Constitution#
This module provides constitutive material
formulations. In FElupe, a constitutive material definition, or so-called umat
(user material), is a class with methods for evaluating gradients and hessians of the strain energy density function with respect to the defined fields in the field container, where the gradient of the first (displacement) field is passed as the deformation gradient. For all following fields, the field values (no gradients) are provided. An attribute x=[np.zeros(statevars_shape)]
has to be added to the class to define the shape of optional state variables. For reasons of performance, FElupe passes the field gradients and values all at once, e.g. the deformation gradient is of shape (3, 3, q, c)
, where q
refers to the number of quadrature points per cell and c
to the number of cells. These last two axes are the so-called trailing axes. Math-functions from felupe.math all support the operation on trailing axes. The constitutive material definition class should be inherited from ConstitutiveMaterial
in order to provide force-stretch curves for elementary deformations. Take this code-block as a template for a two-field \((\boldsymbol{u}, p)\) formulation with the old displacement gradient as a state variable:
import numpy as np
import felupe as fem
# math-functions which support trailing axes
from felupe.math import det, dya, identity, transpose, inv
class MyMaterialFormulation(fem.ConstitutiveMaterial):
def __init__(self):
# provide the shape of state variables without trailing axes
# values are ignored - state variables are always initiated with zeros
self.x = [np.zeros((3, 3))]
def gradient(x):
"Gradients of the strain energy density function."
# extract variables
F, p, statevars = x[0], x[1], x[-1]
# user code
dWdF = None # first Piola-Kirchhoff stress tensor
dWdp = None
# update state variables
# example: the displacement gradient
statevars_new = F - identity(F)
return [dWdF, dWdp, statevars_new]
def hessian(x, **kwargs):
"Hessians of the strain energy density function."
# extract variables
F, p, statevars = x[0], x[1], x[-1]
# user code
d2WdFdF = None # fourth-order elasticity tensor
d2WdFdp = None
d2Wdpdp = None
# upper-triangle items of the hessian
return [d2WdFdF, d2WdFdp, d2Wdpdp]
umat = MyMaterialFormulation()
There are many different pre-defined constitutive material formulations available, including definitions for linear-elasticity, small-strain plasticity, hyperelasticity or pseudo-elasticity. The generation of user materials may be simplified when using frameworks for user-defined functions, like hyperelasticity (with automatic differentiation) or a small-strain based framework with state variables. The most general case is given by a framework with functions for the evaluation of stress and elasticity tensors in terms of the deformation gradient.
View Force-Stretch Curves on Elementary Deformations
Base class for constitutive materials. |
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Create views on normal force per undeformed area vs. |
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Create views on normal force per undeformed area vs. |
Merge Constitutive Materials
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A composite material with two constitutive materials merged. |
Linear-Elasticity
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Isotropic linear-elastic material formulation. |
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Plane-strain isotropic linear-elastic material formulation. |
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Plane-stress isotropic linear-elastic material formulation. |
Isotropic linear-elastic material formulation. |
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Linear-elastic material formulation suitable for large-rotation analyses based on the nearly-incompressible Neo-Hookean material formulation. |
Plasticity
Linear-elastic-plastic material formulation with linear isotropic hardening (return mapping algorithm). |
Hyperelasticity
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Nearly-incompressible isotropic hyperelastic Neo-Hookean material formulation. |
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Compressible isotropic hyperelastic Neo-Hookean material formulation. |
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Neo-Hookean material formulation with deactivated shear modulus. |
Hyperelastic Three-Field-Formulations \((\boldsymbol{u}, p, \bar{J})\)
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Hu-Washizu hydrostatic-volumetric selective \((\boldsymbol{u},p,J)\) - three-field variation for nearly- incompressible material formulations. |
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A nearly-incompressible material formulation to augment the distortional part of the strain energy function by a volumetric part and a constraint equation. |
Pseudo-Elasticity (Isotropic Damage)
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Ogden-Roxburgh Pseudo-Elastic material formulation for an isotropic treatment of the load-history dependent Mullins-softening of rubber-like materials. |
Hyperelastic User-Materials with Automatic Differentation
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A hyperelastic material definition with a given function for the strain energy density function per unit undeformed volume with Automatic Differentiation provided by |
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A user-defined material definition with a given function for the partial derivative of the strain energy function w.r.t. |
Material Model Formulations (Strain Energy Functions) for Hyperelastic
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Strain energy function of the isotropic hyperelastic Saint-Venant Kirchhoff material formulation. |
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Strain energy function of the isotropic hyperelastic Neo-Hookean material formulation. |
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Strain energy function of the isotropic hyperelastic Mooney-Rivlin material formulation. |
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Strain energy function of the isotropic hyperelastic Yeoh material formulation. |
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Strain energy function of the isotropic hyperelastic Third-Order-Deformation material formulation. |
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Strain energy function of the isotropic hyperelastic Ogden material formulation. |
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Strain energy function of the isotropic hyperelastic Arruda-Boyce material formulation. |
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Strain energy function of the isotropic hyperelastic Extended Tube [1] material formulation. |
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Strain energy function of the Van der Waals material formulation. |
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Finite strain viscoelastic material formulation. |
(Small) Strain-based User Materials
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A strain-based user-defined material definition with a given function for the stress tensor and the (fourth-order) elasticity tensor. |
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3D linear-elastic material formulation to be used in |
Linear-elastic-plastic material formulation with linear isotropic hardening (return mapping algorithm) to be used in |
Deformation-Gradient-based User Materials
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A user-defined material definition with given functions for the (first Piola-Kirchhoff) stress tensor \(\boldsymbol{P}\), optional constraints on additional fields (e.g. \(p\) and \(J\)), updated state variables \(\boldsymbol{\zeta}\) as well as the according fourth-order elasticity tensor \(\mathbb{A}\) and the linearizations of the constraint equations. |
Kinematics
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Line Change. |
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Area Change. |
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Volume Change. |
Detailed API Reference
- class felupe.ConstitutiveMaterial[source]#
Base class for constitutive materials.
- plot(incompressible=False, **kwargs)[source]#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)[source]#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)[source]#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- class felupe.ViewMaterial(umat, ux=array([0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1., 1.05, 1.1, 1.15, 1.2, 1.25, 1.3, 1.35, 1.4, 1.45, 1.5, 1.55, 1.6, 1.65, 1.7, 1.75, 1.8, 1.85, 1.9, 1.95, 2., 2.05, 2.1, 2.15, 2.2, 2.25, 2.3, 2.35, 2.4, 2.45, 2.5]), ps=array([1., 1.05, 1.1, 1.15, 1.2, 1.25, 1.3, 1.35, 1.4, 1.45, 1.5, 1.55, 1.6, 1.65, 1.7, 1.75, 1.8, 1.85, 1.9, 1.95, 2., 2.05, 2.1, 2.15, 2.2, 2.25, 2.3, 2.35, 2.4, 2.45, 2.5]), bx=array([1., 1.05, 1.1, 1.15, 1.2, 1.25, 1.3, 1.35, 1.4, 1.45, 1.5, 1.55, 1.6, 1.65, 1.7, 1.75]), statevars=None)[source]#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
umat (class) – A class with methods for the gradient and hessian of the strain energy density function w.r.t. the deformation gradient. See
Material
for further details.ux (ndarray, optional) – Array with stretches for uniaxial tension/compression. Default is
linsteps([0.7, 2.5], num=36)
.ps (ndarray, optional) – Array with stretches for planar shear. Default is
linsteps([1.0, 2.5], num=30)
.bx (ndarray, optional) – Array with stretches for equi-biaxial tension. Default is
linsteps([1.0, 1.75], num=15)`
.statevars (ndarray or None, optional) – Array with state variables (default is None). If None, the state variables are assumed to be initially zero.
Examples
>>> import felupe as fem >>> >>> umat = fem.OgdenRoxburgh(fem.NeoHooke(mu=1, bulk=2), r=3, m=1, beta=0) >>> view = fem.ViewMaterial( >>> umat, >>> ux=fem.math.linsteps([1, 1.5, 1, 2, 1, 2.5, 1], num=15), >>> ps=None, >>> bx=None, >>> ) >>> ax = view.plot(show_title=True, show_kwargs=True)
- biaxial(stretches=None)[source]#
Normal force per undeformed area vs. stretch curve for a equi-biaxial incompressible deformation.
- Parameters:
stretches (ndarray or None, optional) – Array with stretches at which the forces are evaluated (default is None). If None, the stretches from initialization are used.
- Returns:
3-tuple with array of stretches and array of forces and the label.
- Return type:
tuple of ndarray
- evaluate()#
Evaluate normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension. A load case is not included if its array of stretches (attribute
ux
,ps
orbx
) is None.- Returns:
List with 3-tuple of stretch and force arrays and the label string for each load case.
