Anisotropic separable free energy functions for elastic and non-elastic solids

In incompressible isotropic elasticity, the Valanis and Landel strain energy function has certain attractive features from both the mathematical and physical view points. This separable form of strain energy has been widely and successfully used in predicting isotropic elastic deformations. We prove that the Valanis-Landel hypothesis is part of a general form of the isotropic strain energy function. The Valanis-Landel form is extended to take anisotropy into account and used to construct constitutive equations for anisotropic problems including stress-softening Mullins materials. The anisotropic separable forms are expressed in terms of spectral invariants that have clear physical meanings. The elegance and attractive features of the extended form are demonstrated, and its simplicity in analysing anisotropic and stress-softening materials is expressed. The extended anisotropic separable form is able to predict, and compares well with, numerous experimental data available in the literature for different types of materials, such as soft tissues, magneto-sensitive materials and (stress-softening) Mullins materials. The simplicity in handling some constitutive inequalities is demonstrated. The work here sets an alternative direction in formulating anisotropic solids in the sense that it does not explicitly use the standard classical invariants (or their variants) in the governing equations.