Abstract
The problem of vibrations of an ice sheet with a rectilinear crack on the surface of an ideal incompressible fluid of finite depth under the action of a time-periodic local load is solved analytically using the Wiener–Hopf technique. Ice cover is simulated by two thin elastic semi-infinite plates of constant thickness. The thickness of the plates may be different on the opposite sides of the crack. Various boundary conditions on the edges of the plates are considered. For the case of contact of plates of the same thickness, a solution in explicit form is obtained. The asymptotics of the deflection of the plates in the far field is studied. It is shown that in the case of contact of two plates of different thickness, predominant directions of wave propagation at an angle to the crack can be identified in the far field. In the case of contact of plates of the same thickness with free edges and with free overlap, an edge waveguide mode propagating along the crack is excited. It is shown that the edge mode propagates with maximum amplitude if the vertical wall is in contact with the plate. Examples of calculations are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
21 May 2018
In the original publication, the second formula on page 1077 was misspelled. It should read:
References
L. A. Tkacheva, “Behavior of a Semi-Infinite Ice Cover under Periodic Dynamic Impact,” Mekh. Tekh. Fiz. 58 (4), 82–94 (2017) [J. Appl. Mech. Tech. Phys. 58 (4), 641–651 (1994)].
A. V. Marchenko, and A. Yu. Semenov, “Edge Waves in a Shallow Fluid beneath a Fractured Elastic Plate,” Izv. Ros. Akad. Nauk. Mekh. Zhidk. Gaza, No. 4, 185–189 (1994) [Fluid Dyn. 329 (4), 589–592 (1994)].
R. V. Gol’dshtein, A. V. Marchenko, and A. Yu. Semenov, “Edge Waves in a Fluid under an Elastic Plate with a Crack,” Dokl. Akad. Nauk 339 (3), 331–334 (1994) [Dokl. Phys. 39 (11), 813–815 (1994).
A. V. Marchenko, “Natural Vibrations of a Hummock Ridge in an Elastic Ice Sheet Floating on the Surface of an Infinitely Deep Fluid,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 6, 99–105 (1995) [Fluid Dyn., 30 (6), 887–893 (1995)].
D. V. Evans and R. Porter, “Wave Scattering by Narrow Cracks in Ice Sheets Floating on Water of Finite Depth,” J. Fluid Mech. 484, 143–165 (2003).
D. Xia, J. W. Kim, and R. C. Ertekin, “On the Hydroelastic Behavior of Two-Dimensional Articulated Plates,” Marine Structures 13, 261–278 (2000).
H. Chung and C. M. Linton, “Reflection and Transmission of Waves Across a Gap between Two Semi-Infinite Elastic Plates on Water,” Quart. J. Appl. Math. 58 (1), 1–15 (2005).
D. Karmakar and T. Sahoo, “Scattering of Waves by Articulated Floating Elastic Plates in Water of Finite Depth,” Marine Structures 18, 451–471 (2005).
G. L. Waughan, T. D. Williams, and V. A. Squire, “Perfect Transmission and Asymptotic Solutions for Reflection of Ice-Coupled Waves by Inhomogeneities,” Wave Motion 44, 371–384 (2007).
H. Chung and C. Fox, “A Direct Relationship between Bending Waves and Transition Conditions of Floating Plates,” Wave Motion 46, 468–479 (2009).
T. D. Williams and R. Porter, “The Effect of Submergence on the Scattering by the Interface between Two Semi-Infinite Sheets,” J. Fluids Structures 25, 777–793 (2009).
A. V. Marchenko, “Parametric Excitation of Flexural-Gravity Edge Waves in the Fluid beneath an Elastic Ice Sheet with a Crack,” Europ. J. Mech., B: Fluids 18 (3), 511–525 (1999).
L. A. Tkacheva, “Vibrations of an Ice Sheet under the Action of a Local Time-Periodic Load” in Proc. of the 45th All-Russia. Symp. Dedicated to the 70th Anniversary of the Victory, Miass, December 22–24, 2015, Vol. 1: Mechanics and Control (Izd. Ross. Akad. Nauk, Moscow, 2015), pp. 102–113.
B. Noble, Methods Based on the Wiener–Hopf Technique for the Solution of Partial Differential Equations (Pergamon Press, London–New York–Paris, 1958).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © L.A. Tkacheva.
Rights and permissions
About this article
Cite this article
Tkacheva, L.A. Action of a Local Time-Periodic Load on an Ice Sheet with a Crack. J Appl Mech Tech Phy 58, 1069–1082 (2017). https://doi.org/10.1134/S002189441706013X
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S002189441706013X