- Return type:
list of 3-tuple
- planar(stretches=None)[source]#
Normal force per undeformed area vs. stretch curve for a planar shear incompressible deformation.
- Parameters:
stretches (ndarray or None, optional) – Array with stretches at which the forces are evaluated (default is None). If None, the stretches from initialization are used.
- Returns:
3-tuple with array of stretches and array of forces and the label.
- Return type:
tuple of ndarray
- plot(ax=None, show_title=True, show_kwargs=True, **kwargs)#
Plot normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension.
- uniaxial(stretches=None)[source]#
Normal force per undeformed area vs. stretch curve for a uniaxial deformation.
- Parameters:
stretches (ndarray or None, optional) – Array with stretches at which the forces are evaluated (default is None). If None, the stretches from initialization are used.
- Returns:
3-tuple with array of stretches and array of forces and the label.
- Return type:
tuple of ndarray
- class felupe.ViewMaterialIncompressible(umat, ux=array([0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1., 1.05, 1.1, 1.15, 1.2, 1.25, 1.3, 1.35, 1.4, 1.45, 1.5, 1.55, 1.6, 1.65, 1.7, 1.75, 1.8, 1.85, 1.9, 1.95, 2., 2.05, 2.1, 2.15, 2.2, 2.25, 2.3, 2.35, 2.4, 2.45, 2.5]), ps=array([1., 1.05, 1.1, 1.15, 1.2, 1.25, 1.3, 1.35, 1.4, 1.45, 1.5, 1.55, 1.6, 1.65, 1.7, 1.75, 1.8, 1.85, 1.9, 1.95, 2., 2.05, 2.1, 2.15, 2.2, 2.25, 2.3, 2.35, 2.4, 2.45, 2.5]), bx=array([1., 1.05, 1.1, 1.15, 1.2, 1.25, 1.3, 1.35, 1.4, 1.45, 1.5, 1.55, 1.6, 1.65, 1.7, 1.75]), statevars=None)[source]#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
umat (class) – A class with methods for the gradient and hessian of the strain energy density function w.r.t. the deformation gradient. See
Material
for further details.ux (ndarray, optional) – Array with stretches for incompressible uniaxial tension/compression. Default is
linsteps([0.7, 2.5], num=36)
.ps (ndarray, optional) – Array with stretches for incompressible planar shear. Default is
linsteps([1, 2.5], num=30)
.bx (ndarray, optional) – Array with stretches for incompressible equi-biaxial tension. Default is
linsteps([1, 1.75], num=15)
.statevars (ndarray or None, optional) – Array with state variables (default is None). If None, the state variables are assumed to be initially zero.
Examples
>>> import felupe as fem >>> >>> umat = fem.Hyperelastic(fem.extended_tube, Gc=0.2, Ge=0.2, beta=0.2, delta=0.1) >>> preview = fem.ViewMaterialIncompressible(umat) >>> ax = preview.plot(show_title=True, show_kwargs=True)
>>> umat = fem.OgdenRoxburgh(fem.NeoHooke(mu=1), r=3, m=1, beta=0) >>> view = fem.ViewMaterialIncompressible( >>> umat, >>> ux=fem.math.linsteps([1, 1.5, 1, 2, 1, 2.5, 1], num=15), >>> ps=None, >>> bx=None, >>> ) >>> ax = view.plot(show_title=True, show_kwargs=True)
- biaxial(stretches=None)[source]#
Normal force per undeformed area vs. stretch curve for a equi-biaxial incompressible deformation.
- Parameters:
stretches (ndarray or None, optional) – Array with stretches at which the forces are evaluated (default is None). If None, the stretches from initialization are used.
- Returns:
3-tuple with array of stretches and array of forces and the label.
- Return type:
tuple of ndarray
- evaluate()#
Evaluate normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension. A load case is not included if its array of stretches (attribute
ux
,ps
orbx
) is None.- Returns:
List with 3-tuple of stretch and force arrays and the label string for each load case.
- Return type:
list of 3-tuple
- planar(stretches=None)[source]#
Normal force per undeformed area vs. stretch curve for a planar shear incompressible deformation.
- Parameters:
stretches (ndarray or None, optional) – Array with stretches at which the forces are evaluated (default is None). If None, the stretches from initialization are used.
- Returns:
3-tuple with array of stretches and array of forces and the label.
- Return type:
tuple of ndarray
- plot(ax=None, show_title=True, show_kwargs=True, **kwargs)#
Plot normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension.
- uniaxial(stretches=None)[source]#
Normal force per undeformed area vs. stretch curve for a uniaxial incompressible deformation.
- Parameters:
stretches (ndarray or None, optional) – Array with stretches at which the forces are evaluated (default is None). If None, the stretches from initialization are used.
- Returns:
3-tuple with array of stretches and array of forces and the label.
- Return type:
tuple of ndarray
- class felupe.CompositeMaterial(material, other_material)[source]#
A composite material with two constitutive materials merged. State variables are only considered for the first material.
- Parameters:
material (ConstitutiveMaterial) – First constitutive material.
other_material (ConstitutiveMaterial) – Second constitutive material.
Notes
Warning
Do not merge two constitutive materials with the same keys of material parameters. In this case, the values of these material parameters are taken from the first constitutive material.
Examples
>>> import felupe as fem >>> >>> nh = fem.NeoHooke(mu=1.0) >>> vol = fem.Volumetric(bulk=2.0) >>> umat = nh & vol >>> ax = umat.plot()
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- class felupe.NeoHooke(mu=None, bulk=None, parallel=False)[source]#
Nearly-incompressible isotropic hyperelastic Neo-Hookean 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.
- Parameters:
Notes
Note
At least one of the two material parameters must not be None.
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.
\[ \begin{align}\begin{aligned}\mathbb{A} &= \mathbb{A}' + \mathbb{A}_{U}\\\mathbb{A}' &= J^{-2/3} \left(\boldsymbol{I} \overset{ik}{\otimes} \boldsymbol{I} - \frac{1}{3} \left( \boldsymbol{F} \otimes \boldsymbol{F}^{-T} + \boldsymbol{F}^{-T} \otimes \boldsymbol{F} \right ) + \frac{1}{9} (\boldsymbol{F} : \boldsymbol{F}) \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T} \right)\\\mathbb{A}_{U} &= K J \left( (2J - 1) \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T} - (J - 1) \boldsymbol{F}^{-T} \overset{il}{\otimes} \boldsymbol{F}^{-T} \right)\end{aligned}\end{align} \]Examples
>>> import felupe as fem >>> >>> umat = fem.NeoHooke(mu=1.0, bulk=2.0) >>> ax = umat.plot()
- function(x, mu=None, bulk=None)[source]#
Strain energy density function per unit undeformed volume of the Neo-Hookean material formulation.
- gradient(x, mu=None, bulk=None, out=None)[source]#
Gradient of the strain energy density function per unit undeformed volume of the Neo-Hookean material formulation.
- hessian(x, mu=None, bulk=None, out=None)[source]#
Hessian of the strain energy density function per unit undeformed volume of the Neo-Hookean material formulation.
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- class felupe.NeoHookeCompressible(mu=None, lmbda=None, parallel=False)[source]#
Compressible isotropic hyperelastic Neo-Hookean material formulation. The strain energy density function of the Neo-Hookean material formulation is a linear function of the trace of the right Cauchy-Green deformation tensor.
Notes
\[ \begin{align}\begin{aligned}\psi &= \psi(\boldsymbol{C})\\\psi(\boldsymbol{C}) &= \frac{\mu}{2} \text{tr}(\boldsymbol{C}) - \mu \ln(J) + \frac{\lambda}{2} \ln(J)^2\end{aligned}\end{align} \]with
\[J = \text{det}(\boldsymbol{F})\]The first Piola-Kirchhoff stress tensor is evaluated as the gradient of the strain energy density function.
\[ \begin{align}\begin{aligned}\boldsymbol{P} &= \frac{\partial \psi}{\partial \boldsymbol{F}}\\\boldsymbol{P} &= \mu \left( \boldsymbol{F} - \boldsymbol{F}^{-T} \right) + \lambda \ln(J) \boldsymbol{F}^{-T}\end{aligned}\end{align} \]The hessian of the strain energy density function enables the corresponding elasticity tensor.
\[ \begin{align}\begin{aligned}\mathbb{A} &= \frac{\partial^2 \psi}{\partial \boldsymbol{F}\ \partial \boldsymbol{F}}\\\mathbb{A} &= \mu \boldsymbol{I} \overset{ik}{\otimes} \boldsymbol{I} + \left(\mu - \lambda \ln(J) \right) \boldsymbol{F}^{-T} \overset{il}{\otimes} \boldsymbol{F}^{-T} + \lambda \boldsymbol{F}^{-T} {\otimes} \boldsymbol{F}^{-T}\end{aligned}\end{align} \]Examples
>>> import felupe as fem >>> >>> umat = fem.NeoHookeCompressible(mu=1.0, lmbda=2.0) >>> ax = umat.plot()
- function(x, mu=None, lmbda=None)[source]#
Strain energy density function per unit undeformed volume of the Neo-Hookean material formulation.
