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Study on anticrack effect of semirigid base pavement with stress absorbing layer
Journal of Engineering and Applied Science volume 70, Article number: 45 (2023)
Abstract
For the characteristics of semirigid base asphalt pavement prone to cracking, various stressabsorption layers are applied widely by decreasing stress concentration, improving interlayer bonding and waterproofing. Existing studies on the anticracking effect of stress absorbing layers rarely consider topdown fatigue cracking on the outside of the boundary of the vehicle loading zone. Firstly, this paper proposes a finite element model setting a downward expansion of fatigue crack in pavement based on fracture mechanics. The influence law of modulus and thickness of stress absorbing layer on stress intensity factor of crack tip and the fatigue life of the pavement is analyzed furthermore. Numerical calculation results demonstrate that shearing stress intensity factor increases with stress absorbing layer thickness and decreases with stress absorbing layer modulus, while fatigue life of pavement grows with the modulus of the stress absorbing layer and decreases with its thickness. Sensitivity analysis indicates that the modulus of the stress absorbing layer has a greater effect on the fatigue life of the pavement relative to thickness, which offers reference for further promotion and application of stress absorbing layer.
Introduction
Semirigid base asphalt pavement is peculiarly prone to cracks under the action of temperature, humidity, and repeated traffic loads. The circumstances for crack growth and the elements influencing it include a number of aspects, and asphalt pavement cracking can take many distinct shapes in practice. Ameri et al. [1] and Sun et al. [2] both studied topdown cracking in the transverse direction under traffic loading, while Kang et al. [3] comparatively studied longitudinal topdown cracks and base reflective cracks that run transversely. By various modeling methods used in numerical simulation, combining dynamic modulus test and bending test, many scholars have devoted themselves to studying the fracture mechanics, crack paths, propagation phase and fracturing modes of pavement cracks [4,5,6]. The existence and expansion of cracks will not only affect the driving comfort of the road, reduce the durability of the road, but also even trigger the destruction of the entire pavement structure. The study mentioned above demonstrates that the evaluation index based on fracture mechanics is in line with the functioning state of semirigid subgrade pavement with cracks. As a material with some initial defects in general, it is trustworthy to apply the fracture mechanics theory to analyze the cracking issue of asphalt pavement.
On that basis, a technology that can reduce and retard the occurrence and development of cracks in semirigid base asphalt pavements was proposed. By absorbing stress concentration in the crack portion and enhancing interlayer bonding and waterproofing, the setting of stressabsorbing layer can vastly ameliorate the cracking problem of semirigid base asphalt pavement. Normal placement of the stressabsorbing layer is between the asphalt surface and the semirigid base or between the asphalt surface and the old cement concrete panel. From the earliest Strata stress absorbing layer developed by Koch company in the USA, geogrids, geotextiles, modified asphalt geogrids, and modified asphalt are a few of the stressabsorbing layers that have been created over the past 50 years [7,8,9]. The issue of cracking has not been fully solved by any of these methods yet because some performed poorly when put under a variety of field situations [10].
Several tests have verified the effectiveness and applicability of the anticracking properties of certain stress absorbing layers. Pan et al. [11] noted that warm mix rubber modified asphalt mixture was ideal for using as a stress absorbing layer to prevent road cracking in cold regions. Yang, et al. [12] analyzed the anticrack performance of enhanced stress absorbing layer when slow cracking based on fatigue crack propagation tests. Sun et al. [13] compared the crack resistances of high viscous asphalt sand stressabsorption layer, common SBS modified asphalt stress absorbing layer and rubber asphalt stressabsorption layer. The ability of the stress absorbing layer to slow down the crack expansion depends on the kind of stress absorbing layer and external conditions. The former includes additive types, stiffness, and thickness, the latter is mainly load level and temperature [14, 15]. Han et al. [16] discussed the effects of modulus, thickness, and porosity of stress absorbing layer between the old concrete pavement and asphalt pavement on the stress by numerical simulation. Based on the response surface methodology, Nejad et al. [17] investigated the effect of temperature, crack width, geosynthetic type (polypropylene nonwoven and glass grid), and interactions between these parameters on reflective crack propagation were analyzed.
