Note

Go to the end to download the full example code.

Single edge notched shear test#

A well-known benchmark simulation in fracture mechanics is performed, based on the work by [Miehe].

The model consists of a square plate with a notch located halfway up, extending from the left side to the center, as shown in the figure below. The bottom part is fixed in all directions, while the top part can slide horizontally. A horizontal displacement is applied at the top. The geometry and boundary conditions are depicted in the figure. We discretize the model by refining the areas where crack evolution is expected, ensuring that the element size \(h\) is sufficiently small to avoid mesh dependency.

#            u
#            =>=>=>=>=>=>
#            *----------*
#            |          |
#            | a=0.5    |
#            |---       |
#            |          |
#            |          |
#            *----------*
#            /_\/_\/_\/_\
#     |Y    /////////////
#     |
#      ---X
#  Z /

The Young’s modulus, Poisson’s ratio, and the critical energy release rate are given in the table Properties. Young’s modulus \(E\) and Poisson’s ratio \(\nu\) can be represented with the Lamé parameters as: \(\lambda=\frac{E\nu}{(1+\nu)(1-2\nu)}\); \(\mu=\frac{E}{2(1+\nu)}\).

VALUE

UNITS

E

210

kN/mm2

nu

0.3

[-]

Gc

0.0027

kN/mm

l

0.015

mm
[Miehe]

A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits, https://doi.org/10.1016/j.cma.2010.04.011.

Import necessary libraries#

import numpy as np
import matplotlib.pyplot as plt
import pyvista as pv
import dolfinx
import mpi4py
import petsc4py
import os

Import from phasefieldx package#

from phasefieldx.Element.Phase_Field_Fracture.Input import Input
from phasefieldx.Element.Phase_Field_Fracture.solver.solver import solve
from phasefieldx.Boundary.boundary_conditions import bc_xy, bc_y, get_ds_bound_from_marker
from phasefieldx.PostProcessing.ReferenceResult import AllResults

Parameters Definition#

Data is an input object containing essential parameters for simulation setup and result storage:

  • E: Young’s modulus, set to 210 \(kN/mm^2\).

  • nu: Poisson’s ratio, set to 0.3.

  • Gc: Critical energy release rate, set to 0.005 \(kN/mm\).

  • l: Length scale parameter, set to 0.1 \(mm\).

  • degradation: Specifies the degradation type. Options are “isotropic” or “anisotropic”.

  • split_energy: Controls how the energy is split; options include “no” (default), “spectral,” or “deviatoric.” In this case an anisotropic formulation with volumetric-deviatric decomposition is considered.

  • degradation_function: Specifies the degradation function; here, it is “quadratic.”

  • irreversibility: Determines the irreversibility criterion; in this case, set to “miehe.”

  • fatigue: Enables fatigue simulation when set to True.

  • fatigue_degradation_function: Defines the function for fatigue degradation, set to “asymptotic.”

  • fatigue_val: Fatigue parameter value (used only in fatigue simulations, not in this one).

  • k: Stiffness penalty parameter, set to 0.0.

  • min_stagger_iter: Minimum number of staggered iterations, set to 2.

  • max_stagger_iter: Maximum number of staggered iterations, set to 500.

  • stagger_error_tol: Error tolerance for staggered iterations, set to 1e-8.

  • save_solution_xdmf and save_solution_vtu: Specify the file formats to save displacement results. In this case, results are saved as .vtu files.

  • results_folder_name: Name of the folder for saving results. If it exists, it will be replaced with a new empty folder.

Data = Input(E=210.0,    # young modulus
             nu=0.3,     # poisson
             Gc=0.0027,  # critical energy release rate
             l=0.06,     # lenght scale parameter
             degradation="anisotropic",  # "isotropic" "anisotropic"
             split_energy="spectral",    # "spectral" "deviatoric"
             degradation_function="quadratic",
             irreversibility="miehe",  # "miehe"
             fatigue=False,
             fatigue_degradation_function="asymptotic",
             fatigue_val=0.0,
             k=0.0,
             save_solution_xdmf=False,
             save_solution_vtu=True,
             results_folder_name="1712_Single_Edge_Notched_Shear_Test")

Mesh Definition#

The mesh is generated using Gmsh and saved as a ‘mesh.msh’ file. For more details on how to create the mesh, refer to the Examples gmsh “.geo” files examples. The following lines

msh_file = os.path.join("mesh", "mesh_shear.msh")  # Path to the mesh file
gdim = 2                                     # Geometric dimension of the mesh
gmsh_model_rank = 0                          # Rank of the Gmsh model in a parallel setting
mesh_comm = mpi4py.MPI.COMM_WORLD            # MPI communicator for parallel computation

The mesh, cell markers, and facet markers are extracted from the ‘mesh.msh’ file using the read_from_msh function.

mesh_data = dolfinx.io.gmsh.read_from_msh(msh_file, mesh_comm, gmsh_model_rank, gdim)
msh = mesh_data.mesh
cell_markers = mesh_data.cell_tags
facet_markers = mesh_data.facet_tags

fdim = msh.topology.dim - 1 # Dimension of the mesh facets

Facets defined in the .geo file used to generate the ‘mesh.msh’ file are identified here. Each marker variable corresponds to a specific region on the specimen:

  • bottom_facet_marker: Refers to the bottom part of the specimen.

