Comparison of Data Interpretation Procedures for

Road Materials and Pavement Design

ISSN: 1468-0629 (Print) (Online) Journal homepage: http://www.tandfonline.com/loi/trmp20

Comparison of Data Interpretation Procedures for Indirect Tensile Creep Test for Linear Viscoelastic Materials Adam Zofka , Mihai Marasteanu , Lev Khazanovich & Iliya Yut To cite this article: Adam Zofka , Mihai Marasteanu , Lev Khazanovich & Iliya Yut (2010) Comparison of Data Interpretation Procedures for Indirect Tensile Creep Test for Linear Viscoelastic Materials, Road Materials and Pavement Design, 11:sup1, 411-441, DOI: 10.1080/14680629.2010.9690340 To link to this article: http://dx.doi.org/10.1080/14680629.2010.9690340

Published online: 19 Sep 2011.

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Date: 28 June 2016, At: 17:31

Comparison of Data Interpretation Procedures for Indirect Tensile Creep Test for Linear Viscoelastic Materials

Downloaded by [University of Connecticut] at 17:31 28 June 2016

Adam Zofka* — Mihai Marasteanu** — Lev Khazanovich** Iliya Yut* * University of Connecticut 261 Glenbrook Road Unit 2037, Storrs, CT 06269-2037, USA [email protected] [email protected] ** University of Minnesota 500 Pillsbury Drive S.E., Minneapolis, MN, 55414, USA [email protected] [email protected]

ABSTRACT. The current standard in the US to interpret the IDT test results (AASHTO T 32203) is based on the procedure proposed by Roque et al. 1992. The alternative procedure (Zhang-Drescher-Newcomb, ZDN) was proposed by Zhang et al. 1997 and it utilizes elastic – viscoelastic correspondence principle to determine volumetric and deviatoric parts of the creep compliance function. This paper compares both IDT interpretation procedures and proposes improvements to the ZDN method based on the correction factors found in AASHTO. The comparison is done in two ways: first, procedures are applied to the Finite Element Analysis (FEA) results and second all interpretation procedures are applied to the experimental results on two plastic reference materials. For verification, creep compliance for the reference materials is also determined from 3-point bending test. It was concluded that average creep compliance functions determined from the IDT and 3-point bending tests are virtually the same and improved ZDN procedure yields practically the same compliance values as AASHTO standard. Finally, the paper includes distributions of equation coefficients for the improved ZDN procedure that were determined from 178 IDT tests on asphalt mixtures. Example plots of volumetric and deviatoric parts of the creep compliance function are also shown. Improved ZDN procedure can be useful in examination of the bulk behavior of asphalt mixtures and viscoelastic materials in general. KEYWORDS: Creep