Currently, there are relatively mature works on how to install a stress absorbing layer under asphalt concrete overlay on top of existing cement concrete slab, but the impact of placing it directly between the base and the surface layer is still unclear. Additionally, most international studies on how a stress absorbing layer affects crack expansion have been conducted on reflection cracks [18, 19]. The traditional asphalt pavement design method considers that the fatigue effect of tensile stress at the bottom of each structural layer is the primary cause of structural layer cracking. Bottom cracks extend to the surface of the pavement leading to fatigue cracking of the surface. However, the downward expansion of fatigue cracks in semirigid base asphalt pavements is also one of the sources of damage to the pavement structure [20]. The tensile stresses and strains on the outer side of the load area boundary will damage the asphalt surface and cause topdown fatigue cracking of the road surface. Applied to resist reflection cracks, the stress absorbing layer is directly contact with the initial crack and can play a direct function in the upward expansion of reflection cracks. In contrast, the stress absorbing layer is not directly contact with the topdown preliminary crack. Its indirect effect may be quite different. Therefore, it is necessary to analyze the contribution of stress absorbing layer to the topdown preliminary crack in the asphalt structure.
In this paper, the topdown expansion of fatigue cracks in semirigid base asphalt pavements with stress absorbing layer is analyzed. In “Methods” section, the fracture mechanics theory of fatigue cracks is introduced. In “Results and discussion” section, a numerical model of semirigid subgrade pavement with cracks is established using fracture mechanics theory. The influence law of modulus and thickness of stress absorbing layer on stress intensity factor of crack tip and fatigue life of pavement is analyzed and the topdown expansion law of fatigue cracks is studied in “Conclusions” section. The findings can further provide theoretical support and technical guidance for the further promotion and application of composite pavement.
Methods
Fracture mechanics and Paris formula
Cracks are classified into three different types in fracture mechanics based on the stress condition in the cracks: opening mode, shearing mode and tearing mode, as shown in Fig. 1.
The unified formula that can be used to express the stress and displacement fields at the crack tips of the three different types is [22]
where \(\sigma_{ij} \, (i,j = 1,2,3)\) is the stress component; \(u_{i} (i = 1,2,3)\) is the displacement component; \(K_{N} { (N = {\rm I},{\textrm II},{\textrm III})}\) is the stress intensity factor (SIF) and \(N = {\rm I},{\textrm II},{\textrm III}\) denotes the type of crack; \(f_{ij} \left( \theta \right)\) and \(g_{i} \left( \theta \right)\) are both functions of the polar angle \(\theta\); r is the distance to the crack tip.
Equation (1) exhibits the following properties of stress field at crack tip. First above all, the stress field features a singularity that the stress at the crack tip is indefinitely high (r = 0). Secondly, the stress is directly proportional to the parameter \(K_{N}\). As long as \(K_{N}\) is the same, the stress field at the tip will be completely consistent. Evidently, the crack tip stress field singularity is stronger and the probability that the crack would be unstable increases with the stress intensity factor. So, based on the aforementioned properties of the fracture tip stress field, the stress intensity factor \(K_{N} { (N = {\rm I},{\textrm II},{\textrm III})}\), which is used to describe the strength of the stress field at the tip of three different types of cracks, is defined by fracture mechanics as Eq. (2) illustrated:
where \(\sigma_{y}\) is the yaxis normal stress; \(\tau_{xy}\) is the shear stress in xoy plane; \(\tau_{yz}\) is the shear stress in yoz plane.
