裂纹尖端的极坐标 应力强度因子 （Stress Intensity Factors），符号为K 断裂力学 、破壞力学中极其重要的一个参量。在断裂力学中，应力强度因子用于预测由远程载荷或残余应力 引起的裂纹或凹口尖端附近的应力强度，[ 1]
K [ 2] [ 3]
K
=
σ
π
a
f
(
a
/
W
)
{\displaystyle K=\sigma {\sqrt {\pi a}}\,f(a/W)}
其中
f
(
a
/
W
)
{\displaystyle f(a/W)}
a W σ
弹性力学 理论预测，在极坐标 （
r
,
θ
{\displaystyle r,\theta }
σ
i
j
{\displaystyle \sigma _{ij}}
[ 4]
σ
i
j
(
r
,
θ
)
=
K
2
π
r
f
i
j
(
θ
)
+
h
i
g
h
e
r
o
r
d
e
r
t
e
r
m
s
{\displaystyle \sigma _{ij}(r,\theta )={\frac {K}{\sqrt {2\pi r}}}\,f_{ij}(\theta )+\,\,{\rm {higher\,order\,terms}}}
其中K 1/2 ），
f
i
j
{\displaystyle f_{ij}}
r
σ
i
j
{\displaystyle \sigma _{ij}}
∞
{\displaystyle \infty }
[ 5] r 屈服强度 的应力下，此时的线弹性解则不再适用。但如果裂尖的塑性区域相对裂纹长度很小，那么裂尖附近的渐近应力分布则仍然适用。
三种模式的裂纹荷载 1957年，George Rankine Irwin发现，裂纹周围的应力可以用一个称为应力强度因子的比例因子来表达。他发现，裂纹在任意荷载下可以分解为三种线性独立的裂纹模式。[ 6]
对应于三种不同的断裂模式 ，有三个不同的物理量：
K
I
{\displaystyle K_{I}}
K
I
I
{\displaystyle K_{II}}
K
I
I
I
{\displaystyle K_{III}}
[ 7]
K
I
=
lim
r
→
0
2
π
r
σ
y
y
(
r
,
0
)
K
I
I
=
lim
r
→
0
2
π
r
σ
y
x
(
r
,
0
)
K
I
I
I
=
lim
r
→
0
2
π
r
σ
y
z
(
r
,
0
)
.
{\displaystyle {\begin{aligned}K_{\rm {I}}&=\lim _{r\rightarrow 0}{\sqrt {2\pi r}}\,\sigma _{yy}(r,0)\\K_{\rm {II}}&=\lim _{r\rightarrow 0}{\sqrt {2\pi r}}\,\sigma _{yx}(r,0)\\K_{\rm {III}}&=\lim _{r\rightarrow 0}{\sqrt {2\pi r}}\,\sigma _{yz}(r,0)\,.\end{aligned}}}
Equations for stress and displacement fields
在平面应力 条件下，裂纹在纯I型或纯II型荷载下的应变能释放速率（
G
{\displaystyle G}
G
I
=
K
I
2
(
1
E
)
{\displaystyle G_{\rm {I}}=K_{\rm {I}}^{2}\left({\frac {1}{E}}\right)}
G
I
I
=
K
I
I
2
(
1
E
)
{\displaystyle G_{\rm {II}}=K_{\rm {II}}^{2}\left({\frac {1}{E}}\right)}
其中
E
{\displaystyle E}
杨氏模量 ，
ν
{\displaystyle \nu }
泊松比 。假设材料各向同性、分布均匀且为线弹性，则认为裂纹会沿着初始裂纹的方向延伸。
对于平面应变 条件，等效关系为：
G
I
=
K
I
2
(
1
−
ν
2
E
)
{\displaystyle G_{\rm {I}}=K_{\rm {I}}^{2}\left({\frac {1-\nu ^{2}}{E}}\right)\,}
G
I
I
=
K
I
I
2
(
1
−
ν
2
E
)
.