Compliance, Indirect Tensile (IDT) Test, Viscoelastic Materials, Asphalt

Mixture. DOI:10.3166/RMPD.11HS.411-441 © 2010 Lavoisier, Paris

Road Materials and Pavement Design. EATA 2010, pages 411 to 441

412

Road Materials and Pavement Design. EATA 2010

1. Introduction and IDT background


The Indirect Tensile (IDT) test is well-established laboratory procedure to determine primarily creep compliance function and tensile strength of asphalt mixtures. The main advantage of this test is the ability to simulate a state of stress similar to the state of stress induced under the moving vehicle in the asphalt layer of the pavement system (Roque et al., 1992). During the IDT test, the cylindrical specimen is loaded vertically along its length and displacements are measured on both faces of the specimen which leads to the tensile stresses in the center of the specimen. Combining measured displacements and appropriate identification procedure, creep compliance of the specimen can be calculated. The creep compliance describes the low-temperature behavior of asphalt mixes and it serves as primary material input to the thermal cracking prediction model called TCMODEL. TCMODEL was originally developed under the Strategic Highway Research Project (SHRP) (Lytton et al., 1993) and was later updated for inclusion in the Mechanistic-Empirical Pavement Design Guide (M-EPDG) (Hallin et al., 2004). The TCMODEL in the M-EPDG predicts the depth and the amount of thermally induced cracking in asphalt pavements as a function of time. Recent research efforts (Hyunwook et al., 2009) focus on major revisions to TCMODEL to include additional material characteristics such as fracture energy and more sophisticated models than 1-D solution used in the current version of TCMODEL. However, the creep compliance will stay as the material input since it is a fundamental irreplaceable material property. The application of the IDT laboratory setup to the asphalt mixture dates back to early 70’s (Hadley et al., 1969, Gonzales et al., 1975). Since then the IDT setup has been used for variety research purposes including determination of failure strain (Ruth et al., 1977; Leung et al., 1987 ; Birgisson et al., 2007), resilient modulus (Said, 1990; Olard et al., 2006; Ping et al., 2008), Poisson ratio (West et al., 1992, Mohammad et al., 1993; Mirza et al., 1994), tensile strength (Mamlouk et al., 1979; Braz et al., 2000; Xiao et al., 2008), moisture sensitivity (Kennedy et al., 1984, Parker et al., 1988), fatigue cracking (Kennedy et al., 1976; Kim et al., 2002; Cocurullo et al., 2008), dynamic modulus (Roque et al., 1998; Kim et al., 2004; Ping et al., 2008), crack growth rate (Roque et al., 1999; Zhang et al., 2001; Sangpetngam et al., 2003) and fracture energy (Hossain et al., 1999; Kim et al., 2002; Zborowski et al., 2007). All these applications prove universality of the IDT setup for investigating the asphalt mixtures. However, since the Superpave mix design method was implemented by many transportation agencies in the US and recent developments in M-EPDG, the primary application of the IDT setup remains the evaluation of the creep compliance and tensile strength of asphalt mixtures (Roque et al., 2002; Christensen et al., 2004; Zofka et al., 2008). The major developments in the interpretation of the IDT creep test results were accomplished during the SHRP project while working on the pavement performance models and thermal cracking model in particular (Lytton et al., 1993). The solution


Comparison of IDT Data Interpretation Procedures

413

proposed by Roque et al. (1992) has served as foundation for subsequent AASHTO standards including current T 322-03 procedure. According to this standard and Superpave mix design, the IDT creep test is performed at three temperatures 0°C, -10°C and -20°C, regardless of the asphalt binder type in the mixture. The dimensions of the specimen should be 38 to 50 mm high and 150 +/- 9 mm in diameter and the maximum aggregate size of the mixture should be less than 38 mm. The standard duration of the test is 100 s but to obtain better overlaps of the data curves in creating master curve, 1000 s creep test is recommended. According to T 322-03 specimen deformations are measured by the linear variable differential transducers (LVDT) or strain gages mounted on both faces of the specimen (front and back) in both vertical and horizontal directions (Figure 1). The LVDTs are mounted on the magnetic gauge points that are directly glued to the specimen faces. Once loaded, the specimen deforms and LVDTs measure specimen deformations in vertical and horizontal directions on both faces. These measurements are applied next to the identification procedure that allows for determination of creep compliance in given conditions. Further detailed discussion on the IDT testing protocol and its refinements can be found elsewhere (Christensen et al., 2004).

Figure 1. IDT experimental setup

It should be mentioned here that the IDT setup presented in Figure 1 is primarily used in the US. Similar setup used typically in Europe is called the Nottingham Asphalt Tester (NAT). This equipment was developed by Cooper et al. (1989) at the University of Nottingham, UK and it is currently used in several European standards (EN 12697-24, EN 12697-26, EN 13108-20). The main difference between USbased IDT setup and the NAT setup is that the LVDTs in the NAT are mounted horizontally on the side of the specimen and they measure the total specimen deformations in the horizontal direction. The identification procedures discussed in this paper are directly applicable to the IDT creep test results but with minor modifications could be applied to the NAT results.

