Keywords

1 Introduction

Road traffic plays an important role in regional economic growth. In the inspection items of roadbed, subbase and asphalt layer, the weight rating of compaction degree was the highest [1]. It can be seen that the quality of the highway largely depended on the compaction effect. Traditional compaction methods detected a few sections of road after compaction, which is poor representation, time-consuming and destructive structure. With the improvement of construction efficiency, the traditional testing method can not meet the requirements of real-time construction guidance [2]. The development of real-time, accurate and low-cost new compaction monitoring technology has become a research hotspot. Therefore, further study on compaction characteristics of vibratory roller and dynamic response relationship between vibratory drum and soil can also provide a theoretical basis for new compaction detection technology.

Over the years, the interaction between vibrating drum and soil has been studied extensively. Sheng Qiu simulated and analyzed the process of vibration compaction, which showed that the nonlinear response of vibration compaction process can be well simulated by the finite element model of vibration drum-soil [3]. Fu Huang established the dynamics model of vibration pressure system, and obtained the relation between the vibration drum acceleration and the stiffness coefficient and damping coefficient of pavement material [4]. The effect of the sliding rate of steel wheel, the compacting times, the friction coefficient of the steel wheel and the soil and the wheel weight on the deformation of the sticky soil were studied by Maciejewski and Jarzebowski [5]. The results showed that the quality of the steel wheel was the most important factor affecting the degree of soil compaction.

2 Constitutive Model of Soil

2.1 Yielding Criterion

The common yield criterion of geotechnical materials was Mohr–Coulomb yield criterion, whose governing equation was illustrated in Eq. (1) as follows:

$$\frac{{\sigma_{1} { - }\sigma_{3} }}{2}{ + }\frac{{\left( {\sigma_{1} { + }\sigma_{3} } \right)\sin \phi }}{2} - c\cos \phi = 0$$
(1)

where \({\upsigma }_{1}\), \({\upsigma }_{2}\) and \({\upsigma }_{3}\) the first, second and third principal stresses, respectively; c = cohesion; and φ = the internal friction angle.

The Mohr–Coulomb criterion, however, didn’t reflect the effect of intermediate principal stress on yield and failure. Moreover, the yield surface of Mohr–Coulomb criterion was hexagonal conical, and the direction of plastic strain increment was not unique at corner, which made it difficult to converge in plastic analysis and was not good for numerical calculation.

Drucker and Prager constructed the classical Drucker-Prager model to solve the problems above. It was the conical yield surface of the hexagonal cone which was tangent or internally tangent to the Mohr–Coulomb criterion, and the D-P yield criterion was proposed [6, 7]. The governing equation of the yield criterion was illustrated in Eq. (2) as follows:

$$D_{1} I_{1} + \sqrt {J_{2} } - D_{2} = 0$$
(2)
$$I_{1} = \sigma_{1} + \sigma_{2} + \sigma_{3}$$
(3)
$$J_{2} = \frac{1}{6}\left[ {\left( {\sigma_{1} - \sigma_{2} } \right)^{2} + \left( {\sigma_{2} - \sigma_{3} } \right)^{2} + \left( {\sigma_{1} - \sigma_{3} } \right)^{2} } \right]$$
(4)
$$D_{1} = \frac{2\sin \phi }{{\sqrt 3 \left( {3 - \sin \phi } \right)}}$$
(5)
$$D_{2} = \frac{6c\cos \phi }{{\sqrt 3 \left( {3 - \sin \phi } \right)}}$$
(6)

where I1 = the first invariant of the stress tensor; J2 = the second invariants of the stress bias; and the definition of the other parameters are the same as in Eq. (1).

2.2 The Selection of Constitutive Model

Figure 1 shows yielded curve of different yielded criterion on the π plane. A large number of constitutive models for geotechnical materials were provided in ABAQUS. The yield line of the Mohr–Coulomb plastic model, the classical Drucker-Prager plasticity model, and the extended Drucker-Prager plasticity model on the partial plane.

Fig. 1
3 concentric shapes. Outermost, classical Drucker-Prager Model as a smooth, curved line forming an enclosed shape. Middle, developed Drucker-Prager Model as a more complex curve with pointed edges. Innermost, Mohr-Coulomb Model as straight lines forming an angular, polygonal shape.

Yielded curve of different yielded criterion on the π plane

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Through calculation and simulation, authors analyzed the relationship and the applicable conditions between the Mohr–Coulomb model and linear Drucker-Prager model.

As Fig. 1 shows, with the same definition of tension and compression failure for the two models, the linear Drucker-Prager model can be re-expressed as another form [8, 9], which was shown under the condition of triaxial compression and triaxial tensile illustrated as follows: Under the condition of triaxial compression,

$$\sigma_{1} { - }\sigma_{3} { + }\frac{\tan \beta }{{2{ + }\frac{1}{3}\tan \beta }}\left( {\sigma_{1} + \sigma_{3} } \right) - \frac{{1 - \frac{1}{3}\tan \beta }}{{1{ + }\frac{1}{6}\tan \beta }}\sigma_{c}^{0} = 0$$
(7)

where \(\upbeta =\) the friction angle in Drucker-Prager model, and \({\upsigma }_{{\text{c}}}^{0}=\) yield stress.