- gradient(x, mu=None, lmbda=None, out=None)[source]#
Gradient of the strain energy density function per unit undeformed volume of the Neo-Hookean material formulation.
- Parameters:
- hessian(x, mu=None, lmbda=None, out=None)[source]#
Hessian of the strain energy density function per unit undeformed volume of the Neo-Hookean material formulation.
- Parameters:
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- class felupe.Volumetric(bulk, parallel=False)[source]#
Neo-Hookean material formulation with deactivated shear modulus.
- function(x, mu=None, bulk=None)#
Strain energy density function per unit undeformed volume of the Neo-Hookean material formulation.
- gradient(x, mu=None, bulk=None, out=None)#
Gradient of the strain energy density function per unit undeformed volume of the Neo-Hookean material formulation.
- hessian(x, mu=None, bulk=None, out=None)#
Hessian of the strain energy density function per unit undeformed volume of the Neo-Hookean material formulation.
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- class felupe.OgdenRoxburgh(material, r, m, beta)[source]#
Ogden-Roxburgh Pseudo-Elastic material formulation for an isotropic treatment of the load-history dependent Mullins-softening of rubber-like materials.
- Parameters:
material (NeoHooke, Hyperelastic, Material or MaterialAD) – An isotropic hyperelastic (user) material definition.
r (float) – Reciprocal value of the maximum relative amount of softening. i.e.
r=3
means the shear modulus of the base material scales down from \(1\) (no softening) to \(1 - 1/3 = 2/3\) (maximum softening).m (float) – The initial Mullins softening modulus.
beta (float) – Maximum deformation-dependent part of the Mullins softening modulus.
Notes
\[ \begin{align}\begin{aligned}\eta(\psi, \psi_{max}) &= 1 - \frac{1}{r} erf\left( \frac{\psi_{max} - \psi} {m + \beta~\psi_{max}} \right)\\\boldsymbol{P} &= \eta \frac{\partial \psi}{\partial \boldsymbol{F}}\\\mathbb{A} &= \frac{\partial^2 \psi}{\partial \boldsymbol{F} \partial \boldsymbol{F}} + \frac{\partial \eta}{\partial \psi} \frac{\partial \psi} {\partial \boldsymbol{F}} \otimes \frac{\partial \psi}{\partial \boldsymbol{F}}\end{aligned}\end{align} \]Examples
>>> import felupe as fem >>> >>> neo_hooke = fem.NeoHooke(mu=1.0) >>> umat = fem.OgdenRoxburgh(material=neo_hooke, r=3.0, m=1.0, beta=0.0) >>> >>> ax = umat.plot( >>> ux=fem.math.linsteps([1, 1.5, 1, 2, 1, 2.5, 1], num=15), >>> ps=None, >>> bx=None, >>> incompressible=True, >>> )
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- class felupe.LinearElastic(E=None, nu=None)[source]#
Isotropic linear-elastic material formulation.
Notes
\[\begin{split}\begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{12} \\ \sigma_{23} \\ \sigma_{31} \end{bmatrix} = \frac{E}{(1+\nu)(1-2\nu)}\begin{bmatrix} 1-\nu & \nu & \nu & 0 & 0 & 0\\ \nu & 1-\nu & \nu & 0 & 0 & 0\\ \nu & \nu & 1-\nu & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{1-2\nu}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1-2\nu}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1-2\nu}{2} \end{bmatrix} \cdot \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2 \varepsilon_{12} \\ 2 \varepsilon_{23} \\ 2 \varepsilon_{31} \end{bmatrix}\end{split}\]with the strain tensor
\[\boldsymbol{\varepsilon} = \frac{1}{2} \left( \frac{\partial \boldsymbol{u}} {\partial \boldsymbol{X}} + \left( \frac{\partial \boldsymbol{u}} {\partial \boldsymbol{X}} \right)^T \right)\]Examples
>>> import felupe as fem >>> >>> umat = fem.LinearElastic(E=1, nu=0.3) >>> ax = umat.plot()
- gradient(x, E=None, nu=None)[source]#
Evaluate the stress tensor (as a function of the deformation gradient).
- 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 \(boldsymbol{F}\) (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 (tuple of int, optional) – Tuple with shape of the trailing axes (default is (1, 1)).
- Returns:
elasticity tensor (3x3x3x3)
- Return type:
ndarray
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- class felupe.LinearElasticLargeStrain(E=None, nu=None, parallel=False)[source]#
Linear-elastic material formulation suitable for large-rotation analyses based on the nearly-incompressible Neo-Hookean material formulation.
See also
NeoHookeCompressible
Compressible isotropic hyperelastic Neo-Hooke material formulation.
Examples
>>> import felupe as fem >>> >>> umat = fem.LinearElasticLargeStrain(E=1.0, nu=0.3) >>> ax = umat.plot()
- function(x, E=None, nu=None)[source]#
Evaluate the strain energy (as a function of the deformation gradient).
- gradient(x, E=None, nu=None)[source]#
Evaluate the stress tensor (as a function of the deformation gradient).
- hessian(x, E=None, nu=None)[source]#
Evaluate the elasticity tensor (as a function of the deformation gradient).
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- class felupe.constitution.LinearElasticTensorNotation(E=None, nu=None, parallel=False)[source]#
Isotropic linear-elastic material formulation.
Notes
\[ \begin{align}\begin{aligned}\boldsymbol{\sigma} &= 2 \mu \ \boldsymbol{\varepsilon} + \gamma \ \text{tr}(\boldsymbol{\varepsilon}) \ \boldsymbol{I}\\\frac{\boldsymbol{\partial \sigma}}{\partial \boldsymbol{\varepsilon}} &= 2 \mu \ \boldsymbol{I} \odot \boldsymbol{I} + \gamma \ \boldsymbol{I} \otimes \boldsymbol{I}\end{aligned}\end{align} \]with the strain tensor
\[\boldsymbol{\varepsilon} = \frac{1}{2} \left( \frac{\partial \boldsymbol{u}} {\partial \boldsymbol{X}} + \left( \frac{\partial \boldsymbol{u}} {\partial \boldsymbol{X}} \right)^T \right)\]Examples
>>> import felupe as fem >>> >>> umat = fem.constitution.LinearElasticTensorNotation(E=1, nu=0.3) >>> ax = umat.plot()
- gradient(x, E=None, nu=None)[source]#
Evaluate the stress tensor (as a function of the deformation gradient).
- 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) – List with Deformation gradient \(boldsymbol{F}\) (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
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- class felupe.LinearElasticPlaneStress(E, nu)[source]#
Plane-stress isotropic linear-elastic material formulation.
- gradient(x, E=None, nu=None)[source]#
Evaluate the 2d-stress tensor from the deformation gradient.
- Parameters:
- 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 \(boldsymbol{F}\) as first item (default is None)-
E (float, optional) – Young’s modulus (default is None).
nu (float, optional) – Poisson ratio (default is None).
shape (tuple of int, optional) – Tuple with shape of the trailing axes (default is (1, 1)).
- Returns:
In-plane components of elasticity tensor (2x2x2x2).
- Return type:
ndarray
- class felupe.LinearElasticPlaneStrain(E, nu)[source]#
Plane-strain isotropic linear-elastic material formulation.
- gradient(x, E=None, nu=None)[source]#
Evaluate the 2d-stress tensor from the deformation gradient.
- Parameters:
- 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:
- Returns:
In-plane components of elasticity tensor (2x2x2x2)
- Return type:
ndarray
- class felupe.LinearElasticPlasticIsotropicHardening(E, nu, sy, K)[source]#
Linear-elastic-plastic material formulation with linear isotropic hardening (return mapping algorithm).
