import numpy as _np def Cauchy(P,F): """ Return Cauchy stress calculated from first Piola-Kirchhoff stress and deformation gradient. Resulting tensor is symmetrized as the Cauchy stress needs to be symmetric. Parameters ---------- F : numpy.ndarray of shape (...,3,3) Deformation gradient. P : numpy.ndarray of shape (...,3,3) First Piola-Kirchhoff stress. """ sigma = _np.einsum('...,...ij,...kj->...ik',1.0/_np.linalg.det(F),P,F) return symmetric(sigma) def deviatoric_part(T): """ Return deviatoric part of a tensor. Parameters ---------- T : numpy.ndarray of shape (...,3,3) Tensor of which the deviatoric part is computed. """ return T - _np.einsum('...ij,...->...ij',_np.eye(3),spherical_part(T)) def eigenvalues(T_sym): """ Return the eigenvalues, i.e. principal components, of a symmetric tensor. The eigenvalues are sorted in ascending order, each repeated according to its multiplicity. Parameters ---------- T_sym : numpy.ndarray of shape (...,3,3) Symmetric tensor of which the eigenvalues are computed. """ return _np.linalg.eigvalsh(symmetric(T_sym)) def eigenvectors(T_sym,RHS=False): """ Return eigenvectors of a symmetric tensor. The eigenvalues are sorted in ascending order of their associated eigenvalues. Parameters ---------- T_sym : numpy.ndarray of shape (...,3,3) Symmetric tensor of which the eigenvectors are computed. RHS: bool, optional Enforce right-handed coordinate system. Default is False. """ (u,v) = _np.linalg.eigh(symmetric(T_sym)) if RHS: v[_np.linalg.det(v) < 0.0,:,2] *= -1.0 return v def left_stretch(T): """ Return the left stretch of a tensor. Parameters ---------- T : numpy.ndarray of shape (...,3,3) Tensor of which the left stretch is computed. """ return _polar_decomposition(T,'V')[0] def maximum_shear(T_sym): """ Return the maximum shear component of a symmetric tensor. Parameters ---------- T_sym : numpy.ndarray of shape (...,3,3) Symmetric tensor of which the maximum shear is computed. """ w = eigenvalues(T_sym) return (w[...,0] - w[...,2])*0.5 def Mises_strain(epsilon): """ Return the Mises equivalent of a strain tensor. Parameters ---------- epsilon : numpy.ndarray of shape (...,3,3) Symmetric strain tensor of which the von Mises equivalent is computed. """ return _Mises(epsilon,2.0/3.0) def Mises_stress(sigma): """ Return the Mises equivalent of a stress tensor. Parameters ---------- sigma : numpy.ndarray of shape (...,3,3) Symmetric stress tensor of which the von Mises equivalent is computed. """ return _Mises(sigma,3.0/2.0) def PK2(P,F): """ Calculate second Piola-Kirchhoff stress from first Piola-Kirchhoff stress and deformation gradient. Parameters ---------- P : numpy.ndarray of shape (...,3,3) First Piola-Kirchhoff stress. F : numpy.ndarray of shape (...,3,3) Deformation gradient. """ S = _np.einsum('...jk,...kl->...jl',_np.linalg.inv(F),P) return symmetric(S) def right_stretch(T): """ Return the right stretch of a tensor. Parameters ---------- T : numpy.ndarray of shape (...,3,3) Tensor of which the right stretch is computed. """ return _polar_decomposition(T,'U')[0] def rotational_part(T): """ Return the rotational part of a tensor. Parameters ---------- T : numpy.ndarray of shape (...,3,3) Tensor of which the rotational part is computed. """ return _polar_decomposition(T,'R')[0] def spherical_part(T,tensor=False): """ Return spherical (hydrostatic) part of a tensor. Parameters ---------- T : numpy.ndarray of shape (...,3,3) Tensor of which the hydrostatic part is computed. tensor : bool, optional Map spherical part onto identity tensor. Default is false """ sph = _np.trace(T,axis2=-2,axis1=-1)/3.0 return _np.einsum('...jk,...->...jk',_np.eye(3),sph) if tensor else sph def strain_tensor(F,t,m): """ Return strain tensor calculated from deformation gradient. For details refer to https://en.wikipedia.org/wiki/Finite_strain_theory and https://de.wikipedia.org/wiki/Verzerrungstensor Parameters ---------- F : numpy.ndarray of shape (...,3,3) Deformation gradient. t : {‘V’, ‘U’} Type of the polar decomposition, ‘V’ for left stretch tensor and ‘U’ for right stretch tensor. m : float Order of the strain. """ if t == 'V': B = _np.matmul(F,transpose(F)) w,n = _np.linalg.eigh(B) elif t == 'U': C = _np.matmul(transpose(F),F) w,n = _np.linalg.eigh(C) if m > 0.0: eps = 1.0/(2.0*abs(m)) * (+ _np.matmul(n,_np.einsum('...j,...kj->...jk',w**m,n)) - _np.einsum('...jk->...jk',_np.eye(3))) elif m < 0.0: eps = 1.0/(2.0*abs(m)) * (- _np.matmul(n,_np.einsum('...j,...kj->...jk',w**m,n)) + _np.einsum('...jk->...jk',_np.eye(3))) else: eps = _np.matmul(n,_np.einsum('...j,...kj->...jk',0.5*_np.log(w),n)) return eps def symmetric(T): """ Return the symmetrized tensor. Parameters ---------- T : numpy.ndarray of shape (...,3,3) Tensor of which the symmetrized values are computed. """ return (T+transpose(T))*0.5 def transpose(T): """ Return the transpose of a tensor. Parameters ---------- T : numpy.ndarray of shape (...,3,3) Tensor of which the transpose is computed. """ return _np.swapaxes(T,axis2=-2,axis1=-1) def _polar_decomposition(T,requested): """ Singular value decomposition. Parameters ---------- T : numpy.ndarray of shape (...,3,3) Tensor of which the singular values are computed. requested : iterable of str Requested outputs: ‘R’ for the rotation tensor, ‘V’ for left stretch tensor and ‘U’ for right stretch tensor. """ u, s, vh = _np.linalg.svd(T) R = _np.einsum('...ij,...jk->...ik',u,vh) output = [] if 'R' in requested: output.append(R) if 'V' in requested: output.append(_np.einsum('...ij,...kj->...ik',T,R)) if 'U' in requested: output.append(_np.einsum('...ji,...jk->...ik',R,T)) return tuple(output) def _Mises(T_sym,s): """ Base equation for Mises equivalent of a stres or strain tensor. Parameters ---------- T_sym : numpy.ndarray of shape (...,3,3) Symmetric tensor of which the von Mises equivalent is computed. s : float Scaling factor (2/3 for strain, 3/2 for stress). """ d = deviatoric_part(T_sym) return _np.sqrt(s*_np.einsum('...jk->...',d**2.0))