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2. Objectives The main objectives of this study are as follows: – Review two interpretation procedures for the IDT creep test: current AASHTO standard procedure (AAHTO T 322-03) based on the work done by Roque et al. (1992) and procedure proposed by Zhang et al. (1997) (called here ZDN procedure from its authors, Zhang, Drescher and Newcomb). – Propose improvements to the ZDN method based on the correction coefficients from the AASHTO method to determine deviatoric and volumetric parts of creep compliance for asphalt mixtures. – Conduct comparison of the IDT interpretation procedures using Finite Element Analysis (FEA) and laboratory experiments on two reference materials: HighDensity Polyethylene (HDPE) and Ultra-High Molecular Weight Polyethylene (UHMW). Additionally, verify the IDT creep compliance with the results from the simple 3-point bending test.

3. IDT Interpretation procedures The load and displacement measurements recorded during the IDT creep test are used to calculate the creep compliance function D(t) of asphalt mixtures. In order to correctly determine this function an appropriate data interpretation procedure has to be employed. Next sections present two different procedures to calculate D(t) using measured vertical û and horizontal û displacements: current AASHTO standard procedure (AAHTO T 322-03) based on the work done by Roque et al. (1992) and less-known procedure (ZDN) proposed by Zhang et al. (1997). Further section presents proposed improvements to the ZDN method incorporating correction coefficients for bulging of the specimen and for converting 2D plane stress solution to 3D solution. These corrections were originally proposed by Roque et al. (1992) and Buttlar et al. (1994) and they are implemented in the current AASHTO procedure but they were never considered in association with the ZDN procedure.

3.1. AASHTO method The current AASHTO standard (AAHTO T 322-03) to interpret the IDT creep test results is based on the procedure proposed by Roque et al. (1992). It is based on Frocht solution (Frocht, 1948) for stress distributions along horizontal and vertical axes in the IDT specimen. By using elastic-viscoelastic correspondence principle, creep compliance D(t) for the plane stress conditions is derived from the Hooke’s law as follows (Buttlar et al., 1994):



Hx D(t ) V x QV y

'U (t ) CBx CH x GL 2P C 3Q CSy S tD Sx

415

[1]

where is Poisson ratio, t and D are thickness and diameter of the specimen, P is the applied force and GL is the gage length (38 mm for 150 mm specimens). C parameters are corrections coefficients that were introduced to take into account various phenomena during testing: – CBx, CBy coefficients for bulging of specimen faces, applied to measured horizontal deformations ûU and measured vertical deformation ûV, respectively. – C0x, C0y coefficients for converting average strain (derived from corrected deformations ûU and ûV) to horizontal and vertical strains at the point in the center of the specimen. – CSx, CSy coefficients for converting 2D plane stress to 3D solution which are applied to 2D stress solution for 1x and 1y, respectively. After rearranging, the final expression for D(t) is given as D(t )

'U m Davg tavg Pavg GL

CCMPL

[2]

where: ûUm – trimmed mean of the horizontal deformations, tavg, Davg – average specimen thickness and diameter, Pavg – average force, CCMPL – creep compliance parameter computed as CCMPL

1.071S CBx

2 CSx 3Q CSy

[3]

After simplifications, the creep compliance parameter CCMPL can be expressed as the function of ûUm/ûVm where ûVm represents trimmed mean of the vertical deformations. The AASHTO procedure uses a trimmed mean approach on the data from three replicates, i.e. extreme values of displacements ûU and ûV are removed and remaining values are averaged to obtain ûUm and ûVm. The sorting is performed on measurements either in the middle of the creep test or on ‘mid-test’ averages taken from the time window between 460sec and 540sec. It is also possible, though it is not included in the AASHTO standard, to use a simple mean approach. In this approach, trimmed means ûUm and ûVm. in Equations [2] and [3] are replaced by the means of horizontal and vertical deformations taken from one specimen (front and

