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| [[File:cracktipcoords.svg|thumb|300px|Polar coordinates at the crack tip.]]
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| The '''stress intensity factor''', <math>K</math>, is used in [[fracture mechanics]] to predict the [[Stress (mechanics)|stress]] state ("stress intensity") near the tip of a crack caused by a remote [[Structural load|load]] or residual stresses.<ref name=ander>{{cite book|author=Anderson, T.L.|year=2005|title=Fracture mechanics: fundamentals and applications|publisher=CRC Press}}</ref> It is a theoretical construct usually applied to a homogeneous, linear [[Elasticity (physics)|elastic]] material and is useful for providing a failure criterion for [[Brittleness|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 [[yield (engineering)|yield]]ing'' at a crack tip.
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| The magnitude of <math>K</math> depends on sample geometry, the size and location of the crack, and the magnitude and the modal distribution of loads on the material.
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| [[Linear elasticity|Linear elastic]] theory predicts that the stress distribution (<math>\sigma_{ij}</math>) near the crack tip, in [[polar coordinates]] (<math>r,\theta</math>) with origin at the crack tip, has the form <ref>{{cite book |title=The Stress Analysis of Cracks Handbook |first=Hiroshi |last=Tada |coauthors=P. C. Paris; [[George Rankine Irwin]] |publisher=American Society of Mechanical Engineers |edition=3 |date=February 2000}}</ref>
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| :<math>
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| \sigma_{ij}(r, \theta) = \frac {K} {\sqrt{2 \pi r}}\,f_{ij} ( \theta) + \,\,\rm{higher\, order\, terms}
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| </math>
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| where <math>K</math> is the stress intensity factor (with units of stress <math>\times</math> length<sup>1/2</sup>) and <math>f_{ij}</math> is a dimensionless quantity that depends on the load and geometry. This relation breaks down very close to the tip (small <math>r</math>) because as <math>r</math> goes to 0, the stress <math>\sigma_{ij}</math> goes to <math>\infty</math>. [[Plasticity (physics)|Plastic]] distortion typically occurs at high stresses and the linear elastic solution is no longer applicable close to the crack tip. However, if the crack-tip plastic zone is small, it can be assumed that the stress distribution near the crack is still given by the above relation.
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| == Stress intensity factors for various modes == | |
| [[Image:Fracture modes v2.svg|thumb|330px|Mode I, Mode II, and Mode III crack loading.]]
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| 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 stress|tensile]]) mode where the crack surfaces move directly apart. Mode II is a sliding (in-plane [[Shear stress|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.
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| Different subscripts are used to designate the stress intensity factor for the three different modes. The stress intensity factor for mode I is designated <math>K_{\rm I}</math> and applied to the crack opening mode. The mode II stress intensity factor, <math>K_{\rm II}</math>, applies to the crack sliding mode and the mode III stress intensity factor, <math>K_{\rm III}</math>, applies to the tearing mode. These factors are formally defined as <ref name=rooke/>
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| :<math>
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| \begin{align}
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| K_{\rm I} & = \lim_{r\rightarrow 0} \sqrt{2\pi r}\,\sigma_{yy}(r,0) \\
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| K_{\rm II} & = \lim_{r\rightarrow 0} \sqrt{2\pi r}\,\sigma_{yx}(r,0) \\
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| K_{\rm III} & = \lim_{r\rightarrow 0} \sqrt{2\pi r}\,\sigma_{yz}(r,0) \,.
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| \end{align}
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| </math>
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| == Relationship to energy release rate and J-integral ==
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| The [[strain energy release rate]] (<math>G</math>) for a crack under mode I loading is related to the stress intensity factor by
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| :<math>
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| G = K_{\rm I}^2\left(\frac{1-\nu^2}{E}\right)
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| </math>
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| where <math>E</math> is the [[Young's modulus]] and <math>\nu</math> is the [[Poisson's ratio]] of the material. The material is assumed to be an isotropic, homogeneous, and linear elastic. [[Plane strain]] has been assumed and the crack has been assumed to extend along the direction of the initial crack. For [[plane stress]] conditions, the above relation simplifies to
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| :<math>
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| G = K_{\rm I}^2\left(\frac{1}{E}\right)\,.
