Elastic stress distributions for hyperbolic and parabolic notches in round shafts under torsion and uniform antiplane shear loadings

Abstract

Closed-form Solutions are developed for the stress fields induced by circumferential hyperbolic and parabolic notches in axisymmetric shafts under torsion and uniform antiplane shear loading. The boundary Value problem is formulated by using complex potential functions and two different coordinate systems, providing two classes Of solutions. It is also demonstrated that some Solutions of linear elastic fracture and notch mechanics reported ill the literature can be derived as special cases of the general solutions proposed herein. Finally the analytical frame is Used to link the Mode III notch stress intensity factor to the maximum shear stress at the notch tip, as well as to give closed-form expressions for the strain energy averaged over a finite size volume surrounding the notch root

Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading

“…In the close neighbourhood of the notch apex the behaviour of a narrow hyperbolic notch matches that of a parabolic one (Zappalorto et al 2008). While analysing the problem of narrow hyperbolic notches under torsion, Davis and Tuba (1963) found numerically a circular elastic-plastic boundary for small load magnitudes, thus confirming the circular shape obtained in this work.…”

“…Davis and Tuba (1963) recommended not to use Neuber's relation under torsion, due to the absence of a uniformly stressed region which is necessary for determining the stress concentration factor. Coherently with these findings, the present authors wrote that the results obtained under uniform antiplane shear conditions in linear elasticity should not be applied to the torsion case without introducing a correction factor able to take into account the different nominal stress distribution on the transverse sectional area (Lazzarin et al 2007;Zappalorto et al 2008 should be considered valid only as a first approximation when applied to a torsion case.…”

“…The elongated shape of the plastic zone was obtained also in the crack case (Koskinen 1963;Rice 1966Rice , 1967aGdoutos 1990), confirming so the strong analogy characterizing rounded and sharp notches when subjected to antiplane loading. This analogy exists also in linear elastic conditions (Zappalorto et al 2008). Davis and Tuba (1963) recommended not to use Neuber's relation under torsion, due to the absence of a uniformly stressed region which is necessary for determining the stress concentration factor.…”

Stress field equations for U and blunt V-shaped notches in axisymmetric shafts under torsion

“…Recently the elastic stress fields ahead of semi-elliptic and parabolic or hyperbolic circumferential notches in infinite and finite size round bars under torsion were reported by the present authors Zappalorto et al 2008) who also dealt with some analytical expressions for the elastic notch stress intensity factors for sharp notches in round bars under torsion ). While all the above mentioned papers have been developed within the "classical" theory of elasticity, the effect on the near-tip stress state of crack tip blunting has been recently discussed by Fu et al (2008) within the theory of surface elasticity.…”

Fracture behaviour of notched round bars made of PMMA subjected to torsion at −60°C

“…In those papers the authors underlined that no other data were available in the generalised Young's modulus e 3 parameter quantifying the influence of the stress state for the calculation of R c G shear modulus Kt.net stress concentration factor referred to the net area Kmc mode III critical stress intensity factor from cracked specimens M T torque applied to the specimen R notch root radius r 0 distance between the notch tip and the origin of the coordinate system for SED computations r distance between the crack tip and a given point, according to linear elastic fracture mechanics R c radius of the control volume SED strain energy density criterion W c theoretical critical energy density W average strain energy density Greek 2a notch opening angle <j > gross diameter of the specimens t c critical stress under torsion loading T max maximum shear stress calculated on a notched specimen fnom.n nominal shear stress referred to the net area v Poisson's ratio literature. In parallel, many researchers have devoted strong efforts to investigate theoretically the stress distributions of sharp and blunt notches under torsion loading and under linear elastic conditions [52][53][54][55][56][57][58][59].…”