86 lines
2.1 KiB
Python
86 lines
2.1 KiB
Python
import numpy as np
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def Cauchy(F,P):
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"""
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Calculate Cauchy stress from 1st 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 (x,3,3) or (3,3)
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Deformation gradient.
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P : numpy.array of shape (x,3,3) or (3,3)
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1st Piola-Kirchhoff.
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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(F,P)
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return (sigma+sigma.T)*0.5
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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 (sigma + np.transpose(sigma,(0,2,1)))*0.5
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def deviatoric_part(x):
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"""
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Calculate deviatoric part of a tensor.
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Parameters
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----------
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x : numpy.array of shape (x,3,3) or (3,3)
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Tensor.
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"""
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if np.shape(x) == (3,3):
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return x - np.eye(3)*np.trace(x)/3.0
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else:
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return x - np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),[x.shape[0],3,3]),np.trace(x,axis1=1,axis2=2)/3.0)
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def spherical_part(x):
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"""
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Calculate 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 matrix.
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Parameters
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----------
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x : numpy.array of shape (x,3,3) or (3,3)
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Tensor.
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"""
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if np.shape(x) == (3,3):
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return np.trace(x)/3.0
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else:
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return 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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Calculate 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 (x,3,3) or (3,3)
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Symmetric stress tensor.
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"""
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s = deviatoric_part(sigma)
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if np.shape(sigma) == (3,3):
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return np.sqrt(3.0/2.0*np.trace(s))
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else:
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return np.sqrt(np.einsum('ijk->i',s)*3.0/2.0)
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def Mises_strain(epsilon):
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"""
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Calculate the Mises equivalent of a strain tensor.
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Parameters
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----------
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sigma : numpy.array of shape (x,3,3) or (3,3)
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Symmetric strain tensor.
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"""
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s = deviatoric_part(epsilon)
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if np.shape(epsilon) == (3,3):
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return np.sqrt(2.0/3.0*np.trace(s))
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else:
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return np.sqrt(2.0/3.0*np.einsum('ijk->i',s))
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