Phase-Field Fracture#

Input Class#

Input: Phase-Field Fracture#

class phasefieldx.Element.Phase_Field_Fracture.Input.Input(E=210.0, nu=0.3, Gc=0.0027, l=0.015, rho=1.0, degradation='isotropic', split_energy='no', degradation_function='quadratic', irreversibility='no', fatigue=False, fatigue_degradation_function='asymptotic', fatigue_val=0.05625, k=0, save_solution_xdmf=False, save_solution_vtu=True, results_folder_name='results')[source]#

Bases: object

Class for managing phase-field fracture simulation parameters.

This class encapsulates parameters related to phase-field fracture simulations and provides methods for setting, logging, and exporting these parameters.

Attributes:

Efloat: Young’s modulus of the material.
nufloat: Poisson’s ratio of the material.
Gcfloat: Critical energy release rate.
lfloat: Length scale parameter.
lambda_float: Lame’s first parameter calculated from E and nu.
mufloat: Shear modulus calculated from E and nu.
rhofloat: Density of the material.
degradationstr: Type of material degradation (isotropic or anisotropic).
split_energystr: Type of energy splitting (spectral, deviatoric, or none).
degradation_functionstr: Type of degradation function (e.g., quadratic).
irreversibilitystr: Indication of irreversibility of damage.
fatiguebool: Indicates if fatigue degradation is enabled.
fatigue_degradation_functionstr: Fatigue degradation function type.
fatigue_valfloat: Fatigue degradation value.
kint: A numerical parameter.
save_solution_xdmfbool: Indicates whether to save solutions in XDMF format.
save_solution_vtubool: Indicates whether to save solutions in VTU format.
results_folder_namestr: Name of the folder to save simulation results.

Methods

__init__(…)	Initialize the Input class with default parameters.
save_log_info(logger)	Log the simulation parameters using the provided logger.
save_parameters_to_csv(filename=”parameters.input”)	Save the simulation parameters to a two-column text file (tab-separated) for easy loading with pandas.
__str__()	Return a string representation of the simulation parameters.

save_log_info(logger)[source]#

Log the simulation parameters using the provided logger.

Parameters:: logger: An instance of a logging object.

save_parameters_to_csv(filename='parameters.input')[source]#

Save the simulation parameters to a CSV file for easy loading with pandas.

Parameters:: filename (str): The name of the CSV file to save the parameters.

\(g(\phi)\) Degradation functions#

Degradation Functions \(g(\phi)\)#

This module provides various degradation functions and their derivatives, which are used in phase field fracture models.

Examples#

Plot the available degradation functions for illustrative purposes.

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions import g, dg

Define the range for the phase field variable phi.

>>> phi = np.linspace(0.0, 1.0, 100)

Evaluate the degradation functions and their derivatives.

>>> g_quadratic = [g(p, "quadratic") for p in phi]
>>> g_cubic = [g(p, "cubic") for p in phi]
>>> g_quartic = [g(p, "quartic") for p in phi]
>>> dg_quadratic = [dg(p, "quadratic") for p in phi]
>>> dg_cubic = [dg(p, "cubic") for p in phi]
>>> dg_quartic = [dg(p, "quartic") for p in phi]

Plot the evaluated functions and their derivatives.

>>> fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))
>>> _ = ax1.plot(phi, g_quadratic, label="Quadratic")
>>> _ = ax1.plot(phi, g_cubic, label="Cubic")
>>> _ = ax1.plot(phi, g_quartic, label="Quartic")
>>> ax1.set_xlabel(r"$\phi$")
>>> ax1.set_ylabel(r"$g(\phi)$")
>>> ax1.set_title("Phase Field Degradation Functions")
>>> ax1.legend()
>>> ax1.grid(True)
>>> _ = ax2.plot(phi, dg_quadratic, label="Quadratic")
>>> _ = ax2.plot(phi, dg_cubic, label="Cubic")
>>> _ = ax2.plot(phi, dg_quartic, label="Quartic")
>>> ax2.set_xlabel(r"$\phi$")
>>> ax2.set_ylabel(r"$g'(\phi)$")
>>> ax2.set_title("Derivatives of Degradation Functions")
>>> ax2.legend()
>>> ax2.grid(True)
>>> plt.tight_layout()
>>> plt.show()

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.alessi_degradation_derivative(phi)[source]#

Evaluate the derivative of the Alessi degradation function.

