The hydroelastic symmetric response of a floating ice sheet caused by a pressure moving either in the ice lead or on the infinite ice sheet with a crack (a lead of zero width) is investigated. The ice sheet is modeled as a viscoelastic thin plate. The water is of constant depth. The flow under the ice is potential and linear. A boundary integral method (BIM) for the flow under the ice is combined with the finite difference method for the ice plate with free-free edge conditions to solve the coupled problem of linear hydroelasticity. Numerical results for deflections and stress distributions are shown to agree well with the available results by others. The proposed approach can be applied to problems with different edge conditions and different positions of the load with respect to the lead. The ice responses are studied with respect to the speed of the load. The speed can be subcritical, critical, and supercritical with respect to the critical speed for a floating infinite elastic plate. The speeds of the load, which provide maximum deflection, maximum stress, and maximum wave-making resistance, are determined. All these speeds are different and greater than the critical speed for an infinite elastic plate. The effect of the ice thickness, lead width, and load properties on these speeds is discussed.

1.
Batyaev
,
E. A.
, and
Khabakhpasheva
,
T. I.
Hydroelastic waves in a channel covered with a free ice sheet
,”
Fluid Dyn.
50
(
6
),
775
788
(
2015
).
2.
Bonnefoy
,
F.
,
Meylan
,
M. H.
, and
Ferrant
,
P.
, “
Nonlinear higher order spectral solution for a two-dimensional moving load on ice
,”
J. Fluid Mech.
621
,
215
242
(
2009
).
3.
Brocklehurst
,
P.
, “
Hydroelastic waves and their interaction with fixed structures,” Ph.D.
thesis (
University of East Anglia
,
2012
).
4.
Chung
,
H.
, and
Linton
,
C. M.
, “
Reflection and transmission of waves across a gap between two semi-infinite elastic plates on water
,”
Q. J. Mech. Appl. Math.
58
(
1
),
1
15
(
2005
).
5.
Cui
,
P.
,
Zhang
,
A. M.
,
Wang
,
S. P.
, and
Khoo
,
B. C.
, “
Ice breaking by a collapsing bubble
,”
J. Fluid Mech.
841
,
287
309
(
2018
).
6.
Davys
,
J. W.
,
Hosking
,
R. J.
, and
Sneyd
,
A. D.
, “
Waves due to a steadily moving source on a floating ice plate
,”
J. Fluid Mech.
158
(
1
),
269
(
1985
).
7.
Dinvay
,
E.
,
Kalisch
,
H.
, and
Parau
,
E. I.
, “
Fully dispersive models for moving loads on ice sheets
,”
J. Fluid Mech.
876
,
122
149
(
2019
).
8.
Doctors
,
L. J.
, and
Sharma
,
S. M.
, “
The wave resistance of an air-cushion vehicle in steady and accelerated motion
,”
J. Ship Res.
16
(
4
),
248
260
(
1972
).
9.
Ertekin
,
R. C.
, and
Xia
,
D. W.
, “
Hydroelastic response of a floating runway to cnoidal waves
,”
Phys. Fluids
26
(
2
),
027101
(
2014
).
10.
Evans
,
D. V.
, and
Porter
,
R.
, “
Wave scattering by narrow cracks in ice sheets floating on water of finite depth
,”
J. Fluid Mech.
484
,
143
165
(
2003
).
11.
Hess
,
J. L.
, and
Smith
,
A. M. O.
, “
Calculation of nonlifting potential flow about arbitrary three dimensional bodies
,”
J. Ship Res.
8
(
2
),
22
44
(
1964
).
12.
Hosking
,
R. J.
,
Sneyd
,
A. D.
, and
Waugh
,
D. W.
, “
Viscoelastic response of a floating ice plate to a steadily moving load
,”
J. Fluid Mech.
196
(
1
),
409
430
(
1988
).
13.
Khabakhpasheva
,
T. I.
,
Shishmarev
,
K.
, and
Korobkin
,
A. A.
, “
Large-time response of ice cover to a load moving along a frozen channel
,”
Appl. Ocean Res.
86
,
154
165
(
2019
).
14.
