and Analysis of Work Process of a Road Roller With Local Deformation of Roll in Compaction Zone

The paper presents a new simple mathematical model of a fundamental and applied elasticity theory problem, considering the stress-strain state of road-roller working members with elliptic surfaces and the road-building material interacting with them. The derived mathematical relations and the developed calculation theory prove the possibility of adjusting (optimizing) the parameters of contact interaction of flexible working members in roadrollers with elliptic surfaces, with the compacted layer by means of variating the design-constructive dimensions of semi-axles of the ellipseshaped surface of the roll under service conditions, thus adjusting force action on the compacted road-building material. The paper deals with an operation peculiarity of flexible working members with elliptic surfaces, offers a selection of materials and determines geometric parameters of flexible working members with elliptic surfaces. Based on the analysis of stress-strain state of the roll, two cases are considered suitable for constructive realization of the examined mathematical model.


INTRODUCTION
The development of a network of hard motor roads is a topical issue.Rates of motor transport and freight traffic development require intensification of motor roads construction process and increase service life.The increasing scope of motor roads construction and modernization involves, in addition to qualitatively new construction methods, high production machines and complexes.Implementation of these measures based on effective road-building machinery and equipment ensures a significant economy of material, energy and labor resources in construction and public road system.This calls for application of new effective road machinery, including smooth rollers.One important element of the motor road construction process is compaction of the road concrete mix.Compaction is also an important constituent of construction of buildings and structures.Reliability, quality and life time of a completed project depend on how the compacting works are planned and implemented.
For this reason, major attention is paid to the arrangement of a strong and stable road bed which is the foundation of a structure.Compaction process is most important in road building, as the attained compacting factor holistically determines strength and tolerance of the whole structure to the influence of natural climatic and operational factors.Today, unfavorable influence of natural climatic factors gets more and more intensive.Due to an increase in weight and driving speed of transport vehicles, the dynamic loads on road structure constantly increase.Accordingly, requirements to compaction toughen, which in its turn stimulates further theoretical research in this field and, based on their results, perfection of design of road-rollers as one of the most common compacting means.
By their nature, soils are rather varied, and therefore their physical and mechanical properties are different.During construction, soils are compacted in different conditions -on large areas, in fills, slopes, trenches, ditches etc.All this necessitates different requirements to soil-compacting machines.These requirements are often contradictory, and this is why compaction of soils in road building cannot be performed by any one or even two types of existing machines.High density of material is achieved by correct selection of compaction methods, parameters of machines and compaction modes used.
Pressures of compacted medium on contact surface of a machine should never exceed the ultimate strength of the medium.They must gradually increase from passage to passage or from stroke to stroke.Under high pressures, a plastic flow (pressing-out) of the material from under the working members appears on the surfaces of working members contacting with the material.During rolling operations, this causes wave formation.Under insufficient pressures, required density is not achieved, which negatively effects the strength, bearing capacity and life of the road surface.This predetermines the selection of necessary "dosing" of compacting impact, and therefore the method of road surface compaction by rollers usually implies using several nominal sizes of rollers.Using conventional rolls does not allow to design a universal road roller ensuring optimal parameters of road-building materials compaction, even during several passages along a trace.Designing a roller and a roll with a changeable curve would make it possible to adjust its pressure on the compacted material, therefore it is a topical issue.Such a roller would be able to replace several rollers with conventional rolls, which is undoubtedly important with the current equipment costs.
The task to improve road construction quality can be solved under the condition that road rollers used in the construction process will ensure high quality at maximum efficiency.Improving the efficiency of, and intensifying the compaction process are based on the development of theory and calculation methods applied to compacting machines and to rolls of road rollers as conventional multipurpose machines in particular.
This determined the character and structure of theoretical research.Based on the results of the work conducted, the paper describes substantiation of the road roller structure with a locally distorted roll and the method of calculating its parameters.