Parameters:

phifloat: The phase field variable.

Returns:

float: The value of the derivative of the Alessi degradation function.

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.alessi_degradation_function(phi)[source]#

Evaluate the Alessi degradation function.

Parameters:

phifloat: The phase field variable.

Returns:

float: The value of the Alessi degradation function.

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.cubic_degradation_derivative(phi)[source]#

Evaluate the derivative of the cubic degradation function.

Parameters:

phifloat: The phase field variable.

Returns:

float: The value of the derivative of the cubic degradation function.

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.cubic_degradation_function(phi)[source]#

Evaluate the cubic (Borden) degradation function.

Parameters:

phifloat: The phase field variable.

Returns:

float: The value of the cubic degradation function.

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.dg(phi, degradation_type)[source]#

Evaluate the derivative of the degradation function for a given phi and degradation type.

Parameters:

phifloat: The phase field variable.
degradation_typestr: The type of degradation function (‘quadratic’, ‘cubic’, ‘quartic’, ‘alessi’, or ‘sargado’).

Returns:

float: The value of the derivative of the degradation function.

Examples

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions import dg
>>> phi = np.linspace(0.0, 1.0, 100)
>>> dg_quadratic = [dg(p, "quadratic") for p in phi]
>>> dg_cubic = [dg(p, "cubic") for p in phi]
>>> dg_quartic = [dg(p, "quartic") for p in phi]
>>> plt.figure(figsize=(6, 5))
>>> plt.plot(phi, dg_quadratic, label="Quadratic")
>>> plt.plot(phi, dg_cubic, label="Cubic")
>>> plt.plot(phi, dg_quartic, label="Quartic")
>>> plt.xlabel(r"$\phi$")
>>> plt.ylabel(r"$g'(\phi)$")
>>> plt.title("Derivatives of Degradation Functions")
>>> plt.legend()
>>> plt.grid(True)
>>> plt.tight_layout()
>>> plt.show()

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.g(phi, degradation_type)[source]#

Evaluate the degradation function for a given phi and degradation type.

Parameters:

phifloat: The phase field variable.
degradation_typestr: The type of degradation function (‘quadratic’, ‘cubic’, ‘quartic’, ‘alessi’, or ‘sargado’).

Returns:

float: The value of the degradation function.

Examples

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions import g
>>> phi = np.linspace(0.0, 1.0, 100)
>>> g_quadratic = [g(p, "quadratic") for p in phi]
>>> g_cubic = [g(p, "cubic") for p in phi]
>>> g_quartic = [g(p, "quartic") for p in phi]
>>> plt.figure(figsize=(6, 5))
>>> plt.plot(phi, g_quadratic, label="Quadratic")
>>> plt.plot(phi, g_cubic, label="Cubic")
>>> plt.plot(phi, g_quartic, label="Quartic")
>>> plt.xlabel(r"$\phi$")
>>> plt.ylabel(r"$g(\phi)$")
>>> plt.title("Phase Field Degradation Functions")
>>> plt.legend()
>>> plt.grid(True)
>>> plt.tight_layout()
>>> plt.show()

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.quadratic_degradation_derivative(phi)[source]#

Evaluate the derivative of the quadratic degradation function.

Parameters:

phifloat: The phase field variable.

Returns:

float: The value of the derivative of the quadratic degradation function.

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.quadratic_degradation_function(phi)[source]#

Evaluate the quadratic degradation function.

Parameters:

phifloat: The phase field variable.

Returns:

float: The value of the quadratic degradation function.

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.quartic_degradation_derivative(phi)[source]#

Evaluate the derivative of the quartic degradation function.

