felupe.constitution

class felupe.constitution.LinearElastic(E=None, nu=None)

Isotropic linear-elastic material formulation.

\[\begin{split}\begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{12} \\ \sigma_{23} \\ \sigma_{31} \end{bmatrix} = \frac{E}{(1+\nu)(1-2\nu)}\begin{bmatrix} 1-\nu & \nu & \nu & 0 & 0 & 0\\ \nu & 1-\nu & \nu & 0 & 0 & 0\\ \nu & \nu & 1-\nu & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{1-2\nu}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1-2\nu}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1-2\nu}{2} \end{bmatrix} \cdot \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2 \varepsilon_{12} \\ 2 \varepsilon_{23} \\ 2 \varepsilon_{31} \end{bmatrix}\end{split}\]

with the strain tensor

\[\boldsymbol{\varepsilon} = \frac{1}{2} \left( \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{X}} + \left( \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{X}} \right)^T \right)\]

Parameters

• E (float) – Young’s modulus.

• nu (float) – Poisson ratio.

gradient(extract, E=None, nu=None)

Evaluate the stress tensor (as a function of the deformation gradient).

Parameters

• extract (list of ndarray) – List with Deformation gradient F (3x3) as first item

• E (float, optional) – Young’s modulus (default is None)

• nu (float, optional) – Poisson ratio (default is None)

Returns

Stress tensor (3x3)

Return type

ndarray

hessian(extract=None, E=None, nu=None, shape=(1, 1), region=None)

Evaluate the elasticity tensor. The Deformation gradient is only used for the shape of the trailing axes.

Parameters

• extract (list of ndarray, optional) – List with Deformation gradient F (3x3) as first item (default is None)

• E (float, optional) – Young’s modulus (default is None)

• nu (float, optional) – Poisson ratio (default is None)

• shape ((int, int), optional (default is (1, 1))) – Tuple with shape of the trailing axes (default is None)

• region (Region, optional) – A numeric region for shape of the trailing axes (default is None)

Returns

elasticity tensor (3x3x3x3)

Return type

ndarray

class felupe.constitution.LinearElasticTensorNotation(E=None, nu=None, parallel=False)

Isotropic linear-elastic material formulation.

\[ \begin{align}\begin{aligned}\boldsymbol{\sigma} &= 2 \mu \ \boldsymbol{\varepsilon} + \gamma \ \text{tr}(\boldsymbol{\varepsilon}) \ \boldsymbol{I}\\\frac{\boldsymbol{\partial \sigma}}{\partial \boldsymbol{\varepsilon}} &= 2 \mu \ \boldsymbol{I} \odot \boldsymbol{I} + \gamma \ \boldsymbol{I} \otimes \boldsymbol{I}\end{aligned}\end{align} \]

with the strain tensor

\[\boldsymbol{\varepsilon} = \frac{1}{2} \left( \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{X}} + \left( \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{X}} \right)^T \right)\]

Parameters

• E (float) – Young’s modulus.

• nu (float) – Poisson ratio.

gradient(extract=None, E=None, nu=None)

Evaluate the stress tensor (as a function of the deformation gradient).

Parameters

• extract (list of ndarray) – List with Deformation gradient F (3x3) as first item

• E (float, optional) – Young’s modulus (default is None)

• nu (float, optional) – Poisson ratio (default is None)

Returns

Stress tensor (3x3)

Return type

ndarray

hessian(extract=None, E=None, nu=None, shape=(1, 1), region=None)

Evaluate the elasticity tensor. The Deformation gradient is only used for the shape of the trailing axes.

Parameters

• extract (list of ndarray. optional) – List with Deformation gradient F (3x3) as first item (default is None)

• E (float, optional) – Young’s modulus (default is None)

• nu (float, optional) – Poisson ratio (default is None)

• shape ((int, int), optional (default is (1, 1))) – Tuple with shape of the trailing axes (default is None)

• region (Region, optional) – A numeric region for shape of the trailing axes (default is None)

Returns

elasticity tensor (3x3x3x3)

Return type

ndarray

class felupe.constitution.LinearElasticPlaneStress(E, nu)

Plane-stress isotropic linear-elastic material formulation.

