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| {{continuum mechanics|cTopic=[[Solid mechanics]]}}
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| The '''J-[[integral]]''' represents a way to calculate the [[strain energy release rate]], or work ([[energy]]) per unit fracture surface area, in a material.<ref name=VanVliet>[http://www.stellar.mit.edu/S/course/3/fa06/3.032/index.html Van Vliet, Krystyn J. (2006); "3.032 Mechanical Behavior of Materials"]</ref> The theoretical concept of J-integral was developed in 1967 by Cherepanov<ref>G. P. Cherepanov, '' The propagation of cracks in a continuous medium'', Journal of Applied Mathematics and Mechanics, 31(3), 1967, pp. 503-512.</ref> and in 1968 by Jim Rice<ref name=Rice68>J. R. Rice, ''A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks'', Journal of Applied Mechanics, 35, 1968, pp. 379-386.</ref> independently, who showed that an energetic [[contour integral|contour path integral]] (called ''J'') was independent of the path around a [[Fracture|crack]].
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| Later, experimental methods were developed, which allowed measurement of critical fracture properties using laboratory-scale specimens for materials in which sample sizes are too small and for which the assumptions of Linear Elastic [[Fracture Mechanics]] (LEFM) do not hold,<ref name="Lee and Donavon">Lee, R. F., & Donovan, J. A. (1987). J-integral and crack opening displacement as crack initiation criteria in natural rubber in pure shear and tensile specimens. Rubber chemistry and technology, 60(4), 674-688. [http://dx.doi.org/10.5254/1.3536150]</ref> and to infer a critical value of fracture energy ''J''<sub>Ic</sub>. The quantity ''J''<sub>Ic</sub> defines the point at which large-scale [[plasticity (physics)|plastic]] yielding during propagation takes place under mode one loading.<ref name="VanVliet" /><ref name=Meyers>Meyers and Chawla (1999): "Mechanical Behavior of Materials," 445-448.</ref>
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| The J-integral is equal to the [[strain energy release rate]] for a crack in a body subjected to monotonic loading.<ref name="Yoda80">Yoda, M., 1980, ''The J-integral fracture toughness for Mode II'', Int. J. of Fracture, 16(4), pp. R175-R178.</ref> This is generally true, under quasistatic conditions, only for [[linear elasticity|linear elastic]] materials. For materials that experience small-scale [[yield (engineering)|yield]]ing at the crack tip, ''J'' can be used to compute the energy release rate under special circumstances such as monotonic loading in [[fracture mechanics|mode III]] ([[antiplane shear]]). The strain energy release rate can also be computed from ''J'' for pure [[power-law hardening]] [[plasticity (physics)|plastic]] materials that undergo small-scale yielding at the crack tip.
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| The quantity ''J'' is not path-independent for monotonic [[fracture mechanics|mode I]] and [[fracture mechanics|mode II]] loading of elastic-plastic materials, so only a contour very close to the crack tip gives the energy release rate. Also, Rice showed that ''J'' is path-independent in plastic materials when there is no non-[[proportional loading]]. Unloading is a special case of this, but non-proportional plastic loading also invalidates the path-independence. Such non-proportional loading is the reason for the path-dependence for the in-plane loading modes on elastic-plastic materials.
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| == Two-dimensional J-integral ==
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| [[File:Integrale-J 2D.png|200px|right|thumb|Figure 1. Line J-integral around a notch in two dimensions.]]
