- Strain energy density function
A strain energy density function or stored energy density function is a
scalar valued functionthat relates the strain energydensity of a material to the deformation gradient. :where is the (two-point) deformation gradient tensor, is the right Cauchy-Green deformation tensor, and is the left Cauchy-Green deformation tensor.
isotropicmaterial, the deformation gradient can be expressed uniquely in terms of the principal stretches or in terms of the invariantsof 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 materialby postulating that the stress in the material can be obtained by taking the derivativeof with respect to the strain. For an isotropic, hyperelastic material the function relates the energystored 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 relaxationetc.
Examples of strain energy density functions
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.
The strain energy density function for the Odgen model Fact|date=June 2008 is:where are material constants.
Finite strain theory
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