Introduction. The development of the polar areas of the World Ocean and a large number of freezing inland reservoirs drive the need to study the fields of displacement and stresses of the ice cover caused by the action of various types of loads. Numerous papers by domestic and foreign scientists are devoted to solving these problems. Previously, it has been found that the mechanical properties of ice depend on its temperature and water salinity. Much attention was paid to the development of numerical ice models that accurately reflected the interaction of ice and ideal incompressible fluid. In [1][2], the smoothed particle hydrodynamics was used for this purpose, in [3][4] — the method of discrete elements. In [5], ice was modeled by an elastic plate lying on the surface of a stratified fluid. Models that allow cracks were considered in [6][7]. Models of ice strengthened with reinforcing elements were presented in [8][9].

At this, in some papers, an ice cover is considered as an elastic plate lying on the surface of a reservoir [10][11]. At the same time, in [12], on the basis of the conducted research, it is concluded that in some cases, the properties of ice are best described by the Kelvin-Voigt rheological model with one parameter (damping time). Therefore, numerous researchers use a viscoelastic plate when modeling the ice cover [13]. In [14], nonlinear models were used to describe the properties of ice.

In some articles, the effect of a moving load on the ice cover was considered. In [15], the impact of a mobile load on the ice cover in a frozen channel was studied. In [16], the action of a load with an impulsive movement on the ice cover was described. Paper [17] is devoted to the study of the load moving along a frozen riverbed. Here, the rectilinear motion of the load was investigated [18]. However, in real conditions, it is often needed to deal with a load moving in a more complex way. Therefore, the objective of this work is to develop a method for solving problems about the action of a load moving over an ice cover along a complex trajectory. This will provide for a more accurate investigation of the effect of vehicles moving in a complex way on ice.

This work is a continuation of research related to the problems about the effect of a moving load on various objects, whose results are presented in papers [19][20].

Materials and Methods. Setting the task. A reservoir of finite depth with an infinite ice cover (an infinite plate), which is subject to the action of a vertical force moving in an arbitrary way — impulsively, is considered. It is assumed that the reservoir fluid is incompressible and executes irrotational motion.

The problem is reduced to a system of differential equations [15]: where W(x,y,t) — ice cover deflection; Е and μ — Young's modulus and Poisson's ratio of ice, respectively; D = Eh3/12(1–μ2) — cylindrical bending stiffness; h — ice cover thickness; τo — strain relaxation time; ∆o2 = (∂x2+∂y2)2; ∆ = ∂x2+ ∂y2+∂z2; ρл and ρв — density of ice and water, respectively; с-2 = ρлh/D; k = ρвg/D; b = ρв/D; Q(x,y,t) — load acting on the ice surface; F(x,y,z,t) — speed potential. Under boundary conditions at z = 0 (ice-water boundary): At the bottom of the reservoir at z = –H:

In addition, it was assumed that the ice cover and the fluid in the reservoir were resting at the start time. The load was a concentrated unit force (one Newton) Q (x, y, t), which moved arbitrarily along open free-form curve γ. It was assumed that the displacement of the force was given in the form Q = Q(s(t)), where s — arc coordinate measured from some fixed point of trajectory γ. The trajectory of movement was set parametrically in the form

where t — time.

The moving load was approximated by the expression:

where ε — numeric parameter.

After applying the integral Fourier transform with respect to variables x and y, the integral Laplace transform with respect to t, formulas for calculating unknown functions W and F were obtained:

,

,

,

Using the known relations from the theory of thin plates and the theory of the potential ideal flow, it is possible to obtain relations for calculating displacements and stresses in the ice cover, as well as the velocity vector component of fluid particles.

When calculating the improper integral through numerical methods, the approximate relationship

was used, in which value А was chosen so large, that the error estimate

did not exceed the set value.

Thus, for the amount of ice deflection

this estimate has the form:

Similar estimates can be obtained for other calculated quantities. These estimates were used to determine value А.

In the calculations carried out, value А was chosen such, that the estimate

did not exceed

0.001.

When calculating the repeated integral, Simpson's quadrature formula (for variable τ) and Chebyshev quadrature formula with equal weights for two nodes (for variable p) were used. The other values were calculated in the same way.

Research Results. A method has been developed for solving problems on the action of a load moving along the ice cover of a reservoir filled with ideal fluid along a complex trajectory with variable speed. Using this method, calculations were carried out that showed the degree of impact of various parameters on the deformation of the ice cover.

The described method does not impose restrictions on the shape of the trajectory of the concentrated force. In the calculations, a special case of a trajectory consisting of arcs of circles was considered (Fig. 1). The red dot indicates the position of the concentrated force at the time under consideration and the direction of movement of the force.

The following parameter values were taken into account in the calculations: ice cover thickness h = 0.25 m, Young's modulus E = 500,000,000 H/m2, Poisson's ratio of ice μ = 1/3, ice density ρ = 900 kg/m3, fluid density ρ = 1,000 kg/m3, ε = 5. The calculation results are presented below. Figure 2 shows the change in the deflection of the ice cover at speed of force v = 2.5 m/s, tangential acceleration wt = 1 m/s2, reservoir depth H = 25 m, and relaxation time τo = 1s. The law of force motion along the trajectory was taken as: Coefficients a1, a2, a3 were selected in such a way that the force, being at the same point of the trajectory, had the required speed and tangential acceleration.

At other values of these parameters, the qualitative nature of the distribution of ice cover deflections remained almost unchanged.

The fluid motion caused by the action of a moving load at the same values of velocity, tangential acceleration of the load movement, relaxation time, and depth of the reservoir is shown in Figure 3 (the distribution of the velocity vector of fluid particles is shown).

The effect of the tangential acceleration of the force movement on the maximum deflection of the ice cover is shown in Figure 4 a. In this case, the force speed was equal to v = 17.5 m/s, and the relaxation time τo = 1 s. Figure 4 b shows a graph of the dependence of the maximum deflection of the ice cover W on the reservoir depth H. Here, the speed of the load movement was v = 17.5 m/s, tangential acceleration wt = 1 m/s2, and relaxation time τo = 1s.

The dependence of the maximum deflection of the ice cover on the force speed is graphically shown in Figure 5. The depth of the reservoir was assumed to be 25 m, and the tangential acceleration — wt = 1 m/s2. The solid line shows the dependence corresponding to the relaxation time τo = 1 s, the dotted line corresponds to the relaxation time τo = 10 s.

Discussion and Conclusion. The influence of the reservoir depth on the maximum deflection of the ice was studied. A picture of the deflection of the ice cover was obtained due to the action of a concentrated force moving along a complex trajectory with variable speed. Calculations showed that with increasing depth of the reservoir, the maximum deflection of the ice cover decreased (Fig. 2). At the same time, a noticeable dependence of the deflection of the ice cover on the depth of the reservoir H occurred only for H≤25 m. At great depths, the amount of the maximum deflections stabilized near a certain constant value and practically did not change. Thus, if H>25 m, then the depth of reservoirs can be considered infinite when calculating.

An increase in tangential acceleration caused an increase in the deflection of the ice cover. Moreover, the dependence of deflection on tangential acceleration was very close to the linear dependence (Fig. 4).

At low relaxation times τo, the speed of the load movement affected significantly the amount of ice deflection. At large times, the impact of the load movement speed on the deflection of the ice cover was noticeably reduced (Fig. 5).

To study the state of the reservoir fluid, the distribution of the velocity vector of the fluid particles due to the action of the moving force on the ice was determined (Fig. 3).

The developed method of solving problems and the results obtained with its help can be used in the construction of ice roads, design and construction of runways on ice.