Parameters:

phifloat: The phase field variable.

Returns:

float: The value of the derivative of the quartic degradation function.

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.quartic_degradation_function(phi)[source]#

Evaluate the quartic degradation function.

Parameters:

phifloat: The phase field variable.

Returns:

float: The value of the quartic degradation function.

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.sargado_degradation_derivative(phi)[source]#

Evaluate the derivative of the Sargado degradation function.

Parameters:

phifloat: The phase field variable.

Returns:

float: The value of the derivative of the Sargado degradation function.

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.sargado_degradation_function(phi)[source]#

Evaluate the Sargado degradation function.

Parameters:

phifloat: The phase field variable.

Returns:

float: The value of the Sargado degradation function.

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.wu_degradation_derivative(phi)[source]#

Evaluate the derivative of the Wu degradation function.

Parameters:

phifloat: The phase field variable.

Returns:

float: The value of the derivative of the Wu degradation function.

phasefieldx.Element.Phase_Field_Fracture.g_degradation_functions.wu_degradation_function(phi)[source]#

Evaluate the Wu degradation function.

Parameters:

phifloat: The phase field variable.

Returns:

float: The value of the Wu degradation function.

Energy Split#

Energy Decomposition#

Regarding energy degradation, there are two main descriptions of the constitutive free energy functional. The isotropic model degrades all the energy; in the other case, the anisotropic model only degrades the energy related to tension efforts, trying to account that cracks only are generated under tension efforts.

In the Isotropic constitutive free energy functional,

\[E(u,\phi)=\int \left[g(\phi)+k\right] \psi(\boldsymbol \epsilon(\boldsymbol u)) dV,\]

the degradation function \(g(\phi)\) degrades all the energy \(\psi(\boldsymbol \epsilon(\boldsymbol u))\). In the case of the anisotropic constitutive free energy functional,

\[E(u,\phi)=\int \left(\left[g(\phi)+k\right] \psi_a(\boldsymbol \epsilon(\boldsymbol u)) + \psi_b(\boldsymbol \epsilon(\boldsymbol u))\right) dV,\]

the energy \(\psi(\boldsymbol \epsilon(\boldsymbol u))\) is split in two components, \(\psi=\psi_a+\psi_b\), in which the component \(\psi_a\) attains to the energy relative to tensions, so the degradation function only applied to it, avoiding the creation of crack due to compression effort.

There are several methods to carry out the energy slip. PhaseFieldX code implements the Spectral [Miehe] and volumetric-deviatoric [Amor] decomposition.

phasefieldx.Element.Phase_Field_Fracture.split_energy_stress_tangent_functions.psi_a(u, DataSimulation)[source]#

Evaluate the positive part of the strain energy density function based on the specified degradation type.

Parameters:

uufl.Function: The displacement field.
DataSimulationobject: An object containing simulation parameters, including degradation type and split energy type.

Returns:

float: The positive part of the strain energy density.

phasefieldx.Element.Phase_Field_Fracture.split_energy_stress_tangent_functions.psi_a_deviatoric(u, lambda_, mu)[source]#

Evaluate the deviatoric positive part of the strain energy density function.

Parameters:

uufl.Function: The displacement field.
lambda_float: The first Lamé parameter.
mufloat: The second Lamé parameter.

Returns:

float: The deviatoric positive part of the strain energy density.

phasefieldx.Element.Phase_Field_Fracture.split_energy_stress_tangent_functions.psi_a_spectral(u, lambda_, mu)[source]#

Evaluate the spectral positive part of the strain energy density function.

Parameters:

uufl.Function: The displacement field.
lambda_float: The first Lamé parameter.
mufloat: The second Lamé parameter.

Returns:

float: The spectral positive part of the strain energy density.

phasefieldx.Element.Phase_Field_Fracture.split_energy_stress_tangent_functions.psi_b(u, DataSimulation)[source]#

Evaluate the negative part of the strain energy density function based on the specified degradation type.