{\displaystyle G_{\rm {II}}=K_{\rm {II}}^{2}\left({\frac {1-\nu ^{2}}{E}}\right)\,.}
对于纯III型荷载为：
G
I
I
I
=
K
I
I
I
2
(
1
2
μ
)
=
K
I
I
I
2
(
1
+
ν
E
)
{\displaystyle G_{\rm {III}}=K_{\rm {III}}^{2}\left({\frac {1}{2\mu }}\right)=K_{\rm {III}}^{2}\left({\frac {1+\nu }{E}}\right)}
其中
μ
{\displaystyle \mu }
剪切模量 。对于平面应变中的一般荷载，则成立以下线性组合：
G
=
J
=
∫
Γ
(
W
d
x
2
−
t
⋅
∂
u
∂
x
1
d
s
)
.
{\displaystyle G=J=\int _{\Gamma }\left(W~dx_{2}-\mathbf {t} \cdot {\cfrac {\partial \mathbf {u} }{\partial x_{1}}}~ds\right)\,.}
与应力强度因子相对应的表征弹塑性材料的裂纹尖端应力应变场的参量是断裂韧性 J积分，J积分也按照断裂模式分为
J
I
{\displaystyle J_{I}}
J
I
I
{\displaystyle J_{II}}
J
I
I
I
{\displaystyle J_{III}}
应力强度因子
K
{\displaystyle K}
Y
{\displaystyle Y}
K
I
{\displaystyle K_{\mathrm {I} }}
K
I
c
{\displaystyle K_{\mathrm {Ic} }}
K
I
c
{\displaystyle K_{\mathrm {Ic} }}
3/2 ）。
K
I
c
{\displaystyle K_{\mathrm {Ic} }}
K
I
c
{\displaystyle K_{\mathrm {Ic} }}
K
I
c
{\displaystyle K_{\mathrm {Ic} }}
G准则是一种断裂准则，它将临界应力强度因子（或断裂韧性）与三种模式的应力强度因子相关联。这个失效准则可以表述为：[ 8]
K
c
2
=
K
I
2
+
K
I
I
2
+
E
′
2
μ
K
I
I
I
2
{\displaystyle K_{\rm {c}}^{2}=K_{\rm {I}}^{2}+K_{\rm {II}}^{2}+{\frac {E'}{2\mu }}\,K_{\rm {III}}^{2}}
其中，是断裂韧性，
E
′
=
E
/
(
1
−
ν
2
)
{\displaystyle E'=E/(1-\nu ^{2})}
E
′
=
E
{\displaystyle E'=E}
K
c
{\displaystyle K_{\rm {c}}}
在I型加载下的无限大平板中的裂纹 假设长度为
2
a
{\displaystyle 2a}
σ
{\displaystyle \sigma }
[ 5] [ 7]
K
I
=
σ
π
a
{\displaystyle K_{\mathrm {I} }=\sigma {\sqrt {\pi a}}}
无限域内受单轴张力作用的硬币形裂纹 在无限域内一个半径为
a
{\displaystyle a}
[ 9]
K
I
=
2
π
σ
π
a
.
{\displaystyle K_{\rm {I}}={\frac {2}{\pi }}\sigma {\sqrt {\pi a}}\,.}
有限平面在I型加载下的裂缝 如果裂缝位于宽度为
2
b
{\displaystyle 2b}
2
h
{\displaystyle 2h}
[ 7]
K
I
=
σ
π
a
[
1
−
a
2
b
+
0.326
(
a
b
)
2
1
−
a
b
]
.
