Construction of Analytical Eigensolutions for Isotropic Conical Bodies and Their Application for Estimation of Stresses Singularity

Tatyana Korepanova,Igor Shardakov,Valerii Matveenko

doi:10.3221/igf-esis.49.23

Abstract

A complete set of eigensolutions is constructed for different variants of circular conical bodies: homogeneous cone with one lateral surface (solid cone), homogeneous cone with two lateral surfaces (hollow cone) and composite cone for different variants of boundary conditions on the lateral surface. It has been shown that the constructed eigensolutions can be readily applied for estimation of the character of stress singularity at the vertices of conical bodies. The character of stress singularity at the vertex of the solid and hollow cones for different variants of boundary conditions on the lateral surfaces has been defined by direct numerical simulations. Numerical results obtained for solid, hollow and compose cones under different boundary conditions on the lateral surfaces are discussed.

Highlights

One of the main findings of the classical elasticity theory is the possibility of existence of singular solutions due to the occurrence of infinite stresses at the points of surface non-smoothness, changes in the type of boundary conditions, contact of different materials and inside the body at the points of contact of dissimilar materials
W e have considered the analytical method of constructing eigensolutions for circular cones
It has been shown that the proposed analytical relations can be used to construct solutions and to evaluate the character of stress singularity for different conical bodies and different types of boundary conditions on the lateral surfaces and contact surface of dissimilar materials

Summary

INTRODUCTION

One of the main findings of the classical elasticity theory is the possibility of existence of singular solutions due to the occurrence of infinite stresses at the points of surface non-smoothness, changes in the type of boundary conditions, contact of different materials and inside the body at the points of contact of dissimilar materials. At the first stage we define the form of the characteristic equation for the index For this purpose, we retain in the series (25) only the first term with the coefficient N 0 , and in the expressions (24) in front of the function F(x ) and its derivatives in Eqn (23) we retain only the lowest powers of the variable x , which, are not less than the order of the highest derivative (in this case, for Eqn (23) this is the first order). The proposed method allows us to define step by step the types of the generalized power series for all partial solutions of the initial differential equation and in all partial solutions to set apart the regular from irregular solutions (in our case for x 0 ) Such a capability of the method is rather essential for constructing solutions to particular problems, for example, for hollow and composite cones. Vk(5) , v and the obtained relation to derive six partial solutions uk(1) , uk( 2) , uk(3) , uk(4 ) , uk(5) , u

S x 2

CONCLUSION