Saint-Venant torsion of orthotropic bars with a circular cross-section containing multiple cracks

Abstract

In this paper, the problem of the circular orthotropic bars with multiple cracks is investigated based on the Saint-Venant torsion theory. The solution to the problem of an orthotropic bar weakened by a Volterra-type screw dislocation is first obtained by means of the finite Fourier sine transform. In this research, the bar is assumed to be subjected to an axial net torsion when finding the dislocation solution. The closed form solution is then derived for displacement and stress fields in the bar. At the next step, the dislocation solution is employed to derive a set of Cauchy singular integral equations for analysis of the circular bar with curved cracks. The solution to these equations is used to determine the torsional rigidity of the bar and the stress intensity factors of the tips of the cracks. The paper is furnished with several examples of a single crack and multiple cracks.

Torsion of elastic solids with sparse voids parallel to the twist axis

“…The four empty circles shown on the left part for the two crack tips (k = 1 and k = 2) are referred to the corresponding values in Tab. 2, evaluated by [18] for circular cross sections with D/(2a) = 10Υ/3, in the cases Υ = 0.75 (so that D/(2a) = 2.5) and Υ = 1.5 (so that D/(2a) = 5).…”

“…The reliability of the SIFs at the two tips of a 'standard' crack within an infinite matrix (equation (85)) can also be assessed in Table 2 through the comparison with the values reported in Hassani and Faal [24] for circular cross-sections containing multiple cracks, obtained by numerically solving an integral equation. In particular, the numerical values of K III (k)=( ffiffiffi ffi p p mYa…”

“…2 ) for a 'standard' crack noncentred within a circular bar of diameter D = 20Ya=3 are obtained starting from the values reported in Table 1 of Hassani and Faal [24]. Moreover, the four values of SIFs at the two crack tips for the two cases Y = 0:75 and Y = 1:5 are also reported as empty circles in Figure 12.…”

“…cylindrical shafts containing notches of different geometries [39,44,45,46]; (ii.) cylinders with elliptical cross-section containing a single crack [18,37]; (iii.) circular shafts with quasi-regular polygonal voids [5]; (iv.)…”

“…Beside the strong research effort in planar and three-dimensional problems of elasticity, relatively little attention has been devoted to the stress intensification in inhomogeneous solids subject to torsional loading. Within this framework, the research has mainly been focused on (i) cylindrical shafts containing notches of different geometries [20][21][22][23], (ii) cylinders with elliptical cross-sections, each containing a single crack [24,25], (iii) circular shafts with quasi-regular polygonal voids [26], (iv) cross-sections weakened by edge cracks [24,25,27,28], (v) the neutrality of coated cavities of various shapes in cylinders with elliptical cross-sections [29], (vi) multiply connected domains [24,[30][31][32][33][34][35][36] and (vii) star-shaped cracks in square and circular cross-sections [37,38]. However, except for some special geometries, the results are usually obtained through the implementation of numerical techniques; very few analytical expressions are currently available as a 'guide tool' for engineers.…”