The displacement method and the stress method, which compute the SIF value using the nodal displacement or stress close to the crack tip, are two commonly employed direct methods for calculating the stress intensity factor. The displacement approach frequently produces results that are more accurate than those of the stress method because the stress field is generated by the partial derivative of the displacement field. As shown in Fig. 2, the crack tip can acquire singularity by simply shifting the middle node of the 20node isoparametric element to the 1/4 position to make it a degenerative singular element [23, 24]. Thus, the displacement of the node at the crack tip point and 1/4 position may be used to calculate the stress intensity factor, and the equations for the three different types of stress intensity factors can be provided as follows:
where \(\kappa\) is the material parameter related to Young's modulus and Poisson's ratio; \(\mu\) is the shear modulus; \(r_{a  b}\) is the distance between crack tip node and 1/4 node; \(u_{a}\) and \(u_{b}\) are the xdirection displacements of the tip node and 1/4 position node of the crack front, respectively; \(v_{a}\) and \(v_{b}\) are the ydirection displacements of the tip node and 1/4 position node of the crack front, respectively; \(w_{a}\) and \(w_{b}\) are the zdirection displacements of the tip node and 1/4 position node of the crack front, respectively. Note that all displacements here are in the local Cartesian coordinate system.
Unlike the stress intensity factor, which can only characterize the ease of crack expansion, the Paris formula can be used to describe the crack opening and expansion process. If the external load acts cyclically N times and the crack expands as a, then the crack expands as a/N for each cycle of the crack. Paris formula considers the fatigue expansion rate of the crack under repeated loading dc/dN as a function of the increase in SIF as shown in Eq. 4 [25].
Further integration gives the number of load actions:
where dc/dN is the crack expansion rate; \(c_{0}\) and \(c_{f}\) are the initial crack size and the final crack size respectively; \(\Delta K = K_{\max }  K_{\min }\) is the increase of the stress intensity factor; A and n are constants related to the material properties, which is usually determined by conducting fatigue experiments.
The applicability of fracture mechanics principles in predicting the fatigue life of pavements has been confirmed through extensive theoretical analysis and experimental research. It is proved that the development of cracks in asphalt mixes is in accordance with the crack expansion law established by Paris, and the theory of fatigue fracture mechanics can be utilized to forecast the fatigue life of asphalt mixtures [26, 27]. After the stress absorbing layer is added to the road, most of the cracks are still located in the asphalt surface layer and the theory above can still be applied. As a result, SIF will be used as a characterization index for the development of cracks. And it’s also the basis to investigate the effect on the fatigue life of semirigid subgrade pavements when the stress absorbing layer’s parameter changes in the following part.
Based on fracture mechanics and Paris formula, the numerical simulation of anticrack effect of semirigid base pavement with stress absorbing layer can be realized, and the corresponding flow chart is shown in Fig. 3. The whole process consists of four steps. Firstly, the finite element software ANSYS is used to establish the road structure model with crack on the surface of asphalt pavement. By aid of the comparison of the analysis without stressabsorbing layer, the correctness of the numerical simulation method is verified. Secondly, the effect of stress absorbing layer thickness and modulus on SIF is studied. Thirdly, the impact of stress absorbing layer on fatigue life is investigated by curve fitting and Paris formula. Lastly, sensitivity analysis of the effect of modulus and thickness is realized.
Basic assumption
The hypotheses are that each layer of road structure is composed of homogeneous, elastic and isotropic materials. Layers are in constant interaction with one another. The horizontal and downward dimensions of the soil base are assumed to be infinite. Furthermore, cracks only extend downward in the vertical direction.
The most unfavorable load
It is proved that the stress intensity factor at the pavement crack’s tip keeps a negative value under symmetric load, which means that the two faces of the crack move in opposite directions of extrusion and the crack can hardly expand in this case [28]. Therefore, only asymmetric load will be applied on the model with transverse cracks. When the vehicle is moving, the location of the asymmetric loading is obtained based on the maximum probability of occurrence [29], as shown in Fig. 4. It will also be the main reason for the expansion of pavement cracks. According to the current Chinese specification, it is assumed in this paper the tire contact width B is 18.8 cm, the center distance between two wheels is 31.8 cm and the contact length L is 19.0 cm as shown in Fig. 4.
Road structure
A typical representative road structure was adopted with its thickness and material parameters listed in Table 1. The road structure consists of asphalt surface layer, stress absorbing layer, cement stabilized macadam semirigid base and roadbed, where the stress absorbing layer is temporarily not set in the model validation but set in the actual analysis. Asphalt pavement is usually designed based on elastic layer system theory, using Young’s modulus of elasticity as a basic mechanical parameter for calculation. The mechanical parameter for asphalt surface layer and stress absorbing layer can be determined by reference [30], and static modulus is used for semirigid subgrade and road base referring to the recommended practice of AASHTO 2002.
