DAMASK_EICMD/python/damask/mechanics.py

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import numpy as np
def Cauchy(F,P):
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"""
Return Cauchy stress calculated from 1. Piola-Kirchhoff stress and deformation gradient.
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Resulting tensor is symmetrized as the Cauchy stress needs to be symmetric.
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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)
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def deviatoric_part(x):
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"""
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Return deviatoric part of a tensor.
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Parameters
----------
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x : numpy.array of shape (:,3,3) or (3,3)
Tensor of which the deviatoric part is computed.
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"""
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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))
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def eigenvalues(x):
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"""
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Return the eigenvalues, i.e. principal components, of a symmetric tensor.
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The eigenvalues are sorted in ascending order, each repeated according to
its multiplicity.
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Parameters
----------
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x : numpy.array of shape (:,3,3) or (3,3)
Symmetric tensor of which the eigenvalues are computed.
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"""
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return np.linalg.eigvalsh(symmetric(x))
def eigenvectors(x,RHS=False):
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"""
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Return eigenvectors of a symmetric tensor.
The eigenvalues are sorted in ascending order of their associated eigenvalues.
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Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
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Symmetric tensor of which the eigenvectors are computed.
RHS: bool, optional
Enforce right-handed coordinate system. Default is False.
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"""
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(u,v) = np.linalg.eigh(symmetric(x))
if RHS:
if np.shape(x) == (3,3):
if np.linalg.det(v) < 0.0: v[:,2] *= -1.0
else:
v[np.linalg.det(v) < 0.0,:,2] *= -1.0
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return v
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def left_stretch(x):
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"""
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Return the left stretch of a tensor.
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Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the left stretch is computed.
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"""
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return __polar_decomposition(x,'V')[0]
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def maximum_shear(x):
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"""
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Return the maximum shear component of a symmetric tensor.
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Parameters
----------
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x : numpy.array of shape (:,3,3) or (3,3)
Symmetric tensor of which the maximum shear is computed.
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"""
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w = eigenvalues(x)
return (w[0] - w[2])*0.5 if np.shape(x) == (3,3) else \
(w[:,0] - w[:,2])*0.5
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def Mises_strain(epsilon):
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"""
Return the Mises equivalent of a strain tensor.
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Parameters
----------
epsilon : numpy.array of shape (:,3,3) or (3,3)
Symmetric strain tensor of which the von Mises equivalent is computed.
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"""
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return __Mises(epsilon,2.0/3.0)
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def Mises_stress(sigma):
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"""
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Return the Mises equivalent of a stress tensor.
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Parameters
----------
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sigma : numpy.array of shape (:,3,3) or (3,3)
Symmetric stress tensor of which the von Mises equivalent is computed.
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"""
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return __Mises(sigma,3.0/2.0)
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def PK2(F,P):
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"""
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Return 2. Piola-Kirchhoff stress calculated from 1. Piola-Kirchhoff stress and deformation gradient.
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Parameters
----------
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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.
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"""
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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)
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def right_stretch(x):
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"""
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Return the right stretch of a tensor.
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Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the right stretch is computed.
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"""
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return __polar_decomposition(x,'U')[0]
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def rotational_part(x):
"""
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Return the rotational part of a tensor.
Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the rotational part is computed.
"""
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return __polar_decomposition(x,'R')[0]
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def spherical_part(x,tensor=False):
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"""
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Return spherical (hydrostatic) part of a tensor.
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Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the hydrostatic part is computed.
tensor : bool, optional
Map spherical part onto identity tensor. Default is false
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"""
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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)
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def strain_tensor(F,t,m):
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"""
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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
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Parameters
----------
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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.
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"""
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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)
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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))
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return eps.reshape((3,3)) if np.shape(F) == (3,3) else \
eps
def symmetric(x):
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"""
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Return the symmetrized tensor.
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Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the symmetrized values are computed.
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"""
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return (x+transpose(x))*0.5
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def transpose(x):
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"""
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Return the transpose of a tensor.
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Parameters
----------
x : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the transpose is computed.
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"""
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return x.T if np.shape(x) == (3,3) else \
np.transpose(x,(0,2,1))
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def __polar_decomposition(x,requested):
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"""
Singular value decomposition.
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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,
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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)
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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))
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return tuple(output)
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def __Mises(x,s):
"""
Base equation for Mises equivalent of a stres or strain tensor.
Parameters
----------
x : numpy.array 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(x)
return np.sqrt(s*(np.sum(d**2.0))) if np.shape(x) == (3,3) else \
np.sqrt(s*np.einsum('ijk->i',d**2.0))