Introduction. One of the most important challenges facing the Russian road industry is providing a 24-year interrepair life for road pavement. One of the key approaches to solving this is improving approaches to pavement design. In the practice of the Russian Federation, it is common to design road pavement so that the rigidity of the layers increases from bottom to top, which stems from both historical design approaches and certain specific features of stress-strain state calculations using engineering methods and specialized nomograms. However, this approach does not fully reflect the efficiency of reinforced layers, such as subgrades or pavement bases.

In recent years, significant changes have occurred in the technologies and methods for calculating road pavement, associated with the introduction of software packages that implement both accurate and numerical schemes for the direct calculation of layered media [1]. In numerical methods, the most widespread are both classical finite element approaches [2], and more modern methods based on the use of spectral elements [3].

The finite element method is widely used in modeling the stress-strain state of asphalt concrete and cement concrete samples applied for the construction of road pavement and bases [4], as well as in calculating objects of limited size, typical for the practice of road and industrial-civil construction [5]. However, most of the aforementioned software packages implement a predominantly linear-elastic static formulation of the problem of determining the stress-strain state of road pavement. The wider adoption of numerical methods is constrained by the difficulty of their application to environments of unlimited volume [6]. At the same time, the actual deformation of road pavement layers is most accurately described by viscoelastic models or models that take into account the outflow of volumetric waves into infinite space [7].

As a rule, when solving problems in a viscoelastic dynamic formulation, the response of the road structure is considered either to an impact equivalent to the dynamic impact of the calculated load [8], or directly to the load from a wheel moving on the surface of the road pavement [9].

One of the most common modern solutions to this problem is the design of so-called full-depth road pavement structures, consisting entirely of layers reinforced with mineral and complex binders [10]. This approach has a number of advantages and disadvantages. The undoubted advantages include the overall high rigidity of the structures and the possibility of using recycled materials for the construction of base layers and their additional layers [11]. A number of scientific papers devoted to this topic note the good resistance of such structures to the accumulation of plastic deformations [12], the possibility of use under various natural and climatic conditions [13], and high manufacturability [14]. Disadvantages include high cost and, often with excessively increased rigidity of the base layers, a strong probability of cracking [15]. One of the most common types of crack formation for such structures is reflective cracking [16]. Some authors also note the need to take into account the specific properties of such structures in cold areas [17], and the influence of various types of defects on changes in the service life of the coating [18].

The concept of “perpetual” road pavement is well known. It involves increasing the rigidity of the structural layers of the road pavement from top to bottom, that is, when materials with a higher modulus of elasticity than asphalt concrete pavement layers are used as the base layer [19]. Such structures do provide extremely long interrepair life, which allows for the replacement of only the upper wear layers [20]. However, it is obvious that the cost and reliability of such structures are undoubtedly extremely high and often not reasonable [21]. Moreover, issues of providing proper adhesion between structural layers of different rigidity require additional research [22].

Thus, it can be argued that there are numerous approaches to pavement design, which, on the one hand, emphasizes the efficiency of such structures, but on the other, still leaves numerous questions unanswered. One of these is the study on deformation energy dissipation processes in layered media with reinforced layers, and, as a result, an assessment of the possibility of shifting deformation mechanisms to the elastic stage. The amount of energy dissipation is the most complete and physically-based characteristic reflecting the mechanisms of deformation in the structure of a multilayer medium, which can be measured and quantitatively assessed both by mathematical modeling tools and modern measuring equipment. Moreover, the use of this indicator is promising in the context of such a relevant area as the analysis of the life cycle of construction projects [23] and its implementation in information modeling technologies for monitoring the remaining useful life [24]. Thus, the objective of this work is to analyze the dissipation of deformation energy in the structure of road pavement with different options for the arrangement of reinforced layers and to evaluate optimal design solutions that contribute to increasing the durability of the road pavement.

Materials and Methods. The study utilized theoretical and experimental approaches to assess the deformation of the layered pavement environment. The research methodology is presented in Figure 1.

The following road pavement designs are considered during the numerical and experimental studies. Design 1 represents a standard road pavement design in accordance with the current requirements of GOST R 71404: layer 1 is constructed of asphalt concrete, layer 2 is constructed of a crushed stone-sand mixture reinforced with a complex binder, layers 3 and 4 — dry bound macadam, and an antifrost capillary-breaking layer, respectively.