Parameters:

uufl.Function: The displacement field.
DataSimulationobject: An object containing simulation parameters, including degradation type and split energy type.

Returns:

float: The negative part of the strain energy density.

phasefieldx.Element.Phase_Field_Fracture.split_energy_stress_tangent_functions.psi_b_deviatoric(u, lambda_, mu)[source]#

Evaluate the deviatoric negative part of the strain energy density function.

Parameters:

uufl.Function: The displacement field.
lambda_float: The first Lamé parameter.
mufloat: The second Lamé parameter.

Returns:

float: The deviatoric negative part of the strain energy density.

phasefieldx.Element.Phase_Field_Fracture.split_energy_stress_tangent_functions.psi_b_spectral(u, lambda_, mu)[source]#

Evaluate the spectral negative part of the strain energy density function.

Parameters:

uufl.Function: The displacement field.
lambda_float: The first Lamé parameter.
mufloat: The second Lamé parameter.

Returns:

float: The spectral negative part of the strain energy density.

phasefieldx.Element.Phase_Field_Fracture.split_energy_stress_tangent_functions.sigma_a(u, DataSimulation)[source]#

Evaluate the positive part of the stress tensor based on the specified degradation type.

Parameters:

uufl.Function: The displacement field.
DataSimulationobject: An object containing simulation parameters, including degradation type and split energy type.

Returns:

ufl.Expr: The positive part of the stress tensor.

phasefieldx.Element.Phase_Field_Fracture.split_energy_stress_tangent_functions.sigma_a_deviatoric(u, lambda_, mu)[source]#

Evaluate the deviatoric positive part of the stress tensor.

Parameters:

uufl.Function: The displacement field.
lambda_float: The first Lamé parameter.
mufloat: The second Lamé parameter.

Returns:

ufl.Expr: The deviatoric positive part of the stress tensor.

phasefieldx.Element.Phase_Field_Fracture.split_energy_stress_tangent_functions.sigma_a_spectral(u, lambda_, mu)[source]#

Evaluate the spectral positive part of the stress tensor.

Parameters:

uufl.Function: The displacement field.
lambda_float: The first Lamé parameter.
mufloat: The second Lamé parameter.

Returns:

ufl.Expr: The spectral positive part of the stress tensor.

phasefieldx.Element.Phase_Field_Fracture.split_energy_stress_tangent_functions.sigma_b(u, DataSimulation)[source]#

Evaluate the negative part of the stress tensor based on the specified degradation type.

Parameters:

uufl.Function: The displacement field.
DataSimulationobject: An object containing simulation parameters, including degradation type and split energy type.

Returns:

ufl.Expr: The negative part of the stress tensor.

phasefieldx.Element.Phase_Field_Fracture.split_energy_stress_tangent_functions.sigma_b_deviatoric(u, lambda_, mu)[source]#

Evaluate the deviatoric negative part of the stress tensor.

Parameters:

uufl.Function: The displacement field.
lambda_float: The first Lamé parameter.
mufloat: The second Lamé parameter.

Returns:

ufl.Expr: The deviatoric negative part of the stress tensor.

phasefieldx.Element.Phase_Field_Fracture.split_energy_stress_tangent_functions.sigma_b_spectral(u, lambda_, mu)[source]#

Evaluate the spectral negative part of the stress tensor.

Parameters:

uufl.Function: The displacement field.
lambda_float: The first Lamé parameter.
mufloat: The second Lamé parameter.

Returns:

ufl.Expr: The spectral negative part of the stress tensor.

\(f(\bar{\alpha})\) Fatigue Degradation functions#

Fatigue Degradation Functions \(f(\bar{\alpha})\)#

This module provides functions to evaluate fatigue degradation. Fatigue degradation functions describe how the degradation evolves over time or under repeated loading conditions. The degradation is typically quantified using a parameter \(\bar{\alpha}\), which represents the accumulated variable.

phasefieldx.Element.Phase_Field_Fracture.fatigue_degradation_functions.asymptotic(alpha_bar, Data)[source]#

Calculate the asymptotic fatigue degradation function.

