TheJ-integral represents a way to calculate thestrain energy release rate, or work (energy) per unit fracture surface area, in a material.^[1] The theoretical concept of J-integral was developed in 1967 by G. P. Cherepanov^[2] and independently in 1968 byJames R. Rice,^[3] who showed that an energeticcontour path integral (calledJ) was independent of the path around acrack.

Experimental methods were developed using the integral that allowed the measurement of critical fracture properties in sample sizes that are too small for Linear ElasticFracture Mechanics (LEFM) to be valid.^[4] These experiments allow the determination offracture toughness from the critical value of fracture energyJ_Ic, which defines the point at which large-scaleplastic yielding during propagation takes place under mode I loading.^[1][5]

The J-integral is equal to thestrain energy release rate for a crack in a body subjected tomonotonic loading.^[6] This is generally true, under quasistatic conditions, only forlinear elastic materials. For materials that experience small-scaleyielding at the crack tip,J can be used to compute the energy release rate under special circumstances such as monotonic loading inmode III (antiplane shear). The strain energy release rate can also be computed fromJ for pure power-law hardeningplastic materials that undergo small-scale yielding at the crack tip.

The quantityJ is not path-independent for monotonicmode I andmode II loading of elastic-plastic materials, so only a contour very close to the crack tip gives the energy release rate. Also, Rice showed thatJ 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.

The two-dimensional J-integral was originally defined as^[3] (see Figure 1 for an illustration)

${\displaystyle J:={\int }_{\mathrm{\Gamma }}\left(W\mathrm{d}{x}_2-\mathbf{t}\cdot \frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }{x}_1}\mathrm{d}s\right)={\int }_{\mathrm{\Gamma }}\left(W\mathrm{d}{x}_2-t_i\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_1}\mathrm{d}s\right)}$

whereW(x₁,x₂) is the strain energy density,x₁,x₂ are the coordinate directions,t = [σ]n is thesurface traction vector,n is the normal to the curve Γ, [σ] is theCauchy stress tensor, andu is thedisplacement vector. The strain energy density is given by

${\displaystyle W={\int }_0^{[\epsilon ]}[\mathit{\sigma }]:d[\mathit{\epsilon }];[\mathit{\epsilon }]=\frac{1}{2}\left[\mathbf{\nabla }\mathbf{u}+(\mathbf{\nabla }\mathbf{u}{)}^T\right].}$

The J-integral around a crack tip is frequently expressed in a more general form^[7] (and inindex notation) as

${\displaystyle J_i:=\underset{\epsilon \to 0}{lim}{\int }_{{\mathrm{\Gamma }}_{\epsilon }}\left(W(\mathrm{\Gamma })n_i-n_j{\sigma }_{jk}\frac{\mathrm{\partial }{u}_k(\mathrm{\Gamma },x_i)}{\mathrm{\partial }{x}_i}\right)d\mathrm{\Gamma }}$

where${\displaystyle J_i}$ is the component of the J-integral for crack opening in the${\displaystyle x_i}$ direction and${\displaystyle \epsilon }$ is a small region around the crack tip.UsingGreen's theorem we can show that this integral is zero when the boundary${\displaystyle \mathrm{\Gamma }}$ is closed and encloses a region that contains nosingularities and issimply connected. If the faces of the crack do not have anysurface tractions on them then the J-integral is alsopath 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 thestress close to a crack in a nonlinearelastic or elastic-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.

