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)

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