mechanics done
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Samad Vakili
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d365cc9e12
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694b7ec3c5
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@ -8,16 +8,12 @@ def Cauchy(P,F):
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Parameters
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----------
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F : numpy.ndarray of shape (:,3,3) or (3,3)
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F : numpy.ndarray of shape (...,3,3)
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Deformation gradient.
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P : numpy.ndarray of shape (:,3,3) or (3,3)
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P : numpy.ndarray of shape (...,3,3)
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First Piola-Kirchhoff stress.
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"""
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if _np.shape(F) == _np.shape(P) == (3,3):
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sigma = 1.0/_np.linalg.det(F) * _np.dot(P,F.T)
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else:
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#sigma = _np.einsum('i,ijk,ilk->ijl',1.0/_np.linalg.det(F),P,F)
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sigma = _np.einsum('...,...ij,...kj->...ik',1.0/_np.linalg.det(F),P,F)
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return symmetric(sigma)
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@ -44,7 +40,7 @@ def eigenvalues(T_sym):
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Parameters
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----------
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T_sym : numpy.ndarray of shape (:,3,3) or (3,3)
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T_sym : numpy.ndarray of shape (...,3,3)
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Symmetric tensor of which the eigenvalues are computed.
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"""
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@ -59,7 +55,7 @@ def eigenvectors(T_sym,RHS=False):
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Parameters
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----------
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T_sym : numpy.ndarray of shape (:,3,3) or (3,3)
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T_sym : numpy.ndarray of shape (...,3,3)
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Symmetric tensor of which the eigenvectors are computed.
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RHS: bool, optional
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Enforce right-handed coordinate system. Default is False.
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@ -68,9 +64,6 @@ def eigenvectors(T_sym,RHS=False):
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(u,v) = _np.linalg.eigh(symmetric(T_sym))
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if RHS:
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if _np.shape(T_sym) == (3,3):
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if _np.linalg.det(v) < 0.0: v[:,2] *= -1.0
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else:
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v[_np.linalg.det(v) < 0.0,:,2] *= -1.0
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return v
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@ -81,7 +74,7 @@ def left_stretch(T):
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Parameters
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----------
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T : numpy.ndarray of shape (:,3,3) or (3,3)
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T : numpy.ndarray of shape (...,3,3)
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Tensor of which the left stretch is computed.
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"""
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@ -94,13 +87,12 @@ def maximum_shear(T_sym):
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Parameters
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----------
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T_sym : numpy.ndarray of shape (:,3,3) or (3,3)
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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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"""
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w = eigenvalues(T_sym)
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return (w[0] - w[2])*0.5 if _np.shape(T_sym) == (3,3) else \
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(w[...,0] - w[...,2])*0.5
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return (w[...,0] - w[...,2])*0.5
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def Mises_strain(epsilon):
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@ -109,7 +101,7 @@ def Mises_strain(epsilon):
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Parameters
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----------
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epsilon : numpy.ndarray of shape (:,3,3) or (3,3)
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epsilon : numpy.ndarray of shape (...,3,3)
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Symmetric strain tensor of which the von Mises equivalent is computed.
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"""
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@ -122,7 +114,7 @@ def Mises_stress(sigma):
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Parameters
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----------
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sigma : numpy.ndarray of shape (:,3,3) or (3,3)
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sigma : numpy.ndarray of shape (...,3,3)
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Symmetric stress tensor of which the von Mises equivalent is computed.
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"""
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@ -135,15 +127,12 @@ def PK2(P,F):
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Parameters
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----------
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P : numpy.ndarray of shape (...,3,3) or (3,3)
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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) or (3,3)
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F : numpy.ndarray of shape (...,3,3)
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Deformation gradient.
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"""
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if _np.shape(F) == _np.shape(P) == (3,3):
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S = _np.dot(_np.linalg.inv(F),P)
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else:
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S = _np.einsum('...jk,...kl->...jl',_np.linalg.inv(F),P)
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return symmetric(S)
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@ -154,7 +143,7 @@ def right_stretch(T):
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Parameters
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----------
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T : numpy.ndarray of shape (:,3,3) or (3,3)
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T : numpy.ndarray of shape (...,3,3)
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Tensor of which the right stretch is computed.