Parameters

• E (float) – Young’s modulus.

• nu (float) – Poisson ratio.

gradient(extract, E=None, nu=None)

Evaluate the 2d-stress tensor from the deformation gradient.

Parameters

• extract (list of ndarray) – List with In-plane components (2x2) of the Deformation gradient ``F``as first item

• E (float, optional) – Young’s modulus (default is None)

• nu (float, optional) – Poisson ratio (default is None)

Returns

In-plane components of stress tensor (2x2)

Return type

ndarray

hessian(extract=None, E=None, nu=None, shape=(1, 1), region=None)

Evaluate the elasticity tensor from the deformation gradient.

Parameters

• extract (list of ndarray, optional) – List with In-plane components (2x2) of the Deformation gradient ``F``as first item (default is None)

• E (float, optional) – Young’s modulus (default is None)

• nu (float, optional) – Poisson ratio (default is None)

• shape ((int, int), optional (default is (1, 1))) – Tuple with shape of the trailing axes (default is None)

• region (Region, optional) – A numeric region for shape of the trailing axes (default is None)

Returns

In-plane components of elasticity tensor (2x2x2x2)

Return type

ndarray

strain(extract, E=None, nu=None)

Evaluate the strain tensor from the deformation gradient.

Parameters

• extract (list of ndarray) – List with In-plane components (2x2) of the Deformation gradient ``F``as first item

• E (float, optional) – Young’s modulus (default is None)

• nu (float, optional) – Poisson ratio (default is None)

Returns

e – Strain tensor (3x3)

Return type

ndarray

stress(extract, E=None, nu=None)

“Evaluate the 3d-stress tensor from the deformation gradient.

Parameters

• extract (list of ndarray) – List with In-plane components (2x2) of the Deformation gradient ``F``as first item

• E (float, optional) – Young’s modulus (default is None)

• nu (float, optional) – Poisson ratio (default is None)

Returns

Stress tensor (3x3)

Return type

ndarray

class felupe.constitution.LinearElasticPlaneStrain(E, nu)

Plane-strain isotropic linear-elastic material formulation.

Parameters

• E (float) – Young’s modulus.

• nu (float) – Poisson ratio.

gradient(extract, E=None, nu=None)

Evaluate the 2d-stress tensor from the deformation gradient.

Parameters

• extract (list of ndarray) – List with In-plane components (2x2) of the Deformation gradient ``F``as first item

• E (float, optional) – Young’s modulus (default is None)

• nu (float, optional) – Poisson ratio (default is None)

Returns

In-plane components of stress tensor (2x2)

Return type

ndarray

hessian(extract, E=None, nu=None)

Evaluate the 2d-elasticity tensor from the deformation gradient.

Parameters

• extract (list of ndarray) – List with In-plane components (2x2) of the Deformation gradient ``F``as first item

• E (float, optional) – Young’s modulus (default is None)

• nu (float, optional) – Poisson ratio (default is None)

Returns

In-plane components of elasticity tensor (2x2x2x2)

Return type

ndarray

strain(extract, E=None, nu=None)

Evaluate the strain tensor from the deformation gradient.

Parameters

• extract (list of ndarray) – List with In-plane components (2x2) of the Deformation gradient ``F``as first item

• E (float, optional) – Young’s modulus (default is None)

• nu (float, optional) – Poisson ratio (default is None)

Returns

e – Strain tensor (3x3)

Return type

ndarray

stress(extract, E=None, nu=None)

“Evaluate the 3d-stress tensor from the deformation gradient.

Parameters

• extract (list of ndarray) – List with In-plane components (2x2) of the Deformation gradient ``F``as first item

• E (float, optional) – Young’s modulus (default is None)

• nu (float, optional) – Poisson ratio (default is None)

Returns

Stress tensor (3x3)

Return type

ndarray

class felupe.constitution.NeoHooke(mu=None, bulk=None, parallel=False)

Nearly-incompressible isotropic hyperelastic Neo-Hooke material formulation. The strain energy density function of the Neo-Hookean material formulation is a linear function of the trace of the isochoric part of the right Cauchy-Green deformation tensor.