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| The two-dimensional J-integral was originally defined as<ref name="Rice68" /> (see Figure 1 for an illustration)
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| :<math>
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| J := \int_\Gamma \left(W~dx_2 - \mathbf{t}\cdot\cfrac{\partial\mathbf{u}}{\partial x_1}~ds\right)
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| = \int_\Gamma \left(W~dx_2 - t_i\,\cfrac{\partial u_i}{\partial x_1}~ds\right)
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| </math>
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| where ''W''(''x''<sub>1</sub>,''x''<sub>2</sub>) is the [[strain energy density]], ''x''<sub>1</sub>,''x''<sub>2</sub> are the coordinate directions, '''t'''='''n'''.'''''σ''''' is the [[stress (physics)|surface traction]] vector, '''n''' is the normal to the curve Γ, '''''σ''''' is the [[Cauchy stress tensor]], and '''u''' is the [[displacement vector]]. The strain energy density is given by
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| :<math>
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| W = \int_0^\epsilon \boldsymbol{\sigma}:d\boldsymbol{\epsilon} ~;~~
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| \boldsymbol{\epsilon} = \tfrac{1}{2}\left[\boldsymbol{\nabla}\mathbf{u}+(\boldsymbol{\nabla}\mathbf{u})^T\right] ~.
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| </math>
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| The J-Integral around a crack tip is frequently expressed in a more general form (and in [[Einstein notation|index notation]]) as
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| :<math>
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| J_i := \lim_{\epsilon\rightarrow 0} \int_{\Gamma_\epsilon} \left(W n_i - n_j\sigma_{jk}~\cfrac{\partial u_k}{\partial x_i}\right) d\Gamma
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| </math>
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| where <math>J_i</math> is the component of the J-integral for crack opening in the <math>x_i</math> direction and <math>\epsilon</math> is a small region around the crack tip.
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| Using [[Green's theorem]] we can show that this integral is zero when the boundary <math>\Gamma</math> is closed and encloses a region that contains no [[Mathematical singularity|singularities]] and is [[simply connected]]. If the faces of the crack do not have any [[surface traction]]s on them then the J-integral is also [[path independence|path independent]].
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| Rice also showed that the value of the J-integral represents the energy release rate for planar crack growth.
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| The J-integral was developed because of the difficulties involved in computing the [[Stress (mechanics)|stress]] close to a crack in a nonlinear [[Elasticity (physics)|elastic]] or elastic-[[Plasticity (physics)|plastic]] material. Rice showed that if monotonic loading was assumed (without any plastic unloading) then the J-integral could be used to compute the energy release rate of plastic materials too.
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| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
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| !Proof that the J-integral is zero over a closed path
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| |-
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| |To show the path independence of the J-integral, we first have to show that the value of <math>J</math> is zero over a closed contour in a simply connected domain. Let us just consider the expression for <math>J_1</math> which is
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| :<math>
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| J_1 := \int_{\Gamma} \left(W n_1 - n_j\sigma_{jk}~\cfrac{\partial u_k}{\partial x_1}\right) d\Gamma
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| </math>
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| We can write this as
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| :<math>
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| J_1 = \int_{\Gamma} \left(W \delta_{1j} - \sigma_{jk}~\cfrac{\partial u_k}{\partial x_1}\right)n_j d\Gamma
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| </math>
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| From [[Green's theorem]] (or the two-dimensional [[divergence theorem]]) we have
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| :<math>
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| \int_{\Gamma} f_j~n_j~d\Gamma = \int_A \cfrac{\partial f_j}{\partial x_j}~dA
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| </math>
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| Using this result we can express <math>J_1</math> as
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| :<math>
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| \begin{align}
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| J_1 & = \int_{A} \cfrac{\partial}{\partial x_j}\left(W \delta_{1j} - \sigma_{jk}~\cfrac{\partial u_k}{\partial x_1}\right) dA \\
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| & = \int_A \left[\cfrac{\partial W}{\partial x_1} -
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| \cfrac{\partial\sigma_{jk}}{\partial x_j}~\cfrac{\partial u_k}{\partial x_1} -
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| \sigma_{jk}~\cfrac{\partial^2 u_k}{\partial x_1 \partial x_j}\right]~dA
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| \end{align}
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| </math>
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| where <math>A</math> is the area enclosed by the contour <math>\Gamma</math>. Now, if there are '''no body forces''' present, equilibrium (conservation of linear momentum) requires that
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| :<math>
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| \boldsymbol{\nabla}\cdot\boldsymbol{\sigma} = \mathbf{0} \qquad \implies \qquad \cfrac{\partial\sigma_{jk}}{\partial x_j} = 0 ~.