Parameters:

alpha_barnumpy.ndarray: The array of alpha_bar values, representing some measure of material degradation.
Dataobject: An object containing the fatigue degradation properties, specifically fatigue_val.

Returns:

numpy.ndarray: An array containing the calculated degradation values based on the asymptotic function.

phasefieldx.Element.Phase_Field_Fracture.fatigue_degradation_functions.fatigue_degradation(alpha_bar, Data)[source]#

Select and evaluate the appropriate fatigue degradation function.

Parameters:

alpha_barnumpy.ndarray: The array of alpha_bar values, representing some measure of material degradation.
Dataobject: An object containing the fatigue degradation properties, including the type of degradation function.

Returns:

numpy.ndarray: An array containing the calculated degradation values based on the selected degradation function.

Raises:

ValueError: If an invalid degradation type is provided in Data.fatigue_degradation_function.

Solvers#

Solver: Phase-field fracture/fatigue#

This module provides a solver for phase-field fracture and fatigue simulations using the Finite Element Method (FEM).

phasefieldx.Element.Phase_Field_Fracture.solver.solver.solve(Data, msh, final_time, V_u, V_Φ, bc_list_u=[], bc_list_phi=[], update_boundary_conditions=None, f_list_u=None, T_list_u=None, update_loading=None, ds_bound=None, dt=1.0, path=None, bcs_list_u_names=None, min_stagger_iter=2, max_stagger_iter=10000, stagger_error_tol=1e-08)[source]#

Solve the phase-field fracture and fatigue problem.

Parameters:

Dataobject: An object containing the simulation parameters and settings.
mshdolfinx.mesh: The computational mesh for the simulation.
final_timefloat: The final simulation time.
V_udolfinx.FunctionSpace: The function space for the displacement field.
V_Φdolfinx.FunctionSpace: The function space for the phase-field.
bc_list_ulist of dolfinx.DirichletBC, optional: List of Dirichlet boundary conditions for the displacement field.
bc_list_philist of dolfinx.DirichletBC, optional: List of Dirichlet boundary conditions for the phase-field.
update_boundary_conditionscallable, optional: A function to update boundary conditions dynamically.
f_list_ulist, optional: List of body forces.
T_list_ulist of tuple, optional: List of tuples containing traction forces and corresponding measures.
update_loadingcallable, optional: A function to update loading conditions dynamically.
ds_boundnumpy.ndarray, optional: Array of boundary measures for calculating reaction forces.
dtfloat, optional: Time step size.
pathstr, optional: Directory path for saving simulation results. Defaults to current working directory.
min_stagger_iter: int: Minimum stagger iterations for numerical simulations.
max_stagger_iter: int: Maximum stagger iterations for numerical simulations.
stagger_error_tol: float: Tolerance for stagger error in simulations.

Returns:

None

Energy-controlled PFF solver: Variational scheme#

This module implements the variational solver for the phase-field fracture problem presented in Castillón et al.[1].

phasefieldx.Element.Phase_Field_Fracture.solver.solver_ener_variational.solve(Data, msh, final_gamma, V_u, V_Φ, bc_list_u=[], bc_list_phi=[], f_list_u=None, T_list_u=None, ds_bound=None, dtau=0.0005, dtau_min=1e-12, dtau_max=0.1, path=None, bcs_list_u_names=None, c1=1.0, c2=1.0, threshold_gamma_save=None)[source]#

Solve the phase-field fracture problem.

