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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.
J-integral path for the DCB specimen under tensile load. Consider the double cantilever beam specimen shown in the figure, where the crack centered in the beam of height 2 h {\displaystyle 2h} has a length of a {\displaystyle a} , and a load P {\displaystyle P} is applied to open the crack.
The elastic-plastic failure parameter is designated J Ic and is conventionally converted to K Ic using the equation below. Also note that the J integral approach reduces to the Griffith theory for linear-elastic behavior. The mathematical definition of J-integral is as follows:
A path independent integral and the approximate analysis of strain concentration by notches and cracks. Department of Defense Advanced Research Projects Agency. Contract SD-86, Material Research Program, May 1967. Rice, James R. "Mathematical analysis in the mechanics of fracture." Fracture: an advanced treatise 2 (1968): 191–311.
The Integral Institute published the Journal of Integral Theory and Practice, [6] and SUNY Press has published twelve books under the "SUNY series in Integral Theory" [7] in the early 2010s, and a number of texts applying integral theory to various topics have been released by other publishers.
The net electric current I is the surface integral of the electric current density J passing through Σ: =, where dS denotes the differential vector element of surface area S, normal to surface Σ. (Vector area is sometimes denoted by A rather than S , but this conflicts with the notation for magnetic vector potential ).
If the initial momentum of an object is p 1, and a subsequent momentum is p 2, the object has received an impulse J: J = p 2 − p 1 . {\displaystyle \mathbf {J} =\mathbf {p} _{2}-\mathbf {p} _{1}.} Momentum is a vector quantity, so impulse is also a vector quantity: ∑ F × Δ t = Δ p . {\displaystyle \sum \mathbf {F} \times \Delta t=\Delta ...
Marston Morse applied calculus of variations in what is now called Morse theory. [6] Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. [6] The dynamic programming of Richard Bellman is an alternative to the calculus of variations. [7] [8] [9] [c]