364 lines
8.6 KiB
Python
364 lines
8.6 KiB
Python
"""
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Finite-strain continuum mechanics.
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All routines operate on numpy.ndarrays of shape (...,3,3).
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"""
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from typing import Sequence as _Sequence
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import numpy as _np
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from . import tensor as _tensor
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from . import _rotation
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def deformation_Cauchy_Green_left(F: _np.ndarray) -> _np.ndarray:
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r"""
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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, 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, shape (...,3,3)
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Left Cauchy-Green deformation tensor.
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Notes
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-----
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.. math::
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\vb{B} = \vb{F} \vb{F}^\text{T}
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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: _np.ndarray) -> _np.ndarray:
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r"""
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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, 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, shape (...,3,3)
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Right Cauchy-Green deformation tensor.
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Notes
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-----
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.. math::
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\vb{C} = \vb{F}^\text{T} \vb{F}
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"""
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return _np.matmul(_tensor.transpose(F),F)
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def equivalent_strain_Mises(epsilon: _np.ndarray) -> _np.ndarray:
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r"""
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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, 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, shape (...)
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Von Mises equivalent strain of epsilon.
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Notes
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-----
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The von Mises equivalent of a strain tensor is defined as:
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.. math::
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\epsilon_\text{vM} = \sqrt{2/3 \epsilon^\prime_{ij} \epsilon^\prime_{ij}}
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where :math:`\vb*{\epsilon}^\prime` is the deviatoric part
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of the strain tensor.
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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: _np.ndarray) -> _np.ndarray:
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r"""
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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, 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, shape (...)
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Von Mises equivalent stress of sigma.
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Notes
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-----
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The von Mises equivalent of a stress tensor is defined as:
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.. math::
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\sigma_\text{vM} = \sqrt{3/2 \sigma^\prime_{ij} \sigma^\prime_{ij}}
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where :math:`\vb*{\sigma}^\prime` is the deviatoric part
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of the stress tensor.
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"""
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return _equivalent_Mises(sigma,3.0/2.0)
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def maximum_shear(T_sym: _np.ndarray) -> _np.ndarray:
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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, 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, 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 rotation(T: _np.ndarray) -> _rotation.Rotation:
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r"""
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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, 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 : damask.Rotation, shape (...)
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Rotational part of the vector.
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Notes
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-----
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The rotational part is calculated from the polar decomposition:
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.. math::
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\vb{R} = \vb{T} \vb{U}^{-1} = \vb{V}^{-1} \vb{T}
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where :math:`\vb{V}` and :math:`\vb{U}` are the left
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and right stretch tensor, respectively.
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"""
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return _rotation.Rotation.from_matrix(_polar_decomposition(T,'R')[0])
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def strain(F: _np.ndarray,
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t: str,
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m: float) -> _np.ndarray:
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"""
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Calculate strain tensor (Seth–Hill family).
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Parameters
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----------
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F : numpy.ndarray, 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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or '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, shape (...,3,3)
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Strain of F.
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References
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----------
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https://en.wikipedia.org/wiki/Finite_strain_theory
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https://de.wikipedia.org/wiki/Verzerrungstensor
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"""
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if t not in ['V', 'U']: raise ValueError('polar decomposition type not in {V, U}')
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w,n = _np.linalg.eigh(deformation_Cauchy_Green_left(F) if t=='V' else deformation_Cauchy_Green_right(F))
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return 0.5 * _np.einsum('...j,...kj,...lj',_np.log(w),n,n) if m == 0.0 \
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else 0.5/m * (_np.einsum('...j,...kj,...lj', w**m,n,n) - _np.eye(3))
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def stress_Cauchy(P: _np.ndarray,
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F: _np.ndarray) -> _np.ndarray:
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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, shape (...,3,3)
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First Piola-Kirchhoff stress.
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F : numpy.ndarray, 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, 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: _np.ndarray,
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F: _np.ndarray) -> _np.ndarray:
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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, shape (...,3,3)
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First Piola-Kirchhoff stress.
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F : numpy.ndarray, 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, 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: _np.ndarray) -> _np.ndarray:
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r"""
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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, 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, shape (...,3,3)
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Left stretch tensor from Polar decomposition of T.
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Notes
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-----
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The left stretch tensor is calculated from the
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polar decomposition:
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.. math::
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\vb{V} = \vb{T} \vb{R}^\text{T}
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where :math:`\vb{R}` is a rotation.
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"""
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return _polar_decomposition(T,'V')[0]
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def stretch_right(T: _np.ndarray) -> _np.ndarray:
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r"""
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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, 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, shape (...,3,3)
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Left stretch tensor from Polar decomposition of T.
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Notes
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-----
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The right stretch tensor is calculated from the
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polar decomposition:
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.. math::
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\vb{U} = \vb{R}^\text{T} \vb{T}
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where :math:`\vb{R}` is a rotation.
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"""
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return _polar_decomposition(T,'U')[0]
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def _polar_decomposition(T: _np.ndarray,
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requested: _Sequence[str]) -> tuple:
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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, shape (...,3,3)
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Tensor of which the singular values are computed.
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requested : sequence of {'R', 'U', 'V'}
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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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Returns
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-------
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VRU : tuple of numpy.ndarray, shape (...,3,3)
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Requested components of the singular value decomposition.
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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+=[R]
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if 'V' in requested:
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output+=[_np.einsum('...ij,...kj',T,R)]
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if 'U' in requested:
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output+=[_np.einsum('...ji,...jk',R,T)]
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if len(output) == 0:
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raise ValueError('output not in {V, R, U}')
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return tuple(output)
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def _equivalent_Mises(T_sym: _np.ndarray,
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s: float) -> _np.ndarray:
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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, 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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Returns
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-------
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eq : numpy.ndarray, shape (...)
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Scaled second invariant of the deviatoric part of T_sym.
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"""
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d = _tensor.deviatoric(T_sym)
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return _np.sqrt(s*_np.sum(d**2.0,axis=(-1,-2)))
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