"""
Finite-strain continuum mechanics.
All routines operate on numpy.ndarrays of shape (...,3,3).
"""
from typing import Sequence as _Sequence
import numpy as _np
from . import tensor as _tensor
from . import _rotation
[docs]def equivalent_strain_Mises(epsilon: _np.ndarray) -> _np.ndarray:
"""
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.
"""
return _equivalent_Mises(epsilon,2.0/3.0)
[docs]def equivalent_stress_Mises(sigma: _np.ndarray) -> _np.ndarray:
"""
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.
"""
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.
"""
w = _tensor.eigenvalues(T_sym)
return (w[...,0] - w[...,2])*0.5
[docs]def rotation(T: _np.ndarray) -> _rotation.Rotation:
"""
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.
"""
return _rotation.Rotation.from_matrix(_polar_decomposition(T,'R')[0])
[docs]def strain(F: _np.ndarray,
t: str,
m: float) -> _np.ndarray:
"""
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
and ‘U’ for right stretch tensor.
m : float
Order of the strain.
Returns
-------
epsilon : numpy.ndarray, shape (...,3,3)
Strain of F.
References
----------
https://en.wikipedia.org/wiki/Finite_strain_theory
https://de.wikipedia.org/wiki/Verzerrungstensor
"""
if t == 'V':
w,n = _np.linalg.eigh(deformation_Cauchy_Green_left(F))
elif t == 'U':
w,n = _np.linalg.eigh(deformation_Cauchy_Green_right(F))
if m > 0.0:
eps = 1.0/(2.0*abs(m)) * (+ _np.einsum('...j,...kj,...lj',w**m,n,n) - _np.eye(3))
elif m < 0.0:
eps = 1.0/(2.0*abs(m)) * (- _np.einsum('...j,...kj,...lj',w**m,n,n) + _np.eye(3))
else:
eps = _np.einsum('...j,...kj,...lj',0.5*_np.log(w),n,n)
return eps
[docs]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))
[docs]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))
[docs]def stretch_left(T: _np.ndarray) -> _np.ndarray:
"""
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.
"""
return _polar_decomposition(T,'V')[0]
[docs]def stretch_right(T: _np.ndarray) -> _np.ndarray:
"""
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.
"""
return _polar_decomposition(T,'U')[0]
def _polar_decomposition(T: _np.ndarray,
requested: _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.
"""
u, _, vh = _np.linalg.svd(T)
R = _np.einsum('...ij,...jk',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).
"""
d = _tensor.deviatoric(T_sym)
return _np.sqrt(s*_np.sum(d**2.0,axis=(-1,-2)))