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"""
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@ -167,7 +156,7 @@ def rotational_part(T):
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Parameters
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----------
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T : numpy.ndarray of shape (:,3,3) or (3,3)
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T : numpy.ndarray of shape (...,3,3)
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Tensor of which the rotational part is computed.
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"""
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@ -199,7 +188,7 @@ def strain_tensor(F,t,m):
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Parameters
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----------
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F : numpy.ndarray of shape (:,3,3) or (3,3)
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F : numpy.ndarray of 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 and ‘U’ for right stretch tensor.
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@ -207,12 +196,11 @@ def strain_tensor(F,t,m):
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Order of the strain.
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"""
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F_ = F.reshape(1,3,3) if F.shape == (3,3) else F
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if t == 'V':
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B = _np.matmul(F_,transpose(F_))
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B = _np.matmul(F,transpose(F))
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w,n = _np.linalg.eigh(B)
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elif t == 'U':
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C = _np.matmul(transpose(F_),F_)
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C = _np.matmul(transpose(F),F)
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w,n = _np.linalg.eigh(C)
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if m > 0.0:
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@ -225,8 +213,7 @@ def strain_tensor(F,t,m):
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else:
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eps = _np.matmul(n,_np.einsum('...j,...kj->...jk',0.5*_np.log(w),n))
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return eps.reshape(3,3) if _np.shape(F) == (3,3) else \
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eps
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return eps
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def symmetric(T):
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@ -235,7 +222,7 @@ def symmetric(T):
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Parameters
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----------
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T : numpy.ndarray of shape (...,3,3) or (3,3)
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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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"""
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@ -248,12 +235,11 @@ def transpose(T):
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Parameters
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----------
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T : numpy.ndarray of shape (...,3,3) or (3,3)
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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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"""
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return T.T if _np.shape(T) == (3,3) else \
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_np.swapaxes(T,axis2=-2,axis1=-1)
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return _np.swapaxes(T,axis2=-2,axis1=-1)
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def _polar_decomposition(T,requested):
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@ -262,7 +248,7 @@ def _polar_decomposition(T,requested):
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Parameters
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----------
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T : numpy.ndarray of shape (:,3,3) or (3,3)
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T : numpy.ndarray of shape (...,3,3)
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Tensor of which the singular values are computed.
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requested : iterable of str
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Requested outputs: ‘R’ for the rotation tensor,
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@ -270,16 +256,15 @@ def _polar_decomposition(T,requested):
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"""
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u, s, vh = _np.linalg.svd(T)
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R = _np.dot(u,vh) if _np.shape(T) == (3,3) else \
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_np.einsum('...ij,...jk->...ik',u,vh)
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R = _np.einsum('...ij,...jk->...ik',u,vh)
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output = []
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if 'R' in requested:
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output.append(R)
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if 'V' in requested:
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output.append(_np.dot(T,R.T) if _np.shape(T) == (3,3) else _np.einsum('...ij,...kj->...ik',T,R))
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output.append(_np.einsum('...ij,...kj->...ik',T,R))
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if 'U' in requested:
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output.append(_np.dot(R.T,T) if _np.shape(T) == (3,3) else _np.einsum('...ji,...jk->...ik',R,T))
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output.append(_np.einsum('...ji,...jk->...ik',R,T))
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return tuple(output)
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@ -290,12 +275,11 @@ def _Mises(T_sym,s):
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Parameters
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----------
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T_sym : numpy.ndarray of shape (:,3,3) or (3,3)
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T_sym : numpy.ndarray of 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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"""
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d = deviatoric_part(T_sym)
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return _np.sqrt(s*(_np.sum(d**2.0))) if _np.shape(T_sym) == (3,3) else \
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_np.sqrt(s*_np.einsum('...jk->...',d**2.0))
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return _np.sqrt(s*_np.einsum('...jk->...',d**2.0))
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