Longitudinal fracture analysis of inhomogeneous beams with continuously varying sizes of the cross-section along the beam length

Victor Rizov Victor Rizov,Holm Altenbach Holm Altenbach

doi:10.3221/igf-esis.53.04

Abstract

Analyses of longitudinal fracture behavior of inhomogeneous beams which have continuously varying sizes of the cross-section along the beam length are carried-out. Beams of a rectangular cross-section are studied. It is assumed that beams exhibit continuous (smooth) material inhomogeneity along the width and height of the cross-section. A longitudinal crack located arbitrary along the beam height is analyzed. First, a cantilever beam with linearly varying width and height along the beam length is considered. The material of the beam has non-linear elastic mechanical behavior. The external loading consists of one bending moment applied at the free end of the lower crack arm. The fracture behavior is analyzed in terms of the strain energy release rate assuming that the modulus of elasticity is distributed continuously in the beam cross-section. The balance of the energy is considered in order to derive the strain energy release rate. A solution to the strain energy release rate is obtained also by considering the complementary strain energy for verification. The longitudinal fracture behavior of the inhomogeneous non-linear elastic cantilever beam configuration is studied also for the cases when the variation of the width and height of the cross-section is described by sine and power laws.

Highlights

One of the efficient ways to improve the stability and to increase the strength and load bearing capacity of the structures, and at the same time to reduce their weight is to use beams of continuously varying sizes of the cross-section along the beam length
The main novelty of the present paper is that in contrast to previous papers [16 - 19] which deal with longitudinal fracture analysis of inhomogeneous beams with constant cross-section, the inhomogeneous beam considered here has continuously varying height and width in the length direction
It is found that the strain energy release rate decreases with increasing of ht / hn and bt / bn ratios

Summary

Introduction

One of the efficient ways to improve the stability and to increase the strength and load bearing capacity of the structures, and at the same time to reduce their weight is to use beams of continuously varying sizes of the cross-section along the beam length. Various studies of the fracture behavior of functionally graded composite materials have been reviewed in [13] Cracks oriented both parallel and perpendicular to the gradient direction have been analyzed by using methods of linear-elastic fracture mechanics. Analyses of different beam configurations under static or dynamic loading conditions have been carriedout by using methods of linear-elastic fracture mechanics. The aim of the present paper is to analyze the longitudinal fracture behavior of inhomogeneous beam configurations with constantly varying sizes (width and height) of the cross-section along the beam length. It should be mentioned that the previous works of the author are concerned with longitudinal fracture analyses of inhomogeneous (functionally graded) beam configurations with constant sizes of the cross-section along the beam height [16 – 19]. The longitudinal fracture is analyzed in terms of the total strain energy release rate by applying the theory for bending of prismatic beams since this theory can be used for beams with varying cross-section along the beam length provided that the variation is not abrupt and the angle of inclination of the beam edge is small [20]

Objectives

Conclusion