Interface damage and its effect on vibrations of slab track under temperature and vehicle dynamic...

Interface damage and its effect on vibrations of slab track undertemperature and vehicle dynamic loads

Shengyang Zhu a,b,1, Chengbiao Cai a,n

a State Key Laboratory of Traction Power, Southwest Jiaotong University, No. 111 First Section, North of Second Ring Road, Chengdu 610031, PR Chinab Department of Civil and Environmental Engineering, Rice University, 6100 Main Street, Houston, 77005 TX, USA

a r t i c l e i n f o

Article history:Received 14 March 2013Received in revised form6 October 2013Accepted 9 October 2013Available online 17 October 2013

Keywords:Interface damageCohesive zone modelCRTS-II slab trackVehicle–track coupled dynamicsTemperature load

a b s t r a c t

This paper presents a three-dimensional finite element model to investigate the interface damageoccurred between prefabricated slab and CA (cement asphalt) mortar layer in the China Railway TrackSystem (CRTS-II) slab track system. In the finite element model, a cohesive zone model with a non-linearconstitutive law is introduced and utilized to model the damage, cracking and delamination at theinterface. Combining with the temperature field database obtained from the three-dimensional transientheat transfer analysis, the interface damage evolution as a result of temperature change is analyzed.A three-dimensional coupled dynamic model of a vehicle and the slab track is then established tocalculate the varying rail-supporting forces which are utilized as the inputs to the finite element model.The non-linearities of the wheel–rail contact geometry, the wheel–rail normal contact force and thewheel–rail tangential creep force are taken into account in the model. Setting the maximum interfacedamaged state calculated under temperature change as the initial condition, the interface damageevolution and its influence on the dynamic response of the slab track are investigated under the jointaction of the temperature change and vehicle dynamic load. The analysis indicates that the proposedmodel is capable of predicting the initiation and propagation of cracks at the interface. The prefabricatedslab presents lateral warping, resulting in severe interface damage on both the sides of the slab trackalong the longitudinal direction during temperature drop process, while the interface damage level doesnot change significantly under vehicle dynamic loads. The interface damage has great effects on thedynamic responses of the slab track.

1. Introduction

In recent years, high-speed railway lines have been widelyconstructed in the form of slab track owing to several advantagesit offers over ballasted track. Some structural advantages are theimpossibility of rail buckling, a higher lateral and longitudinalpermanent stability, and a reduced sensitivity to uneven settle-ments. Operational advantages of slab tracks are a lower main-tenance (reduction of 70–90% compared with ballasted track [1]),resulting in higher availability and the possibility of longerpossession times, which is important for high-speed connections,the prevention of churning up of ballast particles at high speed,and an increase of riding comfort as well as safety, owing to thehigher track stability and better alignment [2,3]. As an example,42 high-speed passenger railway lines and intercity railways with atotal length of 13,000 km are put into operation by the year 2012 [4].

Fig. 1 shows the CRTS-II slab track and its components. The primarycomponents of the slab track system are the rails, rail pads,prefabricated slab, CA mortar layer, and concrete basfdee.

At present, most of the literatures are focused on slab trackdesign and dynamic behavior of the slab track. Steenbergen [5]studied the design parameters of a slab track railway system from adynamic viewpoint. Auersch [6] calculated the dynamic interactionof the railway slab track in detail through a combined finiteelement and boundary element method. Bezin [7] developed aflexible track systemmodel integrated with a multi-body dynamicssoftware tool to simulate the dynamic interaction between avehicle and two innovative slab track designs, and to comparetheir performance with respect to conventional ballasted track.Gulgou-Carter [8] carried out an analytical and experimental studyof the sleeper SAT S 312 in a slab track Sateba system. Galvin [9]used a general and fully three-dimensional multi-body-finiteelement-boundary element model to study the vibrations due totrain passage non-ballast tracks. Zhai [10] presented a frameworkto investigate the dynamics of overall vehicle and slab tracksystems and with emphasis on theoretical modeling, numericalsimulation and experimental validation. Lei [11] investigated the

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International Journal of Non-Linear Mechanics

n Corresponding author. Tel.: þ86 13618094208.E-mail addresses: syzhu@my.swjtu.edu.cn,

sz21@rice.edu (S. Zhu), cbcai@home.swjtu.edu.cn (C. Cai).1 Tel.: þ1 832 434 4328.

International Journal of Non-Linear Mechanics 58 (2014) 222–232

dynamic behavior for slab track of high-speed railway based on thevehicle and track elements.

