308 lines
8.0 KiB
Python
308 lines
8.0 KiB
Python
import numpy as np
|
||
|
||
def Cauchy(P,F):
|
||
"""
|
||
Return Cauchy stress calculated from first Piola-Kirchhoff stress and deformation gradient.
|
||
|
||
Resulting tensor is symmetrized as the Cauchy stress needs to be symmetric.
|
||
|
||
Parameters
|
||
----------
|
||
F : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Deformation gradient.
|
||
P : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
First Piola-Kirchhoff stress.
|
||
|
||
"""
|
||
if np.shape(F) == np.shape(P) == (3,3):
|
||
sigma = 1.0/np.linalg.det(F) * np.dot(P,F.T)
|
||
else:
|
||
sigma = np.einsum('i,ijk,ilk->ijl',1.0/np.linalg.det(F),P,F)
|
||
return symmetric(sigma)
|
||
|
||
|
||
def deviatoric_part(T):
|
||
"""
|
||
Return deviatoric part of a tensor.
|
||
|
||
Parameters
|
||
----------
|
||
T : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Tensor of which the deviatoric part is computed.
|
||
|
||
"""
|
||
return T - np.eye(3)*spherical_part(T) if np.shape(T) == (3,3) else \
|
||
T - np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),[T.shape[0],3,3]),spherical_part(T))
|
||
|
||
|
||
def eigenvalues(T_sym):
|
||
"""
|
||
Return the eigenvalues, i.e. principal components, of a symmetric tensor.
|
||
|
||
The eigenvalues are sorted in ascending order, each repeated according to
|
||
its multiplicity.
|
||
|
||
Parameters
|
||
----------
|
||
T_sym : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Symmetric tensor of which the eigenvalues are computed.
|
||
|
||
"""
|
||
return np.linalg.eigvalsh(symmetric(T_sym))
|
||
|
||
|
||
def eigenvectors(T_sym,RHS=False):
|
||
"""
|
||
Return eigenvectors of a symmetric tensor.
|
||
|
||
The eigenvalues are sorted in ascending order of their associated eigenvalues.
|
||
|
||
Parameters
|
||
----------
|
||
T_sym : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Symmetric tensor of which the eigenvectors are computed.
|
||
RHS: bool, optional
|
||
Enforce right-handed coordinate system. Default is False.
|
||
|
||
"""
|
||
(u,v) = np.linalg.eigh(symmetric(T_sym))
|
||
|
||
if RHS:
|
||
if np.shape(T_sym) == (3,3):
|
||
if np.linalg.det(v) < 0.0: v[:,2] *= -1.0
|
||
else:
|
||
v[np.linalg.det(v) < 0.0,:,2] *= -1.0
|
||
return v
|
||
|
||
|
||
def left_stretch(T):
|
||
"""
|
||
Return the left stretch of a tensor.
|
||
|
||
Parameters
|
||
----------
|
||
T : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Tensor of which the left stretch is computed.
|
||
|
||
"""
|
||
return __polar_decomposition(T,'V')[0]
|
||
|
||
|
||
def maximum_shear(T_sym):
|
||
"""
|
||
Return the maximum shear component of a symmetric tensor.
|
||
|
||
Parameters
|
||
----------
|
||
T_sym : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Symmetric tensor of which the maximum shear is computed.
|
||
|
||
"""
|
||
w = eigenvalues(T_sym)
|
||
return (w[0] - w[2])*0.5 if np.shape(T_sym) == (3,3) else \
|
||
(w[:,0] - w[:,2])*0.5
|
||
|
||
|
||
def Mises_strain(epsilon):
|
||
"""
|
||
Return the Mises equivalent of a strain tensor.
|
||
|
||
Parameters
|
||
----------
|
||
epsilon : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Symmetric strain tensor of which the von Mises equivalent is computed.
|
||
|
||
"""
|
||
return __Mises(epsilon,2.0/3.0)
|
||
|
||
|
||
def Mises_stress(sigma):
|
||
"""
|
||
Return the Mises equivalent of a stress tensor.
|
||
|
||
Parameters
|
||
----------
|
||
sigma : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Symmetric stress tensor of which the von Mises equivalent is computed.
|
||
|
||
"""
|
||
return __Mises(sigma,3.0/2.0)
|
||
|
||
|
||
def PK2(P,F):
|
||
"""
|
||
Calculate second Piola-Kirchhoff stress from first Piola-Kirchhoff stress and deformation gradient.
|
||
|
||
Parameters
|
||
----------
|
||
P : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
First Piola-Kirchhoff stress.
|
||
F : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Deformation gradient.