In a nearly-incompressible constitutive framework the strain energy density is an additive composition of an isochoric and a volumetric part. While the isochoric part is defined on the distortional part of the deformation gradient, the volumetric part of the strain energy function is defined on the determinant of the deformation gradient.

\[ \begin{align}\begin{aligned}\psi &= \hat{\psi}(\hat{\boldsymbol{C}}) + U(J)\\\hat\psi(\hat{\boldsymbol{C}}) &= \frac{\mu}{2} (\text{tr}(\hat{\boldsymbol{C}}) - 3)\end{aligned}\end{align} \]

with

\[ \begin{align}\begin{aligned}J &= \text{det}(\boldsymbol{F})\\\hat{\boldsymbol{F}} &= J^{-1/3} \boldsymbol{F}\\\hat{\boldsymbol{C}} &= \hat{\boldsymbol{F}}^T \hat{\boldsymbol{F}}\end{aligned}\end{align} \]

The volumetric part of the strain energy density function is a function the volume ratio.

\[U(J) = \frac{K}{2} (J - 1)^2\]

The first Piola-Kirchhoff stress tensor is evaluated as the gradient of the strain energy density function. The hessian of the strain energy density function enables the corresponding elasticity tensor.

\[ \begin{align}\begin{aligned}\boldsymbol{P} &= \frac{\partial \psi}{\partial \boldsymbol{F}}\\\mathbb{A} &= \frac{\partial^2 \psi}{\partial \boldsymbol{F}\ \partial \boldsymbol{F}}\end{aligned}\end{align} \]

A chain rule application leads to the following expression for the stress tensor. It is formulated as a sum of the physical-deviatoric (not the mathematical deviator!) and the physical-hydrostatic stress tensors.

\[ \begin{align}\begin{aligned}\boldsymbol{P} &= \boldsymbol{P}' + \boldsymbol{P}_U\\\boldsymbol{P}' &= \frac{\partial \hat{\psi}}{\partial \hat{\boldsymbol{F}}} : \frac{\partial \hat{\boldsymbol{F}}}{\partial \boldsymbol{F}} = \bar{\boldsymbol{P}} - \frac{1}{3} (\bar{\boldsymbol{P}} : \boldsymbol{F}) \boldsymbol{F}^{-T}\\\boldsymbol{P}_U &= \frac{\partial U(J)}{\partial J} \frac{\partial J}{\partial \boldsymbol{F}} = U'(J) J \boldsymbol{F}^{-T}\end{aligned}\end{align} \]

with

\[ \begin{align}\begin{aligned}\frac{\partial \hat{\boldsymbol{F}}}{\partial \boldsymbol{F}} &= J^{-1/3} \left( \boldsymbol{I} \overset{ik}{\otimes} \boldsymbol{I} - \frac{1}{3} \boldsymbol{F} \otimes \boldsymbol{F}^{-T} \right)\\\frac{\partial J}{\partial \boldsymbol{F}} &= J \boldsymbol{F}^{-T}\\\bar{\boldsymbol{P}} &= J^{-1/3} \frac{\partial \hat{\psi}}{\partial \hat{\boldsymbol{F}}}\end{aligned}\end{align} \]

With the above partial derivatives the first Piola-Kirchhoff stress tensor of the Neo-Hookean material model takes the following form.

\[\boldsymbol{P} = \mu J^{-2/3} \left( \boldsymbol{F} - \frac{1}{3} (\boldsymbol{F} : \boldsymbol{F}) \boldsymbol{F}^{-T} \right) + K (J - 1) J \boldsymbol{F}^{-T}\]

Again, a chain rule application leads to an expression for the elasticity tensor.