  • top_facet_marker: Refers to the top part of the specimen.

  • right_facet_marker: Refers to the right side of the specimen.

  • left_facet_marker: Refers to the left side of the specimen.

bottom_facet_marker = facet_markers.find(9)
top_facet_marker = facet_markers.find(10)
right_facet_marker = facet_markers.find(11)
left_facet_marker = facet_markers.find(12)

The get_ds_bound_from_marker function creates measures for applying boundary conditions on specific facets. These measures are generated for:

  • bottom_facet_marker → Stored in ds_bottom

  • top_facet_marker → Stored in ds_top

  • right_facet_marker → Stored in ds_right

  • left_facet_marker → Stored in ds_left

ds_bottom = get_ds_bound_from_marker(bottom_facet_marker, msh, fdim)
ds_top = get_ds_bound_from_marker(top_facet_marker, msh, fdim)
ds_right = get_ds_bound_from_marker(right_facet_marker, msh, fdim)
ds_left = get_ds_bound_from_marker(left_facet_marker, msh, fdim)

ds_list = np.array([
                   [ds_top, "top"],
                   [ds_bottom, "bottom"],
                   [ds_left, "left"],
                   [ds_right, "right"],
                   ])

Function Space definition

V_u = dolfinx.fem.functionspace(msh, ("Lagrange", 1, (msh.geometry.dim, )))
V_phi = dolfinx.fem.functionspace(msh, ("Lagrange", 1))

Boundary Conditions#

The boundary conditions are applied as follows:

  • The bottom nodes are fixed in both the x and y directions, ensuring the bottom edge remains completely fixed.

  • The top nodes are allowed to slide horizontally, with the x displacement being unconstrained, and the vertical displacement fixed.

  • The left and right boundaries are also constrained in the vertical direction (y displacement), allowing horizontal sliding only.

bc_bottom = bc_xy(bottom_facet_marker, V_u, fdim)
bc_top = bc_xy(top_facet_marker, V_u, fdim)
bc_left = bc_y(left_facet_marker, V_u, fdim)
bc_right = bc_y(right_facet_marker, V_u, fdim)

The bcs_list_u variable is a list that stores all boundary conditions for the displacement field \(\boldsymbol u\). This list facilitates easy management of multiple boundary conditions and can be expanded if additional conditions are needed.

bcs_list_u = [bc_top, bc_bottom, bc_left, bc_right]
bcs_list_u_names = ["top", "bottom", "left", "right"]

Function: update_boundary_conditions#

The update_boundary_conditions function updates the displacement boundary conditions dynamically at each time step. This enables quasi-static analysis by incrementally adjusting the displacements applied to specific degrees of freedom.

Parameters:

  • bcs: A list of boundary conditions, where each element corresponds to a boundary condition applied to a specific facet of the mesh.

  • time: A scalar representing the current time step in the analysis.

Function Details:

  • The displacement value val is computed as a function of time:

  • val = dt0 * time, where dt0 is a small time step factor (10^-4), representing a gradual displacement applied along the x-axis. This displacement increases linearly over time.

  • The calculated value is assigned to the x-component of the displacement field for the boundary condition specified in bcs_list_u[0] by modifying bcs_list_u[0].g.value[0].

Return Value:

  • A tuple (val, 0, 0) is returned, representing the incremental displacement vector:

  • The first element (val) is the calculated x-displacement.

  • The second element (0) indicates no update for the y-displacement.

  • The third element (0) indicates no update for the z-displacement, applicable in 2D simulations.

Purpose:

  • This function facilitates quasi-static analysis by applying controlled, time-dependent boundary displacements. It is essential for simulations that involve gradual loading or unloading, with a continuous linear displacement evolution along the x-direction over time.

def update_boundary_conditions(bcs, time):
    dt0 = 10**-4
    val = time * dt0
    bcs_list_u[0].g.value[0] = petsc4py.PETSc.ScalarType(val)
    return val, 0, 0


T_list_u = None
update_loading = None
f = None
T = dolfinx.fem.Constant(msh, petsc4py.PETSc.ScalarType((0.0, 0.0)))

Boundary Conditions for phase field

bcs_list_phi = []

Solver Call for a Phase-Field Fracture Problem#

This section sets up and calls the solver for a phase-field fracture problem.

Key Points:

  • The simulation is run for a final time of 150, with a time step of 1.0.

  • The solver will manage the mesh, boundary conditions, and update the solution over the specified time steps.

Parameters:

  • dt: The time step for the simulation, set to 1.0.

  • final_time: The total simulation time, set to 150.0, which determines how long the problem will be solved.

  • path: Optional parameter for specifying the folder where results will be saved; here it is set to None, meaning results will be saved to the default location.

Function Call:

The solve function is invoked with the following arguments:

  • Data: Contains the simulation parameters and configurations.

  • msh: The mesh representing the domain for the problem.

