Simulation of asphalt concrete cracking using Cohesive Zone Model

Construction and Building Materials 38 (2013) 1097–1106

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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

Simulation of asphalt concrete cracking using Cohesive Zone Model Li Chang ⇑, Niu Kaijian Dept. of Highway and Railway Engineering, Univ. of Southeast, Jiangsu 210096, China

h i g h l i g h t s " Thermal Stress Restrained Specimen Test (TSRST) is simulated. " Crack of asphalt concrete pavement under abrupt temperature drop is predicted. " Cohesive Zone Model and viscoelastic model are used. " Possible ways to control top-down crack are suggested.

a r t i c l e

i n f o

Article history: Received 19 June 2012 Received in revised form 17 September 2012 Accepted 19 September 2012 Available online 7 November 2012 Keywords: Asphalt pavements Low temperature cracking Mathematical models Simulation

a b s t r a c t The top-down crack of asphalt pavement is very common in northern China. Its main reason is the abrupt temperature drop in winter. Cohesive Zone Model (CZM) is a new development in fracture mechanics. Recently, it is introduced to simulate the fracture procedure of Asphalt Concrete (AC) materials. This model has its advantages in considering non-linear property, investigating fracture path and crack propagation rule. Using CZM to simulate the fracture part of AC, combined with viscoelastic constitutive model, cracks produced by abrupt temperature drop are investigated. The Thermal Stress Restrained Specimen Test (TSRST) is selected to verify the CZM model. Then, this model is used to a two-dimension (2D) pavement structure. Through parameter sensitivity analysis, it is found that the relaxation of AC materials retards the fracture procedure. The modulus and Poisson’s ratio are the key parameters to avoid crack during abrupt temperature dropping. It provides a potential way to improve the properties of AC used in cold areas for resisting top-down crack. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Low temperature cracking is the major distress observed on asphalt pavements in cold regions, such as northern China. Fracture mechanics is introduced to explain the fracture mechanism of asphalt mixtures at low temperature. Cracking can occur as a result of a single severe temperature drop (single event) or multiple cycles of less severe temperature change (thermal fatigue). Low temperature cracking has been known for many years [1,2], but fracture mechanics has only been applied to this topic for several years. There are two typical fracture models: a long-term cyclic fracture procedure and a short-term (such as in 1 day) one. The former is fatigue fracture problem; its main reasons are cyclic loading or temperature changing. The latter is one time fracture failure problem, usually, it is because of the abrupt dropping of temperature and the contraction stresses. This paper deals with the latter problem. ⇑ Corresponding author. Tel.: +86 13851514929. E-mail address: [email protected] (L. Chang). 0950-0618/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2012.09.063

The Cohesive Zone Model (CZM) was introduced by Dugdale [3] and Barenblatt [4]. This model has been used for studying the fracture in various materials, such as metals, polymers, ceramics, and geomaterials. Compared with traditional fracture mechanics, CZM has its advantages: (1) the non-linear property around the crack tip zone can be considered; (2) fracture path needs not be set in advance, and the fracture mechanism can be investigated; (3) the fracture zone length under different loads can be attained, and the crack propagation rule can be analyzed. CZM ideally localizes the physical fracture into a small zone, defined by two imaginary surfaces and the bulk; material outside this zone is still undamaged. Once the traction transferred across surfaces exceeds certain threshold, these two surfaces start to open and crack initiates. A zero-thickness interface element is used to implement the CZM into a finite element scheme. Recently, CZM has been used by researchers to investigate the fracture of AC pavement. Jenq and Perng [5] developed a CZM model for asphalt mixtures. In this model, the material beyond the cohesive zone was considered as linear elastic. The parameters of the model at different temperatures were determined by Indirect Tensile Test (IDT) and

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Fig. 1. Illustration of CZM.

