 The material is assumed to be incompressible.The strain energy density function for the incompressible Arruda–Boyce model is given by
where formula_2 is the number of chain segments, formula_3 is the Boltzmann constant, formula_4 is the temperature in kelvins, formula_5 is the number of chains in the network of a cross-linked polymer,
where formula_7 is the first invariant of the left Cauchy–Green deformation tensor, and formula_8 is the inverse Langevin function which can be approximated by
For small deformations the Arruda–Boyce model reduces to the Gaussian network based neo-Hookean solid model.It can be shown that the Gent model is a simple and accurate approximation of the Arruda–Boyce model.