Spaces

Spaces are full or partial deformations on which a given material formulation may be projected to, e.g. to the distortional (part of the deformation) space.

class hyperelastic.spaces.Deformation(material, parallel=False, finalize=True, force=None, area=0)

The deformation space.

This class takes a Total-Lagrange material formulation and applies it on the deformation space.

(1)\[\psi = \psi(\boldsymbol{C}(\boldsymbol{F}))\]

The gradient of the strain energy function is carried out w.r.t. the Green Lagrange strain tensor. Hence, the work-conjugate stress tensor here refers to the second Piola-Kirchhoff stress tensor.

(2)\[\boldsymbol{S} = \frac{\partial \psi}{\partial \frac{1}{2}\boldsymbol{C}}\]

The hessian of the strain energy function is carried out w.r.t. the Green-Lagrange strain tensor. Hence, the work-conjugate elasticity tensor here refers to the fourth-order Total-Lagrangian elasticity tensor.

(3)\[\mathbb{C} = \frac{\partial^2 \psi}{\partial \frac{1}{2}\boldsymbol{C}~ \frac{1}{2}\boldsymbol{C}}\]

Given a Total-Lagrange material formulation, for the variation and linearization of the virtual work of internal forces, the output quantities have to be transformed: The second Piola-Kirchhoff stress tensor is converted into the deformation gradient work-conjugate first Piola-Kirchhoff stress tensor, along with its fourth-order elasticity tensor. Also, the so-called geometric tangent stiffness component (initial stress matrix) is added to the fourth-order elasticity tensor.

(4)\[ \begin{align}\begin{aligned}\delta W_{int} &= - \int_V \boldsymbol{P} : \delta \boldsymbol{F} ~ dV\\\Delta \delta W_{int} &= - \int_V \delta \boldsymbol{F} : \mathbb{A} : \Delta \boldsymbol{F} ~ dV\end{aligned}\end{align} \]

where

(5)\[ \begin{align}\begin{aligned}\boldsymbol{P} &= \boldsymbol{F} \boldsymbol{S}\\\mathbb{A}_{iJkL} &= F_{iI} F_{kK} \mathbb{C}_{IJKL} + \delta_{ik} S_{JL}\end{aligned}\end{align} \]

gradient(x)

The gradient as the partial derivative of the strain energy function w.r.t. the deformation gradient.

hessian(x)

The hessian as the second partial derivative of the strain energy function w.r.t. the deformation gradient.

piola(F, S, detF=None, C4=None, invC=None)

Convert the Total-Lagrange stress or elasticity tensor to the chosen configurations for the differential force and area vectors by applying a Piola-transformation.

class hyperelastic.spaces.Dilatational(material, parallel=False, finalize=True, force=None, area=0)

gradient(x)

hessian(x)

piola(F, S, detF=None, C4=None, invC=None)

Convert the Total-Lagrange stress or elasticity tensor to the chosen configurations for the differential force and area vectors by applying a Piola-transformation.

class hyperelastic.spaces.Distortional(material, parallel=False, finalize=True, force=None, area=0)

The distortional (part of the deformation) space is a partial deformation with constant volume. For a given deformation map \(\boldsymbol{x}(\boldsymbol{X})\) and its deformation gradient \(\boldsymbol{F}\), the distortional part of the deformation gradient \(\hat{\boldsymbol{F}}\) is obtained by a multiplicative (consecutive) split into a volume-changing (dilatational) and a constant-volume (distortional) part of the deformation gradient. Due to the fact that the dilatational part is proportional to the unit tensor, the order of these partial deformations is not unique.

(6)\[\boldsymbol{F} = \overset{\circ}{\boldsymbol{F}} \hat{\boldsymbol{F}} = \hat{\boldsymbol{F}} \overset{\circ}{\boldsymbol{F}}\]

This class takes a Total-Lagrange material formulation and applies it only on the distortional space.

(7)\[\hat{\psi} = \psi(\hat{\boldsymbol{C}}(\boldsymbol{F}))\]

The distortional (unimodular) part of the right Cauchy-Green deformation tensor is evaluated by the help of its third invariant (the determinant). The determinant of a distortional (an unimodular) tensor equals to one.

(8)\[\hat{\boldsymbol{C}} = I_3^{-1/3} \boldsymbol{C}\]

The gradient of the strain energy function is carried out w.r.t. the Green Lagrange strain tensor. Hence, the work-conjugate stress tensor used in this space projection refers to the second Piola-Kirchhoff stress tensor.

(9)\[\boldsymbol{S}' = \frac{\partial \hat{\psi}}{\partial \frac{1}{2}\boldsymbol{C}}\]

The distortional space projection leads to a physically deviatoric second Piola-Kirchhoff stress tensor, evaluated by the application of the chain rule.

(10)\[\hat{\boldsymbol{S}} = \frac{\partial \hat{\psi}} {\partial \frac{1}{2}\hat{\boldsymbol{C}}}\]

The (phyiscally) deviatoric projection is obtained by the partial derivative of the distortional part of the right Cauchy-Green deformation tensor w.r.t. the right Cauchy-Green deformation tensor.