Parameters:

Dataobject: Container for simulation parameters (material constants, Gc, l, degradation_function, save flags, results folder name, logging helpers, and methods like save_log_info).
mshdolfinx.mesh.Mesh: The computational mesh.
final_gammafloat: Target crack surface measure at which the simulation stops.
V_udolfinx.FunctionSpace: Function space for the displacement field.
V_Φdolfinx.FunctionSpace: Function space for the phase-field.
bc_list_ulist, optional: List of dolfinx DirichletBC objects for the displacement DOFs (default: []).
bc_list_philist, optional: List of dolfinx DirichletBC objects for the phase-field DOFs (default: []).
f_list_ulist or None, optional: List of body forces (not required by this solver; default: None).
T_list_ulist of tuple or None, optional: List of traction definitions as tuples (traction_vector, measure). The solver currently uses T_list_u[0] for external work and Lagrange coupling (default: None).
ds_boundarray-like or None, optional: Boundary measure(s) for reaction force calculation (default: None).
dtaufloat, optional: Initial pseudo-time increment / load parameter increment (default: 5e-4).
dtau_minfloat, optional: Minimum allowed dtau before aborting (default: 1e-12).
dtau_maxfloat, optional: Maximum allowed dtau (default: 0.1).
pathstr or None, optional: Path where results folder will be created; if None uses current working dir (default: None).
bcs_list_u_nameslist of str or None, optional: Names used when saving reaction files for displacement boundary conditions (default: None).
c1float, optional: Coefficient multiplying the crack measure (default: 1.0).
c2float, optional: Coefficient multiplying the external energy (default: 1.0).
threshold_gamma_savefloat or None, optional: Threshold on change in gamma to trigger Paraview output. If None, treated as 0.0 (default: None).

Notes

The docstring parameters correspond to the function signature. Some inputs (like T_list_u) are accessed by index in the implementation and must be provided in that form.

Energy-controlled PFF solver: Non-variational scheme#

This module implements the non-variational solver for the phase-field fracture problem presented in Castillón et al.[1].

phasefieldx.Element.Phase_Field_Fracture.solver.solver_ener_non_variational.solve(Data, msh, final_gamma, V_u, V_Φ, bc_list_u=[], bc_list_phi=[], f_list_u=None, T_list_u=None, ds_bound=None, dtau=0.0005, dtau_min=1e-12, dtau_max=0.1, path=None, bcs_list_u_names=None, c1=1.0, c2=1.0, threshold_gamma_save=None, case='AT2', rho=0.0)[source]#

Solve the phase-field fracture problem.

Parameters:

Dataobject: Container for simulation parameters (material constants, Gc, l, degradation_function, save flags, results folder name, logging helpers, and methods like save_log_info).
mshdolfinx.mesh.Mesh: The computational mesh.
final_gammafloat: Target crack surface measure at which the simulation stops.
V_udolfinx.FunctionSpace: Function space for the displacement field.
V_Φdolfinx.FunctionSpace: Function space for the phase-field.
bc_list_ulist, optional: List of dolfinx DirichletBC objects for the displacement DOFs (default: []).
bc_list_philist, optional: List of dolfinx DirichletBC objects for the phase-field DOFs (default: []).
f_list_ulist or None, optional: List of body forces (not required by this solver; default: None).
T_list_ulist of tuple or None, optional: List of traction definitions as tuples (traction_vector, measure). The solver currently uses T_list_u[0] for external work and Lagrange coupling (default: None).
ds_boundarray-like or None, optional: Boundary measure(s) for reaction force calculation (default: None).
dtaufloat, optional: Initial pseudo-time increment / load parameter increment (default: 5e-4).
dtau_minfloat, optional: Minimum allowed dtau before aborting (default: 1e-12).
dtau_maxfloat, optional: Maximum allowed dtau (default: 0.1).
pathstr or None, optional: Path where results folder will be created; if None uses current working dir (default: None).
bcs_list_u_nameslist of str or None, optional: Names used when saving reaction files for displacement boundary conditions (default: None).
c1float, optional: Coefficient multiplying the crack measure (default: 1.0).
c2float, optional: Coefficient multiplying the external energy (default: 1.0).
threshold_gamma_savefloat or None, optional: Threshold on change in gamma to trigger Paraview output. If None, treated as 0.0 (default: None).

Notes

The docstring parameters correspond to the function signature. Some inputs (like T_list_u) are accessed by index in the implementation and must be provided in that form.