Abstract
ObjectivesTo study the problem of determining the degree of stress at the apex of a wedge-shaped area in cases where the sides (or one of them) are covered with a thin flexible coating.MethodIt is assumed that the coating is not stretchable. On the other side of the wedge-shaped area, the same coating is assumed to be present; it is either fixed, stress-free or in smooth contact with a rigid base. Mathematically, the problem is reduced to the task of determining the roots of characteristic transcendental equations arising from the existence of a nontrivial solution to the system of linear homogeneous equations.ResultsValues for the specific characteristics of the radial component of a stress tensor are determined for different combinations of boundary conditions and solution angles. In particular, the angles at which the singular behaviour of stresses occurs are determined. The case is considered when a special boundary condition is given on the edge surface, simulating the overlay. Characteristic equations are obtained to determine the index of the degree dependency of the asymptotic solution in its vicinity for four variants of boundary conditions. In two cases, transcendental equations are obtained, which are solved numerically.ConclusionCalculations of the first positive roots of the equations depending on the angle of the edge solution and Poisson's ratio are presented. The values of the angles, at which the singular behaviour of stresses occurs, are determined. In the case of a combination of boundary conditions (III – IV), the singular stress behaviour is observed for the angle ???? = ????/8, while in the case of (III – III) this value is equal to ????/4.
Highlights
Objectives To study the problem of determining the degree of stress at the apex
Method It is assumed that the coating is
the problem is reduced to the task of determining the roots
Summary
В работе [1] показано, что проблема исследования особенности напряжений в окрестности трехмерного ребра изотропного упругого тела может быть разделена на плоскую и анти плоскую задачи теории упругости. В монографии [2] для плоской задачи предложен метод определения показателей при особенности напряжений в вершине клиновидной области при различных граничных условиях на ее сторонах. Определение этих показателей (не только меньших по величине) может быть использовано при построении асимптотического ряда в окрестности нерегулярной точки границы. Она сформулирована на основе асимптотического анализа решения задачи для упругой полосы [8] и представляет собой специальное граничное условие.
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