- Parameters:
See also
MaterialStrain
A strain-based user-defined material definition with a given function for the stress tensor and the (fourth-order) elasticity tensor.
linear_elastic_plastic_isotropic_hardening
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)#
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- 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 \(\boldsymbol{A}_{(\bullet,\bullet)}\).
\[ \begin{align}\begin{aligned}\delta_{\boldsymbol{u}} \Pi_{int} &= \int_V \boldsymbol{P} : \delta \boldsymbol{F} \ dV = \int_V \left( \frac{\partial \psi} {\partial \overline{\boldsymbol{F}}} : \frac{\partial \overline{\boldsymbol{F}}} {\partial \boldsymbol{F}} + p J \boldsymbol{F}^{-T} \right) : \delta \boldsymbol{F} \ dV\\\delta_{p} \Pi_{int} &= \int_V f_{p} \ \delta p \ dV = \int_V (J - \overline{J}) \ \delta p \ dV\\\delta_{\overline{J}} \Pi_{int} &= \int_V f_{\overline{J}} \ \delta \overline{J} \ dV = \int_V \left( \frac{\partial \psi}{\partial \overline{\boldsymbol{F}}} : \frac{\partial \overline{\boldsymbol{F}}}{\partial \overline{J}} - p \right) : \delta \overline{J} \ dV\end{aligned}\end{align} \]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.
\[ \begin{align}\begin{aligned}\boldsymbol{P} &= \overline{\overline{\boldsymbol{P}}} - \frac{1}{3} \left( \overline{\overline{\boldsymbol{P}}} : \boldsymbol{F} \right) \boldsymbol{F}^{-T}\\f_p &= J - \overline{J}\\f_{\overline{J}} &= \frac{1}{3 \overline{J}} \left( \overline{\overline{\boldsymbol{P}}} : \boldsymbol{F} \right) - p\end{aligned}\end{align} \]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 (ConstitutiveMaterial) – A hyperelastic material definition for the strain energy density function with methods for the
gradient
and thehessian
w.r.t the deformation gradient tensor.parallel (bool, optional) – A flag to invoke parallel (threaded) math operations (default is False).
- gradient(x)[source]#
Return a list of variations of the total potential energy w.r.t. the fields displacements, pressure and volume ratio.
- Parameters:
x (list of ndarray) – List of extracted field values with the Deformation gradient tensor \(\boldsymbol{F}\) as first, the hydrostatic pressure \(p\) as second and the volume ratio \(\bar{J}\) as third list item.
- Returns:
List of gradients w.r.t. the input variables \(\boldsymbol{F}\), \(p\) and \(\bar{J}\).
- 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]
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- class felupe.NearlyIncompressible(material, bulk, parallel=False, dUdJ=<function NearlyIncompressible.<lambda>>, d2UdJdJ=<function NearlyIncompressible.<lambda>>)[source]#
A nearly-incompressible material formulation to augment the distortional part of the strain energy function by a volumetric part and a constraint equation.
Notes
The total potential energy of internal forces is given in Eq. (5).
(1)#\[\Pi_{int}(\boldsymbol{F}, p, \bar{J}) = \int_V \hat{\psi}(\boldsymbol{F})\ dV + \int_V U(\bar{J})\ dV + \int_V p (J - \bar{J})\ dV\]The volumetric part of the strain energy density function is denoted in Eq. (9) along with its first and second derivatives.
(2)#\[ \begin{align}\begin{aligned}\bar{U} &= \frac{K}{2} \left( \bar{J} - 1 \right)^2\\\bar{U}' &= K \left( \bar{J} - 1 \right)\\\bar{U}'' &= K\end{aligned}\end{align} \]- Parameters:
material (ConstitutiveMaterial) – A hyperelastic material definition for the distortional part of the strain energy density function \(\hat{\psi}(\boldsymbol{F})\) with methods for the
gradient
\(\partial_\boldsymbol{F}(\hat{\psi})\) and thehessian
\(\partial_\boldsymbol{F}[\partial_\boldsymbol{F}(\hat{\psi})]\) w.r.t the deformation gradient tensor \(\boldsymbol{F}\).bulk (float) – The bulk modulus \(K\) for the volumetric part of the strain energy function.
parallel (bool, optional) – A flag to invoke parallel (threaded) math operations (default is False).
dUdJ (callable, optional) – A function which evaluates the derivative of the volumetric part of the strain energy function \(\bar{U}'\) w.r.t. the volume ratio \(\bar{J}\). Function signature must be
lambda J, bulk: dUdJ
. Default is \(\bar{U}' = K (\bar{J} - 1)\) orlambda J, bulk: bulk * (J - 1)
.d2UdJdJ (callable, optional) – A function which evaluates the second derivative of the volumetric part of the strain energy function \(\bar{U}''\) w.r.t. the volume ratio \(\bar{J}\). Function signature must be
lambda J, bulk: d2UdJdJ
. Default is \(\bar{U}'' = K\) orlambda J, bulk: bulk
.
Examples
>>> import felupe as fem >>> >>> field = fem.FieldsMixed(fem.RegionHexahedron(fem.Cube(n=6)), n=3) >>> boundaries, loadcase = fem.dof.uniaxial(field, clamped=True) >>> umat = fem.NearlyIncompressible(fem.NeoHooke(mu=1), bulk=5000) >>> solid = fem.SolidBody(umat, field) >>> job = fem.Job(steps=[fem.Step(items=[solid], boundaries=boundaries)]).evaluate()
See also
ThreeFieldVariation
Hu-Washizu hydrostatic-volumetric selective three-field variation for nearly-incompressible material formulations.
- gradient(x, out=None)[source]#
Return a list with the gradient of the strain energy density function w.r.t. the fields displacements, pressure and volume ratio.
- Parameters:
x (list of ndarray) – List of extracted field values with the deformation gradient tensor \(\boldsymbol{F}\) as first, the pressure \(p\) as second and the volume ratio \(\bar{J}\) as third list item. Initial state variables are stored in the last (fourth) list item.
out (ndarray or None, optional) – A location into which the result is stored (default is None).
- Returns:
List of gradients w.r.t. the input variables \(\boldsymbol{F}\), \(p\) and \(\bar{J}\). The last item of the list contains the updated state variables.
- Return type:
list of ndarrays
Notes
\[ \begin{align}\begin{aligned}\delta_\boldsymbol{u}(\Pi_{int}) &= \int_V \left( \frac{\partial \hat{\psi}}{\partial \boldsymbol{F}} + p\ J \boldsymbol{F}^{-T} \right) : \delta\boldsymbol{F}\ dV\\\delta_p(\Pi_{int}) &= \int_V \left( J - \bar{J} \right)\ \delta p\ dV\\\delta_\bar{J}(\Pi_{int}) &= \int_V \left( \bar{U}' - p \right)\ \delta \bar{J}\ dV\end{aligned}\end{align} \]
- hessian(x, out=None)[source]#
Return a list with the hessian of the strain energy density function w.r.t. the fields displacements, pressure and volume ratio.
- Parameters:
x (list of ndarray) – List of extracted field values with the deformation gradient tensor \(\boldsymbol{F}\) as first, the pressure \(p\) as second and the volume ratio \(\bar{J}\) as third list item. Initial state variables are stored in the last (fourth) list item.
out (ndarray or None, optional) – A location into which the result is stored (default is None).
- Returns:
List of the hessian w.r.t. the input variables \(\boldsymbol{F}\), \(p\) and \(\bar{J}\). The upper-triangle items of the hessian are returned as the items of the list.
- Return type:
list of ndarrays
Notes
\[ \begin{align}\begin{aligned}\Delta_\boldsymbol{u}\delta_\boldsymbol{u}(\Pi_{int}) &= \int_V \delta\boldsymbol{F} : \left[ \frac{\partial^2 \hat{\psi}} {\partial\boldsymbol{F}\ \partial\boldsymbol{F}} + p\ J \left( \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T} - \boldsymbol{F}^{-T} \overset{il}{\odot} \boldsymbol{F}^{-T} \right) \right] : \Delta\boldsymbol{F}\ dV\\\Delta_p\delta_\boldsymbol{u}(\Pi_{int}) &= \int_V \delta\boldsymbol{F} : J \boldsymbol{F}^{-T}\ \Delta p\ dV\\\Delta_\bar{J}\delta_\boldsymbol{u}(\Pi_{int}) &= \int_V \delta\boldsymbol{F} : \boldsymbol{0}\ \Delta \bar{J}\ dV\\\Delta_p\delta_p(\Pi_{int}) &= \int_V \delta p\ (0)\ \Delta p\ dV\\\Delta_p\delta_\bar{J}(\Pi_{int}) &= \int_V \delta \bar{J}\ (-1)\ \Delta p\ dV\\\Delta_\bar{J}\delta_\bar{J}(\Pi_{int}) &= \int_V \delta \bar{J}\ \bar{U}''\ \Delta \bar{J}\ dV\end{aligned}\end{align} \]
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- class felupe.Material(stress, elasticity, nstatevars=0, **kwargs)[source]#
A user-defined material definition with given functions for the (first Piola-Kirchhoff) stress tensor \(\boldsymbol{P}\), optional constraints on additional fields (e.g. \(p\) and \(J\)), updated state variables \(\boldsymbol{\zeta}\) as well as the according fourth-order elasticity tensor \(\mathbb{A}\) and the linearizations of the constraint equations. Both functions take a list of the 3x3 deformation gradient \(\boldsymbol{F}\) and optional vector of state variables \(\boldsymbol{\zeta}_n\) as the first input argument. The stress-function must return the updated state variables \(\boldsymbol{\zeta}\).
