325 lines
7.5 KiB
Python
325 lines
7.5 KiB
Python
"""Finite-strain continuum mechanics."""
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from . import tensor
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import numpy as _np
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def deformation_Cauchy_Green_left(F):
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"""
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Calculate left Cauchy-Green deformation tensor (Finger deformation tensor).
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Parameters
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----------
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F : numpy.ndarray of shape (...,3,3)
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Deformation gradient.
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Returns
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-------
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B : numpy.ndarray of shape (...,3,3)
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Left Cauchy-Green deformation tensor.
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"""
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return _np.matmul(F,tensor.transpose(F))
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def deformation_Cauchy_Green_right(F):
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"""
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Calculate right Cauchy-Green deformation tensor.
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Parameters
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----------
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F : numpy.ndarray of shape (...,3,3)
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Deformation gradient.
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Returns
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-------
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C : numpy.ndarray of shape (...,3,3)
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Right Cauchy-Green deformation tensor.
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"""
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return _np.matmul(tensor.transpose(F),F)
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def stress_Cauchy(P,F):
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"""
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Calculate the Cauchy stress (true stress).
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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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P : numpy.ndarray of shape (...,3,3)
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First Piola-Kirchhoff stress.
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F : numpy.ndarray of shape (...,3,3)
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Deformation gradient.
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Returns
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-------
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sigma : numpy.ndarray of shape (...,3,3)
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Cauchy stress.
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"""
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sigma = _np.einsum('...,...ij,...kj',1.0/_np.linalg.det(F),P,F)
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return tensor.symmetric(sigma)
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def deviatoric_part(T):
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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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T : numpy.ndarray of shape (...,3,3)
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Tensor of which the deviatoric part is computed.
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Returns
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-------
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T' : numpy.ndarray of shape (...,3,3)
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Deviatoric part of T.
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"""
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return T - _np.einsum('...ij,...',_np.eye(3),spherical_part(T))
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def maximum_shear(T_sym):
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"""
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Calculate the maximum shear component of a symmetric tensor.
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Parameters
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----------
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T_sym : numpy.ndarray of shape (...,3,3)
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Symmetric tensor of which the maximum shear is computed.
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Returns
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-------
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gamma_max : numpy.ndarray of shape (...)
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Maximum shear of T_sym.
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"""
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w = tensor.eigenvalues(T_sym)
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return (w[...,0] - w[...,2])*0.5
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def equivalent_strain_Mises(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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epsilon : numpy.ndarray of shape (...,3,3)
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Symmetric strain tensor of which the von Mises equivalent is computed.
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Returns
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-------
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epsilon_vM : numpy.ndarray of shape (...)
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Von Mises equivalent strain of epsilon.
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"""
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return _equivalent_Mises(epsilon,2.0/3.0)
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def equivalent_stress_Mises(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.ndarray of shape (...,3,3)
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Symmetric stress tensor of which the von Mises equivalent is computed.
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Returns
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-------
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sigma_vM : numpy.ndarray of shape (...)
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Von Mises equivalent stress of sigma.
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"""
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return _equivalent_Mises(sigma,3.0/2.0)
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def stress_second_Piola_Kirchhoff(P,F):
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"""
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Calculate the second Piola-Kirchhoff stress.
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Resulting tensor is symmetrized as the second Piola-Kirchhoff stress
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needs to be symmetric.
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Parameters
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----------
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P : numpy.ndarray of shape (...,3,3)
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First Piola-Kirchhoff stress.
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F : numpy.ndarray of shape (...,3,3)
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Deformation gradient.
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Returns
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-------
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S : numpy.ndarray of shape (...,3,3)
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Second Piola-Kirchhoff stress.
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"""
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S = _np.einsum('...jk,...kl',_np.linalg.inv(F),P)
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return tensor.symmetric(S)
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def rotational_part(T):
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"""
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Calculate the rotational part of a tensor.
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Parameters
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----------
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T : numpy.ndarray of shape (...,3,3)
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Tensor of which the rotational part is computed.
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Returns
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-------
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R : numpy.ndarray of shape (...,3,3)
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Rotational part.
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"""
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return _polar_decomposition(T,'R')[0]
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def spherical_part(T,tensor=False):
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"""
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Calculate spherical (hydrostatic) part of a tensor.
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Parameters
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----------
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T : numpy.ndarray of shape (...,3,3)
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Tensor of which the hydrostatic part is computed.
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tensor : bool, optional
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Map spherical part onto identity tensor. Defaults to false
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Returns
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-------
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p : numpy.ndarray of shape (...)
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unless tensor == True: shape (...,3,3)
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Spherical part of tensor T, e.g. the hydrostatic part/pressure
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of a stress tensor.
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"""
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sph = _np.trace(T,axis2=-2,axis1=-1)/3.0
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return _np.einsum('...jk,...',_np.eye(3),sph) if tensor else sph
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def strain(F,t,m):
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"""
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Calculate strain tensor (Seth–Hill family).
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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.ndarray of shape (...,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
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and ‘U’ for right stretch tensor.
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m : float
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Order of the strain.
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Returns
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-------
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epsilon : numpy.ndarray of shape (...,3,3)
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Strain of F.
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"""
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if t == 'V':
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w,n = _np.linalg.eigh(deformation_Cauchy_Green_left(F))
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elif t == 'U':
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w,n = _np.linalg.eigh(deformation_Cauchy_Green_right(F))
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if m > 0.0:
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eps = 1.0/(2.0*abs(m)) * (+ _np.einsum('...j,...kj,...lj',w**m,n,n) - _np.eye(3))
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elif m < 0.0:
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eps = 1.0/(2.0*abs(m)) * (- _np.einsum('...j,...kj,...lj',w**m,n,n) + _np.eye(3))
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else:
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eps = _np.einsum('...j,...kj,...lj',0.5*_np.log(w),n,n)
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return eps
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def stretch_left(T):
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"""
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Calculate left stretch of a tensor.
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Parameters
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----------
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T : numpy.ndarray of shape (...,3,3)
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Tensor of which the left stretch is computed.
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Returns
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-------
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V : numpy.ndarray of shape (...,3,3)
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Left stretch tensor from Polar decomposition of T.
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"""
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return _polar_decomposition(T,'V')[0]
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def stretch_right(T):
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"""
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Calculate right stretch of a tensor.
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Parameters
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----------
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T : numpy.ndarray of shape (...,3,3)
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Tensor of which the right stretch is computed.
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Returns
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-------
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U : numpy.ndarray of shape (...,3,3)
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Left stretch tensor from Polar decomposition of T.
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"""
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return _polar_decomposition(T,'U')[0]
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def _polar_decomposition(T,requested):
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"""
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Perform singular value decomposition.
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Parameters
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----------
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T : numpy.ndarray of shape (...,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, _, vh = _np.linalg.svd(T)
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R = _np.einsum('...ij,...jk',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.einsum('...ij,...kj',T,R))
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if 'U' in requested:
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output.append(_np.einsum('...ji,...jk',R,T))
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return tuple(output)
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def _equivalent_Mises(T_sym,s):
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"""
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Base equation for Mises equivalent of a stress or strain tensor.
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Parameters
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----------
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T_sym : numpy.ndarray of shape (...,3,3)
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Symmetric tensor of which the von Mises equivalent is computed.
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s : float
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Scaling factor (2/3 for strain, 3/2 for stress).
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"""
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d = deviatoric_part(T_sym)
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return _np.sqrt(s*_np.sum(d**2.0,axis=(-1,-2)))
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# for compatibility
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strain_tensor = strain
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