On the Stress State of the Semi-Strip with a Longitudinal Crack

Natalya Vaysfeld,Zinaida Zhuravlova

doi:10.32732/jmo.2018.10.2.104

Abstract

The stress state of the elastic semi-strip is investigated in the paper. The lateral sides of the semi-strip are fixed and the semi-strip’s short edge is under the mechanical load. The longitudinal crack is located inside the semi-strip. The problem is reduced to the one-dimensional problem with the help of Fourier sin-, cos- transformation, which was applied directly to the Lame’s equilibrium equations and the boundary conditions. The one-dimensional problem is formulated is a vector form. Its solution is constructed with the help of the matrix differential calculation and the Green matrix-function, which was constructed in the bilinear form. The solution of the problem is reduced to the solving of three singular integral equations. The first equation in this system contains two fixed singularities in its kernel. To consider them the corresponding transcendental equation is constructed, and its roots are found. The special generalized method is applied to solve the system of singular integral equations. The stress intensity factors are calculated.

Highlights

The plane elasticity problems are important as model examples for more complicated problems
The investigation of the stress state of the elastic semi-strip with a longitudinal crack can be used for the solving of the discontinuous problems in areas that contain angles and defects
The new approach was used for the solving of the plane mixed elasticity problem for a semi-strip

Summary

Introduction

The plane elasticity problems are important as model examples for more complicated problems. The investigation of the stress state of the elastic semi-strip with a longitudinal crack can be used for the solving of the discontinuous problems in areas that contain angles and defects. There are three classed of methods that can be applied for the solving of the plane problems of elasticity for: analytical, numeric and analytically-numeric. The plane problems of elasticity for a strip and a semi-strip were solved in the following works. The solving of the problem for the infinite strip with a semi-infinite crack was reduced to the solving of the singular integral equation by the use of simple layer and double layer potentials in [4]. In the proposed work the plane mixed problem for a semi-strip with a longitudinal crack was solved by the analytically-numeric approach. The system of singular integral equations was solved with consideration of the fixed singularities in its kernel

The statement of the problem

The general solving scheme

The construction of the partial and discontinuous solutions

The solving of the singular integral equations

Conclusions