import numpy as np
def Cauchy(F,P):
"""
Return Cauchy stress calculated from 1. Piola-Kirchhoff stress and deformation gradient.
Resulting tensor is symmetrized as the Cauchy stress needs to be symmetric.
Parameters
----------
F : numpy.array of shape (:,3,3) or (3,3)
Deformation gradient.
P : numpy.array of shape (:,3,3) or (3,3)
1. 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 PK2(F,P):
"""
Return 2. Piola-Kirchhoff stress calculated from 1. Piola-Kirchhoff stress and deformation gradient.
Parameters
----------
F : numpy.array of shape (:,3,3) or (3,3)
Deformation gradient.
P : numpy.array of shape (:,3,3) or (3,3)
1. Piola-Kirchhoff stress.
"""
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 S
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.array 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 deviatoric_part(x):
"""
Return deviatoric part of a tensor.
Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
Tensor of which the deviatoric part is computed.
"""
return x - np.eye(3)*spherical_part(x) if np.shape(x) == (3,3) else \
x - np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),[x.shape[0],3,3]),spherical_part(x))
def spherical_part(x,tensor=False):
"""
Return spherical (hydrostatic) part of a tensor.
Parameters
----------
x : numpy.array 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 x.shape == (3,3):
sph = np.trace(x)/3.0
return sph if not tensor else np.eye(3)*sph
else:
sph = np.trace(x,axis1=1,axis2=2)/3.0
if not tensor:
return sph
else:
return np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),(x.shape[0],3,3)),sph)
def Mises_stress(sigma):
"""
Return the Mises equivalent of a stress tensor.
Parameters
----------
sigma : numpy.array of shape (:,3,3) or (3,3)
Symmetric stress tensor of which the von Mises equivalent is computed.
"""
s = deviatoric_part(sigma)
return np.sqrt(3.0/2.0*(np.sum(s**2.0))) if np.shape(sigma) == (3,3) else \
np.sqrt(3.0/2.0*np.einsum('ijk->i',s**2.0))
def Mises_strain(epsilon):
"""
Return the Mises equivalent of a strain tensor.
Parameters
----------
epsilon : numpy.array of shape (:,3,3) or (3,3)
Symmetric strain tensor of which the von Mises equivalent is computed.
"""
s = deviatoric_part(epsilon)
return np.sqrt(2.0/3.0*(np.sum(s**2.0))) if np.shape(epsilon) == (3,3) else \
np.sqrt(2.0/3.0*np.einsum('ijk->i',s**2.0))
def symmetric(x):
"""
Return the symmetrized tensor.
Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
Tensor of which the symmetrized values are computed.
"""
return (x+transpose(x))*0.5
def maximum_shear(x):
"""
Return the maximum shear component of a symmetric tensor.
Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
Symmetric tensor of which the maximum shear is computed.
"""
w = np.linalg.eigvalsh(symmetric(x)) # eigenvalues in ascending order
return (w[2] - w[0])*0.5 if np.shape(x) == (3,3) else \
(w[:,2] - w[:,0])*0.5
def principal_components(x):
"""
Return the principal components of a symmetric tensor.
The principal components (eigenvalues) are sorted in descending order, each repeated according to
its multiplicity.
Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
Symmetric tensor of which the principal compontents are computed.
"""
w = np.linalg.eigvalsh(symmetric(x)) # eigenvalues in ascending order
return w[::-1] if np.shape(x) == (3,3) else \
w[:,::-1]
def transpose(x):
"""
Return the transpose of a tensor.
Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
Tensor of which the transpose is computed.
"""
return x.T if np.shape(x) == (3,3) else \
np.transpose(x,(0,2,1))
def rotational_part(x):
"""
Return the rotational part of a tensor.
Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
Tensor of which the rotational part is computed.
"""
return __polar_decomposition(x,'R')[0]
def left_stretch(x):
"""
Return the left stretch of a tensor.
Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
Tensor of which the left stretch is computed.
"""
return __polar_decomposition(x,'V')[0]
def right_stretch(x):
"""
Return the right stretch of a tensor.
Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
Tensor of which the right stretch is computed.
"""
return __polar_decomposition(x,'U')[0]
def __polar_decomposition(x,requested):
"""
Singular value decomposition.
Parameters
----------
x : numpy.array 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(x)
R = np.dot(u,vh) if np.shape(x) == (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(x,R.T) if np.shape(x) == (3,3) else np.einsum('ijk,ilk->ijl',x,R))
if 'U' in requested:
output.append(np.dot(R.T,x) if np.shape(x) == (3,3) else np.einsum('ikj,ikl->ijl',R,x))
return tuple(output)