Single-Edge notched Beam (SEB) test. The model was used to simulate the low temperature fracture of asphalt overlay on old PCC road. Paulino et al. [6] proposed a built-in CZM model for AC concrete. In this model, the material out of the cohesive zone was considered as linear elastic, and crack propagation behavior had nothing to do with temperature and loading rate. Through IDT and SEB test, the strength and cohesive energy of the material were attained. Using ABAQUS, with the parameters validated by SEB test, the crack propagation of IDT specimen was simulated. The CZM interface elements were distributed within a possible crack zone. Soares et al. [7] investigated the crack propagation of IDT specimen in Superpave project by using the CZM constitutive relation. The crack propagation behavior was independent of temperature and loading rate. Asphalt mixture was considered as a two-phase material: gravel and asphalt. They were considered as linear elastic materials. Song et al. [8] used CZM to predict the mixed-mode crack trajectory, and found it was in close agreement with experimental results. Kim et al. [9,10] developed a computational constitutive model to predict damage and fracture failure of AC. Complex heterogeneity and inelastic mechanical behavior were addressed by the model. It used non-linear viscoelastic finite-element methods and elastic–viscoelastic constitutive relations. Computational simulations demonstrated that damage evolution and failure of AC were dependent on the mechanical properties of AC mixtures. This approach was suggested for the relative evaluation of AC mixtures by simply employing material properties and fracture properties of mixture components. Aragão et al. [11] presented a computational model to predict fracture behavior of heterogeneous viscoelastic asphalt mixtures.

(a) Bilinear type

Geometric model inputs could be realistically taken into account. Finite-element meshes reproduced actual AC samples through digital imaging techniques. A computational modeling framework was presented through virtual testing simulations. Model simulations were compared to performance test results for calibration. Simulation using this model could provide qualitative and/or quantitative insights into the effects of mixture design variables on overall mixture performance without relying on intensive laboratory performance tests. CZM is an explicit expression of fracture state in materials. It represents the relationship of force and displacement. A numeric approach is provided by this model to simulate the behavior of fracture within the crack area, as shown in Fig. 1. Tc is the strength of the material, df is the maximum displacement between two fracture planes. When the displacement larger than df, the attraction force between two fracture planes is zero. Since the CZM model is put forward, several types are developed, such as bilinear type, exponential type and polynomial type. They are mainly different in the t–d curves, corresponding to different materials. ABAQUS provides two types of cohesive model: bilinear type and exponential type, they are suitable for brittle or quasi-brittle materials, as shown in Fig. 2. Gc is the area under the curve. When used to asphalt material, the two CZM types have no significant difference. Bilinear CZM is selected in this study. Although, several studies have been done to investigate the fracture of AC using CZM, most of them focus on test simulation. Their conclusions are mainly about the feasibility of using CZM on AC, when the specimens are cracked by loading. This paper deals with the low temperature cracking of AC pavement. TSRST is used to validate CZM. This test is directly correlated to low temperature cracking of AC pavement, which has not been used for validating CZM in the open literature. Then, verified CZM

(b) Exponential type

Fig. 2. t–d curves of different CZM types.

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constitutive relation is adopted for AC pavement analysis, and the principles of cracking under abrupt temperature dropping condition are attained. 2. Objectives and method 2.1. Objectives The top-down crack of asphalt pavement has been found and investigated in cold regions in China for years. Previous fracture theories did not do well, and CZM provides a new approach to improve. The goal of this paper is to find the principles in the procedure of low temperature fracture. ABAQUS is used as a software instrument, because of its build-in CZM constitutive models and its ability to be customized. CZM elements are distributed at the possible fracture zone. AC has obvious viscous property. So, except for the CZM zone, other parts of the AC are expressed by viscoelastic constitutive relation. Through the models above, a new computation method can be constructed. Using it, the fracture principles of AC pavement under abrupt temperature dropping condition can be analyzed. The possibilities of fracture prediction and material control for avoiding this kind of crack become practical.

In TSRST equipment, the length of specimen is restricted to constant, and the load can be recorded continuously. So, the elastic modulus can be measured by artificial operation. Then, initial strain can be attained by initial stress and elastic modulus. For accurately measuring the elastic modulus, repeated loading method is used. Loading time is no more than 2 s, and unloading time can be more. An intermission of no less than 10 min is left before the next experiment step. The elastic modulus at given temperature is computed by load Fi, strain ei, and area A

E0 ¼

This research uses finite elements numerical method as the basic approach. Based on CZM and viscoelastic models, two computation objects are analyzed. The first is the TSRST specimen. TSRST is used to evaluate the low temperature performance of AC. By inputting the parameters determined from parameter tests for CZM, the thermal–stress curve can be computed. This curve is compared with TSRST test results. If they are close, it means the models used are feasible for analyzing actual AC pavement under low temperature. The main purpose of this research is to investigate the low temperature fracture of actual AC pavement. So, an actual pavement structure is introduced and analyzed under typical temperature dropping condition. For convenience, AC layers are considered as one integrated layer. Viscoelastic constitutive model is used for this AC layer, and other underlying base courses and subgrade are considered as elastic layers. CZM elements are distributed at the possible fracture zone. Using the models verified above, pavement structure with changing temperature field can be analyzed. Parameter sensibility analysis is conducted, four main factors are investigated. It provides a new approach for pavement fracture prediction and material control of AC layer. Usage of this method will give guidance for adopting suitable engineering approach to avoid this kind of top-down crack in cold areas.