Although the slab track has gained successful improvementsand applications in high-speed railway lines compared withtraditional ballasted tracks, slab track still represent some dis-advantages including the high initial investment costs, moreimportantly, the premature structural damage caused by a varietyof factors such as vehicle dynamic loads due to wheel–railinteractions associated with the wheel–rail irregularities, and thetemperature loads attributed to temperature change. In practice, itwas recently found that severe interface damages and cracksappeared at the interface between prefabricated slab and CAmortar layer on Chinese railway lines after a short time operation,as shown in Fig. 2. Such damage mode might cause significant lossof structural integrity of the slab track system and, therefore,strongly affect the dynamic behavior of slab track and produce thepotential for failure of the slab track. Consequently, it willinfluence the running behavior of railway vehicles in terms ofmotion stability, riding comfort and derailment safety. Therefore,modern railway designers as well as maintenance engineers haverequirements for increasing fundamental understanding of howthe interface damage produces and how it effects the vibrations ofslab track. However, so far very few studies have been publishedon the slab track damage and its effect on the dynamic response ofslab track. Zhu [12] developed a vertical dynamic coupled model ofvehicle and double-block ballastless track with roadbed cracksusing the finite element method on the basis of theories of linearelastic fracture mechanics and vehicle–track coupled dynamics,and investigated the value of stress intensity factor in the crack tipfield varying with the dynamic loads as well as the influence ofroadbed cracks on roadbed dynamic responses. Zhu [13] adoptedthe concrete damaged plasticity model to describe the mechanicalcharacteristics of concrete base of double-block ballastless track,and analyzed the inherent damage of track bed slab and itsinfluence on dynamic behavior of track bed slab under the joint

action of change temperature and vehicle dynamic load. Colla [14]reported a test methodology for investigating the bond conditionbetween sleepers and slab on two different railway slab trackconstructions.

The objective of the present paper is to investigate the damageoccurred at the interface between prefabricated slab and CAmortar layer on the CRTS-II slab track under temperature andvehicle dynamic loads, and also to study the effects of interfacedamage on the dynamic behavior of the slab track. The damage,cracks or delamination at the interface is investigated by usingcohesive elements in a three-dimensional FE model of the slabtrack. The interface damage evolution under the condition oftemperature change is analyzed based on the temperature fielddatabase obtained from the three-dimensional transient heattransfer analysis. Then, a three-dimensional vehicle–track coupleddynamic model is then built to calculate the dynamic rail-supporting forces which are utilized as the inputs to the finiteelement model. Setting the maximum interface damaged statecalculated under temperature change as the initial condition, theinterface damage evolution and its influence on dynamicresponses of the slab track are investigated under the joint actionof temperature change and vehicle dynamic loads. The numericalresults obtained are very useful in the design and maintenance ofslab tracks and the evaluation of material deterioration.

2. Finite element modeling

2.1. Cohesive zone model

Cohesive zone model (CZM) introduced by Dugdale [15] andBarrenblatt [16,17] has gained considerable attention over the pastyears, as it represents a computationally efficient technique andhas the simplicity and the unification of crack initiation andgrowth within one model for the fracture studies. This approachis attractive because it combines the classical fracture mechanicsconcept of a fracture toughness criterion for crack propagationwith the damage mechanics assumption of a zone ahead of thecrack tip. Pioneering works on the FE implementations of CZMtechnique can be found in Refs. [18,19] while applications to themodeling of delamination in fiber-reinforced composites areintroduced in Refs. [20–25].

In this paper, a cohesive element in Fig. 3, with zero thicknessin normal direction and a 8-node, is applied to analyze interfacedamage by embedding the cohesive element at the interfacebetween prefabricated slab and CA mortar layer. Normally, stressthat occurs inside the cohesive element is calculated throughrelative displacements between two bordering elements.

Before crack initiation, the cohesive element is assumed to havea high stiffness in order to prevent separation of two borderingelements, and is also provided for preventing node interpenetra-tion when the cohesive element is under compression. Damage isassumed to be initiated when a quadratic interaction functioninvolving the nominal stress ratios reaches a value of one, as

Prefabricated slabCA mortar layer

Concrete base

Subgrade

Fig. 1. CRTS-II slab track.

Prefabricated slab

CA mortar layer

Concrete base

Interface damage

Fig. 2. Delamination at the interface between the prefabricated slab and CAmortar layer.

Zero thickness

Fig. 3. Cohesive element with zero thickness.

S. Zhu, C. Cai / International Journal of Non-Linear Mechanics 58 (2014) 222–232 223

defined in the following equation:

⟨tn⟩t0n

þ tst0s

þ ttt0t

¼ 1 ð1Þ

where t0n, t0s and t0t represent the peak values of the nominal stress

when the deformation is either purely normal to the interface orpurely in the first or the second shear direction, respectively. tn, ts,and tt represent the normal and the two shear stress under mixedmodes of tension and shear, respectively.

