import pytest import numpy as np from scipy import linalg from damask import tensor from damask import mechanics from damask import Rotation def stress_Cauchy(P,F): sigma = 1.0/np.linalg.det(F) * np.dot(P,F.T) return symmetric(sigma) def eigenvalues(T_sym): return np.linalg.eigvalsh(symmetric(T_sym)) def maximum_shear(T_sym): w = eigenvalues(T_sym) return (w[0] - w[2])*0.5 def equivalent_strain_Mises(epsilon): return equivalent_Mises(epsilon,2.0/3.0) def equivalent_stress_Mises(sigma): return equivalent_Mises(sigma,3.0/2.0) def stress_second_Piola_Kirchhoff(P,F): S = np.dot(np.linalg.inv(F),P) return symmetric(S) def rotation(T): return polar_decomposition(T,'R')[0] def strain(F,t,m): if t == 'V': B = np.matmul(F,F.T) w,n = np.linalg.eigh(B) elif t == 'U': C = np.matmul(F.T,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.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.eye(3)) else: eps = np.matmul(n,np.einsum('j,kj->jk',0.5*np.log(w),n)) return eps def stretch_left(T): return polar_decomposition(T,'V')[0] def stretch_right(T): return polar_decomposition(T,'U')[0] def symmetric(T): return (T+T.T)*0.5 def polar_decomposition(T,requested): u, s, vh = np.linalg.svd(T) R = np.dot(u,vh) output = [] if 'R' in requested: output.append(R) if 'V' in requested: output.append(np.dot(T,R.T)) if 'U' in requested: output.append(np.dot(R.T,T)) return tuple(output) def equivalent_Mises(T_sym,s): return np.sqrt(s*(np.sum(deviatoric(T_sym)**2.0))) def deviatoric(T): return T - np.eye(3)*np.trace(T)/3.0 class TestMechanics: n = 1000 c = np.random.randint(n) @pytest.mark.parametrize('vectorized,single',[(mechanics.maximum_shear, maximum_shear), (mechanics.equivalent_stress_Mises, equivalent_stress_Mises), (mechanics.equivalent_strain_Mises, equivalent_strain_Mises), (mechanics.stretch_left, stretch_left), (mechanics.stretch_right, stretch_right), ]) def test_vectorize_1_arg(self,vectorized,single): epsilon = np.random.rand(self.n,3,3) epsilon_vec = np.reshape(epsilon,(self.n//10,10,3,3)) for i,v in enumerate(np.reshape(vectorized(epsilon_vec),vectorized(epsilon).shape)): assert np.allclose(single(epsilon[i]),v) def test_vectorize_rotation(self): epsilon = Rotation.from_random(self.n).as_matrix() epsilon_vec = np.reshape(epsilon,(self.n//10,10,3,3)) for i,v in enumerate(np.reshape(mechanics.rotation(epsilon_vec).as_matrix(), mechanics.rotation(epsilon).as_matrix().shape)): assert np.allclose(rotation(epsilon[i]),v) @pytest.mark.parametrize('vectorized,single',[(mechanics.stress_Cauchy, stress_Cauchy), (mechanics.stress_second_Piola_Kirchhoff, stress_second_Piola_Kirchhoff) ]) def test_vectorize_2_arg(self,vectorized,single): P = np.random.rand(self.n,3,3) F = np.random.rand(self.n,3,3) P_vec = np.reshape(P,(self.n//10,10,3,3)) F_vec = np.reshape(F,(self.n//10,10,3,3)) for i,v in enumerate(np.reshape(vectorized(P_vec,F_vec),vectorized(P,F).shape)): assert np.allclose(single(P[i],F[i]),v) @pytest.mark.parametrize('vectorized,single',[(mechanics.strain,strain)]) def test_vectorize_strain(self,vectorized,single): F = np.random.rand(self.n,3,3) F = np.einsum('...ij,...jk',F,F) # positive determinant F_vec = np.reshape(F,(self.n//10,10,3,3)) t = ['V','U'][np.random.randint(0,2)] m = np.random.random()*10.0 -5.0 for i,v in enumerate(np.reshape(vectorized(F_vec,t,m),vectorized(F,t,m).shape)): if np.linalg.det(F[i]) < 1.e-7: continue assert np.allclose(single(F[i],t,m),v) @pytest.mark.parametrize('function',[mechanics.stress_Cauchy, mechanics.stress_second_Piola_Kirchhoff, ]) def