ð1Þ

where n is the number of steps. Usually it is 3. When loading the initial load F0, the experiment control system starts. Under fixed supports, the specimen temperature and stress are recorded every 10 s, with initial strain e0

e0 ¼ 2.2. Method

n 1 X Fi nA i¼1 ei

F0 A  E0

ð2Þ

The relaxation modulus E(t) can be expressed by F(t)

EðtÞ ¼

FðtÞ Ae

ð3Þ

Prony series model is used in ABAQUS software to express the relaxation characteristic. It is a numerical method, and it means using Prony series to fit the test data of relaxation modulus. The Prony series model has the same mathematical expression as generalized Maxwell model. By TSRST experiment, the relaxation modulus at different temperatures can be attained. Through time–temperature equivalent formula, the relaxation curve at reference temperature T0 can be attained. The following step is to use Prony series model to fit the curve, and the relaxation modulus is

EðtÞ ¼ E1 þ

  N X t Ei exp 

ð4Þ

si

i¼1

3. Experiment and computing parameters Before any modeling and computing work, the mechanical parameters of materials must be attained. There are two objects, the parameters needed are different. For TSRST specimen, single material of AC is used. Thus parameters about this AC type are needed. For pavement structure, besides AC layers, there are base courses and subgrade, and their parameters should be determined too. In a viscoelastic system, the parameters about creep and relaxation of AC should be tested. To simplify, the AC layers are assumed as a single layer, and its parameters are defined by the AC-20 gradation. This kind of AC is widely used as middle layer of a threelayer AC pavement in China. It has a nominal maximum aggregate size of 20 mm (exact sieve size is 19.5 mm). 3.1. Viscoelastic parameters The viscous parameters of AC are important to evaluate the low temperature fracture characteristic of this material. It provides more accurate expression of AC material and the whole pavement structure. TSRST equipment is used to attain viscous parameters in this research. The specimen is prism shape and has the dimensions of 200 mm  30 mm  30 mm. To ensure the specimens with uniform temperature, they were kept in the chamber for at least 5 h after the temperature reaches the required value.

Where Static modulus of elasticity (Long-term relaxation modulus) E1 can be computed by residual stress r1

E1 ¼

r1 e0

ð5Þ

For fitting the Prony series, it begins with selecting a positive time sequence of s1, s2, . . . , sn. Based on this sequence, the fitting problem is simplified to determine the constant Ei (Ei > 0). It makes deviation the least between E(t) and relaxation modulus E(tn)

Eðt n Þ  Einfty ¼ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} fAg

  t exp  Ei ; i¼1 s |{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffli ffl} fCg

XN

n ¼ 1; 2; 3; . . .

ð6Þ

½B

The problem above is converted to a linear programming problem

MINIMIZEj½BfCg  fAgj WherefCg > 0

ð7Þ

This is a Nonnegative Least Squares Problem. Solving function of linear programming in mathematic software can be used to attain the matrix {C}, namely Ei. There can be different numbers of elements adopted in a generalized Prony series. Theoretically, when the number of elements grows, the relaxation behavior of AC material can be expressed more precisely. Actually, when this number is larger than 6, the Prony series results will fit the relaxation curve very well.

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3.2. Experiments and parameters for CZM

Table 1 CZM model parameters from experiments.

CZM model can reflect the fracture propagation of the material, and its model parameters come from several classical experiments. The two most usually used experiments are SEB and IDT test. The basic principle of the former experiment is the conservation of energy. Under static load, the energy produced by load will be fully converted to interfacial fracture energy. The physical meaning of this experiment is very clear. Full and continuous load– displacement curve can be attained if well controlled and the specimen can fracture steadily. During experiment, the load P and mid-span deflection are recorded and P–d curve attained. UTM-25 tester (manufactured by IPC Company) is used to conduct the experiment in this study. The beam specimen used in SEB test has the dimensions of 30 cm  6 cm  6 cm. Firstly, the AC mixture should be compressed by tire-rolling equipment; the specimen has dimensions of 30 cm in length, 30 cm in width and 6 cm in height. Then it is cut into 4 narrower specimens. An artificial notch is made at the mid-span, and the depth is 1 cm. The span of the specimen is 24 cm, it is four times of its height. Because the fracture failure is more common under low temperature, the experiment temperature is set to be 10 °C. Before testing, the specimens are kept in refrigerator for 6 h. Experiment is operated in an environmental chamber. Loading mode is to increase the deflection d at a constant speed, 1 mm/min. The load and deflection can be recorded continuously as P–d curve. Fracture Energy GF represents the energy needed to produce one unit of fracture area. It can be attained by calculating the area under the P–d curve. Based on the geometrical shape, load mode and typical P–d curve, the energy needed to produce fully fracture on a notched beam (notch height is a0) is

