MATHEMATICAL MODELS OF WAVE PROPAGATION ON A SOIL LAYER WITH THE PROPERTY OF NON-LINEARLY COMPRESSIBLE AND IRREVERSIBLE UNLOADING WITH THE BASE UNDER THE INFLUENCE OF A MOVING LOAD
Abstract (English):

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

Formulation of the problem. Let us consider the problem of the effect of a moving load on a two-layer medium consisting of a soft soil layer and an elastic-yielding pad with thicknesses  and densities . The soil is modeled by a nonlinearly compressible medium, and the pad, which has a weaker, than a soil with a stiffness and a density with a Winkler base. The lower boundary of the two-layer medium is solid and non-deformable. According to the accepted assumptions, the wave process in the spacer is neglected, and the compressed wave  at  from the contact surface of the two media is reflected in the form of the unloading wave of a strong rupture, and the behavior of the soil in regions 1, 2, 3, etc. is determined by the unloading branches of the  diagram.

The problem is of practical importance in assessing the levels of reduction of dynamic loads on various underground structures using a bulk screen with a resilient pad.

The solution to the problem is constructed analytically in both reverse and direct ways. Let's proceed with the presentation of these decisions. In the course of this task, the load profile , was determined, which in the future, when constructing solutions to the problem for areas 2 and 3, is considered given [1,2-5].

Taking into account that the medium in region 2 is in a state of unloading, then to solve the problem with respect to the velocity potential  we have the equation

,           (1)

with the following boundary conditions

at ,                      (2)

,     at ,                          (3)

Where    , horizontal and vertical components of speed; medium pressure in area 2; the angle of inclination of the reflected wave with the line ; Young's modulus spacers.

It is known that equation (1) for  admits a solution of the form

.                                            (4)

Hence

(5)

Substituting (5) into (2), after some transformations, we obtain

Substitute (5) into (3). Then we have

,       (7)

where the dash above means the derivative with respect to the argument.

Equation (7) has a solution of the form

.(5.6.8),

where

Obtain

.                                             (9)

Thus, the solution to the problem in region 2 is expressed by the formulas

,  (10)

(11)

(12)

where

Now let's start solving the problem in region 3. For this we have the equation [6,7-10]

,                                                    (13)

and boundary conditions

at              (14)

at  .           (15)

We represent the solution of equation (6.6.6) in the form

.                           (16)

Then, substituting (16) into (14) and (15) to find the required functions  and  we get the formulas

(17)

(18)

So, to determine the components of the velocity and pressure of the medium in region 3, we have the formulas

(19)

(20)

.                                                   (21)

where

The solution to the problem for the subsequent areas is not given, since it is constructed in a similar way. If the gasket material has a rigid plastic property, i.e.  then for the solution of the problem in the region of 2 substitutions (3) we have the condition [1-4,10]

at                        (22)

In this case, the unknown function , in contrast to (8), is found using the formula

.                                               (23)

Then the velocity components  and  in region 2 are represented as

(24)

(25)

where

In order to study the effect of laying on soil parameters, it is necessary to carry out a series of calculations on a PC for areas 2 and 3.

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