Construction and Building Materials 240 (2020) 117822
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Properties of rubber under-ballast mat used as ballastless track isolation layer in high-speed railway Xing-Wang Sheng a, Wei-Qi Zheng a,⇑, Zhi-Hui Zhu a,b, Tian-Jing Luo a,c, Yan-Huang Zheng a a
School of Civil Engineering, Central South University, Changsha, Hunan 410075, China National Engineering Laboratory for High Speed Railway Construction, Changsha, Hunan 410075, China c Railway Engineering Consulting Group CO., LTD, Beijing 100020, China b
h i g h l i g h t s A variety of static, dynamic and fatigue tests for the rubber UBMs were carried out. The stiffness of the rubber UBM changed noticeably with the change of loading level. The rubber UBM had a large loss factor. The rubber UBM performed well under the repeated load. The rubber UBM can be used as the isolation layer owing to its reliable mechanical characteristics.
a r t i c l e
i n f o
Article history: Received 8 April 2019 Received in revised form 24 November 2019 Accepted 6 December 2019
Keywords: High-speed railway Ballastless track Isolation layer Under-ballast mat (UBM) Rubber material Experimental research Mechanical performance Viscoelastic characteristics
a b s t r a c t Rubber under-ballast mats (UBM) commonly used in metro traffic in China are applied in high-speed railway ballastless tracks as the isolation layer to improve the operating conditions of high-speed trains. Due to the different operating conditions between the high-speed railway and the metro traffic, the rubber UBM, used as the isolation layer in ballastless tracks, needs to be investigated furtherly. In this paper, a finite element analysis of the China Railway Track System (CRTS) III ballastless track was initially carried out, and the stress state and distribution of the rubber UBM used as the isolation layer were obtained at the service condition. Furthermore, the static, dynamic and fatigue tests of the rubber UBM samples were designed and carried out based on the analysis results. The main conclusions are as follows: under normal service conditions, the stress level of the rubber UBM is low, and it can be considered that the rubber UBM is still in the stage of linear elasticity. The material properties of the rubber UBM can be characterized by the equivalent elastic modulus, which has been determined to be 0.6 MPa by the experimental research. Furthermore, the dynamic characteristic of the rubber isolation layer exhibits a hysteresis phenomenon. The equivalent dynamic stiffness is larger than the equivalent static stiffness due to the viscoelastic characteristics of rubber material. The rubber UBM has large internal friction, high energy absorption rate and good damping performance characterized by its large loss factor. In addition, following the application of 3,000,000 repeated load, which is 1.5 times of a train axle load, the equivalent static stiffness only changes by 2%, indicating that the rubber UBM used as an isolation layer can maintain good mechanical performance under repeated loading. As such, the appropriate selection of rubber UBMs for the isolation layer in ballastless tracks can be made following the analysis and investigation of its performance characteristics. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, high-speed railway systems have developed rapidly due to the reductions in travel time, high quality service provided and being an environmentally friendly form of transport ⇑ Corresponding author. E-mail address:
[email protected] (W.-Q. Zheng). https://doi.org/10.1016/j.conbuildmat.2019.117822 0950-0618/Ó 2019 Elsevier Ltd. All rights reserved.