- Parameters:
stress (callable) – A constitutive material definition which returns a list containting the (first Piola-Kirchhoff) stress tensor, optional additional constraints as well as the state variables. The state variables must always be included even if they are None. See template code-blocks for the required function signature.
elasticity (callable) – A constitutive material definition which returns a list containing the fourth- order elasticity tensor as the jacobian of the (first Piola-Kirchhoff) stress tensor w.r.t. the deformation gradient, optional linearizations of the additional constraints. The state variables must not be returned. See template code-blocks for the required function signature.
nstatevars (int, optional) – Number of internal state variable components (default is 0). State variable components must always be concatenated into a 1d-array.
Notes
Note
The first item in the list of the input arguments always contains the gradient of the (displacement) field \(\boldsymbol{u}\) w.r.t. the undeformed coordinates \(\boldsymbol{X}\). The identity matrix \(\boldsymbol{1}\) is added to this gradient, i.e. the first item of the list
x
contains the deformation gradient \(\boldsymbol{F} = \boldsymbol{1} + \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{X}}\). All other fields are provided as interpolated values (no gradients evaluated).For \((\boldsymbol{u})\) single-field formulations, the callables for
stress
andelasticity
must return the gradient and hessian of the strain energy density function \(\psi(\boldsymbol{F})\) w.r.t. the deformation gradient tensor \(\boldsymbol{F}\).\[\begin{split}\text{stress}(\boldsymbol{F}, \boldsymbol{\zeta}_n) = \begin{bmatrix} \frac{\partial \psi}{\partial \boldsymbol{F}} \\ \boldsymbol{\zeta} \end{bmatrix}\end{split}\]The callable for
elasticity
(hessian) must not return the updated state variables.\[\text{elasticity}(\boldsymbol{F}, \boldsymbol{\zeta}_n) = \begin{bmatrix} \frac{\partial^2 \psi}{\partial \boldsymbol{F}\ \partial \boldsymbol{F}} \end{bmatrix}\]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 = Material(stress, elasticity, **kwargs)
For \((\boldsymbol{u}, p, J)\) mixed-field formulations, the callables for
stress
andelasticity
must return the gradients and hessians of the (augmented) strain energy density function w.r.t. the deformation gradient and the other fields.\[\begin{split}\text{stress}(\boldsymbol{F}, p, J, \boldsymbol{\zeta}_n) = \begin{bmatrix} \frac{\partial \psi}{\partial \boldsymbol{F}} \\ \frac{\partial \psi}{\partial p} \\ \frac{\partial \psi}{\partial J} \\ \boldsymbol{\zeta} \end{bmatrix}\end{split}\]For the hessians, the upper-triangle blocks have to be provided.
\[\begin{split}\text{elasticity}(\boldsymbol{F}, p, J, \boldsymbol{\zeta}_n) = \begin{bmatrix} \frac{\partial^2 \psi}{\partial \boldsymbol{F}\ \partial \boldsymbol{F}} \\ \frac{\partial^2 \psi}{\partial \boldsymbol{F}\ \partial p} \\ \frac{\partial^2 \psi}{\partial \boldsymbol{F}\ \partial J} \\ \frac{\partial^2 \psi}{\partial p\ \partial p} \\ \frac{\partial^2 \psi}{\partial p\ \partial J} \\ \frac{\partial^2 \psi}{\partial J\ \partial J} \end{bmatrix}\end{split}\]For \((\boldsymbol{u}, p, J)\) mixed-field formulations, take this code-block as template:
def gradient(x, **kwargs): "Gradients of the strain energy density function." # extract variables F, p, J, statevars = x[0], x[1], x[2], x[-1] # user code dWdF = None # first Piola-Kirchhoff stress tensor dWdp = None dWdJ = None # update state variables statevars_new = None return [dWdF, dWdp, dWdJ, statevars_new] def hessian(x, **kwargs): "Hessians of the strain energy density function." # extract variables F, p, J, statevars = x[0], x[1], x[2], x[-1] # user code d2WdFdF = None # fourth-order elasticity tensor d2WdFdp = None d2WdFdJ = None d2Wdpdp = None d2WdpdJ = None d2WdJdJ = None # upper-triangle items of the hessian return [d2WdFdF, d2WdFdp, d2WdFdJ, d2Wdpdp, d2WdpdJ, d2WdJdJ] umat = Material(gradient, hessian, **kwargs)
Examples
The compressible isotropic hyperelastic Neo-Hookean material formulation is given by the strain energy density function.
\[ \begin{align}\begin{aligned}\psi &= \psi(\boldsymbol{C})\\\psi(\boldsymbol{C}) &= \frac{\mu}{2} \text{tr}(\boldsymbol{C}) - \mu \ln(J) + \frac{\lambda}{2} \ln(J)^2\end{aligned}\end{align} \]with
\[J = \text{det}(\boldsymbol{F})\]The first Piola-Kirchhoff stress tensor is evaluated as the gradient of the strain energy density function.
\[ \begin{align}\begin{aligned}\boldsymbol{P} &= \frac{\partial \psi}{\partial \boldsymbol{F}}\\\boldsymbol{P} &= \mu \left( \boldsymbol{F} - \boldsymbol{F}^{-T} \right) + \lambda \ln(J) \boldsymbol{F}^{-T}\end{aligned}\end{align} \]The hessian of the strain energy density function enables the corresponding elasticity tensor.
\[ \begin{align}\begin{aligned}\mathbb{A} &= \frac{\partial^2 \psi}{\partial \boldsymbol{F}\ \partial \boldsymbol{F}}\\\mathbb{A} &= \mu \boldsymbol{I} \overset{ik}{\otimes} \boldsymbol{I} + \left(\mu - \lambda \ln(J) \right) \boldsymbol{F}^{-T} \overset{il}{\otimes} \boldsymbol{F}^{-T} + \lambda \boldsymbol{F}^{-T} {\otimes} \boldsymbol{F}^{-T}\end{aligned}\end{align} \]>>> import numpy as np >>> import felupe as fem >>> from felupe.math import ( >>> cdya_ik, >>> cdya_il, >>> det, >>> dya, >>> identity, >>> inv, >>> transpose >>> )
>>> def stress(x, mu, lmbda): >>> F, statevars = x[0], x[-1] >>> J = det(F) >>> lnJ = np.log(J) >>> iFT = transpose(inv(F, J)) >>> dWdF = mu * (F - iFT) + lmbda * lnJ * iFT >>> return [dWdF, statevars]
>>> def elasticity(x, mu, lmbda): >>> F = x[0] >>> J = det(F) >>> iFT = transpose(inv(F, J)) >>> eye = identity(F) >>> return [ >>> mu * cdya_ik(eye, eye) + lmbda * dya(iFT, iFT) + >>> (mu - lmbda * np.log(J)) * cdya_il(iFT, iFT) >>> ]
>>> umat = fem.Material(stress, elasticity, mu=1.0, lmbda=2.0)
The material formulation is tested in a minimal example of non-homogeneous uniaxial tension.
>>> mesh = fem.Cube(n=6) >>> region = fem.RegionHexahedron(mesh) >>> field = fem.FieldContainer([fem.Field(region, dim=3)]) >>> boundaries, loadcase = fem.dof.uniaxial(field, clamped=True, move=0.5) >>> solid = fem.SolidBodyNearlyIncompressible(umat, field, bulk=5000) >>> step = fem.Step(items=[solid], boundaries=boundaries) >>> job = fem.Job(steps=[step]).evaluate()
See also
felupe.NeoHookeCompressible
Nearly-incompressible isotropic hyperelastic Neo-Hookean material formulation.
- gradient(x)[source]#
Return the evaluated gradient of the strain energy density function.
- Parameters:
x (list of ndarray) – The list with input arguments. These contain the extracted fields of a
FieldContainer
along with the old vector of state variables,[*field.extract(), statevars_old]
.- Returns:
A list with the evaluated gradient(s) of the strain energy density function and the updated vector of state variables.
- Return type:
list of ndarray
- hessian(x)[source]#
Return the evaluated upper-triangle components of the hessian(s) of the strain energy density function.
- Parameters:
x (list of ndarray) – The list with input arguments. These contain the extracted fields of a
FieldContainer
along with the old vector of state variables,[*field.extract(), statevars_old]
.- Returns:
A list with the evaluated upper-triangle components of the hessian(s) of the strain energy density function.