416


back) and D(t) is thus calculated separately for each specimen. Such approach was implemented in this study when three replicates were not applicable (FEA) or not applicable (reference testing). The AASHTO procedure outlined above does not require the direct solution of Poisson ratio . Instead the procedure uses ûUm/ûVm parameter that is inherently built into CCMPL parameter. However, the solution for Poisson ratio is available and it can be calculated if necessary from the following formula (Buttlar et al. 1994): 2


Q

2

§ 'U · § t · § 'U · 0.100 1.480 ¨ ¸ 0.778 ¨ ¸ ¨ ¸ © 'V ¹ © D ¹ © 'V ¹

2

[4]

Finally, it should be mentioned that correction coefficients CBx, CBy, CSx, and CSy used in AASHTO method depend on the Poisson ratio and specimen dimensions t and D as presented in Figure 2. The appropriate equations can be found in Roque et al. (1992) and Buttlar et al. (1994).

Figure 2. Coefficients CBx, CBy, CSx, and CSy in AASHTO procedure


417

3.2. Zhang-Drescher-Newcomb (ZDN) method


Zhang et al. (1997) was the first to present an alternative to AASHTO standard method of calculating creep compliance D(t) from the IDT test. For simplicity this method is referred to as the ZDN method from the names of its authors: Zhang, Drescher and Newcomb. In general, ZDN method uses correspondence principle to determine creep compliance D(t) from the IDT elastic solution proposed by Hondros (1959). Similar approach was presented later by Wen et al. (2002) and Kim et al. (2002) however these studies lead to similar equations as ZDN procedure and thus are not presented here. Brief outline of the ZDN method is presented below whereas details can be found in Zhang et al. (1997). Hondros’s solutions for vertical (1yy) and horizontal (1xx) stresses along major axes in the IDT specimen can be expressed in the following form:

V xx ( x, 0)

2P § x · f D, S aL ¨© R ¸¹

V yy ( x, 0)

2P § x · g D, S aL ¨© R ¸¹

[5.1]

V xx (0, y )

2P § y · h D, S aL ¨© R ¸¹

V yy (0, y )

2P § y · k D, S aL ¨© R ¸¹

[5.2]

where: R, L – radius and length (thickness) of the IDT specimen, p – radial pressure applied to loading strip, 2. – central angle of loading strip, a = 2Rsin . – width of loading strip, P = paL – force applied to loading strip, f, g, h, k – algebraic functions. The above equations are free from material constants and thus in light of the elastic - viscoelastic correspondence principle, stresses for both elastic and viscoelastic bodies remain the same. This is not true for strains which are calculated from stresses using appropriate constitutive equations. Considering asphalt mixture as a continuous elastic body, the change in length of the centrally situated sections of the horizontal (ûU2mR=ûU/2R*m) and vertical (ûV2nR=ûV/2R*n) axes can be expressed as follows: mR

'U 2mR

³

mR

nR

H xx ( x, 0)dx 'V2nR

³

H yy (0, y )dy

[6]

nR

m and n are the parameters between 0 and 1 that define considered sections of specimen axes. If m = n = 1 above equations express the change in length of the entire horizontal and vertical diameters, respectively. When testing 150 mm

418


specimens with gage length (GL) of 38 mm, m and n parameters are equal to 38/150 = 0.25. Considering Hooke’s law for plane stress conditions

H xx

1 V xx QV yy and H yy E

[7]

P I 4 Q I 3 EL

[8]

1 V yy QV xx E

Equation [6] can be re-written in the following form P I1 Q I 2 'V2nR EL


'U 2mR

Ii (i = 1…4) represent dimensionless factors that depend on m, n, and a/2R through Equations [5]:

