267 lines
7.2 KiB
Python
267 lines
7.2 KiB
Python
import numpy as np
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def Cauchy(F,P):
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"""
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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
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----------
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F : numpy.array of shape (:,3,3) or (3,3)
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Deformation gradient.
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P : numpy.array of shape (:,3,3) or (3,3)
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1. Piola-Kirchhoff stress.
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"""
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if np.shape(F) == np.shape(P) == (3,3):
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sigma = 1.0/np.linalg.det(F) * np.dot(P,F.T)
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else:
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sigma = np.einsum('i,ijk,ilk->ijl',1.0/np.linalg.det(F),P,F)
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return symmetric(sigma)
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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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----------
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F : numpy.array of shape (:,3,3) or (3,3)
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Deformation gradient.
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P : numpy.array of shape (:,3,3) or (3,3)
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1. Piola-Kirchhoff stress.
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"""
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if np.shape(F) == np.shape(P) == (3,3):
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S = np.dot(np.linalg.inv(F),P)
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else:
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S = np.einsum('ijk,ikl->ijl',np.linalg.inv(F),P)
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return S
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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.
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For details refer to https://en.wikipedia.org/wiki/Finite_strain_theory and
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https://de.wikipedia.org/wiki/Verzerrungstensor
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Parameters
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----------
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F : numpy.array of shape (:,3,3) or (3,3)
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Deformation gradient.
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t : {‘V’, ‘U’}
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Type of the polar decomposition, ‘V’ for left stretch tensor and ‘U’ for right stretch tensor.
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m : float
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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
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if t == 'U':
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B = np.matmul(F_,transpose(F_))
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w,n = np.linalg.eigh(B)
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elif t == 'V':
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C = np.matmul(transpose(F_),F_)
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w,n = np.linalg.eigh(C)
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if m > 0.0:
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eps = 1.0/(2.0*abs(m)) * (+ np.matmul(n,np.einsum('ij,ikj->ijk',w**m,n))
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- np.broadcast_to(np.eye(3),[F_.shape[0],3,3]))
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elif m < 0.0:
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eps = 1.0/(2.0*abs(m)) * (- np.matmul(n,np.einsum('ij,ikj->ijk',w**m,n))
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+ np.broadcast_to(np.eye(3),[F_.shape[0],3,3]))
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else:
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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 \
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eps
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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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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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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 \
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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 spherical_part(x):
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"""
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Return spherical (hydrostatic) part of a tensor.
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A single scalar is returned, i.e. the hydrostatic part is not mapped on the 3rd order identity
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matrix.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the hydrostatic part is computed.
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"""
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return np.trace(x)/3.0 if np.shape(x) == (3,3) else \
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np.trace(x,axis1=1,axis2=2)/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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----------
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sigma : numpy.array of shape (:,3,3) or (3,3)
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Symmetric stress tensor of which the von Mises equivalent is computed.
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"""
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s = deviatoric_part(sigma)
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return np.sqrt(3.0/2.0*(np.sum(s**2.0))) if np.shape(sigma) == (3,3) else \
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np.sqrt(3.0/2.0*np.einsum('ijk->i',s**2.0))
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def Mises_strain(epsilon):
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"""
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Return the Mises equivalent of a strain tensor.
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Parameters
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----------
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epsilon : numpy.array of shape (:,3,3) or (3,3)
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Symmetric strain tensor of which the von Mises equivalent is computed.
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"""
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s = deviatoric_part(epsilon)
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return np.sqrt(2.0/3.0*(np.sum(s**2.0))) if np.shape(epsilon) == (3,3) else \
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np.sqrt(2.0/3.0*np.einsum('ijk->i',s**2.0))
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def symmetric(x):
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"""
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Return the symmetrized tensor.
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Parameters
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----------
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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 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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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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Symmetric tensor of which the maximum shear is computed.
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"""
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w = np.linalg.eigvalsh(symmetric(x)) # eigenvalues in ascending order
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return (w[2] - w[0])*0.5 if np.shape(x) == (3,3) else \
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(w[:,2] - w[:,0])*0.5
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def principal_components(x):
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"""
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Return the principal components of a symmetric tensor.
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The principal components (eigenvalues) are sorted in descending order, each repeated according to
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its multiplicity.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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Symmetric tensor of which the principal compontents are computed.
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"""
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w = np.linalg.eigvalsh(symmetric(x)) # eigenvalues in ascending order
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return w[::-1] if np.shape(x) == (3,3) else \
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w[:,::-1]
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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
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----------
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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 \
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np.transpose(x,(0,2,1))
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def rotational_part(x):
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"""
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Return the rotational part of a tensor.
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Parameters
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----------
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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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"""
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return __polar_decomposition(x,'R')[0]
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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
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----------
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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 right_stretch(x):
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"""
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Return the right stretch of a tensor.
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Parameters
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----------
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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 __polar_decomposition(x,requested):
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"""
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Singular value decomposition.
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Parameters
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----------
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x : numpy.array of shape (:,3,3) or (3,3)
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Tensor of which the singular values are computed.
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requested : iterable of str
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Requested outputs: ‘R’ for the rotation tensor,
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‘V’ for left stretch tensor and ‘U’ for right stretch tensor.
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"""
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u, s, vh = np.linalg.svd(x)
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R = np.dot(u,vh) if np.shape(x) == (3,3) else \
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np.einsum('ijk,ikl->ijl',u,vh)
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output = []
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if 'R' in requested:
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output.append(R)
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if 'V' in requested:
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output.append(np.dot(x,R.T) if np.shape(x) == (3,3) else np.einsum('ijk,ilk->ijl',x,R))
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if 'U' in requested:
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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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