\[ \begin{align}\begin{aligned}\mathbb{A} &= \mathbb{A}' + \mathbb{A}_{U}\\\mathbb{A}' &= \bar{\mathbb{A}} - \frac{1}{3} \left( (\bar{\mathbb{A}} : \boldsymbol{F}) \otimes \boldsymbol{F}^{-T} + \boldsymbol{F}^{-T} \otimes (\boldsymbol{F} : \bar{\mathbb{A}}) \right ) + \frac{1}{9} (\boldsymbol{F} : \bar{\mathbb{A}} : \boldsymbol{F}) \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T}\\\mathbb{A}_{U} &= (U''(J) J + U'(J)) J \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T} - U'(J) J \boldsymbol{F}^{-T} \overset{il}{\otimes} \boldsymbol{F}^{-T}\end{aligned}\end{align} \]

with

\[\bar{\mathbb{A}} = J^{-1/3} \frac{\partial^2 \hat\psi}{\partial \hat{\boldsymbol{F}}\ \partial \hat{\boldsymbol{F}}} J^{-1/3}\]

With the above partial derivatives the (physical-deviatoric and -hydrostatic) parts of the elasticity tensor associated to the first Piola-Kirchhoff stress tensor of the Neo-Hookean material model takes the following form.

\[ \begin{align}\begin{aligned}\mathbb{A} &= \mathbb{A}' + \mathbb{A}_{U}\\\mathbb{A}' &= J^{-2/3} \left(\boldsymbol{I} \overset{ik}{\otimes} \boldsymbol{I} - \frac{1}{3} \left( \boldsymbol{F} \otimes \boldsymbol{F}^{-T} + \boldsymbol{F}^{-T} \otimes \boldsymbol{F} \right ) + \frac{1}{9} (\boldsymbol{F} : \boldsymbol{F}) \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T} \right)\\\mathbb{A}_{U} &= K J \left( (2J - 1) \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T} - (J - 1) \boldsymbol{F}^{-T} \overset{il}{\otimes} \boldsymbol{F}^{-T} \right)\end{aligned}\end{align} \]

Parameters

• mu (float) – Shear modulus

• bulk (float) – Bulk modulus

function(extract, mu=None, bulk=None)

Strain energy density function per unit undeformed volume of the Neo-Hookean material formulation.

Parameters

• extract (list of ndarray) – List with the Deformation gradient F (3x3) as first item

• mu (float, optional) – Shear modulus (default is None)

• bulk (float, optional) – Bulk modulus (default is None)

gradient(extract, mu=None, bulk=None)

Gradient of the strain energy density function per unit undeformed volume of the Neo-Hookean material formulation.

Parameters

• extract (list of ndarray) – List with the Deformation gradient F (3x3) as first item

• mu (float, optional) – Shear modulus (default is None)

• bulk (float, optional) – Bulk modulus (default is None)

hessian(extract, mu=None, bulk=None)

Hessian of the strain energy density function per unit undeformed volume of the Neo-Hookean material formulation.

Parameters

• extract (list of ndarray) – List with the Deformation gradient F (3x3) as first item

• mu (float, optional) – Shear modulus (default is None)

• bulk (float, optional) – Bulk modulus (default is None)

class felupe.constitution.ThreeFieldVariation(material, parallel=False)

Hu-Washizu hydrostatic-volumetric selective \((\boldsymbol{u},p,J)\) - three-field variation for nearly- incompressible material formulations. The total potential energy for nearly-incompressible hyperelasticity is formulated with a determinant-modified deformation gradient. Pressure and volume ratio fields should be kept one order lower than the interpolation order of the displacement field, e.g. linear displacement fields should be paired with element-constant (mean) values of pressure and volume ratio.