Structure 2 is an example of a full-depth road structure in which all layers are reinforced with organic or complex binders, which inevitably increases their elastic modulus. However, due to the use of expensive stabilizing and strengthening additives, as well as the inability to reduce the thickness of the structural layers within the current regulatory framework, the cost of this structure will significantly exceed that of Structure 1, which contains layers untreated with binders Structure 3 is similar in its parameters to Structure 1, but layers 2 and 5 (subgrade soil) are reinforced with a complex binder. Structure 4 involves strengthening only the subgrade soil with a binder. The structures under consideration are shown schematically in Figure 2.

The theoretical approach consists in determining the stress-strain state of a layered medium under the impact of a dynamic load from a falling weight in an axisymmetric setting (Fig. 3).

The equations of motion are as follows: [25][26]:

(1)

where λ, μ — Lamé coefficients; ur, uz — radial and vertical components of the displacement vector; ρ — density of the material.

(2)

where r — radial coordinate of the point at which the displacement is found.

The Fourier transformed system of equations (1) takes the form [27]:

(3)

where сj1, сj2 — reduced frequencies of oscillations.

(4)

where ω — oscillation frequency, rad/s; R — radius of the loading area; Vpj, Vsj — velocities of longitudinal and transverse waves in the body.

This form of recording allows using the principle of elastic-viscoelastic correspondence, according to which the Lamé coefficients become complex, which, in turn, leads to the complex-valued nature of the reduced frequencies сj1, сj2.

Since the displacement u can be expressed through the scalar φ and vector (vortex) ψ components, the solution will be further considered in the form of the Lamé representation as:

(5)

The boundary conditions of this problem in stresses are formulated as follows:

(6)

At the boundaries of the layers, conditions of rigid adhesion are set, requiring the continuity of displacements ur(r, z, t) and uz(r, z, t), and stresses σz(r, z, t), τrz(r, z, t).

At infinity, the conditions for all stress and displacement components to tend to zero are satisfied. The initial conditions of this problem are defined as follows:

(7)

The further solution is constructed through applying the properties of the Hankel and Fourier integral transforms to the system of equations (3) and is presented in [28]. Since the solution to a non-stationary problem is considered, the method of discrete harmonic analysis is applied [29].

Experimental approach to assessing the deformation of layered road surfaces involves instrumental measurements of road pavement deformation parameters. The FWD PRIMAX 1500 shock loading unit is used for these measurements [30][31] (Fig. 4). This unit consists of a trailer with a shock loading mechanism and a measuring beam mounted on it. This beam records the vertical component of the displacement speed, which is subsequently converted into absolute values of vertical displacement. Deformation characteristics are recorded by measuring sensors. Figure 5 shows the distances from them to the dynamic loading stamp of the unit. For this study, only sensor D1, located in the center of the stamp, is used. However, in the future, other sensors could be used to evaluate deformation energy distribution processes.

The difference between FWD and other shock loading units is the possibility of practically continuous recording, with a sampling spacing of 0.002 s, of the amplitude-time characteristics of displacements on the coating surface and the amplitude-time characteristics of the shock loading pulse, based on which it is possible to construct an experimental hysteresis loop.

This study examines road pavement structures with different rigidity ratios of the structural layers. The mechanical parameters are presented in Table 1.

Research Results. The problem of determining the dynamic stress-strain state is solved and hysteresis loops are constructed on the surface of each of them using mathematical modeling for the above structures. The solution to the problem for a layered medium allows specifying the amplitude-time characteristics of the displacements u(t) on the surface of the layered medium, and the amplitude-time characteristic of the impulse F(t), from which the dynamic hysteresis loop, whose area determines the energy irreversibly dissipated in the medium, can be reconstructed [32][33]. This curve is specified parametrically in accordance with dependence (8):

(8)

In the case of setting the dynamic hysteresis loop in the form of a data series containing information on discrete values of the actual load (Fi) and vertical displacements (ui), corresponding to a given load, the area of the hysteresis curve is determined in accordance with (9):

(9)

Function F(t) is described by the equation:

(10)

where F — design load (taken equal to 57.5 kN); t — time of observation of the object deformation (t = 0.1 s); tимп — pulse time (tимп = 0.03 s).

The possibility of constructing the amplitude-time characteristic of a sawtooth pulse and a triangular pulse, which are often used to approximate the impact loading reproduced by FWD, is also implemented.

The design load for the simulation was a 57.5 kN drop load distributed over a 30 cm diameter area, which complied with road industry regulatory requirements. The deformation energy dissipation (W) was calculated at the impact point.

