Effect of residual stresses on the oscillations of a welded cylindrical shell

Translated from Fiziko-Khimicheskaya Mekhanika Materialov, No. 6, pp. 69–72, November–December, 1988.     

Redistribution of residual stresses of the first kind in cylindrical specimens with hardened surface layers

The nature of the distribution in axial residual stresses of the first kind was determined for specimens from which successive surface layers had been removed. It was established that the complete removal (by machining) of surface hardened layers, whose extent is determined by metallographic examination and microhardness measurements, does not eliminate residual stresses which extend deeper than the work-hardened zone and beyond the confines of etch-resistant surface layers. The character of residual stress distribution in specimen subsurface regions is retained after the removal of metal surface layers.     

Field of residual stresses in a coating with crack

We consider an object with coating deposited on its plane surface. The coating is subjected to the action of known uniform uniaxial tensile residual stresses. We analyze the field of residual stresses formed in the coating as a result of the formation of a long flat crack perpendicular both to the direction of action of tensile stresses and to the surface of the coating. The problem is studied in the two-dimensional statement. The stress field is investigated in the sections perpendicular both to the crack plane and to the surface of the coating. Two cases are analyzed, namely, the case of a through crack (through the entire thickness of the coating) and the case of a nonthrough crack with plastic zone on its continuation. The performed numerical analysis enables us to estimate the possibility of subsequent fracture of the coating after crack initiation. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 2, pp. 80–88, March–April, 2006.     

Stresses in a cylindrical shell with an arbitrarily oriented crack

An analysis is presented of the problem of the stressed state of a cylindrical shell with an arbitrarily oriented crack for the general case of symmetrical and asymmetrical loads. Basic solutions and formulas are obtained for the stress intensity coefficients in crack tip regions.     

An axial crack in a pressurized cylindrical shell

Following an earlier analysis of a line crack in a spherical cap, the stresses in a cylindrical shell containing artaxial crack are presented. The inverse square root singular behavior of the stresses peculiar to crack problems is obtained in both the extensional and bending components. This singularity may be related to that found in an initially flat plate by $$\frac{{{}^\sigma shell}}{{{}^\sigma plate}} \approx 1 + (a + b\ell n\frac{c}{{\sqrt {Rh} }})\frac{{c^2 }}{{Rh}} + ...$$ where the quantity in parentheses is positive. An approximate fracture criterion, based on Griffit 's Theory, is also deduced, and bending-stretching interaction curves for this case are presented.     

An axial crack in a pressurized cylindrical shell

Following an earlier analysis of a line crack in a spherical cap, the stresses in a cylindrical shell containing an axial crack are presented. The inverse square root singular behavior of the stresses peculiar to crack problems is obtained in both the extensional and bending components. This singularity may be related to that found in an initially flat plate by

$$\frac{{{}^\sigma {\text{shell}}}}{{{}^\sigma {\text{plate}}}} \approx 1 + ({\text{a + b}}\ell {\text{n}}\frac{{\text{c}}}{{\sqrt {{\text{Rh}}} }})\frac{{{\text{c}}^{\text{2}} }}{{{\text{Rh}}}} + ...$$     

A circumferential crack in a pressurized cylindrical shell

Following an earlier analysis of an axial line crack in a cylindrical shell, the stresses for a finite circumferential crack are presented. The inverse square root singularity of the stresses peculiar to crack problems is obtained in both the extensional and bending components. Furthermore, the initial curvature may be related to that found in an initially flat plate by a factor of the form

$$\frac{{{}^\sigma shell}}{{{}^\sigma plate}} \approx 1 + (a + b \ell n \frac{c}{{\sqrt {Rh} }})\frac{{c^2 }}{{Rh}} + ...$$     

Redistribution of residual welding stresses in pulsed electromagnetic treatment

Experimental results demonstrate that in an AMg6 alloy specimen after pulsed electromagnetic treatment, a significant decrease and redistribution of residual welding stresses are observed. Numerical simulation corroborates that the treatment effect is not associated with a thermal field influence.     

A circumferential crack in a cylindrical shell under torsion

The fully anti-symmetric problem for a cylindrical shell with a circumferential crack is considered. The solution of the problem is reduced to that of a system of singular integral equations of the first kind. As an example the torsion of the cylinder is discussed and membrane and bending components of the stress intensity factor ratio are given.     

A circumferential crack in a cylindrical shell under tension

A closed cylindrical shell under uniform internal pressure has a slit around a portion of its circumference. Linear shallow shell theory predicts inverse square-root-type singularities in certain of the stresses at the crack tips. This paper reports the computed strength of these singularities for different values of a dimensionless parameter based on crack length, shell radius and shell thickness.     

