Mechanical and mathematical research of local deformations of a steel roller shell with a variable geometry of contact surface

Abstract The article is devoted to solving the fundamental and applied problem of nonlinear structural mechanics of machines by introducing into the drum two additional stop cylinders with supporting rollers at the end and adjustable length, providing a given elliptical or circular shape of a flexible shell with a smoothly variable geometry in the area of its contact with compacted pavement material. Compaction of soil, gravel and asphalt concrete in the sphere of road is not only an integral part of the technological process of the roadbed, road foundation and surface construction, but it is actually the main operation to ensure their strength, stability and durability. The quality, cost and speed of road construction, the possibility of using fundamentally new technologies, structures and materials is largely determined by the availability of modern road machinery.


Introduction
One of the main directions of scientific and technological progress is the creation and introduction of new equipment, the improvement of the existing one, as well as the improvement of its quality, reliability, durability with minimal material consumption and cost per unit of power.
A summary of the issue of quantifying the carrying capacity of the shell can only be done after joint mathematical modeling of the physical process of interaction between the roller and the material of the sealing coating and solving the problem of local elastic stability of a thin-walled cylindrical shell (shell) under local contact pressure" (Abdeev et al., 2011;Sakimov et al., 2018).
The article presents the analysis of the behavior of the deformable roller shell of the road roller, and the material to be compacted under the compacting roller of the road roller (Dudkin et al., 2006), in which the rigid circular shell of the roller is replaced by a forcefully deformable elliptical shape, which, unlike the circular design, allows variation, adjustment and optimization of the impact of the road roller on the material to be compacted.