[1]. Both ballasted and ballastless track structures are widely used in high-speed railway construction. Compared to ballasted track structure, ballastless track structure has obvious advantages in environmental friendliness, stability and durability [2]. As such, its wide application in high-speed railway is of great significance, especially for the China Railway Track System (CTRS) III ballastless track [3–5]. Increases in the train running speed and the weight of transported loads have incurred much higher dynamic forces on
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the ballastless track structures as well as increased vibrations and noise [6–8]. Considering the ever-growing concern for the environment and passenger comfort, some new components are applied not only to improve the track quality and travel comfort but also to limit the vibration and noise generated by running trains [9]. Primarily being employed in ballasted track structures, UBMs (under-ballast mat) have been growing as a feasible and effective solution to lower the track stiffness and reduce the ground-borne vibrations [10]. Similarly, for the ballastless track structure, the same purpose could be achieved by adjusting the mechanical characteristics of its isolation layer. Therefore, it is possible to use appropriate UBMs as the isolation layer of the ballastless track structure to improve the mechanical behaviors and vibration isolation performances of the ballastless track structures. Many published research papers focus on the elastic element in tracks, including rail pads [11–15], under-sleeper pads [16,17], UBMs [18,19] and new materials or recycled material used as the elastic element [20–24]. However, not much attention has been given to the mechanical properties of the rubber UBM used as the isolation layer in a new type of ballastless track of the highspeed railway system in China. At present, the commonly used isolation layers in the CRTS III ballastless track include the ethylene propylene diene monomer (EPDM) isolation layer, geotextile isolation layer and polyethylene film isolation layer, shown in Fig. 1. The thickness, materials and technical requirements of these isolation layers differ across different systems. Based on the research and analysis of isolation layers used in ballastless tracks of highspeed railway systems in terms of types and technical requirements, the main technical indicators of these isolation layers are
a) EPDM isolaon layer
specified in relevant specifications [25–27]. However, the thickness of these isolation layers is very small, and they can hardly be compressed during the operation process. As a result, the elastic properties of these isolation layers are not obvious, and the vibration reduction performance of these isolation layers can almost be neglected [19,28]. Therefore, it can be considered that they mainly play the role of interlayer separation, and they have no regulating effect on the deformation of the superstructures. Currently, a series of elastic elements have been incorporated into the high-speed railway systems as the standard practice in order to reduce the effect of the increase of rail traffic loads and train running speed. UBM is a type of elastic element used as an essential component in applications such as bridges and tunnels [29]. UBMs are installed on the substructure in the case of ballasted tracks, and underneath the concrete track slab in the case of ballastless tracks, as seen in Fig. 2. The deformation capacity and mechanical energy dissipation of the rubber UBM depend on its thickness and density, as well as on its size and type of material. It is envisaged to use the rubber UBMs as the isolation layer of the ballastless tracks in high-speed railway systems. Due to the increases in the operating speed and the axle trainload, higher requirements are placed on the UBM used as the isolation layer of ballastless track in high-speed railway systems. When the UBMs are used as the isolation layer of the CRTS III ballastless track, their function and working state are different to those used in ballasted tracks. Based on this, the rubber UBMs used as isolation layers in this research were selected as the research focus, and a series of static, dynamic, and fatigue tests were carried out to investigate their properties.
b) Geotexle isolaon layer
c) Polyethylene film isolaon layer
Fig. 1. The commonly used isolation layers.
Fig. 2. The UBMs in ballasted track and ballastless track.
X.-W. Sheng et al. / Construction and Building Materials 240 (2020) 117822 2. Methodology 2.1. Ballastless track and rubber UBMs The CRTS III ballastless track is a new type of ballastless track, which is developed on the basis of the commonly used ballastless tracks in China. In detail, the CRTS III ballastless track is a unit-plate type ballastless track structure [30], which consists of rails, a fastener system, a concrete track slab, self-compacting concrete, an isolation layer and a concrete base plate (shown in Fig. 3). The track slab is a piece of C60 prestressed concrete slab (60 MPa), with a thickness of 210 mm. A series of connecting steel bars at the bottom surface of the track slab serve as reinforcement, and the self-compacting concrete is poured underneath the track slab, creating a composite-slab structure formed by the track slab and the selfcompacting concrete layer. An isolation layer is set between the composite-slab and the base plate, and a pair of limit structures are set on the base plate of the ballastless track. The isolation layer is one of the key components of the CRTS III ballastless track structure. Besides, as one of the most common elastic elements in the railway track system, the main functions of the isolation layer are to control the deformation between the lower structures and the track slab, facilitate maintenance by serving as an isolation layer between the track slab and the lower foundation, and reduce vibration and noise. Elastic UBMs usually have a thickness of 15–30 mm, whereas their horizontal dimensions depend on the technique developed during construction. Currently, there are various product specifications depending on the thickness and rigidity of the rubber mats. In this research, a series of rubber UBMs with a thickness of 27 mm and a stiffness of 0.025 N/mm3 were selected as the research specimens. The stiffness was obtained from the standard test specimens of its constituent material, and not the actual stiffness of this special UBM. The UBMs used in these tests were composed of two parts: (i) a 10 mm distribution layer with a platelike rubber structure to uniformly distribute loads, and (ii) a series of elastic nail structures to dampen loads, comprised of a set of evenly-arranged circular truncated rubber cones with a thickness of 17 mm. The structure of the rubber UBM is shown in Fig. 4. The two parts of the rubber UBM were cast in the mold as a single body. As the acting load increases in the process of the train movement, the rubber circular cone structures are compressed. The UBMs take advange of this unique structure, once the train load is too heavy, the UBMs only produce a small compressive deformation, so as to ensure the smooth running of the trains [31].