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${\displaystyle J}$ is zero over a closed contour in a simply connected domain. Let us just consider the expression for${\displaystyle J_1}$ which is ${\displaystyle J_1:={\int }_{\mathrm{\Gamma }}\left(W{n}_1-n_j{\sigma }_{jk}\frac{\mathrm{\partial }{u}_k}{\mathrm{\partial }{x}_1}\right)d\mathrm{\Gamma }}$ We can write this as ${\displaystyle J_1={\int }_{\mathrm{\Gamma }}\left(W{\delta }_{1j}-{\sigma }_{jk}\frac{\mathrm{\partial }{u}_k}{\mathrm{\partial }{x}_1}\right)n_{j}d\mathrm{\Gamma }}$ FromGreen's theorem (or the two-dimensionaldivergence theorem) we have ${\displaystyle {\int }_{\mathrm{\Gamma }}{f}_j{n}_{j}d\mathrm{\Gamma }={\int }_A\frac{\mathrm{\partial }{f}_j}{\mathrm{\partial }{x}_j}dA}$ Using this result we can express${\displaystyle J_1}$ as where${\displaystyle A}$ is the area enclosed by the contour${\displaystyle \mathrm{\Gamma }}$. Now, if there areno body forces present, equilibrium (conservation of linear momentum) requires that ${\displaystyle \mathbf{\nabla }\cdot \mathit{\sigma }=\mathbf{0}⟹\frac{\mathrm{\partial }{\sigma }_{jk}}{\mathrm{\partial }{x}_j}=0.}$ Also, ${\displaystyle [\mathit{\epsilon }]=\frac{1}{2}\left[\mathbf{\nabla }\mathbf{u}+(\mathbf{\nabla }\mathbf{u}{)}^T\right]⟹{\epsilon }_{jk}=\frac{1}{2}\left(\frac{\mathrm{\partial }{u}_k}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_k}\right).}$ Therefore, ${\displaystyle {\sigma }_{jk}\frac{\mathrm{\partial }{\epsilon }_{jk}}{\mathrm{\partial }{x}_1}=\frac{1}{2}\left({\sigma }_{jk}\frac{{\mathrm{\partial }}^2{u}_k}{\mathrm{\partial }{x}_1\mathrm{\partial }{x}_j}+{\sigma }_{jk}\frac{{\mathrm{\partial }}^2{u}_j}{\mathrm{\partial }{x}_1\mathrm{\partial }{x}_k}\right)}$ From the balance of angular momentum we have${\displaystyle {\sigma }_{jk}={\sigma }_{kj}}$. Hence, ${\displaystyle {\sigma }_{jk}\frac{\mathrm{\partial }{\epsilon }_{jk}}{\mathrm{\partial }{x}_1}={\sigma }_{jk}\frac{{\mathrm{\partial }}^2{u}_j}{\mathrm{\partial }{x}_1\mathrm{\partial }{x}_k}}$ The J-integral may then be written as ${\displaystyle J_1={\int }_A\left[\frac{\mathrm{\partial }W}{\mathrm{\partial }{x}_1}-{\sigma }_{jk}\frac{\mathrm{\partial }{\epsilon }_{jk}}{\mathrm{\partial }{x}_1}\right]dA}$ Now, for an elastic material the stress can be derived from the stored energy function${\displaystyle W}$ using ${\displaystyle {\sigma }_{jk}=\frac{\mathrm{\partial }W}{\mathrm{\partial }{\epsilon }_{jk}}}$ Then, if the elastic modulus tensor is homogeneous, using thechain rule of differentiation, ${\displaystyle {\sigma }_{jk}\frac{\mathrm{\partial }{\epsilon }_{jk}}{\mathrm{\partial }{x}_1}=\frac{\mathrm{\partial }W}{\mathrm{\partial }{\epsilon }_{jk}}\frac{\mathrm{\partial }{\epsilon }_{jk}}{\mathrm{\partial }{x}_1}=\frac{\mathrm{\partial }W}{\mathrm{\partial }{x}_1}}$ Therefore, we have${\displaystyle J_1=0}$ for a closed contour enclosing a simply connected region without any elastic inhomogeneity, such as voids and cracks.

Proof that the J-integral is path-independent

Consider the contour${\displaystyle \mathrm{\Gamma }={\mathrm{\Gamma }}_1+{\mathrm{\Gamma }}^{+}+{\mathrm{\Gamma }}_2+{\mathrm{\Gamma }}^{-}}$. Since this contour is closed and encloses a simply connected region, the J-integral around the contour is zero, i.e. ${\displaystyle J=J_{(1)}+J^{+}-J_{(2)}-J^{-}=0}$ assuming that counterclockwise integrals around the crack tip have positive sign. Now, since the crack surfaces are parallel to the${\displaystyle x_1}$ axis, the normal component${\displaystyle n_1=0}$ on these surfaces. Also, since the crack surfaces are traction free,${\displaystyle t_k=0}$. Therefore, ${\displaystyle J^{+}=J^{-}={\int }_{\mathrm{\Gamma }}\left(W{n}_1-t_k\frac{\mathrm{\partial }{u}_k}{\mathrm{\partial }{x}_1}\right)d\mathrm{\Gamma }=0}$ Therefore, ${\displaystyle J_{(1)}=J_{(2)}}$ and the J-integral is path independent.

For isotropic, perfectly brittle, linear elastic materials, the J-integral can be directly related to thefracture toughness if the crack extends straight ahead with respect to its original orientation.^[6]

For plane strain, underMode I loading conditions, this relation is

${\displaystyle J_{\mathrm{I}\mathrm{c}}=G_{\mathrm{I}\mathrm{c}}=K_{\mathrm{I}\mathrm{c}}^2\left(\frac{1-{\nu }^2}{E}\right)}$

where${\displaystyle G_{\mathrm{I}\mathrm{c}}}$ is the critical strain energy release rate,${\displaystyle K_{\mathrm{I}\mathrm{c}}}$ is the fracture toughness in Mode I loading,${\displaystyle \nu }$ is the Poisson's ratio, andE is theYoung's modulus of the material.