The total potential energy of internal forces is defined with a strain energy density function in terms of a determinant-modified deformation gradient and an additional control equation.

\[ \begin{align}\begin{aligned}\Pi &= \Pi_{int} + \Pi_{ext}\\\Pi_{int} &= \int_V \psi(\boldsymbol{F}) \ dV \qquad \rightarrow \qquad \Pi_{int}(\boldsymbol{u},p,J) = \int_V \psi(\overline{\boldsymbol{F}}) \ dV + \int_V p (J-\overline{J}) \ dV\\\overline{\boldsymbol{F}} &= \left(\frac{\overline{J}}{J}\right)^{1/3} \boldsymbol{F}\end{aligned}\end{align} \]

The variations of the total potential energy w.r.t. \((\boldsymbol{u},p,J)\) lead to the following expressions. We denote first partial derivatives as \(\boldsymbol{f}_{(\bullet)}\) and second partial derivatives as \(\boldsymbol{A}_{(\bullet,\bullet)}\).

\[ \begin{align}\begin{aligned}\delta_{\boldsymbol{u}} \Pi_{int} &= \int_V \boldsymbol{f}_{\boldsymbol{u}} : \delta \boldsymbol{F} \ dV = \int_V \left( \frac{\partial \psi}{\partial \overline{\boldsymbol{F}}} : \frac{\partial \overline{\boldsymbol{F}}}{\partial \boldsymbol{F}} + p J \boldsymbol{F}^{-T} \right) : \delta \boldsymbol{F} \ dV\\\delta_{p} \Pi_{int} &= \int_V f_{p} \ \delta p \ dV = \int_V (J - \overline{J}) \ \delta p \ dV\\\delta_{\overline{J}} \Pi_{int} &= \int_V f_{\overline{J}} \ \delta \overline{J} \ dV = \int_V \left( \frac{\partial \psi}{\partial \overline{\boldsymbol{F}}} : \frac{\partial \overline{\boldsymbol{F}}}{\partial \overline{J}} - p \right) : \delta \overline{J} \ dV\end{aligned}\end{align} \]

The projection tensors from the variations lead the following results.

\[ \begin{align}\begin{aligned}\frac{\partial \overline{\boldsymbol{F}}}{\partial \boldsymbol{F}} &= \left(\frac{\overline{J}}{J}\right)^{1/3} \left( \boldsymbol{I} \overset{ik}{\odot} \boldsymbol{I} - \frac{1}{3} \boldsymbol{F} \otimes \boldsymbol{F}^{-T} \right)\\\frac{\partial \overline{\boldsymbol{F}}}{\partial \overline{J}} &= \frac{1}{3 \overline{J}} \overline{\boldsymbol{F}}\end{aligned}\end{align} \]

The double-dot products from the variations are now evaluated.

\[ \begin{align}\begin{aligned}\overline{\boldsymbol{P}} &= \frac{\partial \psi}{\partial \overline{\boldsymbol{F}}} = \overline{\overline{\boldsymbol{P}}} - \frac{1}{3} \left( \overline{\overline{\boldsymbol{P}}} : \boldsymbol{F} \right) \boldsymbol{F}^{-T} \qquad \text{with} \qquad \overline{\overline{\boldsymbol{P}}} = \left(\frac{\overline{J}}{J}\right)^{1/3} \frac{\partial \psi}{\partial \overline{\boldsymbol{F}}}\\\frac{\partial \psi}{\partial \overline{\boldsymbol{F}}} : \frac{1}{3 \overline{J}} \overline{\boldsymbol{F}} &= \frac{1}{3 \overline{J}} \overline{\overline{\boldsymbol{P}}} : \boldsymbol{F}\end{aligned}\end{align} \]

We now have three formulas; one for the first Piola Kirchhoff stress and two additional control equations.

\[ \begin{align}\begin{aligned}\boldsymbol{f}_{\boldsymbol{u}} (= \boldsymbol{P}) &= \overline{\overline{\boldsymbol{P}}} - \frac{1}{3} \left( \overline{\overline{\boldsymbol{P}}} : \boldsymbol{F} \right) \boldsymbol{F}^{-T}\\f_p &= J - \overline{J}\\f_{\overline{J}} &= \frac{1}{3 \overline{J}} \left( \overline{\overline{\boldsymbol{P}}} : \boldsymbol{F} \right) - p\end{aligned}\end{align} \]

A linearization of the above formulas gives six equations (only results are given here).