The track slab and the base plate were simulated by 3D solid elements, with the material property set as C60 concrete; the special rubber isolation layer was simplified into a plate-like structure with a thickness of 27 mm, simulated by 3D solid elements. The elastic modulus of the rubber material was obtained by using the values
Fig. 5. Finite element model of CRTS III ballastless track structure.
2.2. Stress distribution of the rubber UBM The UBM bears different loads when it acts as the isolation layer of the CRTS III ballastless track in high-speed railway. The effective stress distribution of the isolation layer is different when different train axle loads are acting on the rails. In order to understand the stress level and distribution of the UBMs used as isolation layers under different load conditions, a finite element model of the CRTS III ballastless track structure with a length of 6 m was established based on the ANSYS platform.
a) Stress Level A
Fig. 3. The system of CRTS III ballastless track.
b) Stress Level B Fig. 4. The structure of the rubber UBM used as isolation layer.
3
Fig. 6. The stress levels of the rubber isolation layer.
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of the stiffness and the dimensions of the rubber UBM tested in this paper. A fixed restraint was applied at the bottom of the base plate, and the axle load acted directly on the position of the fastener of the track slab. The finite element model of the CRTS III ballastless track is shown in Fig. 5. The focus of investigation concentrates on the stress state and distribution of the rubber UBM under the actual operation state. The stress levels of the rubber UBM under different working conditions are shown in Fig. 6. A close analysis of Fig. 6a) shows the stress level of the rubber UBM under the deadweight of its superstructures such as the concrete track slab, rails, and fastener systems, while Fig. 6b) shows the stress level of the rubber UBM under the superstructure deadweight and the typical axle load of the running trains. The typical axle load of the running train was set to the fatigue checking load in China, which is 255 kN. That is 1.5 times of a train axle load [26]. It can be determined from Fig. 6 that the maximum compressive stress of the rubber UBM is about 0.007 MPa under the deadweight of its superstructures, and the maximum compressive stress of the rubber isolation layer is about 0.11 MPa under the combined action of superstructure deadweight and the train axle load of 255 kN. In this research, the stress level of the UBM ranged from 0.007 MPa to 0.11 MPa under normal service conditions and the effective distribution area of the maximum stress was about 200 mm 200 mm. 2.3. Testing specimens The sizes of the test specimens were determined based on the maximum stress distribution area, and the load level of the dynamic and fatigue tests were determined by the stress range under normal service conditions. Therefore, three sets of differently sized testing specimens were randomly selected from factory coil products to carry out static, dynamic and fatigue tests in this investigation. The three types of specimens were labeled as type A, type B, and type C. The area of type A specimens was 129 mm 129 mm, which is the smallest test specimen in this work, while the size of type B was close to the distribution area of the actual maximum stress, and the size of type C was largest, used for comparison. Detailed parameters of the above-mentioned testing specimens are shown in Table 1. 2.4. Testing planning At present, there is no standard testing method for researching the mechanical properties of the rubber isolation layer in ballastless tracks [18,19]. On the basis of
referring to the testing of mechanical characteristics of rail pads and rubber floating plates [32,33], the loading range of the static tests of various UBM test specimens were determined based on the size specifications of various specimens. In this experimental procedure, all sets of test specimens were initially selected to undergo the static tests, while only type B and type C specimens were chosen to carry out dynamic tests. Furthermore, a piece of type B specimen was selected for the fatigue test considering the limited test funds and the long-time costs of fatigue test. During the whole testing procedure, the room temperature was kept at 20 . In the static loading test, each testing specimen was statically loaded with a universal testing machine. The loading speed was no greater than 1 kN/s, and the loading values for each set of tests were increased to values exceeding the normal load values of the specimens. All specimens were tested under a similar stress level, thus different normal service load ranges were used according to the dimensions of the specimen tested. After the above process, the static load-displacement curve of the whole loading process of each specimen is obtained. The procedure of the static test is shown in Fig. 7. Following the completion of the static tests, the above-mentioned type B and type C test specimens were used to perform the dynamic tests. The dynamic cyclic loads for type B and type C test specimens were also different, due to their different sizes. The dynamic cyclic load was applied to the testing specimens by