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| </math>
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| For pure mode II loading, we have similar relations
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| :<math>
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| G = K_{\rm II}^2\left(\frac{1-\nu^2}{E}\right) \quad \text{or} \quad
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| G = K_{\rm II}^2\left(\frac{1}{E}\right) \,.
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| </math>
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| For pure mode III loading,
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| :<math>
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| G = K_{\rm III}^2\left(\frac{1}{2\mu}\right)
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| </math>
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| where <math>\mu</math> is the [[shear modulus]]. For general loading in plane strain, the relationship between the strain energy and the stress intensity factors for the three modes is
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| :<math>
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| G = K_{\rm I}^2\left(\frac{1-\nu^2}{E}\right) + K_{\rm II}^2\left(\frac{1-\nu^2}{E}\right) + K_{\rm III}^2\left(\frac{1}{2\mu}\right)\,.
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| </math>
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| A similar relation is obtained for plane stress by adding the contributions for the three modes.
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| The above relations can also be used to connect the [[J integral]] to the stress intensity factor because
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| :<math>
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| G = J = \int_\Gamma \left(W~dx_2 - \mathbf{t}\cdot\cfrac{\partial\mathbf{u}}{\partial x_1}~ds\right) \,.
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| </math>
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| ==Critical stress intensity factor==
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| The stress intensity factor, <math>K</math>, is a parameter that amplifies the magnitude of the applied stress that includes the geometrical parameter <math>Y</math> (load type). Stress intensity in any mode situation is directly proportional to the applied load on the material. If a very sharp crack can be made in a material, the minimum value of <math>K_\mathrm{I}</math> 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 [[Infinitesimal strain theory#Plane strain|plane strain]] is referred to as the critical fracture toughness (<math>K_\mathrm{Ic}</math>) of the material. <math>K_\mathrm{Ic}</math> has units of stress times the root of a distance. The units of <math>K_\mathrm{Ic}</math> imply that the fracture stress of the material must be reached over some critical distance in order for <math>K_\mathrm{Ic}</math> to be reached and crack propagation to occur. The Mode I critical stress intensity factor, <math>K_\mathrm{Ic}</math>, 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{{fact|date=November 2012}}.
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| === G–criterion ===
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| The '''G-criterion''' is a [[material failure theory|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<ref name=sih>{{citation|title=Fracture mechanics applied to engineering problems-strain energy density fracture criterion| author=Sih, G. C. and Macdonald, B.| journal=Engineering Fracture Mechanics| volume=6| number=2| pages=361–386| year=1974}}</ref>
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| :<math>
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| K_{\rm Ic}^2 = K_{\rm I}^2 + K_{\rm II}^2 + \frac{E'}{2\mu}\,K_{\rm III}^2
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| </math>
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| where <math>K_{\rm Ic}</math> is the mode I fracture toughness, <math>E' = E/(1-\nu^2)</math> for [[plane strain]] and <math>E' = E</math> for [[plane stress]]. The critical stress intensity factor for ''plane stress'' is often written as <math>K_{\rm c}</math>.
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| == Examples of stress intensity factors ==
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| {|
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| |- align = "left" valign = "top"
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| | ''' Uniform uniaxial stress '''
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| The stress intensity factor for a through crack of length <math>2a</math>, at right angles, in an infinite plane, to a uniform stress field <math>\sigma</math> is <ref name=rooke>{{cite book| title=Compendium of stress intensity factors|author=Rooke, D.P. and Cartwright, D.J.|publisher=HMSO Ministry of Defence. Procurement Executive|year=1976}}</ref>
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| :<math>
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| K_\mathrm{I}=\sigma \sqrt{\pi a}
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| </math>
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| If the crack is located centrally in a finite plate of width <math>2b</math> and height <math>2h</math>, an approximate relation for the stress intensity factor is <ref name=rooke/>
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| :<math>
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| 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] \,.
