# Stress intensity factor

The stress intensity factor, ${\displaystyle K}$, is used in fracture mechanics to predict the stress state ("stress intensity") near the tip of a crack or notch caused by a remote load or residual stresses.[1] It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials, and is a critical technique in the discipline of damage tolerance. The concept can also be applied to materials that exhibit small-scale yielding at a crack tip.

The magnitude of ${\displaystyle K}$ depends on sample geometry, the size and location of the crack or notch, and the magnitude and the modal distribution of loads on the material.

Linear elastic theory predicts that the stress distribution (${\displaystyle \sigma _{ij}}$) near the crack tip, in polar coordinates (${\displaystyle r,\theta }$) with origin at the crack tip, has the form [2]

${\displaystyle \sigma _{ij}(r,\theta )={\frac {K}{\sqrt {2\pi r}}}\,f_{ij}(\theta )+\,\,{\rm {higher\,order\,terms}}}$

where ${\displaystyle K}$ is the stress intensity factor (with units of stress ${\displaystyle \times }$ length1/2) and ${\displaystyle f_{ij}}$ is a dimensionless quantity that varies with the load and geometry. Theoretically, as ${\displaystyle r}$ goes to 0, the stress ${\displaystyle \sigma _{ij}}$ goes to ${\displaystyle \infty }$ resulting in a stress singularity.[3] Practically however, this relation breaks down very close to the tip (small ${\displaystyle r}$) because plasticity typically occurs at stresses exceeding the material's yield strength and the linear elastic solution is no longer applicable. Nonetheless, if the crack-tip plastic zone is small in comparison to the crack length, the asymptotic stress distribution near the crack tip is still applicable.

## Stress intensity factors for various modes

Three linearly independent cracking modes are used in fracture mechanics. These load types are categorized as Mode I, II, or III as shown in the figure. Mode I, shown to the left, is an opening (tensile) mode where the crack surfaces move directly apart. Mode II is a sliding (in-plane shear) mode where the crack surfaces slide over one another in a direction perpendicular to the leading edge of the crack. Mode III is a tearing (antiplane shear) mode where the crack surfaces move relative to one another and parallel to the leading edge of the crack. Mode I is the most common load type encountered in engineering design.

Different subscripts are used to designate the stress intensity factor for the three different modes. The stress intensity factor for mode I is designated ${\displaystyle K_{\rm {I}}}$ and applied to the crack opening mode. The mode II stress intensity factor, ${\displaystyle K_{\rm {II}}}$, applies to the crack sliding mode and the mode III stress intensity factor, ${\displaystyle K_{\rm {III}}}$, applies to the tearing mode. These factors are formally defined as:[4]

{\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}}}

## Relationship to energy release rate and J-integral

In plane stress conditions, the strain energy release rate (${\displaystyle G}$) for a crack under pure mode I, or pure mode II loading is related to the stress intensity factor by:

${\displaystyle G_{\rm {I}}=K_{\rm {I}}^{2}\left({\frac {1}{E}}\right)}$
${\displaystyle G_{\rm {II}}=K_{\rm {II}}^{2}\left({\frac {1}{E}}\right)}$

where ${\displaystyle E}$ is the Young's modulus and ${\displaystyle \nu }$ is the Poisson's ratio of the material. The material is assumed to be an isotropic, homogeneous, and linear elastic. The crack has been assumed to extend along the direction of the initial crack

For plane strain conditions, the equivalent relation is a little more complicated:

${\displaystyle G_{\rm {I}}=K_{\rm {I}}^{2}\left({\frac {1-\nu ^{2}}{E}}\right)\,}$
${\displaystyle G_{\rm {II}}=K_{\rm {II}}^{2}\left({\frac {1-\nu ^{2}}{E}}\right)\,.}$

${\displaystyle G_{\rm {III}}=K_{\rm {III}}^{2}\left({\frac {1}{2\mu }}\right)=K_{\rm {III}}^{2}\left({\frac {1+\nu }{E}}\right)}$

where ${\displaystyle \mu }$ is the shear modulus. For general loading in plane strain, the linear combination holds:

${\displaystyle G=G_{\rm {I}}+G_{\rm {II}}+G_{\rm {III}}\,.}$

A similar relation is obtained for plane stress by adding the contributions for the three modes.

The above relations can also be used to connect the J-integral to the stress intensity factor because

${\displaystyle G=J=\int _{\Gamma }\left(W~dx_{2}-\mathbf {t} \cdot {\cfrac {\partial \mathbf {u} }{\partial x_{1}}}~ds\right)\,.}$

## Critical stress intensity factor

The stress intensity factor, ${\displaystyle K}$, is a parameter that amplifies the magnitude of the applied stress that includes the geometrical parameter ${\displaystyle Y}$ (load type). Stress intensity in any mode situation is directly proportional to the applied load on the material. If a very sharp crack, or a V-notch can be made in a material, the minimum value of ${\displaystyle K_{\mathrm {I} }}$ can be empirically determined, which is the critical value of stress intensity required to propagate the crack. This critical value determined for mode I loading in plane strain is referred to as the critical fracture toughness (${\displaystyle K_{\mathrm {Ic} }}$) of the material. ${\displaystyle K_{\mathrm {Ic} }}$ has units of stress times the root of a distance (e.g. MN/m3/2). The units of ${\displaystyle K_{\mathrm {Ic} }}$ imply that the fracture stress of the material must be reached over some critical distance in order for ${\displaystyle K_{\mathrm {Ic} }}$ to be reached and crack propagation to occur. The Mode I critical stress intensity factor, ${\displaystyle K_{\mathrm {Ic} }}$, is the most often used engineering design parameter in fracture mechanics and hence must be understood if we are to design fracture tolerant materials used in bridges, buildings, aircraft, or even bells.