Evolution of residual stresses with fatigue crack growth in integral structures with crack retarders

Bonded straps are investigated for their ability to retard a growing fatigue crack in metallic structures. The evolution of the residual stresses in the vicinity of the strap with fatigue crack growth has been studied. Cracks were grown in single edge-notched tension (SEN(T)) specimens reinforced with either a titanium or a carbon fibre reinforced plastics (CFRP) strap. The residual stress evolution has been measured in situ during crack growth using neutron diffraction, and modelled with a finite element approach. The peak residual stresses induced by the mismatch of the coefficient of thermal expansion between the strap and plate materials were seen to be fairly constant with crack growth. Good correlation between the experimental and the modelling results was found, except at very long crack lengths for a specimen that exhibited considerable fracture surface roughness at long crack lengths. The difference was attributed to wedging of the fracture surface changing the expected stress state, rather than any effect of the strap.     

Interlaminar stresses in antisymmetric angle-ply cylindrical shell panels

Layerwise theory of Reddy is utilized for investigating free-edge effects in antisymmetric angle-ply laminated shell panels under uniform axial extension. Following some physical arguments, governing displacement field is divided into local and global parts. The former is discretized through the shell thickness by a zig-zag interpolation function while the latter is calculated by a first-order shear deformation theory. Local equilibrium equations are then solved through a state space approach. Accuracy of the proposed technical solution is subsequently verified by a novel analytical elasticity solution. For this end, the problem is analytically solved for specific boundary conditions along the edges. The numerical results show excellent agreement between two theories for various composite shell panels.     

Stresses around an axial crack in a pressurized cylindrical shell

Perturbation solutions are obtained for stresses around the tip of an axial crack in a pressurized circular cylindrical shell. The analysis, using a differential equation approach, involves perturbation in a curvature parameter relating crack length to shell diameter and thickness. The results are valid for cracks of small length and agree completely with those obtained earlier by Folias by the method of integral equations.     

A longitudinal crack in a cylindrical shell under internal pressure

The study relates to the strength of a cracked shell. Linear thin shell theory is employed to obtain solutions for internal pressure and uniform circumferential bending. A shallow shell approximation is used, and shown valid providing crack length and shell thickness are small enough in comparison with shell radius. Initial formulation as a boundary value problem is shown equivalent to two coupled singular integral equations. These were solved numerically using a computer. Stresses are found to display singularities as inverse square-root of distance from a crack tip, as with the flat plate. The main results are graphs of normal and bending stress singularity strengths against a curvature/crack-length dimensionless parameter.

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Zusammenfassung Diese Arbeit behandelt die Stärke von einer gerissenen Schale. Die lineare Theorie dünner Schalen ist benutzt um dieses Problem für den Innendruck und gleichmäßige Umkreis-biegung zu lösen. Die Annäherungen der mäßig gekrümte Schalen sind benutzt welche zulassig sind nur wenn die Rißlänge und die Dicke der Schale klein sind im Vergleich zu dem Schalradius. Die anfängliche Formelierung als ein Randwertproblem ergibt sich als gleichwertig zwei gekuppelten singulären Integralgleichungen. Diese sind numerisch gelöst mit Hilfe einer Rechenautomat. Die Spannungen zeigen, in Übereinstimmung mit der flachen Platte, umgekehrte Quadratwurzelsingularitäten. Die Resultaten der Stärke der Normal- und Biegung-spannungsingularitäten sind graphisch als Funktion eines dimensionslosen Krümmungs-Rißlängeparameter gegeben.

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Résumé Cette étude est sur la résistance d'une monocque fêlée longitudinalement. À l'aide de la théorie linéare des toques minces élastiques l'on obtient la solution de deux cas: 1) coque sous pression intérieure et 2) flexion circonférentielle uniforme. On fait l'approximation que la coque est de faible profondeur et l'on démontre que les solutions sont valables pourvu que la longeur de la fêlure et l'épaisseur de la coque soient assez petites comparées au rayon de la coque. On formule d'abord un système elliptique des équations aux dérivées partielles, qui se transforme en deux équations intégrales simultanées. Celles-ci sent résolues numériquement à l'aide d'un ordinateur. On trouve que la contrainte possède des singularités proportionelles à l'inverse de la racine carrée de la distance d'une extrémité de la fêlure: ce résultat est identique à celui qui existe pour une plaque. Les résultats principaux que l'on présente sont des courbes de l'amplitude de ces singularités en fonction d'un paramètre charactéristique de la grandeur de la fêlure et du rayon de la coque.

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Notation c half crack length, inch - E Young's modulus of elasticity, psi - h thickness of shell, inch - M _t twisting moment resultant - M circumferential bending moment resultant far from crack - M _x , M _y , M _xy bending and twisting moment resultants - N _x , N _y , N _xy normal and shear stress resultants - p _i internal pressure, psi - p ₀ reference pressure, psi - Q, Q _x , Q _y transverse shear stress resultants - R radius of shell, inch - x, y Cartesian coordinates for shallow shell formulation, inch - w outward radial displacement of cylindrical shell, inch - Euler constant (0.5772...) - dimensionless parameter, ²=[3(1–v ²)/4]^1/2(c ²/Rh) - v Poisson's ratio This work was supported in part by the Office of Naval Research under Contract Nonr 1866(02), in part by the National Aeronautics and Space Administration under Grant NsG-559, and by the Division of Engineering and Applied Physics, Harvard University.