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, 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')

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.

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)

Log the simulation parameters using the provided logger.

Parameters:

logger: An instance of a logging object.

save_parameters_to_csv(filename='parameters.input')

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.

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.

\(f(\bar{\alpha})\) 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.

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)

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].

[1] (1,2)

M. Castillón, I. Romero, and J. Segurado. A phase-field approach to fatigue analysis: bridging theory and simulation. International Journal of Fatigue, 205:109397, 2026. URL: https://www.sciencedirect.com/science/article/pii/S0142112325005948, doi:https://doi.org/10.1016/j.ijfatigue.2025.109397.

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)

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)

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.