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The J-integral represents a way to calculate the strain energy release rate, or work 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 by James R. Rice, [3] who showed that an energetic contour path integral (called J) was independent of the path around a crack.
A discontinuity may exist as a single feature (e.g. fault, isolated joint or fracture) and in some circumstances, a discontinuity is treated as a single discontinuity although it belongs to a discontinuity set, in particular if the spacing is very wide compared to the size of the engineering application or to the size of the geotechnical unit.
Strain energy release rate per unit fracture surface area is calculated by J-integral method which is a contour path integral around the crack tip where the path begins and ends on either crack surfaces. J-toughness value signifies the resistance of the material in terms of amount of stress energy required for a crack to grow.
The first integral is over the surface of the material, and the second is over its volume . The figure on the right shows the plot of an external force P {\displaystyle P} vs. the load-point displacement q {\displaystyle q} , in which the area under the curve is the strain energy.
Toughness is the strength with which the material opposes rupture. One definition of material toughness is the amount of energy per unit volume that a material can absorb before rupturing . This measure of toughness is different from that used for fracture toughness , which describes the capacity of materials to resist fracture. [ 2 ]
For most normal-scale applications to metals and fine-grained ceramics, except for micrometer scale devices, the size is large enough for the Weibull theory to apply (but not for coarse-grained materials such as concrete). From Eq. 2 one can show that the mean strength and the coefficient of variation of strength are obtained as follows:
Hence the integral is path independent and the compatibility condition is sufficient to ensure a unique field, provided that the body is simply connected. Compatibility of the deformation gradient [ edit ]
The concept can also be applied to materials that exhibit small-scale yielding at a crack tip. The magnitude of K depends on specimen geometry, the size and location of the crack or notch, and the magnitude and the distribution of loads on the material. It can be written as: [2] [3] = (/)