"""
Finite-strain continuum mechanics.
All routines operate on numpy.ndarrays of shape (...,3,3).
"""
from typing import Sequence as _Sequence, Union as _Union, Literal as _Literal
import numpy as _np
from . import tensor as _tensor
from . import _rotation
[docs]
def equivalent_strain_Mises(epsilon: _np.ndarray) -> _np.ndarray:
r"""
Calculate the von Mises equivalent of a strain tensor.
Parameters
----------
epsilon : numpy.ndarray, shape (...,3,3)
Symmetric strain tensor of which the von Mises equivalent is computed.
Returns
-------
epsilon_vM : numpy.ndarray, shape (...)
Von Mises equivalent strain of epsilon.
See Also
--------
equivalent_stress_Mises : Calculate the von Mises equivalent
of a stress tensor.
Notes
-----
The von Mises equivalent of a strain tensor is defined as:
.. math::
\epsilon_\text{vM} = \sqrt{\frac{2}{3}\,\epsilon^\prime_{ij} \epsilon^\prime_{ij}}
where :math:`\vb*{\epsilon}^\prime` is the deviatoric part
of the strain tensor.
"""
return _equivalent_Mises(epsilon,2.0/3.0)
[docs]
def equivalent_stress_Mises(sigma: _np.ndarray) -> _np.ndarray:
r"""
Calculate the von Mises equivalent of a stress tensor.
Parameters
----------
sigma : numpy.ndarray, shape (...,3,3)
Symmetric stress tensor of which the von Mises equivalent is computed.
Returns
-------
sigma_vM : numpy.ndarray, shape (...)
Von Mises equivalent stress of sigma.
See Also
--------
equivalent_strain_Mises : Calculate the von Mises equivalent
of a strain tensor.
Notes
-----
The von Mises equivalent of a stress tensor is defined as:
.. math::
\sigma_\text{vM} = \sqrt{\frac{3}{2}\,\sigma^\prime_{ij} \sigma^\prime_{ij}}
where :math:`\vb*{\sigma}^\prime` is the deviatoric part
of the stress tensor.
"""
return _equivalent_Mises(sigma,3.0/2.0)
[docs]
def maximum_shear(T_sym: _np.ndarray) -> _np.ndarray:
"""
Calculate the maximum shear component of a symmetric tensor.
Parameters
----------
T_sym : numpy.ndarray, shape (...,3,3)
Symmetric tensor of which the maximum shear is computed.
Returns
-------
gamma_max : numpy.ndarray, shape (...)
Maximum shear of T_sym.
Notes
-----
The maximum shear component is half of the difference
between the maximum and minium eigenvalue.
"""
w = _tensor.eigenvalues(T_sym)
return (w[...,0] - w[...,2])*0.5
[docs]
def rotation(T: _np.ndarray) -> _rotation.Rotation:
r"""
Calculate the rotational part of a tensor.
Parameters
----------
T : numpy.ndarray, shape (...,3,3)
Tensor of which the rotational part is computed.
Returns
-------
R : damask.Rotation, shape (...)
Rotational part of the tensor.
See Also
--------
damask.Rotation : Rotation with functionality for
conversion between different representations.
Notes
-----
The rotational part is calculated from the polar decomposition:
.. math::
\vb{R} = \vb{T} \vb{U}^{-1} = \vb{V}^{-1} \vb{T}
where :math:`\vb{V}` and :math:`\vb{U}` are the left
and right stretch tensor, respectively.
"""
return _rotation.Rotation.from_matrix(_polar_decomposition(T,'R')[0])
[docs]
def strain(F: _np.ndarray,
t: _Literal['V', 'U'], # noqa: F821
m: float) -> _np.ndarray:
r"""
Calculate strain tensor (Seth–Hill family).
Parameters
----------
F : numpy.ndarray, shape (...,3,3)
Deformation gradient.
t : {'V', 'U'}
Type of the polar decomposition, 'V' for left stretch tensor
or 'U' for right stretch tensor.
m : float
Order of the strain.
Returns
-------
epsilon : numpy.ndarray, shape (...,3,3)
Strain of F.
Notes
-----
The strain is defined as:
.. math::
\vb*{\epsilon}_V^{(m)} = \begin{cases}
\ln (\vb{V}) \text{ if }m = 0\\
\frac{1}{2m} (\vb{V}^{2m} - \vb{I}) \text{ else}
\end{cases} \\\\
\vb*{\epsilon}_U^{(m)} = \begin{cases}
\ln (\vb{U}) \text{ if }m = 0\\
\frac{1}{2m} (\vb{U}^{2m} - \vb{I}) \text{ else}
\end{cases}
The presence of rotational parts in the elastic and plastic deformation
gradient calls for the use of
material/Lagragian strain measures (based on 'U') for plastic strains and
spatial/Eulerian strain measures (based on 'V') for elastic strains
when calculating averages.
References
----------
| https://en.wikipedia.org/wiki/Finite_strain_theory
| https://de.wikipedia.org/wiki/Verzerrungstensor
"""
if t not in ['V', 'U']: raise ValueError('polar decomposition type not in {V, U}')
w,n = _np.linalg.eigh(deformation_Cauchy_Green_left(F) if t=='V' else deformation_Cauchy_Green_right(F))
w = _np.clip(w,1e-16,None)
return 0.5 * _np.einsum('...j,...kj,...lj',_np.log(w),n,n) if m == 0.0 \
else 0.5/m * (_np.einsum('...j,...kj,...lj',w**m, n,n) - _np.eye(3))
[docs]
def stress_Cauchy(P: _np.ndarray,
F: _np.ndarray) -> _np.ndarray:
r"""
Calculate the Cauchy stress (true stress).
