Skip to content

LEFM

Module for computing stress intensity factor (SIF) influence functions and energy release rates.

This module provides functions to compute the functions F_ij and G_i as described in Amestoy and Leblond (1992). These functions are used to calculate the SIFs at the tip of an infinitesimal straight crack extension with a given bifurcation angle.

References

Amestoy, M., & Leblond, J. B. (1992). Crack paths in plane situations—II. Detailed form of the expansion of the stress intensity factors. International Journal of Solids and Structures, 29(4), 465–501. https://doi.org/10.1016/0020-7683(92)90210-K

F11(m)

Computes F11 function for a given normalized crack angle.

Parameters:

Name Type Description Default
m float

Normalized crack angle, defined as (phi - phi0) / pi.

 required

Returns:

Name Type Description
float float

The value of the F11 for the given normalized crack angle.
Source code in src/gcrack/lefm.py 
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
@jit
def F11(m: float) -> float:
    """Computes F11 function for a given normalized crack angle.

    Args:
        m (float): Normalized crack angle, defined as (phi - phi0) / pi.

    Returns:
        float: The value of the F11 for the given normalized crack angle.
    """
    return (
        1
        - 3 * pi**2 / 8 * m**2
        + (pi**2 - 5 * pi**4 / 128) * m**4
        + (pi**2 / 9 - 11 * pi**4 / 72 + 119 * pi**6 / 15_360) * m**6
        + 5.07790 * m**8
        - 2.88312 * m**10
        - 0.0925 * m**12
        + 2.996 * m**14
        - 4.059 * m**16
        + 1.63 * m**18
        + 4.1 * m**20
    )

F12(m)

Computes F12 function for a given normalized crack angle.

Parameters:

Name Type Description Default
m float

Normalized crack angle, defined as (phi - phi0) / pi.

 required

Returns:

Name Type Description
float float

The value of the F12 for the given normalized crack angle.
Source code in src/gcrack/lefm.py 
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
@jit
def F12(m: float) -> float:
    """Computes F12 function for a given normalized crack angle.

    Args:
        m (float): Normalized crack angle, defined as (phi - phi0) / pi.

    Returns:
        float: The value of the F12 for the given normalized crack angle.
    """
    return (
        -3 * pi / 2 * m
        + (10 * pi / 3 + pi**3 / 16) * m**3
        + (-2 * pi - 133 * pi**3 / 180 + 59 * pi**5 / 1280) * m**5
        + 12.313906 * m**7
        - 7.32433 * m**9
        + 1.5793 * m**11
        + 4.0216 * m**13
        - 6.915 * m**15
        + 4.21 * m**17
        + 4.56 * m**19
    )

F21(m)

Computes F21 function for a given normalized crack angle.

Parameters:

Name Type Description Default
m float

Normalized crack angle, defined as (phi - phi0) / pi.

 required

Returns:

Name Type Description
float float

The value of the F21 for the given normalized crack angle.
Source code in src/gcrack/lefm.py 
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
@jit
def F21(m: float) -> float:
    """Computes F21 function for a given normalized crack angle.

    Args:
        m (float): Normalized crack angle, defined as (phi - phi0) / pi.

    Returns:
        float: The value of the F21 for the given normalized crack angle.
    """
    return (
        pi / 2 * m
        - (4 * pi / 3 + pi**3 / 48) * m**3
        + (-2 * pi / 3 + 13 * pi**3 / 30 - 59 * pi**5 / 3840) * m**5
        - 6.176023 * m**7
        + 4.44112 * m**9
        - 1.5340 * m**11
        - 2.0700 * m**13
        + 4.684 * m**15
        - 3.95 * m**17
        - 1.32 * m**19
    )

F22(m)

Computes F22 function for a given normalized crack angle.

Parameters:

Name Type Description Default
m float

Normalized crack angle, defined as (phi - phi0) / pi.

 required

Returns:

Name Type Description
float float

The value of the F22 for the given normalized crack angle.
Source code in src/gcrack/lefm.py 
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
@jit
def F22(m: float) -> float:
    """Computes F22 function for a given normalized crack angle.

    Args:
        m (float): Normalized crack angle, defined as (phi - phi0) / pi.

