Arruda–Boyce model

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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 material is assumed to be incompressible.

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

where is the number of chain segments, is the Boltzmann constant, is the temperature in kelvins, 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 approximated by

For small deformations the Arruda–Boyce model reduces to the Gaussian network based neo-Hookean solid model. It can be shown[3] that the Gent model is a simple and accurate approximation of the Arruda–Boyce model.

Later on, Tehrani and Sarvestani developed a new model based on the Arruda-Boyce model and eliminated some major assumptions such as monodispersity and simple special homogenization[4].. In results, the Tehrani-Sarvestani model can predict mechanical behaviour and failure of polydisperse materials[5].

Alternative expressions for the Arruda–Boyce model[edit]

An alternative form of the Arruda–Boyce model, using the first five terms of the inverse Langevin function, is[6]

where is a material constant. The quantity can also be interpreted as a measure of the limiting network stretch.

If 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[7] extension has been found to be reasonably accurate. With that extension, the Arruda-Boyce strain energy density function can be expressed as

where is a material constant and . For consistency with linear elasticity, we must have where is the bulk modulus.

Consistency condition[edit]

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 ,

Substituting in the values of leads to the consistency condition

Stress-deformation relations[edit]

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

Uniaxial extension[edit]

Stress-strain curves under uniaxial extension for Arruda–Boyce model compared with various hyperelastic material models.

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 . The engineering stress is

Equibiaxial extension[edit]

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 . The engineering stress is

Planar extension[edit]

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 . The engineering stress is

Simple shear[edit]

The deformation gradient for a simple shear deformation has the form[8]

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[edit]

The Arruda–Boyce model is based on the statistical mechanics of polymer chains. In this approach, each macromolecule is described as a chain of segments, each of length . 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 is a constant. The total entropy in a network of chains is therefore

where an affine deformation has been assumed. Therefore the strain energy of the deformed network is

where is the temperature.

Notes and references[edit]

  1. ^ Arruda, E. M. and Boyce, M. C., 1993, A three-dimensional model for the large stretch behavior of rubber elastic materials,, J. Mech. Phys. Solids, 41(2), pp. 389–412.
  2. ^ Bergstrom, J. S. and Boyce, M. C., 2001, Deformation of Elastomeric Networks: Relation between Molecular Level Deformation and Classical Statistical Mechanics Models of Rubber Elasticity, Macromolecules, 34 (3), pp 614–626, doi:10.1021/ma0007942.
  3. ^ Horgan, C. O. and Saccomandi, G., 2002, A molecular-statistical basis for the Gent constitutive model of rubber elasticity, Journal of Elasticity, 68(1), pp. 167–176.
  4. ^ Tehrani, M.; Sarvestani, A. (2017). "Effect of chain length distribution on mechanical behavior of polymeric networks". European Polymer Journal. 87: 136–146. doi:10.1016/j.eurpolymj.2016.12.017. 
  5. ^ Tehrani, M.; Sarvestani, A. (2017). "Effect of chain length distribution on mechanical behavior of polymeric networks". European Polymer Journal. 87: 136–146. doi:10.1016/j.eurpolymj.2016.12.017. 
  6. ^ Hiermaier, S. J., 2008, Structures under Crash and Impact, Springer.
  7. ^ Kaliske, M. and Rothert, H., 1997, On the finite element implementation of rubber-like materials at finite strains, Engineering Computations, 14(2), pp. 216–232.
  8. ^ Ogden, R. W., 1984, Non-linear elastic deformations, Dover.

See also[edit]