def first_piola_kirchhoff_stress(cls, material, deformation_gradient, dimension=3, test=False):
"""Compute the first Piola-Kirchhoff stress for the material from the deformation gradient under
the specified assumptions.
:param material: material to which the deformation gradient applies
:param numpy.ndarray deformation_gradient: 3x3 matrix describing the deformation of the body
:param int dimension: desired dimension of the returned matrix
:param bool test: whether to perform the verification test for the stress result
"""
result = (
(material.first_lame_parameter * math.log(numpy.linalg.det(deformation_gradient)) - material.shear_modulus)
* operations.inverse_transpose(deformation_gradient)
+ material.shear_modulus * deformation_gradient)
# Verify the correctness of this result by comparing to numerical differentiation
if test:
tests.numerical_differentiation_first_piola_kirchhoff_stress(constitutive_model=cls,
material=material,
deformation_gradient=deformation_gradient,
first_piola_kirchhoff_stress=result)
# Return a 2x2 matrix if requested for plane stress:
if dimension == 2:
return result[0:2, 0:2]
# Otherwise return the full 3x3 matrix
else:
return result
def error_testing():
# NOTE: this test will result in an error for the default tolerance value, because this function is intended
# to violate the tolerance for the purposes of showing the behavior of the error as a function of h.
deformation_gradient = operations.generate_random_deformation_gradient() # uncomment for "bad" F: - numpy.eye(3)
material = materials.Custom(name='custom material', first_lame_parameter=5, shear_modulus=3)
constitutive_model = constitutive_models.Neohookean()
(strain_energy_density,
first_piola_kirchhoff_stress,
tangent_moduli) = constitutive_model.calculate_all(material=material,
deformation_gradient=deformation_gradient)
h_values = list(numpy.logspace(-2, -10, 100))
p_errors = []
c_errors = []
for h_value in h_values:
# NOTE these functions don't normally return these values
p_error = tests.numerical_differentiation_first_piola_kirchhoff_stress(constitutive_model=constitutive_model,
material=material,
deformation_gradient=deformation_gradient,
first_piola_kirchhoff_stress=first_piola_kirchhoff_stress,
h=h_value)
c_error = tests.numerical_differentiation_tangent_moduli(constitutive_model=constitutive_model,
material=material,
deformation_gradient=deformation_gradient,
tangent_moduli=tangent_moduli,
h=h_value)
p_errors.append(p_error)
c_errors.append(c_error)
plt.figure()
plt.plot(h_values, p_errors, 'b', label='P error')
plt.plot(h_values, c_errors, 'r', label='C error')
plt.xscale('log')
plt.yscale('log')
plt.ylim(10e-10, 10e-3)
plt.title('3-point formula errors for "good" deformation gradient')
plt.xlabel('h')
plt.ylabel('error')
plt.legend(loc='best')
plt.show()