    Returns:
        float: The value of the F22 for the given normalized crack angle.
    """
    return (
        1
        - (4 + 3 / 8 * pi**2) * m**2
        + (8 / 3 + 29 / 18 * pi**2 - 5 / 128 * pi**4) * m**4
        + (-32 / 15 - 4 / 9 * pi**2 - 1159 / 7200 * pi**4 + 119 / 15_360 * pi**6) * m**6
        + 10.58254 * m**8
        - 4.78511 * m**10
        - 1.8804 * m**12
        + 7.280 * m**14
        - 7.591 * m**16
        + 0.25 * m**18
        + 12.5 * m**20
    )

Fmat(m)

Construct the matrix F containing the Fij functions of Amestoy-Leblond.

Parameters:

Name Type Description Default
m float

Normalized crack angle, defined as (phi - phi0) / pi.

 required

Returns:

Type Description
ndarray

jnp.ndarray: The 2x2 matrix F for the given normalized crack angle.
Source code in src/gcrack/lefm.py 
115
116
117
118
119
120
121
122
123
124
125
@jit
def Fmat(m: float) -> jnp.ndarray:
    """Construct the matrix F containing the Fij functions of Amestoy-Leblond.

    Args:
        m (float): Normalized crack angle, defined as (phi - phi0) / pi.

    Returns:
        jnp.ndarray: The 2x2 matrix F for the given normalized crack angle.
    """
    return jnp.array([[F11(m), F12(m)], [F21(m), F22(m)]])

G1(m)

Computes G1 function for a given normalized crack angle.

Parameters:

Name Type Description Default
m float

Normalized crack angle, defined as (phi - phi0) / pi.

 required

Returns:

Name Type Description
float float

The value of the G1 for the given normalized crack angle.
Source code in src/gcrack/lefm.py 
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
@jit
def G1(m: float) -> float:
    """Computes G1 function for a given normalized crack angle.

    Args:
        m (float): Normalized crack angle, defined as (phi - phi0) / pi.

    Returns:
        float: The value of the G1 for the given normalized crack angle.
    """
    return (
        (2 * pi) ** (3 / 2) * m**2
        - 47.933390 * m**4
        + 63.665987 * m**6
        - 50.70880 * m**8
        + 26.66807 * m**10
        - 6.0205 * m**12
        - 7.314 * m**14
        + 10.947 * m**16
        - 2.85 * m**18
        - 13.7 * m**20
    )

G2(m)

Computes G2 function for a given normalized crack angle.

Parameters:

Name Type Description Default
m float

Normalized crack angle, defined as (phi - phi0) / pi.

 required

Returns:

Name Type Description
float float

The value of the G2 for the given normalized crack angle.
Source code in src/gcrack/lefm.py 
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
@jit
def G2(m: float) -> float:
    """Computes G2 function for a given normalized crack angle.

    Args:
        m (float): Normalized crack angle, defined as (phi - phi0) / pi.

    Returns:
        float: The value of the G2 for the given normalized crack angle.
    """
    return (
        -2 * jnp.sqrt(2 * pi) * m
        + 12 * jnp.sqrt(2 * pi) * m**3
        - 59.565733 * m**5
        + 61.174444 * m**7
        - 39.90249 * m**9
        + 15.6222 * m**11
        + 3.0343 * m**13
        - 12.781 * m**15
        + 9.69 * m**17
        + 6.62 * m**19
    )

G_star(phi, phi0, KI, KII, T, Ep, s)

Computes the energy release rate G* after a infinitesimal kink of angle.

This function computes the energy release rate G* using the Irwin formula. The SIFs are calculated as described in Amestoy and Leblond (1992).

Parameters:

Name Type Description Default
phi float

Current crack angle.

 required
phi0 float

Initial crack angle.

 required
KI float

Mode I stress intensity factor.

 required
KII float

Mode II stress intensity factor.

 required
T float

T-stress.

 required
Ep float

Plane strain/stress modulus.

 required
s float

Internal length associated with T-stress.

 required

Returns:

Name Type Description
float float

The energy release rate G* for the given crack angle and stress intensity factors.
Source code in src/gcrack/lefm.py 
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
@jit
def G_star(
    phi: float, phi0: float, KI: float, KII: float, T: float, Ep: float, s: float
) -> float:
    """Computes the energy release rate G* after a infinitesimal kink of angle.