{\displaystyle K_{\rm {I}}=\sigma {\sqrt {\pi a}}\left[{\cfrac {1-{\frac {a}{2b}}+0.326\left({\frac {a}{b}}\right)^{2}}{\sqrt {1-{\frac {a}{b}}}}}\right]\,.}
如果裂纹不位于宽度的中心位置，即
d
≠
b
{\displaystyle d\neq b}
A 处的应力强度因子可以通过级数展开来近似得到：[ 7] [ 10]
K
I
A
=
σ
π
a
[
1
+
∑
n
=
2
M
C
n
(
a
b
)
n
]
{\displaystyle K_{\rm {IA}}=\sigma {\sqrt {\pi a}}\left[1+\sum _{n=2}^{M}C_{n}\left({\frac {a}{b}}\right)^{n}\right]}
从应力强度曲线的拟合中可以基于不同
d
{\displaystyle d}
C
n
{\displaystyle C_{n}}
[ 7] B 点可以得到一个类似（但不完全相同）的表达式。在A点和B点的应力强度因子的替代表达式是：[ 11]
K
I
A
=
σ
π
a
Φ
A
,
K
I
B
=
σ
π
a
Φ
B
{\displaystyle K_{\rm {IA}}=\sigma {\sqrt {\pi a}}\,\Phi _{A}\,\,,K_{\rm {IB}}=\sigma {\sqrt {\pi a}}\,\Phi _{B}}
其中
Φ
A
:=
[
β
+
(
1
−
β
4
)
(
1
+
1
4
sec
α
A
)
2
]
sec
α
A
Φ
B
:=
1
+
[
sec
α
A
B
−
1
1
+
0.21
sin
{
8
tan
−
1
[
(
α
A
−
α
B
α
A
+
α
B
)
0.9
]
}
]
{\displaystyle {\begin{aligned}\Phi _{A}&:=\left[\beta +\left({\frac {1-\beta }{4}}\right)\left(1+{\frac {1}{4{\sqrt {\sec \alpha _{A}}}}}\right)^{2}\right]{\sqrt {\sec \alpha _{A}}}\\\Phi _{B}&:=1+\left[{\frac {{\sqrt {\sec \alpha _{AB}}}-1}{1+0.21\sin \left\{8\,\tan ^{-1}\left[\left({\frac {\alpha _{A}-\alpha _{B}}{\alpha _{A}+\alpha _{B}}}\right)^{0.9}\right]\right\}}}\right]\end{aligned}}}
其中
β
:=
sin
(
π
α
B
α
A
+
α
B
)
,
α
A
:=
π
a
2
d
,
α
B
:=
π
a
4
b
−
2
d
;
α
A
B
:=
4
7
α
A
+
3
7
α
B
.
{\displaystyle \beta :=\sin \left({\frac {\pi \alpha _{B}}{\alpha _{A}+\alpha _{B}}}\right)~,~~\alpha _{A}:={\frac {\pi a}{2d}}~,~~\alpha _{B}:={\frac {\pi a}{4b-2d}}~;~~\alpha _{AB}:={\frac {4}{7}}\,\alpha _{A}+{\frac {3}{7}}\,\alpha _{B}\,.}
在上述表达试中，
d
{\displaystyle d}
d
=
b
{\displaystyle d=b}
单轴应力下有限平面中的边缘裂纹 对于一个尺寸为
2
h
×
b
{\displaystyle 2h\times b}
a
{\displaystyle a}
h
/
b
≥
0.5
{\displaystyle h/b\geq 0.5}
a
/
b
≤
0.6
{\displaystyle a/b\leq 0.6}
σ
{\displaystyle \sigma }
[ 5]
K
I
=
σ
π
a
[
1.122
−
0.231
(
a
b
)
+
10.55
(
a
b
)
2
−
21.71
(
a
b
)
3
+
30.382
(
a
b
)
4
]
.
