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.

$W = \hat{W}(\boldsymbol{C}) = \hat{W}(\boldsymbol{F}^T\cdot\boldsymbol{F}) =\bar{W}(\boldsymbol{F}) = \bar{W}(\boldsymbol{B}^{1/2}\cdot\boldsymbol{R})=\tilde{W}(\boldsymbol{B},\boldsymbol{R})$

Equivalently,

$W = \hat{W}(\boldsymbol{C}) = \hat{W}(\boldsymbol{R}^T\cdot\boldsymbol{B}\cdot\boldsymbol{R}) =\tilde{W}(\boldsymbol{B},\boldsymbol{R})$

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,

$W = \hat{W}(\lambda_1,\lambda_2,\lambda_3) = \tilde{W}(I_1,I_2,I_3) = \bar{W}(\bar{I}_1,\bar{I}_2,J) = U(I_1^c, I_2^c, I_3^c)$

with

$\begin{align} \bar{I}_1 & = J^{-2/3}~I_1 ~;~~ I_1 = \lambda_1^2 + \lambda_2 ^2+ \lambda_3 ^2 ~;~~ J = \det(\boldsymbol{F}) \\ \bar{I}_2 & = J^{-4/3}~I_2 ~;~~ I_2 = \lambda_1^2 \lambda_2^2 + \lambda_2^2 \lambda_3^2 + \lambda_3^2 \lambda_1^2 \end{align}$

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 isothermal elastic processes, the strain energy density function relates to the Helmholtz free energy function ,

$W = \rho_0 \psi \;.$

For isentropic elastic processes, the strain energy density function relates to the internal energy function ,

$W = \rho_0 u \;.$

Read more about Strain Energy Density Function: Examples of Strain Energy Density Functions

Famous quotes containing the words strain, energy and/or function:

“O Master, let me walk with Thee
In lowly paths of service free;
Tell me Thy secret, help me bear
The strain of toil, the fret of care.”
—Washington Gladden (1836–1918)

“All my energy and attention were devoted to trying to help her solve her problems. Unfortunately I didn’t have much success.”
—Arthur Miller (b. 1915)

“The fact remains that the human being in early childhood learns to consider one or the other aspect of bodily function as evil, shameful, or unsafe. There is not a culture which does not use a combination of these devils to develop, by way of counterpoint, its own style of faith, pride, certainty, and initiative.”
—Erik H. Erikson (1904–1994)