Abstract
The analytical solution to the problem of the behavior of an ice sheet with rectilinear crack under the action of uniformly moving rectangular external pressure zone simulating an air-cushion vehicle is obtained using the Wiener–Hopf technique. The ice cover is simulated by thin elastic semi-infinite plates of constant thickness floating on the surface of an incompressible fluid of finite depth. Two configurations are considered: 1) two semi-infinite plates with free edges (whose thicknesses can be different) are separated by a crack; 2) fluid is bounded by the vertical wall and the ice cover edge can be both free and frozen to the wall. In the case of the contact of plates of the same thickness, as well as in the presence of the wall, the solution is obtained in the explicit form. It is shown that in the case of the contact of identical plates with free edges, edge waveguide modes traveling along the crack are excited when the load moves at a supercritical speed. Both the wave forces acting on the moving body and the deflections of the plates are investigated for various values of the plate thicknesses and the load velocity in the sub- and supercritical regimes.
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References
D. E. Kheisin, “Movement of a load on an elastic plate floating on the surface of an ideal fluid,” Izv. Akad. Nauk SSSR, OTN, Mekh. i Mashinostr. 1, 178–180 (1963).
S. F. Dotsenko, “Steady flexural-gravity three-dimensional waves initiated by traveling perturbations,” in: Tsunami and Internal Waves (Marine Hydrophysical Institute of the Academy of Sciences of the Ukrainian SSR, Sevastopol, 1976), pp. 144–155 [in Russian].
D. Y. Kheisin, Dynamics of Floating Ice Covers (Gidrometeorologicheskoe Izdatel’stvo, Leningrad, 1967; Technical Translation FSTC-HT-23-485-69, US Army Foreign Science and Technology Center, 1967) [in Russian, English translation].
L. V. Cherkesov, Surface and Internal Waves (Naukova Dumka, Kiev, 1973) [in Russian].
V. A. Squire, R. J. Hosking, A. D. Kerr, and P. J. Langhorne, Moving Loads on Ice Plates (Kluwer, Dordrecht, 1996).
V. M. Kozin, V. D. Zhestkaya, A. V. Pogorelova, S. D. Chizhiumov, M. P. Dzhabrailov, V. S. Morozov, and A. N. Kustov, Applied Problems of Ice Sheet Dynamics (Izd-vo “Akademiya Estestvoznaniya,” Moscow, 2008) [in Russian].
A. E. Bukatov and V. V. Zharkov, “Formation of the ice cover’s flexural oscillations by action of surface and internal ship waves. part i. surface waves,” Intern. J. Offshore Polar Engng. 7 (1), 1–12 (1997).
V. A. Squire, “Synergies between VLFS Hydroelasticity and Sea Ice Research,” Int. J. Offshore Polar Eng. ISOPE. 18 (4), 241–253 (2008).
V. M. Kozin, A. V. Pogorelova, V. L. Zemlyak, V. Yu. Vereshchagin, E. G. Rogozhnikova, D. Yu. Kipin, and A. A. Matyushina, Experimental and Theoretical Investigations of the Dependence of Parameters of the Flexural-Gravity Waves Propagating in a Floating Plate on their Excitation Conditions (Press of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 2016) [in Russian].
T. Takizawa, “Deflection of a floating sea ice sheet induced by moving load,” Cold Regions Sci. Tech. 11, 171–180 (1985).
J. W. Davys, R. J. Hosking, and A. D. Sneyd, “Waves due to steadily moving source on a floating ice plate,” J. Fluid Mech. 158, 269–287 (1985).
F. Milinazzo, M. Shinbrot, and N. W. Evans, “A mathematical analysis of the steady response of floating ice to the uniform motion of a rectangular load,” J. Fluid Mech. 287, 173–197 (1995).
R. M. S. M. Schulkes, R. J. Hosking, and A. D. Sneyd, “Waves due to a steadily moving source on a floating ice plate. Part 2,” J. Fluid Mech. 180, 297–318 (1987).
R. W. Yeung and J. W. Kim, “Effects of a translating load on a floating plate-structural drag and plate deformation,” J. Fluids Struct. 14 (7), 993–1011 (2000).
V. M. Kozinand A. V. Pogorelova, “Wave resistance of amphibian air-cushion vehicles during motion on ice fields,” J. Appl. Mech. Techn. Phys. 44 (2), 193–197 (2003).
A. V. Pogorelova, “Wave resistance of an air-cushion vehicle in unsteady motion on ice sheet,” J. Appl. Mech. Techn. Phys. 49 (1), 71–79 (2008).
R. V. Goldshtein, A. V. Marchenko, and A. Yu. Semenov, “Edge waves in a fluid beneath a fractured elastic plate,” Dokl. Ross. Akad. Nauk 339 (3), 331–334 (1994).
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).
A. V. Marchenko, “Natural vibrations of a hummock ridge in an elastic ice sheet floating on the surface of an infinitely deep fluid,” Fluid Dynamics 30 (6), 887–893 (1995).
H. Chung and C. M. Linton, “Reflection and transmission of waves across a gap between two semi-infinite elastic plates on water,” Q. J. Appl. Math. 58 (1), 1–15 (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. of Fluids and Structures 25, 777–793 (2009).
L. A. Tkacheva, “Action of a local time-periodic load on an ice sheet with a crack,” J. Appl. Mech. Techn. Phys. 58 (6), 1069–1082 (2017).
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).
V. D. Zhestkaya and V. M. Kozin, “Stress-deformed state of a semi-infinite ice sheet under the action of a moving load,” J. Appl. Mech. Techn. Phys. 35 (5), 745–749 (1994).
M. Kashiwagi, “Transient responses of a vlfs during landing and take-off of an airplane,” J. Mar. Sci. Technol. 9 (1), 14–23 (2004).
P. Brocklehurst, “Hydroelastic waves and their interaction with fixed structures,” PhD Thesis (University of East Anglia, UK, 2012).
K. Shishmarev, T. Khabakhpasheva, and A. Korobkin, “The response of ice cover to a load moving along a frozen channel,” Applied Ocean Research 59, 313–326 (2016).
I. V. Sturova and L. A. Tkacheva, “Wave motion in a fluid under an inhomogeneous ice cover,” J. Phys.: Conf. Ser. 894, 1–8 (2017). https://doi.org/10.1088/1742-6596/894/1/012092
B. Noble, Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations (Pergamon Press, London, 1958; Izd-vo Inostrannaya Literatura, Moscow, 1962).
I. M. Gel’fand and G. E. Shilov, Distributions and Operations on Them (Fizmatgiz, Moscow, 1959) [in Russian].
M. J. Lighthill, Waves in Fluids (Cambridge University Press, Cambridge, 2001; Mir, Moscow, 1981).
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Russian Text © The Author(s), 2019, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2019, No. 1, pp. 17–35.
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Tkacheva, L.A. Wave Motion in an Ice Sheet with Crack under Uniformly Moving Load. Fluid Dyn 54, 14–32 (2019). https://doi.org/10.1134/S0015462819010154
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DOI: https://doi.org/10.1134/S0015462819010154