{\displaystyle K_{\rm {I}}=\sigma {\sqrt {\pi a}}\left[1.122-0.231\left({\frac {a}{b}}\right)+10.55\left({\frac {a}{b}}\right)^{2}-21.71\left({\frac {a}{b}}\right)^{3}+30.382\left({\frac {a}{b}}\right)^{4}\right]\,.}
对于
h
/
b
≥
1
{\displaystyle h/b\geq 1}
a
/
b
≥
0.3
{\displaystyle a/b\geq 0.3}
K
I
=
σ
π
a
[
1
+
3
a
b
2
π
a
b
(
1
−
a
b
)
3
/
2
]
.
{\displaystyle K_{\rm {I}}=\sigma {\sqrt {\pi a}}\left[{\frac {1+3{\frac {a}{b}}}{2{\sqrt {\pi {\frac {a}{b}}}}\left(1-{\frac {a}{b}}\right)^{3/2}}}\right]\,.}
双轴加载平面中的倾斜裂缝。 对于一个长度为
2
a
{\displaystyle 2a}
y
{\displaystyle y}
σ
{\displaystyle \sigma }
x
{\displaystyle x}
α
σ
{\displaystyle \alpha \sigma }
[ 7] [ 8]
K
I
=
σ
π
a
(
cos
2
β
+
α
sin
2
β
)
K
I
I
=
σ
π
a
(
1
−
α
)
sin
β
cos
β
{\displaystyle {\begin{aligned}K_{\rm {I}}&=\sigma {\sqrt {\pi a}}\left(\cos ^{2}\beta +\alpha \sin ^{2}\beta \right)\\K_{\rm {II}}&=\sigma {\sqrt {\pi a}}\left(1-\alpha \right)\sin \beta \cos \beta \end{aligned}}}
β
{\displaystyle \beta }
x
{\displaystyle x}
在局部作用力
F
x
{\displaystyle F_{x}}
F
y
{\displaystyle F_{y}}
一个尺寸为
2
h
×
2
b
{\displaystyle 2h\times 2b}
2
a
{\displaystyle 2a}
x
,
y
{\displaystyle x,y}
F
x
{\displaystyle F_{x}}
F
y
{\displaystyle F_{y}}
在平面比裂纹尺寸大，且力的位置相对靠近裂纹的情况下，即
h
≫
a
{\displaystyle h\gg a}
b
≫
a
{\displaystyle b\gg a}
x
≪
b
{\displaystyle x\ll b}
y
≪
h
{\displaystyle y\ll h}
x
=
a
{\displaystyle x=a}
F
x
{\displaystyle F_{x}}
[ 8] [ 12]
K
I
=
F
x
2
π
a
(
κ
−
1
κ
+
1
)
[
G
1
+
1
κ
−
1
H
1
]
K
I
I
=
F
x
2
π
a
[
G
2
+
1
κ
+
1
H
2
]
{\displaystyle {\begin{aligned}K_{\rm {I}}&={\frac {F_{x}}{2{\sqrt {\pi a}}}}\left({\frac {\kappa -1}{\kappa +1}}\right)\left[G_{1}+{\frac {1}{\kappa -1}}H_{1}\right]\\K_{\rm {II}}&={\frac {F_{x}}{2{\sqrt {\pi a}}}}\left[G_{2}+{\frac {1}{\kappa +1}}H_{2}\right]\end{aligned}}}
其中
G
1
=
1
−
Re
[
a
+
z
z
2
−
a
2
]
,
G
2
=
−
Im
[
a
+
z
z
2
−
a
2
]
H
1
=
Re
[
a
(
z
¯
−
z
)
(
z
¯
−
a
)
z
¯
2
−
a
2
]
,
H
2
=
−
Im
[
a
(
z
¯
−
z
)
(
z
¯
−
a
)
z
¯
2
−
a
2
]