I1

L P

mR

³

V xx ( x, 0)dx

L P

I2

mR

mR

³

V yy ( x, 0)dx

[9.1]

mR

and I3

L P

nR

³

V xx (0, y )dy

nR

I4

L P

nR

³ V yy (0, y)dy

[9.2]

nR

By manipulating Equation [8], E and can be calculated as follows

E

Q

P 'U 2mR L

I1 Q I 2

P 'V2nR L

I 4 Q I 3

I 4 'U 2mR I1'V2nR I 3 'U 2mR I 2 'V2nR

[10.1]

[10.2]

Equations [10.1] and [10.2] were used in obsolete ASTM D 4123 standard to determine the resilient modulus and Poisson ratio of asphalt mixtures. Next, ZDN method applies the correspondence principle to Equations [8] and uses representations of elastic modulus E and Poisson ratio in the Laplace domain which are expressed in terms of deviatoric Dd and volumetric Dv parts of creep compliance function. As the results, fictitious elastic solutions in Laplace s-domain as can be presented as 'U 2mR ( s )

K1

P( s) P(s) sD d ( s ) K 2 sD v ( s ) L L

[11.1]


'V 2 nR ( s )

K3

P(s) P(s) sD d ( s ) K 4 sDv ( s ) L L

419

[11.2]

Ki (i = 1…4) are defined as dimensionless factors and can be expressed in terms of previously introduced Ii factors. Equation [11] can be now written in time domain using the convolution theorem and inverse Laplace transform: t


'U 2mR (t )

wD (W ) K1 K P (t W ) d dW 2 wW L L

³

³ P(t W )

0

0

wDv (W ) dW wW

[12.1]

K1 K J d (0) P (t ) 2 J v (0) P (t ) L L t

'V2nR (t )

t

wD (W ) K3 K P (t W ) d dW 4 wW L L

t

³

³ P(t W )

0

0

wDv (W ) dW wW

[12.2]

K K 3 J d (0) P (t ) 4 J v (0) P (t ) L L

where P(t) represents arbitrary time-dependent load history. In case of the creep test, P(t) = P0 H(t) and Equation [12] reduces to 'U 2mR (t ) 'V2nR (t )

K1 K P0 Dd (t ) 2 P0 Dv (t ) L L

[13.1]

K3 K P0 Dd (t ) 4 P0 Dv (t ) L L

[13.2]

or in more useful form Dd (t )

L 'U 2mR (t ) K 4 'V2nR (t ) K 2 P0 K1K 4 K 2 K3

Dv (t )

L 'U 2 mR (t ) K3 'V2 nR (t ) K1 P0 K1K 4 K 2 K3

[14.1]

[14.2]

The original ZDN procedure presented in Zhang et al. (1997) ends with Equation [14] that allows to calculate deviatoric Dd and volumetric Dv parts of creep compliance function from the IDT measurements.

420



3.3. Improved ZDN method ZDN method presented in the previous section does not include any corrections coefficients found in the AASHTO procedure. CBx and CBy coefficients for bulging and CSx and CSy coefficients for converting 2D plane stress to 3D solution should potentially improve the ZDN method since this method uses the same displacement measurements as AASHTO procedure and it is also based on the plane stress solution. However, C0x and C0y coefficients for converting average strains to the point strains are not conceptually applicable to the ZDN method. In ZDN method, strains are integrated over a gauge length (GL) (Equation [6]) and converting them to the values in the center of the specimen face would be logically unacceptable. In light of this discussion, the following “improved” ZDN method is proposed for the interpretation of the creep test results: 1) Obtain average ûU/ûV from displacement measurements and calculate Poisson ratio from Equation [4]. Knowing the dimensions of the specimen determine correction coefficients CBx, CBy, CSx and CSy from the equations found in Roque et al. (1992) and Buttlar et al. (1994). 2) Use CSx and CSy to calculate a new Iic factors using the following expressions (compare with Equation [9]) I1c I3c

L P

CSx

CSx I3

mR

³

V xx ( x, 0)dx CSx I1 I 2c

CSy I 2

[15.1]

mR

I 4c

[15.2]

