327 lines
7.6 KiB
Python
327 lines
7.6 KiB
Python
"""Finite-strain continuum mechanics."""
|
||
|
||
from . import tensor
|
||
|
||
import numpy as _np
|
||
|
||
|
||
def Cauchy_Green_deformation_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 Cauchy_Green_deformation_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 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.
|
||
|
||
"""
|
||
sigma = _np.einsum('...,...ij,...kj->...ik',1.0/_np.linalg.det(F),P,F)
|
||
return tensor.symmetric(sigma)
|
||
|
||
|
||
def deviatoric_part(T):
|
||
"""
|
||
Calculate deviatoric part of a tensor.
|
||
|
||
Parameters
|
||
----------
|
||
T : numpy.ndarray of shape (...,3,3)
|
||
Tensor of which the deviatoric part is computed.
|
||
|
||
Returns
|
||
-------
|
||
T' : numpy.ndarray of shape (...,3,3)
|
||
Deviatoric part of T.
|
||
|
||
"""
|
||
return T - _np.einsum('...ij,...->...ij',_np.eye(3),spherical_part(T))
|
||
|
||
|
||
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 Mises_strain(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 _Mises(epsilon,2.0/3.0)
|
||
|
||
|
||
def Mises_stress(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 _Mises(sigma,3.0/2.0)
|
||
|
||
|
||
def PK2(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.
|
||
|
||
"""
|
||
S = _np.einsum('...jk,...kl->...jl',_np.linalg.inv(F),P)
|
||
return tensor.symmetric(S)
|
||
|
||
|
||
def rotational_part(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 : numpy.ndarray of shape (...,3,3)
|
||
Rotational part.
|
||
|
||
"""
|
||
return _polar_decomposition(T,'R')[0]
|
||
|
||
|
||
def spherical_part(T,tensor=False):
|
||
"""
|
||
Calculate spherical (hydrostatic) part of a tensor.
|
||
|
||
Parameters
|
||
----------
|
||
T : numpy.ndarray of shape (...,3,3)
|
||
Tensor of which the hydrostatic part is computed.
|
||
tensor : bool, optional
|
||
Map spherical part onto identity tensor. Defaults to false
|
||
|
||
Returns
|
||
-------
|
||
p : numpy.ndarray of shape (...)
|
||
unless tensor == True: shape (...,3,3)
|
||
Spherical part of tensor T, e.g. the hydrostatic part/pressure
|
||
of a stress tensor.
|
||
|
||
"""
|
||
sph = _np.trace(T,axis2=-2,axis1=-1)/3.0
|
||
return _np.einsum('...jk,...->...jk',_np.eye(3),sph) if tensor else sph
|
||
|
||
|
||
def strain(F,t,m):
|
||
"""
|
||
Calculate strain tensor (Seth–Hill 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(Cauchy_Green_deformation_left(F))
|
||
elif t == 'U':
|
||
w,n = _np.linalg.eigh(Cauchy_Green_deformation_right(F))
|
||
|
||
if m > 0.0:
|
||
eps = 1.0/(2.0*abs(m)) * (+ _np.matmul(n,_np.einsum('...j,...kj->...jk',w**m,n))
|
||
- _np.einsum('...jk->...jk',_np.eye(3)))
|
||
|
||
elif m < 0.0:
|
||
eps = 1.0/(2.0*abs(m)) * (- _np.matmul(n,_np.einsum('...j,...kj->...jk',w**m,n))
|
||
+ _np.einsum('...jk->...jk',_np.eye(3)))
|
||
else:
|
||
eps = _np.matmul(n,_np.einsum('...j,...kj->...jk',0.5*_np.log(w),n))
|
||
|
||
return eps
|
||
|
||
|
||
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->...ik',u,vh)
|
||
|
||
output = []
|
||
if 'R' in requested:
|
||
output.append(R)
|
||
if 'V' in requested:
|
||
output.append(_np.einsum('...ij,...kj->...ik',T,R))
|
||
if 'U' in requested:
|
||
output.append(_np.einsum('...ji,...jk->...ik',R,T))
|
||
|
||
return tuple(output)
|
||
|
||
|
||
def _Mises(T_sym,s):
|
||
"""
|
||
Base equation for Mises equivalent of a stres 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 = deviatoric_part(T_sym)
|
||
return _np.sqrt(s*_np.einsum('...jk->...',d**2.0))
|
||
|
||
|
||
# for compatibility
|
||
strain_tensor = strain
|