def test_elastostatics(self):
eps = 1e-3
for C11, C12, C44, surface_energy, k1 in self.materials:
crack = CubicCrystalCrack([1,0,0], [0,1,0], C11, C12, C44)
k = crack.k1g(surface_energy)*k1
for i in range(10):
x = i+1
y = i+1
# Finite difference approximation of the stress divergence
sxx0x, syy0x, sxy0x = crack.stresses(x-eps, y, 0, 0, k)
sxx0y, syy0y, sxy0y = crack.stresses(x, y-eps, 0, 0, k)
sxx1x, syy1x, sxy1x = crack.stresses(x+eps, y, 0, 0, k)
sxx1y, syy1y, sxy1y = crack.stresses(x, y+eps, 0, 0, k)
divsx = (sxx1x-sxx0x)/(2*eps) + (sxy1y-sxy0y)/(2*eps)
divsy = (sxy1x-sxy0x)/(2*eps) + (syy1y-syy0y)/(2*eps)
# Check that divergence of stress is zero (elastostatic
# equilibrium)
self.assertAlmostEqual(divsx, 0.0, places=3)
self.assertAlmostEqual(divsy, 0.0, places=3)
def test_elastostatics(self):
eps = 1e-3
for C11, C12, C44, surface_energy, k1 in self.materials:
crack = CubicCrystalCrack([1, 0, 0], [0, 1, 0], C11, C12, C44)
k = crack.k1g(surface_energy) * k1
for i in range(10):
x = i + 1
y = i + 1
# Finite difference approximation of the stress divergence
sxx0x, syy0x, sxy0x = crack.stresses(x - eps, y, 0, 0, k)
sxx0y, syy0y, sxy0y = crack.stresses(x, y - eps, 0, 0, k)
sxx1x, syy1x, sxy1x = crack.stresses(x + eps, y, 0, 0, k)
sxx1y, syy1y, sxy1y = crack.stresses(x, y + eps, 0, 0, k)
divsx = (sxx1x - sxx0x) / (2 * eps) + (sxy1y - sxy0y) / (2 *
eps)
divsy = (sxy1x - sxy0x) / (2 * eps) + (syy1y - syy0y) / (2 *
eps)
# Check that divergence of stress is zero (elastostatic
# equilibrium)
self.assertAlmostEqual(divsx, 0.0, places=3)
self.assertAlmostEqual(divsy, 0.0, places=3)
def test_consistency_of_deformation_gradient_and_stress(self):
for C11, C12, C44, surface_energy, k1 in self.materials:
crack = CubicCrystalCrack([1, 0, 0], [0, 1, 0], C11, C12, C44)
k = crack.k1g(surface_energy) * k1
for i in range(10):
x = np.random.uniform(-10, 10)
y = np.random.uniform(-10, 10)
F = crack.deformation_gradient(x, y, 0, 0, k)
eps = (F + F.T) / 2 - np.eye(2)
sig_x, sig_y, sig_xy = crack.stresses(x, y, 0, 0, k)
eps_xx = crack.crack.a11*sig_x + \
crack.crack.a12*sig_y + \
crack.crack.a16*sig_xy
eps_yy = crack.crack.a12*sig_x + \
crack.crack.a22*sig_y + \
crack.crack.a26*sig_xy
# Factor 1/2 because elastic constants and matrix product are
# expressed in Voigt notation.
eps_xy = (crack.crack.a16*sig_x + \
crack.crack.a26*sig_y + \
crack.crack.a66*sig_xy)/2
self.assertAlmostEqual(eps[0, 0], eps_xx)
self.assertAlmostEqual(eps[1, 1], eps_yy)
self.assertAlmostEqual(eps[0, 1], eps_xy)
def test_consistency_of_deformation_gradient_and_stress(self):
for C11, C12, C44, surface_energy, k1 in self.materials:
crack = CubicCrystalCrack([1,0,0], [0,1,0], C11, C12, C44)
k = crack.k1g(surface_energy)*k1
for i in range(10):
x = np.random.uniform(-10, 10)
y = np.random.uniform(-10, 10)
F = crack.deformation_gradient(x, y, 0, 0, k)
eps = (F+F.T)/2-np.eye(2)
sig_x, sig_y, sig_xy = crack.stresses(x, y, 0, 0, k)
eps_xx = crack.crack.a11*sig_x + \
crack.crack.a12*sig_y + \
crack.crack.a16*sig_xy
eps_yy = crack.crack.a12*sig_x + \
crack.crack.a22*sig_y + \
crack.crack.a26*sig_xy
# Factor 1/2 because elastic constants and matrix product are
# expressed in Voigt notation.
eps_xy = (crack.crack.a16*sig_x + \
crack.crack.a26*sig_y + \
crack.crack.a66*sig_xy)/2
self.assertAlmostEqual(eps[0, 0], eps_xx)
self.assertAlmostEqual(eps[1, 1], eps_yy)
self.assertAlmostEqual(eps[0, 1], eps_xy)