Abstract
Using an integral formulation, the problem of a spherical shell containing a through crack of length 2c and subjected to periodic transverse vibrations of frequency ω is solved for the in-plane and Kirchhoff bending stresses. The usual inverse square root singular behavior characteristic to crack problems is recovered. Furthermore, it is found that the transverse vibrations reduce the stresses in the vicinity of the crack tip, except when the forcing frequency ω reaches the natural frequency of the uncracked shell in which case they become infinite.
Résumé
On analyse le problème de l'enveloppe sphérique ayant une fissure de longeur 2c de part en part de son épaisseur, et sujette à des vibrations transversales de pulsation ω; on résoud ré problème à l'aide de fonctions intégrales, pour les contraintes coplanaires et les contraintes de flexion de Kirchhoff.
On retrouve le comportement singulier habituel d'ordre 1/2, caractéristique des problèmes de fissuration. En outre, on trouve que des vibrations transversales ont tendance à réduire les contraintes au voisinage de l'extrémitée des fissures, sous réserve que leur fréquence ω atteigne la fréquence naturelle de l'enveloppe non fissurée; dans ces conditions les contraintes deviennent en effet infinies.
Zusammenfassung
Das Problem einer sphärischen Hülle, mit einem sich über die gesamte Dicke der Hülle hinziehenden Riß der Länge 2c, welche Querschwingungen mit einer Pulsierung unterworfen ist, wurde für die Fälle von koplanaren and von Kirchhoff-Biegebeanspruchungen mit Hilfe einer Integralformulierung gelöst. Hierbei ergab sich wiederum das fur Riß-probleme charakteristische Gesetz der umgekehrten Quadratwurzel.
Außerdem zeigte sich, das Querschwingungen die Spannungen in der Umgebung der Rißspitze vermindern, ausgenommen der Fall, wo die Frequenz ω der aufgezwungenen Schwingung mit der Eigenfrequenz der unbeschädigten Hülle übereinstimmt, wo sie dann ins Unendliche ansteigen.
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Abbreviations
- c :
-
half crack length
- D :
-
Eh 3/[12(1 − v 2)] = flexural rigidity
- E :
-
Young's modulus of elasticity
- F (x, y, t), \(\tilde F\)(x,y):
-
stress functions
- \(\tilde F\) (c)(x,y):
-
complementary stress function
- G :
-
shear modulus
- h :
-
thickness
- K n :
-
modified Bessel function of the third kind of order n
- L i :
-
kernels as defined in text
- m 0 :
-
constant as defined in text
- M (c) x , M (c) y , M (c) xy :
-
complementary bending forces
- M (p) x , M (p) y , M (p) xy :
-
particular bending forces
- n 0 :
-
constant as defined in text
- N (c) x , N (c) y , N (c) xy :
-
complementary membrane forces
- N (p) x , N (p) y , N (p) xy :
-
Particular membrane forces
- N n :
-
Newman function of order n
- q(x, y, t), q(x, y):
-
internal pressure
- r :
-
\(\{ (X - 1)^2 + Y^2 \} ^{\tfrac{1}{2}} \)
- R :
-
radius of curvature of the shell
- R:
-
\(\{ (X^2 + Y^2 \} ^{\tfrac{1}{2}} \)
- t :
-
time variables
- U(s−λ), U(λ−s):
-
the unit step function
- V y :
-
equivalent shear
- W(x, y, t), \(\tilde W\)(x, y, t):
-
displacement functions
- \(\tilde W\) (c)(x,y):
-
complementary displacement function
- \(x = \frac{X}{c}, y = \frac{Y}{c}, z = \frac{Z}{c}\) :
-
dimensionless rectangular coordinates
- X, Y, Z :
-
rectangular cartesian coordinates
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Do, S.H., Folias, E.S. On the steady-state transverse vibrations of a cracked spherical shell. Int J Fract 7, 23–37 (1971). https://doi.org/10.1007/BF00236481
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DOI: https://doi.org/10.1007/BF00236481