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| </math>
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| If the crack is not located centrally along the width, i.e., <math>d \ne b</math>, the stress intensity factor at location '''A''' can be approximated by the series expansion<ref name=rooke/><ref name=isida66>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.</ref>
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| :<math>
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| K_{\rm IA} = \sigma \sqrt{\pi a}\left[1 + \sum_{n=2}^{M} C_n\left(\frac{a}{b}\right)^n\right]
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| </math>
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| where the factors <math>C_n</math> can be found from fits to stress intensity curves<ref name=rooke/>{{rp|6}} for various values of <math>d</math>. 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 <ref name=kathir>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.</ref>{{rp|175}}
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| :<math>
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| K_{\rm IA} = \sigma\sqrt{\pi a}\,\Phi_A \,\, , K_{\rm IB} = \sigma\sqrt{\pi a}\,\Phi_B
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| </math>
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| where
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| :<math>
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| \begin{align}
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| \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} \\
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| \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]
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| \end{align}
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| </math>
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| with
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| :<math>
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| \beta := \sin\left(\frac{\pi\alpha_B}{\alpha_A+\alpha_B}\right) ~,~~ \alpha_A := \frac{\pi a}{2 d}
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| ~,~~ \alpha_B := \frac{\pi a}{4b - 2d} ~;~~ \alpha_{AB} := \frac{4}{7}\,\alpha_A + \frac{3}{7}\,\alpha_B \,.
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| </math>
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| If the above expressions <math>d</math> is the distance from the center of the crack to the boundary closest to point '''A'''. Note that when <math>d=b</math> the above expressions do ''not'' simplify into the approximate expression for a centered crack.
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| |[[File:CrackFinitePlate.svg|thumb|175px|left|Crack in a finite plate under mode I loading.]]
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| <!--
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| |{{multiple image
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| |width1 = 150
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| |image1 = crackInfinitePlate.svg
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| |caption1=Crack in an infinite plate under mode I loading.
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| |width2 = 175
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| |image2 = crackFinitePlate.svg
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| |caption2=Crack in a finite plate under mode I loading.
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| }}
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| -->
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| |- align = "left" valign = "top"
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| | '''Edge crack in a plate under uniaxial stress'''
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| For a plate of dimensions <math>h \times b</math> containing an edge crack of length <math>a</math>, if the dimensions
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| of the plate are such that <math>h/b \ge 1</math> and <math>a/b \le 0.6</math>, the stress intensity factor at the
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| crack tip under an uniaxial stress <math>\sigma</math> is
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| :<math>
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| 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
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| - 21.7\left(\frac{a}{b}\right)^3 + 30.4\left(\frac{a}{b}\right)^4\right] \,.
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| </math>
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| For the situation where <math>h/b \ge 1</math> and <math>a/b \ge 0.3</math>, the stress intensity factor can be approximated
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| by
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| :<math>
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| 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] \,.
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| </math>
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| Specimens of this configuration are commonly used in [[fracture toughness]] testing.<ref name=gross>{{cite book|title=Fracture mechanics: with an introduction to micromechanics|author=Gross, D. and Seelig, T.|year=2011|publisher=Springer}}</ref>
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| |[[File:CrackEdgeFinitePlate.svg|thumb|175px|left|Edge crack in a finite plate under uniaxial stress.]]
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| |- align = "left" valign = "top"
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| | '''Slanted crack in a biaxial stress field '''
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| For a slanted crack of length <math>2a</math> in a biaxial stress field with stress <math>\sigma</math> in the <math>y</math>-direction and <math>\alpha\sigma</math> in the <math>x</math>-direction, the stress intensity factors are <ref name=rooke/><ref name=spe62/>
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| :<math>
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| \begin{align}
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| K_{\rm I} & = \sigma\sqrt{\pi a}\left(\cos^2\beta + \alpha \sin^2\beta\right) \\
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| K_{\rm II} & = \sigma\sqrt{\pi a}\left(1- \alpha\right)\sin\beta\cos\beta
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| \end{align}
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| </math>
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| where <math>\beta</math> is the angle made by the crack with the <math>x</math>-axis.
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| | [[File:CrackSlantedPlateBiaxialLoad.svg|thumb|left|260px|A slanted crack in a thin plate under biaxial load.]]