the MTS test machine to obtain the dynamic load-displacement curves. The dynamic test loading frequency was 5 Hz, which is close to the frequency of the train bogie load acting on the UBM at the actual working state. The total loading time was 600 s, and the test data of the last 10 s was extracted for further research. At last, using the specially designed loading device system and the MTS testing machine, a piece of type B specimen was chosen to carry out the compresscompress fatigue test. The loading control force of the fatigue testing machine is the same as the load in used in the dynamic test, and the frequency of fatigue test was also set to 5 Hz. Thermal aging caused by fatigue loading can also result in degrading conditions for rubber materials, and the test specimen cannot be loaded continuously, to avoid the influence of excessively high temperature on the test results. During the repeat loading progress, the loading was periodically paused and unloaded for half an hour every 100,000 times until the fatigue cycle reaches 3,000,000 times. The static stiffness was tested prior to the start of the fatigue test (cycle 0), and then at the 1,000,000th, 2,000,000th, 2,500,000th and 3,000,000th fatigue cycles. In the post-fatigue static loading test, the static load was loaded sequentially from zero to a sufficiently large static load Pmax, which is larger than Fmax. In this work, Fmax was the force exerted from the stress under a load of 255 kN (1.5 times of the train axle load), Pmax was the force exerted from the stress
Table 1 Parameters of testing specimens. Type of specimens
Type A
Type B
Type C
Dimensions Number of nail structures Area of the top surface of a nail structure Central distance of nail structures Number of specimens Structures of testing specimens
129 mm 129 mm 4 226.865 mm2 65 mm 9
193 mm 193 mm 9 226.865 mm2 65 mm 9
258 mm 258 mm 16 226.865 mm2 65 mm 9
a) Stac test
b) Loading of stac test
Fig. 7. Experimental procedure of static tests.
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Fig. 8. Progress of fatigue loading.
a) Stac test results of type A specimens
Fig. 9. Experimental setup of the dynamic test and fatigue test.
b) Stac test results of type B specimens when a load of 340 kN (2.0 times of the train axle load) is applied to the test specimen. The complete loading procedure of the test is shown in Fig. 8, and the experimental setup of the dynamic and fatigue tests is shown in Fig. 9.
3. Test results 3.1. Static test results The static load-deformation curves of the UBM specimens of type A, type B, and type C are shown in Fig. 10. It can be observed from the curves in Fig. 10 that during the whole loading process, the load-displacement curves show obvious non-linear characteristics, but good linear characteristics can be observed at low stress levels. Physically, the circular truncated cone rubber structures are compressed during the loading process, and the contact area between the rubber nail structures and other structures increases. As a result, the equivalent compressive rigidity of the rubber UBM changes according to the change of the loading level, which has the characteristics of ‘‘low load, low stiffness, high load, and high stiffness” [31]. Furthermore, the initial deformation, such as warpage of the test specimens, can hardly be restored during the manufacture of the specimens, indicating that the boundary conditions have a certain influence on the loaddisplacement curve of the rubber specimen. Therefore, the test data present a particular degree of dispersion, and the discreteness of the type A specimens is the most evident, while the influence of the boundary conditions on type B and type C specimens are not apparent. Additionally, the range between the two dashed lines in Fig. 10 represents the normal service condition of the UBM testing specimens, bounded by the upper and lower limits of the load values for the equivalent static stiffness. As such, the slope of the secant in the normal service range of the load-displacement curve of the UBM specimen is defined as its equivalent elastic modulus, calculated by Eq. (1):
K¼
Fa Fb Da Db
ð1Þ
c) Stac test results of type C specimens Fig. 10. Load-deformation curves of UBM testing specimens.
Where F a and F b are the upper and lower limit loads of the normal service state of testing specimens, which are converted from the size and stress level of the testing specimens. Da and Db are the upper and lower limits of the deformation of the testing specimens, respectively. Based on the static test results and Eq. (1), the calculated equivalent static stiffnesses of testing specimens are shown in Table 2. For each set of testing specimens, the secant stiffness is different because of the size difference between the specimens. Therefore, in order to facilitate the comparative study of all the sets of specimens, the secant stiffness of the specimens is converted into the equivalent unit stiffness, which is the equivalent static unit stiffness of the rubber UBM. The equation used to calculate the equivalent static unit stiffness K T is shown in Eq. (2).
KT ¼
K Fa Fb ¼ S0 S0 ðDa Db Þ
ð2Þ
Where S0 is the area of the test specimen. The equivalent static stiffness values of various testing specimens are subsequently summarized in Table 3.
