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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:
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.
This means j provides a bijection from the set of elliptic curves over C to the complex plane. [2] As a Riemann surface, the fundamental region has genus 0, and every modular function is a rational function in j; and, conversely, every rational function in j is a modular function. In other words, the field of modular functions is C(j).
Linear elastic theory predicts that the stress distribution near the crack tip, in polar coordinates (,) with origin at the crack tip, has the form [4] σ i j ( r , θ ) = K 2 π r f i j ( θ ) + h i g h e r o r d e r t e r m s {\displaystyle \sigma _{ij}(r,\theta )={\frac {K}{\sqrt {2\pi r}}}\,f_{ij}(\theta )+\,\,{\rm {higher\,order\,terms}}}
From January 2008 to December 2012, if you bought shares in companies when James D. Plummer joined the board, and sold them when he left, you would have a -21.5 percent return on your investment, compared to a -2.8 percent return from the S&P 500.
The Kirkwood–Buff (KB) solution theory, due to John G. Kirkwood and Frank P. Buff, links macroscopic (bulk) properties to microscopic (molecular) details. Using statistical mechanics , the KB theory derives thermodynamic quantities from pair correlation functions between all molecules in a multi-component solution. [ 1 ]