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For isothermal elastic processes, the strain energy density function relates to the specific Helmholtz free energy function , [4] W = ρ 0 ψ . {\displaystyle W=\rho _{0}\psi \;.} For isentropic elastic processes, the strain energy density function relates to the internal energy function u {\displaystyle u} ,
The Gent hyperelastic material model [1] is a phenomenological model of rubber elasticity that is based on the concept of limiting chain extensibility. In this model, the strain energy density function is designed such that it has a singularity when the first invariant of the left Cauchy-Green deformation tensor reaches a limiting value .
In continuum mechanics, a Mooney–Rivlin solid [1] [2] is a hyperelastic material model where the strain energy density function is a linear combination of two invariants of the left Cauchy–Green deformation tensor.
The primary, and likely most widely employed, strain-energy function formulation is the Mooney-Rivlin model, which reduces to the widely known neo-Hookean model. The strain energy density function for an incompressible Mooney—Rivlin material is = + (); =
Yeoh model prediction versus experimental data for natural rubber. Model parameters and experimental data from PolymerFEM.com. The Yeoh hyperelastic material model [1] is a phenomenological model for the deformation of nearly incompressible, nonlinear elastic materials such as rubber.
A hyperelastic or Green elastic material [1] is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density function. The hyperelastic material is a special case of a Cauchy elastic material .
For rubber and biological materials, more sophisticated models are necessary. Such materials may exhibit a non-linear stress–strain behaviour at modest strains, or are elastic up to huge strains. These complex non-linear stress–strain behaviours need to be accommodated by specifically tailored strain-energy density functions.
If the rubber is compressible, a dependence on = can be introduced into the strain energy density; being the deformation gradient. Several possibilities exist, among which the Kaliske–Rothert [5] extension has been found to be reasonably accurate. With that extension, the Arruda-Boyce strain energy density function can be expressed as