Constitution#

class felupe.NeoHooke(mu=None, bulk=None, parallel=False)[source]#

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(x, mu=None, bulk=None)[source]#

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

Parameters

  • x (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(x, mu=None, bulk=None)[source]#

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

Parameters

  • x (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(x, mu=None, bulk=None)[source]#

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

Parameters

  • x (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.OgdenRoxburgh(material, r, m, beta)[source]#

A Pseudo-Elastic material formulation for an isotropic treatment of the load-history dependent Mullins-softening of rubber-like materials.

\[ \begin{align}\begin{aligned}\eta(W, W_{max}) &= 1 - \frac{1}{r} erf\left( \frac{W_{max} - W}{m + \beta~W_{max}} \right)\\\boldsymbol{P} &= \eta \frac{\partial \psi}{\partial \boldsymbol{F}}\\\mathbb{A} &= \frac{\partial^2 \psi}{\partial \boldsymbol{F} \partial \boldsymbol{F}} + \frac{\partial \eta}{\partial \psi} \frac{\partial \psi}{\partial \boldsymbol{F}} \otimes \frac{\partial \psi}{\partial \boldsymbol{F}}\end{aligned}\end{align} \]
gradient(x)[source]#
hessian(x)[source]#
class felupe.LinearElastic(E=None, nu=None)[source]#

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(x, E=None, nu=None)[source]#

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

Parameters

  • x (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(x=None, E=None, nu=None, shape=(1, 1))[source]#

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

Parameters

  • x (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, ...), optional) – Tuple with shape of the trailing axes (default is (1, 1))

Returns

elasticity tensor (3x3x3x3)

Return type

ndarray

class felupe.LinearElasticPlaneStress(E, nu)[source]#

Plane-stress isotropic linear-elastic material formulation.

Parameters

  • E (float) – Young’s modulus.

  • nu (float) – Poisson ratio.

gradient(x, E=None, nu=None)[source]#

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

Parameters

  • x (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(x=None, E=None, nu=None, shape=(1, 1))[source]#

Evaluate the elasticity tensor from the deformation gradient.

Parameters

  • x (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, ...), optional) – Tuple with shape of the trailing axes (default is (1, 1))

Returns

In-plane components of elasticity tensor (2x2x2x2)

Return type

ndarray

strain(x, E=None, nu=None)[source]#

Evaluate the strain tensor from the deformation gradient.

Parameters

  • x (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(x, E=None, nu=None)[source]#

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

Parameters

  • x (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.LinearElasticPlaneStrain(E, nu)[source]#

Plane-strain isotropic linear-elastic material formulation.

Parameters

  • E (float) – Young’s modulus.

  • nu (float) – Poisson ratio.

gradient(x, E=None, nu=None)[source]#

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

Parameters

  • x (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(x, E=None, nu=None)[source]#

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

Parameters

  • x (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(x, E=None, nu=None)[source]#

Evaluate the strain tensor from the deformation gradient.

Parameters

  • x (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(x, E=None, nu=None)[source]#

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

Parameters

  • x (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.LinearElasticPlasticIsotropicHardening(E, nu, sy, K)[source]#

Linear-elastic-plastic material formulation with linear isotropic hardening (return mapping algorithm).

extract(x)#

Extract the input and evaluate strains, stresses and state variables.

gradient(x)#
hessian(x)#
class felupe.ThreeFieldVariation(material, parallel=False)[source]#

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

class UserMaterial(stress, elasticity, nstatevars=0, **kwargs)#

A user-defined material definition with given functions for the (first Piola-Kirchhoff) stress tensor and the according fourth-order elasticity tensor. Both functions take a list of the deformation gradient and optional state variables as the first input argument. The stress-function also returns the updated state variables.

Take this code-block as template:

def stress(x, **kwargs):
    "First Piola-Kirchhoff stress tensor."

    # extract variables
    F, statevars = x[0], x[-1]

    # user code for (first Piola-Kirchhoff) stress tensor
    P = None

    # update state variables
    statevars_new = None

    return [P, statevars_new]

def elasticity(x, **kwargs):
    "Fourth-order elasticity tensor."

    # extract variables
    F, statevars = x[0], x[-1]

    # user code for fourth-order elasticity tensor
    # according to the (first Piola-Kirchhoff) stress tensor
    dPdF = None

    return [dPdF]

umat = UserMaterial(stress, elasticity, **kwargs)
Members

Undoc-members

Inherited-members

gradient(x)[source]#

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

x (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(x)[source]#

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.UserMaterialStrain(material, dim=3, statevars=(0,), **kwargs)[source]#

A strain-based user-defined material definition with a given functions for the stress tensor and the (fourth-order) elasticity tensor.