Kheisin
,
D. Y.
, “
Moving load on an elastic plate which floats on the surface of an ideal fluid (in Russian)
,”
Ivz. AN SSSR, Otd, Tekh. i Mashinostroenie
1
,
178
180
(
1963
).
15.
Korobkin
,
A. A.
,
Khabakhpasheva
,
T. I.
, and
Papin
,
A. A.
, “
Waves propagating along a channel with ice cover
,”
Eur. J. Mech. B
47
,
166
175
(
2014
).
16.
Kozin
,
V. M.
, and
Pogorelova
,
A. V.
, “
Wave resistance of amphibian aircushion vehicles during motion on ice fields
,”
J. Appl. Mech. Tech. Phys.
44
(
2
),
193
197
(
2003
).
17.
Kozin
,
V. M.
, and
Pogorelova
,
A. V.
, “
Effect of the viscosity properties of ice on the deflection of an ice sheet subjected to a moving load
,”
J. Appl. Mech. Tech. Phys.
50
(
3
),
484
492
(
2009
).
18.
Letcher
,
J. S.
, “
Properties of finite-difference operators for the steady-wave problem
,”
J. Ship Res.
37
(
1
),
1
7
(
1993
).
19.
Li
,
Y.
,
Liu
,
J.
,
Hu
,
M.
, and
Zhang
,
Z.
, “
Numerical modeling of ice-water system response based on rankine source method and finite difference method
,”
Ocean Eng.
138
,
1
8
(
2017
).
20.
Li
,
Z. F.
,
Shi
,
Y. Y.
, and
Wu
,
G. X.
, “
Interaction of wave with a body floating on a wide polynya
,”
Phys. Fluids
29
,
097104
(
2017
).
21.
Li
,
Z. F.
,
Shi
,
Y. Y.
, and
Wu
,
G. X.
, “
Interaction of waves with a body floating on polynya between two semi-infinite ice sheets
,”
J. Fluids Struct.
78
,
86
108
(
2018a
).
22.
Li
,
Z. F.
,
Wu
,
G. X.
, and
Ji
,
C. Y.
, “
Wave radiation and diffraction by a circular cylinder submerged below an ice sheet with a crack
,”
J. Fluid Mech.
845
,
682
712
(
2018b
).
23.
Liu
,
J. B.
,
Zhang
,
Z. H.
,
Zhang
,
L. Y.
, and
Yao
,
J.
, “
Application of mixed BEM and FDM in numerical simulation of ice-breaking by air cushion vehicle (in Chinese)
,”
J. Naval Univ. Eng.
25
(
3
),
50
55
(
2013
).
24.
Marchenko
,
A. V.
, “
Resonance interactions of waves in an ice channel
,”
J. Appl. Mathematics Mech.
61
(
6
),
931
940
(
1997
).
25.
Mase
,
G. E.
Theory and Problems of Continuum Mechanics
(
McGraw-Hill
,
New York
,
1970
).
26.
Milinazzo
,
F.
,
Shinbrot
,
M.
, and
Evans
,
N. W.
, “
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
).
27.
Ni
,
B. Y.
,
Han
,
D. F.
,
Di
,
S. C.
, and
Xue
,
Y. Z.
, “
On the development of ice-water-structure interaction
,”
J. Hydrodynamics
32
(
4
),
629
652
(
2020
).
28.
Parau
,
E. I.
, and
Dias
,
F.
, “
Nonlinear effects in the response of a floating ice plate to a moving load
,”
J. Fluid Mech.
460
,
281
305
(
2002
).
29.
Parau
,
E. I.
, and
Vanden-Broeck
,
J. M.
, “
Three-dimensional waves beneath an ice sheet due to a steadily moving pressure
,”
Philos. Trans. R. Soc. A
369
(
1947
),
2973
2988
(
2011
).
30.
Parau
,
E. I.
,
Vanden-Broeck
,
J. M.
, and
Cooker
,
M. J.
, “
Three-dimensional capillary-gravity waves generated by a moving disturbance
,”
Phys. Fluids
19
(
8
),
082102
(
2007
).
31.
Pogorelova
,
A. V.
, and
Kozin
,
V. M.
, “
Flexural-gravity waves due to unsteady motion of point source under a floating plate in fluid of finite depth
,”
J. Hydrodynamics
22
(
5-supp-S1
),
71
76
(
2010
).