{\displaystyle {\begin{aligned}G_{1}&=1-{\text{Re}}\left[{\frac {a+z}{\sqrt {z^{2}-a^{2}}}}\right]\,,\,\,G_{2}=-{\text{Im}}\left[{\frac {a+z}{\sqrt {z^{2}-a^{2}}}}\right]\\H_{1}&={\text{Re}}\left[{\frac {a({\bar {z}}-z)}{({\bar {z}}-a){\sqrt {{\bar {z}}^{2}-a^{2}}}}}\right]\,,\,\,H_{2}=-{\text{Im}}\left[{\frac {a({\bar {z}}-z)}{({\bar {z}}-a){\sqrt {{\bar {z}}^{2}-a^{2}}}}}\right]\end{aligned}}}
z
=
x
+
i
y
{\displaystyle z=x+iy}
z
¯
=
x
−
i
y
{\displaystyle {\bar {z}}=x-iy}
κ
=
3
−
4
ν
{\displaystyle \kappa =3-4\nu }
κ
=
(
3
−
ν
)
/
(
1
+
ν
)
{\displaystyle \kappa =(3-\nu )/(1+\nu )}
ν
{\displaystyle \nu }
泊松比 。
在B端点
F
y
{\displaystyle F_{y}}
K
I
=
F
y
2
π
a
[
G
2
−
1
κ
+
1
H
2
]
K
I
I
=
−
F
y
2
π
a
(
κ
−
1
κ
+
1
)
[
G
1
−
1
κ
−
1
H
1
]
.
{\displaystyle {\begin{aligned}K_{\rm {I}}&={\frac {F_{y}}{2{\sqrt {\pi a}}}}\left[G_{2}-{\frac {1}{\kappa +1}}H_{2}\right]\\K_{\rm {II}}&=-{\frac {F_{y}}{2{\sqrt {\pi a}}}}\left({\frac {\kappa -1}{\kappa +1}}\right)\left[G_{1}-{\frac {1}{\kappa -1}}H_{1}\right]\,.\end{aligned}}}
点A（
x
=
−
a
{\displaystyle x=-a}
(
x
,
y
)
{\displaystyle (x,y)}
F
x
{\displaystyle F_{x}}
K
I
(
−
a
;
x
,
y
)
=
−
K
I
(
a
;
−
x
,
y
)
,
K
I
I
(
−
a
;
x
,
y
)
=
K
I
I
(
a
;
−
x
,
y
)
.
{\displaystyle K_{\rm {I}}(-a;x,y)=-K_{\rm {I}}(a;-x,y)\,,\,\,K_{\rm {II}}(-a;x,y)=K_{\rm {II}}(a;-x,y)\,.}
F
y
{\displaystyle F_{y}}
K
I
(
−
a
;
x
,
y
)
=
K
I
(
a
;
−
x
,
y
)
,
K
I
I
(
−
a
;
x
,
y
)
=
−
K
I
I
(
a
;
−
x
,
y
)
.
{\displaystyle K_{\rm {I}}(-a;x,y)=K_{\rm {I}}(a;-x,y)\,,\,\,K_{\rm {II}}(-a;x,y)=-K_{\rm {II}}(a;-x,y)\,.}
平面裂纹荷载 如果裂纹受到位于
y
=
0
{\displaystyle y=0}
−
a
<
x
<
a
{\displaystyle -a<x<a}
F
y
{\displaystyle F_{y}}
[ 7]
K
I
=
F
y
2
π
a
a
+
x
a
−
x
,
K
I
I
=
−
F
x
2
π
a
(
κ
−
1
κ
+
1
)
.
{\displaystyle K_{\rm {I}}={\frac {F_{y}}{2{\sqrt {\pi a}}}}{\sqrt {\frac {a+x}{a-x}}}\,,\,\,K_{\rm {II}}=-{\frac {F_{x}}{2{\sqrt {\pi a}}}}\left({\frac {\kappa -1}{\kappa +1}}\right)\,.}
如果力在
−
a
<
x
<
a
{\displaystyle -a<x<a}
K
I
=
1
2
π
a
∫
−
a
a
F
y
(
x
)
a
+
x
a
−
x
d
x
,
K
I
I
=
−
1
2
π
a
(
κ
−
1
κ
+
1
)
∫
−
a
a
F
y
(
x
)
d
x
,
.