    This function computes the energy release rate G* using the Irwin formula.
    The SIFs are calculated as described in Amestoy and Leblond (1992).

    Args:
        phi (float): Current crack angle.
        phi0 (float): Initial crack angle.
        KI (float): Mode I stress intensity factor.
        KII (float): Mode II stress intensity factor.
        T (float): T-stress.
        Ep (float): Plane strain/stress modulus.
        s (float): Internal length associated with T-stress.

    Returns:
        float: The energy release rate G* for the given crack angle and stress intensity factors.
    """
    # Store the SIFs in an array
    k = jnp.array([KI, KII])
    # Calculate m
    m = (phi - phi0) / pi
    # Compute the Amestoy-Leblond functions
    f_mat = Fmat(m)
    g_vec = Gvec(m)
    # Apply Amestoy-Leblond formula
    ks = f_mat @ k + g_vec * T * jnp.sqrt(s)
    # Compute the G star
    return 1 / Ep * jnp.dot(ks, ks)

G_star_coupled(phi, phi0, KI1, KII1, T1, KI2, KII2, T2, Ep, s)

Computes the coupled energy release rate G* for two sets of stress intensity factors.

It is used to evaluate the energy release rate for two interacting loading conditions.

Parameters:

Name Type Description Default
phi float

Current crack angle.

 required
phi0 float

Initial crack angle.

 required
KI1 float

Mode I stress intensity factor for the first load.

 required
KII1 float

Mode II stress intensity factor for the first load.

 required
T1 float

T-stress for the first load.

 required
KI2 float

Mode I stress intensity factor for the second load.

 required
KII2 float

Mode II stress intensity factor for the second load.

 required
T2 float

T-stress for the second load.

 required
Ep float

Plane strain modulus.

 required
s float

Internal length associated with T-stress.

 required

Returns:

Name Type Description
float float

The coupled energy release rate G* for the given crack angle and stress intensity factors.
Source code in src/gcrack/lefm.py 
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
@jit
def G_star_coupled(
    phi: float,
    phi0: float,
    KI1: float,
    KII1: float,
    T1: float,
    KI2: float,
    KII2: float,
    T2: float,
    Ep: float,
    s: float,
) -> float:
    """Computes the coupled energy release rate G* for two sets of stress intensity factors.

    It is used to evaluate the energy release rate for two interacting loading conditions.

    Args:
        phi (float): Current crack angle.
        phi0 (float): Initial crack angle.
        KI1 (float): Mode I stress intensity factor for the first load.
        KII1 (float): Mode II stress intensity factor for the first load.
        T1 (float): T-stress for the first load.
        KI2 (float): Mode I stress intensity factor for the second load.
        KII2 (float): Mode II stress intensity factor for the second load.
        T2 (float): T-stress for the second load.
        Ep (float): Plane strain modulus.
        s (float): Internal length associated with T-stress.

    Returns:
        float: The coupled energy release rate G* for the given crack angle and stress intensity factors.
    """
    # Calculate m
    m = (phi - phi0) / pi
    # Compute F^T * F
    F = Fmat(m)
    FT_F = F.T @ F
    # Store the SIFs in an array
    k1 = jnp.array([KI1, KII1])
    k2 = jnp.array([KI2, KII2])
    # Compute the G star
    return 2 / Ep * jnp.einsum("i,ij,j->", k1, FT_F, k2)

Gvec(m)

Construct the vector G containing the Gi functions of Amestoy-Leblond.

Parameters:

Name Type Description Default
m float

Normalized crack angle, defined as (phi - phi0) / pi.

 required

Returns:

Type Description
ndarray

jnp.ndarray: The vector G for the given normalized crack angle.
Source code in src/gcrack/lefm.py 
176
177
178
179
180
181
182
183
184
185
186
@jit
def Gvec(m: float) -> jnp.ndarray:
    """Construct the vector G containing the Gi functions of Amestoy-Leblond.

    Args:
        m (float): Normalized crack angle, defined as (phi - phi0) / pi.

    Returns:
        jnp.ndarray: The vector G for the given normalized crack angle.
    """
    return jnp.array([G1(m), G2(m)])