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Table 2 Static stiffness of various testing specimens. Types of specimens
A01
A02
A03
A04
A05
A06
A07
A08
A09
Static stiffness (N/mm) Specimen type Static stiffness (N/mm) Specimen type Static stiffness (N/mm)
228.5 B01 811.9 C01 1385.4
356.4 B02 725.0 C02 1506.8
368.8 B03 758.8 C03 1410.6
388.6 B04 765.9 C04 1469.7
543.1 B05 761.4 C05 1563.2
515.9 B06 770.2 C06 1602.0
332.2 B07 730.0 C07 1380.6
511.6 B08 767.4 C08 1497.7
329.6 B09 788.6 C09 1463.8
Table 3 Summary of testing equivalent static body stiffness values. Equivalent static stiffness
Type A
Type B
Type C
Average value
Design value
Equivalent static stiffness of various testing specimens KðN=mmÞ Equivalent static unit stiffness KT ðN=mm3 Þ
397.19 0.0239
759.01 0.0204
1475.52 0.0222
— 0.0222
— 0.0250
It can be seen in Table 3 that the equivalent static unit stiffness of various specimens are similar, with an average static unit stiffness of 0.0222 N/mm3, while the design equivalent stiffness of the standard specimen is 0.025 N/mm3. The empirical value of the equivalent static unit stiffness of the UBM is about 12.6% lower than the design value of the standard specimen of the same rubber material. In the finite element model of the ballastless track structure, it is difficult to simulate the structural characteristics of this type of rubber UBM. Therefore, it is necessary to simplify the structurally complex rubber insulation layer into a plate-like rubber structure. The material property of elastic modulus Ei used in the modeling process is calculated by Eq. (3):
Ej ¼ K T h
ð3Þ
Where h is the total thickness of the test specimen. The average static stiffness obtained from the static tests of various specimens is 0.0222 N/mm3, and the resulting elastic modulus of the equivalent plate-like layered rubber structure is calculated to be 0.6 MPa. This value of elastic modulus can be conveniently applied to finite element models of the ballastless track structures. 3.2. Dynamic test results Following the static tests, type B and type C specimens were selected for dynamic tests. Based on this, the dynamic loaddeformation curves are shown in Fig. 11. In Fig. 12, the dynamic load-displacement curves of the UBM present elliptical dynamic hysteretic characteristics, which is consistent with the material properties of rubber. As rubber is a homogeneous polymer material, the movement of rubber molecules
during deformation is not instantaneous as the attraction between molecules must be overcome by the vibrational energy of atoms. Applying the aforementioned conversion method of static stiffness and equivalent elastic modulus and taking the upper and low load limit of the normal service state of testing specimen in the dynamic test, the dynamic stiffness and equivalent dynamic elastic modulus of testing specimens are summarized in Table 4. The average dynamic stiffness of type B and type C specimens are 0.0311 N/mm3 and 0.0282 N/mm3, respectively. In comparison with the static stiffness, the value of the dynamic stiffness is larger. When the deformation of the rubber material is slow, its deformation can be easily recovered over time. If the deformation speed increases, there is insufficient time for the rubber molecules to rearrange themselves, causing an obvious and simultaneous decrease of the elasticity of rubber, and resulting in a higher value of the dynamic stiffness compared to the static stiffness. In addition, the equivalent dynamic elastic modulus is obtained, which can also be used in the finite element analysis.
3.3. Fatigue test results Investigation into the fatigue characteristics of the UBM samples was performed by subjecting type B samples to repeated fatigue loading, as its area is close to the actual stress distribution. The appearances of the testing specimen before and after 3,000,000 times of the fatigue loading are shown in Fig. 12. At present, it is generally believed that fatigue failure occurs when the apparent failure or stiffness of rubber changes to a certain extent. From Fig. 12, it can be observed that following 3,000,000 times of fatigue loading in this test, only a small amount of wear appears on the surface of the test specimen, which is inevitable. There is no
a) Type B specimens Fig. 11. Dynamic load-deformation curves.
b) Type C specimens
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a) UBM sample before fague test
b) UBM sample a er 3,000,000 mes fa gue loadings
Fig. 12. UBM sample for fatigue test.