The bilinear relationship between stress and relative displace-ment for the cohesive element is shown in Fig. 4 where bothnormal stress ðtnÞ and shear stress ðts; ttÞ are dependent on relativedisplacement in normal direction ðδnÞ and in sliding directionðδs; δtÞ, respectively. After damage initiation, tensile and shearstresses decrease or undergo a softening process as shown inFig. 4. The cohesive element completely fails when maximumrelative displacement ðδfn; δfs ; δft Þ is attained. An evolution of thedamage variable D, proposed by Camanho and Davila [26], is usedto describe the linear softening process and can be express as

D¼ δfmðδmaxm �δ0mÞ

δmaxm ðδfm�δ0mÞ

and the non-linear constitutive relation for the CZM model can bedescribed by the following equation:

ti ¼Kiδi for δirδ0i

ð1�DÞKiδi for δ0i oδirδfi

8<: i¼ n; s; t ð3Þ

In Eqs. (2) and (3), δmaxm refers to the maximum value of the

effective displacement attained during the loading history, δfmdenotes the effective displacement at complete failure, δ0m is theeffective displacement at damage initiation. Ki is the so-calledpenalty stiffness.

During the fracture process, the area under the curve of stressvs. relative displacement is assumed to be equal to fracturetoughness ðGC

n ; GCs ; G

Ct Þ or the critical energy release rate [18–25]

which is defined as the crack propagation criterion of the cohesiveelement in the study. Experimental results indicate that the failureunder mixed-mode conditions is governed by a power law inter-action of the energies required to cause failure in the individual(normal and two shear) modes [27]. The following failure criterionis thus adopted in this study:

¼ 1 ð4Þ

where Gn, Gs, and Gt refer to the work done by the stress and itsconjugate relative displacement in the normal, the first, and thesecond shear directions, respectively. GC

n , GCs , and GC

t which refer tothe fracture toughness required to cause failure in the normal, thefirst, and the second shear directions, respectively, needed to be

specified according to the characteristic of materials at theinterface.

2.2. Finite element model of the slab track

In order to study the interface damage occurred betweenprefabricated slab and CA mortar layer, a three-dimensional finiteelement model of the CRTS-II slab track system is built based onthe commercial software ABAQUS, as shown in Fig. 5. The slabtrack system is represented by 7808 solid elements, 1612 cohesiveelements, and 15,006 degrees of freedom, and the mesh has a totallength of 10 m. Due to the influence of rails on the interfacedamage analysis is extremely small, the rails are not considered inthe model.

In this study, mechanical analysis under temperature andvehicle dynamic loads is combined with the transient heat transferanalysis to investigate the damage and delamination at the inter-face. The transient heat transfer analysis is first conducted toobtain the temperature field database by using the heat transferelement DC3D8, and two user defined subroutines named DFLUXand FILM are used in this analysis to simulate the solar radiationand convective heat transfer around the surface of the slab tracksystem, respectively.

After the completion of the transient heat transfer analysis,transient temperatures of the slab track system are transferredinto the mechanical analysis. This is accomplished by creating adatabase file as the predefined temperature field in each step ofthe mechanical analysis. In this analysis, all the components of theslab track system are modeled as element C3D8R, all the displace-ments of the nodes at the bottom of the slab track system arerestricted, and all the nodes at both the ends of the slab track are

Nominal Stress

Stress Relativedisplacement

Fig. 4. Interfacial bilinear softening law.

Prefabricated slabZero thickness

cohesive element CA mortar layerConcrete base

Subgrade

Fig. 5. Finite element model. (a) The entire finite element mesh, (b) end view of theFE model, and (c) cohesive elements with zero thickness.

S. Zhu, C. Cai / International Journal of Non-Linear Mechanics 58 (2014) 222–232224

applied as symmetry constraints. A relatively fine mesh is usedadjacent to interface where the material volume undergoes theprocess of damage and delamination. In the present FE model, theCZM described above is utilized at the interface, the othercomponents are assumed to behave pure elastically. As the initialstiffness of the interface is treated as a penalty parameter, its valuehere is taken to be 1.0�1013 N/m, which is arbitrarily high valuesbut not so high as to cause numerical instability. Mahaboonpachai[28] showed that the interface fracture toughness between con-crete and PCM in the external wall tile structure is shown to be2–16 J/m2, when the phase angle is in the range of 0–841. Lim [29]reported that the interface between concrete and fiber-reinforcedconcrete (FRC) has the interface fracture toughness equal to2–24 J/m2 when the phase angle is 0–751, and the interfacebetween concrete and engineered cementitious composites (ECC)has the interface fracture toughness equal to 3–34 J/m2 when thephase angle is 0–601. Based on the conclusions, the fracturetoughness at the interface between prefabricated slab andCA mortar layer of the slab track is conservatively assumed to be10–50 J/m2 when the phase angle is 0–901. The normal and shearbonding strength for the CZM used here are both selected as5.4�106 Pa [30]. In Table 1, the thermal and mechanical proper-ties of CRTS-II slab track components are provided [31].

3. Coupled dynamic model of a vehicle and the slab track

Fig. 6 illustrates the three-dimensional coupled dynamic modelof a vehicle and the CRTS-II slab track which is used to predict thevarying rail-supporting forces when a vehicle passes through theslab track.