W ¼ W0 þ W1 þ W2 þ W3

W ¼ Alig

R dmax 0

PðdÞdd þ mgdmax Alig

ð9Þ

P H2

TC (kPa)

Value

120

2900

Table 2 Initial elastic moduli of AC at different temperatures (MPa). Temperature (°C)

20

15

10

5

0

5

10

Initial elastic modulus (MPa)

19,760

18,426

16,806

13,797

10,466

6976

4263

Table 3 Relaxation moduli at different temperatures (MPa). Data collection time (s)

Temperature of experiments (°C) 20

15

10

5

0

5

10

10 20 40 70 100 200 400 700 1000 2000 4000 7000 10,000 20,000

19,689 19,646 19,560 19,250 19,304 18,877 18,022 16,737 15,454 12,434 9696 8420 7635 6470

18,375 18,030 17,771 17,600 17,397 16,930 16,110 14,918 13,753 11,054 8618 7469 6760 5709

16,750 16,005 14,764 14,143 13,066 12,220 10,475 9286 8176 6827 5122 3626 2786 2045

13,750 13,545 13,421 12,156 10,156 8401 6687 6361 5665 4465 3029 2199 2000 1607

10,427 7966 5246 3258 2653 2436 2265 2048 1878 1487 1376 1341 1334 1309

6927 5740 4499 3447 2858 2470 2064 1994 1751 1380 1100 947 904 822

4241 3183 2550 2338 2021 1705 1438 1185 1017 782 610 475 389 253

Table 4 Generalized Prony series parameters of AC layer. Modulus Ei (MPa)

Relax time si (s)

628.4 1005.5 5020.3 6655.9 2806.9 719.0 735.5 372.7

0.2 2 20 200 2000 20,000 200,000 2,000,000

Elastic modulus E0 = 18,000 MPa.

where mg is the weight of the beam between the two supports; dmax is the maximum deflection at mid-span, and Alig is the area of beam fracture scope, which equals to b(h  a0). For eliminating the difference of compaction, cubical shape specimens are also used in IDT test. They are cut directly from the rutting specimen. And they have the dimensions of 6 cm  6 cm  6 cm. Before testing, the specimens are kept in refrigerator for 6 h, and the experiment should be finished in 1 min after being taken out. The tensile strength is

T C ¼ 0:637

GC (J/m2)

ð8Þ

Rd where W0 is the energy produced by load P(d), W 0 ¼ 0 max PðdÞdd; W1 + W2 are the energy produced by the weight of the beam between two supports, which is separated into two parts(W1 and W2, W1  W2) by the fracture; W3 is the energy produced by the mass of the part of the loading arrangement, not attached to the machine, that follows the beam during failure, it is very small and can be ignored. Because W 1  W 2 ¼ 12 mgdmax , the fracture energy per unit area can be expressed as

GC ¼

Parameters

ð10Þ

where TC is the tensile strength of AC, kPa; P is the maximum load, kN; H is the dimension of the cube, m.

3.3. Other parameters In this research, because the focus is the low temperature fracture of AC, 20 °C is selected as the standard temperature for experiments. Parameters are all measured at this temperature. SEB and IDT test are used to measure the fracture energy and tension strength. Data from former researchers are shown in Table 1 [12]. They are used as computing parameters in this study. Seven different temperatures are selected to perform relaxation experiments. They are: 20 °C, 15 °C, 10 °C, 5 °C, 0 °C, 5 °C and 10 °C. The initial moduli at different temperatures are shown in Table 2. Other parameters needed for considering viscoelastic property of asphalt layers are relaxation moduli. They are measured at different temperatures. Zheng et al. [13] have measured

L. Chang, N. Kaijian / Construction and Building Materials 38 (2013) 1097–1106 Table 5 Elastic parameters of base courses and subgrade. Layer

Thickness (cm)

Modulus E (MPa)

Poisson’s ratio

Temperature contraction coefficient a (1/°C)

Semirigid base Subbase Subgrade

40

4000

0.15

7  106

20 —

1000 60

0.20 0.35

1  105 2  105

the relaxation stresses at different temperatures, and calculated the relaxation moduli, as shown in Table 3. Suppose the Poisson’s ratio keeps constant (l = 0.3). When brought into ABAQUS software, the relaxation moduli of AC from Table 3 are expressed by generalized Prony series. They are shown in Table 4. Because the heat insulation effect of AC, temperature changings in layers beneath are very small. And the stress and strain in these layers produced by load decrease quickly as the depth increases. So, in this study, the semi-rigid base courses and subgrade in pavement structure are all assumed as elastic materials. Their mechanical parameters used for computing are listed in Table 5.

