Publisher Summary In recent years, the interest in the problem of brittle fracture and, in particular, in the theory of cracks has grown appreciably in connection with various technical applications. Numerous investigations have been carried out, enlarging in essential points the classical concepts of cracks and methods of analysis. The qualitative features of the problems of cracks, associated with their peculiar nonlinearity as revealed in these investigations, makes the theory of cracks stand out distinctly from the whole range of problems in terms of the theory of elasticity. The chapter presents a unified view of the way basic problems in the theory of equilibrium cracks are formulated and discusses the results obtained thereby. The object of the theory of equilibrium cracks is the study of the equilibrium of solids in the presence of cracks. However, there exists a fundamental distinction between these two problems, The form of a cavity undergoes only slight changes even under a considerable variation in the load acting on a body, while the cracks whose surface also constitutes a part of the body boundary can expand even with small increase of the load to which the body is subjected.

Singular Integral Equations

The mathematical theory of equilibrium cracks in brittle fracture

In this paper a pair of Gauss-Chebyshev integration formulas for singular integrals are developed. Using these formulas a simple numerical method for solving a system of singular integral equations is described. To demonstrate the effectiveness of the method, a numerical example is given. In order to have a basis of comparison, the example problem is solved also by using an alternate method.

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On the numerical solution of singular integral equations

What is a classical r-matrix?

This work investigates how magnetic reconnection affects the acceleration of coronal mass ejections (CMEs) and how the acceleration in turn affects the reconnection pro-cess. To model the CME process, we use a two-dimensional flux rope model, which drives the ejection by means of a catastrophic loss of mechanical equilibrium. Our model provides a method for relating the motion of the ejected material to the reconnection rate in the current sheet created by the erupting field. In the complete absence of reconnection the tension force associated with the current sheet is always strong enough to prevent the flux rope from escaping from the Sun. However, our results imply that even a fairly small reconnection rate is sufficient to allow the flux rope to escape. Specifically, for a coronal density model that decreases exponentially with height we find that average Alfven Mach number M-A for the inflow into the reconnection site can be as small as M-A = 0.005 and still be fast enough to give a plausible eruption. The best fit to observations is obtained by assuming an inflow rate on the order of M-A approximate to 0.1. With this value the energy output matches the temporal behavior inferred for the long duration events often associated with CMEs. The model also suggests an explanation for the peculiar motion of giant X-ray arches reported by Svestka et al. [1995, 1997]. X-ray arches are the large loops associated with CMEs which are similar in form to "post"-flare loops, but they have an upward motion that is often different. Instead of continually slowing with time, the arches move upward at a rate that remains nearly constant or may even increase with time. Here we show how the difference can be explained by reversal of the gradient of the coronal Alfven speed with height.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/1999JA900477

Effects of reconnection on the coronal mass ejection process

The analogy between potential theory and classical elasticity suggests an extension of the powerful method of integral equations to the boundary value problems of elasticity. A vector boundary formula relating the boundary values of displacement and traction for the general equilibrated stress state is derived. The vector formula itself is shown to generate integral equations for the solution of the traction, displacement, and mixed boundary value problems of plane elasticity. However, an outstanding conceptual advantage of the formulation is that it is not restricted to two dimensions. This distinguishes it from the methods of Muskhelishvili and most other familiar integral equation methods. The presented approach is a real variable one and is applicable, without inherent restriction, to multiply connected domains. More precisely, no difficulty of the order of determining a mapping function is present and unwanted Volterra type dislocation solutions are eliminated a priori. An indication of techniques necessary to effect numerical solution of the resulting integral equations is presented with numerical data from a set of test problems.

/pdf/an-integral-equation-approach-to-boundary-value-problems-of-57bd9bqjh6.pdf

Singular Integral Equations

Citations

The mathematical theory of equilibrium cracks in brittle fracture

On the numerical solution of singular integral equations

What is a classical r-matrix?

Effects of reconnection on the coronal mass ejection process

An integral equation approach to boundary value problems of classical elastostatics

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