Evaluations of the solution to the homogeneous plane problem of the theory of elasticity in the neighborhood of an irregular boundary point

Ludmila Frishter,S Jemioło,A Zbiciak,M Gajewski,M Krzemiński,M Mitew-Czajewska

doi:10.1051/matecconf/201711700047

Abstract

The most complex stress-strain state (SSS) occurs in the domain of stress concentration due to the shape of the boundary (“the geometric factor”) and the finite discontinuities of the specified forced deformations, the mechanical properties, emerging at the irregular point of the boundary of the domain. In this paper, there is a review of the methods for analyzing the particular qualities of the solution to the problem of the theory of elasticity, due to the shape of the boundary or “the geometric factor”. The features of the stress-strain state of constructions and structures, possessing “constructive heterogeneity” under the action of discontinuous forced deformations, are stress concentrators, determined on the polymer models of the photoelastic method. For interpreting and decoding of experimentally obtained local SSS in stress concentration domains of structures there are given estimations of the solution to homogeneous plane problem of the elasticity theory in neighborhood of an irregular boundary point.

Highlights

The stress-strain state (SSS) of complex structures is characterized by a significant concentration of stresses in the domains of conjugation of elements with different options for the constructive design of the boundary: singular lines, incoming angle, etc
The research of the stress state of complex structures in the domains of conjugation of elements made from materials with different mechanical properties under the action of forced deformations, that are discontinuous along the contact line of the elements, is an actual problem in the practice of engineering design
The research of the stress-strain state of complex structures in zones of constructive heterogeneity is focused on the consideration of the problem of elasticity theory in the neighborhood of the irregular point of the domain boundary, which includes the discontinuity of forced deformations

Summary

Introduction

The stress-strain state (SSS) of complex structures is characterized by a significant concentration of stresses in the domains of conjugation of elements with different options for the constructive design of the boundary: singular lines, incoming angle, etc. The fundamental work of V.A. Kondratiev [1] proves that the solution to the general elliptic boundary value problem in the neighborhood of irregular points of the domain boundary is represented in the form of an asymptotic expansion and an infinitely differentiable function. It is possible to show that the representation of the solution to an elastic problem in the neighborhood of an irregular point on a singular boundary line in the form of two homogeneous plane problems is valid in the case when: a) given forced deformations and body forces are continuous along the domain of the elastic body; b) given forced deformations and body forces are piecewise-continuous functions, and the jump in the values of the forced deformations and body forces along the inner surface of the contact of the domains goes to the specific boundary line of the body; c) the contact surface of domains V1 and V2 of the elastic composite body V , having different mechanical characteristics Ei ,νi , i = 1, 2 , goes to the specific boundary line of the body. 2) In an infinitesimal neighborhood of a singular point the solution to the correct boundary value problem of the theory of elasticity behaves as an asymptotically largest in absolute value eigenfunction of the corresponding canonical singular problem

Statement of the problem and estimations of the solution

Conclusions