.4.3-5), the pressure is Equation (2.4.3-3) can be further developed by applying the chain rule to the first term: where the fictitious second Piola-Kirchoff tensor  is defined by, and is the deviator operator in the reference frame: The Cauchy stress can then be obtained from equation (2.3.3-5) : where The following expression will be useful in the following development. Notice that the contraction with a symmetric tensor results in, The elasticity tensor, defined in (2.3.3-8), takes on the following form. where The spatial elasticity tensor follows from where