THEORETICAL RESEARCH OF THE INTER-ACTION PROCESS OF A LOCALLY-DISTORTED ROLL WITH COMPACTED MATERIAL IN THE COMPACTION ZONE
Issues of stress-strain state of working members in road rollers and of compacted medium belong to a constantly researched field of road-building industry.With regard to resolving these issues, the paper introduces a new simple mathematical model of a fundamental and applied problem of elasticity theory, previously not described in literature.
The scientific and technical literature [1, 2] considers a general case of flat contact distortion, when the contact of compressed bodies happens along a straight line perpendicular to plane ХОУ (Fig. 1a), and functions f 1 (x) and f 2 (x) that determine configuration of cylindrical surfaces have continuous first and second derivatives in the area of point х = у = 0. Directing the axis Ох at a common tangent to curves f 1 (x) and f 2 (x) that limit the elastic bodies, we obtain: The sum of the second derivatives 1 is considered here as different from zero and, assuming that elastic displacements are small, we approximate as follows [1,3]: Regarding distributed contact forces q = q(x), we introduce a supposition that their resulting force, perpendicular to axis Ox, is directed to point O of the beginning of the interacting surfaces contact, that is, to the origin of coordinates.As initial clearance between the contacting bodies, according to (2), is symmetric with respect to axis Оу, pressure q on cylindrical surfaces is also an axially symmetric elliptic function by argument x (Hertz-Staierman function), which, in compliance with [1], looks as follows: where m q , c q are, respectively, the maximum and average values of functional dependence q (х) (Fig. 1a), determined by the following formula: where c is half-width of the contact area [1].
In order to re-express and adapt the fundamental correspondences ( 3) -( 6), we cite the necessary analytical relations (Fig. 2) [3,5]: -functions of the lower part of cylindrical surface of the roll f 1 (x) and its second derivative f 1 ''(х) from equation of an ellipse (see Fig. 1b): -value of f 1 ''(0) at х = 0: -curvature radii R = R(x) and R(0) of the elliptical directrix of the cylinder (8), considering expressions ( 9) -(10): -formula connecting specific linear force P (N/m) with the width В of the roll and with the vertical load G B , applied to its centre: -depth h of the roller dipping into the compacted material layer (height of segment КОК', Fig. 2), which is found from ellipse equation (Fig. 1b), where х = ± с and у = h: where c is the value of semichord of the curve КОК' (Fig. 2).Using formulae ( 8) -( 15), we render expressions (3) -( 6) and (15) concrete, after the above substitutions 2 1.875 ( 1) From the physical-technical point of view, correctness of formulae ( 16) -( 21) based on classic relations (3) -( 6) [1] results primarily from axial symmetry calculation models (Fig. 1) and initial contact of interacting bodies along axis z ⊥ xОy, passing through point x = y = 0 (Fig. 1b).Thereat, the model of flat deformed state underlying the correspondences ( 16) -( 21) is adequate in the context of the given structural (non-classic) problem [7] for contact areas of the roll cylindrical surface with the compacted material, when adhering to specification (see ( 12) and [1]) when congruence (2) is fulfilled, confirming the possibility of crude approximation of function ( 8) in compliance with (10) and [3], i. e.: c where (Figures 1b and 2).Practical implementation of analytical correspondences ( 16) -( 21) calls for introduction of an additional prerequisite on invariable (of longitudinal non-deformability) length S of elliptical cylinder generatrix in the process of its transformation into a circle with radius R = const.In this connection and with the purpose of comparability of successive calculated values at different axes а, b of the ellipse, we introduce a method of calculating S and linear dimension 2l of КОК' contact arc (Fig. 2).
For mathematical formulation of the given procedure, it is more convenient to write down the canonical equation of the same ellipse (Fig. 1b and 2): (2) for determination of the elliptic half-arc length l at Э х с = (formula (20), (26) and Fig. 3) Differential dS of arc S (Fig. 3) is as follows: where is the eccentricity of the ellipse [3, 5]  Taking into consideration (29), we present unknown dimensions l and S by elliptic integrals ( , of the second type in Legendre form [5,6] (complete and incomplete, respectively), which, as is known [6], cannot be expressed via elementary functions and are not taken in their final shape, while reference tables [3,8] are compiled for their calculation.
For different values of ξ (in general case 0 ≤ ξ ≤ 1) function E(ϕ,ξ) changes from E(ϕ,ξ) = ϕ at ξ = 0 (for a circle, Fig. 3) to E(ϕ,ξ), if ξ =1.When ϕ = 0, we have E(0,ξ) = 0, while at 2 we obtain a complete elliptic integral E(ξ).The proposed and fully described theoretical model can be used for rolls with a circular outline and outer radius R, after replacing the corresponding letter symbols: As a result, expressions (15), ( 17), ( 18), ( 20) -( 22), (28), and (32) are converted as follows: The derived formulae and the developed calculation theory confirm the assumption by A. F. Zubkov [9], that the length of arc "c" is equal to the length of its contracting chord l, and prove that it is possible to adjust (rationalize) the parameters of contact interaction of a roller with compacted material by varying the design-structural dimensions of semi-axes of the ellipseshaped surface of the flexible shell of a road roller roll in service conditions, thus increasing the quality of compacted road carpet.
In order to determine correspondences between the roller parameters and characteristics of the road surface compacted layer, let us consider the known model of compacting efforts distribution during the roller roll movement on the low-density base shown in Fig. 4 [7].Contact pressure buildup continues until the resistance to friction and coupling between solid phase particles, and to pressing-out (with filtration) of air through the layer pores is overcome.