(11)\[\frac{\partial \hat{\boldsymbol{C}}}{\partial \boldsymbol{C}} = \frac{\partial I_3^{-1/3} \boldsymbol{C}}{\partial \boldsymbol{C}} = I_3^{-1/3} \left( \boldsymbol{1} \odot \boldsymbol{1} - \frac{1}{3} \boldsymbol{C} \otimes \boldsymbol{C}^{-1} \right)\]

This partial derivative is used to perform the distortional space projection of the second Piola-Kirchhoff stress tensor. Instead of asserting the determinant-scaling to the fourth-order projection tensor, this factor is combined with the second Piola-Kirchhoff stress tensor in the distortional space. Hence, the stress tensor in the distortional space, scaled by \(I_3^{-1/3}\), is introduced as a new (frequently re-used) variable, denoted by an overset bar.

(12)\[ \begin{align}\begin{aligned}\boldsymbol{S}' &= \mathbb{P} : \bar{\boldsymbol{S}}\\\bar{\boldsymbol{S}} &= I_3^{-1/3} \hat{\boldsymbol{S}}\\\mathbb{P} &= \boldsymbol{1} \odot \boldsymbol{1} - \frac{1}{3} \boldsymbol{C}^{-1} \otimes \boldsymbol{C}\end{aligned}\end{align} \]

The evaluation of the double-dot product for the distortional space projection leads to the mathematical deviator of the product between the scaled distortional space stress tensor and the right Cauchy-Green deformation tensor, right multiplied by the inverse of the right Cauchy-Green deformation tensor.

(13)\[\boldsymbol{S}' = \bar{\boldsymbol{S}} - \frac{\bar{\boldsymbol{S}}:\boldsymbol{C}}{3} \boldsymbol{C}^{-1} = \text{dev}(\bar{\boldsymbol{S}} \boldsymbol{C}) \boldsymbol{C}^{-1}\]

The hessian of the strain energy function is carried out w.r.t. the Green-Lagrange strain tensor. Hence, the work-conjugate elasticity tensor used in this space projection refers to the fourth-order Total-Lagrangian elasticity tensor.

(14)\[\mathbb{C}' = \frac{\partial^2 \hat{\psi}}{\partial \frac{1}{2}\boldsymbol{C}~ \frac{1}{2}\boldsymbol{C}}\]

The evaluation of this second partial derivative leads to the elasticity tensor of the distortional space projection. The remaining determinant scaling terms of the projection tensor are included in the determinant-modified fourth-order elasticity tensor, denoted with an overset bar.

(15)\[\mathbb{C}' = \mathbb{P} : \bar{\mathbb{C}} : \mathbb{P}^T + \frac{2}{3} \left( \left( \bar{\boldsymbol{S}}:\boldsymbol{C} \right) \boldsymbol{C}^{-1} \odot \boldsymbol{C}^{-1} - \bar{\boldsymbol{S}} \otimes \boldsymbol{C}^{-1} - \boldsymbol{C}^{-1} \otimes \bar{\boldsymbol{S}} + \frac{1}{3} \left( \bar{\boldsymbol{S}}:\boldsymbol{C} \right) \boldsymbol{C}^{-1} \otimes \boldsymbol{C}^{-1} \right)\]

(16)\[\bar{\mathbb{C}} = I_3^{-2/3} \hat{\mathbb{C}}\]

For the variation and linearization of the virtual work of internal forces, the output quantities have to be transformed: The second Piola-Kirchhoff stress tensor is converted into the deformation gradient work-conjugate first Piola-Kirchhoff stress tensor, along with its fourth-order elasticity tensor. Also, the so-called geometric tangent stiffness component (initial stress matrix) is added to the fourth-order elasticity tensor.

(17)\[ \begin{align}\begin{aligned}\delta W_{int} &= - \int_V \boldsymbol{P} : \delta \boldsymbol{F} ~ dV\\\Delta \delta W_{int} &= - \int_V \delta \boldsymbol{F} : \mathbb{A} : \Delta \boldsymbol{F} ~ dV\end{aligned}\end{align} \]

where

(18)\[ \begin{align}\begin{aligned}\boldsymbol{P} &= \boldsymbol{F} \boldsymbol{S}\\\mathbb{A}_{iJkL} &= F_{iI} F_{kK} \mathbb{C}_{IJKL} + \delta_{ik} S_{JL}\end{aligned}\end{align} \]

gradient(x)

The gradient as the partial derivative of the strain energy function w.r.t. the deformation gradient.

hessian(x)

The hessian as the second partial derivative of the strain energy function w.r.t. the deformation gradient.

piola(F, S, detF=None, C4=None, invC=None)

Convert the Total-Lagrange stress or elasticity tensor to the chosen configurations for the differential force and area vectors by applying a Piola-transformation.