Stress intensity factor

The mode I stress field expressed in terms of ${\displaystyle K_{\rm {I}}}$ is^[6]

${\displaystyle \left\{{\begin{aligned}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{aligned}}\right\}={\frac {K_{\rm {I}}}{\sqrt {2\pi r}}}\cos {\frac {\theta }{2}}\left\{{\begin{aligned}1-\sin {\frac {\theta }{2}}\sin {\frac {3\theta }{2}}\\1+\sin {\frac {\theta }{2}}\sin {\frac {3\theta }{2}}\\\sin {\frac {\theta }{2}}\cos {\frac {3\theta }{2}}\end{aligned}}\right\}}$,

and

${\displaystyle \left\{{\begin{aligned}\sigma _{rr}\\\sigma _{\theta \theta }\\\sigma _{r\theta }\end{aligned}}\right\}={\frac {K_{\rm {I}}}{\sqrt {2\pi r}}}\cos {\frac {\theta }{2}}\left\{{\begin{aligned}1+\sin ^{2}{\frac {\theta }{2}}\\\cos ^{2}{\frac {\theta }{2}}\\\sin {\frac {\theta }{2}}\cos {\frac {\theta }{2}}\end{aligned}}\right\}}$.

${\displaystyle \sigma _{zz}=\nu _{1}(\sigma _{xx}+\sigma _{yy})=\nu _{1}(\sigma _{rr}+\sigma _{\theta \theta })}$,

${\displaystyle \sigma _{xz}=\sigma _{yz}=\sigma _{rz}=\sigma _{\theta z}=0}$.

The displacements are

${\displaystyle \left\{{\begin{aligned}u_{x}\\u_{y}\end{aligned}}\right\}={\frac {K_{\rm {I}}}{2E}}{\sqrt {\frac {r}{2\pi }}}\left\{{\begin{aligned}(1+\nu )\left[(2\kappa -1)\cos {\frac {\theta }{2}}-\cos {\frac {3\theta }{2}}\right]\\(1+\nu )\left[(2\kappa +1)\sin {\frac {\theta }{2}}-\sin {\frac {3\theta }{2}}\right]\end{aligned}}\right\}}$

${\displaystyle \left\{{\begin{aligned}u_{r}\\u_{\theta }\end{aligned}}\right\}={\frac {K_{\rm {I}}}{2E}}{\sqrt {\frac {r}{2\pi }}}\left\{{\begin{aligned}(1+\nu )\left[(2\kappa -1)\cos {\frac {\theta }{2}}-\cos {\frac {3\theta }{2}}\right]\\(1+\nu )\left[-(2\kappa -1)\sin {\frac {\theta }{2}}+\sin {\frac {3\theta }{2}}\right]\end{aligned}}\right\}}$

${\displaystyle u_{z}=-\left({\frac {\nu _{2}z}{E}}\right)(\sigma _{xx}+\sigma _{yy})=-\left({\frac {\nu _{2}z}{E}}\right)(\sigma _{rr}+\sigma _{\theta \theta })}$

Where, for plane stress conditions

${\displaystyle \kappa ={\frac {(3-\nu )}{(1+\nu )}}}$, ${\displaystyle \nu _{1}=0}$, ${\displaystyle \nu _{2}=\nu }$,

and for plane strain

${\displaystyle \kappa =(3-4\nu )}$, ${\displaystyle \nu _{1}=\nu }$, ${\displaystyle \nu _{2}=0}$.