{\displaystyle K_{\rm {I}}={\frac {1}{2{\sqrt {\pi a}}}}\int _{-a}^{a}F_{y}(x)\,{\sqrt {\frac {a+x}{a-x}}}\,{\rm {d}}x\,,\,\,K_{\rm {II}}=-{\frac {1}{2{\sqrt {\pi a}}}}\left({\frac {\kappa -1}{\kappa +1}}\right)\int _{-a}^{a}F_{y}(x)\,{\rm {d}}x,\,.}
如果裂纹间距远远大于裂纹长度（
h
≫
a
{\displaystyle h\gg a}
K
I
=
σ
π
a
{\displaystyle {\begin{aligned}K_{\rm {I}}&=\sigma {\sqrt {\pi a}}\end{aligned}}}
如果裂纹长度远大于间距（
a
≫
h
{\displaystyle a\gg h}
K
I
=
σ
h
{\displaystyle {\begin{aligned}K_{\rm {I}}&=\sigma {\sqrt {h}}\end{aligned}}}
用于断裂韧性测试的紧凑拉伸样品 紧凑拉伸试样裂纹尖端的应力强度因子为：[ 15]
K
I
=
P
B
π
W
[
16.7
(
a
W
)
1
/
2
−
104.7
(
a
W
)
3
/
2
+
369.9
(
a
W
)
5
/
2
−
573.8
(
a
W
)
7
/
2
+
360.5
(
a
W
)
9
/
2
]
{\displaystyle {\begin{aligned}K_{\rm {I}}&={\frac {P}{B}}{\sqrt {\frac {\pi }{W}}}\left[16.7\left({\frac {a}{W}}\right)^{1/2}-104.7\left({\frac {a}{W}}\right)^{3/2}+369.9\left({\frac {a}{W}}\right)^{5/2}\right.\\&\qquad \left.-573.8\left({\frac {a}{W}}\right)^{7/2}+360.5\left({\frac {a}{W}}\right)^{9/2}\right]\end{aligned}}}
其中
P
{\displaystyle P}
B
{\displaystyle B}
a
{\displaystyle a}
W
{\displaystyle W}
用于断裂韧性测试的单边缺口弯曲试样（也称为三点弯曲试样） 单边缺口弯曲试样裂纹尖端的应力强度因子为：[ 15]
K
I
=
4
P
B
π
W
[
1.6
(
a
W
)
1
/
2
−
2.6
(
a
W
)
3
/
2
+
12.3
(
a
W
)
5
/
2
−
21.2
(
a
W
)
7
/
2
+
21.8
(
a
W
)
9
/
2
]
{\displaystyle {\begin{aligned}K_{\rm {I}}&={\frac {4P}{B}}{\sqrt {\frac {\pi }{W}}}\left[1.6\left({\frac {a}{W}}\right)^{1/2}-2.6\left({\frac {a}{W}}\right)^{3/2}+12.3\left({\frac {a}{W}}\right)^{5/2}\right.\\&\qquad \left.-21.2\left({\frac {a}{W}}\right)^{7/2}+21.8\left({\frac {a}{W}}\right)^{9/2}\right]\end{aligned}}}
其中
P
{\displaystyle P}
B
{\displaystyle B}
a
{\displaystyle a}
W
{\displaystyle W}
^ Anderson, T. L. Fracture mechanics: fundamentals and applications. CRC Press. 2005. ^ Soboyejo, W. O. 11.6.2 Crack Driving Force and Concept of Similitude. Mechanical properties of engineered materials . Marcel Dekker. 2003 [2024-03-28 ] . ISBN 0-8247-8900-8OCLC 300921090 存档 于2021-05-31）. ^ Janssen, M. (Michael). Fracture mechanics . Zuidema, J. (Jan), Wanhill, R. J. H. 2nd. London: Spon Press. 2004: 41. ISBN 0-203-59686-2OCLC 57491375 ^ Hiroshi Tada ; P. C. Paris ; George R. Irwin . The Stress Analysis of Cracks Handbook 3rd. American Society of Mechanical Engineers. February 2000. ^ 5.0 5.1 5.2 Liu, M.; et al. An improved semi-analytical solution for stress at round-tip notches (PDF) . Engineering Fracture Mechanics. 