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A strain energy density function or stored energy density function is a scalar-valued function that relates the strain energy density of a material to the deformation gradient.
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.
While the neo-Hookean material model can be stable for contact without sliding, sliding often leads to instability. To address this, regularization techniques are applied to the strain energy density function. Regularization is typically achieved by adding a regularization term to the strain energy density function of the material model.
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 = + (); =
The concept of strain is used to evaluate how much a given displacement differs locally from a rigid body displacement. [1] [8] [9] One of such strains for large deformations is the Lagrangian finite strain tensor, also called the Green-Lagrangian strain tensor or Green–St-Venant strain tensor, defined as
[2] [4] Tytus Maksymilian Huber (1904), in a paper written in Polish, anticipated to some extent this criterion by properly relying on the distortion strain energy, not on the total strain energy as his predecessors. [5] [6] [7] Heinrich Hencky formulated the same criterion as von Mises independently in 1924. [8]
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 .
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 .