r"""
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()
"""
from math import exp
[docs]
def quadratic_degradation_function(phi):
"""
Evaluate the quadratic degradation function.
Parameters
----------
phi : float
The phase field variable.
Returns
-------
float
The value of the quadratic degradation function.
"""
return (1.0 - phi) ** 2.0
[docs]
def quadratic_degradation_derivative(phi):
"""
Evaluate the derivative of the quadratic degradation function.
Parameters
----------
phi : float
The phase field variable.
Returns
-------
float
The value of the derivative of the quadratic degradation function.
"""
return -2.0 * (1.0 - phi)
[docs]
def cubic_degradation_function(phi):
"""
Evaluate the cubic (Borden) degradation function.
Parameters
----------
phi : float
The phase field variable.
Returns
-------
float
The value of the cubic degradation function.
"""
return 3.0 * (1.0 - phi)**2 - 2.0 * (1.0 - phi)**3
[docs]
def cubic_degradation_derivative(phi):
"""
Evaluate the derivative of the cubic degradation function.
Parameters
----------
phi : float
The phase field variable.
Returns
-------
float
The value of the derivative of the cubic degradation function.
"""
return -6.0 * phi * (1.0 - phi)
[docs]
def quartic_degradation_function(phi):
"""
Evaluate the quartic degradation function.
Parameters
----------
phi : float
The phase field variable.
Returns
-------
float
The value of the quartic degradation function.
"""
return 4.0 * (1.0 - phi)**3 - 3.0 * (1.0 - phi)**4
[docs]
def quartic_degradation_derivative(phi):
"""
Evaluate the derivative of the quartic degradation function.
Parameters
----------
phi : float
The phase field variable.
Returns
-------
float
The value of the derivative of the quartic degradation function.
"""
return -12.0 * (1.0 - phi) ** 2 + 12.0 * (1.0 - phi)**3
[docs]
def alessi_degradation_function(phi):
"""
Evaluate the Alessi degradation function.
Parameters
----------
phi : float
The phase field variable.
Returns
-------
float
The value of the Alessi degradation function.
"""
k = 100.0
Q = 1.0 - ((1.0 - phi) * (1.0 - phi))
return (1.0 - phi) * (1.0 - phi) / (1.0 + (k - 1.0) * Q)
[docs]
def alessi_degradation_derivative(phi):
"""
Evaluate the derivative of the Alessi degradation function.
Parameters
----------
phi : float
The phase field variable.
Returns
-------
float
The value of the derivative of the Alessi degradation function.
"""
k = 100.0
return (2.0 * k * (phi - 1.0) * (k * (phi - 1.0) * phi - 1.0)) / \
((k * phi - 1.0) ** 2.0)
[docs]
def sargado_degradation_function(phi):
"""
Evaluate the Sargado degradation function.
Parameters
----------
phi : float
The phase field variable.
Returns
-------
float
The value of the Sargado degradation function.
"""
k = 100.0
Q = 1.0 - ((1.0 - phi) * (1.0 - phi))
return (1.0 - phi) * (1.0 - phi) / (1.0 + (k - 1.0) * Q)
[docs]
def sargado_degradation_derivative(phi):
"""
Evaluate the derivative of the Sargado degradation function.
Parameters
----------
phi : float
The phase field variable.
Returns
-------
float
The value of the derivative of the Sargado degradation function.
"""
k = 100.0
return -2.0 * k * (phi - 1.0) * k * (k * (phi - 2.0) *
phi - 1.0) / ((k * phi - 1.0) ** 2.0 * exp(k * k))
[docs]
def wu_degradation_function(phi):
"""
Evaluate the Wu degradation function.
Parameters
----------
phi : float
The phase field variable.
Returns
-------
float
The value of the Wu degradation function.
"""
k = 100.0
Q = 1.0 - ((1.0 - phi) * (1.0 - phi))
return (1.0 - phi) * (1.0 - phi) / (1.0 + (k - 1.0) * Q)
[docs]
def wu_degradation_derivative(phi):
"""
Evaluate the derivative of the Wu degradation function.
Parameters
----------
phi : float
The phase field variable.
Returns
-------
float
The value of the derivative of the Wu degradation function.
"""
k = 100.0
return -2.0 * k * (phi - 1.0) * k * (k * (phi - 2.0) *
phi - 1.0) / ((k * phi - 1.0) ** 2.0 * exp(k * k))
[docs]
def g(phi, degradation_type):
"""
Evaluate the degradation function for a given phi and degradation type.
Parameters
----------
phi : float
The phase field variable.
degradation_type : str
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()
"""
if degradation_type == "quadratic":
return quadratic_degradation_function(phi)
elif degradation_type == "cubic":
return cubic_degradation_function(phi)
elif degradation_type == "quartic":
return quartic_degradation_function(phi)
elif degradation_type == "alessi":
return alessi_degradation_function(phi)
elif degradation_type == "sargado":
return sargado_degradation_function(phi)
else:
raise ValueError(
"Invalid degradation_type. Please choose from 'quadratic', 'cubic', 'quartic', 'alessi', or 'sargado'.")
[docs]
def dg(phi, degradation_type):
"""
Evaluate the derivative of the degradation function for a given phi and degradation type.
Parameters
----------
phi : float
The phase field variable.
degradation_type : str
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()
"""
if degradation_type == "quadratic":
return quadratic_degradation_derivative(phi)
elif degradation_type == "cubic":
return cubic_degradation_derivative(phi)
elif degradation_type == "quartic":
return quartic_degradation_derivative(phi)
elif degradation_type == "alessi":
return alessi_degradation_derivative(phi)
elif degradation_type == "sargado":
return sargado_degradation_derivative(phi)
else:
raise ValueError(
"Invalid degradation_derivative_type. Please choose from 'quadratic', 'cubic', 'quartic', 'alessi', or 'sargado'.")
