- Strain energy density function
A strain energy density function or stored energy density function is a
scalar valued function that relates thestrain energy density of a material to thedeformation gradient . :where is the (two-point) deformation gradienttensor , is the right Cauchy-Green deformation tensor, and is the left Cauchy-Green deformation tensor.For an
isotropic material, the deformation gradient can be expressed uniquely in terms of the principal stretches or in terms of theinvariants of the left Cauchy-Green deformation tensor or right Cauchy-Green deformation tensor and we have:A strain energy density function is used to define a
hyperelastic material by postulating that the stress in the material can be obtained by taking thederivative of with respect to the strain. For an isotropic, hyperelastic material the function relates theenergy stored in an elastic material, and thus the stress-strain relationship, only to the three strain (elongation) components, thus disregarding the deformation history, heat dissipation,stress relaxation etc.Examples of strain energy density functions
Examples of strain energy density functions are the Neo-Hookean, Mooney-Rivlin and Ogden models.
Generalized Neo-Hookean solid
The strain energy density function for a generalized Neo-Hookean solid Fact|date=June 2008 can be written as:where and are material constants.
Generalized Mooney-Rivlin solid
The generalized Mooney-Rivlin modelFact|date=June 2008 can be derived from the following strain energy function::where are material constants.
Polynomial rubber elasticity model
For the polynomial rubber model, the strain energy density function may be expressed as :where are material constants.
Ogden model
The strain energy density function for the Odgen model Fact|date=June 2008 is:where are material constants.
References
See also
*
Hyperelastic material
*Finite strain theory
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