Figure 4. Distribution of compacting zones, pressures and residue during movement of a locally-deformed roll on a low-density base
Zone 1 predetermines the compaction process connected with pressing out the air, the latter being actively pressed out in zone 2. While in the area of zone 1, along with vertical deformations, shear deformation can happen as well, in zone 2, the volume of material under the roll is mostly subject to vertical deformations.In zone 2, a creeping process related to pressing out the air is observed.Zone 3 is related to the unloading process and pressure drop.
Alignment of the maximum pressure line ВК with the vertical axis of the roll means cessation of the process of compacting with the roll in the given conditions.The character of contact pressures distribution reflects the degree to which the roll is used in the compaction process.
In order to maintain connection between the roller parameters and compacted material properties, it is not sufficient to maintain connection between pressure and density.It is also necessary to maintain it between energy (power) consumption or pressure pulse and density increment -decrease in the amount and volume of air.
Let us look at the operation scheme of a locally deformed roll during its interaction with a compacted road-building material (Fig. 5) [3].While the metal roll passes, indentation resistance forces appear, which causes deformation of the compacted material.The resulting contact zone of the locally-deformed roll and compacted material has a certain value of contact area Fk and average value of contact stresses σк.At depth Z of compacted material, in any point, the stress state occurs, and its value depends on values Fk and σк and on the thickness of the compacted layer.Due to elastic-viscous-plastic relations in the compacted material, deformation does not proceed immediately, but requires a certain period of time.The period the material remains stressed is determined by the traveling speed of the locally deformed roll, ceteris paribus.Therefore, the compaction effect depends on the stress state that occurs in every elementary volume of material and on its duration.The stress state value and time of duration are determined, apparently, by the value of the average contact pressure and the contact area, character of stress distribution on the contact area and the traveling speed.
The main contact parameters of the locally deformed roll are as follows: Fk -contact area; σк -contact pressures.
Average pressure on the contact surface of the locally deformed roll is determined from the expression: where Q is load on the roll, Н; F k is the contact area, m 2 .Therefore, ( ) In order to determine F k , let us look at Fig. 6, from where we find: where B is the width of the roll, m; R is the radius of the roll, m; α is the angle of contact of the roll with the compacted medium.
Having inserted the received expression (43) into equation (40), we obtain: Having inserted expression (44) into formula (38), we receive the law of contact pressures σ, MPa for the smooth metal roll with local deformation in compaction zone: Please use this document as a template to prepare your manuscript.
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ESTIMATE CALCULATIONS OF FLEXIBLE ROLL
Figure 7 represents a flexible roll and shows its main dimensions.Tolerance limit of ordinary steels varies between ±200 and ±400 MPa.In special steels, e. g. composed of С = 0.3%, Mn = 0.56%, Ni = 4.3%, Cr = 1.4%, it can reach ±500 … ±700 MPa.Let us find the sheet thickness that can be rolled from square to circular with a radius of 1 m.
As is known, when a beam is bent with a moment (pure bending), curvature radius depends on the moment 1 M E J R = ⋅ . Having inserted the expressions for stress and section inertia moment, we obtain: 2 700 1000 7 2 10 Let us find to what radius these thicknesses allow bending the roll from the first curvature meter.
In this case, curvature change is determined by the formula:  Herewith, deformations occur along the vertical diameter:  Now, let us find the roll curvature for these two directions: Along the vertical diameter:

Figure 1 .
Figure 1.General case of flat contact distortion of a road roller roll

Figure 2 .
Figure 2. Diagram of locally-distorted roll contact with compacted material in the compaction zone In coordinate system x э О э y э and in parametric form [the parameter ϕ is clear from Fig.3, where ' ANA is a semicircle of radius а and point N taken on the same vertical line with point М of the ellipse, on the same side of axis ' AA .In the current problem, angle ϕ has two numeric values: (1) for calculation of the fourth part of parameter S, where in (26)

Figure 3 .
Figure 3. Determination of elliptic roll half-arc length For the circle (а = b), being a subcase of an ellipse, ξ= 0.Taking into consideration (29), we present unknown dimensions l and S by elliptic integrals ( , )

Figure 5 .
Figure 5. Stress state of an elementary volume of compacted materialAs a result of the stress state, deformation of the compacted material occurs, accompanied, first of all, by displacement of the material particles into a new position, and then -by modification of internal relations.Due to elastic-viscous-plastic relations in the compacted material, deformation does not proceed immediately, but requires a certain period of time.The period the material remains stressed is determined by the traveling speed of the locally deformed roll, ceteris paribus.Therefore, the compaction effect depends on the stress state that occurs in every elementary volume of material and on its duration.The stress state value and time of duration are determined, apparently, by the value of the average contact pressure and the contact area, character of stress distribution on the contact area and the traveling speed.

Figure 6 .
Figure 6.Main parameters of contact of a locally deformed roll with a compacted medium Accepting ground deformation depth h, from Fig. 6 we find: cos R h h α − = (41)

Figure 7 .
Figure 7. Symbolic notations of roll dimensions Based on these stress values, let us determine the necessary roll thickness that would allow to change the curvature radius by 10-15 times without exceeding the specified stresses.Let us find the sheet thickness that can be rolled from square to circular with a radius of 1 m.As is known, when a beam is bent with a moment (pure bending), curvature radius depends on the moment 1 M E J R = ⋅.Having inserted the expressions for stress Let us examine what forces cause, at h = 7 mm, a change of curvature from r = 1 m to r = 0.5 m. (Fig.8).

Figure 8 .
Figure 8. Roll deformation caused by different forces As already noted, stress in the roll is equal to [σ] = 700 MPa [ ]