Table 4 Summary of stiffness values of testing specimens. Specimen types
Dynamic stiffness Kd (N/mm)
Dynamic unit stiffness KdT (N/mm3)
Equivalent dynamic elastic modulusEd (MPa)
B01 B02 B03 B04 B05 B06 B07 B08 B09 Average C01 C02 C03 C04 C05 C06 C07 C08 C09 Average
1191.72 1193.47 1151.95 1142.98 1124.59 1115.62 1157.92 1182.70 1176.76 1159.74 1881.06 1947.14 1681.89 2036.46 2008.88 1892.19 1766.15 1911.19 1723.10 1872.01
0.0320 0.0320 0.0309 0.0307 0.0302 0.0300 0.0311 0.0318 0.0316 0.0311 0.0283 0.0293 0.0253 0.0306 0.0302 0.0284 0.0265 0.0287 0.0259 0.0282
0.864 0.865 0.835 0.828 0.815 0.809 0.839 0.857 0.853 0.841 0.764 0.791 0.683 0.826 0.815 0.767 0.716 0.775 0.699 0.761
Fig. 13. Static stiffness before and after a certain number of fatigue loadings.
Table 5 Equivalent static stiffness during fatigue test.
obvious failure or damage found in the specimen, and the changes in stiffness during the fatigue loading process are used for further study. The changes in the specimen’s static stiffness before and after a certain number of the fatigue loadings are shown in Fig. 13. Taking into account the conditions imposed on the isolation layer in ballastless track structures during actual service, Eq. (1) is used to calculate the equivalent static stiffness, which is defined as the static stiffness of the UBM when it is subjected to a certain number of fatigue loads. This is taken as the secant stiffness, according to Fig. 13, and the values of equivalent static stiffness of the fatigue testing specimen are shown in Table 5. The results in Table 5 reveal that the equivalent static stiffness of the rubber UBM specimens under 3,000,000 times of fatigue loading does not change significantly, and the rate of reduction of equivalent stiffness is only about 2%. This indicates that the impact of 3,000,000 repeated loading on the rubber UBM is not significant, and the UBM used as an isolation layer in ballastless track structure can maintain good mechanical performance under the action of a number of repeated loads.
Number of fatigue loads on specimens (times)
Equivalent static stiffness (N/mm)
Bed stiffness (N/mm3)
Equivalent elastic modulus (MPa)
0 1,000,000 2,000,000 2,500,000 3,000,000
775.28 789.47 780.54 754.92 750.00
0.021 0.021 0.021 0.020 0.020
0.562 0.572 0.566 0.547 0.544
4. Discussion 4.1. Static behaviors The stress-strain curves of the rubber materials of type B and type C specimens can be obtained, and the behavior and characteristics of the rubber UBMs are further analyzed by the stress-strain curves. The stress-strain curves of the rubber materials are shown in Fig. 14
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Fig. 14. The stress-strain curve of the rubber material.
Under a low stress state, the stress-strain curves of rubber in Fig. 14 behave linearly, suggesting that they are in the phase of linear elastic deformation, where the stress-strain relationship can be expressed as Eq. (4).
r ¼ ce ð0 < e < 0:1Þ
ð4Þ
Where c is a constant, obtained by the least square method using the test data. In this case, c ¼ 0:5953. At higher stress levels, the stress-strain relationships of rubber are non-linear, and they can be defined by the exponential curves. The stress-strain relationship of rubber can be expressed in Eq. (5).
r ¼ aebe ð0:1 < eÞ
ð5Þ
Where both a and b are constants, obtained by the least square method using the test data. In this case, a ¼ 0:0215 and b ¼ 10:17. Since the stress-strain relationship is a continuous curve, when e ¼ 0:1, the following expression in Eq. (6) can be derived.
c ¼ 10ae
0:1b
ð6Þ
Subsequently, the stress-strain relationship of rubber is summarized in Eq. (7),
(
r ¼ 0:2153e1:017 e ¼ 0:5953e ð0 < e < 0:1Þ r ¼ 0:02153e10:17e ð0:1 < eÞ
Fig. 15. The load-displacement curve of type C specimen.