Resulting tensor is symmetrized as the Cauchy stress is
symmetric by definition.
Parameters
----------
P : numpy.ndarray, shape (...,3,3)
First Piola-Kirchhoff stress.
F : numpy.ndarray, shape (...,3,3)
Deformation gradient.
Returns
-------
sigma : numpy.ndarray, shape (...,3,3)
Cauchy stress.
See Also
--------
stress_second_Piola_Kirchhoff : Calculate the second Piola-Kirchhoff
stress.
Notes
-----
The Cauchy stress is defined as:
.. math::
\vb*{\sigma} = \vb{P} \vb{F}^\text{T}/\operatorname{det}(\vb{F})
References
----------
J. Bonet and R. D. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis
Cambridge University Press, 2008
https://doi.org/10.1017/CBO9780511755446
"""
return _tensor.symmetric(_np.einsum('...,...ij,...kj',1.0/_np.linalg.det(F),P,F))
[docs]
def stress_second_Piola_Kirchhoff(P: _np.ndarray,
F: _np.ndarray) -> _np.ndarray:
r"""
Calculate the second Piola-Kirchhoff stress.
Resulting tensor is symmetrized as the second Piola-Kirchhoff
stress is symmetric by definition.
Parameters
----------
P : numpy.ndarray, shape (...,3,3)
First Piola-Kirchhoff stress.
F : numpy.ndarray, shape (...,3,3)
Deformation gradient.
Returns
-------
S : numpy.ndarray, shape (...,3,3)
Second Piola-Kirchhoff stress.
See Also
--------
stress_Cauchy : Calculate the Cauchy stress (true stress).
Notes
-----
The second Piola-Kirchhoff stress is defined as:
.. math::
\vb{S} = \vb{F}^{-1} \vb{P}
which is the definition in nonlinear continuum mechanics.
As such, no intermediate configuration, for instance that reached
by :math:`\vb{F}_\text{p}`, is taken into account.
References
----------
J. Bonet and R. D. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis
Cambridge University Press, 2008
https://doi.org/10.1017/CBO9780511755446
"""
return _tensor.symmetric(_np.einsum('...ij,...jk',_np.linalg.inv(F),P))
[docs]
def stretch_left(T: _np.ndarray) -> _np.ndarray:
r"""
Calculate left stretch of a tensor.
Parameters
----------
T : numpy.ndarray, shape (...,3,3)
Tensor of which the left stretch is computed.
Returns
-------
V : numpy.ndarray, shape (...,3,3)
Left stretch tensor from Polar decomposition of T.
See Also
--------
stretch_right : Calculate right stretch of a tensor.
Notes
-----
The left stretch tensor is calculated from the
polar decomposition:
.. math::
\vb{V} = \vb{T} \vb{R}^\text{T}
where :math:`\vb{R}` is a rotation.
"""
return _polar_decomposition(T,'V')[0]
[docs]
def stretch_right(T: _np.ndarray) -> _np.ndarray:
r"""
Calculate right stretch of a tensor.
Parameters
----------
T : numpy.ndarray, shape (...,3,3)
Tensor of which the right stretch is computed.
Returns
-------
U : numpy.ndarray, shape (...,3,3)
Left stretch tensor from Polar decomposition of T.
See Also
--------
stretch_left : Calculate right left of a tensor.
Notes
-----
The right stretch tensor is calculated from the
polar decomposition:
.. math::
\vb{U} = \vb{R}^\text{T} \vb{T}
where :math:`\vb{R}` is a rotation.
"""
return _polar_decomposition(T,'U')[0]
def _polar_decomposition(T: _np.ndarray,
requested: _Union[str, _Sequence[str]]) -> tuple:
"""
Perform singular value decomposition.
Parameters
----------
T : numpy.ndarray, shape (...,3,3)
Tensor of which the singular values are computed.
requested : sequence of {'R', 'U', 'V'}
Requested outputs: 'R' for the rotation tensor,
'V' for left stretch tensor, and 'U' for right stretch tensor.
Returns
-------
VRU : tuple of numpy.ndarray, shape (...,3,3)
Requested components of the singular value decomposition.
"""
u, _, vh = _np.linalg.svd(T)
R = u @ vh
output = []
if 'R' in requested:
output+=[R]
if 'V' in requested:
output+=[_np.einsum('...ij,...kj',T,R)]
if 'U' in requested:
output+=[_np.einsum('...ji,...jk',R,T)]
if len(output) == 0 or len(set(['V','R','U']).union(requested))> 3:
raise ValueError(f'requested invalid dataset {requested}')
return tuple(output)
def _equivalent_Mises(T_sym: _np.ndarray,
s: float) -> _np.ndarray:
"""
Base equation for Mises equivalent of a stress or strain tensor.
Parameters
----------
T_sym : numpy.ndarray, shape (...,3,3)
Symmetric tensor of which the von Mises equivalent is computed.
s : float
Scaling factor (2/3 for strain, 3/2 for stress).
Returns
-------
eq : numpy.ndarray, shape (...)
Scaled second invariant of the deviatoric part of T_sym.
"""
d = _tensor.deviatoric(T_sym)
return _np.sqrt(s*_np.sum(d**2.0,axis=(-1,-2)))