3.1. Equations of motion of vehicle subsystem

The vehicle is considered as a rigid multi-body model in whichthe car body is supported on two double-axle bogies with theprimary and the secondary suspension systems. As shown in Fig. 6,each component of the vehicle has five degrees of freedom (DOFs):the vertical displacement Z, the lateral displacement Y, the rollangle Φ, the yaw angle ψ and the pitch angle β. As a result, the totalDOFs of the vehicle are 35, as listed in Table 2. Based on the systemof coordinates moving along the track at a constant speed of thevehicle, the equations of the vehicle subsystem can be easilydescribed in the form of second-order differential equations inthe time domain:

MVAVþCVðVVÞVVþKVðXVÞXV ¼ FVðXV;VV;XT ;VTÞ ð5Þ

where MV is the mass matrix of the vehicle. CV and KV are thedamping and the stiffness matrices, depending on the currentstate of the vehicle subsystem to describe the non-linearitieswithin the suspension. XV , VV and AV are the vectors of displace-ments, velocities and accelerations of the vehicle subsystem,respectively. XT and VT are the vectors of displacements andvelocities of the track subsystem. FVðXV ;VV;XT ;VTÞ is the systemload vector representing the non-linear wheel–rail contact forces

which are the function of the motions XV and VV of the vehicleand XT and VT of the track.

3.2. Equations of motion of the slab track subsystem

For the modeling of the slab track, each rail is treated as acontinuous Bernoulli–Euler beam resting on rail pads, and thevertical, lateral, and torsion motions of the rails are simultaneouslytaken into account. According to Ritz's method, the fourth-orderpartial differential equations of the rails can be given in the form ofsecond-order ordinary differential equations.

Table 1Thermal and mechanical properties of CRTS-II slab track components.

Components Thermalconductivity(J/mh 1C)

Heatcapacity(J/kg 1C)

Young'smodulus (Pa)

Poisson ratio Density(kg/m3)

Thermal expansioncoefficient (1C�1)

Prefabricated slab 6264 900 3.6�1010 0.2 2400 1.0�10�5

CA mortar 3348 800 8�109 0.2 2400 1.5�10�5

Concrete base 3600 830 2.55�1010 0.2 2400 1.5�10�5

Fig. 6. Three-dimensional coupled dynamic model of a vehicle and the CRTS-II slabtrack: (a) elevation and (b) end view.

Table 2Degrees of freedom of the vehicle model.

Vehicle component Type of motion

Lateral Vertical Roll Yaw Pitch

Wheelset1 Yw1 Zw1 Φw1 ψw1 βw1

Front bogie frame Yb1 Zb1 Φb1 ψb1 βb1Rear bogie frame Yb2 Zb2 Φb2 ψb2 βb2Car body Yc Zc Φc ψc βc

For the vertical vibration:

€qVkðtÞþEIYmr

� �4

qVkðtÞ ¼ � ∑N

i ¼ 1PrViZkðxsiÞþ ∑

j ¼ 1PjZkðxwjÞ;

k¼ 1�NV ð6ÞFor the lateral vibration:

€qLkðtÞþEIZmr

� �4

qLkðtÞ ¼ � ∑N

i ¼ 1PrLiYkðxsiÞþ ∑

j ¼ 1QjYkðxwjÞ;

k¼ 1�NL ð7ÞFor the torsional vibration:

€qTkðtÞþGKρI0

� �2

qTkðtÞ ¼ � ∑N

i ¼ 1MsiΘkðxsiÞþ ∑

j ¼ 1MwjΘkðxwjÞ;

k¼ 1�NT ð8Þwhere qVkðtÞ, qLkðtÞ and qTkðtÞ are the generalized coordinates,describing the vertical, lateral and torsional motions of the rail,respectively. EIY , EIZ and GK are the vertical bending, lateralbending and torsional stiffness of the rail. mr is the mass per unitlongitudinal length. I0 is the torsional inertia of the rail. ρ is thedensity of the rail. l is the calculation length of the rail which ischosen to be 100 m. Zk, Yk and Θk are the kth mode functions ofvertical bending, lateral bending, and torsion of the rail, respec-tively. xsi are the coordinates of the rail-supporting points, and xwj

are the coordinates of the contact points of the wheel and rail. NV,NL and NT are the total numbers of the rail mode functions whichare all selected to be 100 in the calculation. Pj and Qj are thevertical and lateral wheel–rail forces. Msi and Mwj are the equiva-lent calculation moments on the rail. Pr Vi and Pr Li are the verticaland lateral rail-supporting forces. When qVkðtÞ, qLkðtÞ and qTkðtÞ areobtained through the dynamic calculation, the vertical, lateral andtorsional displacements of the rail can be written as

Zrðx; tÞ ¼ ∑NV

k ¼ 1ZkðxÞqVkðtÞ ð9Þ

Yrðx; tÞ ¼ ∑NL

k ¼ 1YkðxÞqLkðtÞ ð10Þ

ϕrðx; tÞ ¼ ∑NT

k ¼ 1ΘkðxÞqTkðtÞ ð11Þ

For the rail-supporting forces, take the right rail for example,it can be written as

Pr Vi ¼ Kpv½Zr�bϕr�Zs�þCpv½ _Zr�b _ϕr� _Zs�þKpv½Zrþbϕr�Zs�þCpv½ _Zrþb _ϕr� _Zs�Pr Li ¼ KplðYr�Ys�aϕrÞþCplð _Yr� _Ys�a _ϕrÞ

ð12Þwhere Zr and ϕr are the vertical and torsional displacements of therail, respectively. Zs is the vertical displacement of the slab. Kpv andCpv are the damping and stiffness of the rail fastener in the verticaldirection, respectively. Kpl and Cpl are the damping and stiffness ofthe rail fastener in the lateral direction, respectively. a is thevertical distance between the rail torsional center and the lateralforce from the fastening system. b is half of the distance betweentwo vertical forces from the fastening system.