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| |- align = "left" valign = "top"
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| | ''' Penny-shaped crack in an infinite domain '''
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| The stress intensity factor at the tip of a penny-shaped crack of radius <math>a</math> in an infinite domain under uniaxial tension <math>\sigma</math> is <ref name=sned>{{citation | title=The distribution of stress in the neighbourhood of a crack in an elastic solid|author=Sneddon, I. N.|journal=[[Proceedings of the Royal Society A]]| volume=187| number=1009| pages=229| year=1946}}</ref>
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| :<math>
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| K_{\rm I} = 2\sigma\sqrt{\frac{a}{\pi}} \,.
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| </math>
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| |[[File:PennyShapedCrack.svg|thumb|200px|left|Penny-shaped crack in an infinite domain under uniaxial tension.]]
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| |- align = "left" valign = "top"
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| | '''Crack in a plate under point in-plane force '''
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| Consider a plate with dimensions <math>2h \times 2b</math> containing a crack of length <math>2a</math>. A point force with components <math>F_x</math> and <math>F_y</math> is applied at the point (<math>x,y</math>) of the plate.
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| 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., <math>h \gg a</math>, <math>b \gg a</math>, <math>x \ll b</math>, <math>y \ll h</math>, the plate can be considered infinite. In that case, for the stress intensity factors for <math>F_x</math> at crack tip ''' B''' (<math>x = a</math>) are <ref name=spe62>{{citation|author=Sih, G. C., Paris, P. C. and Erdogan, F.|year=1962|title=Crack-tip stress intensity factors for the plane extension and plate bending problem|journal=Journal of Applied Mechanics|volume=29|pages=306–312|bibcode = 1962JAM....29..306S |doi = 10.1115/1.3640546 }}</ref><ref name=erdo62>{{citation|title=On the stress distribution in plates with collinear cuts under arbitrary loads|author=Erdogan, F.|journal=Proceedings of the Fourth US National Congress of Applied Mechanics|volume=1|pages=547–574|year=1962}}</ref>
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| :<math>
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| \begin{align}
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| K_{\rm I} & = \frac{F_x}{2\sqrt{\pi a}}\left(\frac{\kappa -1}{\kappa+1}\right)
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| \left[G_1 + \frac{1}{\kappa-1} H_1\right] \\
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| K_{\rm II} & = \frac{F_x}{2\sqrt{\pi a}}
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| \left[G_2 + \frac{1}{\kappa+1} H_2\right]
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| \end{align}
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| </math>
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| where
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| :<math>
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| \begin{align}
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| G_1 & = 1 - \text{Re}\left[\frac{a+z}{\sqrt{z^2-a^2}}\right] \,,\,\,
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| G_2 = - \text{Im}\left[\frac{a+z}{\sqrt{z^2-a^2}}\right] \\
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| H_1 & = \text{Re}\left[\frac{a(\bar{z}-z)}{(\bar{z}-a)\sqrt{{\bar{z}}^2-a^2}}\right] \,,\,\,
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| H_2 = -\text{Im}\left[\frac{a(\bar{z}-z)}{(\bar{z}-a)\sqrt{{\bar{z}}^2-a^2}}\right]
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| \end{align}
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| </math>
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| with <math>z = x + iy</math>, <math>\bar{z} = x - iy</math>, <math> \kappa = 3-4\nu</math> for [[plane strain]], <math>\kappa= (3-\nu)/(1+\nu)</math> for [[plane stress]], and <math>\nu</math> is the [[Poisson's ratio]].
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| The stress intensity factors for <math>F_y</math> at tip ''' B''' are
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| :<math> | |
| \begin{align}
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| K_{\rm I} & = \frac{F_y}{2\sqrt{\pi a}}
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| \left[G_2 - \frac{1}{\kappa+1} H_2\right] \\
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| K_{\rm II} & = -\frac{F_y}{2\sqrt{\pi a}}\left(\frac{\kappa -1}{\kappa+1}\right)
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| \left[G_1 - \frac{1}{\kappa-1} H_1\right] \,.
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| \end{align}
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| </math>
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| The stress intensity factors at the tip ''' A''' (<math>x = -a</math>) can be determined from the above relations. For the load <math>F_x</math> at location <math>(x,y)</math>,
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| :<math>
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| K_{\rm I}(-a; x,y) = -K_{\rm I}(a; -x,y) \,,\,\,
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| K_{\rm II}(-a; x,y) = K_{\rm II}(a; -x,y) \,.
