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{{Short description|Calculation of strain energy release rate}}
{{continuum mechanics|cTopic=[[Solid mechanics]]}}
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>[https://web.archive.org/web/20191217044608/http://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 G. P. 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 independently in 1968 by [[James R. 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> who showed that an energetic [[contour integral|contour path integral]] (called ''J'') was independent of the path around a [[Fracture|crack]].

Experimental methods were developed using the integral that allowed the measurement of critical fracture properties in sample sizes that are too small for Linear Elastic [[Fracture Mechanics]] (LEFM) to be valid.<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. [https://dx.doi.org/10.5254/1.3536150]</ref> These experiments allow the determination of [[fracture toughness]] from the critical value of fracture energy ''J''<sub>Ic</sub>, which defines the point at which large-scale [[plasticity (physics)|plastic]] yielding during propagation takes place under mode I loading.<ref name="VanVliet" /><ref name=Meyers>Meyers and Chawla (1999): "Mechanical Behavior of Materials," 445–448.</ref>

The J-integral is equal to the [[strain energy release rate]] for a crack in a body subjected to [[Monotonic function|monotonic]] loading.<ref name="Yoda80">Yoda, M., 1980, ''The J-integral fracture toughness for Mode II'', Int. J. 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.

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.

== Two-dimensional J-integral ==
[[File:Integrale-J 2D.png|200px|right|thumb|Figure 1. Line J-integral around a notch in two dimensions.]]
The two-dimensional J-integral was originally defined as<ref name="Rice68" /> (see Figure 1 for an illustration)
:<math>
   J := \int_\Gamma \left(W~\mathrm{d}x_2 - \mathbf{t}\cdot\cfrac{\partial\mathbf{u}}{\partial x_1}~\mathrm{d}s\right)
        = \int_\Gamma \left(W~\mathrm{d}x_2 - t_i\,\cfrac{\partial u_i}{\partial x_1}~\mathrm{d}s\right)
 </math>
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'''&nbsp;=&nbsp;['''σ''']'''n''' 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
:<math>
  W = \int_0^{[\varepsilon]} [\boldsymbol{\sigma}]:d[\boldsymbol{\varepsilon}] ~;~~ 
  [\boldsymbol{\varepsilon}] = \tfrac{1}{2}\left[\boldsymbol{\nabla}\mathbf{u}+(\boldsymbol{\nabla}\mathbf{u})^T\right] ~.
 </math>
The J-integral around a crack tip is frequently expressed in a more general form<ref name="COMSOL Doc">{{cite web |title=J-integral Theory |url=https://web.archive.org/web/20250502151359/https://doc.comsol.com/5.6/doc/com.comsol.help.sme/sme_ug_theory.06.70.html |website=COMSOL Documentation |publisher=COMSOL, Inc. |access-date=2 May 2025}}</ref> (and in [[Einstein notation|index notation]]) as 
:<math>
   J_i := \lim_{\varepsilon\rightarrow 0} \int_{\Gamma_\varepsilon} \left(W(\Gamma) n_i - n_j\sigma_{jk}~\cfrac{\partial u_k(\Gamma, x_i)}{\partial x_i}\right) \, d\Gamma
 </math> 
where <math>J_i</math> is the component of the J-integral for crack opening in the <math>x_i</math> direction and <math>\varepsilon</math> is a small region around the crack tip.
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]].

Rice also showed that the value of the J-integral represents the energy release rate for planar crack growth.
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.

