Strain energy density function

Strain energy density function

A strain energy density function or stored energy density function is a scalar valued function that relates the strain energy density of a material to the deformation gradient. : W = ar{W}(oldsymbol{F}) = hat{W}(oldsymbol{C}) = ilde{W}(oldsymbol{B}) where oldsymbol{F} is the (two-point) deformation gradient tensor, oldsymbol{C} is the right Cauchy-Green deformation tensor, and oldsymbol{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 the invariants 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 the derivative of W 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.

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.


See also

*Hyperelastic material
*Finite strain theory

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