2015, 149 : 134–143 [2024-03-28 ] . S2CID 51902898 doi:10.1016/j.engfracmech.2015.10.004 存档 (PDF) 于2018-07-13）. ^ Suresh, S. Fatigue of Materials. Cambridge University Press. 2004. ISBN 978-0-521-57046-6 ^ 7.0 7.1 7.2 7.3 7.4 7.5 7.6 Rooke, D. P.; Cartwright, D. J. Compendium of stress intensity factors. HMSO Ministry of Defence. Procurement Executive. 1976. ^ 8.0 8.1 8.2 Sih, G. C.; Macdonald, B., Fracture mechanics applied to engineering problems-strain energy density fracture criterion, Engineering Fracture Mechanics, 1974, 6 (2): 361–386, doi:10.1016/0013-7944(74)90033-2 ^ Anderson, T. L. Fracture mechanics: fundamentals and applications. CRC Press. 2005. ^ Isida, M., 1966, Stress intensity factors for the tension of an eccentrically cracked strip , Transactions of the ASME Applied Mechanics Section, v. 88, p.94.
^ Kathiresan, K.; Brussat, T. R.; Hsu, T. M. Advanced life analysis methods. Crack Growth Analysis Methods for Attachment Lugs. Flight Dynamics Laboratory, Air Force Wright Aeronautical Laboratories, AFSC W-P Air Force Base, Ohio. 1984. ^ Erdogan, F., On the stress distribution in plates with collinear cuts under arbitrary loads, Proceedings of the Fourth US National Congress of Applied Mechanics, 1962, 1 : 547–574 ^ Kundu, Tribikram. Fundamentals of Fracture Mechanics . 2008-01-30. ISBN 9780429110054doi:10.1201/9781439878057 ^ 范, 天佑. 紧凑拉伸试样的K_I . Chinese Science Bulletin. 1976-07-01, 21 (Z1). ISSN 0023-074X doi:10.1360/csb1976-21-z1-494 ^ 15.0 15.1 Bower, A. F. Applied mechanics of solids. CRC Press. 2009.
![{\displaystyle K_{\rm {I}}=\sigma {\sqrt {\pi a}}\left[{\cfrac {1-{\frac {a}{2b}}+0.326\left({\frac {a}{b}}\right)^{2}}{\sqrt {1-{\frac {a}{b}}}}}\right]\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25cc4dadb6371c8239b7ebca823e25b13d2ffeb3)
![{\displaystyle K_{\rm {IA}}=\sigma {\sqrt {\pi a}}\left[1+\sum _{n=2}^{M}C_{n}\left({\frac {a}{b}}\right)^{n}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71eaed2d5214af8609ee2614e072a831dff9bb27)
![{\displaystyle {\begin{aligned}\Phi _{A}&:=\left[\beta +\left({\frac {1-\beta }{4}}\right)\left(1+{\frac {1}{4{\sqrt {\sec \alpha _{A}}}}}\right)^{2}\right]{\sqrt {\sec \alpha _{A}}}\\\Phi _{B}&:=1+\left[{\frac {{\sqrt {\sec \alpha _{AB}}}-1}{1+0.21\sin \left\{8\,\tan ^{-1}\left[\left({\frac {\alpha _{A}-\alpha _{B}}{\alpha _{A}+\alpha _{B}}}\right)^{0.9}\right]\right\}}}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97a81ceb7c9a5bbd82590c76afd49d6690a9c74a)