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Proof that the J-integral is zero over a closed path
|-
|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
:<math>
   J_1 := \int_\Gamma \left(W n_1 - n_j\sigma_{jk}~\cfrac{\partial u_k}{\partial x_1}\right) \, d\Gamma
 </math>
We can write this as
:<math>
   J_1 = \int_\Gamma \left(W \delta_{1j} - \sigma_{jk}~\cfrac{\partial u_k}{\partial x_1}\right)n_j \, d\Gamma
 </math>
From [[Green's theorem]] (or the two-dimensional [[divergence theorem]]) we have
:<math>
  \int_{\Gamma} f_j~n_j~d\Gamma = \int_A \cfrac{\partial f_j}{\partial x_j}~dA
 </math>
Using this result we can express <math>J_1</math> as
:<math>
   \begin{align}
   J_1 & = \int_A \cfrac{\partial}{\partial x_j}\left(W \delta_{1j} - \sigma_{jk}~\cfrac{\partial u_k}{\partial x_1}\right) dA \\
     & = \int_A \left[\cfrac{\partial W}{\partial x_1} - 
                \cfrac{\partial\sigma_{jk}}{\partial x_j}~\cfrac{\partial u_k}{\partial x_1} -
                \sigma_{jk}~\cfrac{\partial^2 u_k}{\partial x_1 \partial x_j}\right]~dA
   \end{align}
 </math>
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
:<math>
   \boldsymbol{\nabla}\cdot\boldsymbol{\sigma} = \mathbf{0} \qquad \implies \qquad  \cfrac{\partial\sigma_{jk}}{\partial x_j} = 0 ~.
 </math>
Also,
:<math>
   [\boldsymbol{\varepsilon}] = \tfrac{1}{2}\left[\boldsymbol{\nabla}\mathbf{u}+(\boldsymbol{\nabla}\mathbf{u})^T\right] 
  \qquad \implies \qquad
  \varepsilon_{jk} = \tfrac{1}{2}\left(\cfrac{\partial u_k}{\partial x_j} + \cfrac{\partial u_j}{\partial x_k}\right) ~.
 </math>
Therefore,
:<math>
   \sigma_{jk}\cfrac{\partial\varepsilon_{jk}}{\partial x_1} = 
    \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)
 </math>
From the balance of angular momentum we have <math>\sigma_{jk} = \sigma_{kj}</math>.  Hence, 
:<math>
   \sigma_{jk}\cfrac{\partial\varepsilon_{jk}}{\partial x_1} = 
    \sigma_{jk}\cfrac{\partial^2 u_j}{\partial x_1 \partial x_k}
 </math>
The J-integral may then be written as
:<math>
   J_1 = \int_A \left[\cfrac{\partial W}{\partial x_1} - 
                \sigma_{jk}~\cfrac{\partial\varepsilon_{jk}}{\partial x_1}\right]~dA
 </math>
Now, for an elastic material the stress can be derived from the stored energy function <math>W</math> using 
:<math>
   \sigma_{jk} = \cfrac{\partial W}{\partial\varepsilon_{jk}}
 </math>
Then, if the elastic modulus tensor is homogeneous, using the [[chain rule]] of differentiation,
:<math>
   \sigma_{jk}~\cfrac{\partial\varepsilon_{jk}}{\partial x_1} = \cfrac{\partial W}{\partial\varepsilon_{jk}}~\cfrac{\partial\varepsilon_{jk}}{\partial x_1} = \cfrac{\partial W}{\partial x_1}
 </math>
Therefore, we have <math>J_1 = 0 </math> for a closed contour enclosing a simply connected region without any elastic inhomogeneity, such as voids and cracks.
|}

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Proof that the J-integral is path-independent
|-
|[[Image:J integralPathIndep.png|200px|right|thumb|Figure 2. Integration paths around a notch in two dimensions.]]
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.
:<math>
   J = J_{(1)} + J^{+} - J_{(2)} - J^{-} = 0
 </math>
assuming that counterclockwise integrals around the crack tip have positive sign.  Now, since the crack surfaces are parallel to the <math>x_1</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,
:<math>
  J^{+} = J^{-} = \int_{\Gamma} \left(W n_1 - t_k~\cfrac{\partial u_k}{\partial x_1}\right) \, d\Gamma = 0
 </math>
Therefore,
:<math>
  J_{(1)} = J_{(2)}
 </math>
and the J-integral is path independent.
|}

==J-integral and fracture toughness==

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" />

For plane strain, under [[fracture mechanics|Mode I]] loading conditions, this relation is
:<math>
  J_{\rm Ic} = G_{\rm Ic} = K_{\rm Ic}^2 \left(\frac{1-\nu^2}{E}\right)
</math>
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.