The prefabricated slab and concrete base of the slab track aredescribed as elastic rectangle plates supported on viscoelasticfoundation. As shown in Fig. 6, the three layers of discrete springsand dampers represent the elasticity and damping effects of therail fastener, the CA mortar layer and the subgrade, respectively. Byusing Ritz's method, the vertical vibrations of the slab can beexpressed as a series of second-order ordinary differential equa-tions in terms of the generalized coordinates, as expressed byEq. (13), which can be solved with the time-stepping integration

method:

€TmnðtÞþCs

ρshs_TmnðtÞþ

B3B2þ2B4B5þB1B6

B1B2TmnðtÞ

¼ 1ρshsB1B2

i ¼ 1PrViðtÞXmðxpiÞYnðypiÞ� ∑

j ¼ 1FsVjðtÞXmðxbiÞYnðybiÞ

ð13Þwhere

B1 ¼R Ls0 X2

mðxÞ dxB2 ¼

0 Y2nðyÞ dy

B3 ¼R Ls0 Xm

0000 ðxÞXmðxÞ dxB4 ¼

R Ls0 Xm

00 ðxÞXmðxÞ dxB5 ¼

0 Yn00 ðyÞYnðyÞ dy

B6 ¼R Ls0 Yn

0000 ðyÞYnðyÞ dy

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð14Þ

According to the elastic thin slab theory, the solution of thevertical motions of the slab can be assumed as

wðx; y; tÞ ¼ ∑Nx

m ¼ 1∑Ny

n ¼ 1XmðxÞYnðyÞTmnðtÞ ð15Þ

Regarding the lateral motions, the slab is considered as a rigidbody due to its large bending stiffness in the lateral direction.Thus, the lateral vibrations can be written as

ρsLsWshs €yðtÞ ¼ ∑Np

i ¼ 1PrLiðtÞ� ∑

i ¼ 1FsLjðtÞ ð16Þ

in Eqs. (13)–(16), XmðxÞ and YnðyÞ are the mode functions of theslab with x and y coordinates, respectively. TmnðtÞ are the general-ized coordinates and describe the vertical motion of the slab. mand n are the mode numbers of XmðxÞ and YnðyÞ, respectively, andNx ¼Ny ¼ 5. Cs is the damping of the slab, Ds is the verticalbending stiffness of the slab. ρs, Ls, W and hs are the density,length, width and thickness of the slab, respectively. FsVjðtÞ andFsLjðtÞ are the vertical and lateral dynamic forces at the jthsupporting point under the slab, respectively. Np and Nb are,respectively, the total number of rail fasteners on the slab andthe total number of discrete supporting points under one slab usedin the calculation.

Similarly, the equations of motions of the concrete base can beeasily obtained and are omitted in the paper for brevity. Detaileddescriptions of the equations of the vehicle–track coupleddynamic system can be found in Ref. [10,32].

3.3. Model of wheel–rail contact

As shown in Fig. 6, the vehicle and the slab track are spatiallycoupled by the wheel–rail interface. The wheel–rail contactmodeling is fundamental to accurately investigate the interactionbetween wheels and rails. Many sophisticated models have beendeveloped for the wheel–rail contact problem in recent years.Shabana [33] proposed a method in which four parameters areapplied to describe the wheel and the rail surfaces, which allowsfor multiple points of contact between the wheel and the rail byusing an optimized search for all possible contact points. Pombo[34] presented a contact model which can be applied to study thetwo points of contact scenario, and it allows analyzing lead and lagflange contact configurations. Meli and Falomi [35–37] developedtwo semi-analytic procedures for the detection of the wheel–railcontact points (named the DIST and the DIFF methods), In bothcases, the original problem can be reduced analytically to a simpleone-dimensional-problem that can be easily solved numerically.Falomi and Malvezzi [38] presented some innovative methods toevaluate the position of contact points, including the semi-analytic

procedures (named DIST and DIFF methods) and the neural net-work model. The details of the proposed methods and theperformances, in terms of computation time and accuracy, werecompared with those of the conventional algorithms used bycommercial software, showing their reliability and low computa-tional burden.

In this paper, when calculating the dynamic response of thevehicle–track, the trace curve method [39] is adopted to locate thewheel–rail spatial contact points which are only on a curve namedtrace curve, therefore, two-dimensional scanning can be replacedby one-dimensional scanning through a trace curve.