|
||
|
||
"""
|
||
if np.shape(F) == np.shape(P) == (3,3):
|
||
S = np.dot(np.linalg.inv(F),P)
|
||
else:
|
||
S = np.einsum('ijk,ikl->ijl',np.linalg.inv(F),P)
|
||
return symmetric(S)
|
||
|
||
|
||
def right_stretch(T):
|
||
"""
|
||
Return the right stretch of a tensor.
|
||
|
||
Parameters
|
||
----------
|
||
T : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Tensor of which the right stretch is computed.
|
||
|
||
"""
|
||
return __polar_decomposition(T,'U')[0]
|
||
|
||
|
||
def rotational_part(T):
|
||
"""
|
||
Return the rotational part of a tensor.
|
||
|
||
Parameters
|
||
----------
|
||
T : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Tensor of which the rotational part is computed.
|
||
|
||
"""
|
||
return __polar_decomposition(T,'R')[0]
|
||
|
||
|
||
def spherical_part(T,tensor=False):
|
||
"""
|
||
Return spherical (hydrostatic) part of a tensor.
|
||
|
||
Parameters
|
||
----------
|
||
T : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Tensor of which the hydrostatic part is computed.
|
||
tensor : bool, optional
|
||
Map spherical part onto identity tensor. Default is false
|
||
|
||
"""
|
||
if T.shape == (3,3):
|
||
sph = np.trace(T)/3.0
|
||
return sph if not tensor else np.eye(3)*sph
|
||
else:
|
||
sph = np.trace(T,axis1=1,axis2=2)/3.0
|
||
if not tensor:
|
||
return sph
|
||
else:
|
||
return np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),(T.shape[0],3,3)),sph)
|
||
|
||
|
||
def strain_tensor(F,t,m):
|
||
"""
|
||
Return strain tensor calculated from deformation gradient.
|
||
|
||
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) or (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.
|
||
|
||
"""
|
||
F_ = F.reshape((1,3,3)) if F.shape == (3,3) else F
|
||
if t == 'V':
|
||
B = np.matmul(F_,transpose(F_))
|
||
w,n = np.linalg.eigh(B)
|
||
elif t == 'U':
|
||
C = np.matmul(transpose(F_),F_)
|
||
w,n = np.linalg.eigh(C)
|
||
|
||
if m > 0.0:
|
||
eps = 1.0/(2.0*abs(m)) * (+ np.matmul(n,np.einsum('ij,ikj->ijk',w**m,n))
|
||
- np.broadcast_to(np.eye(3),[F_.shape[0],3,3]))
|
||
elif m < 0.0:
|
||
eps = 1.0/(2.0*abs(m)) * (- np.matmul(n,np.einsum('ij,ikj->ijk',w**m,n))
|
||
+ np.broadcast_to(np.eye(3),[F_.shape[0],3,3]))
|
||
else:
|
||
eps = np.matmul(n,np.einsum('ij,ikj->ijk',0.5*np.log(w),n))
|
||
|
||
return eps.reshape((3,3)) if np.shape(F) == (3,3) else \
|
||
eps
|
||
|
||
|
||
def symmetric(T):
|
||
"""
|
||
Return the symmetrized tensor.
|
||
|
||
Parameters
|
||
----------
|
||
T : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Tensor of which the symmetrized values are computed.
|
||
|
||
"""
|
||
return (T+transpose(T))*0.5
|
||
|
||
|
||
def transpose(T):
|
||
"""
|
||
Return the transpose of a tensor.
|
||
|
||
Parameters
|
||
----------
|
||
T : numpy.ndarray of shape (:,3,3) or (3,3)
|
||
Tensor of which the transpose is computed.
|
||
|
||
"""
|
||
return T.T if np.shape(T) == (3,3) else \
|
||
np.transpose(T,(0,2,1))
|
||
|
||
|
||
def __polar_decomposition(T,requested):
|
||
"""
|
||
Singular value decomposition.
|
||
|
||
Parameters
|
||
----------
|
||
T : numpy.ndarray of shape (:,3,3) or (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, s, vh = np.linalg.svd(T)
|
||
R = np.dot(u,vh) if np.shape(T) == (3,3) else \
|
||
np.einsum('ijk,ikl->ijl',u,vh)
|
||
|
||
output = []
|
||
if 'R' in requested:
|
||
output.append(R)
|
||
if 'V' in requested:
|
||
output.append(np.dot(T,R.T) if np.shape(T) == (3,3) else np.einsum('ijk,ilk->ijl',T,R))
|
||
if 'U' in requested:
|
||
output.append(np.dot(R.T,T) if np.shape(T) == (3,3) else np.einsum('ikj,ikl->ijl',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) or (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.sum(d**2.0))) if np.shape(T_sym) == (3,3) else \
|
||
np.sqrt(s*np.einsum('ijk->i',d**2.0))
|