CSy I 4

3) Calculate a set of new Kic factors using improved Iic factors from above step. 4) Apply CBx and CBy to correct changes in horizontal and vertical axis sections as follows (compare with Equation [13]) 'U 2cmR (t )

'V2cnR (t )

CBx 'U 2mR (t )

CBy 'V2nR (t )

K §K · CBx ¨ 1 P0 Dd (t ) 2 P0 Dv (t ) ¸ L © L ¹ K §K · CBy ¨ 3 P0 Dd (t ) 4 P0 Dv (t ) ¸ L © L ¹

[16.1]

[16.2]

5) Calculate deviatoric Dd and volumetric Dv parts of the creep compliance function from Equation [14] using Kic, 'U 2cmR and 'V2cnR . If b = t = 38mm, D = 150mm and 2m = 2n = 2*38/150 = 0.5 then Equation [14] can be presented as: Dd (t )

L > A'U 0.5R (t ) B'V0.5R (t )@ P0

[17.1]


Dv (t )

L >C 'U 0.5R (t ) D'V0.5R (t )@ P0

421

[17.2]

where coefficients A, B, C and D can be easily calculated from Kic, 'U 2cmR and 'V2cnR and Equation [14].

6) If desired, calculate creep compliance D(t) from the following relationship


D(t )

2 1 Dd (t ) Dv (t ) 3 3

[18]

3.4. 3-Point bending In the next sections, the IDT interpretation procedures will be compared against each other as well as the resultant creep compliance functions will be verified with the results from the 3-point bending test. What follows is very brief description of the 3-point bending theory. Differential equation of the beam deflection curve under arbitrary load, known as the Bernoulli-Euler law of elementary bending theory, can be found in any book on elasticity and it has the following form (Gere et al., 1990): dT dx

d 2Q dx

2

M EI

[19]

where is the angle of rotation, is the beam deflection, M is the bending moment. There are several important assumptions for Equation [19]: plane sections remain plane, plane stress mode is valid, deflections and angle are small and material is isotropic and linear. By employing method of successive integrations with appropriate boundary conditions including a single force in the mid-span, one can derive the maximum beam defection /max located under the force as

G max

Pl 3 48 EI

[20]

where P is a concentrated force and l is the beam span. Applying the correspondence principle to Equation [20], the creep compliance function D(t) can be found as D(t )

48 I G (t ) Pl 3

[21]

422


It should be mentioned that Equation [21] does not considers deflections due to shear. It can be easily shown that neglecting shear effect for the beam proportions considered in this study (1 thickness x 2 width x 16 length) leads to the error in deflections below 1% regardless of the Poisson ratio. More discussion on this issue can be found elsewhere (Zofka, 2007).


4. Comparison of IDT interpretation procedures by Finite Element Analysis (FEA) Two experimental setups – the IDT and 3-point bending were modeled in the Finite Element Analysis (FEA) software (Abaqus, 2006). In both cases, 10 s creep test was simulated on the linear viscoelastic material with known creep compliance function D(t)INPUT. Once simulations were completed, simulated specimen displacements were taken from the appropriate model nodes and applied to the interpretation procedures presented in the previous sections to calculate D(t) function. So determined D(t) functions were then compared with the FEA material input function D(t)INPUT and thus the accuracy of each interpretation procedures could be evaluated.

