MATHEMATICAL MODELS OF THE PROPAGATION OF A PLASTIC WAVE IN A HALF-SPACE WITH THE PROPERTY OF LINEAR COMPRESSIBILITY AND LINEAR IRREVERSIBLE UNLOADING  Abstract (English):

Keywords:
Mathematical models, moving load, propagation, plastic wave, soil, half-space, wave front, ideal fluid, linear compressibility, irreversible unloading equation of motion, continuity, states of the environment

Formulation of the problem. The problem of the effect of a moving load of an arbitrarily decreasing profile on a soil layer of finite thickness , lying on a rigid horizontal foundation is considered.

The soil is modeled by an ideal compressible medium, in which the relationship between pressure and volumetric deformation under loading and during unloading of the medium is linear and irreversible.

The load is applied to the upper surface of the layer and moves at a superseismic speed . Since in this case the modulus of volumetric compression of the modulus of unloading of the medium, in the physical plane the characteristic has a greater in comparison with the speed of the reflected wave , and as a result, regions 2, 3, 4 appear, which are separated by the characteristic of the positive direction and the front reflected wave . The parameters of the environment in region 1 are known from the solution of the problem about . Note that this problem is stationary, and therefore all the parameters of the medium depend on two moving coordinates , and the motion of the medium in regions 2 and 3 of loading and unloading is described by the wave equation of the potential of the velocity we have the wave equation [1-4] in plane deformation.

We represent solutions in research areas in the form (1)

where velocity potentials.

To find the unknown functions and i.e. to solve the problem in region 2, we have the conditions for the continuity of the velocities on the characteristic and the condition that at different horizontal levels the pressure of the medium in front of the reflected wave is equal to the pressure at the front of the incident wave. This means that the state of the medium in region 1 is on the unloading branches of the diagram, and after the arrival of perturbations from the rigid boundary using the characteristic , in region 2 the pressure increases continuously to values determined by the points of intersection of the unloading and loading branches of the diagram. Subsequently, under the action of the reflected plastic wave an abrupt increase in pressure occurs. This means that the reduced media in regions 2 and 3, according to the hydrostatic compressed, obeys Prandtal's scheme. A similar picture takes place in the rod theory with the difference that, in this case, the loading of the medium starts from the perturbed unloading region 1.

Thus, to solve the problem in domain 2 in the case of an exponential load we have the conditions

on characteristic , i.e. when  ,                                               (2) ,                                             (3)

where  in the section of the reflected wave front, i.e. when [5-10] .                                                 (4)

where cm.

Then, substituting (1) into (2), (4) taking into account (3), we obtain the expressions (5) (6) .                                      (7)

where  So, the solution to the problem when using (5) and (6) will finally be written in the form ,                                             (8) ,                                            (9)

where        Now let's start solving the problem in region 3. For this, we have the following conditions: at the front of the reflected wave AE, i.e. at [5,6-10], , ,                                   (10)

on a rigid boundary at  .                                                                                  (11)

Given that (12)

from (10) we obtain .                                                           (13)

From the second equation in (1), taking into account (10) and (11) with respect to the function and we obtain a system of equations in the form (14) ,                                               (15)

where   Solving equation (15), by the method of successive approximations, it is easy to obtain the formula .                 (16)

Thus, for region 3 we have a solution to the problem in the form ,                                (17) ,                     (18)

where  The pressure in the region is determined by the formula .                                                                    (19)

Conclusion. An analytical solution to the problem of the propagation of a plastic wave in a half-space is constructed 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 of the half-space has a monotonically decreasing profile, then in the disturbance 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 of the half-space, decreases slowly than on the free surface. In the case when the dependence between and during 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.

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