"""Finite-strain continuum mechanics.""" from . import tensor import numpy as _np def Cauchy_Green_deformation_left(F): """ Calculate left Cauchy-Green deformation tensor (Finger deformation tensor). Parameters ---------- F : numpy.ndarray of shape (...,3,3) Deformation gradient. Returns ------- B : numpy.ndarray of shape (...,3,3) Left Cauchy-Green deformation tensor. """ return _np.matmul(F,tensor.transpose(F)) def Cauchy_Green_deformation_right(F): """ Calculate right Cauchy-Green deformation tensor. Parameters ---------- F : numpy.ndarray of shape (...,3,3) Deformation gradient. Returns ------- C : numpy.ndarray of shape (...,3,3) Right Cauchy-Green deformation tensor. """ return _np.matmul(tensor.transpose(F),F) def Cauchy(P,F): """ Calculate the Cauchy (true) stress. Resulting tensor is symmetrized as the Cauchy stress needs to be symmetric. Parameters ---------- P : numpy.ndarray of shape (...,3,3) First Piola-Kirchhoff stress. F : numpy.ndarray of shape (...,3,3) Deformation gradient. Returns ------- sigma : numpy.ndarray of shape (...,3,3) Cauchy stress. """ sigma = _np.einsum('...,...ij,...kj->...ik',1.0/_np.linalg.det(F),P,F) return tensor.symmetric(sigma) def deviatoric_part(T): """ Calculate deviatoric part of a tensor. Parameters ---------- T : numpy.ndarray of shape (...,3,3) Tensor of which the deviatoric part is computed. Returns ------- T' : numpy.ndarray of shape (...,3,3) Deviatoric part of T. """ return T - _np.einsum('...ij,...->...ij',_np.eye(3),spherical_part(T)) def maximum_shear(T_sym): """ Calculate 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. Returns ------- gamma_max : numpy.ndarray of shape (...) Maximum shear of T_sym. """ w = tensor.eigenvalues(T_sym) return (w[...,0] - w[...,2])*0.5 def Mises_strain(epsilon): """ Calculate 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. Returns ------- epsilon_vM : numpy.ndarray of shape (...) Von Mises equivalent strain of epsilon. """ return _Mises(epsilon,2.0/3.0) def Mises_stress(sigma): """ Calculate 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. Returns ------- sigma_vM : numpy.ndarray of shape (...) Von Mises equivalent stress of sigma. """ return _Mises(sigma,3.0/2.0) def PK2(P,F): """ Calculate the second Piola-Kirchhoff stress. Resulting tensor is symmetrized as the second Piola-Kirchhoff stress needs to be symmetric. Parameters ---------- P : numpy.ndarray of shape (...,3,3) First Piola-Kirchhoff stress. F : numpy.ndarray of shape (...,3,3) Deformation gradient. Returns ------- S : numpy.ndarray of shape (...,3,3) Second Piola-Kirchhoff stress. """ S = _np.einsum('...jk,...kl->...jl',_np.linalg.inv(F),P) return tensor.symmetric(S) def rotational_part(T): """ Calculate the rotational part of a tensor. Parameters ---------- T : numpy.ndarray of shape (...,3,3) Tensor of which the rotational part is computed. Returns ------- R : numpy.ndarray of shape (...,3,3) Rotational part. """ return _polar_decomposition(T,'R')[0] def spherical_part(T,tensor=False): """ Calculate 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. Defaults to false Returns ------- p : numpy.ndarray of shape (...) unless tensor == True: shape (...,3,3) Spherical part of tensor T, e.g. the hydrostatic part/pressure of a stress tensor. """ 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): """ Calculate strain tensor 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. Returns ------- epsilon : numpy.ndarray of shape (...,3,3) Strain of F. """ if t == 'V': w,n = _np.linalg.eigh(Cauchy_Green_deformation_left(F)) elif t == 'U': w,n = _np.linalg.eigh(Cauchy_Green_deformation_right(F)) 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 stretch_left(T): """ Calculate left stretch of a tensor. Parameters ---------- T : numpy.ndarray of shape (...,3,3) Tensor of which the left stretch is computed. Returns ------- V : numpy.ndarray of shape (...,3,3) Left stretch tensor from Polar decomposition of T. """ return _polar_decomposition(T,'V')[0] def stretch_right(T): """ Calculate right stretch of a tensor. Parameters ---------- T : numpy.ndarray of shape (...,3,3) Tensor of which the right stretch is computed. Returns ------- U : numpy.ndarray of shape (...,3,3) Left stretch tensor from Polar decomposition of T. """ return _polar_decomposition(T,'U')[0] def _polar_decomposition(T,requested): """ Perform 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, _, 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))