DAMASK_EICMD/python/damask/mechanics.py

287 lines
6.6 KiB
Python
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

"""
Finite-strain continuum mechanics.
All routines operate on numpy.ndarrays of shape (...,3,3).
"""
from . import tensor as _tensor
from . import _rotation
import numpy as _np
def deformation_Cauchy_Green_left(F):
"""
Calculate left Cauchy-Green deformation tensor (Finger deformation tensor).
Parameters
----------
F : numpy.ndarray of shape (...,3,3)
Deformation gradient.
Returns
-------
B : numpy.ndarray of shape (...,3,3)
Left Cauchy-Green deformation tensor.
"""
return _np.matmul(F,_tensor.transpose(F))
def deformation_Cauchy_Green_right(F):
"""
Calculate right Cauchy-Green deformation tensor.
Parameters
----------
F : numpy.ndarray of shape (...,3,3)
Deformation gradient.
Returns
-------
C : numpy.ndarray of shape (...,3,3)
Right Cauchy-Green deformation tensor.
"""
return _np.matmul(_tensor.transpose(F),F)
def equivalent_strain_Mises(epsilon):
"""
Calculate the Mises equivalent of a strain tensor.
Parameters
----------
epsilon : numpy.ndarray of shape (...,3,3)
Symmetric strain tensor of which the von Mises equivalent is computed.
Returns
-------
epsilon_vM : numpy.ndarray of shape (...)
Von Mises equivalent strain of epsilon.
"""
return _equivalent_Mises(epsilon,2.0/3.0)
def equivalent_stress_Mises(sigma):
"""
Calculate the Mises equivalent of a stress tensor.
Parameters
----------
sigma : numpy.ndarray of shape (...,3,3)
Symmetric stress tensor of which the von Mises equivalent is computed.
Returns
-------
sigma_vM : numpy.ndarray of shape (...)
Von Mises equivalent stress of sigma.
"""
return _equivalent_Mises(sigma,3.0/2.0)
def maximum_shear(T_sym):
"""
Calculate the maximum shear component of a symmetric tensor.
Parameters
----------
T_sym : numpy.ndarray of shape (...,3,3)
Symmetric tensor of which the maximum shear is computed.
Returns
-------
gamma_max : numpy.ndarray of shape (...)
Maximum shear of T_sym.
"""
w = _tensor.eigenvalues(T_sym)
return (w[...,0] - w[...,2])*0.5
def rotation(T):
"""
Calculate the rotational part of a tensor.
Parameters
----------
T : numpy.ndarray of shape (...,3,3)
Tensor of which the rotational part is computed.
Returns
-------
R : damask.Rotation of shape (...)
Rotational part of the vector.
"""
return _rotation.Rotation.from_matrix(_polar_decomposition(T,'R')[0])
def strain(F,t,m):
"""
Calculate strain tensor (SethHill family).
For details refer to https://en.wikipedia.org/wiki/Finite_strain_theory and
https://de.wikipedia.org/wiki/Verzerrungstensor
Parameters
----------
F : numpy.ndarray of 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 of shape (...,3,3)
Strain of F.
"""
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
def stress_Cauchy(P,F):
"""
Calculate the Cauchy stress (true stress).
Resulting tensor is symmetrized as the Cauchy stress needs to be symmetric.
Parameters
----------
P : numpy.ndarray of shape (...,3,3)
First Piola-Kirchhoff stress.
F : numpy.ndarray of shape (...,3,3)
Deformation gradient.
Returns
-------
sigma : numpy.ndarray of 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,F):
"""
Calculate the second Piola-Kirchhoff stress.
Resulting tensor is symmetrized as the second Piola-Kirchhoff stress
needs to be symmetric.
Parameters
----------
P : numpy.ndarray of shape (...,3,3)
First Piola-Kirchhoff stress.
F : numpy.ndarray of shape (...,3,3)
Deformation gradient.
Returns
-------
S : numpy.ndarray of shape (...,3,3)
Second Piola-Kirchhoff stress.
"""
return _tensor.symmetric(_np.einsum('...ij,...jk',_np.linalg.inv(F),P))
def stretch_left(T):
"""
Calculate left stretch of a tensor.
Parameters
----------
T : numpy.ndarray of shape (...,3,3)
Tensor of which the left stretch is computed.
Returns
-------
V : numpy.ndarray of shape (...,3,3)
Left stretch tensor from Polar decomposition of T.
"""
return _polar_decomposition(T,'V')[0]
def stretch_right(T):
"""
Calculate right stretch of a tensor.
Parameters
----------
T : numpy.ndarray of shape (...,3,3)
Tensor of which the right stretch is computed.
Returns
-------
U : numpy.ndarray of shape (...,3,3)
Left stretch tensor from Polar decomposition of T.
"""
return _polar_decomposition(T,'U')[0]
def _polar_decomposition(T,requested):
"""
Perform singular value decomposition.
Parameters
----------
T : numpy.ndarray of shape (...,3,3)
Tensor of which the singular values are computed.
requested : iterable of str
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.append(R)
if 'V' in requested:
output.append(_np.einsum('...ij,...kj',T,R))
if 'U' in requested:
output.append(_np.einsum('...ji,...jk',R,T))
if len(output) == 0:
raise ValueError('output needs to be out of V, R, U')
return tuple(output)
def _equivalent_Mises(T_sym,s):
"""
Base equation for Mises equivalent of a stress or strain tensor.
Parameters
----------
T_sym : numpy.ndarray of 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)))