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.array of shape (:,3,3) or (3,3) Deformation gradient. P : numpy.array of shape (:,3,3) or (3,3) 1. Piola-Kirchhoff stress. """ if np.shape(F) == np.shape(P) == (3,3): sigma = 1.0/np.linalg.det(F) * np.dot(P,F.T) else: sigma = np.einsum('i,ijk,ilk->ijl',1.0/np.linalg.det(F),P,F) return symmetric(sigma) def deviatoric_part(T): """ Return deviatoric part of a tensor. Parameters ---------- T : numpy.array of shape (:,3,3) or (3,3) Tensor of which the deviatoric part is computed. """ return T - np.eye(3)*spherical_part(T) if np.shape(T) == (3,3) else \ T - np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),[T.shape[0],3,3]),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.array of shape (:,3,3) or (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.array of shape (:,3,3) or (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: if np.shape(T_sym) == (3,3): if np.linalg.det(v) < 0.0: v[:,2] *= -1.0 else: 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.array of shape (:,3,3) or (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.array of shape (:,3,3) or (3,3) Symmetric tensor of which the maximum shear is computed. """ w = eigenvalues(T_sym) return (w[0] - w[2])*0.5 if np.shape(T_sym) == (3,3) else \ (w[:,0] - w[:,2])*0.5 def Mises_strain(epsilon): """ Return the Mises equivalent of a strain tensor. Parameters ---------- epsilon : numpy.array of shape (:,3,3) or (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.array of shape (:,3,3) or (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.array of shape (:,3,3) or (3,3) 1. Piola-Kirchhoff stress. F : numpy.array of shape (:,3,3) or (3,3) Deformation gradient. """ if np.shape(F) == np.shape(P) == (3,3): S = np.dot(np.linalg.inv(F),P) else: S = np.einsum('ijk,ikl->ijl',np.linalg.inv(F),P) return symmetric(S) def right_stretch(T): """ Return the right stretch of a tensor. Parameters ---------- T : numpy.array of shape (:,3,3) or (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.array of shape (:,3,3) or (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.array of shape (:,3,3) or (3,3) Tensor of which the hydrostatic part is computed. tensor : bool, optional Map spherical part onto identity tensor. Default is false """ if T.shape == (3,3): sph = np.trace(T)/3.0 return sph if not tensor else np.eye(3)*sph else: sph = np.trace(T,axis1=1,axis2=2)/3.0 if not tensor: return sph else: return np.einsum('ijk,i->ijk',np.broadcast_to(np.eye(3),(T.shape[0],3,3)),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.array of shape (:,3,3) or (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. """ F_ = F.reshape((1,3,3)) if F.shape == (3,3) else F 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('ij,ikj->ijk',w**m,n)) - np.broadcast_to(np.eye(3),[F_.shape[0],3,3])) elif m < 0.0: eps = 1.0/(2.0*abs(m)) * (- np.matmul(n,np.einsum('ij,ikj->ijk',w**m,n)) + np.broadcast_to(np.eye(3),[F_.shape[0],3,3])) else: eps = np.matmul(n,np.einsum('ij,ikj->ijk',0.5*np.log(w),n)) return eps.reshape((3,3)) if np.shape(F) == (3,3) else \ eps def symmetric(T): """ Return the symmetrized tensor. Parameters ---------- T : numpy.array of shape (:,3,3) or (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.array of shape (:,3,3) or (3,3) Tensor of which the transpose is computed. """ return T.T if np.shape(T) == (3,3) else \ np.transpose(T,(0,2,1)) def __polar_decomposition(T,requested): """ Singular value decomposition. Parameters ---------- T : numpy.array of shape (:,3,3) or (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.dot(u,vh) if np.shape(T) == (3,3) else \ np.einsum('ijk,ikl->ijl',u,vh) output = [] if 'R' in requested: output.append(R) if 'V' in requested: output.append(np.dot(T,R.T) if np.shape(T) == (3,3) else np.einsum('ijk,ilk->ijl',T,R)) if 'U' in requested: output.append(np.dot(R.T,T) if np.shape(T) == (3,3) else np.einsum('ikj,ikl->ijl',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.array of shape (:,3,3) or (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.sum(d**2.0))) if np.shape(T_sym) == (3,3) else \ np.sqrt(s*np.einsum('ijk->i',d**2.0))