Take this code-block from the linear-elastic material formulation

from felupe.math import identity, cdya, dya, trace

def linear_elastic(dε, εn, σn, ζn, λ, μ, **kwargs):
    '''3D linear-elastic material formulation.

    Arguments
    ---------
    dε : ndarray
        Incremental strain tensor.
    εn : ndarray
        Old strain tensor.
    σn : ndarray
        Old stress tensor.
    ζn : ndarray
        Old state variables.
    λ : float
        First Lamé-constant.
    μ : float
        Second Lamé-constant (shear modulus).
    '''

    # change of stress due to change of strain
    I = identity(dε)
    dσ = 2 * μ * dε + λ * trace(dε) * I

    # update stress and evaluate elasticity tensor
    σ = σn + dσ
    dσdε = 2 * μ * cdya(I, I) + λ * dya(I, I)

    # update state variables (not used here)
    ζ = ζn

    return dσdε, σ, ζ

umat = UserMaterialStrain(material=linear_elastic, μ=1, λ=2)

or this minimal header as template:

def fun(dε, εn, σn, ζn, **kwargs):
    return dσdε, σ, ζ

umat = UserMaterialStrain(material=fun, **kwargs)
extract(x)[source]#

Extract the input and evaluate strains, stresses and state variables.

gradient(x)[source]#
hessian(x)[source]#
felupe.constitution.linear_elastic(dε, εn, σn, ζn, λ, μ, **kwargs)[source]#

3D linear-elastic material formulation.

  1. Given state in point x (σn, ζn=[εpn, αn]) (valid).

  2. Given strain increment dε, so that ε = εn + dε.

  3. Evaluation of the stress σ and the algorithmic consistent tangent modulus dσdε.

    dσdε = λ 1 ⊗ 1 + 2μ 1 ⊙ 1

    σ = σn + dσdε : dε

Parameters

  • dε (ndarray) – Strain increment.

  • εn (ndarray) – Old strain tensor.

  • σn (ndarray) – Old stress tensor.

  • ζn (list) – List of old state variables.

  • λ (float) – First Lamé-constant.

  • μ (float) – Second Lamé-constant (shear modulus).

felupe.constitution.linear_elastic_plastic_isotropic_hardening(dε, εn, σn, ζn, λ, μ, σy, K, **kwargs)[source]#

Linear-elastic-plastic material formulation with linear isotropic hardening (return mapping algorithm).

  1. Given state in point x (σn, ζn=[εpn, αn]) (valid).

  2. Given strain increment dε, so that ε = εn + dε.

  3. Evaluation of the hypothetic trial state:

    dσdε = λ 1 ⊗ 1 + 2μ 1 ⊙ 1

    σ = σn + dσdε : dε

    s = dev(σ)

    εp = εpn

    α = αn

    f = ||s|| - sqrt(2/3) (σy + K α)

  4. If f ≤ 0, then elastic step:

    Set y = yn + dy, y=(σ, ζ=[εp, α]),

    algorithmic consistent tangent modulus dσdε.

    Else:

    dγ = f / (2μ + 2/3 K)

    n = s / ||s||

    σ = σ - 2μ dγ n

    εp = εpn + dγ n

    α = αn + sqrt(2 / 3) dγ

    Algorithmic consistent tangent modulus:

    dσdε = dσdε - 2μ / (1 + K / 3μ) n ⊗ n - 2μ dγ / ||s|| ((2μ 1 ⊙ 1 - 1/3 1 ⊗ 1) - 2μ n ⊗ n)

Parameters

  • dε (ndarray) – Strain increment.

  • εn (ndarray) – Old strain tensor.

  • σn (ndarray) – Old stress tensor.

  • ζn (list) – List of old state variables.

  • λ (float) – First Lamé-constant.

  • μ (float) – Second Lamé-constant (shear modulus).

  • σy (float) – Initial yield stress.

  • K (float) – Isotropic hardening modulus.

class felupe.LineChange(parallel=False)[source]#

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)[source]#

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)[source]#

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.AreaChange(parallel=False)[source]#

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)[source]#

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)[source]#

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.VolumeChange(parallel=False)[source]#

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)[source]#

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)[source]#

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)[source]#

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