As a multilayered structure, the stress intensity factor will change abruptly between layers so that the cracking and transmission mechanism between layers will be significantly complicated with the increase of pavement layers. To further facilitate in finite element calculation, top surface course, intermediate layer, and the following layer are simplified into a unified layer based on the modulus and Poisson’s ratio conversion equations derived from the membrane force equivalence in Eq. 6 and Eq. 7 [31]. The thickness of top layer is 0.04 m, the thickness of middle layer is 0.06 m, and the thickness of bottom layer is 0.08 m. The asphalt mixture is converted into one layer. The equivalent modulus is 10622 MPa and the total thickness remains 0.18 m.
The Poisson ratio of the equivalent structure is expressed as
The modulus of the equivalent structure is expressed as
where h is the total thickness of structural layers;\(E_{i}\), \(h_{i}\), and \(v_{i}\) are respectively modulus, thickness and Poisson’s ratio of the ith layer.
Finite element modeling
As shown in Fig. 5, ANSYS APDL is used to produce a finite element model of preinstalled fractures on the top of an asphalt mixture to investigate the topdown cracks propagation rule based on the theory of linear elastic fracture mechanics. This is a 3D model employing solid45 and solid95 elements. Since the crack is only hypothesized to develop downward (in the vertical direction), it is only modeled in the crack length region. Secondly, the downward expansion of road cracks under vertical vehicle loads is an example of inplane stressing in fracture mechanics. It means the probability of cracks breaking along the length direction is very small and K_{III} is usually not considered [32, 33]. So a thin model is adopted.
The beginning depth of the crack is fixed at 0.007 m since initial cracks are frequently present before operation [34]. The model width is taken as 10 m, and the thickness of the roadbed is taken as 7 m. The direct modeling method is employed in this bottomup modeling starting from the crack front to provide better control of the cell accuracy especially in the area close to the crack. As stated in the previous section, it is required to set the first cell at the crack tip as Singular elements to describe its singularity of the stress and strain field when modeling. A macro program offsets the center nodes of the isoparametric cells surrounding the original tip by 1/4 to produce threedimensional singular elements automatically while Solid45 solid elements are used in other area. To increase calculation accuracy, the transitional elements are placed close to the full crack model tip. The computational model of the road is large in size. However, the crack size is small, which is as low as 10^{−3} m in one dimension. In the XOY plane, i.e., the vertical section of the road, the edge length of singular elements is always kept 0.01 times of the crack depth for calculation. The tip model is composed of 16 singular pieces to make sure that there is no less than one element within 30°. The crack length direction is defined as being along the Zaxis direction and the depth direction is defined as the Yaxis direction under the global coordinate system. Depending on the basic assumptions above, the cracks only extend along the Yaxis direction.
The boundary conditions of the model simulate how the road structure is actually constrained. Complete restraint is applied to the bottom surface of the soil base and no displacement is allowed. The Xaxis displacement is limited in the road crosssectional direction, while the Zaxis displacement is limited in the road’s length direction. The pavement’s surface serves as a free surface for the application of axle load. Vertical displacement is allowed besides the bottom surface of the structure.
Results and discussion
Model validation
The static load method is the most commonly used method to calculate the fatigue life [1, 32]. Therefore, the static load method is used to calculate the stress intensity factor. The nodes displacement of crack tip can be obtained by aid of the postprocessing of ANSYS. Then SIF can be directly calculated based on Eq. (3). For the sake of comparing, the stress intensity factor of crack tip in the model without stress absorbing layer is calculated. The thickness of asphalt mixture surface layer is 0.18 m. The thickness of rigid base layer is 0.36 m. But the stress absorbing layer is not installed [29]. The road parameters listed in Table 1 are exactly the same as those of the road model in the literature. The loading in the literature is consistent with Fig. 4. The tire contact width is 18.8 cm, the center distance between two wheels is 31.8 cm and the contact length is 19.0 cm. When the crack length is 0.12 mm and the modulus of cement stabilized macadam base varies from 500 to 6000 MPa as shown in Fig. 6, the simulation results show considerable consistency with the data of the reference. The relative errors of the calculations are within 5% and the accuracy of the modeling and calculation techniques used in this study is proved. A snapshot of the von Mises stress distribution map at the crack tip is shown in Fig. 7. Obviously, the stress value increases as one gets closer to the fracture tip, which on the other hand proves the effectiveness of singular elements.
