Gradient model in the problem of stress-concentration around a circular hole in two-component stochastic composites

Abstract

The gradient model of stochastically inhomogeneous media is used to study the stress concentration around a circular hole in a two-component elastic composite. The study is based on a general solution of the system of equilibrium equations expressed in terms of harmonic functions and functions that satisfy the Helmholtz equation. This solution is used to solve problems for an infinite plane with a circular hole under uniform and uniaxial tension. The results obtained are compared with the solutions found using the theory of effective moduli, which is simpler Introduction. Wide application of composite materials in various fields of modern engineering and construction calls for comprehensive analysis of problems associated with the selection of components and structure of materials, modeling and prediction of their mechanical behavior and properties, and calculation of the load-bearing capacity of structures. Such studies are based on certain mathematical models that allow for specific features of materials and describe the behavior of loads at the macro, micro, and nano-levels [1,3,5,10,11,16,17,19,20]. One of the first and the most developed and widely used continuum theories in the mechanics of composites is the theory of effective moduli, which deals with dynamic and kinematic parameters averaged over elementary macrovolumes and macroareas. Effective constants in this theory are determined in necessarily simple and feasible experimental programs or from some mathematical models. Either deterministic [2,12,13] or stochastic [15,18] approaches can be used. A shortage of this theory is that it disregards gradients of external loads at distances comparable with the dimensions of structural members.Mixture theory [6, 8] is more exact and free from this shortage. It deals with dynamic and kinematic parameters averaged over components of elementary macrovolumes and macroareas. However, here the number of constants relating the dynamic and kinematic parameters is much greater than in the theory of effective moduli, and there is a need to substantiate the equations of state and to correctly determine (experimentally and theoretically) the corresponding constants for composites. It should be noted that the concept of a mechanical two-component mixture applied to describe the mechanical behavior of coupled charges in a dielectric [9] makes it possible to set up coupled equations of electromagnetomechanics of dielectrics invariant to the Galilean transformations from which the Maxwell equations follow as a special case.One of the few constructive theories free from the shortcomings of the theory of effective moduli and mixture theory is the static gradient model [4,7] for two-component particulate and unidirectional fibrous composites of stochastic structure under static loads. The zero approximation of this model is the second-order equations of the theory of effective moduli, the first approximation is the refined equations of the theory of effective moduli, and the second approximation is the fourth-order equations for...