ForMode II loading, the relation between the J-integral and the mode II fracture toughness (${\displaystyle K_{\mathrm{I}\mathrm{I}\mathrm{c}}}$) is

${\displaystyle J_{\mathrm{I}\mathrm{I}\mathrm{c}}=G_{\mathrm{I}\mathrm{I}\mathrm{c}}=K_{\mathrm{I}\mathrm{I}\mathrm{c}}^2\left[\frac{1-{\nu }^2}{E}\right]}$

ForMode III loading, the relation is

${\displaystyle J_{\mathrm{I}\mathrm{I}\mathrm{I}\mathrm{c}}=G_{\mathrm{I}\mathrm{I}\mathrm{I}\mathrm{c}}=K_{\mathrm{I}\mathrm{I}\mathrm{I}\mathrm{c}}^2\left(\frac{1+\nu }{E}\right)}$

Hutchinson, Rice and Rosengren^[8][9] subsequently showed that J characterizes thesingular 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 materialconstitutive law of the form suggested byW. Ramberg and W. Osgood:^[10]

${\displaystyle \frac{\epsilon }{{\epsilon }_y}=\frac{\sigma }{{\sigma }_y}+\alpha {\left(\frac{\sigma }{{\sigma }_y}\right)}^n}$

whereσ is thestress in uniaxial tension,σ_y is ayield stress,ε is thestrain, andε_y =σ_y/E is the corresponding yield strain. The quantityE is the elasticYoung's modulus of the material. The model is parametrized byα, a dimensionless constant characteristic of the material, andn, the coefficient ofwork 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 nounloading.

If a far-field tensile stressσ_far is applied to the body shown in the adjacent figure, the J-integral around the path Γ₁ (chosen to be completely inside the elastic zone) is given by

${\displaystyle J_{{\mathrm{\Gamma }}_1}=\pi ({\sigma }_{\text{far}}{)}^2.}$

Since the total integral around the crack vanishes and the contributions along the surface of the crack are zero, we have

${\displaystyle J_{{\mathrm{\Gamma }}_1}=-J_{{\mathrm{\Gamma }}_2}.}$

If the path Γ₂ is chosen such that it is inside the fully plastic domain, Hutchinson showed that

${\displaystyle J_{{\mathrm{\Gamma }}_2}=-\alpha K^{n+1}{r}^{(n+1)(s-2)+1}I}$

whereK is a stress amplitude, (r,θ) is apolar coordinate system with origin at the crack tip,s is a constant determined from an asymptotic expansion of the stress field around the crack, andI is a dimensionless integral. The relation between the J-integrals around Γ₁ and Γ₂ leads to the constraint

${\displaystyle s=\frac{2n+1}{n+1}}$

and an expression forK in terms of the far-field stress

${\displaystyle K={\left(\frac{\beta \pi }{\alpha I}\right)}^{\frac{1}{n+1}}({\sigma }_{\text{far}}{)}^{\frac{2}{n+1}}}$

whereβ = 1 forplane stress andβ = 1 −ν² forplane strain (ν is thePoisson'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:

${\displaystyle {\sigma }_{ij}={\sigma }_y{\left(\frac{EJ}{r\alpha {\sigma }_y^{2}I}\right)}^{\frac{1}{n+1}}{\tilde{\sigma }}_{ij}(n,\theta )}$

${\displaystyle {\epsilon }_{ij}=\frac{\alpha {\epsilon }_y}{E}{\left(\frac{EJ}{r\alpha {\sigma }_y^{2}I}\right)}^{\frac{n}{n+1}}{\tilde{\epsilon }}_{ij}(n,\theta )}$

where${\displaystyle {\tilde{\sigma }}_{ij}}$ and${\displaystyle {\tilde{\epsilon }}_{ij}}$ are dimensionless functions.

These expressions indicate thatJ can be interpreted as a plastic analog to thestress intensity factor (K) that is used in linear elastic fracture mechanics, i.e., we can use a criterion such asJ >J_Ic as a crack growth criterion.

• Fracture toughness
• Toughness
• Fracture mechanics
• Stress intensity factor
• Nature of the slip band local field

• 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.
• Van Vliet, Krystyn J. (2006); "3.032 Mechanical Behavior of Materials",[2]
• X. Chen (2014), "Path-Independent Integral", In: Encyclopedia of Thermal Stresses, edited by R. B. Hetnarski, Springer,ISBN 978-9400727380.
• Nonlinear Fracture Mechanics Notes by Prof. John Hutchinson (from Harvard University)
• Notes on Fracture of Thin Films and Multilayers by Prof. John Hutchinson (from Harvard University)
• Mixed mode cracking in layered materials by Profs. John Hutchinson and Zhigang Suo (from Harvard University)
• Fracture Mechanics by Piet Schreurs (from TU Eindhoven, The Netherlands)
• Introduction to Fracture Mechanics by Dr. C. H. Wang (DSTO - Australia)
• Fracture mechanics course notes by Prof. Rui Huang (from Univ. of Texas at Austin)
• HRR solutions by Ludovic Noels (University of Liege)

• Failure
• Solid mechanics
• Materials testing
• Mechanics
• Fracture mechanics

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