Effect of stress absorbing layer on stress intensity factor
The road structure is remodeled and recomputed in the same way based on the road structure added a stress absorbing layer shown in Fig. 4, intending to research how SIFs change when stress absorbing layer modulus and thickness vary. When the modulus of stress absorber layer is 4000 MPa and its thickness is 0.05 m. The depth of crack is 0.03 m. The stress intensity factors of cracks with varying crack length are displayed using quadrant graphs with scatter plots in Fig. 8. Figure 9 shows the SIFs with various depths of cracks. When the length of crack is 0.01 m, the modulus of the stress absorbing layer is 4000 MPa and the thickness of the stress absorber layer is 0.05 cm. Combining these two pictures, it reveals that \(K_{{{\rm III}}}\) (tearing type) is always negative while \(K_{\rm I}\) (opening type) maintains 0 regardless of changes in the length and height of the crack. This means that the probability of crack opening in the XOY plane under asymmetrical load as well as shearing along the Zaxis is 0, which is consistent with the basic assumptions proposed previously. As a result, \(K_{{{\rm II}}}\) is the topic of discussion below.
Figure 10 shows that \(K_{{{\rm II}}}\) at the crack tip basically steps down as the crack length grows. On the contrary, \(K_{{{\rm II}}}\) becomes clearly larger as the crack depth becomes larger shown in Fig. 11. The likelihood that a fracture may become unstable and enlarge grows as the crack’s depth rises. As all the four graphs illustrated, \(K_{{{\rm II}}}\) increases with stress absorbing layer thickness and decreases with stress absorbing layer modulus, which is less obvious in the pictures of Fig. 10 but clearer in Fig. 11. The variation pattern of \(K_{{{\rm II}}}\) with increasing crack depth is also shown in Fig. 11b for the road structure without stress absorbing layer. When the stressabsorbing layer has been laid, the stress intensity factors at the crack tips are all lower.
Effect of stress absorbing layer on fatigue life
To clarify how the parameters of the stress absorbing layer actually affect the fatigue life of pavement, the SIFs are factored as a function of crack length and depth using the polynomial form, which is stated in the Eq. (8).
where \(p_{1}\), \(p_{2}\), \(p_{3}\), \(p_{4}\), and \(p_{5}\) are polynomial coefficients obtained by fitting, \(h_{{\text{d}}}\) is the depth of crack, \(K_{{{\rm II}}}\) is the shearing stress intensity factor.
When taking into account the possibility of crack expansion in the length and depth directions, the function fitting is utilized based on Fig. 11. The pavement service life can be obtained by first substituting the fitted function expression into the Paris formula and then integrating it over the range of crack depth variation. Since no experiments were conducted, the parameters \(A\) and \(n\) in the Paris equation used in this paper were obtained by referring to the reference [29], taking A = 1.72 × 10^{−5} and n = 1.60. Figure 12 depicts the influence of the stressabsorbing layer's modulus and thickness on fatigue life. The diagram shows that in the specified range, the crack fatigue propagation life grows with the modulus of the stress absorbing layer and decreases with its thickness. Similarly, the SIFs at the crack tip without stress absorbing layer in Fig. 11b are also fitted to the depth to further calculate the fatigue life. The fatigue life of the pavement without a stress absorbing layer, with other structural layers of the pavement having the same parameters, is 0.14 × 10^{6}. Obviously, regardless of the parameters chosen, installing a stress absorbing layer between the semirigid base and the asphalt surface layer significantly improves the fatigue life of the pavement. Its setting is beneficial to retard the topdown expansion of fatigue cracks on the road surface. However, it should be noted that in order to obtain better crack relief, an excessively thick stress absorbing layer should not be installed. Besides, the recalculated fatigue life is 0.14 × 10^{6} and it is lower than the fatigue life from Fig. 12b assuming the thickness is 0. The main reason for this phenomenon is that the model is valid only for a certain thickness interval. The thickness of stress absorbing layer is usually larger than 0.01 m in practical engineering. The proposed calculated model based on polynomial function is only applicable to the stress absorbing layer with thickness greater than 0.01 m. A new model has to be investigated, which could be applicable in the case of thin stress absorbing layers.