Polishing cannot detect a crack. Typically, if a crack can be seen it is very close to the critical stress state predicted by the stress intensity factor.

### G–criterion

The G-criterion is a fracture criterion that relates the critical stress intensity factor (or fracture toughness) to the stress intensity factors for the three modes. This failure criterion is written as[5]

${\displaystyle K_{\rm {c}}^{2}=K_{\rm {I}}^{2}+K_{\rm {II}}^{2}+{\frac {E'}{2\mu }}\,K_{\rm {III}}^{2}}$

where ${\displaystyle K_{\rm {c}}}$ is the fracture toughness, ${\displaystyle E'=E/(1-\nu ^{2})}$ for plane strain and ${\displaystyle E'=E}$ for plane stress. The critical stress intensity factor for plane stress is often written as ${\displaystyle K_{\rm {c}}}$.

## Examples

### Infinite plate: Uniform uniaxial stress

 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 [3][4] ${\displaystyle K_{\mathrm {I} }=\sigma {\sqrt {\pi a}}}$

### Penny-shaped crack in an infinite domain

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

### Finite plate: Uniform uniaxial stress

 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 [4] ${\displaystyle K_{\rm {I}}=\sigma {\sqrt {\pi a}}\left[{\cfrac {1-{\frac {a}{2b}}+0.326\left({\frac {a}{b}}\right)^{2}}{\sqrt {1-{\frac {a}{b}}}}}\right]\,.}$ 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[4][6] ${\displaystyle K_{\rm {IA}}=\sigma {\sqrt {\pi a}}\left[1+\sum _{n=2}^{M}C_{n}\left({\frac {a}{b}}\right)^{n}\right]}$ where the factors ${\displaystyle C_{n}}$ can be found from fits to stress intensity curves[4]: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 [7]: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.

### Edge crack in a plate under uniaxial stress

 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 an uniaxial stress ${\displaystyle \sigma }$ is [3] ${\displaystyle K_{\rm {I}}=\sigma {\sqrt {\pi a}}\left[1.122-0.231\left({\frac {a}{b}}\right)+10.55\left({\frac {a}{b}}\right)^{2}-21.71\left({\frac {a}{b}}\right)^{3}+30.382\left({\frac {a}{b}}\right)^{4}\right]\,.}$ For the situation where ${\displaystyle h/b\geq 1}$ and ${\displaystyle a/b\geq 0.3}$, the stress intensity factor can be approximated by ${\displaystyle K_{\rm {I}}=\sigma {\sqrt {\pi a}}\left[{\frac {1+3{\frac {a}{b}}}{2{\sqrt {\pi {\frac {a}{b}}}}\left(1-{\frac {a}{b}}\right)^{3/2}}}\right]\,.}$ Specimens of this configuration are commonly used in fracture toughness testing.[8]

### Infinite plate: Slanted crack in a biaxial stress field

 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 [4][9] {\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.

### Crack in a plate under point in-plane force

 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 [9][10] {\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 ${\displaystyle F_{y}}$, ${\displaystyle K_{\rm {I}}(-a;x,y)=K_{\rm {I}}(a;-x,y)\,,\,\,K_{\rm {II}}(-a;x,y)=-K_{\rm {II}}(a;-x,y)\,.}$

### Loaded crack in a plate

 If the crack is loaded by a point force ${\displaystyle F_{y}}$ located at ${\displaystyle y=0}$ and ${\displaystyle -a, the stress intensity factors at point B are[4] ${\displaystyle K_{\rm {I}}={\frac {F_{y}}{2{\sqrt {\pi a}}}}{\sqrt {\frac {a+x}{a-x}}}\,,\,\,K_{\rm {II}}=-{\frac {F_{x}}{2{\sqrt {\pi a}}}}\left({\frac {\kappa -1}{\kappa +1}}\right)\,.}$ If the force is distributed uniformly between ${\displaystyle -a, then the stress intensity factor at tip B is ${\displaystyle K_{\rm {I}}={\frac {1}{2{\sqrt {\pi a}}}}\int _{-a}^{a}F_{y}(x)\,{\sqrt {\frac {a+x}{a-x}}}\,{\rm {d}}x\,,\,\,K_{\rm {II}}=-{\frac {1}{2{\sqrt {\pi a}}}}\left({\frac {\kappa -1}{\kappa +1}}\right)\int _{-a}^{a}F_{y}(x)\,{\rm {d}}x,\,.}$

## Stress intensity factors for fracture toughness tests

### Compact tension specimen

 The stress intensity factor at the crack tip of a compact tension specimen is[11] {\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.

### Single edge notch bending specimen

 The stress intensity factor at the crack tip of a single edge notch bending specimen is[11] {\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.

## References

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3. Liu, M.; et al. (2015). "An improved semi-analytical solution for stress at round-tip notches" (PDF). Engineering Fracture Mechanics. 149: 134–143.
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7. Kathiresan, K., Brussat, T. R., & Hsu, T. M. (1984). "Advanced life analysis methods. Crack Growth Analysis Methods for Attachment Lugs," Flight Dynamics Laboratory, Air Force Wright Aeronautical Laboratories, AFSC W-P Air Forec Base, Ohio, Report No. AFWAL-TR-84-3080.
8. Gross, D. & Seelig, T. (2011). Fracture mechanics: with an introduction to micromechanics. Springer.
9. Sih, G. C.; Paris, P. C. & Erdogan, F. (1962), "Crack-tip stress intensity factors for the plane extension and plate bending problem", Journal of Applied Mechanics, 29: 306–312, Bibcode:1962JAM....29..306S, doi:10.1115/1.3640546
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