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The Flory theory of rubber elasticity suggests that rubber elasticity has primarily entropic origins. By using the following basic equations for Helmholtz free energy and its discussion about entropy, the force generated from the deformation of a rubber chain from its original unstretched conformation can be derived.
The model is based on Ronald Rivlin's observation that the elastic properties of rubber may be described using a strain energy density function which is a power series in the strain invariants,, of the Cauchy-Green deformation tensors. [2] The Yeoh model for incompressible rubber is a function only of .
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 .
The polynomial hyperelastic material model [1] is a phenomenological model of rubber elasticity. In this model, the strain energy density function is of the form of a polynomial in the two invariants , of the left Cauchy-Green deformation tensor. The strain energy density function for the polynomial model is [1]
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.
In continuum mechanics, an Arruda–Boyce model [1] is a hyperelastic constitutive model used to describe the mechanical behavior of rubber and other polymeric substances. This model is based on the statistical mechanics of a material with a cubic representative volume element containing eight chains along the diagonal directions.
For soft materials, [1] such as rubber, where the bulk modulus is much higher than the shear modulus, Poisson's ratio is near 0.5. For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend to collapse in compression. Many typical solids have Poisson's ratios in the range of 0.2 to 0.3.
Elastic response of rubber-like materials are often modeled based on the Mooney–Rivlin model. The constants , are determined by fitting the predicted stress from the above equations to the experimental data. The recommended tests are uniaxial tension, equibiaxial compression, equibiaxial tension, uniaxial compression, and for shear, planar ...