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| </math>
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| Also,
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| :<math>
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| \boldsymbol{\epsilon} = \tfrac{1}{2}\left[\boldsymbol{\nabla}\mathbf{u}+(\boldsymbol{\nabla}\mathbf{u})^T\right]
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| \qquad \implies \qquad
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| \epsilon_{jk} = \tfrac{1}{2}\left(\cfrac{\partial u_k}{\partial x_j} + \cfrac{\partial u_j}{\partial x_k}\right) ~.
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| </math>
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| Therefore,
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| :<math>
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| \sigma_{jk}\cfrac{\partial\epsilon_{jk}}{\partial x_1} =
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| \tfrac{1}{2}\left(\sigma_{jk}\cfrac{\partial^2 u_k}{\partial x_1 \partial x_j} + \sigma_{jk}\cfrac{\partial^2 u_j}{\partial x_1 \partial x_k}\right)
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| </math>
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| From the balance of angular momentum we have <math>\sigma_{jk} = \sigma_{kj}</math>. Hence,
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| :<math>
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| \sigma_{jk}\cfrac{\partial\epsilon_{jk}}{\partial x_1} =
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| \sigma_{jk}\cfrac{\partial^2 u_j}{\partial x_1 \partial x_k}
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| </math>
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| The J-integral may then be written as
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| :<math>
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| J_1 = \int_A \left[\cfrac{\partial W}{\partial x_1} -
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| \sigma_{jk}~\cfrac{\partial\epsilon_{jk}}{\partial x_1}\right]~dA
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| </math>
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| Now, for an elastic material the stress can be derived from the stored energy function <math>W</math> using
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| :<math>
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| \sigma_{jk} = \cfrac{\partial W}{\partial\epsilon_{jk}}
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| </math>
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| Then, using the [[chain rule]] of differentiation,
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| :<math>
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| \sigma_{jk}~\cfrac{\partial\epsilon_{jk}}{\partial x_1} = \cfrac{\partial W}{\partial\epsilon_{jk}}~\cfrac{\partial\epsilon_{jk}}{\partial x_1} = \cfrac{\partial W}{\partial x_1}
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| </math>
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| Therefore we have <math>J_1 = 0 </math> for a closed contour enclosing a simply connected region without any stress singularities.
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| |}
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| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
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| !Proof that the J-integral is path-independent
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| |-
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| |[[Image:J integralPathIndep.png|200px|right|thumb|Figure 2. Integration paths around a notch in two dimensions.]]
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| Consider the contour <math>\Gamma = \Gamma_1 + \Gamma^{+} + \Gamma_2 + \Gamma^{-} </math>. Since this contour is closed and encloses a simply connected region, the J-integral around the contour is zero, i.e.
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| :<math>
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| J = J_{(1)} + J^{+} - J_{(2)} - J^{-} = 0
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| </math>
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| assuming that counterclockwise integrals around the crack tip have positive sign. Now, since the crack surfaces are parallel to the <math>x_2</math> axis, the normal component <math>n_1 = 0</math> on these surfaces. Also, since the crack surfaces are traction free, <math>t_k = 0 </math>. Therefore,
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| :<math>
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| J^{+} = J^{-} = \int_{\Gamma} \left(W n_1 - t_k~\cfrac{\partial u_k}{\partial x_1}\right) d\Gamma = 0
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| </math>
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| Therefore,
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| :<math>
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| J_{(1)} = J_{(2)}
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| </math>
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| and the J-integral is path independent.
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| |}
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| ==J-integral and fracture toughness==
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| For isotropic, perfectly brittle, linear elastic materials, the J-integral can be directly related to the [[fracture toughness]] if the crack extends straight ahead with respect to its original orientation.<ref name="Yoda80" />
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| For plane strain, under [[fracture mechanics|Mode I]] loading conditions, this relation is
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| :<math>
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| J_{\rm Ic} = G_{\rm Ic} = K_{\rm Ic}^2 \left(\frac{1-\nu^2}{E}\right)
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| </math>
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| where <math>G_{\rm Ic}</math> is the critical strain energy release rate, <math> K_{\rm Ic}</math> is the fracture toughness in Mode I loading, <math>\nu</math> is the Poisson's ratio, and ''E'' is the [[Young's modulus]] of the material.
