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{{Expert-subject|Physics|date=November 2008}}

A '''strain energy density function''' or '''stored energy density function''' is a [[scalar valued]] [[Function (mathematics)|function]] that relates the [[strain energy]] density of a material to the [[deformation gradient]].
:<math>
W = \hat{W}(\boldsymbol{C}) = \hat{W}(\boldsymbol{F}^T\cdot\boldsymbol{F}) =\bar{W}(\boldsymbol{F}) = \bar{W}(\boldsymbol{B}^{1/2}\cdot\boldsymbol{R})=\tilde{W}(\boldsymbol{B},\boldsymbol{R})
</math>
Equivalently,
:<math>
W = \hat{W}(\boldsymbol{C}) = \hat{W}(\boldsymbol{R}^T\cdot\boldsymbol{B}\cdot\boldsymbol{R}) =\tilde{W}(\boldsymbol{B},\boldsymbol{R})
</math>
where <math>\boldsymbol{F}</math> is the (two-point) deformation gradient [[tensor]], <math>\boldsymbol{C}</math> is the [[Finite strain theory#The_Right_Cauchy-Green_deformation_tensor|right Cauchy-Green deformation tensor]], <math>\boldsymbol{B}</math> is the [[Finite strain theory#The_Left_Cauchy-Green_deformation_tensor|left Cauchy-Green deformation tensor]],<ref name=Bower>{{cite book
|title=Applied Mechanics of Solids
|last=Bower |first=Allan
|year=2009
|publisher=CRC Press
|isbn=1-4398-0247-5
|url=http://solidmechanics.org/
|accessdate=January 2010}}</ref><ref name=Ogden>{{cite book
|title=Nonlinear Elastic Deformations
|author=Ogden, R. W.
|year=1998
|publisher=Dover
|isbn=0-486-69648-0}}</ref>
and <math>\boldsymbol{R}</math> is the rotation tensor from the polar decomposition of <math>\boldsymbol{F}</math>.

For an anisotropic material, the strain energy density function <math>\hat{W}(\boldsymbol{C})</math> 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, <math>\tilde{W}(\boldsymbol{B},\boldsymbol{R})</math> must further depend explicitly on the polar rotation tensor <math>\boldsymbol{R}</math> 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 <math>\boldsymbol{C}</math> (or, equivalently, the invariants of <math>\boldsymbol{B}</math> since both have the same eigenvalues). In other words, the strain energy density function can be expressed uniquely in terms of the [[Finite strain theory#Spectral_decompositions|principal stretches]] or in terms of the [[Invariant (mathematics)|invariants]] of the [[Finite strain theory#The_Left_Cauchy-Green_deformation_tensor|left Cauchy-Green deformation tensor]] or [[Finite strain theory#The_Right_Cauchy-Green_deformation_tensor|right Cauchy-Green deformation tensor]] and we have:

For isotropic materials,
:<math>
W = \hat{W}(\lambda_1,\lambda_2,\lambda_3) = \tilde{W}(I_1,I_2,I_3) = \bar{W}(\bar{I}_1,\bar{I}_2,J) = U(I_1^c, I_2^c, I_3^c)
</math>
with
:<math>
\begin{align}
\bar{I}_1 & = J^{-2/3}~I_1 ~;~~ I_1 = \lambda_1^2 + \lambda_2 ^2+ \lambda_3 ^2 ~;~~ J = \det(\boldsymbol{F}) \\
\bar{I}_2 & = J^{-4/3}~I_2 ~;~~ I_2 = \lambda_1^2 \lambda_2^2 + \lambda_2^2 \lambda_3^2 + \lambda_3^2 \lambda_1^2
\end{align}
</math>
A strain energy density function is used to define a [[hyperelastic material]] by postulating that the [[stress (physics)|stress]] in the material can be obtained by taking the [[derivative]] of <math>W</math> with respect to the [[strain (physics)|strain]]. For an isotropic, hyperelastic material the function relates the [[energy]] stored in an [[Elasticity (physics)|elastic material]], and thus the stress-strain relationship, only to the three [[strain (materials science)|strain]] (elongation) components, thus disregarding the deformation history, heat dissipation, [[stress relaxation]] etc.

For isothermal elastic processes, the strain energy density function relates to the [[Helmholtz free energy]] function <math>\psi</math>,<ref name=Wriggers>{{cite book
|title=Nonlinear Finite Element Methods
|author=Wriggers, P.
|year=2008
|publisher=Springer-Verlag
|isbn=978-3-540-71000-4}}</ref>

:<math>
W = \rho_0 \psi \;.
</math>
For isentropic elastic processes, the strain energy density function relates to the internal energy function <math>u</math>,
:<math>
W = \rho_0 u \;.
</math>

== Examples of strain energy density functions ==
Some examples of hyperelastic [[constitutive equations]] are<ref>Muhr, A. H. (2005). Modeling the stress-strain behavior of rubber. Rubber chemistry and technology, 78(3), 391-425. [http://dx.doi.org/10.5254/1.3547890]</ref>
*[[Hyperelastic material#Saint Venant–Kirchhoff model|Saint Venant–Kirchhoff]]
*[[Neo-Hookean solid|Neo-Hookean]]
*[[Polynomial (hyperelastic model)|Generalized Rivlin]]
*[[Mooney–Rivlin solid|Mooney–Rivlin]]
*[[Ogden (hyperelastic model)|Ogden]]
*[[Yeoh (hyperelastic model)|Yeoh]]
*[[Arruda–Boyce model]]
*[[Gent (hyperelastic model)|Gent]]

== References ==
<references/>

== See also ==
{{wikiversity|Continuum mechanics/Thermoelasticity}}
*[[v:Continuum mechanics/Thermoelasticity#Helmholtz and Gibbs free energy|Helmholtz and Gibbs free energy in thermoelasticity]]
*[[Hyperelastic material]]
*[[Finite strain theory]]

[[Category:Continuum mechanics]]
[[Category:Rubber properties]]
[[Category:Solid mechanics]]

[[ja:ひずみエネルギー]]

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