374 lines
8.9 KiB
Python
374 lines
8.9 KiB
Python
"""
|
||
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
|
||
|
||
|
||
def deformation_Cauchy_Green_left(F: _np.ndarray) -> _np.ndarray:
|
||
r"""
|
||
Calculate left Cauchy-Green deformation tensor (Finger deformation tensor).
|
||
|
||
Parameters
|
||
----------
|
||
F : numpy.ndarray, shape (...,3,3)
|
||
Deformation gradient.
|
||
|
||
Returns
|
||
-------
|
||
B : numpy.ndarray, shape (...,3,3)
|
||
Left Cauchy-Green deformation tensor.
|
||
|
||
Notes
|
||
-----
|
||
.. math::
|
||
|
||
\vb{B} = \vb{F} \vb{F}^\text{T}
|
||
|
||
"""
|
||
return _np.matmul(F,_tensor.transpose(F))
|
||
|
||
|
||
def deformation_Cauchy_Green_right(F: _np.ndarray) -> _np.ndarray:
|
||
r"""
|
||
Calculate right Cauchy-Green deformation tensor.
|
||
|
||
Parameters
|
||
----------
|
||
F : numpy.ndarray, shape (...,3,3)
|
||
Deformation gradient.
|
||
|
||
Returns
|
||
-------
|
||
C : numpy.ndarray, shape (...,3,3)
|
||
Right Cauchy-Green deformation tensor.
|
||
|
||
Notes
|
||
-----
|
||
.. math::
|
||
|
||
\vb{C} = \vb{F}^\text{T} \vb{F}
|
||
|
||
"""
|
||
return _np.matmul(_tensor.transpose(F),F)
|
||
|
||
|
||
def equivalent_strain_Mises(epsilon: _np.ndarray) -> _np.ndarray:
|
||
r"""
|
||
Calculate the 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.
|
||
|
||
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)
|
||
|
||
|
||
def equivalent_stress_Mises(sigma: _np.ndarray) -> _np.ndarray:
|
||
r"""
|
||
Calculate the 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.
|
||
|
||
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)
|
||
|
||
|
||
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.
|
||
|
||
"""
|
||
w = _tensor.eigenvalues(T_sym)
|
||
return (w[...,0] - w[...,2])*0.5
|
||
|
||
|
||
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 vector.
|
||
|
||
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])
|
||
|
||
|
||
def strain(F: _np.ndarray,
|
||
#t: _Literal['V', 'U'], should work, but rejected by SC
|
||
t: str,
|
||
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)} = \frac{1}{2m} (\vb{V}^{2m} - \vb{I}) \\\\
|
||
\vb*{\epsilon}_U^{(m)} = \frac{1}{2m} (\vb{U}^{2m} - \vb{I})
|
||
|
||
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))
|
||
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))
|
||
|
||
|
||
def stress_Cauchy(P: _np.ndarray,
|
||
F: _np.ndarray) -> _np.ndarray:
|
||
"""
|
||
Calculate the Cauchy stress (true stress).
|
||
|
||
Resulting tensor is symmetrized as the Cauchy stress needs to be symmetric.
|
||
|
||
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.
|
||
|
||
"""
|
||
return _tensor.symmetric(_np.einsum('...,...ij,...kj',1.0/_np.linalg.det(F),P,F))
|
||
|
||
|
||
def stress_second_Piola_Kirchhoff(P: _np.ndarray,
|
||
F: _np.ndarray) -> _np.ndarray:
|
||
"""
|
||
Calculate the second Piola-Kirchhoff stress.
|
||
|
||
Resulting tensor is symmetrized as the second Piola-Kirchhoff stress
|
||
needs to be symmetric.
|
||
|
||
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.
|
||
|
||
"""
|
||
return _tensor.symmetric(_np.einsum('...ij,...jk',_np.linalg.inv(F),P))
|
||
|
||
|
||
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.
|
||
|
||
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]
|
||
|
||
|
||
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.
|
||
|
||
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:
|
||
raise ValueError('output not in {V, R, U}')
|
||
|
||
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)))
|