Mathematical Model
This article is devoted to solving this fundamental and applied problem of nonlinear structural mechanics of machines (Birger et al., 1979) by introducing into the drum two additional stop cylinders with supporting rollers at the end and adjustable length, providing a given elliptical or circular shape of a flexible shell with a smoothly variable geometry (Dudkin et al, 2006;Abdeev et al., 2012et al., 2018 in the area of its contact with compacted pavement material, based on: 1) the closest and symmetrical positioning at the same distance lр≥С of roller support-cylinders, the length of which in the design scheme of figure 1 is almost identical to the width B of the roller (figure 2), and their rigidity, according to size, is much more than deformability of the flexible shell having the thickness ≪ р -the diameter of the rollers; 2) the distribution locality of contact pressure functions С = С ( 1 ),q p= qp(x1),, at which the lengths of the corresponding projections С,Cp pologic shell arcs in contact with the material of the coating being compacted are comparable to its thickness δ, and in this connection, as is known (Ponomarev and Andreeva, 1980), in small areas of the simulated thinwalled system bounded by 2C×B areas (stationary roller, figure 1) and CП× В (rotating movable roller, figure 1) obvious conditions are observed (Sakimov et al., 2018;Bostanov et al., 2018) confirming the low probability of clipping, since within the constraints (1) the shell will deform like a cylindrical flat shell-panel of relatively large thickness δ and a small bend (within the limits of size δ), in which (Kolkunov, 1972) At the same time, in order to guarantee the tuning of the considered structure in order to exclude the possibility of local deformations from the reactive distributed forces с , (n), it is necessary to solve the stability test proble. For its mathematical formulation, with a margin of rigidity mand bearing capacity, an idealized model of a linearly elastic homogeneous isotropic rectangular plate-panel with a plan size 2 р × В (Fig. 2) is used with an initial bend 1 = ( 1 ), approximated by circle or ellipse functions and fixed projections р ≥ С = , symmetrically located arcs Sр. At the same time, for the calculated pressure р ( 1 ) with the greatest extremum q=max (Fig. 2) the functional expression of Hertz-Shtererman is taken (Sakimov et al., 2018) changing on a plane − р ≤ 1 ≤ р , −0,5 ≤ 1 ≤ 0,5 by the same even law as addiction с = с ( 1 ), in the case of a stationary drum ( Fig. 1) while limiting the maximum value < кв regulatory minimum safety factor � у � = 1,5 for an ideal original shell surface (no dents) in relation to the upper critical load , corresponding to the unacceptable phenomenon of flipping the shell to a new stationary equilibrium state with the formation of a local bend (Temirbekov et al., 2019) Each mechanical system with the loss of stability can behave differently. A transition to a certain new equilibrium state usually takes place, which in the overwhelming majority of cases, is accompanied by large displacements, the appearance of plastic (residual) deformations or complete destruction. In relation to the considered road roller with flexible elastic shell (Dudkin et al., 2006;Abdeev et al., 2012), the listed negative factors lead to the impossibility of its further operation or to a significant reduction in the quality indicators of the fulfillment of its functional purpose. On this basis, the topic of this article is extremely relevant, innovative and promising.
The corresponding applied physical and mathematical problem of an elastically deformable solid body is based on the fundamental system of two nonlinear fourth-order differential equations (Fig. 2). (5) where Е, µ -respectively, the modulus of elasticity and Poisson's ratio of the shell material (structural spring steel (Abdeev et al., 2012)); 1 -Support rollers at the end of the hydraulic cylinders, creating and supporting with the help of forces P, an elliptical shape of the shell, 2 -Hydraulic cylinders with roller bearings of cylindrical type, providing a predetermined shape of the shell and increasing its bearing capacity (local stability) in the area of action of reactive distributed loads , -the greatest linear geometrical parameters within which contact pressures act с , п for stationary (C) and rolling (P) rollers (Sakimov et al., 2018): − arbitrary coordinate fixing the natural ("smooth") outline of a shallow elliptical cylindrical shell when р = 0 ( Fig. 2): = ( 1 , 1 ) -additional deflection, measured parallel to the axis 1 from the original middle surface (10) of the shell; Е к -specified normalized modulus of soil deformation or pavement, compacted to the termination of residual displacements (Sakimov et al., 2018); = ( ), = ( ) -the dimensions of the semi-axes of the ellipse presented in Figure 1 and described by the wellknown equation: depending on eccentricity (Abdeev et al., 2012.): -the calculated mass of the roller and the corresponding part of the frame; = ( 1 , 1 ) -desired function of normal 1 , 1 and tangent local stresses (Kolkunov, 1972)  (13) (Kolkunov, 1972;Ponomarev and Andreeva, 1980).
In relation to the roller circular shape with a radius of the middle surface = should, in the above relations (10) -(12), make a replacement = = , as a module =0, according to (12).
In the reserve of local deformability of the shell and in order to simplify the integration of system (5), (6), due to the design features of the design schemes of figures 1 and 2, it is followed by the description of the final state of the element of the shallow shell ( Figure 2) by the mechanicalmathematical model of the cylindrical bend of a rectangular plate with the initial curvature y1 (10) (Kolkunov, 1972. Ponomarev andAndreeva, 1980), when the desired functions ( 1 , 1 ) and ( 1 , 1 ), take the form: and the stresses (13) in the middle surface become equal under the assumption that the internal force components 1 is considered positive under compression. In this case, the equilibrium condition (5) is transformed into an ordinary differential equation and relation (6) is identically satisfied 0≡0.
It is necessary to supplement dependence (17) by Hooke's law for constant linear relative deformation and geometrically nonlinear Cauchy formula (Kolkunov, 1972): where = ( 1 ) -displacement function in the direction of the coordinate axis 1 (Fig. 3) The equation (17) is integrated approximately using the Bubnov-Galerkin method of sufficiently effective and accurate variational analytical method Doudkin et al., 2019). The function ( 1 ) is approximated by a polynomial of the fourth degree (Kolkunov, 1972. Ponomarev andAndreeva, 1980).
satisfying the boundary conditions of Figure 3: where the maximum calculated chord length р , identical to the distance between roller bearings (Fig. 1), is found, according to (8) and guided by (Sakimov and Eleukenov, 2012;Sakimov et al., 2018).