Steady-state rectilinear motion of a load on an ice sheet modeled by a viscoelastic plate is considered. The viscoelastic properties of ice are described using the linear Maxwell and Kelvin-Voigt models and a generalized Maxwell-Kelvin model. Calculated vertical displacements and strains of the ice plate are compared with available experimental data.Key words: incompressible liquid, viscoelastic plate, steady-state motion of load, ice sheet.
1.The hydrodynamic problem of a load moving on a continuous ice is modeled by a system of surface pressures which moves above a floating ice plate [1,2].We consider an infinite region covered with continuous ice on which a given system of surface pressures q moves at a velocity u. The coordinate system attached to the load is arranged as follows: the plane xOy coincides with the ice-water unperturbed interface, the x direction coincides with the direction of motion of the load, and the z axis is directed vertically upward. It is assumed that water is an ideal incompressible liquid of density ρ 2 and the motion of the liquid is potential. The ice sheet is modeled by a viscoelastic homogeneous isotropic plate which is initially not stressed. The linear viscoelastic material simulating ice is described using the Maxwell and Kelvin-Voigt models and a generalized Maxwell-Kelvin model [3]. In the case of pure shear, the behavior of a Maxwell body can be described by a mechanical model consisting of a spring and a viscous damper connected in series, and the behavior of a Kelvin-Voigt body by a model consisting of a spring and a viscous damper connected in parallel. In the case of a one-dimensional state, the generalized Maxwell-Kelvin model considered in the present paper represents Maxwell and Kelvin units connected in series. This model takes into account instantaneous elastic response, viscous flow and delayed elastic response.By analogy with [1,4], using the generalized model of a viscoelastic Maxwell-Kelvin material to describe the deformation of an ice sheet under the action of a moving load, it is possible to derive the following equation of small oscillations of a floating viscoelastic plate:are the shear elastic moduli of ice corresponding to Maxwell and Kelvin materials, E M and E K are Young moduli for Maxwell and Kelvin bodies, respectively, ν is the Poisson ratio, τ M and τ K are stress and strain relaxation times of ice, respectively, h(x, y) is the ice thickness, ρ 1 (x, y) is the ice density, w(x, y) vertical displacement of ice, and Φ = Φ(x, y, z) is a function of the liquid velocity potential which satisfies the Laplace equation ΔΦ = 0. Below, it is assumed that ρ 1 and h are constants. As the calculated 1