Taking into account the assumption of the need for the road pavement to operate in the elastic stage, allowed in domestic regulatory documents, as well as the fact that the energy value W is essentially a function of both the mechanical and geometric parameters of the studied medium, the problem of optimal design can be reduced to the equation:

(11)

where Ej — elastic moduli of the materials of the pavement layers; hj — thickness of the pavement layers; γj — loss angles or other viscosity characteristics of the material of the layers; vj — Poisson's ratios of the materials of the layers.

The damping properties of individual layers are taken into account through introducing the loss tangent tgγ, determined on the basis of the given oscillation frequencies:

Figure 6 shows the calculated amplitude-time characteristics of the displacements on the surface of the simulated road pavements. Figure 7 presents various shock loading pulse shapes, in particular, sinusoidal, sawtooth, and triangular, corresponding to different shock load application scenarios. In this study, the sinusoidal shape is considered, as it best corresponds to the experimental loading reproduced by the FWD.

The results of constructing dynamic hysteresis loops and the results of determining the dynamic deformation energy are presented in Figure 8 and in Table 2, respectively.

To experimentally confirm the results obtained, measurements are carried out using the FWD PRIMAX 1500 shock loading unit on the real operating road pavement, whose design is similar to structures 1, 3 and 4 shown in Table 1. The obtained experimental shapes of the dynamic hysteresis loops are shown in Figure 9, and the experimental values of the deformation energy dissipated in the structure are presented in Table 3.

Design options for road pavement similar to design 2 are currently undergoing feasibility studies for their applicability and have not been implemented under the field conditions of the Russian road network.

Discussion. Analyzing the data in Tables 2 and 3, we can conclude that the highest value of deformation energy dissipation is characteristic of the traditional design option 1, in which the rigidity of the structural layers increases from bottom to top. The lowest value of energy dissipation is undoubtedly characteristic of the most rigid design, 2, which assumes the construction of all layers using binder reinforcement. However, it should be noted that a nearly identical effect can be reached by the reinforcement of the working subgrade layer in the first turn. The numerical modeling has shown that strengthening only the subgrade layer, even without installing a reinforced base layer beneath the asphalt concrete, reduces the dissipated deformation energy from 9.43 to 3.56 J/m³. It can also be concluded that the elastic modulus of the underlying half-space, which simulates the roadbed, has the greatest impact on the amount of dissipated energy. Therefore, the greatest effect, both technical and economic, can be reached through strengthening the top of the roadbed while preserving the loose layers at the base of the road structure (similar to structures 3 and 4). This solution will bring the road pavement performance closer to the elastic stage while reducing the risk of cracks appearing on the road pavement surface due to an excessively rigid reinforced base layer. These modeling results were generally validated by the results of a full-scale experiment, which found that the experimental deformation energy on the surface of unreinforced structure 1, which was 9.44 J/m³, decreased to 4.56 J/m³ in the presence of a reinforced subgrade layer, and to 3.74 J/m³ in the presence of a reinforced base layer under the asphalt concrete and a reinforced subgrade layer. This fact validates the qualitative agreement between the results of in-situ and computational experiments. The research results can be applied to substantiating competing road surface design options and also be developed for use in the road maintenance industry in assessing their residual service life [34][35]. An important conclusion is the establishment of the greatest impact on the magnitude of deformation energy dissipation from the strengthening of the subgrade layer. In recent years, the issue of using various soil strengthening additives has been actively addressed at various levels [36–38]. Once more data on the strengthening of the roadbed using appropriate additives and stabilization is received, the results obtained can be applied to prove the effect of their introduction. Undoubtedly, this will require conducting research using dynamic loading and stamp testing equipment [39][40], which makes it possible to directly register dynamic hysteresis loops and load-unloading curves, and obtain the necessary information about the design characteristics of such layers. The approach presented in the study can also be efficiently developed at accelerated testing sites for road pavement [41], and used to calibrate models when their operational condition deteriorates [42].

Conclusion. Thus, as a result of the analysis performed within the framework of this research on the dissipation of deformation energy in the structure of road pavement with various options for the arrangement of reinforced layers and the assessment of the most efficient options for the arrangement of reinforced layers in the structure of the road pavement, it was established that the greatest impact on the magnitude of the dissipation of deformation energy is exerted by the strengthening of the roadbed, modeled as an elastic half-space, unlimited in thickness. This conclusion was validated by experimental studies, which revealed a similar qualitative pattern of changes in the dissipated deformation energy in a structure consisting of only unreinforced layers; a structure with a reinforced subgrade layer; and a structure with a reinforced base layer and a reinforced subgrade layer. It has been shown that the use of reinforced base layers reduces the amount of deformation energy dissipated in the road pavement structure by more than 2–3 times.