32.
Porter
,
R.
, “
Trapping of waves by thin floating ice floes
,”
Quart. J. Mech. Appl. Math.
71
(
4
),
463
483
(
2018
).
33.
Ren
,
K.
,
Wu
,
G. X.
, and
Thomas
,
G. A.
, “
Wave excited motion of a body floating on water confined between two semi-infinite ice sheets
,”
Phys. Fluids
28
(
12
),
127101
(
2016
).
34.
Ren
,
K.
,
Wu
,
G. X.
, and
Li
,
Z. F.
, “
Hydroelastic waves propagating in an ice-covered channel
,”
J. Fluid Mech.
886
(
A18
),
1
24
(
2020
).
35.
Raven
,
H.
, “
A solution method for the nonlinear ship wave resistance problem
,” Ph.D. thesis (
TU Delft
,
1996
).
36.
Schulkes
,
R. M. S. M.
, and
Sneyd
,
A. D.
, “
Time-dependent response of floating ice to a steadily moving load
,”
J. Fluid Mech.
186
(
1
),
25
46
(
1988
).
37.
Shi
,
Y. Y.
,
Li
,
Z. F.
, and
Wu
,
G. X.
, “
Interaction of wave with multiple wide polynyas
,”
Phys. Fluids
31
,
042106
(
2019
).
38.
Shishmarev
,
K.
,
Khabakhpasheva
,
T. I.
, and
Korobkin
,
A. A.
, “
The response of ice cover to a load moving along a frozen channel
,”
Appl. Ocean Res.
59
,
313
326
(
2016
).
39.
Shishmarev
,
K.
,
Khabakhpasheva
,
T. I.
, and
Korobkin
,
A. A.
, “
Ice response to an underwater body moving in a frozen channel
,”
Appl. Ocean Res.
91
,
101877
(
2019
).
40.
Squire
,
V. A.
, “
Of ocean waves and sea-ice revisited
,”
Cold Regions Sci. Technol.
49
(
2
),
110
133
(
2007
).
41.
Squire
,
V. A.
,
Hosking
,
R. J.
,
Kerr
,
A. D.
, and
Langhorne
,
P. J.
,
Moving Loads on Ice Plates
(
Kluwer Academic Publishers
,
1996
).
42.
Squire
,
V. A.
,
Robinson
,
W. H.
,
Haskell
,
T. G.
, and
Moore
,
S. C.
, “
Dynamic strain response of lake and sea ice to moving loads
,”
Cold Regions Sci. Technol.
11
(
2
),
123
139
(
1985
).
43.
Sturova
,
I. V.
, “
Motion of an external load over a semi-infinite ice sheet in the subcritical regime
,”
Fluid Dyn.
53
(
1
),
49
58
(
2018
).
44.
Sturova
,
I. V.
, and
Tkacheva
,
L. A.
, “
The motion of pressure distribution over a free surface near the edge of ice sheet
,”
IOP Conf. Ser.
193
(
1
),
012065
(
2018
).
45.
Sturova
,
I. V.
, and
Tkacheva
,
L. A.
, “
Movement of external load over free surface of fluid in the ice channel
,”
J. Phys.: Conf. Ser.
1268
,
012066
(
2019
).
46.
Takizawa
,
T.
, “
Deflection of a floating sea ice sheet induced by a moving load
,”
Cold Regions Sci. Technol.
11
(
2
),
171
180
(
1985
).
47.
Tkacheva
,
L. A.
, “
Edge waves produced by the motion of a vessel in an ice channel
,”
J. Appl. Mech. Tech. Phys.
60
(
5
),
850
864
(
2019a
).
48.
Tkacheva
,
L. A.
, “
Wave motion in an ice sheet with crack under uniformly moving Load
,”
Fluid Dyn.
54
(
1
),
14
32
(
2019b
).
49.
Yuan
,
G. Y.
,
Ni
,
B. Y.
,
Wu
,
Q. G.
,
Xue
,
Y. Z.
, and
Zhang
,
A. M.
, “
An experimental study on the dynamics and damage capabilities of a bubble collapsing in the neighborhood of a floating ice cake
,”
J. Fluids Struct.
92
,
102833
(
2020
).
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