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| For [[fracture mechanics|Mode II]] loading, the relation between the J-integral and the mode II fracture toughness (<math>K_{\rm IIc}</math>) is
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| :<math>
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| J_{\rm IIc} = G_{\rm IIc} = K_{\rm IIc}^2 \left[\frac{1-\nu^2}{E}\right]
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| </math>
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| For [[fracture mechanics|Mode III]] loading, the relation is
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| :<math>
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| J_{\rm IIIc} = G_{\rm IIIc} = K_{\rm IIIc}^2 \left(\frac{1+\nu}{E}\right)
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| </math>
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| == Elastic-plastic materials and the HRR solution ==
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| [[File:HRRSingularity plain.svg|300px|right|thumb|Paths for J-integral calculation around a notch in a two-dimensional elastic-plastic material.]]
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| Hutchinson, Rice and Rosengren <ref>{{Citation|last=Hutchinson|first= J. W. |year=1968|title= Singular behaviour at the end of a tensile crack in a hardening material|journal= Journal of the Mechanics and Physics of Solids|volume=16|number=1|pages= 13–31.|url=http://www.seas.harvard.edu/hutchinson/papers/312.pdf}}</ref><ref>{{Citation|last1=Rice|first1=J. R.|first2=G. F.|last2= Rosengren|title=Plane strain deformation near a crack tip in a power-law hardening material|journal=Journal of the Mechanics and Physics of Solids|volume= 16|number=1|year=1968|pages= 1–12.|url=http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0659300}}</ref> subsequently showed that J characterizes the [[Mathematical singularity|singular]] stress and strain fields at the tip of a crack in nonlinear (power law hardening) elastic-plastic materials where the size of the plastic zone is small compared with the crack length. Hutchinson used a material [[constitutive law]] of the form suggested by [[Ramberg-Osgood plasticity|Ramberg and Osgood]]:<ref>{{Citation|last1=Ramberg|first1=Walter|first2=William R.|last2=Osgood|title= Description of stress-strain curves by three parameters|journal= US National Advisory Committee for Aeronautics|volume=902|year=1943}}</ref>
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| :<math>\frac{\varepsilon}{\varepsilon_y}=\frac{\sigma}{\sigma_y}+\alpha\left(\frac{\sigma}{\sigma_y}\right)^n</math>
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| where ''σ'' is the [[Cauchy stress tensor|stress]] in uniaxial tension, ''σ''<sub>y</sub> is a [[yield (engineering)|yield stress]], ''ε'' is the [[infinitesimal strain theory|strain]], and ''ε''<sub>y</sub> = ''σ''<sub>y</sub>/''E'' is the corresponding yield strain. The quantity ''E'' is the elastic [[Young's modulus]] of the material. The model is parametrized by ''α'', a dimensionless constant characteristic of the material, and ''n'', the coefficient of [[work hardening]]. This model is applicable only to situations where the stress increases monotonically, the stress components remain approximately in the same ratios as loading progresses (proportional loading), and there is no [[flow plasticity theory|unloading]].
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| If a far-field tensile stress ''σ''<sub>far</sub> is applied to the body shown in the adjacent figure, the J-integral around the path Γ<sub>1</sub> (chosen to be completely inside the elastic zone) is given by
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| :<math>
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| J_{\Gamma_1} = \pi\,(\sigma_{\text{far}})^2 \,.
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| </math>
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| Since the total integral around the crack vanishes and the contributions along the surface of the crack are zero, we have
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| :<math>
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| J_{\Gamma_1} = -J_{\Gamma_2} \,.