test_stress_measures(self,function): """Ensure that all stress measures are equivalent for no deformation.""" P = np.random.rand(self.n,3,3) assert np.allclose(function(P,np.broadcast_to(np.eye(3),(self.n,3,3))),tensor.symmetric(P)) def test_polar_decomposition_identity(self): """F = RU = VR.""" F = np.broadcast_to(np.eye(3),[self.n,3,3])*np.random.rand(self.n,3,3) R = mechanics.rotation(F).as_matrix() V = mechanics.stretch_left(F) U = mechanics.stretch_right(F) assert np.allclose(np.matmul(R,U), np.matmul(V,R)) @pytest.mark.parametrize('side',[('left','V'),('right','U')]) def test_polar_decomposition(self,side): F = np.random.rand(self.n,3,3) F = F @ F # positive determinant F_vec = np.reshape(F,(self.n//10,10,3,3)) p = mechanics._polar_decomposition(F_vec,side[1]) for p_,F_ in zip(np.reshape(p,F.shape),F): if np.linalg.det(F_) < 1.e-7: continue assert np.allclose(p_,linalg.polar(F_,side[0])[1]) @pytest.mark.parametrize('m',[0.0,np.random.random()*10.,np.random.random()*-10.]) def test_strain_no_rotation(self,m): """Ensure that left and right stretch give same results for no rotation.""" F = np.broadcast_to(np.eye(3),[self.n,3,3])*np.random.rand(self.n,3,3) assert np.allclose(mechanics.strain(F,'U',m), mechanics.strain(F,'V',m)) @pytest.mark.parametrize('m',[0.0,np.random.random()*2.5,np.random.random()*-2.5]) def test_strain_rotation_equivalence(self,m): """Ensure that left and right strain differ only by a rotation.""" F = np.broadcast_to(np.eye(3),[self.n,3,3]) + (np.random.rand(self.n,3,3)*0.5 - 0.25) assert np.allclose(np.linalg.det(mechanics.strain(F,'U',m)), np.linalg.det(mechanics.strain(F,'V',m))) @pytest.mark.parametrize('m',[0.0,np.random.random(),np.random.random()*-1.]) @pytest.mark.parametrize('t',['V','U']) def test_strain_rotation(self,m,t): """Ensure that pure rotation results in no strain.""" F = Rotation.from_random(self.n).as_matrix() assert np.allclose(mechanics.strain(F,t,m), 0.0) def test_rotation_determinant(self): """ Ensure that the determinant of the rotational part is +- 1. Should be +1, but random F might contain a reflection. """ x = np.random.rand(self.n,3,3) assert np.allclose(np.abs(np.linalg.det(mechanics._polar_decomposition(x,'R')[0])), 1.0) def test_deviatoric_Mises(self): """Ensure that Mises equivalent stress depends only on deviatoric part.""" x = np.random.rand(self.n,3,3) full = mechanics.equivalent_stress_Mises(x) dev = mechanics.equivalent_stress_Mises(tensor.deviatoric(x)) assert np.allclose(full, dev) @pytest.mark.parametrize('Mises_equivalent',[mechanics.equivalent_strain_Mises, mechanics.equivalent_stress_Mises]) def test_spherical_Mises(self,Mises_equivalent): """Ensure that Mises equivalent strain/stress of spherical strain is 0.""" x = np.random.rand(self.n,3,3) assert np.allclose(Mises_equivalent(tensor.spherical(x,True)), 0.0) def test_Mises(self): """Ensure that equivalent stress is 3/2 of equivalent strain.""" x = np.random.rand(self.n,3,3) assert np.allclose(mechanics.equivalent_stress_Mises(x)/mechanics.equivalent_strain_Mises(x), 1.5) def test_spherical_no_shear(self): """Ensure that sherical stress has max shear of 0.0.""" A = tensor.spherical(tensor.symmetric(np.random.rand(self.n,3,3)),True) assert np.allclose(mechanics.maximum_shear(A),0.0) def test_invalid_decomposition(self): with pytest.raises(ValueError): mechanics._polar_decomposition(np.random.rand(10,3,3),'A') def test_invalid_strain(self): with pytest.raises(ValueError): mechanics.strain(np.random.rand(10,3,3),'A',0)