Abstract
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.
Résumé
On considère une enveloppe cylindrique fermée soumise à pression interne et comportant une saignée sur une portion de sa circonférence.
D'après la théorie linéaire des enveloppes courtes, on peut prédire des singularités d'exposant — 1/2 dans certains états de contrainte aux extrémités de l'entaille. Le mémoire fournit une évaluation numérique de ces singularités pour diverses valeurs d'un paramètre sans dimensions caractéristique de la longueur de la fissure, du rayon de courbure de l'enveloppe et de son épaisseur.
Zusammenfassung
Eine einem gleichförmigen internen Druck ausgesetzte geschlossene zylindrische Hülle hat einen an einem Teil des Umfangs entlanglaufénden Schlitz. Die lineare Theorie kurzer Hüllen sagt für verschiedene Spannungen an der Rißspitze Singularitäten des Typs der umgekehrten Quadratwurzel voraus. Vorliegender Bericht gibt für verschiedene Werte eines auf Rißlänge, Hüllenumfang und Hüllenlänge begründeten dimensionslosen Parameters, die berechneten Werte dieser Singularitäten.
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Abbreviations
- b :
-
Rh 1/2[3/4(1−v2)]−1/2
- c :
-
half crack length
- E, v :
-
Young's modulus and Poisson's ratio
- h :
-
thickness of shell
- \(\overline M _x ,{\text{ }}\overline M _y ,{\text{ }}\overline M _{xy} {\text{ = }}\left( {\frac{{Po^{b^2 } }}{8}} \right){\text{ (}}\overline M _x ,{\text{ }}\overline M _y ,{\text{ }}\overline M _{xy} )\) :
-
bending stress measures
- \(\overline N _x ,{\text{ }}\overline N _y ,{\text{ }}\overline N _{xy} {\text{ = (}}P_0 {\text{ }}R){\text{ (}}\overline N _x ,{\text{ }}\overline N _y ,{\text{ }}\overline N _{xy} )\) :
-
(p 0 R)(N x , N y , N xy ) membrane stress measures
- \(\overline Q _x ,{\text{ }}\overline Q _y {\text{ = }}\left( {\frac{{P_0 ^b }}{{8\lambda }}} \right){\text{ (}}Q_x ,{\text{ }}Q_y )\) :
-
\(\overline u ,{\text{ }}\overline v {\text{ = }}\frac{{P_0 ^{Rc} }}{{2Eh}}{\text{ (}}u,{\text{ }}v)\) transverse shear stress measures
- V :
-
Q x + M xy, y effective transverse shear stress
- p 0 :
-
pressure inside shell
- R :
-
radius of shell
- ū, \(\overline x {\text{,}}\overline y {\text{ = }}c(x,y)\) :
-
tangential displacements
- \(\bar w = \frac{{2p_0 R^2 \lambda ^2 }}{{Eh}}\) :
-
normal displacement
- \(\overline x {\text{,}}\overline y {\text{ = }}c(x,y)\) :
-
c(x, y) axial and circumferential coordinates
- λ:
-
c/b
- ϕ:
-
dimensionless stress function
- ψ:
-
w + iϕ complex stress function
References
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This work was supported in part by the National Aeronautics and Space Administration under Grant NGL 22-007-012, and by the Division of Engineering and Applied Physics, Harvard University. The first author was also supported in part by a Fellowship from the New Zealand Federation of University Women.
Formerly Graduate Student, Division of Engineering and Applied Physics, Harvard University.
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Duncan-Fama, M.E., Sanders, J.L. A circumferential crack in a cylindrical shell under tension. Int J Fract 8, 15–20 (1972). https://doi.org/10.1007/BF00185194
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DOI: https://doi.org/10.1007/BF00185194