Under the condition of triaxial tensile,

$${\sigma }_{1}-{\sigma }_{3}+\frac{\mathit{tan}\beta }{\frac{2}{K}-\frac{1}{3}\mathit{tan}\beta }\left({\sigma }_{1}+{\sigma }_{3}\right)-\frac{1-\frac{1}{3}\mathit{tan}\beta }{\frac{1}{K}-\frac{1}{6}\mathit{tan}\beta }{\sigma }_{c}^{0}=0$$
(8)

For all (\(\sigma_{1} ,\,\,\sigma_{3}\)), in order to make Eqs. (7) and (8) be consistent with the control equation of the Mohr–Coulomb model (1), Coefficient K was described in Eqs. (9), (10), (11), and (12) as follows:

$$K = \frac{1}{{1 + \frac{1}{3}\tan \beta }}$$
(9)
$$\tan \beta = \frac{6\sin \phi }{{3 - \sin \phi }}$$
(10)
$$\sigma_{c}^{0} = 2c\frac{\cos \phi }{{1 - \sin \phi }}$$
(11)

According to the analysis and calculation above, the results can be obtained.

$$K = \frac{3 - \sin \phi }{{3 + \sin \phi }}$$
(12)

Where K is the ratio of triaxial tensile strength to triaxial compression strength.

When K = 1, the yield surface is Von Mises circle in the deflection stress plane, and the yield stress of triaxial tension and compression is the same. In order to guarantee the yield surface protruding, it is required that K ≥ 0.778. According to Eq. (12), it is required φ ≤  22°. That is to say, when φ ≤  22°, the two models can be well fitted. At this time, the Drucker-Prager model is applied to analyze and calculate. When φ ≥ 22°, the fitting effect of the two models is not good, so it is not suitable for parameter conversion. Mohr–Coulomb plastic model should be adopted directly at this time.

The soil parameters of the Mohr–Coulomb model can be measured by different conventional experiments, which is convenient and practical. The Drucker-Prager model overcomes the disadvantages of Mohr–Coulomb model, but its model parameters need to be converted from Mohr–Coulomb model. In a comprehensive view, the Drucker-Prager model is more suitable for the soil unit in this paper, but it should satisfy the condition that the friction angle is less than or equal to 22°. When the friction angle is greater than 22°, the Mohr–Coulomb model should be more useful.

3 The Establishment of a Finite Element Model

3.1 The Determination of Size Parameters

The principle of vibration roller is that the centrifugal force is driven by the hydraulic device to drive the eccentric block in the steel wheel to rotate at high speed, and the centrifugal force compels the steel wheel to vibrate to achieve the purpose of compaction [10]. In finite-element analysis, complex structure of vibratory drum did not only lead to modeling difficulties, but also increased the amount of calculation and was not easy to converge, so the model should be simplified.

In the simplified model, the quality of the front wheel distribution of the vibratory roller was directly reflected by the vibration drum, and the excitation force acted directly on the center of mass. Considering the analysis time of the model, the running speed of the vibratory roller and the geometric parameters of the actual compacted soil, the finite-element model size of the soil in this paper was taken as follows: 20 m*8 m*0.8 m. This paper chose the small vibratory roller of XD82E of Xugong.

3.2 Determination of Material Parameters

Based on the project of Jiangxi Provincial Communications Department, the parameters of soil under different compaction times in a compaction operation were determined by combining experimental research with experience. The main parameters of the soil were shown in Table 1.

Table 1 The main parameters of soil
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It is found that the soil has elastoplastic deformation at the beginning of compaction. With the soil compacted, the plastic deformation of the soil was decreased gradually. When the soil is completely compacted, it is only elastic deformation. Therefore, the soil is elastic–plastic model, the elastic part of the soil is defined by the linear elastic model, and the plastic part linear Drucker-Prager plastic model.

3.3 Division of Units

The unit partition directly affects the speed, the convergence, and the accuracy of analysis and calculation. The soil was composed of 120,000 C3D8R units with reduced integral. Vibration drum adopts C3D4 element, with a total of 2000 units. The soil belongs to semi space infinite body, but the finite-element model can only calculate the finite problem. In modeling, the soil is truncated so that the pressure wave can not propagate to infinity, but the truncation is reflected back, which affects the accuracy of the calculation. In order to reduce the influence of truncating infinite space, a higher precision viscous artificial boundary is applied at the truncation point. The “vibratory drum-soil” model was shown in Fig. 2.

Fig. 2
An illustration has a rectangular surface with a small cylindrical component on the left. The structures have mesh patterns.

The finite element model of drum-soil

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4 Simulation Analysis of Finite-Element Model

4.1 Specificity Analysis of Soil Stress Distribution Under Vibration Drum

According to the finite-element calculation of the established model parameters and material parameters, the cloud chart of soil stress distribution was obtained shown in Fig. 3.

