On the solution of three-dimensional boundary-value problems for layered bodies with nonplanar interfaces

Abstract

The interfaces play an important role in various buildup bodies, and also in the composite materials and structural elements. Special monographs [7, 8] have been devoted to this question, presenting the results of scientific studies of the physical and chemical phenomena on the interfaces, the mechanical behavior, and the role of the interfaces in the damage processes, and also their influence on the basic mechanical properties of the composites. In many cases the interfaces deviate from the ideal geometric shapes: planar (in the layered composites), circular cylindrical (in the fibrous composites), and spherical (in the granular composites). Numerous theoretical and experimental studies confirm this. Thus, in the explosive welding of metals (and nonmetals) there form wavy surfaces, the sections of which may be close to sinusoids, for example in the welding of niobium and copper [9]. If the densities of the materials differ significantly, then the sinusoidal nature of the interface distorts as illustrated in [12] for the example of the welding of lead and steel. In addition, in view of the nature of the technological processes [10] the interfaces may become curved in the layered composite materials and deviate locally or periodically from the ideal coordinate planes. Theoretical and experimental studies have shown that the shape of the interface has a significant influence on the physical and mechanical processes and phenomena (bond strength, stress concentration, wave diffraction, thermal conduction, and so on). Numerous publications that are cited in the survey works [1, 3, 11] confirm this. A second variant of the boundary shape perturbation method was developed in [4, 5] for the solution of the three-dimensional boundary-value problems for nonorthogonal surfaces that are close to the coordinate planes. It was assumed that the equations of the interfaces are linear relative to the small parameter characterizing the degree of deviation from the coordinate planes. This narrowed significantly the class of the examined boundary-value problems and their practical importance. In the present work we examine the three-dimensional boundary-value problems of the mechanics of layered bodies with interfaces that are described by nonlinear equations relative to a small parameter. We construct in general form the recurrence relations and the differential operators of the boundary conditions, making it possible to solve the three-dimensional boundary-value problems with the accuracy that is required for applications. We examine particular cases and present one of the possible criteria for evaluating the accuracy of the approximate solutions that are obtained with the aid of the described variant of the boundary shape perturbation method.