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| </math>
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| If the path Γ<sub>2</sub> is chosen such that it is inside the fully plastic domain, Hutchinson showed that
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| :<math>
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| J_{\Gamma_2} = -\alpha\,K^{n+1}\,r^{(n+1)(s-2)+1}\,I
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| </math>
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| where ''K'' is a stress amplitude, (''r'',''θ'') is a [[polar coordinate system]] with origin at the crack tip, ''s'' is a constant determined from an asymptotic expansion of the stress field around the crack, and ''I'' is a dimensionless integral. The relation between the J-integrals around Γ<sub>1</sub> and Γ<sub>2</sub> leads to the constraint
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| :<math>
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| s = \frac{2n+1}{n+1}
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| </math>
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| and an expression for ''K'' in terms of the far-field stress
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| :<math>
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| K = \left(\frac{\beta\,\pi}{\alpha\,I}\right)^{\frac{1}{n+1}}\,(\sigma_{\text{far}})^{\frac{2}{n+1}}
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| </math>
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| where ''β'' = 1 for [[plane stress]] and ''β'' = 1 - ''ν''<sup>2</sub> for [[plane strain]] (''ν'' is the [[Poisson's ratio]]).
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| The asymptotic expansion of the stress field and the above ideas can be used to determine the stress and strain fields in terms of the J-integral:
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| :<math>\sigma_{ij}= \sigma_y \left (\frac{EJ}{r\,\alpha \sigma_y^2 I} \right )^{{1}\over{n+1}}\tilde{\sigma}_{ij}(n,\theta)</math>
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| :<math>\varepsilon_{ij}=\frac{\alpha \varepsilon_y}{E} \left (\frac{EJ}{r\,\alpha \sigma_y^2 I} \right )^{{n}\over{n+1}}\tilde{\varepsilon}_{ij}(n,\theta)</math>
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| where <math>\tilde{\sigma}_{ij} </math> and <math> \tilde{\varepsilon}_{ij}</math> are dimensionless functions.
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| These expressions indicate that ''J'' can be interpreted as a plastic analog to the [[stress intensity factor]] (''K'') that is used in linear elastic fracture mechanics, i.e., we can use a criterion such as ''J'' > ''J''<sub>Ic</sub> as a crack growth criterion.
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| ==See also==
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| * [[Fracture toughness]]
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| * [[Toughness]]
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| * [[Fracture mechanics]]
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| * [[Stress intensity factor]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| * J. R. Rice, "[http://esag.harvard.edu/rice/015_Rice_PathIndepInt_JAM68.pdf A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks]", Journal of Applied Mechanics, 35, 1968, pp. 379–386.
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| * Van Vliet, Krystyn J. (2006); "3.032 Mechanical Behavior of Materials", [http://www.stellar.mit.edu/S/course/3/fa06/3.032/index.html]
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| *[http://hdl.handle.net/1813/3075 Fracture Mechanics Notes] by Prof. Alan Zehnder (from Cornell University)
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| *[http://imechanica.org/node/755 Nonlinear Fracture Mechanics Notes] by Prof. John Hutchinson (from Harvard University)
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| *[http://imechanica.org/node/903 Notes on Fracture of Thin Films and Multilayers] by Prof. John Hutchinson (from Harvard University)
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| *[http://www.seas.harvard.edu/hutchinson/papers/416.pdf Mixed mode cracking in layered materials] by Profs. John Hutchinson and Zhigang Suo (from Harvard University)
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| *[http://www.mate.tue.nl/~piet/edu/frm/sht/bmsht.html Fracture Mechanics] by Prof. Piet Schreurs (from TU Eindhoven, Netherlands)
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| *[http://www.dsto.defence.gov.au/publications/1880/DSTO-GD-0103.pdf Introduction to Fracture Mechanics] by Dr. C. H. Wang (DSTO - Australia)
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| *[http://imechanica.org/node/2621 Fracture mechanics course notes] by Prof. Rui Huang (from Univ. of Texas at Austin)
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| *[http://www.ltas-cm3.ulg.ac.be/MECA0058-1/MecaRuptNLFMHRRField.pdf HRR solutions] by Ludovic Noels (University of Liege)
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| [[Category:Failure]]
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| [[Category:Solid mechanics]]
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| [[Category:Materials science]]
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| [[Category:Materials testing]]
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| [[Category:Mechanics]]
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