# J integral

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J integral

The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material.Van Vliet, Krystyn J. (2006); "3.032 Mechanical Behavior of Materials", [http://www.stellar.mit.edu/S/course/3/fa06/3.032/index.html] ] The theoretical concept of J-integral was developed in 1967 by Cherepanov [G. P. Cherepanov, The propagation of cracks in a continuous medium, Journal of Applied Mathematics and Mechanics, 31(3), 1967, pp. 503-512.] and in 1968 by Jim RiceJ. 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.] independently, who showed that an energetic contour path integral (called J) was independent of the path around a crack.

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, and to infer a critical value of fracture energy $J_\left\{Ic\right\}$. The quantity $J_\left\{1c\right\}$ defines the point at which large-scale plastic yielding during propagation takes place under mode one loading. ] Physically the J-integral is related to the area under a curve of load versus load point displacement.Meyers and Chawla (1999): "Mechanical Behavior of Materials," 445-448.] .

Two-dimensional J-Integral

The two-dimensional J-integral was originally defined as ] (see Figure 1 for an illustration):$J := int_\left\{Gamma\right\} left\left(W~dx_2 - mathbf\left\{t\right\}cdotcfrac\left\{partialmathbf\left\{u\left\{partial x_1\right\}~ds ight\right)$where $W\left(x_1,x_2\right)$ is the strain energy density, $x_1, x_2$ are the coordinate directions, is the traction vector, $mathbf\left\{n\right\}$ is the normal to the curve $Gamma$, $sigma$ is the Cauchy stress tensor, and $mathbf\left\{u\right\}$ is the displacement vector. The strain energy density is given by:The J-Integral around a crack tip is frequently expressed in a more general form (and in index notation) as:$J_i := lim_\left\{epsilon ightarrow 0\right\} int_\left\{Gamma_epsilon\right\} left\left(W n_i - n_jsigma_\left\{jk\right\}~cfrac\left\{partial u_k\right\}\left\{partial x_i\right\} ight\right) dGamma$ where $J_i$ is the component of the J-integral for crack opening in the $x_i$ direction and $epsilon$ is a small reqion around the crack tip.Using Green's theorem we can show that this integral is zero when the boundary $Gamma$ is closed and encloses a region that contains no singularities and is simply connected. If the faces of the crack do not have any tractions on them then the J-integral is also 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 close to a crack in a nonlinear elastic 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.

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J-Integral and Fracture Toughness

The J-integral can be described as follows ]

:$J=oint_\left\{C\right\} frac \left\{F\right\}\left\{A\right\}frac\left\{du\right\}\left\{dl_0\right\}=int_\left\{\right\}^\left\{\right\}sigma dvarepsilon,$

where
* "F" is the force applied at the crack tip
* "A" is the area of the crack tip
* "$frac\left\{du\right\}\left\{dl_0\right\}$" is the change in energy per unit length
* "$sigma$" is the stress
* "$dvarepsilon$" is the change in the strain caused by the stress

Fracture toughness is then calculated from the following equation ]

:$J_\left\{1c\right\} = K_\left\{1c\right\}^2\left(frac\left\{1-v^2\right\}\left\{E\right\}\right)$

where
*"$K_\left\{1c\right\}$" is the fracture toughness in mode one loading
*"v" is the Poisson's ratio
*"E" is the Young's Modulus of the material

ee also

* Fracture toughness
* Toughness
* Fracture Mechanics

References

* 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.
* Van Vliet, Krystyn J. (2006); "3.032 Mechanical Behavior of Materials", [http://www.stellar.mit.edu/S/course/3/fa06/3.032/index.html]
* [http://hdl.handle.net/1813/3075 Fracture Mechanics Notes] by Prof. Alan Zehnder (from Cornell University)
* [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/suo/papers/17.pdf Mixed mode cracking in layered materials] by Profs. John Hutchinson and Zhigang Suo (from Harvard University)
* [http://www.mate.tue.nl/~piet/edu/frm/sht/bmsht.html Fracture Mechanics] by Prof. Piet Schreurs (from TU Eindhoven, Netherlands)
* [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)

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