4. Structure, hypotheses and model 4.1. Simulation of TSRST TSRST is used to evaluate the anti-crack property of AC material under low temperature. The specimen is settled in the chamber and held firmly to avoid shrinkage. The temperature is lowered down at a fixed rate during testing. Because the specimen remains a constant length during the test, the tensile stress is accumulated

1101

when the temperature keeps going down. When the tensile stress exceeds the strength of the material, the specimen will initiate tensile fracture. The tensile force, its corresponding temperature and elapsed time are recorded. The materials can be compared and evaluated by the fracture failure temperature. This paper selects the TSRST as the first simulation object, because the material, shape and testing conditions are very simple. But the experiment has its value in evaluating the low temperature fracture failure of AC. Its result can be used to calibrate the model and parameters before the simulation of actual AC pavement. ABAQUS is used as analyzing instrument. In the material database, elastic, viscoelastic and expansion parameters are defined. The coefficient of thermal expansion is assumed as constant (2.0  105/°C). CZM elements are used near the center of its 200 mm length (‘‘C’’ point, see Fig. 3). The time-domain analysis method is used and time–temperature equivalent parameter is assigned. Eight elements general Prony series model is used for AC. Time–temperature equivalence uses classical WLF formula. The specimen is a prism, and has the dimensions of 200 mm  30 mm  30 mm. It has rectangle section and the length is 200 mm. The prism is divided as elements with basic dimensions of 5 mm  5 mm  5 mm, as shown in Fig. 3. The element type is C3D20R, it is a 3D solid element with 20 nodes and second order (quadratic) reduced integration element. While computing, a viscoelastic analysis step is constructed, and the step time is the same as the real time. The start temperature is set to be 10 °C, and the cooling rate of the temperature is 5 °C/h. The final stable temperature is 40 °C. The whole cooling time for analyzing is 36,000 s. Initial and maximum increments are all set to be 10s, and the CETOL is set to be 1  106 to define the precision requirement. The top and bottom of the specimen are fixed. This ensures that the TSRST specimen has ‘‘zero displacement’’ in length direction. During the whole temperature cooling period (from 10 °C to

Fig. 3. Mesh of the TSRST specimen.

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40 °C), a linear temperature changing curve is used to simplify the model. The temperature of every point in the specimen is assumed to be the same. Heat transferring procedure is not considered. 4.2. Fracture simulation of AC pavement under abrupt temperature drop condition When temperature abruptly drops in winter, AC pavement has the possibility of fracture. Although this paper mainly deals with this short-term fracture failure problem, it is still very complicate and need to adopt some hypotheses. They are: (1) Every layer of the pavement structure is made of uniform, isotropic, and single material. (2) AC is assumed as viscoelastic material, and the base course layers and subgrade are assumed as linear elastic materials. (3) The interfaces between any two layers are fully continuous. (4) In horizontal plane, the temperature is the same at any point, only the temperature difference in vertical direction is considered; (5) The elastic modulus, Poisson’s ratio and coefficient of thermal expansion are assumed as constant for base course layers and subgrade. (6) The drop of temperature is assumed as linear. (7) The weight of the material is not considered. Computation object has dimensions of 5 m in length and 3 m in depth, and the thickness of each layer is shown in Fig. 4a, this pave-

ment structure is used widely in China. The pavement structure includes AC surface layer, semi-rigid base course, subbase layer and subgrade. They form the four-layer pavement system. Fig. 4a shows the structure layers and their dimensions, and Fig. 4b gives the meshing details of this structure in numeric computation. There are totally 19 rows and 56 columns. The distribution interface of ‘‘coh elements’’ is shown in both figures by a bold lines at the middle position of asphalt pavement in the horizontal direction. The ‘‘coh elements’’ (CZM model) with zero thickness are inserted only at the symmetrical axis of asphalt pavement in horizontal direction. Two-dimension (2D) plane strain elements with viscoelastic property are used for bulk of asphalt pavement. Former researcher [12] tried to insert cohesive elements in a potential area to investigate the propagation path of crack in SEB specimen. It is found that, numerical convergence is the main difficulty to solve this kind of problem. For simplifying, this study assumes the fracture plane is the symmetrical plane. Only one column cohesive elements are inserted at this plane. The temporal and spatial distribution of temperature is called temperature field, which can be expressed as following

