In continuum mechanics, an Arruda–Boyce model 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 material is assumed to be incompressible. The model is named after Ellen Arruda and Mary Cunningham Boyce, who published it in 1993. The strain energy density function for the incompressible Arruda–Boyce model is given by where n is the number of chain segments, k_B is the Boltzmann constant, \theta is the temperature in kelvins, N is the number of chains in the network of a cross-linked polymer, where I_1 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. It can be shown that the Gent model is a simple and accurate approximation of the Arruda–Boyce model.

Alternative expressions for the Arruda–Boyce model

An alternative form of the Arruda–Boyce model, using the first five terms of the inverse Langevin function, is where C_1 is a material constant. The quantity N can also be interpreted as a measure of the limiting network stretch. If \lambda_m 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 and 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 extension has been found to be reasonably accurate. With that extension, the Arruda-Boyce strain energy density function can be expressed as where D_1 is a material constant and. For consistency with linear elasticity, we must have where \kappa is the bulk modulus.

Consistency condition

For the incompressible Arruda–Boyce model to be consistent with linear elasticity, with \mu 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 I_1 = 3, Substituting in the values of \alpha_i leads to the consistency condition

Stress-deformation relations

The Cauchy stress for the incompressible Arruda–Boyce model is given by

Uniaxial extension

For uniaxial extension in the -direction, the principal stretches are. From incompressibility. Hence. 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, we have Therefore, The engineering strain is \lambda-1,. The engineering stress is

Equibiaxial extension

For equibiaxial extension in the and directions, the principal stretches are. From incompressibility. Hence. 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 \lambda-1,. The engineering stress is

Planar extension

Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the directions with the direction constrained, the principal stretches are. From incompressibility. Hence. 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 \lambda-1,. The engineering stress is

Simple shear

The deformation gradient for a simple shear deformation has the form 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

Statistical mechanics of polymer deformation

The Arruda–Boyce model is based on the statistical mechanics of polymer chains. In this approach, each macromolecule is described as a chain of N segments, each of length l. 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 around the origin will contain the other end of the chain,, assuming a Gaussian probability density function, is The configurational entropy of a single chain from Boltzmann statistical mechanics is where c is a constant. The total entropy in a network of n chains is therefore where an affine deformation has been assumed. Therefore the strain energy of the deformed network is where \theta is the temperature.

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