  • final_time: The total duration of the simulation (200.0).

  • V_u: Function space for the displacement field, \(\boldsymbol{u}\).

  • V_phi: Function space for the phase field, \(\phi\).

  • bcs_list_u: List of Dirichlet boundary conditions for the displacement field.

  • bcs_list_phi: List of boundary conditions for the phase field (empty in this case).

  • update_boundary_conditions: Function to update boundary conditions for the displacement field.

  • f: The body force applied to the domain (if any).

  • T_list_u: Time-dependent loading parameters for the displacement field.

  • update_loading: Function to update loading parameters for the quasi-static analysis.

  • ds_list: Boundary measures for integration over the domain boundaries.

  • dt: The time step for the simulation.

  • path: Directory for saving results (if specified).

This setup provides a framework for solving static problems with specified boundary conditions and loading parameters.

dt = 1.0
final_time = 150.0

Uncomment the following lines to run the solver with the specified parameters.

# solve(Data,
#       msh,
#       final_time,
#       V_u,
#       V_phi,
#       bcs_list_u,
#       bcs_list_phi,
#       update_boundary_conditions,
#       f,
#       T_list_u,
#       update_loading,
#       ds_list,
#       dt,
#       path=None,
#       bcs_list_u_names=bcs_list_u_names,
#       min_stagger_iter=2,
#       max_stagger_iter=500,
#       stagger_error_tol=1e-8)

Load results#

Once the simulation finishes, the results are loaded from the results folder. The AllResults class takes the folder path as an argument and stores all the results, including logs, energy, convergence, and DOF files. Note that it is possible to load results from other results folders to compare results. It is also possible to define a custom label and color to automate plot labels.

S = AllResults(Data.results_folder_name)
S.set_label('Simulation')
S.set_color('b')

Plot: phase-field \(\phi\)#

The phase-field result saved in the .vtu file is shown. For this, the file is loaded using PyVista.

file_vtu = pv.read(os.path.join(Data.results_folder_name, "paraview-solutions_vtu", "phasefieldx_p0_000097.vtu"))

file_vtu.plot(scalars='phi', cpos='xy', show_scalar_bar=True, show_edges=False)
plot 1712

Plot: displacement \(\boldsymbol u\)#

The displacements results saved in the .vtu file are shown.

file_vtu = pv.read(os.path.join(Data.results_folder_name, "paraview-solutions_vtu", "phasefieldx_p0_000097.vtu"))
file_vtu.plot(scalars='u', cpos='xy', show_scalar_bar=True, show_edges=False)

plt.show()
plot 1712

Plot: Displacement vs Fracture Energy#

The vertical displacement is saved in S.dof_files[“top.dof”][“Uy”].

displacement = S.dof_files["top.dof"]["Ux"]

fig, energyW = plt.subplots()

energyW.plot(displacement, S.energy_files['total.energy']["W"], 'b-', linewidth=2.0, label=r'$W$')
energyW.plot(displacement, S.energy_files['total.energy']["W_phi"], 'y-', linewidth=2.0, label=r'$W_{\phi}$')
energyW.plot(displacement, S.energy_files['total.energy']["W_gradphi"], 'g-', linewidth=2.0, label=r'$W_{\nabla \phi}$')

energyW.grid(color='k', linestyle='-', linewidth=0.3)
energyW.set_xlabel('displacement - u $[mm]$')
energyW.set_ylabel('Energy')
energyW.legend()
plot 1712
<matplotlib.legend.Legend object at 0x734277951450>

Plot: Force vs Vertical Displacement#

Miehe = np.loadtxt(os.path.join("reference_solutions", "miehe_solution_shear.csv"))

fig, ax_reaction = plt.subplots()

ax_reaction.plot(Miehe[:, 0], Miehe[:, 1], 'g-', linewidth=2.0, label='Miehe')
ax_reaction.plot(displacement, -S.reaction_files['bottom.reaction']["Rx"], 'k.', linewidth=2.0, label=S.label)
ax_reaction.grid(color='k', linestyle='-', linewidth=0.3)
ax_reaction.set_xlabel('displacement - u $[mm]$')
ax_reaction.set_ylabel('reaction force - F $[kN]$')
ax_reaction.set_title('Reaction Force vs Vertical Displacement')
ax_reaction.legend()
Reaction Force vs Vertical Displacement
<matplotlib.legend.Legend object at 0x73426e79feb0>

Plot: Staggered Iterations vs Vertical Displacement#

fig, ax_convergence = plt.subplots()

ax_convergence.plot(displacement, S.convergence_files["phasefieldx.conv"]["stagger"], 'k.', linewidth=2.0, label='Stagger iterations')

ax_convergence.grid(color='k', linestyle='-', linewidth=0.3)
ax_convergence.set_xlabel('displacement - u $[mm]$')
ax_convergence.set_ylabel('stagger iterations - []')
ax_convergence.set_title('Stagger iterations vs vertical displacement')
ax_convergence.legend()

plt.show()
Stagger iterations vs vertical displacement

Total running time of the script: (0 minutes 4.789 seconds)

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