Abstract
A general solution is developed for the symmetric bending stress distribution at the tip of a crack in a plate taking shear deformation into account through Reissner's theory. The solution is obtained in terms of polar coordinates at the crack tip and includes the complete class of solutions satisfying all the three boundary conditions along the crack. The solution has arbitrary multiplicative constants and in specific problems, these constants can be determined from conditions on the exterior boundary by well-known numerical techniques such as collocation, successive integration. Results of a numerical solution for a square plate with a central crack subject to uniaxial bending are presented along with a critical discussion of the sensitivity of the numerical solution which is associated with the exponential character of Bessel terms in this higher order analytical solution.
Résumé
On développe une solution générale à la distribution symétrique de contraintes de flexion à l'extrémité d'une entaille dans une plaque soumise à une déformation de cisaillement, en tenant compte de la théorie de Reissner. La solution est obtenue sous forme de coordonnées polaires à l'extrémité de la fissure et comporte l'ensemble des solutions satisfaisant à toutes les conditions de frontières le long de la fissure. La solution comporte des constantes multiplicatives arbitraires et, dans des problèmes spécifiques, ces constantes peuvent être déterminées à partir des conditions de confinement extérieur à l'aide de techniques numériques bien connues telles que la collocation ou l'intégration successive. Les résultats d'une solution numérique dans le cas d'une plaque carrée comportant une fissure centrale soumise à flexion uniaxiale sont présentés en même temps qu'une discussion critique de la sensibilité de la solution numérique associée au caractère exponentiel des termes de Bessel dans cette solution analytique d'une ordre supérieur.
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Abbreviations
- 2a :
-
crack length
- h :
-
plate thickness
- k 2 :
-
\(k^2 = \frac{5}{2}\left( {\frac{a}{h}} \right)^2 \)
- E :
-
Young's Modulus
- v :
-
Poisson's ratio
- D :
-
bending rigidity of plate,Eh 3/12(1-v 2)
- M 0 :
-
reference bending moment
- σ0 :
-
extreme fibre stress due toM 0, 6M 0/h 2
- r, θ:
-
polar coordinates with crack tip as the origin (Fig. 1), r being nondimensionalized with respect toa
- W :
-
normal displacement, nondimensionalized with respect toM 0 a 2/D
- M r ,M θ ,M rθ :
-
bending and twisting moments per unit length of plate element (Fig. 1), nondimensionalized with respect toM 0
- σ r , σθ :
-
transverse shear forces per unit length of plate element (Fig. I), nondimensionalized with respect toM 0/a
- σ r σθ τ rθ :
-
stresses on that surface where a positive moment produces tension, nondimensionalized with respect to σ0
- χ:
-
a stress function in Reissner's theory, nondimensionalized with respect toM 0
- K (b) :
-
bending stress intensity factor
- K(b) :
-
Modified Bessel Functions of first and second kinds respectively
- ϕ(μ,m):
-
(μ + 1)(μ + 2)...(μ +m)form5≠ 0 = 1 form = 0
- θ:
-
Gamma function
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Structures Division
Materials Division
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Murthy, M.V.V., Raju, K.N. & Viswanath, S. On the bending stress distribution at the tip of a stationary crack from Reissner's theory. Int J Fract 17, 537–552 (1981). https://doi.org/10.1007/BF00681555
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DOI: https://doi.org/10.1007/BF00681555