MATHEMATICAL MODELS OF PROPAGATION AND REFLECTION OF A PLASTIC WAVE IN A BAND LYING ON AN ELASTIC HALF-SPACE, WHICH HAS THE PROPERTY OF LINEAR COMPRESSIBILITY AND LINEAR IRREVERSIBLE UNLOADING  Abstract (English):

Keywords:
Mathematical models, propagation of a plastic wave, half-space, analytical solution, wave front, ideal fluid, linear compressibility, irreversible unloading. equation of motion, continuity, states of the environment

Formulation of the problem. Let us consider the problem of the propagation of a plastic wave in a two-layer medium with a plane-parallel interface under the action of an intense load of a falling profile moving along its upper boundary with a constant super seismic velocity D.

A two-layer medium consists of a soft soil layer of thickness h with an elastic deformable base. The soil is modeled by an inelastic ideal medium with linear compressibility and linear irreversible unloading. Consequently, the shear resistance of the medium is neglected.

According to this statement, the influence of the deformability of the base and the load profile on the distribution of the dynamic parameters of the layer and the contact surface was investigated. The results of the numerical calculation are compared with the results of the acoustic layer and the layer with a rigid base. The solution of the problem is constructed in series, and their convergence is proved.

Let a monotonically decreasing normal load move along the upper boundary of the layer with an elastic base with a speed D exceeding the speed of wave propagation. The layer material has such a property that, under loading and unloading, the relationship between pressure and volumetric deformation is linear and irreversible, the slope of the unloading branch of the diagram exceeds the slope of the loading branch, i.e. .

Under the action of the above load, a compression wave , first propagates in the layer, which is reached by the contact line of the media, induces a reflected plastic wave , in the layer, and in the second medium a system of elastic (longitudinal and transverse) waves and . At he speed of propagation of the AD  characteristic is greater than the speed of the front , therefore, as in the previous section, regions 2, 3, 4, etc. appear. On the and system, the layer material is instantly loaded, and then in areas 1, 2, 3, the medium is unloaded. Taking into account that the solution of the problem in domains 1 and 2 was obtained in the previous section, below we propose a solution to the problem only in domains 3 of layers and of the elastic half-plane. For the joint problem of the domain 3, (5.1.7) holds and the equations for the displacement potentials , the elastic half-plane , ,                                     (1) .

and according to the d'Alembert formula, their solutions are represented in the form .  (2)

where initial density and Lame coefficients of an elastic medium.

The boundary conditions for this problem are as follows:

at the front of the reflected wave at  ,                                                     (3)

on contact AE of two media at  .               (4)

Here, stress components in an elastic medium. To find the function from (3) and (4), taking into account (2), we obtain the functional equation ,                        (5)

where , , .

The solution to equation (5) is constructed by the method of successive approximations. Indeed, taking as the zero approximation .

for the first approximation we have .

Then, continuing the iteration process, we obtain a recurrent formula of the form .      (6)

Research has shown that and are monotonically decreasing functions.

Consequently, according to the d'Alembert criterion, series (6) converges absolutely, and one can set the radius of its convergence. Then the solution of the problem taking into account (6) takes the form (7) .                                                       (8)

where  In this case, the normal stress of the elastic half-plane in the regions and is determined by the formulas  (5.5.9)

at  (10)

At ,   If then , and from (7), (8) for the case of a layer with an absolutely rigid base we have (11) (12)

where  In the future, based on formula (7)-(12), it is necessary to carry out some calculations on a PC and analyze them.

Note that the above technique allows us to solve the problem of the effect of a moving load on a nonlinearly compressible strip lying on an elastic half-space.

Conclusion. The problem of propagation, reflection and a two-dimensional stationary plastic wave in a two-layer medium with densities is investigated for the case when the state diagram of the first medium (soil) is shock and under loading has the form , and the second medium (black rock of a rock or pad) - elastic or rigid plastic. The problem is solved analytically by both direct and inverse methods, taking into account wave processes in the second medium and without taking them into account. Analysis of the results obtained on the PC shows that at taking into account the elastic - plastic properties of the second medium (spacer), modeled by a half-space, leads mainly to a decrease in the maximum values of stresses (pressure) at the contact of two media. At a stress concentration appears on the contact surface, and the pressure acquires the highest value in the case of an acoustic layer lying on a rigid foundation. The qualitative and quantitative picture of changes in the values of pressure and kinematic parameters depends not only on the stiffness characteristics of the media, but also on the ratio of their densities.

Thus, the above studies on the study of the two-dimensional stress-strain state of a homogeneous, inhomogeneous and layered medium under intense action of a mobile load on the boundary of a multilayer half-space confirm the need and importance of taking into account nonlinear, irreversible, wave processes.

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