![{\displaystyle K_{\rm {I}}=\sigma {\sqrt {\pi a}}\left[1.122-0.231\left({\frac {a}{b}}\right)+10.55\left({\frac {a}{b}}\right)^{2}-21.71\left({\frac {a}{b}}\right)^{3}+30.382\left({\frac {a}{b}}\right)^{4}\right]\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d8ceaef43b7ab34b12662c8847e580f260cb701)
![{\displaystyle K_{\rm {I}}=\sigma {\sqrt {\pi a}}\left[{\frac {1+3{\frac {a}{b}}}{2{\sqrt {\pi {\frac {a}{b}}}}\left(1-{\frac {a}{b}}\right)^{3/2}}}\right]\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a0366ab9b177d5a2d15ce70349ef83e9456f1f)
![{\displaystyle {\begin{aligned}K_{\rm {I}}&={\frac {F_{x}}{2{\sqrt {\pi a}}}}\left({\frac {\kappa -1}{\kappa +1}}\right)\left[G_{1}+{\frac {1}{\kappa -1}}H_{1}\right]\\K_{\rm {II}}&={\frac {F_{x}}{2{\sqrt {\pi a}}}}\left[G_{2}+{\frac {1}{\kappa +1}}H_{2}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9df6165ba8ba56686979fe76299274dd25540945)
![{\displaystyle {\begin{aligned}G_{1}&=1-{\text{Re}}\left[{\frac {a+z}{\sqrt {z^{2}-a^{2}}}}\right]\,,\,\,G_{2}=-{\text{Im}}\left[{\frac {a+z}{\sqrt {z^{2}-a^{2}}}}\right]\\H_{1}&={\text{Re}}\left[{\frac {a({\bar {z}}-z)}{({\bar {z}}-a){\sqrt {{\bar {z}}^{2}-a^{2}}}}}\right]\,,\,\,H_{2}=-{\text{Im}}\left[{\frac {a({\bar {z}}-z)}{({\bar {z}}-a){\sqrt {{\bar {z}}^{2}-a^{2}}}}}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0255e5aec3179e43fd82a04e723ef65dff683f6f)
![{\displaystyle {\begin{aligned}K_{\rm {I}}&={\frac {F_{y}}{2{\sqrt {\pi a}}}}\left[G_{2}-{\frac {1}{\kappa +1}}H_{2}\right]\\K_{\rm {II}}&=-{\frac {F_{y}}{2{\sqrt {\pi a}}}}\left({\frac {\kappa -1}{\kappa +1}}\right)\left[G_{1}-{\frac {1}{\kappa -1}}H_{1}\right]\,.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f91850cc6735cb02154d8d82ca2b502090d7a67)
![{\displaystyle {\begin{aligned}K_{\rm {I}}&={\frac {P}{B}}{\sqrt {\frac {\pi }{W}}}\left[16.7\left({\frac {a}{W}}\right)^{1/2}-104.7\left({\frac {a}{W}}\right)^{3/2}+369.9\left({\frac {a}{W}}\right)^{5/2}\right.\\&\qquad \left.-573.8\left({\frac {a}{W}}\right)^{7/2}+360.5\left({\frac {a}{W}}\right)^{9/2}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a74d53e379ebf47cf3ea5f1a45619a1e78dcc4ad)
![{\displaystyle {\begin{aligned}K_{\rm {I}}&={\frac {4P}{B}}{\sqrt {\frac {\pi }{W}}}\left[1.6\left({\frac {a}{W}}\right)^{1/2}-2.6\left({\frac {a}{W}}\right)^{3/2}+12.3\left({\frac {a}{W}}\right)^{5/2}\right.\\&\qquad \left.-21.2\left({\frac {a}{W}}\right)^{7/2}+21.8\left({\frac {a}{W}}\right)^{9/2}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/466d7f6e915a160e93963497a9016099b626cae6)