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Martin Diehl
parent
5a5dd24687
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1c07152b96
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@ -1,6 +1,6 @@
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"""Finite-strain continuum mechanics."""
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from . import tensor
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from . import tensor as _tensor
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import numpy as _np
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@ -17,10 +17,10 @@ def deformation_Cauchy_Green_left(F):
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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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Left Cauchy-Green deformation _tensor.
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"""
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return _np.matmul(F,tensor.transpose(F))
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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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@ -35,32 +35,10 @@ def deformation_Cauchy_Green_right(F):
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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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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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return tensor.symmetric(_np.einsum('...,...ij,...kj',1.0/_np.linalg.det(F),P,F))
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return _np.matmul(_tensor.transpose(F),F)
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def deviatoric_part(T):
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@ -81,25 +59,6 @@ def deviatoric_part(T):
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return T - spherical_part(T,tensor=True)
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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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@ -136,27 +95,23 @@ def equivalent_stress_Mises(sigma):
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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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def maximum_shear(T_sym):
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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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Calculate the maximum shear component of a symmetric tensor.
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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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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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S : numpy.ndarray of shape (...,3,3)
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Second Piola-Kirchhoff stress.
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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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return tensor.symmetric(_np.einsum('...ij,...jk',_np.linalg.inv(F),P))
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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 rotational_part(T):
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@ -186,14 +141,14 @@ def spherical_part(T,tensor=False):
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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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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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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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@ -213,7 +168,7 @@ def strain(F,t,m):
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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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and ‘U’ for right stretch _tensor.
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m : float
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Order of the strain.
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@ -239,9 +194,55 @@ def strain(F,t,m):
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return eps
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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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return _tensor.symmetric(_np.einsum('...,...ij,...kj',1.0/_np.linalg.det(F),P,F))
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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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return _tensor.symmetric(_np.einsum('...ij,...jk',_np.linalg.inv(F),P))
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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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Calculate left stretch of a _tensor.
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Parameters
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----------
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@ -259,7 +260,7 @@ def stretch_left(T):
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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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Calculate right stretch of a _tensor.
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Parameters
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----------
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@ -11,42 +11,6 @@ to operate on numpy.ndarrays of shape (...,3,3).
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import numpy as _np
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def symmetric(T):
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"""
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Symmetrize 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 symmetrized values are computed.
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Returns
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-------
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T_sym : numpy.ndarray of shape (...,3,3)
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Symmetrized tensor T.
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"""
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return (T+transpose(T))*0.5
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def transpose(T):
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"""
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Transpose 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 transpose is computed.
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Returns
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-------
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T.T : numpy.ndarray of shape (...,3,3)
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Transpose of tensor T.
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"""
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return _np.swapaxes(T,axis2=-2,axis1=-1)
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def eigenvalues(T_sym):
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"""
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Eigenvalues, i.e. principal components, of a symmetric tensor.
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@ -89,3 +53,39 @@ def eigenvectors(T_sym,RHS=False):
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if RHS:
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v[_np.linalg.det(v) < 0.0,:,2] *= -1.0
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return v
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def symmetric(T):
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"""
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Symmetrize 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 symmetrized values are computed.
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Returns
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-------
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T_sym : numpy.ndarray of shape (...,3,3)
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Symmetrized tensor T.
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"""
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return (T+transpose(T))*0.5
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def transpose(T):
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"""
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Transpose 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 transpose is computed.
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Returns
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-------
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T.T : numpy.ndarray of shape (...,3,3)
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Transpose of tensor T.
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"""
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return _np.swapaxes(T,axis2=-2,axis1=-1)
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