\[ \begin{align}\begin{aligned}\mathbb{A}_{\boldsymbol{u},\boldsymbol{u}} &= \overline{\overline{\mathbb{A}}} + \frac{1}{9} \left( \boldsymbol{F} : \overline{\overline{\mathbb{A}}} : \boldsymbol{F} \right) \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T} - \frac{1}{3} \left( \boldsymbol{F}^{-T} \otimes \left( \overline{\overline{\boldsymbol{P}}} + \boldsymbol{F} : \overline{\overline{\mathbb{A}}} \right) + \left( \overline{\overline{\boldsymbol{P}}} + \overline{\overline{\mathbb{A}}} : \boldsymbol{F} \right) \otimes \boldsymbol{F}^{-T} \right)\\&+\left( p J + \frac{1}{9} \overline{\overline{\boldsymbol{P}}} : \boldsymbol{F} \right) \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T} - \left( p J - \frac{1}{3} \overline{\overline{\boldsymbol{P}}} : \boldsymbol{F} \right) \boldsymbol{F}^{-T} \overset{il}{\odot} \boldsymbol{F}^{-T}\\A_{p,p} &= 0\\A_{\overline{J},\overline{J}} &= \frac{1}{9 \overline{J}^2} \left( \boldsymbol{F} : \overline{\overline{\mathbb{A}}} : \boldsymbol{F} \right) - 2 \left( \overline{\overline{\boldsymbol{P}}} : \boldsymbol{F} \right)\\\boldsymbol{A}_{\boldsymbol{u},p} &= \boldsymbol{A}_{p, \boldsymbol{u}} = J \boldsymbol{F}^{-T}\\\boldsymbol{A}_{\boldsymbol{u},\overline{J}} &= \boldsymbol{A}_{\overline{J}, \boldsymbol{u}} = \frac{1}{3 \overline{J}} \left( \boldsymbol{P}' + \boldsymbol{F} : \overline{\overline{\mathbb{A}}} - \frac{1}{3} \left( \boldsymbol{F} : \overline{\overline{\mathbb{A}}} : \boldsymbol{F} \right) \boldsymbol{F}^{-T} \right)\\A_{p,\overline{J}} &= A_{\overline{J}, p} = -1\end{aligned}\end{align} \]

with

\[\overline{\overline{\mathbb{A}}} = \left(\frac{\overline{J}}{J}\right)^{1/3} \frac{\partial^2 \psi}{\partial \overline{\boldsymbol{F}} \partial \overline{\boldsymbol{F}}} \left(\frac{\overline{J}}{J}\right)^{1/3}\]

as well as

\[\boldsymbol{P}' = \boldsymbol{P} - p J \boldsymbol{F}^{-T}\]

Parameters

material (Material) – A material definition with gradient and hessian methods.

fun_P

Method for gradient evaluation

Type

function

fun_A

Method for hessian evaluation

Type

function

detF

Determinant of deformation gradient

Type

ndarray

iFT

Transpose of inverse of the deformation gradient

Type

ndarray

Fb

Determinant-modified deformation gradient

Type

ndarray

Pb

First Piola-Kirchhoff stress tensor (in determinant-modified framework)

Type

ndarray

Pbb

Determinant-modification multiplied by Pb

Type

ndarray

PbbF

Double-dot product of Pb and the deformation gradient

Type

ndarray

gradient(extract)

List of variations of total potential energy w.r.t displacements, pressure and volume ratio.

δ_u(Π_int) = ∫_V (∂ψ/∂F + p cof(F)) : δF dV
δ_p(Π_int) = ∫_V (det(F) - J) δp dV
δ_J(Π_int) = ∫_V (∂U/∂J - p) δJ dV

Parameters

extract (list of ndarray) – List of extracted field values with Deformation gradient F as first, the hydrostatic pressure p as second and the volume ratio J as third item.