T ¼ Tðx; y; z; tÞ

ð11Þ

In this research, it is simplified to one dimension steady temperature field. When a temperature dropping happens at the pavement surface, the temperature at depth y is supposed to change as following

TðyÞ ¼ Pi exp½bi ðy  hi Þ i ¼ 1; 2; 3; 4 Distribution Interface of coh Elements Asphalt Pavement (18cm) Semi-rigid Base (40cm) + Subbase (20cm)

3m Subgrade (222cm)

5m

(a) Pavement structure 18 Columns

20 Columns

18 Columns

9 Rows 6 Rows

4 Rows

(b) Meshing of pavement Fig. 4. Pavement structure and its meshing.

ð12Þ

L. Chang, N. Kaijian / Construction and Building Materials 38 (2013) 1097–1106

1103

where Pi is the temperature dropping at the surface of the ith layer; hi is the y coordinate of the surface of the ith layer (see the coordinate in Fig. 4a). bi is a coefficient used to control the cooling rate of the temperature along the y direction. Usually, b1 = 5, for others bi+1 = bi1, and

perature field is set, and it is defined as dropping linearly in the viscoelastic analysis step.

Piþ1 ¼ Pi expðbi g i Þ

5.1. TSRST test simulation

ð13Þ

where gi is the thickness of the ith layer. The coefficient of thermal expansion (COE) of AC depends on several parameters. Many studies show that it changes not only with the gradation of aggregate, characteristic of asphalt, but also the instantaneous temperature and its changing rate. Different gradations and asphalt contents influence the COE remarkably. In this research, to simplify, the asphalt layers of the pavement are assumed as an integrated layer, and its COE a is simplified as

a ¼ aðT; T’Þ

ð14Þ 0

0

where T = T(y), is the temperature at any depth; T = T (y), is the changing rate of the temperature. Pavement has semi-rigid base course, which has lower temperature changing rate than AC, and its COE can be assumed as constant. CZM area selects the COH2D4 type as cohesive element in ABAQUS, whose nonconforming finite element type is CPE4I. The mesh of the structure is shown in Fig. 4b, which has a depth of 3 m (y coordinate), and a length of 5 m (x coordinate). The x coordinate is parallel with the driving direction. For the element in the CZM area, the modulus E is set to 6500 MPa, and Poisson’s ratio l to 0.3. The failure rules follow ‘‘maximum nominal stress criterion’’. When the maximum nominal stress reaches value 1, the interface is considered damaged

( ) ht n i ts tt ¼1 max ; ; t 0n t0s t0t

ð15Þ

where t 0n , t 0s and t 0t express separately the nominal stress on the damage starting plane. They have three directions: normal, the first shear direction and the second shear direction; ‘‘<>’’ is the Macaulay n bracket, whose definition is ht n i ¼ jtn jþt , which expresses that the 2 pure compressing situation produces no damage. Damage propagation law based on principle of energy dissipation is used to confirm the crack propagation tendency. The damage developing criterion can be expressed as following

(

Gn

GCn

)a

( þ

Gs

GCs

)a

( þ

Gt

GCt

)a ¼1

ð16Þ

where Gn, Gs and Gt separately express the energy produced in normal, the first shear direction and the second shear direction; GCn , GCs and GCt is the corresponding critical fracture energy in the three directions above; a defines the correlation under the condition of mixed crack propagation mode. For linear developing damage, a damage factor D is introduced

D¼

  dfm dmax  d0m m  dfm  d0m dmax m

ð17Þ

where dfm ¼ 2GC =T 0eff , and T 0eff is the effective stress when the damage starts; dmax is the effective maximum displacement that can m be reached during the loading procedure. When simulation begins, a viscoelastic analyzing step is established. Its time step equals to real time; initial and maximum time step are all 20 s. This method discretes the whole analysis time period. The boundary conditions are: horizontal constraints are set to the left and right boundaries; horizontal and vertical constraints are set to bottom boundary. In the initial analysis, a beginning tem-