The non-linear Hertzian elastic contact theory is used tocalculate the wheel–rail normal contact forces:

PNðtÞ ¼1GδNðtÞ

� �3=2ð17Þ

where G is the wheel–rail contact constant, and δZNðtÞ is thenormal compressing deformation at wheel–rail contact point. Theleft and right wheel–rail normal compression deformation can bedefined as

δNL ¼ δZLcos ðδL þϕwÞ

δNR ¼ δZRcos ðδR �ϕwÞ

8<: ð18Þ

where δL and δR are the left and right wheel–rail contact angles,respectively. ϕw denotes the wheelset roll angle. δZL and δZR arethe left and right wheel–rail vertical relative displacements,respectively, which depend on the motions of wheels and rails.

The tangential wheel–rail creep forces are calculated first byusing Kalker's linear creep theory and then modified by the Shen–Hedrick–Elkins non-linear model [40], as described in the follow-ing equations:

Fx ¼ � f 11ξxFy ¼ � f 22ξy� f 23ξϕMz ¼ f 23ξy� f 33ξϕ

8><>: ð19Þ

F′x ¼ FF′ Fx

F′y ¼ FF′ Fy

M′z ¼ FF′Mz

8>><>>: ð20Þ

in which

F ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF2x þF2y

qð21Þ

F′¼f N F

fN � 13

� �2þ 1

� �3� �

for ðFr3f NÞ

f N for ðF43f NÞ

8><>: ð22Þ

where Fx, Fy, and Mz are the longitudinal, lateral creep forcesand spin creep torque before non-linear modification, respectively.

f ij denotes the creep coefficients. ξx, ξy, and ξϕ are the longitudinal,lateral and spin creepage, respectively. F′x, F′y, and M′z are themodified longitudinal, lateral creep forces and spin creep torque,respectively. f is the wheel–rail contact coefficient.

If rail surface irregularities are known, the dynamic responsesincluding the rail-supporting forces can be obtained with the helpof the coupled dynamic model of vehicle–track interaction solvedby means of a new explicit integration method [41]. For simplicity,detailed descriptions of the equations of the wheel–rail contactmodel can be found in Ref. [32,39].

4. Results and discussion

4.1. Transient heat transfer analysis

It is well known that there is a big difference of atmospherictemperature between the daytime and the nighttime, whichpresents periodic variation each day in some regions due to theeffect of solar radiation. In the present analysis, the measuredtemperature and meteorological data from a certain region in hightemperature season [42], as list in Table 3, are used here toconduct the transient heat transfer analysis in which the effectof solar radiation and convective heat transfer around the surfaceof the slab track system are taken into account. The cosinefunction in Eq. (23) and the combination of two sinusoidalfunctions in Eq. (25) are used to describe the effect of solarradiation variance and the influence of convective heat transfer[42], respectively

qðtÞ ¼ a02

þ ∑1

k ¼ 1ak cos

kπðt�12Þ12

ð23Þ

a0 ¼ 2q0mπ

ak ¼q0π

1mþk sin ðmþkÞ π

2m þ π2m

h iq0π

1mþk sin ðmþkÞ π

2m þ 1m�k sin ðm�kÞ π

h i8><>:

8>>>><>>>>:

ð24Þ

where q0 represents the maximum solar radiation. m¼ 12=c,where c is the effective sunshine time. k is selected as 30 whichis sufficient to satisfy the accuracy requirement

Ta ¼ TaþTm 0:96 sin ωðt�t0Þþ0:14 sin 2ωðt�t0Þ½ � ð25Þwhere Ta is the daily average temperature. Tm is the variationamplitude of daily temperature. t0 is the initial phase.

Fig. 7 shows the temperature variations of the slab track systemduring a whole day. It can be found that the top surface ofprefabricated slab has a significant change in temperature whichgoes through a cycle of temperature rising and falling. Themaximum temperature occurs approximately at the time of

Table 3Representative atmospheric temperature data of a whole day in high temperature seasons.

Time(h) Temperature (1C) Time (h) Temperature (1C) Time (h) Temperature (1C) Time (h) Temperature (1C)

0.5 25.2 6.5 24.6 12.5 34.9 18.5 32.11 24.7 7 25.4 13 35.3 19 31.51.5 24.2 7.5 26.2 13.5 35.5 19.5 30.92 23.7 8 27.2 14 35.6 20 30.32.5 23.3 8.5 28.2 14.5 35.5 20.5 29.83 23.1 9 29.2 15 35.3 21 29.23.5 22.9 9.5 30.2 15.5 35.1 21.5 28.64 22.8 10 31.2 16 34.7 22 28.14.5 22.9 10.5 32.2 16.5 34.2 22.5 27.55 23.1 11 33.0 17 33.7 23 26.95.5 23.5 11.5 33.8 17.5 33.2 23.5 26.36 24.0 12 34.4 18 32.6 24 25.8

14:00 and the minimum temperature occurs approximately at thetime of 6:00. It is also clearly observed that the temperaturevariation of bottom surface of prefabricated slab is not noticeable.As a result, the peak value of temperature difference between thetop and the bottom surface of prefabricated slab can reach at16.9 1C. Such big difference between prefabricated slab and CAmortar layer may lead to severe damage or fracture at theirinterface, which will be analyzed in details in the next section.

