GENERAL INTEGRAL EQUATIONS OF MICROMECHANICS OF HETEROGENEOUS MATERIALS

2015 ◽  
Vol 13 (1) ◽  
pp. 11-53 ◽  
Author(s):  
Valeriy A. Buryachenko
Keyword(s):  
Integral Equations ◽  
Heterogeneous Materials ◽  
General Integral

Micromechanics of Random Heterogeneous Materials: New Background, Opportunities and Prospects

2016 ◽  
Author(s):  
Valeriy A. Buryachenko
Keyword(s):  
Integral Equations ◽  
Heterogeneous Media ◽  
Effective Field ◽  
Local Fields ◽  
Homogeneous Field ◽  
Multi Scale ◽  
General Integral ◽  
Operator Form ◽  
Mechanics Of Heterogeneous Media ◽  
Improved Accuracy

One considers a linear heterogeneous media (e.g. filtration in porous media, composite materials, CMs, nanocomposites, peristatic CMs, and rough contacted surfaces). The idea of the effective field hypothesis (EFH, H1, see for references and details [1], [2]) dates back to Mossotti (1850) who pioneered the introduction of the effective field concept as a local homogeneous field acting on the inclusions and differing from the applied macroscopic one. It is proved that a concept of the EFH (even if this term is not mentioned) is a (first) background of all four groups of analytical methods in physics and mechanics of heterogeneous media (model methods, perturbation methods, self-consistent methods, and variational ones, see for refs. [1]). An operator form of the general integral equations (GIEs) is obtained which connects the driving fields and fluxes in a point being considered. Either the volume integral equations or boundary ones are used for these GIEs, new concept of the interface polarisation tensors are introduced. New GIEs present in fact the new (second) background (which does not use the EFH) of multi-scale analysis offering the opportunities for a fundamental jump in multiscale research of random heterogeneous media with drastically improved accuracy of local field estimations (with possible change of sign of predicted local fields).

A Note on the Integral Equation Formulation of Dynamical Theory

1972 ◽  
Vol 27 (3) ◽  
pp. 434-436 ◽  
Author(s):  
Jon Gjønnes
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Perturbation Methods ◽  
Dynamical Theory ◽  
Forward Scattering ◽  
Equation Formulation ◽  
General Integral ◽  
Scattering Approximation ◽  
Integral Equation Formulation

AbstractThe coupled integral equations for dynamical scattering are developed from the general integral equation. The results are given in the forward scattering approximation. Extension to bade scattering is briefly mentioned. Expressions for distorted crystals are derived both in the column approximation and beyond. The formulation is suggested to be very useful as a basis for perturbation methods.

scholarly journals Integral equations for Lamé functions

1942 ◽  
Vol 7 (1) ◽  
pp. 3-15 ◽  
Author(s):  
A. Erdélyi
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Integral Representations ◽  
Linear Differential Equations ◽  
Mathieu Functions ◽  
General Integral ◽  
Mathieu’S Equation ◽  
Linear Differential ◽  
Heun’S Equation ◽  
Ordinary Linear Differential Equations

In the theory of ordinary linear differential equations with three regular singularities and in the theory of their special and limiting cases, integral representations of the solutions are known to be very important. It seems that there is no corresponding simple integral representation of the solutions of ordinary linear differential equations with four regular singularities (Heun's equation) or of particular (e.g. Lamé's equation) or limiting (e.g. Mathieu's equation) cases of such equations. It has been suggested (Whittaker 1915 c) that the theorems corresponding in these latter cases to integral representations of the hypergeometric functions involve integral equations of the second kind. Such integral equations have been discovered for Mathieu functions (Whittaker 1912, cf. also Whittaker and Watson 1927 pp. 407–409 and 426) as well as for Lame functions (Whittaker 1915 a and b, cf. also Whittaker and Watson 1927 pp. 564–567) and polynomial or “quasi-algebraic” solutions of Heun's equation (Lambe and Ward 1934). Ince (1921–22) investigated general integral equations connected with periodic solutions of linear differential equations.

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