Returns

List of gradients w.r.t. the input variables F, p and J

Return type

list of ndarrays

hessian(extract)

List of linearized variations of total potential energy w.r.t displacements, pressure and volume ratio (these expressions are symmetric; A_up = A_pu if derived from a total potential energy formulation). List entries have to be arranged as a flattened list from the upper triangle blocks:

Δ_u(δ_u(Π_int)) = ∫_V δF : (∂²ψ/(∂F∂F) + p ∂cof(F)/∂F) : ΔF dV
Δ_p(δ_u(Π_int)) = ∫_V δF : J cof(F) Δp dV
Δ_J(δ_u(Π_int)) = ∫_V δF :  ∂²ψ/(∂F∂J) ΔJ dV
Δ_p(δ_p(Π_int)) = ∫_V δp 0 Δp dV
Δ_J(δ_p(Π_int)) = ∫_V δp (-1) ΔJ dV
Δ_J(δ_J(Π_int)) = ∫_V δJ ∂²ψ/(∂J∂J) ΔJ dV

[[0 1 2],
 [  3 4],
 [    5]] --> [0 1 2 3 4 5]

Parameters

extract (list of ndarray) – List of extracted field values with Deformation gradient F as first, the hydrostatic pressure p as second and the volume ratio J as third item.

Returns

List of hessians in upper triangle order

Return type

list of ndarrays

class felupe.constitution.LineChange(parallel=False)

Line Change.

\[d\boldsymbol{x} = \boldsymbol{F} d\boldsymbol{X}\]

Gradient:

\[\frac{\partial \boldsymbol{F}}{\partial \boldsymbol{F}} = \boldsymbol{I} \overset{ik}{\otimes} \boldsymbol{I}\]

function(extract)

Line change.

Parameters

extract (list of ndarray) – List of extracted field values with Deformation gradient as first item.

Returns

F – Deformation gradient

Return type

ndarray

gradient(extract, parallel=None)

Gradient of line change.

Parameters

extract (list of ndarray) – List of extracted field values with Deformation gradient as first item.

Returns

Gradient of line change

Return type

ndarray

class felupe.constitution.AreaChange(parallel=False)

Area Change.

\[d\boldsymbol{a} = J \boldsymbol{F}^{-T} d\boldsymbol{A}\]

Gradient:

\[\frac{\partial J \boldsymbol{F}^{-T}}{\partial \boldsymbol{F}} = J \left( \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T} - \boldsymbol{F}^{-T} \overset{il}{\otimes} \boldsymbol{F}^{-T} \right)\]

function(extract, N=None, parallel=None)

Area change.

Parameters

• extract (list of ndarray) – List of extracted field values with Deformation gradient as first item.

• N (ndarray or None, optional) – Area normal vector (default is None)

Returns

Cofactor matrix of the deformation gradient

Return type

ndarray

gradient(extract, N=None, parallel=None)

Gradient of area change.

Parameters

• extract (list of ndarray) – List of extracted field values with Deformation gradient as first item.

• N (ndarray or None, optional) – Area normal vector (default is None)

Returns

Gradient of cofactor matrix of the deformation gradient

Return type

ndarray

class felupe.constitution.VolumeChange(parallel=False)

Volume Change.

\[d\boldsymbol{v} = \text{det}(\boldsymbol{F}) d\boldsymbol{V}\]

Gradient and hessian (equivalent to gradient of area change):

\[ \begin{align}\begin{aligned}\frac{\partial J}{\partial \boldsymbol{F}} &= J \boldsymbol{F}^{-T}\\\frac{\partial^2 J}{\partial \boldsymbol{F}\ \partial \boldsymbol{F}} &= J \left( \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T} - \boldsymbol{F}^{-T} \overset{il}{\otimes} \boldsymbol{F}^{-T} \right)\end{aligned}\end{align} \]

function(extract)

Gradient of volume change.

Parameters

extract (list of ndarray) – List of extracted field values with Deformation gradient as first item.

Returns

J – Determinant of the deformation gradient

Return type

ndarray

gradient(extract)

Gradient of volume change.

Parameters

F (ndarray) – Deformation gradient

Returns

Gradient of the determinant of the deformation gradient

Return type

ndarray

hessian(extract, parallel=None)

Hessian of volume change.

Parameters

extract (list of ndarray) – List of extracted field values with Deformation gradient as first item.

Returns

Hessian of the determinant of the deformation gradient

Return type

ndarray