4.2. Interface damage under temperature variation

In this analysis, the interface damage during the temperature dropprocess is analyzed by using the ABAQUS implicit code. As the initialstress field has influence on the stress history and stress level, the timeof 14:00 with the maximum temperature difference is selected as thestarting time, and the analysis ends at the time of 6:00 with theminimum temperature difference, as shown in Fig. 7.

Fig. 8 shows the prefabricated slab presenting lateral warping,damage initiation and propagation at the interface betweenprefabricated slab and CA mortar layer on both the sides of theslab track along the longitudinal direction (shown in red). Suchdamage mode is exactly the same as that observed in high-speedrailway lines (shown in Fig. 2).

The normal and the two shear stress contours of the interfaceelement at the end step of this analysis are shown in Fig. 9. It can befound that the interface damage is mainly caused by the presence ofnormal and the lateral shear stresses due to the combination ofopening displacement and lateral sliding at the interface (as presentedin Fig. 10). It is worth noticing that the stresses in the areas with a

width of 464.58 mm (indicated in Fig. 9(a)) along the longitudinaldirection are all found to be zero, which indicates that interfaceelements in these areas are completely failed, eventually resulting inthe interface cracks which is symmetric across the width of theslab track.

Fig. 11 shows the normal stresses of three representative nodes (asshown in Fig. 9a) varying with the time. The results indicate thatdelamination process starts to take place at a certain threshold ofnormal stress which is approximately 0.53 MPa. After that point, dueto the substantial interface damage shown in Fig. 12, the normal stressbegins to decrease dramatically until it becomes zero when cohesiveelements at the interface are fully delaminated. From Fig. 12, it can be

0:00 4:00 8:00 12:00 16:00 20:00 24:00-10

50 Top surface Bottom surface Difference

Time/h

Fig. 7. Temperature variation of prefabricated slab in a whole day.

Fig. 8. Deformation and damage distribution of the slab track system as a result oftemperature change (deformation scale factor is 300). (For interpretation of thereferences to color in this figure caption, the reader is referred to the web version ofthis paper.)

Fig. 9. Stresses contours of the interface element at the end step of the analysis.(a) Normal stress, (b) lateral shear stress, and (c) longitudinal shear stress(deformation scale factor is 300).

known that the damage of node A is initiated at about 4.5 h aftertemperature starting to drop, and then the crack propagates to thenode B and node C at the interface until the crack length reaches464.58 mm. Also, the normal stresses of the three representativenodes as a function of corresponding opening displacement areplotted in Fig. 13. The results indicate the fact that although theirresponses curves are different from each other, the area under therelationship between stress and relative displacement in each case,assumed to be equal to the fracture toughness, is nearly the same.It can also be seen from Fig. 13 that the normal stress of each nodedecreases to be zero when the relative displacement is around0.08 mm, and keeps unchanged as the relative displacement increases.

Based on the above analysis, it is worth to point out that thecohesive zone model is capable of predicting the initiation andpropagation of cracks at the interface. And it can be predicted thatthe premature interface damage would be much deteriorated as itallows rainwater to be permeated into the interface.

4.3. Interface damage under joint action of temperature and vehicledynamic loads

In the present analysis, the axle load of the vehicle is 15.5 t, andthe vehicle speed is 300 km/h. Typical rail irregularities character-ized by wavelengths between 1 and 30 m are selected to excite thevehicle–track coupled system.

Fig. 14a and b shows the typical time histories of the vertical andthe lateral rail-supporting forces when the vehicle passes through theslab track, respectively. It is clearly observed that the maximum valueof the vertical rail-supporting force is about 66 kN. This force isgenerated when the axle is above the rail fastening system and ismuch larger than that of the lateral rail-supporting force of approxi-mately 1.9 kN. For simplicity, the other rail-supporting forces are notlisted below.

In order to investigate the interface damage evolution underthe joint action of temperature change and vehicle dynamic load,

Fig. 10. Displacements contours of the interface element at the end step of theanalysis. (a) Normal displacement, (b) lateral displacement, and (c) longitudinaldisplacement (deformation scale factor is 300).

0 2 4 6 8 10 12 14-0.4

Node A Node B Node C

Time/h

Fig. 11. Normal stress varying with time.

0 2 4 6 8 10 12 140.0

Time/s

Fig. 12. Damage evolution as a result of temperature dropping.

