Strain energy density function
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where is the (two-point) deformation gradient tensor, is the right Cauchy-Green deformation tensor, is the left Cauchy-Green deformation tensor, and is the rotation tensor from the polar decomposition of .
For an anisotropic material, the strain energy density function depends implicitly on reference vectors or tensors (such as the initial orientation of fibers in a composite) that characterize internal material texture. The spatial representation, must further depend explicitly on the polar rotation tensor to provide sufficient information to convect the reference texture vectors or tensors into the spatial configuration.
For an isotropic material, consideration of the principle of material frame indifference leads to the conclusion that the strain energy density function depends only on the invariants of (or, equivalently, the invariants of since both have the same eigenvalues). In other words, the strain energy density function can be expressed uniquely in terms of the principal stretches or in terms of the invariants of the left Cauchy-Green deformation tensor or right Cauchy-Green deformation tensor and we have:
For isotropic materials,
For linear isotropic materials undergoing small strains, the strain energy density function specializes to
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 the derivative of with respect to the strain. For an isotropic, hyperelastic material the function relates the energy 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.
For isentropic elastic processes, the strain energy density function relates to the internal energy function ,
Examples of strain energy density functions
- Saint Venant–Kirchhoff
- Generalized Rivlin
- Arruda–Boyce model
- Bower, Allan (2009). Applied Mechanics of Solids. CRC Press. ISBN 1-4398-0247-5. Retrieved January 2010.
- Ogden, R. W. (1998). Nonlinear Elastic Deformations. Dover. ISBN 0-486-69648-0.
- Sadd, Martin H. (2009). Elasticity Theory, Applications and Numerics. Elsevier. ISBN 978-0-12-374446-3.
- Wriggers, P. (2008). Nonlinear Finite Element Methods. Springer-Verlag. ISBN 978-3-540-71000-4.
- Muhr, A. H. (2005). Modeling the stress-strain behavior of rubber. Rubber chemistry and technology, 78(3), 391-425. 
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