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.

Conclusion. Mathematical models of wave propagation under the influence of a moving load on a nonlinearly compressible and irreversible unloading soil layer with a base have been built. An analytical solution is obtained for the problem of the propagation of a plastic wave in the case when the relationship between pressure and volumetric deformation during loading and unloading is linear, but different. Based on the analysis of the calculation results, it is shown that if the moving load acting on the boundary has a monotonically decreasing profile, then in the perturbation region, the medium is unloaded and an oblique compression wave is obtained by the load-unloading wave. The pressure of the medium against the background of this wave, depending on the depth, decreases slowly than on the free surface. In the case when the dependence between and under loading of the medium is assumed to be nonlinear and shock, which corresponds to the propagation of a shock wave in the medium, the pressure in the perturbed region, in comparison with the linear case, is somewhat overestimated.

1. Rakhmatulin Kh.A., Demyanov Yu.A. Durability under intense short-term loads. Moscow, Logos. 2009. –P. 512 P.

2. A. Aydosov, G. A. Aidosov, E.S. Temirbekov, S.N. Toybaev. Mathematical modeling of shock load propagation in continuous deformable media and interaction of two deformable media under dynamic moving loads - Almaty, 2015, - 208 P. - ISBN 928-601-263-327-6.

3. A. Aydosov, E. S. Temirbekov. Impact theory - Study guide. - Almaty: ATU, 2015. - 64 P.- ISBN 978-601-263-323-8.

4. A. Aydosov, G. A. Aidosov, M.N. Kalimoldaev, S.N. Toybaev, Mathematical modeling of the interaction of a beam (plates, plates, strips) with a deformable base under dynamic loads - Monograph / - Almaty, 2015, - 208 P.

5. Aidosov G.A., Aidosov A.A., Toybaev S.N. Modeling the interaction of a beam (plates, slabs, strips) of variable thickness, lying on a non-uniform foundation. Formulation of the problem. Bulletin of K.I. Satpayev KazNTU, №4, Almaty: 2009. – P. 41–45. 0,25

6. Aidosov A.A., Aidosov G.A., Toybaev S.N. Modeling the interaction of a beam (plates, slabs, strips) of variable thickness, lying on an inhomogeneous foundation. General equations. Bulletin of Al-Farabi KazNU (60), Almaty: 2009. – P. 48 53.

7. Aidosov A.A., Aidosov G.A. Toybaev S.N. The main conclusions of modeling the propagation of blast waves in a multilayer inhomogeneous half-space Science news of Kazakhstan Scientific and technical collection. Issue 2 (101), Almaty: 2009. – P. 56- 60.

8. Aidosov, A., Mamadaliev, N., Khakimov, U. Effect of a mobile load on a nonlinearly compressed strip with a rigid foundation (Article) - Journal of Applied Mechanics and Technical Physics Volume 27, Issue 3, May 1986, Pages 441-445 - Scopus – 36 pr.

9. Aidosov A.A., Aidosov G.A., Toybaev S.N. Modeling the interaction of a beam (plates, slabs, strips) of variable thickness, lying on a non-uniform foundation. Formulation of the problem// Bulletin of K.I. Satpayev KazNTU. – Almaty, 2009. - №4. – P.41–46.

10. Aidosov A.A., Aidosov G.A., Toybaev S.N. Modeling the interaction of a beam (plates, slabs, strips) of variable thickness lying on an inhomogeneous foundation // Approximate equation. KazNU Bulletin. – Almaty, 2009. - № 2 (61). – P.51-56.