- Return type:
list of ndarray
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- class felupe.MaterialStrain(material, dim=3, statevars=(0,), **kwargs)[source]#
A strain-based user-defined material definition with a given function 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 = MaterialStrain(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 = MaterialStrain(material=fun, **kwargs)
See also
linear_elastic
3D linear-elastic material formulation
linear_elastic_plastic_isotropic_hardening
Linear-elastic-plastic material formulation with linear isotropic hardening (return mapping algorithm).
LinearElasticPlasticIsotropicHardening
Linear-elastic-plastic material formulation with linear isotropic hardening (return mapping algorithm).
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- felupe.linear_elastic(dε, εn, σn, ζn, λ, μ, **kwargs)[source]#
3D linear-elastic material formulation to be used in
MaterialStrain
.- Parameters:
- Returns:
dσdε (ndarray) – Elasticity tensor.
σ (ndarray) – (New) stress tensor.
ζ (list) – List of new state variables.
Notes
Given state in point \(\boldsymbol{x} (\boldsymbol{\sigma}_n)\) (valid).
Given strain increment \(\Delta\boldsymbol{\varepsilon}\), so that \(\boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}_n + \Delta\boldsymbol{\varepsilon}\).
Evaluation of the stress \(\boldsymbol{\sigma}\) and the algorithmic consistent tangent modulus \(\mathbb{C}\) (=``dσdε``).
\[ \begin{align}\begin{aligned}\mathbb{C} &= \lambda \ \boldsymbol{1} \otimes \boldsymbol{1} + 2 \mu \ \boldsymbol{1} \odot \boldsymbol{1}\\\boldsymbol{\sigma} &= \boldsymbol{\sigma}_n + \mathbb{C} : \Delta\boldsymbol{\varepsilon}\end{aligned}\end{align} \]
Examples
>>> import felupe as fem >>> >>> umat = fem.MaterialStrain(material=fem.linear_elastic, λ=2.0, μ=1.0)
See also
MaterialStrain
A strain-based user-defined material definition with a given function for the stress tensor and the (fourth-order) elasticity tensor.
- felupe.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) to be used in
MaterialStrain
.- 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.
- Returns:
dσdε (ndarray) – Algorithmic consistent elasticity tensor.
σ (ndarray) – (New) stress tensor.
ζ (list) – List of new state variables.
Notes
Given state in point \(x (\sigma_n, \zeta_n=[\varepsilon^p_n, \alpha_n])\) (valid).
Given strain increment \(\Delta\varepsilon\), so that \(\varepsilon = \varepsilon_n + \Delta\varepsilon\).
Evaluation of the hypothetic trial state:
\[ \begin{align}\begin{aligned}\mathbb{C} &= \lambda\ \boldsymbol{1} \otimes \boldsymbol{1} + 2 \mu\ \boldsymbol{1} \odot \boldsymbol{1}\\\sigma &= \sigma_n + \mathbb{C} : \Delta\varepsilon\\s &= \text{dev}(\sigma)\\\varepsilon^p &= \varepsilon^p_n\\\alpha &= \alpha_n\\f &= ||s|| - \sqrt{\frac{2}{3}}\ (\sigma_y + K \alpha)\end{aligned}\end{align} \]If \(f \le 0\), then elastic step:
Set \(y = y_n + \Delta y, y=(\sigma, \zeta=[\varepsilon^p, \alpha])\),
algorithmic consistent tangent modulus \(d\sigma d\varepsilon\).
\[d\sigma d\varepsilon = \mathbb{C}\]Else:
\[ \begin{align}\begin{aligned}d\gamma &= \frac{f}{2\mu + \frac{2}{3} K}\\n &= \frac{s}{||s||}\\\sigma &= \sigma - 2\mu \Delta\gamma n\\\varepsilon^p &= \varepsilon^p_n + \Delta\gamma n\\\alpha &= \alpha_n + \sqrt{\frac{2}{3}}\ \Delta\gamma\end{aligned}\end{align} \]Algorithmic consistent tangent modulus:
\[d\sigma d\varepsilon = \mathbb{C} - \frac{2 \mu}{1 + \frac{K}{3 \mu}} n \otimes n - \frac{2 \mu \Delta\gamma}{||s||} \left[ 2 \mu \left( \boldsymbol{1} \odot \boldsymbol{1} - \frac{1}{3} \boldsymbol{1} \otimes \boldsymbol{1} - n \otimes n \right) \right]\]
Examples
>>> import felupe as fem >>> >>> umat = fem.MaterialStrain( >>> material=fem.linear_elastic_plastic_isotropic_hardening, >>> λ=2.0, >>> μ=1.0, >>> σy=1.0, >>> K=0.1, >>> dim=3, >>> statevars=(1, (3, 3)), >>> )
See also
MaterialStrain
A strain-based user-defined material definition with a given function for the stress tensor and the (fourth-order) elasticity tensor.
- class felupe.MaterialAD(fun, nstatevars=0, parallel=False, **kwargs)[source]#
A user-defined material definition with a given function for the partial derivative of the strain energy function w.r.t. the deformation gradient tensor with Automatic Differentiation provided by tensortrax.
- Parameters:
fun (callable) – A gradient of the strain energy density function w.r.t. the deformation gradient tensor \(\boldsymbol{F}\). Function signature must be
fun = lambda F, **kwargs: P
for functions without state variables andfun = lambda F, statevars, **kwargs: [P, statevars_new]
for functions with state variables. The deformation gradient tensor will be atensortrax.Tensor
when the function is evaluated. It is important to only use differentiable math-functions fromtensortrax.math
!nstatevars (int, optional) – Number of state variables (default is 0).
parallel (bool, optional) – A flag to invoke threaded gradient of strain energy density function evaluations (default is False). May introduce additional overhead for small-sized problems.
**kwargs (dict, optional) – Optional keyword-arguments for the gradient of the strain energy density function.
Notes
The gradient of the strain energy density function \(\frac{\partial \psi}{\partial \boldsymbol{F}}\) must be given in terms of the deformation gradient tensor \(\boldsymbol{F}\).
Warning
It is important to only use differentiable math-functions from
tensortrax.math
!\[\boldsymbol{P} = \frac{\partial \psi(\boldsymbol{F}, \boldsymbol{\zeta})}{ \partial \boldsymbol{F}}\]Take this code-block as template
import tensortrax.math as tm def neo_hooke(F, mu): "First Piola-Kirchhoff stress of the Neo-Hookean material formulation." C = tm.dot(tm.transpose(F), F) Cu = tm.linalg.det(C) ** (-1/3) * C return mu * F @ tm.special.dev(Cu) @ tm.linalg.inv(C) umat = fem.MaterialAD(neo_hooke, mu=1)
and this code-block for material formulations with state variables:
import tensortrax.math as tm def viscoelastic(F, Cin, mu, eta, dtime): "Finite strain viscoelastic material formulation." # unimodular part of the right Cauchy-Green deformation tensor C = tm.dot(tm.transpose(F), F) Cu = tm.linalg.det(C) ** (-1 / 3) * C # update of state variables by evolution equation Ci = tm.special.from_triu_1d(Cin, like=C) + mu / eta * dtime * Cu Ci = tm.linalg.det(Ci) ** (-1 / 3) * Ci # second Piola-Kirchhoff stress tensor S = mu * tm.special.dev(Cu @ tm.linalg.inv(Ci)) @ tm.linalg.inv(C) # first Piola-Kirchhoff stress tensor and state variable return F @ S, tm.special.triu_1d(Ci) umat = fem.MaterialAD( viscoelastic, mu=1, eta=1, dtime=1, nstatevars=6 )
Note
See the documentation of tensortrax for further details.
Examples
>>> import felupe as fem >>> import tensortrax.math as tm >>> >>> def neo_hooke(F, mu): >>> C = tm.dot(tm.transpose(F), F) >>> S = mu * tm.special.dev(tm.linalg.det(C)**(-1/3) * C) @ tm.linalg.inv(C) >>> return F @ S >>> >>> umat = fem.MaterialAD(neo_hooke, mu=1) >>> ax = umat.plot(incompressible=True)
- gradient(x)#
Return the evaluated gradient of the strain energy density function.
- Parameters:
x (list of ndarray) – The list with input arguments. These contain the extracted fields of a
FieldContainer
along with the old vector of state variables,[*field.extract(), statevars_old]
.- Returns:
A list with the evaluated gradient(s) of the strain energy density function and the updated vector of state variables.
- Return type:
list of ndarray
- hessian(x)#
Return the evaluated upper-triangle components of the hessian(s) of the strain energy density function.
