Arruda–Boyce model

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.

The model is named after Ellen Arruda and Mary Cunningham Boyce, who published it in 1993.

[1] The strain energy density function for the incompressible Arruda–Boyce model is given by[2] where

is the number of chains in the network of a cross-linked polymer, where

is the first invariant of the left Cauchy–Green deformation tensor, and

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.

An alternative form of the Arruda–Boyce model, using the first five terms of the inverse Langevin function, is[4] where

can also be interpreted as a measure of the limiting network stretch.

is the stretch at which the polymer chain network becomes locked, we can express the Arruda–Boyce strain energy density as We may alternatively express the Arruda–Boyce model in the form where

If the rubber is compressible, a dependence on

can be introduced into the strain energy density;

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 where

For the incompressible Arruda–Boyce model to be consistent with linear elasticity, with

as the shear modulus of the material, the following condition has to be satisfied: From the Arruda–Boyce strain energy density function, we have, Therefore, at

leads to the consistency condition The Cauchy stress for the incompressible Arruda–Boyce model is given by For uniaxial extension in the

Therefore, The left Cauchy–Green deformation tensor can then be expressed as If the directions of the principal stretches are oriented with the coordinate basis vectors, we have If

The engineering stress is For equibiaxial extension in the

Therefore, The left Cauchy–Green deformation tensor can then be expressed as If the directions of the principal stretches are oriented with the coordinate basis vectors, we have The engineering strain is

The engineering stress is Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction.

direction constrained, the principal stretches are

The engineering stress is The deformation gradient for a simple shear deformation has the form[6] where

are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by In matrix form, the deformation gradient and the left Cauchy–Green deformation tensor may then be expressed as Therefore, and the Cauchy stress is given by The Arruda–Boyce model is based on the statistical mechanics of polymer chains.

In this approach, each macromolecule is described as a chain of

If we assume that the initial configuration of a chain can be described by a random walk, then the initial chain length is If we assume that one end of the chain is at the origin, then the probability that a block of size

, assuming a Gaussian probability density function, is The configurational entropy of a single chain from Boltzmann statistical mechanics is where

chains is therefore where an affine deformation has been assumed.

Therefore the strain energy of the deformed network is where