For mode II

${\displaystyle \left\{{\begin{aligned}\sigma _{xx}\\\sigma _{yy}\\\sigma _{xy}\end{aligned}}\right\}={\frac {K_{\rm {II}}}{\sqrt {2\pi r}}}\left\{{\begin{aligned}-\sin {\frac {\theta }{2}}(2+\cos {\frac {\theta }{2}}\cos {\frac {3\theta }{2}})\\\sin {\frac {\theta }{2}}\cos {\frac {\theta }{2}}\sin {\frac {3\theta }{2}}\\\cos {\frac {\theta }{2}}(1-\sin {\frac {\theta }{2}}\sin {\frac {3\theta }{2}})\end{aligned}}\right\}}$

and

${\displaystyle \left\{{\begin{aligned}\sigma _{rr}\\\sigma _{\theta \theta }\\\sigma _{r\theta }\end{aligned}}\right\}={\frac {K_{\rm {II}}}{\sqrt {2\pi r}}}\left\{{\begin{aligned}\sin {\frac {\theta }{2}}(1-3\sin ^{2}{\frac {\theta }{2}})\\-3\sin {\frac {\theta }{2}}\cos ^{2}{\frac {\theta }{2}}\\\cos {\frac {\theta }{2}}(1-3\sin ^{2}{\frac {\theta }{2}})\end{aligned}}\right\}}$,

${\displaystyle \sigma _{zz}=\nu _{1}(\sigma _{xx}+\sigma _{yy})=\nu _{1}(\sigma _{rr}+\sigma _{\theta \theta })}$,

${\displaystyle \sigma _{xz}=\sigma _{yz}=\sigma _{rz}=\sigma _{\theta z}=0}$.

${\displaystyle \left\{{\begin{aligned}u_{x}\\u_{y}\end{aligned}}\right\}={\frac {K_{\rm {II}}}{2E}}{\sqrt {\frac {r}{2\pi }}}\left\{{\begin{aligned}(1+\nu )\left[(2\kappa +3)\sin {\frac {\theta }{2}}+\sin {\frac {3\theta }{2}}\right]\\-(1+\nu )\left[(2\kappa -3)\cos {\frac {\theta }{2}}+\cos {\frac {3\theta }{2}}\right]\end{aligned}}\right\}}$

${\displaystyle \left\{{\begin{aligned}u_{r}\\u_{\theta }\end{aligned}}\right\}={\frac {K_{\rm {II}}}{2E}}{\sqrt {\frac {r}{2\pi }}}\left\{{\begin{aligned}(1+\nu )\left[-(2\kappa -1)\sin {\frac {\theta }{2}}+3\sin {\frac {3\theta }{2}}\right]\\(1+\nu )\left[-(2\kappa +1)\cos {\frac {\theta }{2}}+3\cos {\frac {3\theta }{2}}\right]\end{aligned}}\right\}}$

${\displaystyle u_{z}=-\left({\frac {\nu _{2}z}{E}}\right)(\sigma _{xx}+\sigma _{yy})=-\left({\frac {\nu _{2}z}{E}}\right)(\sigma _{rr}+\sigma _{\theta \theta })}$

And finally, for mode III

${\displaystyle \left\{{\begin{aligned}\sigma _{xz}\\\sigma _{yz}\end{aligned}}\right\}={\frac {K_{\rm {III}}}{\sqrt {2\pi r}}}\left\{{\begin{aligned}-\sin {\frac {\theta }{2}}\\\cos {\frac {\theta }{2}}\end{aligned}}\right\}}$

${\displaystyle \left\{{\begin{aligned}\sigma _{rz}\\\sigma _{\theta z}\end{aligned}}\right\}={\frac {K_{\rm {III}}}{\sqrt {2\pi r}}}\left\{{\begin{aligned}\sin {\frac {\theta }{2}}\\\cos {\frac {\theta }{2}}\end{aligned}}\right\}}$

with ${\displaystyle \sigma _{xx}=\sigma _{yy}=\sigma _{rr}=\sigma _{\theta \theta }=\sigma _{zz}=\sigma _{xy}=\sigma _{r\theta }=0}$.

${\displaystyle u_{z}={\frac {2K_{\rm {III}}}{E}}{\sqrt {\frac {r}{2\pi }}}\left\{2(1+\nu )\sin {\frac {\theta }{2}}\right\}}$,

${\displaystyle u_{x}=u_{y}=u_{r}=u_{\theta }=0}$.

