Free Energy: Mixing and Elasticity

A free energy function specifies the material response of the gel body and allows us to compute chemical potential and stresses. The free energy is assumed to be split into a mixing and an elastic component: chemical potential depends on the mixing part and stresses depend on the elastic part. The mixing component has the Flory–Huggins representation form and can be expressed as a function of the polymer fraction or, equivalently, of the liquid concentration or of the Jacobian of the deformation. The elastic component is defined by different elastic models; here, we present the neo-Hookean, Arruda–Boyce and Gent elastic models for isotropic gels, and an extended neo-Hookean elastic model for transversely isotropic gels. The corresponding constitutive equations for the First Piola–Kirchhoff, Second Piola–Kirchhoff and Cauchy stress tensor are derived.