For [[fracture mechanics|Mode II]] loading, the relation between the J-integral and the mode II fracture toughness (<math>K_{\rm IIc}</math>) is
:<math>
  J_{\rm IIc} = G_{\rm IIc} = K_{\rm IIc}^2 \left[\frac{1-\nu^2}{E}\right]
</math>

For [[fracture mechanics|Mode III]] loading, the relation is 
:<math>
  J_{\rm IIIc} = G_{\rm IIIc} = K_{\rm IIIc}^2 \left(\frac{1+\nu}{E}\right)
</math>

== Elastic-plastic materials and the HRR solution ==
[[File:HRRSingularity plain.svg|upright=1.15|thumb|Paths for J-integral calculation around a crack in a two-dimensional elastic-plastic material.]]
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|doi=10.1016/0022-5096(68)90014-8|bibcode= 1968JMPSo..16...13H }}</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|archive-url=https://web.archive.org/web/20130904065144/http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0659300|url-status=dead|archive-date=September 4, 2013|doi=10.1016/0022-5096(68)90013-6|bibcode=1968JMPSo..16....1R}}</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 relationship|W. Ramberg and W. 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>
:<math>\frac{\varepsilon}{\varepsilon_y}=\frac{\sigma}{\sigma_y}+\alpha\left(\frac{\sigma}{\sigma_y}\right)^n</math>
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]].

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
:<math>
    J_{\Gamma_1} = \pi\,(\sigma_{\text{far}})^2 \,.
 </math>
Since the total integral around the crack vanishes and the contributions along the surface of the crack are zero, we have
:<math>
   J_{\Gamma_1} = -J_{\Gamma_2} \,.
 </math>
If the path Γ<sub>2</sub> is chosen such that it is inside the fully plastic domain, Hutchinson showed that
:<math>
   J_{\Gamma_2} = -\alpha\,K^{n+1}\,r^{(n+1)(s-2)+1}\,I
 </math>
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
:<math>
   s = \frac{2n+1}{n+1} 
 </math>
and an expression for ''K'' in terms of the far-field stress
:<math>
  K = \left(\frac{\beta\,\pi}{\alpha\,I}\right)^{\frac{1}{n+1}}\,(\sigma_{\text{far}})^{\frac{2}{n+1}}
 </math>
where ''β'' = 1 for [[plane stress]] and ''β'' = 1 − ''ν''<sup>2</sup> for [[plane strain]] (''ν'' is the [[Poisson's ratio]]).

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:
:<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>
:<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>
where <math>\tilde{\sigma}_{ij} </math> and <math> \tilde{\varepsilon}_{ij}</math> are dimensionless functions.

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.

==See also==
* [[Fracture toughness]]
* [[Toughness]]
* [[Fracture mechanics]]
* [[Stress intensity factor]]
* [[Slip bands in metals#Nature of the non-cyclic slip band local field|Nature of the slip band local field]]

==References==
{{reflist}}

==External links==
* 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.&nbsp;379–386.
* Van Vliet, Krystyn J. (2006); "3.032 Mechanical Behavior of Materials", [https://web.archive.org/web/20191217044608/http://stellar.mit.edu/S/course/3/fa06/3.032/index.html]
* X. Chen (2014), "Path-Independent Integral", In: Encyclopedia of Thermal Stresses, edited by  R. B. Hetnarski, Springer, {{ISBN|978-9400727380}}.
*[http://imechanica.org/node/755 Nonlinear Fracture Mechanics Notes] by Prof. John Hutchinson (from Harvard University)
*[http://imechanica.org/node/903 Notes on Fracture of Thin Films and Multilayers] by Prof. John Hutchinson (from Harvard University)
*[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)
*[https://pjgschreurs.nl/htm/psfrm.html Fracture Mechanics] by Piet Schreurs (from TU Eindhoven, The Netherlands)
*[https://web.archive.org/web/20080916040340/http://www.dsto.defence.gov.au/publications/1880/DSTO-GD-0103.pdf Introduction to Fracture Mechanics] by Dr. C. H. Wang (DSTO - Australia)
*[http://imechanica.org/node/2621 Fracture mechanics course notes] by Prof. Rui Huang (from Univ. of Texas at Austin)
*[http://www.ltas-cm3.ulg.ac.be/MECA0058-1/MecaRuptNLFMHRRField.pdf HRR solutions] by Ludovic Noels (University of Liege)

{{Topics in continuum mechanics}}

[[Category:Failure]]
[[Category:Solid mechanics]]
[[Category:Materials testing]]
[[Category:Mechanics]]
[[Category:Fracture mechanics]]