The stress intensity factor for an assumed straight crack of length ${\displaystyle 2a}$ perpendicular to the loading direction, in an infinite plane, having a uniform stress field ${\displaystyle \sigma }$ is ^[5][7]

${\displaystyle K_{\mathrm {I} }=\sigma {\sqrt {\pi a}}}$

The stress intensity factor at the tip of a penny-shaped crack of radius ${\displaystyle a}$ in an infinite domain under uniaxial tension ${\displaystyle \sigma }$ is ^[1]

${\displaystyle K_{\rm {I}}={\frac {2}{\pi }}\sigma {\sqrt {\pi a}}\,.}$

If the crack is located centrally in a finite plate of width ${\displaystyle 2b}$ and height ${\displaystyle 2h}$, an approximate relation for the stress intensity factor is ^[7]

${\displaystyle K_{\rm {I}}=\sigma {\sqrt {\pi a}}\left[{\cfrac {1-{\frac {a}{2b}}+0.326\left({\frac {a}{b}}\right)^{2}}{\sqrt {1-{\frac {a}{b}}}}}\right]\,.}$

If the crack is not located centrally along the width, i.e., ${\displaystyle d\neq b}$, the stress intensity factor at location A can be approximated by the series expansion^[7][9]

${\displaystyle K_{\rm {IA}}=\sigma {\sqrt {\pi a}}\left[1+\sum _{n=2}^{M}C_{n}\left({\frac {a}{b}}\right)^{n}\right]}$

where the factors ${\displaystyle C_{n}}$ can be found from fits to stress intensity curves^[7]: 6 for various values of ${\displaystyle d}$. A similar (but not identical) expression can be found for tip B of the crack. Alternative expressions for the stress intensity factors at A and B are ^[10]: 175

${\displaystyle K_{\rm {IA}}=\sigma {\sqrt {\pi a}}\,\Phi _{A}\,\,,K_{\rm {IB}}=\sigma {\sqrt {\pi a}}\,\Phi _{B}}$

where

${\displaystyle {\begin{aligned}\Phi _{A}&:=\left[\beta +\left({\frac {1-\beta }{4}}\right)\left(1+{\frac {1}{4{\sqrt {\sec \alpha _{A}}}}}\right)^{2}\right]{\sqrt {\sec \alpha _{A}}}\\\Phi _{B}&:=1+\left[{\frac {{\sqrt {\sec \alpha _{AB}}}-1}{1+0.21\sin \left\{8\,\tan ^{-1}\left[\left({\frac {\alpha _{A}-\alpha _{B}}{\alpha _{A}+\alpha _{B}}}\right)^{0.9}\right]\right\}}}\right]\end{aligned}}}$

with

${\displaystyle \beta :=\sin \left({\frac {\pi \alpha _{B}}{\alpha _{A}+\alpha _{B}}}\right)~,~~\alpha _{A}:={\frac {\pi a}{2d}}~,~~\alpha _{B}:={\frac {\pi a}{4b-2d}}~;~~\alpha _{AB}:={\frac {4}{7}}\,\alpha _{A}+{\frac {3}{7}}\,\alpha _{B}\,.}$

In the above expressions ${\displaystyle d}$ is the distance from the center of the crack to the boundary closest to point A. Note that when ${\displaystyle d=b}$ the above expressions do not simplify into the approximate expression for a centered crack.

For a plate having dimensions ${\displaystyle 2h\times b}$ containing an unconstrained edge crack of length ${\displaystyle a}$, if the dimensions of the plate are such that ${\displaystyle h/b\geq 0.5}$ and ${\displaystyle a/b\leq 0.6}$, the stress intensity factor at the crack tip under a uniaxial stress ${\displaystyle \sigma }$ is ^[5]

${\displaystyle K_{\rm {I}}=\sigma {\sqrt {\pi a}}\left[1.122-0.231\left({\frac {a}{b}}\right)+10.55\left({\frac {a}{b}}\right)^{2}-21.71\left({\frac {a}{b}}\right)^{3}+30.382\left({\frac {a}{b}}\right)^{4}\right]\,.}$