ð7Þ
The above results show the hyperelastic properties of rubber UBMs and represent the mechanical behavior of rubber, which can be used as a reference in practical engineering. 4.2. Dynamic characteristics The periodic dynamic load-displacement curves of the test specimens always take the form of an ellipse, indicating that their dynamic behavior is consistent with that of linear viscoelastic materials. The loading frequency has an effect on the mechanical properties of the rubber material, which has a certain dynamic viscoelasticity and internal friction [11,34]. Based on this, the UBM is further analyzed with a loading frequency of 5 Hz, and its loss factor, storage stiffness and other indicators were subsequently derived. A section of the load-displacement curve of a piece of type C specimen is shown in Fig. 15. The force applied to the specimen in the dynamic test shown in Fig. 15 is loaded in the form of sine curve. The deformation curve is
similar to the loading curve, and a lag between the deformation and loading phases can be observed. The force-time and the displacement-time curves can be expressed as Eq. (8) and Eq. (9):
F ¼ A þ F 0 sinðxt þ DT 1 Þ
ð8Þ
S ¼ B þ S0 sinðxt þ DT 2 Þ
ð9Þ
Where F 0 is the maximum amplitude of the periodic external force, S0 is the maximum amplitude of the periodic displacement, x is the angular velocity at the test frequency, DT 1 is the phase of the external force, DT 2 is the phase of the displacement, A and Bare undetermined constants. The values of variables F 0 , S0 , x, A, B, DT 1 and DT 2 can be obtained from the above force-time curve and the displacementtime curve, resulting in Eqs. (10) and (11).
F ¼ 3799:36 þ 2275:94 sinð10pt þ 214:70pÞ
ð10Þ
S ¼ 22:85 þ 1:3368 sinð10pt þ 214:46pÞ
ð11Þ
The phase differenced refers to the difference between the load and the corresponding displacement, as indicated in Eq. (12) below.
d ¼ jDT 1 DT 2 j ¼ 0:24p rad ¼ 43:2
ð12Þ
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Subsequently, the loss factorgof the rubber UBM can be obtained by Eq. (13).
g ¼ tand ¼ 0:94
ð13Þ
From the results of the above analyses, the complex stiffness K*, the storage stiffness K0 , and the loss stiffness K00 of the rubber UBM can be calculated by Eqs. (14)–(16).
K0 ¼ K 00 ¼ K ¼
F0 cos d ¼ 1241:09 N=mm S0
ð14Þ
F0 sin d ¼ 1165:46 N=mm S0
ð15Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðK 0 Þ þ ðK 00 Þ ¼ 1702:53 N=mm
9
& editing, Supervision. Tian-Jing Luo: Validation, Funding acquisition. Yan-Huang Zheng: Data curation. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements
ð16Þ
The rubber UBM has a large loss factor, which indicates that it has large internal friction, high energy absorption rate and good damping performance. In addition, the empirically obtained complex stiffness of 1702.53 N/mm is close to the dynamic stiffness calculated by Eq. (1), which has an average value of 1872.01 N/mm. This verifies the reliability of the test results, suggesting that the material characteristic parameters obtained in this work can be used to evaluate the performance of rubber UBMs and also provide a reference for further research. 5. Conclusions The work undertaken in this research aimed to investigate the application of UBMs in ballastless tracks in high-speed railway systems as the isolation layers, which are commonly used in the metro traffic. Of particular importance is that the relevant mechanical properties of the UBMs used as isolation layers must be reliable and resilient at the conditions they are subjected to. Therefore, a series of static, dynamic, and fatigue tests were designed and carried out to obtain the basic material characteristics of the rubber UBMs. The main conclusions are as follows: (1) The stiffness of the rubber UBM changes noticeably with the change of loading level. Under normal service conditions, the stress level of the rubber UBM is low, and it can be considered to still be in the linear elastic stage, and the material properties can be characterized by the equivalent elastic modulus. (2) The dynamic characteristics of rubber UBM are hysteretic, and the equivalent dynamic stiffness is larger than that of the equivalent static stiffness due to its viscoelastic characteristics. (3) The rubber UBM has a large loss factor, which indicates that it has significant internal friction, a high energy absorption rate and good damping performance. (4) Under the action of 3,000,000 times of fatigue loading, the equivalent static stiffness of the rubber UBM decreases by 2%, indicating that the impact of the fatigue loads on the rubber UBM is not significant. (5) The appropriate selection of the rubber UBM can enable its use as an isolation layer in the ballastless tracks, owing to its reliable mechanical characteristics. CRediT authorship contribution statement Xing-Wang Sheng: Conceptualization, Methodology, Validation, Project administration. Wei-Qi Zheng: Conceptualization, Software, Investigation, Resources, Writing - original draft, Visualization. Zhi-Hui Zhu: Validation, Formal analysis, Writing - review
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