Stress Intensity Factor - Examples of Stress Intensity Factors

Examples of Stress Intensity Factors

Uniform uniaxial stress

The stress intensity factor for a through crack of length, at right angles, in an infinite plane, to a uniform stress field is

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

If the crack is located centrally in a finite plate of width and height, an approximate relation for the stress intensity factor is

K_{\rm I} = \sigma \sqrt{\pi a}\left \,.

If the crack is not located centrally along the width, i.e., the stress intensity factor at location A can be approximated by the series expansion

K_{\rm IA} = \sigma \sqrt{\pi a}\left

where the factors can be found from fits to stress intensity curves for various values of . 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

K_{\rm IA} = \sigma\sqrt{\pi a}\,\Phi_A \,\, K_{\rm IB} = \sigma\sqrt{\pi a}\,\Phi_B

where

\begin{align} \Phi_A &:= \left\sqrt{\sec\alpha_A} \\ \Phi_B &:= 1 + \left\right\}}\right] \end{align}

with

\beta := \sin\left(\frac{\pi\alpha_B}{\alpha_A+\alpha_B}\right) ~,~~ \alpha_A := \frac{\pi a}{2 d} ~,~~ \alpha_B := \frac{\pi a}{4b - 2d} ~;~~ \alpha_{AB} := \frac{4}{7}\,\alpha_A + \frac{3}{7}\,\alpha_B \,.

If the above expressions is the distance from the center of the crack to the boundary closest to point A. Note that when 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 of dimensions containing an edge crack of length, if the dimensions of the plate are such that and, the stress intensity factor at the crack tip under an uniaxial stress is

K_{\rm I} = \sigma\sqrt{\pi a}\left[1.12 - 0.23\left(\frac{a}{b}\right) + 10.6\left(\frac{a}{b}\right)^2 - 21.7\left(\frac{a}{b}\right)^3 + 30.4\left(\frac{a}{b}\right)^4\right] \,.

For the situation where and, the stress intensity factor can be approximated by

K_{\rm I} = \sigma\sqrt{\pi a}\left \,.

Specimens of this configuration are commonly used in fracture toughness testing.
Slanted crack in a biaxial stress field

For a slanted crack of length in a biaxial stress field with stress in the -direction and in the -direction, the stress intensity factors are

\begin{align} 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{align}

where is the angle made by the crack with the -axis.
Penny-shaped crack in an infinite domain

The stress intensity factor at the tip of a penny-shaped crack of radius in an infinite domain under uniaxial tension is

K_{\rm I} = 2\sigma\sqrt{\frac{a}{\pi}} \,.
Crack in a plate under point in-plane force

Consider a plate with dimensions containing a crack of length . A point force with components and is applied at the point 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., the plate can be considered infinite. In that case, for the stress intensity factors for at crack tip B are

\begin{align} K_{\rm I} & = \frac{F_x}{2\sqrt{\pi a}}\left(\frac{\kappa -1}{\kappa+1}\right) \left \\ K_{\rm II} & = \frac{F_x}{2\sqrt{\pi a}} \left \end{align}

where

\begin{align} G_1 & = 1 - \text{Re}\left \,,\,\, G_2 = - \text{Im}\left \\ H_1 & = \text{Re}\left \,,\,\, H_2 = -\text{Im}\left \end{align}

with, for plane strain, for plane stress, and is the Poisson's ratio. The stress intensity factors for at tip B are

\begin{align} K_{\rm I} & = \frac{F_y}{2\sqrt{\pi a}} \left \\ K_{\rm II} & = -\frac{F_y}{2\sqrt{\pi a}}\left(\frac{\kappa -1}{\kappa+1}\right) \left \,. \end{align}

The stress intensity factors at the tip A can be determined from the above relations. For the load at location ,

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 ,

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 located at and, the stress intensity factors at point B are

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, then the stress intensity factor at tip B is

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, \,.

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