For the situation where ${\displaystyle h/b\geq 1}$ and ${\displaystyle a/b\geq 0.3}$, the stress intensity factor can be approximated by

${\displaystyle K_{\rm {I}}=\sigma {\sqrt {\pi a}}\left[{\frac {1+3{\frac {a}{b}}}{2{\sqrt {\pi {\frac {a}{b}}}}\left(1-{\frac {a}{b}}\right)^{3/2}}}\right]\,.}$

For a slanted crack of length ${\displaystyle 2a}$ in a biaxial stress field with stress ${\displaystyle \sigma }$ in the ${\displaystyle y}$-direction and ${\displaystyle \alpha \sigma }$ in the ${\displaystyle x}$-direction, the stress intensity factors are ^[7][11]

${\displaystyle {\begin{aligned}K_{\rm {I}}&=\sigma {\sqrt {\pi a}}\left(\cos ^{2}\beta +\alpha \sin ^{2}\beta \right)\\K_{\rm {II}}&=\sigma {\sqrt {\pi a}}\left(1-\alpha \right)\sin \beta \cos \beta \end{aligned}}}$

where ${\displaystyle \beta }$ is the angle made by the crack with the ${\displaystyle x}$-axis.

Consider a plate with dimensions ${\displaystyle 2h\times 2b}$ containing a crack of length ${\displaystyle 2a}$. A point force with components ${\displaystyle F_{x}}$ and ${\displaystyle F_{y}}$ is applied at the point (${\displaystyle x,y}$) of the plate.

For the situation where the plate is large compared to the size of the crack and the location of the force is relatively close to the crack, i.e., ${\displaystyle h\gg a}$, ${\displaystyle b\gg a}$, ${\displaystyle x\ll b}$, ${\displaystyle y\ll h}$, the plate can be considered infinite. In that case, for the stress intensity factors for ${\displaystyle F_{x}}$ at crack tip B (${\displaystyle x=a}$) are ^[11][12]

${\displaystyle {\begin{aligned}K_{\rm {I}}&={\frac {F_{x}}{2{\sqrt {\pi a}}}}\left({\frac {\kappa -1}{\kappa +1}}\right)\left[G_{1}+{\frac {1}{\kappa -1}}H_{1}\right]\\K_{\rm {II}}&={\frac {F_{x}}{2{\sqrt {\pi a}}}}\left[G_{2}+{\frac {1}{\kappa +1}}H_{2}\right]\end{aligned}}}$

where

${\displaystyle {\begin{aligned}G_{1}&=1-{\text{Re}}\left[{\frac {a+z}{\sqrt {z^{2}-a^{2}}}}\right]\,,\,\,G_{2}=-{\text{Im}}\left[{\frac {a+z}{\sqrt {z^{2}-a^{2}}}}\right]\\H_{1}&={\text{Re}}\left[{\frac {a({\bar {z}}-z)}{({\bar {z}}-a){\sqrt {{\bar {z}}^{2}-a^{2}}}}}\right]\,,\,\,H_{2}=-{\text{Im}}\left[{\frac {a({\bar {z}}-z)}{({\bar {z}}-a){\sqrt {{\bar {z}}^{2}-a^{2}}}}}\right]\end{aligned}}}$

with ${\displaystyle z=x+iy}$, ${\displaystyle {\bar {z}}=x-iy}$, ${\displaystyle \kappa =3-4\nu }$ for plane strain, ${\displaystyle \kappa =(3-\nu )/(1+\nu )}$ for plane stress, and ${\displaystyle \nu }$ is the Poisson's ratio. The stress intensity factors for ${\displaystyle F_{y}}$ at tip B are