When the crack is located between the semirigid base and stress absorbing layer, the effectiveness of preventing reflection crack increases with the decrease of the modulus of the stress absorbing layer [35]. However, the law of the topdown expansion of fatigue crack that located in asphalt structure is different. As the modulus of the stress absorbing layer increases, the stress of crack tip decreases and the fatigue life increases.
Equation (9) defines the sensitivity factor that can be used to evaluate the effect magnitude of the thickness and modulus of the stress absorbing layer on fatigue crack extension.
where \(S_{AF}\) is the sensitivity factor; \(\Delta F/F\) is the rate of change of the factor F; \(\Delta A/A\) is the rate of change of fatigue life as an evaluation indicator when the uncertainty factor \(F\) changes by \(\Delta F\). The sensitivity of the modulus of the stress absorbing layer to fatigue life is 0.269 based on Eq. (9). The sensitivity of the thickness of the stress absorbing layer to fatigue life is − 0.096. It indicates that the modulus of the stress absorbing layer has a greater effect on the fatigue life of the pavement relative to thickness.
Conclusions
The effect of stress absorbing layer’s setting on the fatigue life of semirigid base asphalt pavement with initial cracks is analyzed. The topdown expansion of the nearsurface cracks at the boundary of the wheelloaded area within the asphalt surface layer is taken as the source of damage. A numerical model of road with cracks with stress intensity factor as the evaluation index is established based on fracture mechanics in this paper. The accuracy of the numerical calculation approach based on finite elements is confirmed. Some conclusions are as follows:

(1) The effect of stress absorbing layer on stress intensity factor is analyzed. The numerical results show that SIFs of the \(K_{{{\rm III}}}\) type (opening type) is negative while \(K_{\rm I}\) maintains 0, \(K_{{{\rm II}}}\) becomes clearly larger as the crack depth becomes larger. Besides, \(K_{{{\rm II}}}\) increases with stress absorbing layer thickness and decreases with stress absorbing layer modulus.

(2) The effect of the stress absorbing layer on fatigue life is analyzed. The results show that with increase of the modulus of stress absorbing layer, the fatigue life increases gradually. With increase of the thickness of stress absorbing layer, the fatigue life decreases gradually. Based on sensitivity analysis, the modulus of the stress absorbing layer has a greater effect on the fatigue life of the pavement relative to thickness.
There are still some limitations in this study. In future work, the anticrack effect of semirigid base asphalt pavement with stress absorbing layer under the temperature and load frequency will be conducted.
Availability of data and materials
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 SIF:

Stress intensity factor
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This research was funded by the Education and Scientific Research Project for Middleaged and Young Teachers in Fujian Province (Grant No. JAT191532), Research on Vocational Education Teaching Reform of Longyan City (Grant No. ZJKT2022004), Science and Technology Research Project of Henan Province (Grant No. 222102230010) and Key Research Project of Henan Higher Education Institutions (Grant No. 22A560005).
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DN was the major contributor in writing the manuscript. SW conducted the visualization and comparison. PS conducted the conceptualization, formal analysis and writing—review. CH provided some data. The authors read and approved the final manuscript.
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Nie, D., Wang, S., Sun, P. et al. Study on anticrack effect of semirigid base pavement with stress absorbing layer. J. Eng. Appl. Sci. 70, 45 (2023). https://doi.org/10.1186/s44147023002175
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DOI: https://doi.org/10.1186/s44147023002175
Keywords
 Stress absorbing layer
 Fatigue crack
 Stress intensity factor
 Asphalt pavement
 Fatigue life