def calculate_jacobian(self):
"""Compute the value for the Jacobian when the deformation gradient is updated,
and confirm that it has a positive value.
"""
jacobian = numpy.linalg.det(self.deformation_gradient)
tests.deformation_gradient_physical(jacobian=jacobian)
return jacobian
def homework1_part1():
"""Curvilinear Kinematics: Uniaxial deformation of a cylinder"""
# Position at which to evaluate
radius = 10
angle = math.pi / 4
# Constants
lambda1 = 2
lambda2 = 3
# Lab frame
lab_vector1 = numpy.array([math.cos(angle), math.sin(angle), 0])
lab_vector2 = numpy.array([-math.sin(angle), math.cos(angle), 0])
lab_vector3 = numpy.array([0, 0, 1])
lab_frame = configurations.Basis(vector1=lab_vector1,
vector2=lab_vector2,
vector3=lab_vector3,
type=constants.LAB)
# Reference configuration
reference_vector1_covariant = numpy.array(lab_vector1)
reference_vector2_covariant = numpy.array(radius * lab_vector2)
reference_vector3_covariant = numpy.array(lab_vector3)
reference_configuration_covariant = configurations.Basis(vector1=reference_vector1_covariant,
vector2=reference_vector2_covariant,
vector3=reference_vector3_covariant,
type=constants.COVARIANT)
reference_vector1_contravariant = numpy.array(lab_vector1)
reference_vector2_contravariant = numpy.array(1 / radius * lab_vector2)
reference_vector3_contravariant = numpy.array(lab_vector3)
reference_configuration_contravariant = configurations.Basis(vector1=reference_vector1_contravariant,
vector2=reference_vector2_contravariant,
vector3=reference_vector3_contravariant,
type=constants.CONTRAVARIANT)
reference_configuration = configurations.ReferenceConfiguration(covariant_basis=reference_configuration_covariant,
contravariant_basis=reference_configuration_contravariant)
# Deformed configuration
deformed_vector1_covariant = numpy.array(lambda1 * lab_vector1)
deformed_vector2_covariant = numpy.array(lambda1 * radius * lab_vector2)
deformed_vector3_covariant = numpy.array(lambda2 * lab_vector3)
deformed_configuration_covariant = configurations.Basis(vector1=deformed_vector1_covariant,
vector2=deformed_vector2_covariant,
vector3=deformed_vector3_covariant,
type=constants.COVARIANT)
deformed_vector1_contravariant = numpy.array(1 / lambda1 * lab_vector1)
deformed_vector2_contravariant = numpy.array(1 / (lambda1 * radius) * lab_vector2)
deformed_vector3_contravariant = numpy.array(1 / lambda2 * lab_vector3)
deformed_configuration_contravariant = configurations.Basis(vector1=deformed_vector1_contravariant,
vector2=deformed_vector2_contravariant,
vector3=deformed_vector3_contravariant,
type=constants.CONTRAVARIANT)
deformed_configuration = configurations.DeformedConfiguration(covariant_basis=deformed_configuration_covariant,
contravariant_basis=deformed_configuration_contravariant)
# Compute deformation gradient and other kinematic quantities
deformation_gradient_matrix = kinematics.deformation_gradient(deformed_configuration_covariant,
reference_configuration_contravariant)
tests.deformation_gradient_physical(numpy.linalg.det(deformation_gradient_matrix))
right_cauchy_green_deformation = kinematics.right_cauchy_green_deformation_tensor(
deformation_gradient_matrix)
left_cauchy_green_deformation = kinematics.left_cauchy_green_deformation_tensor(
deformation_gradient_matrix)
green_lagrange_strain = kinematics.green_lagrange_strain(right_cauchy_green_deformation)