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:

“Realistic about how much one person can accomplish in a given day, women expect to have to make some trade-offs between work and family. Families, however, have absorbed all the stress and strain they possibly can. The entire responsibility for accommodation is taking place on the home side of the equation.”
—Deborah J. Swiss (20th century)

“Because humans are not alone in exhibiting such behavior—bees stockpile royal jelly, birds feather their nests, mice shred paper—it’s possible that a pregnant woman who scrubs her house from floor to ceiling [just before her baby is born] is responding to a biological imperative . . . . Of course there are those who believe that . . . the burst of energy that propels a pregnant woman to clean her house is a perfectly natural response to their mother’s impending visit.”
—Mary Arrigo (20th century)

“The information links are like nerves that pervade and help to animate the human organism. The sensors and monitors are analogous to the human senses that put us in touch with the world. Data bases correspond to memory; the information processors perform the function of human reasoning and comprehension. Once the postmodern infrastructure is reasonably integrated, it will greatly exceed human intelligence in reach, acuity, capacity, and precision.”
—Albert Borgman, U.S. educator, author. Crossing the Postmodern Divide, ch. 4, University of Chicago Press (1992)