0.0 0.1 0.2 0.3 0.4 0.5-0.4

Normal displacement/mm

Fig. 13. Normal stress as a function of normal displacement.

the maximum interface damaged state calculated in Section 4.2 isset as the initial condition of the analysis. The varying rail-supporting forces are applied on the prefabricated slab to conductnon-linear dynamic analysis by using the ABAQUS dynamicimplicit code. Fig. 15 and 16 show the time histories of the lateralshear stress and normal stress of the node near the crack tip at theinterface due to the vehicle passing, respectively. It can be foundthat the lateral shear stress increases quickly from about 0.09 MPato 0.3 MPa, and the normal stress decreases from about 0.22 MPato 0.08 MPa, when the vehicle passes the slab track. Therefore, it isthe presence of lateral shear stress rather than the normal stressthat contributes to the increase of interface damage in this case.Meanwhile the damage level of the interface increases slightlyfrom 0.82 to 0.85 as shown in Fig. 17.

4.4. Effect of interface damage on the dynamic behavior of the slabtrack

In this analysis, another similar FE model but without cohesiveelements at the interface is built, and all the nodes at the interfacein this model are tied together to ensure delamination will not

occur under the joint action of temperature and vehicle dynamicload. Therefore, the influence of interface damage on the dynamicbehavior of the slab track can be investigated by comparing resultsobtained from the model with and without cohesive elements.The effects of interface damage on the vertical accelerationand displacement of prefabricated slab are shown in Fig. 18and 19, respectively. The results shown in Fig. 18 indicate thatwhen damages occur at the interface as described in the abovesections, the maximum vertical acceleration of prefabricated slabis 4.86 m/s2, as a result of temperature and vehicle dynamic loads.This maximum vertical acceleration is about 3.2 times as large asthat without damage at the interface. It also can be clearly seenfrom Fig. 19 that the vertical dynamic displacements of prefabri-cated slab with and without interface damage are 0.04 mm and0.11 mm, respectively, which indicate about 1.75 times increasedue to the significant loss of structural integrity of the slab tracksystem caused by the interface damage. Therefore, it can beconcluded that the damage occurred at the interface betweenprefabricated slab and CA mortar layer has a great effect on thedynamic behavior of the slab track, and it can be predicted that the

0.00 0.09 0.18 0.27 0.36 0.45 0.54-2

2 Left rail Right rail

Time/s

0.00 0.09 0.18 0.27 0.36 0.45 0.54-75

Left rail Right rail

Time/s

Fig. 14. Time histories of rail-supporting forces: (a) vertical direction and (b) lateraldirection.

0.00 0.09 0.18 0.27 0.36 0.45 0.54-0.35

r stre

Time/s

Fig. 15. Shear stress of the node near the crack tip at the interface.

0.00 0.09 0.18 0.27 0.36 0.45 0.540.04

Time/s

Fig. 16. Normal stress of the node near the crack tip at the interface.

0.00 0.09 0.18 0.27 0.36 0.45 0.540.81

Time/s

Fig. 17. Damage evolution when the vehicle passing.

0.00 0.09 0.18 0.27 0.36 0.45 0.54-6

With interface damage Without interface damageV

Time/s

Fig. 18. Comparison of vertical accelerations of prefabricated slab.

service life of the slab track will be much shorten under therepeated vehicle dynamic loads. Hence, the current design stan-dard for the interface connection needs to be further improved.

5. Conclusions

A cohesive zone model is introduced in a three-dimensional FEmodel of CRTS-II slab track and utilized to investigate the damage,fracture or delamination at the interface between prefabricatedslab and CA mortar layer on the slab track. The interface damageevolution under the condition of temperature change is analyzedbased on the temperature field database obtained from the three-dimensional transient heat transfer analysis. The analysis revealsthat the prefabricated slab presents lateral warping resulting insevere interface damage on both the sides of the slab track alongthe longitudinal direction, and the use of cohesive element iscapable of predicting the initiation and propagation of cracks atthe interface. Taking the dynamic results of rail-supporting forcesobtained with the vehicle–track coupled dynamic model as theinputs to finite element model, and setting the maximum interfacedamaged state calculated under temperature change as the initialcondition, the interface damage evolution and its influence ondynamic responses of the slab track are investigated under thejoint action of temperature changing and vehicle dynamic load.The results show that the interface damage level does not changesignificantly with vehicle dynamic loads. However, such damagehas great effects on the dynamic responses of the slab track.

Future developments of present work will focus on experi-mental activities in terms of interface material properties, theinterface damage evolution and its effect on dynamics of vehicle–track system. And further analysis will be conducted based on theexperimental data in order to validate the whole model and tobetter investigate the interface damage of the slab track system

Acknowledgments

The present work is supported by the National Basic ResearchProgram of China (973 Program) under Grant 2013CB036202, theNational Hi-tech Research and Development Program of China(863 Program) under Grant 2011AA11A103-3-1-1, the NationalNatural Science Foundation of China (NSFC) under Grant51008254, the Program for Changjiang Scholars and InnovativeResearch Team in University under Grant IRT1178, the Funds fromChina Scholarship Council, the Doctoral Innovation Funds fromSouthwest Jiaotong University, and the Fundamental ResearchFunds for the Central Universities.

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