${\displaystyle {\begin{aligned}K_{\rm {I}}&={\frac {F_{y}}{2{\sqrt {\pi a}}}}\left[G_{2}-{\frac {1}{\kappa +1}}H_{2}\right]\\K_{\rm {II}}&=-{\frac {F_{y}}{2{\sqrt {\pi a}}}}\left({\frac {\kappa -1}{\kappa +1}}\right)\left[G_{1}-{\frac {1}{\kappa -1}}H_{1}\right]\,.\end{aligned}}}$

The stress intensity factors at the tip A (${\displaystyle x=-a}$) can be determined from the above relations. For the load ${\displaystyle F_{x}}$ at location ${\displaystyle (x,y)}$,

${\displaystyle K_{\rm {I}}(-a;x,y)=-K_{\rm {I}}(a;-x,y)\,,\,\,K_{\rm {II}}(-a;x,y)=K_{\rm {II}}(a;-x,y)\,.}$

Similarly for the load ${\displaystyle F_{y}}$,

${\displaystyle K_{\rm {I}}(-a;x,y)=K_{\rm {I}}(a;-x,y)\,,\,\,K_{\rm {II}}(-a;x,y)=-K_{\rm {II}}(a;-x,y)\,.}$

If the crack is loaded by a point force ${\displaystyle F_{y}}$ located at ${\displaystyle y=0}$ and ${\displaystyle -a<x<a}$, the stress intensity factors at point B are^[7]

${\displaystyle K_{\rm {I}}={\frac {F_{y}}{2{\sqrt {\pi a}}}}{\sqrt {\frac {a+x}{a-x}}}\,,\,\,K_{\rm {II}}=-{\frac {F_{x}}{2{\sqrt {\pi a}}}}\left({\frac {\kappa -1}{\kappa +1}}\right)\,.}$

If the force is distributed uniformly between ${\displaystyle -a<x<a}$, then the stress intensity factor at tip B is

${\displaystyle K_{\rm {I}}={\frac {1}{2{\sqrt {\pi a}}}}\int _{-a}^{a}F_{y}(x)\,{\sqrt {\frac {a+x}{a-x}}}\,{\rm {d}}x\,,\,\,K_{\rm {II}}=-{\frac {1}{2{\sqrt {\pi a}}}}\left({\frac {\kappa -1}{\kappa +1}}\right)\int _{-a}^{a}F_{y}(x)\,{\rm {d}}x,\,.}$

The stress intensity factor at the crack tip of a compact tension specimen is^[14]

${\displaystyle {\begin{aligned}K_{\rm {I}}&={\frac {P}{B}}{\sqrt {\frac {\pi }{W}}}\left[16.7\left({\frac {a}{W}}\right)^{1/2}-104.7\left({\frac {a}{W}}\right)^{3/2}+369.9\left({\frac {a}{W}}\right)^{5/2}\right.\\&\qquad \left.-573.8\left({\frac {a}{W}}\right)^{7/2}+360.5\left({\frac {a}{W}}\right)^{9/2}\right]\end{aligned}}}$

where ${\displaystyle P}$ is the applied load, ${\displaystyle B}$ is the thickness of the specimen, ${\displaystyle a}$ is the crack length, and ${\displaystyle W}$ is the width of the specimen.

The stress intensity factor at the crack tip of a single-edge notch-bending specimen is^[14]

${\displaystyle {\begin{aligned}K_{\rm {I}}&={\frac {4P}{B}}{\sqrt {\frac {\pi }{W}}}\left[1.6\left({\frac {a}{W}}\right)^{1/2}-2.6\left({\frac {a}{W}}\right)^{3/2}+12.3\left({\frac {a}{W}}\right)^{5/2}\right.\\&\qquad \left.-21.2\left({\frac {a}{W}}\right)^{7/2}+21.8\left({\frac {a}{W}}\right)^{9/2}\right]\end{aligned}}}$

where ${\displaystyle P}$ is the applied load, ${\displaystyle B}$ is the thickness of the specimen, ${\displaystyle a}$ is the crack length, and ${\displaystyle W}$ is the width of the specimen.