- Parameters:
x (list of ndarray) – The list with input arguments. These contain the extracted fields of a
FieldContainer
along with the old vector of state variables,[*field.extract(), statevars_old]
.- Returns:
A list with the evaluated upper-triangle components of the hessian(s) of the strain energy density function.
- Return type:
list of ndarray
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- class felupe.Hyperelastic(fun, nstatevars=0, parallel=False, **kwargs)[source]#
A hyperelastic material definition with a given function for the strain energy density function per unit undeformed volume with Automatic Differentiation provided by
tensortrax
.- Parameters:
fun (callable) – A strain energy density function in terms of the right Cauchy-Green deformation tensor \(\boldsymbol{C}\). Function signature must be
fun = lambda C, **kwargs: psi
for functions without state variables andfun = lambda C, statevars, **kwargs: [psi, statevars_new]
for functions with state variables. The right Cauchy-Green deformation tensor will be atensortrax.Tensor
when the function is evaluated. It is important to only use differentiable math-functions fromtensortrax.math
!nstatevars (int, optional) – Number of state variables (default is 0).
parallel (bool, optional) – A flag to invoke threaded strain energy density function evaluations (default is False). May introduce additional overhead for small-sized problems.
**kwargs (dict, optional) – Optional keyword-arguments for the strain energy density function.
Notes
The strain energy density function \(\psi\) must be given in terms of the right Cauchy-Green deformation tensor \(\boldsymbol{C} = \boldsymbol{F}^T \boldsymbol{F}\).
Warning
It is important to only use differentiable math-functions from
tensortrax.math
!Take this minimal code-block as template
\[\psi = \psi(\boldsymbol{C})\]import tensortrax.math as tm def neo_hooke(C, mu): "Strain energy function of the Neo-Hookean material formulation." return mu / 2 * (tm.linalg.det(C) ** (-1/3) * tm.trace(C) - 3) umat = fem.Hyperelastic(neo_hooke, mu=1)
and this code-block for material formulations with state variables.
\[\psi = \psi(\boldsymbol{C}, \boldsymbol{\zeta})\]import tensortrax.math as tm def viscoelastic(C, Cin, mu, eta, dtime): "Finite strain viscoelastic material formulation." # unimodular part of the right Cauchy-Green deformation tensor Cu = tm.linalg.det(C) ** (-1 / 3) * C # update of state variables by evolution equation Ci = tm.special.from_triu_1d(Cin, like=C) + mu / eta * dtime * Cu Ci = tm.linalg.det(Ci) ** (-1 / 3) * Ci # first invariant of elastic part of right Cauchy-Green deformation tensor I1 = tm.trace(Cu @ tm.linalg.inv(Ci)) # strain energy function and state variable return mu / 2 * (I1 - 3), tm.special.triu_1d(Ci) umat = fem.Hyperelastic( viscoelastic, mu=1, eta=1, dtime=1, nstatevars=6 )
Note
See the documentation of tensortrax for further details.
Examples
View force-stretch curves on elementary incompressible deformations.
>>> import felupe as fem >>> import tensortrax.math as tm >>> >>> def neo_hooke(C, mu): >>> "Strain energy function of the Neo-Hookean material formulation." >>> return mu / 2 * (tm.linalg.det(C) ** (-1/3) * tm.trace(C) - 3) >>> >>> umat = fem.Hyperelastic(neo_hooke, mu=1) >>> ax = umat.plot(incompressible=True)
See also
saint_venant_kirchhoff
Strain energy function of the Saint Venant-Kirchhoff material formulation.
neo_hooke
Strain energy function of the Neo-Hookean material formulation.
mooney_rivlin
Strain energy function of the Mooney-Rivlin material formulation.
yeoh
“Strain energy function of the Yeoh material formulation.
third_order_deformation
Strain energy function of the Third-Order-Deformation material formulation.
ogden
Strain energy function of the Ogden material formulation.
arruda_boyce
Strain energy function of the Arruda-Boyce material formulation.
extended_tube
Strain energy function of the Extended-Tube material formulation.
van_der_waals
Strain energy function of the Van der Waals material formulation.
finite_strain_viscoelastic
Finite strain viscoelastic material formulation.
- gradient(x)#
Return the evaluated gradient of the strain energy density function.
- Parameters:
x (list of ndarray) – The list with input arguments. These contain the extracted fields of a
FieldContainer
along with the old vector of state variables,[*field.extract(), statevars_old]
.- Returns:
A list with the evaluated gradient(s) of the strain energy density function and the updated vector of state variables.
- Return type:
list of ndarray
- hessian(x)#
Return the evaluated upper-triangle components of the hessian(s) of the strain energy density function.
- Parameters:
x (list of ndarray) – The list with input arguments. These contain the extracted fields of a
FieldContainer
along with the old vector of state variables,[*field.extract(), statevars_old]
.- Returns:
A list with the evaluated upper-triangle components of the hessian(s) of the strain energy density function.
- Return type:
list of ndarray
- plot(incompressible=False, **kwargs)#
Return a plot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- screenshot(filename='umat.png', incompressible=False, **kwargs)#
Save a screenshot with normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
filename (str, optional) – The filename of the screenshot (default is “umat.png”).
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- view(incompressible=False, **kwargs)#
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- Parameters:
incompressible (bool, optional) – A flag to enforce views on incompressible deformations (default is False).
**kwargs (dict, optional) – Optional keyword-arguments for
ViewMaterial
orViewMaterialIncompressible
.
- Return type:
See also
felupe.ViewMaterial
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
felupe.ViewMaterialIncompressible
Create views on normal force per undeformed area vs. stretch curves for the elementary homogeneous incompressible deformations uniaxial tension/compression, planar shear and biaxial tension of a given isotropic material formulation.
- felupe.saint_venant_kirchhoff(C, mu, lmbda)[source]#
Strain energy function of the isotropic hyperelastic Saint-Venant Kirchhoff material formulation.
- Parameters:
Notes
\[\psi = \mu I_2 + \lambda \frac{I_1^2}{2}\]With the first and second invariant of the Green-Lagrange strain tensor \(\boldsymbol{E} = \frac{1}{2} (\boldsymbol{C} - \boldsymbol{1})\).
\[ \begin{align}\begin{aligned}I_1 &= \text{tr}\left( \boldsymbol{E} \right)\\I_2 &= \boldsymbol{E} : \boldsymbol{E}\end{aligned}\end{align} \]Examples
Warning
The Saint-Venant Kirchhoff material formulation is unstable for large strains.
>>> import felupe as fem >>> >>> umat = fem.Hyperelastic(fem.saint_venant_kirchhoff, mu=1.0, lmbda=20.0) >>> ax = umat.plot(incompressible=False) UserWarning: Uniaxial data with volume ratio det(F) <= 0 included. UserWarning: Planar Shear data with volume ratio det(F) <= 0 included. UserWarning: Biaxial data with volume ratio det(F) <= 0 included.
- felupe.neo_hooke(C, mu)[source]#
Strain energy function of the isotropic hyperelastic Neo-Hookean material formulation.
- Parameters:
C (tensortrax.Tensor) – Right Cauchy-Green deformation tensor.
mu (float) – Shear modulus.
Notes
\[\psi = \frac{\mu}{2} \left(\text{tr}\left(\hat{\boldsymbol{C}}\right) - 3\right)\]Examples
>>> import felupe as fem >>> >>> umat = fem.Hyperelastic(fem.neo_hooke, mu=1.0) >>> ax = umat.plot(incompressible=True)
- felupe.mooney_rivlin(C, C10, C01)[source]#
Strain energy function of the isotropic hyperelastic Mooney-Rivlin material formulation.
- Parameters:
Notes
\[\psi = C_{10} \left(\hat{I}_1 - 3 \right) + C_{01} \left(\hat{I}_2 - 3 \right)\]With the first and second main invariant of the distortional part of the right Cauchy-Green deformation tensor.
\[ \begin{align}\begin{aligned}\hat{I}_1 &= J^{-2/3} \text{tr}\left( \boldsymbol{C} \right)\\\hat{I}_2 &= J^{-4/3} \frac{1}{2} \left( \text{tr}\left(\boldsymbol{C}\right)^2 - \text{tr}\left(\boldsymbol{C}^2\right) \right)\end{aligned}\end{align} \]The doubled sum of both material parameters is equal to the shear modulus \(\mu\).
\[\mu = 2 \left( C_{10} + C_{01} \right)\]Examples
>>> import felupe as fem >>> >>> umat = fem.Hyperelastic(fem.mooney_rivlin, C10=0.3, C01=0.8) >>> ax = umat.plot(incompressible=True)
- felupe.yeoh(C, C10, C20, C30)[source]#
Strain energy function of the isotropic hyperelastic Yeoh material formulation.