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| </math>
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| Similarly for the load <math>F_y</math>,
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| :<math>
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| K_{\rm I}(-a; x,y) = K_{\rm I}(a; -x,y) \,,\,\,
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| K_{\rm II}(-a; x,y) = -K_{\rm II}(a; -x,y) \,.
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| </math>
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| |[[File:CrackFinitePlatePointForce.svg|thumb|200px|left|A crack in a plate under the action of a localized force with components <math>F_x</math> and <math>F_y</math>.]]
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| |- align = "left" valign = "top"
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| | '''Loaded crack in a plate'''
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| If the crack is loaded by a point force <math>F_y</math> located at <math>y=0</math> and <math>-a < x < a</math>, the stress intensity factors at point '''B''' are<ref name=rooke/>
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| :<math>
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| K_{\rm I} = \frac{F_y}{2\sqrt{\pi a}}\sqrt{\frac{a+x}{a-x}}\,,\,\,
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| K_{\rm II} = -\frac{F_x}{2\sqrt{\pi a}}\left(\frac{\kappa -1}{\kappa+1}\right) \,.
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| </math>
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| If the force is distributed uniformly between <math>-a < x < a</math>, then the stress intensity factor at tip '''B''' is
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| :<math>
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| K_{\rm I} = \frac{1}{2\sqrt{\pi a}}\int_{-a}^a F_y(x)\,\sqrt{\frac{a+x}{a-x}}\,{\rm d}x\,,\,\,
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| 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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| </math>
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| |[[File:loadedCrackPlate.svg|thumb|200px|left|A loaded crack in a plate.]]
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| |}
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| == Stress intensity factors for fracture toughness tests ==
| |
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| {|
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| |- align = "left" valign = "top"
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| | ''' Compact tension specimen '''
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| The stress intensity factor at the crack tip of a [[compact tension specimen]] is<ref name=bower>{{cite book|title=Applied mechanics of solids|author=Bower, A. F.| year=2009| publisher=CRC Press.}}</ref>
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| :<math> | |
| \begin{align}
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| 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}
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| + 369.9\left(\frac{a}{W}\right)^{5/2} \right.\\
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| & \qquad \left.- 573.8\left(\frac{a}{W}\right)^{7/2} + 360.5\left(\frac{a}{W}\right)^{9/2} \right]
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| \end{align}
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| </math>
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| where <math>P</math> is the applied load, <math>B</math> is the thickness of the specimen, <math>a</math> is the crack length, and
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| <math>W</math> is the width of the specimen.
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| |[[File:CompactTensionSpecimen.svg|thumb|left|200px|Compact tension specimen for fracture toughness testing.]]
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| |- align = "left" valign = "top"
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| | '''Single edge notch bending specimen'''
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| The stress intensity factor at the crack tip of a [[three point flexural test|single edge notch bending specimen]] is<ref name=bower/>
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| :<math>
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| \begin{align}
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| 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}
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| + 12.3\left(\frac{a}{W}\right)^{5/2} \right.\\
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| & \qquad \left.- 21.2\left(\frac{a}{W}\right)^{7/2} + 21.8\left(\frac{a}{W}\right)^{9/2} \right]
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| \end{align}
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| </math>
| |
| where <math>P</math> is the applied load, <math>B</math> is the thickness of the specimen, <math>a</math> is the crack length, and
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| <math>W</math> is the width of the specimen.
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| |[[File:SingleEdgeNotchBending.svg|thumb|left|300px|Single edge notch bending specimen (also called three point bending specimen) for fracture toughness testing.]]
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| |}
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| == See also ==
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| * [[Fracture mechanics]]
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| * [[Fracture toughness]]
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| * [[Strain energy release rate]]
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| * [[J integral]]
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| * [[Material failure theory]]
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| ==References==
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| {{reflist|2}}
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| == External links ==
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| * [http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA150420| Kathiresan, K. ; Hsu, T. M. ; Brussat, T. R., 1984, Advanced Life Analysis Methods. Volume 2. Crack Growth Analysis Methods for Attachment Lugs]
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| [[Category:Fracture mechanics]]
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