Source code for hyperelastic.lab._load_case
import numpy as np
[docs]
class IncompressibleHomogeneousStretch:
r"""An incompressible homogeneous stretch load case with a longitudinal stretch and
perpendicular transverse stretches in principal directions. This class is intended
to be subclassed by another class with a `.defgrad()` method for the evaluation of
the deformation gradient as utilized by the :class:`Uniaxial <.lab.Uniaxial>`,
:class:`Planar <.lab.Planar>` and :class:`Biaxial <.lab.Biaxial>` load cases.
Notes
-----
The Cauchy stress for an incompressible material is given by
.. math::
\boldsymbol{\sigma} = \boldsymbol{\sigma}' + p \boldsymbol{I}
where the Cauchy stress is converted to the first Piola-Kirchhoff stress tensor.
.. math::
\boldsymbol{P} = J \boldsymbol{\sigma} \boldsymbol{F}^{-T}
The deformation gradient and its determinant are evaluated for the
homogeneous incompressible deformation.
.. math::
\boldsymbol{F} &= \text{diag} \left(\begin{bmatrix}
\lambda_1 & \lambda_2 & \lambda_3 \end{bmatrix} \right)
J &= \lambda_1 \lambda_2 \lambda_3 = 1
This enables the evaluation of the normal force per undeformed area, where
quantities in the traction-free transverse direction are denoted with a subscript
:math:`(\bullet)_t`.
.. math::
\frac{N}{A} = P - P_t \frac{\lambda_t}{\lambda}
"""
[docs]
def stress(self, F, P, axis=0, traction_free=-1):
r"""Normal force per undeformed area for a given deformation gradient of an
incompressible deformation and the first Piola-Kirchhoff stress tensor.
Parameters
----------
F : ndarray
The deformation gradient.
P : ndarray
The first Piola-Kirchhoff stress tensor.
axis : int, optional
The primary axis where the longitudinal stretch is applied on (default is
0).
traction_free : int, optional
The secondary axis where the traction-free transverse stretch results from
the constraint of incompressibility (default is -1).
Returns
-------
ndarray
The one-dimensional normal force per undeformed area.
"""
i = axis
j = traction_free
return P[i, i] - P[j, j] * F[j, j] / F[i, i]
[docs]
class Uniaxial(IncompressibleHomogeneousStretch):
r"""Incompressible uniaxial tension/compression load case.
.. math::
\boldsymbol{F} = \text{diag} \left(\begin{bmatrix}
\lambda & \frac{1}{\sqrt{\lambda}} & \frac{1}{\sqrt{\lambda}} \end{bmatrix}
\right)
"""
def __init__(self, label=None):
if label is None:
label = "Uniaxial Tension"
self.label = label
[docs]
def defgrad(self, stretch):
"Return the Deformation Gradient tensor from given stretches."
x = stretch
y = 1 / np.sqrt(stretch)
z = np.zeros_like(stretch)
return np.array([[x, z, z], [z, y, z], [z, z, y]])
[docs]
class Biaxial(IncompressibleHomogeneousStretch):
r"""Incompressible biaxial tension/compression load case.
.. math::
\boldsymbol{F} = \text{diag} \left(\begin{bmatrix}
\lambda & \lambda & \frac{1}{\lambda^2} \end{bmatrix}
\right)
"""
def __init__(self, label=None):
if label is None:
label = "Biaxial Tension"
self.label = label
[docs]
def defgrad(self, stretch):
"Return the Deformation Gradient tensor from given stretches."
x = stretch
y = 1 / stretch**2
z = np.zeros_like(stretch)
return np.array([[x, z, z], [z, x, z], [z, z, y]])
[docs]
class Planar(IncompressibleHomogeneousStretch):
r"""Incompressible planar (shear) tension/compression load case.
.. math::
\boldsymbol{F} = \text{diag} \left(\begin{bmatrix}
\lambda & 1 & \frac{1}{\lambda} \end{bmatrix}
\right)
"""
def __init__(self, label=None):
if label is None:
label = "Planar Tension"
self.label = label
[docs]
def defgrad(self, stretch):
"Return the Deformation Gradient tensor from given stretches."
x = stretch
y = 1 / stretch
o = np.ones_like(stretch)
z = np.zeros_like(stretch)
return np.array([[x, z, z], [z, o, z], [z, z, y]])