- Parameters:
C (tensortrax.Tensor) – Right Cauchy-Green deformation tensor.
C10 (float) – Material parameter associated to the linear term of the first invariant.
C20 (float) – Material parameter associated to the quadratic term of the first invariant.
C30 (float) – Material parameter associated to the cubic term of the first invariant.
Notes
\[\psi = C_{10} \left(\hat{I}_1 - 3 \right) + C_{20} \left(\hat{I}_1 - 3 \right)^2 + C_{30} \left(\hat{I}_1 - 3 \right)^3\]With the first main invariant of the distortional part of the right Cauchy-Green deformation tensor.
\[\hat{I}_1 = J^{-2/3} \text{tr}\left( \boldsymbol{C} \right)\]The \(C_{10}\) material parameter is equal to half the initial shear modulus \(\mu\).
\[\mu = 2 C_{10}\]Examples
>>> import felupe as fem >>> >>> umat = fem.Hyperelastic(fem.yeoh, C10=0.5, C20=-0.1, C30=0.02) >>> ax = umat.plot(incompressible=True)
- felupe.third_order_deformation(C, C10, C01, C11, C20, C30)[source]#
Strain energy function of the isotropic hyperelastic Third-Order-Deformation material formulation.
- Parameters:
C (tensortrax.Tensor) – Right Cauchy-Green deformation tensor.
C10 (float) – Material parameter associated to the linear term of the first invariant.
C01 (float) – Material parameter associated to the linear term of the second invariant.
C11 (float) – Material parameter associated to the mixed term of the first and second invariant.
C20 (float) – Material parameter associated to the quadratic term of the first invariant.
C30 (float) – Material parameter associated to the cubic term of the first invariant.
Notes
\[ \begin{align}\begin{aligned}\psi &= C_{10} \left(\hat{I}_1 - 3 \right) + C_{01} \left(\hat{I}_2 - 3 \right) + C_{11} \left(\hat{I}_1 - 3 \right) \left(\hat{I}_2 - 3 \right)\\ &+ C_{20} \left(\hat{I}_1 - 3 \right)^2 + C_{30} \left(\hat{I}_1 - 3 \right)^3\end{aligned}\end{align} \]With the first and second main invariant of the distortional part of the right Cauchy-Green deformation tensor.
\[ \begin{align}\begin{aligned}\hat{I}_1 &= J^{-2/3} \text{tr}\left( \boldsymbol{C} \right)\\\hat{I}_2 &= J^{-4/3} \frac{1}{2} \left( \text{tr}\left(\boldsymbol{C}\right)^2 - \text{tr}\left(\boldsymbol{C}^2\right) \right)\end{aligned}\end{align} \]The doubled sum of the material parameters \(C_{10}\) and \(C_{01}\) is equal to the initial shear modulus \(\mu\).
\[\mu = 2 \left( C_{10} + C_{01} \right)\]Examples
>>> import felupe as fem >>> >>> umat = fem.Hyperelastic( >>> fem.third_order_deformation, C10=0.5, C01=0.1, C11=0.01, C20=-0.1, C30=0.02 >>> ) >>> ax = umat.plot(incompressible=True)
- felupe.ogden(C, mu, alpha)[source]#
Strain energy function of the isotropic hyperelastic Ogden material formulation.
- Parameters:
Notes
\[\psi = \sum_i \frac{2 \mu_i}{\alpha^2_i} \left( \lambda_1^{\alpha_i} + \lambda_2^{\alpha_i} + \lambda_3^{\alpha_i} - 3 \right)\]The sum of the moduli \(\mu_i\) is equal to the initial shear modulus \(\mu\).
\[\mu = \sum_i \mu_i\]Examples
>>> import felupe as fem >>> >>> umat = fem.Hyperelastic(fem.ogden, mu=[1, 0.2], alpha=[1.7, -1.5]) >>> ax = umat.plot(incompressible=True)
- felupe.arruda_boyce(C, C1, limit)[source]#
Strain energy function of the isotropic hyperelastic Arruda-Boyce material formulation.
- Parameters:
Notes
\[\psi = C_1 \sum_{i=1}^5 \alpha_i \beta^{i-1} \left( \hat{I}_1^i - 3^i \right)\]With the first main invariant of the distortional part of the right Cauchy-Green deformation tensor
\[\hat{I}_1 = J^{-2/3} \text{tr}\left( \boldsymbol{C} \right)\]and :math_`alpha_i` and :math`beta`.
\[ \begin{align}\begin{aligned}\begin{split}\boldsymbol{\alpha} &= \begin{bmatrix} \frac{1}{2} \\ \frac{1}{20} \\ \frac{11}{1050} \\ \frac{19}{7000} \\ \frac{519}{673750} \end{bmatrix}\end{split}\\\beta &= \frac{1}{\lambda_m^2}\end{aligned}\end{align} \]The initial shear modulus is a function of both material parameters.
\[\mu = C_1 \left( 1 + \frac{3}{5 \lambda_m^2} + \frac{99}{175 \lambda_m^4} + \frac{513}{875 \lambda_m^6} + \frac{42039}{67375 \lambda_m^8} \right)\]Examples
>>> import felupe as fem >>> >>> umat = fem.Hyperelastic(fem.arruda_boyce, C1=1.0, limit=3.2) >>> ax = umat.plot(incompressible=True)
- felupe.extended_tube(C, Gc, delta, Ge, beta)[source]#
Strain energy function of the isotropic hyperelastic Extended Tube [1] material formulation.
- Parameters:
C (tensortrax.Tensor) – Right Cauchy-Green deformation tensor.
Gc (float) – Cross-link contribution to the initial shear modulus.
delta (float) – Finite extension parameter of the polymer strands.
Ge (float) – Constraint contribution to the initial shear modulus.
beta (float) – Global rearrangements of cross-links upon deformation (release of topological constraints).
Notes
\[\psi = \frac{G_c}{2} \left[ \frac{\left( 1 - \delta^2 \right) \left( \hat{I}_1 - 3 \right)}{1 - \delta^2 \left( \hat{I}_1 - 3 \right)} + \ln \left( 1 - \delta^2 \left( \hat{I}_1 - 3 \right) \right) \right] + \frac{2 G_e}{\beta^2} \left( \hat{\lambda}_1^{-\beta} + \hat{\lambda}_2^{-\beta} + \hat{\lambda}_3^{-\beta} - 3 \right)\]With the first main invariant of the distortional part of the right Cauchy-Green deformation tensor
\[\hat{I}_1 = J^{-2/3} \text{tr}\left( \boldsymbol{C} \right)\]and the principal stretches, obtained from the distortional part of the right Cauchy-Green deformation tensor
\[\hat{\lambda}_\alpha = J^{-1/3} \lambda_\alpha\]The initial shear modulus results from the sum of the cross-link and the constraint contributions to the total initial shear modulus.
\[\mu = G_e + G_c\]Examples
>>> import felupe as fem >>> >>> umat = fem.Hyperelastic( >>> fem.extended_tube, Gc=0.1867, Ge=0.2169, beta=0.2, delta=0.09693 >>> ) >>> ax = umat.plot(incompressible=True)
References
[1] M. Kaliske and G. Heinrich, “An Extended Tube-Model for Rubber Elasticity: Statistical-Mechanical Theory and Finite Element Implementation”, Rubber Chemistry and Technology, vol. 72, no. 4. Rubber Division, ACS, pp. 602–632, Sep. 01, 1999. doi: 10.5254/1.3538822.
- felupe.van_der_waals(C, mu, limit, a, beta)[source]#
Strain energy function of the Van der Waals material formulation.
Examples
>>> import felupe as fem >>> >>> umat = fem.Hyperelastic(fem.van_der_waals, mu=1.0, beta=0.1, a=0.5, limit=5.0) >>> ax = umat.plot(incompressible=True)
- felupe.finite_strain_viscoelastic(C, Cin, mu, eta, dtime)[source]#
Finite strain viscoelastic material formulation.
Examples
>>> import felupe as fem >>> >>> umat = fem.Hyperelastic( >>> fem.finite_strain_viscoelastic, mu=1.0, eta=1.0, dtime=1.0, nstatevars=6 >>> ) >>> ax = umat.plot( >>> incompressible=True, >>> ux=fem.math.linsteps([1, 1.5, 1, 2, 1, 2.5, 1], num=15), >>> ps=None, >>> bx=None, >>> )
- class felupe.LineChange(parallel=False)[source]#
Line Change.
\[d\boldsymbol{x} = \boldsymbol{F} d\boldsymbol{X}\]Gradient:
\[\frac{\partial \boldsymbol{F}}{\partial \boldsymbol{F}} = \boldsymbol{I} \overset{ik}{\otimes} \boldsymbol{I}\]
- 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