Fig. 3
Two cloud charts of the stress distribution characteristics on a drum width direction having a rectangular structure and various stress intensities and a drum traveling direction having a spherical structure with varying stress intensities.

Cloud chart of vertical stress distribution characteristics along the width and moving direction of the vibration drum

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As Fig. 3 shown, the region with larger vertical stress was similar to the ellipsoid extending from the vibration drum axis to the downward direction, and along the direction of the ellipsoid, the vertical stress of the soil becomes smaller and smaller from the contact surface of the vibration wheel to the soil. The maximum vertical stress of the soil appears on the vertical line of the vibrating drum shaft. The vertical stress is symmetrically distributed along the width of the vibration drum, and the vertical stress is asymmetrical along the direction of the vibration drum, and it is larger in the direction of the drum moving forward.

4.2 Influence of Soil Parameters on Vertical Acceleration of Vibrating Drum

The parameters of the vibration drum were consistent. Finite-element analysis of soil I, soil II, and soil III was carried out, respectively, which obtained the acceleration curve of the result shown in Fig. 4.

Fig. 4
Three graphs labeled acceleration curves of soils 1, 2, and 3 plot acceleration versus time have a curve that exhibits consistent, dense oscillations, indicating uniform vibrational behavior across all soil samples.

Vertical acceleration curve of drum

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As Fig. 4 shown, the parameters of the vibration drum were consistent. Finite-element analysis of soil I, soil II, and soil III was carried out, respectively, which obtained the acceleration curve of the result showed that the vibration wheel produced a large positive and negative acceleration at the beginning of vibration, and then the vibration acceleration increased gradually, and then tended to be relatively stable. With the change of soil parameters, the amplitude of vertical acceleration of vibration drum changed, and the amplitude of vertical acceleration was positively related to compaction times. There was obvious irregularity in the acceleration curve corresponding to the soil III, which inferred that the soil compaction degree has reached the optimum value of this condition due to the increasement of compaction times, and the vibration wheel became irregularly.

5 Test Verification

In order to verify the correctness of the finite element model, the experiments were carried out. The main items of test were: working speed of roller, vertical acceleration of vibration drum, and compaction degree of roadbed.

Because the period of vibration acceleration, usually 0.02 to 0.05 s, was short, and the speed of the roller was slow, therefore, 20 periodic signals were taken as an testing object period in this paper. After filtering the vibration acceleration signal and eliminating the trend item, the effective acceleration value of 20 period signals was calculated, and then the average value of the acceleration effective value in one compaction operation was taken as the acceleration value of this compaction. After each compaction, the compaction degree of roadbed was measured by sand filling method. The relationship between acceleration, compactness and compaction times was shown in Fig. 5, and the relation between acceleration and compaction in Fig. 6.

Fig. 5
A graph of compactness versus compaction times and acceleration has some following values. Compactness, (1, 70), (4, 91), (7, 100). Acceleration, (1, 15), (4, 35), (7, 45). Values are estimated.

Correlation between measured acceleration, compaction degree and compaction times

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Fig. 6
A graph of compactness versus acceleration has data points along an increasing regression line with y = 0.8665 x plus 61.339 and R squared = 0.9684.

Correlation between measured acceleration and compaction degree

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As Fig. 5 shown, the vertical acceleration of the vibration drum was positively correlated with compacted times, which was basically consistent with the conclusion of the model. The compaction degree also increased by the increase of compaction times. With the further analysis of acceleration and compaction degree, Eq. (13) can be obtained by the linear fitting of points in Fig. 6.

$${\text{y = 0}}{\text{.8665x + 61}}{.339,}\,\,\,\,{\text{R}}^{2} = 0.9684$$
(13)

where y = compaction degree, x = acceleration, and R2 = correlation coefficient.

The correlation coefficient of Eq. (13) was 0.9684, which indicates that the correlation between the vertical vibration acceleration of the vibration drum and the compaction degree were well correlated. Therefore, the acceleration were monitored, and compactness value was obtained by using correlation formula between the acceleration of vibration wheel and compaction degree.

6 Conclusion

  1. (1)

    The discussion of the relationship and the applicable conditions between Mohr–Coulomb model and linear Drucker-Prager model showed that when the friction angle was less than 22°, Drucker-Prager model was more suitable for modelling the soil unit; otherwise, the Mohr–Coulomb model should be used.

  2. (2)

    The finite-element analysis showed that the vertical stress of the soil was symmetrically distributed along the axial direction of the vibration drum and shifted along forward in the direction of the vibration drum, and the vertical stress of the soil decreased sharply with the increasement of the soil depth; In the reasonable working condition range of the roller, the acceleration of the vibration drum was positively correlated with the compaction times.

  3. (3)

    The test results showed that it was feasible to simulate the mechanical properties of soil under vibration drum by Drucker-Prager elastoplastic model, and the finite-element model of vibration drum-soil was basically useful. In addition, there was also a good positive correlation between the effective value of acceleration and the degree of compaction, which provided a new train of thought for compaction degree monitoring system.