5. Computation results and discussions

In this simulation, the start temperature of TSRST is set to be 10 °C, temperature dropping rate is 5 °C/h, and the final temperature is 40 °C. The fracture positions in experiments are located very close to the middle of the specimen. The point ‘‘C’’ located at the middle is taken as a reference position (as shown in Fig. 5a). The S22 stress is the vertical tension stress. Fig. 5a is the nephogram of vertical tension stress after 7 h low temperature. It can be seen that there is stress concentration at the top and bottom, because of the fixed supports. Within the most part along the height, the stress values are nearly the same. The simulation result of the TSRST is shown in Fig. 5b. The thermal stress increases as the temperature drops. In the initial period, because of the relaxation of viscoelastic AC material, the increasing speed is very slow. In the following period, because the temperature is lower, the relaxation effect of AC is weakened. AC material becomes more brittle, and the increasing in stress speeds up. The whole curve is very close to the actual experimental data. It suggests that the model and its parameters used are feasible to simulate AC material. And the next step is to use these in fracture simulation of actual pavement. 5.2. Pavement fracture simulation 5.2.1. Simulation results AC pavement has the possibility of fracture when facing abrupt dropping of temperature. Because the computing structure is axial–symmetrical, the fracture position should be located at the center point of the 5 m width (see Fig. 4a). So, the CZM elements are distributed around this position. Stress increases when the temperature keeps dropping. At some moment, the fracture initiates, and the index ‘SDEG’ is used to evaluate whether the CZM element has been fractured or not. SDEG is 0 when the material has no damage, and 1 when fractured. Through judging the SDEG index, the fracture depth can be confirmed. In this simulation, initial temperature is 0 °C, and several typical fracture depths are selected as special fracture situations. They are: 0 (critical situation of fracture), 2 cm, 4 cm, 6 cm, and 18 cm (fully fractured, the whole height of AC layers is reached). For every special fracture depth situation, it is the combination influence of two parameters: dropping temperature and the evolving time. When one of the two parameters is fixed, the relationship of other parameter with fracture depth can be extracted. Suppose temperature dropping time is 5 h, the relationship of temperature dropping and fracture depth is shown in Fig. 6a. It can be seen from Fig. 6a: (1) When the temperature drops from 0 °C to 19.6 °C in 5 h, the surface of AC reaches the critical point of fracture. (2) With the further dropping, the fracture begins to develop. It means the thermal stress has produced enough damage to the AC material, and when it exceeds the anti-crack strength, the fracture propagates. (3) When the temperature drops to 22.8 °C in 5 h, the AC layer is fully fractured. The fracture depth is zero if the temperature drops to 19.6 °C and 6 cm to 22.6 °C. It means the fracture of AC layer is very sudden. There are two main reasons: firstly, under low temperature condition, the stiffness

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Fracture depth (cm)

Temperature dropping ( -19 0 2 4 6 8 10 12 14 16 18

-20

-21

) -22

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(a) Relationship of temperature dropping and fracture depth (5 hours)

Fracture Depth (cm)

Temperature Dropping ( )

(a) Vertical tensile stress nephogram of specimen after 7h (uint: Pa, “C” is the center point)

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Viscoelastic Elastic

(b) Fracture temperature with or without consideration of viscous property Fig. 6. Basic simulation results.

(b) Comparison of TSRST and Simulation Result Fig. 5. Simulation and test results of TSRST.

of AC material is higher and inclined to be more brittle; secondly, when the initial crack coming into being, the stress concentration effect speeds up the fracture procedure. For understanding the influence of viscous property of AC material in the procedure of low temperature fracture, the calculating results using elastic and viscoelastic constitutive models are compared in Fig. 6b. The initial temperature is set to 0 °C, and temperature dropping time is 5 h. It is shown that: (1) For reaching critical point of fracture, viscoelastic model has lower temperature than elastic model. The elastic model begins to fracture at 15.7 °C, and it is 19.6 °C for viscoelastic model. It means the viscous property retards the fracture of AC. (2) For the elastic model, the corresponding temperatures to critical fracture point and fully fracture are 15.7 °C and 16.6 °C respectively; it is 19.6 °C and 22.8 °C for the viscoelastic model. It means, under similar temperature drop-

Fig. 7. Critical fracture temperature under different elastic moduli and Possion’s ratios.

ping situation, the fracture procedure for viscoealstic model is obviously longer than elastic model. It is more accurate to represent the formation of fracture in actual AC.

