# -*- coding: utf-8 -*-
"""
This file is part of FElupe.
FElupe is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
FElupe is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with FElupe. If not, see <http://www.gnu.org/licenses/>.
"""
import numpy as np
from ._tensor import dot, eigh, eigvalsh
from ._tensor import tovoigt as tensor_to_vector
from ._tensor import transpose
def displacement(field, dim=3, n=0):
"Return the 3d-values of the n-th field."
u = field[n].values
return np.pad(u, ((0, 0), (0, dim - u.shape[1])))
def right_cauchy_green_deformation(field, n=0):
"Return the right Cauchy-Green deformation tensor of the n-th field."
F = deformation_gradient(field, n=n)
return dot(transpose(F), F)
[docs]
def strain_stretch_1d(stretch, k=0):
r"""Compute the Seth-Hill strains.
Parameters
----------
stretch : ndarray of shape (M, ...)
The array of stretches.
k : float, optional
The strain-exponent of the Seth-Hill strain-stretch relation (default is 0).
If 0, the logarithmic strains are computed.
Returns
-------
ndarray of shape (M, ...)
The computed strains.
Notes
-----
.. math::
E(\lambda, k) = \frac{1}{k} \left( \lambda^k - 1 \right)
"""
if k == 0:
strain = np.log(stretch)
else:
strain = (stretch**k - 1) / k
return strain
[docs]
def strain(
field, C=None, fun=strain_stretch_1d, tensor=True, asvoigt=False, n=0, **kwargs
):
r"""Return Lagrangian strain tensor or its principal values of the n-th field.
Parameters
----------
field : FieldContainer or None
A field container with a displacement field from which the deformation gradient
tensor is extracted if ``def_grad`` is None.
C : ndarray of shape (N, N, ...) or None, optional
Optional array of right Cauchy-Green deformation tensors. If None, the
right Cauchy-Green deformation tensor is obtained from the field. Default is
None.
fun : callable, optional
A callable for the one-dimensional strain-stretch relation. Function signature
must be ``lambda stretch, **kwargs: strain`` (default is the log. strain,
:func:`~felupe.math.strain_stretch_1d` with ``k=0``).
tensor : bool, optional
Assemble and return the strain tensors if True or return their principal values
only if False. Default is True.
asvoigt : bool, optional
Return the symmetric strain tensors in reduced vector (Voigt) storage. Default
is False.
n : int, optional
The index of the displacement field (default is 0).
**kwargs : dict, optional
Optional keyword-arguments are passed to ``fun(stretch, **kwargs)``.
Returns
-------
ndarray of shape (N, N, ...) tensor, (N!, ...) asvoigt or (N, ...) princ. values
The strain tensors or their principal values.
Notes
-----
The right Cauchy-Green deformation tensor is defined by the dot-products of the
column vectors of the deformation gradient tensor and enables a quadratic length
measure, see Eq. :eq:`right-cauchy-green-deformation-tensor`.
.. math::
:label: right-cauchy-green-deformation-tensor
\boldsymbol{C} = \boldsymbol{F}^T \boldsymbol{F}
The principal stretches are obtained as the square-roots of the eigenvalues of the
right Cauchy-Green deformation tensor, see Eq. :eq:`principal-stretches`.
.. math::
:label: principal-stretches
\lambda^2_\alpha, \boldsymbol{N}_\alpha = \text{eig}\left( \boldsymbol{C} \right)
The Lagrangian strain tensor is assembled with the one-dimensional strain-stretch
relation from Eq. :eq:`principal-strain` and the eigenbases as dyadic products of
the eigenvectors, see Eq. :eq:`strain-tensor`.
.. math::
:label: principal-strain
E_\alpha = \text{f}\left( \lambda_\alpha \right)
.. math::
:label: strain-tensor
\boldsymbol{E} = \sum_\alpha E_\alpha \
\boldsymbol{N}_\alpha \otimes \boldsymbol{N}_\alpha
See Also
-------
math.strain_stretch_1d : Compute the Seth-Hill strains.
"""
if C is None:
F = deformation_gradient(field)
C = dot(transpose(F), F)
if tensor:
w, N = eigh(C)
stretch = np.sqrt(w)
tensor = np.einsum("a...,ia...,ja...->ij...", fun(stretch, **kwargs), N, N)
if asvoigt:
# double the off-diagonal items in Voigt-notation for strain tensors
tensor = tensor_to_vector(tensor, strain=True)
return tensor
else:
stretch = np.sqrt(eigvalsh(C))
return fun(stretch, **kwargs)
[docs]
def values(field):
"Return values of a field or a tuple of fields."
return np.concatenate([f.values.ravel() for f in field.fields])
[docs]
def norm(array, axis=None):
"Calculate the norm of an array or the norms of a list of arrays."
if isinstance(array, list):
return np.array([np.linalg.norm(arr, axis=axis) for arr in array])
else:
return np.linalg.norm(array, axis=axis)
[docs]
def interpolate(field):
"Interpolate method of a field."
return field.interpolate()
[docs]
def grad(x, **kwargs):
"Return the gradient of a field or the gradient of a basis-array."
if callable(x.grad):
return x.grad(**kwargs)
else:
return x.grad
def hess(x, **kwargs):
"Return the hessian of a field or the hessian of a basis-array."
if callable(x.hess):
return x.hess(**kwargs)
else:
return x.hess
