- J integral
integralrepresents 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 . The quantity 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.] .
The two-dimensional J-integral was originally defined as ] (see Figure 1 for an illustration):where is the strain energy density, are the coordinate directions, is the traction vector, is the normal to the curve , is the Cauchy stress tensor, and 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: where is the component of the J-integral for crack opening in the direction and is a small reqion around the crack tip.Using
Green's theoremwe can show that this integral is zero when the boundary is closed and encloses a region that contains no singularitiesand is simply connected. If the faces of the crack do not have any tractionson 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
stressclose 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.
J-Integral and Fracture Toughness
The J-integral can be described as follows ]
* "F" is the force applied at the crack tip
* "A" is the area of the crack tip
* "" is the change in energy per unit length
* "" is the stress
* "" is the change in the strain caused by the stress
Fracture toughness is then calculated from the following equation ]
*"" is the fracture toughness in mode one loading
*"v" is the Poisson's ratio
*"E" is the Young's Modulus of the material
* 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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