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5.2.2.1. Elastic modulus and Poisson’s ratio. Elastic modulus in the Prony series expression represents the elastic property of AC. It influences greatly the initial response of AC. For understanding its influence, the elastic modulus is changed from 8000 MPa to 28,000 MPa. And their corresponding fracture temperatures are shown in Fig. 7a. It can be seen that: the higher the elastic modulus of AC is, the higher the critical fracture temperature will be. It is because the higher the elastic modulus is, the higher the accumulated thermal stress is produced under the same temperature dropping situation. Consequently, more severe damage is produced. For understanding the influence of Poisson’s ratio, its value is changed from 0.1 to 0.4. The critical fracture temperature is shown in Fig. 7b. It can be seen from Fig. 7b that, the higher the Poisson’s ratio of AC, the higher the critical fracture temperature is. This is because the accumulated stress increases when the Poisson’s ratio increases. 5.2.2.2. Initial temperature. For understanding the influence of initial temperature, its value is changed from 5 °C to 5 °C. The result is shown in Fig. 8a). It can be seen in Fig. 8a that, when the initial temperature changes from 5 °C to 5 °C, the fracture temperature drops too. But the shapes of curves under different initial temperature are very similar. They are nearly parallel with each other. The higher the initial temperature is, the higher the fracture temperature is.

Fracture Depth (cm)

Temperature Dropping ( -15 0 2 4 6 8 10 12 14 16 18

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Temperature Dropping (

Fracture Depth (cm)

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1 hour dropping 3 hours dropping 5 hours dropping 7 hours dropping 9 hours dropping

Fig. 9. Influence of temperature dropping time.

Converting the fracture temperature data in Fig. 8a to temperature dropping value, Fig. 8b can be attained. It can be seen from Fig. 8b that, for reaching the same situation of fracture, the higher the initial temperature is, the sharper the temperature dropping is needed. It means the relaxation effect becomes more effective when the initial temperature is higher. More accumulated thermal stresses are released by relaxation effect. 5.2.2.3. Temperature dropping time. Suppose the dropping time is fixed, the relation of temperature dropping and the fracture depth can be determined. It is shown in Fig. 9. The initial temperature is 0 °C, and other conditions keep the same. It can be seen from Fig. 9 that, when the temperature dropping time changes from 1 h to 9 h, the basic principle is the same: the shorter the temperature dropping time is, the higher the critical fracture temperature is. Because the relaxation effect can be fully developed when the dropping speed is slower, the accumulated stress can be released more. When temperature dropping speed increases, the viscous effect develops slowly, elastic strain increases more quickly, and the AC material shows more elastic characteristic. As a result, the AC pavement begins to fracture at a higher temperature. 6. Summary and conclusions Viscoelastic and CZM constitutive models are used in this paper to simulate the behavior of AC materials. Two objects with AC materials are simulated. The first one is the TSRST specimen, its experimental procedure is simulated. The result shows that the models are feasible to simulate AC material. The second object is the actual pavement, and parameter sensitivity analysis is done. Some conclusion can be attained from these simulations:

Dropping from 5 Dropping from 2 Dropping from 0 Dropping from -2 Dropping from -5

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)

(a) Fracture temperature under different initial temperature

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Temperature Dropping (

Fracture Depth (cm)

5.2.2. Parameter sensitivity analysis Based on the basic object and parameters above, parameter sensitivity analysis is carried out. It includes: elastic modulus, Poisson’s ratio, initial temperature and temperature dropping time.

)

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Dropping from 5 Dropping from 2 Dropping from 0 Dropping from -2 Dropping from -5

(b) Temperature dropping under different initial temperatures Fig. 8. Fracture temperatures at different initial temperatures.

(1) The simulation result of TSRST shows, the thermal stress increases as temperature drops. When the temperature is still high, because of the relaxation of AC material, the increasing speed is very slow. When the temperature becomes lower, the relaxation effect of AC is weakened. AC material becomes more brittle, and the increasing of stress speeds up. The computed curve is very close to the actual experimental data. It suggests that the CZM constitutive models and parameters are feasible to simulate AC materials. (2) For reaching critical point of fracture, viscoelastic model has lower temperature than elastic model. It means the viscous property retards the fracture of AC. The fracture procedure for viscoelastic model is obviously longer than elastic model. It is more similar to the actual fracture procedure in AC pavement. (3) Parameter sensitivity analysis shows that, elastic modulus and Poisson’s ratio, initial temperature, and temperature

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dropping time, the four parameters all have influence on the fracture temperature of AC. The combination of these four parameters decides the fracture possibility of AC layer in actual pavement. The former two parameters are the keys to avoid crack during abrupt temperature dropping. The AC layers with lower modulus and Possion’s ratio at low temperature are better to resist the top-down crack.

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