- Strain energy density function
A

**strain energy density function**or**stored energy density function**is ascalar valued function that relates thestrain energy density of a material to thedeformation gradient . :$W\; =\; ar\{W\}(\backslash boldsymbol\{F\})\; =\; hat\{W\}(\backslash boldsymbol\{C\})\; =\; ilde\{W\}(\backslash boldsymbol\{B\})$where $\backslash boldsymbol\{F\}$ is the (two-point) deformation gradienttensor , $\backslash boldsymbol\{C\}$ is the right Cauchy-Green deformation tensor, and $\backslash boldsymbol\{B\}$ 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:$W\; =\; hat\{W\}(lambda\_1,lambda\_2,lambda\_3)\; =\; ilde\{W\}(I\_1,I\_2,I\_3)\; =\; ar\{W\}(ar\{I\}\_1,ar\{I\}\_2,J)\; =\; U(I\_1^c,\; I\_2^c,\; I\_3^c)$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 $W$ 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:$ar\{W\}\; =\; frac\{mu\}\{2\}~(ar\{I\}\_1\; -\; 3)\; +\; frac\{kappa\}\{2\}~(J-1)^2$where $mu$ and $kappa$ are material constants.

**Generalized Mooney-Rivlin solid**The generalized Mooney-Rivlin modelFact|date=June 2008 can be derived from the following strain energy function::$ar\{W\}\; =\; frac\{mu\_1\}\{2\}(ar\{I\}\_1-3)\; +\; frac\{mu\_2\}\{2\}(ar\{I\_2\}-3)\; +\; frac\{kappa\}\{2\}(J-1)^2$where $mu\_1,mu\_2,kappa$ are material constants.

**Polynomial rubber elasticity model**For the polynomial rubber model, the strain energy density function may be expressed as :$ar\{W\}\; =\; sum\_\{i+j=1\}^n\; C\_\{ij\}~(ar\{I\}\_1-3)^i~(ar\{I\}\_2-3)^j\; +\; sum\_\{i=1\}^n\; frac\{kappa\_i\}\{2\}(J-1)^\{2i\}$where $C\_\{ij\},\; kappa\_i$ are material constants.

**Ogden model**The strain energy density function for the Odgen model Fact|date=June 2008 is:$U\; =\; sum\_\{i=1\}^n\; cfrac\{2mu\_i\}\{alpha\_i^2\}~left\; [left(cfrac\{lambda\_1\}\{J^\{1/3\; ight)^\{alpha\_i\}+left(cfrac\{lambda\_2\}\{J^\{1/3\; ight)^\{alpha\_i\}+left(cfrac\{lambda\_3\}\{J^\{1/3\; ight)^\{alpha\_i\}-3\; ight]\; +\; frac\{kappa\}\{2\}(J-1)^2$where $mu\_i,\; alpha\_i,\; kappa$ are material constants.

**References****See also***

Hyperelastic material

*Finite strain theory

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