4.1. FEA details The IDT setup was modeled in the FEA software based on the configuration and dimensions found in AASHTO T 313-02. The IDT specimen mesh used in the simulations is presented in Figure 3. Only 1/8 of the full IDT specimen (defined as 38 mm thick and 150 mm diameter) was considered to save computational time and to allow for finer mesh. Appropriate boundary conditions were assumed on the faces of the specimen and the load was applied on the top of loading strip at each node in the form of a vertical concentrated force. The steel loading strip was modeled as elastic, isotropic and homogenous material with Young modulus equal to 300 GPa and Poisson ratio equal to 0.3. Full contact was assumed between the IDT specimen and loading strip since it has a little influence on the state of stress around the center of the specimen as reported by Roque et al. (1992). The geometry of 3-point bending was modeled following AASHTO T 313-05 which standardizes the Bending Beam Rheometer (BBR). The specimen was modeled as simply supported beam without overhanging parts beyond the supports. Forces were applied at nodes on the top of the beam in the middle of the span. The FEA mesh for the beam specimen is shown in Figure 4. Both specimens presented in Figures 3 and 4 were modeled using 3-element Generalized Maxwell Model (GMM). The material was assumed as linear, isotropic and viscoelastic with creep compliance function D(t)INPUT and constant Poisson ratio equal to 0.3. For the assumed GMM, the FEA requires normalized shear gi and bulk ki relaxation moduli which are presented in Table 1.



423

Figure 3. IDT mesh in the FEA software (24,010 wedge elements C3D15)

Figure 4. Beam mesh in the FEA software (7,200 brick elements C3D20R)

Table 1. GMM inputs for FEA software Normalized shear relaxation modulus, gi [-]

Normalized bulk relaxation modulus, ki [-]

Relaxation time, !i [sec]

0.3542

0.3542

2.8889

0.2114

0.2114

33.0071

0.2417

0.2417

334.0924

In order to determine gi and ki from D(t)INPUT the following procedure was followed: 1) Assume example creep compliance data and fit to it 3-term Prony series to describe D(t)INPUT function. Fitting was done using a two-step optimization

424


procedure: the Levenberg-Marquardt method for non-linear coefficients and the Least-Squares (LS) method for linear coefficients. 2) Interconvert D(t)INPUT function to the extensional relaxation modulus E(t)INPUT in the Laplace domain using procedure outlined by Park et al. (1999). Resultant E(t)INPUT function is expressed as E t

3

Ef ¦ Ei 1 et / Ui i 1

[22.1]


where Ef – equilibrium modulus, 3

Ef

E0 ¦ Ei

[22.2]

i 1

E0 – initial modulus, !i – relaxation time for spring element i, Ei – modulus for spring element i. The graphical representation of the GMM is shown in Figure 5.

Figure 5. Generalized Maxwell Model (GMM) 3) Calculate normalized shear relaxation moduli gi as

gi

Gi G0

i 1, 2,3 Gi

Ei 2 1 Q

[23]

Since Poisson ratio was assumed constant, normalized bulk relaxation moduli gi were equal to corresponding shear relaxation moduli gi as shown in Table 1. More discussion on the FEA simulations can be found in Zofka (2007).


425

4.2. Interpretation procedures for FEA comparison


For the IDT simulations, the following five interpretation procedures were employed to determine D(t): 1. AASHTO (simple mean approach). 2. AASHTO without CBx and CBy. 3. ZDN (without any correction coefficients C). 4. ZDN with CSx and CSy only. 5. ZDN with CSx, CSy, CBx and CBy. Similar to the real IDT experiments, the displacements of gage points were obtained from the FEA output at appropriate nodes. The locations of these virtual gage points corresponded to the locations of steel buttons glued to both faces of the specimen used to mount either LVDTs or the strain gages. The relative node displacements are equal in fact the change in length of the centrally situated sections ûU0.5R and ûV0.5R which were introduced earlier. For the fair comparison, the bulging coefficients CBx and CBy were eliminated from the alternative version of AASHTO procedure. These factors were originally introduced by Roque et al. (1992) to correct measurements from the LVDTs. In the FEA simulations the displacements are taken directly from the nodes and incorporating CBx and CBy can potentially deteriorate the procedure. CCMPL (Equation [3]) for alternative AASHTO was equal to 0.9104 versus 0.8773 value for the full version of AASHTO procedure.

Table 2. Correction coefficients for ZDN method t* [mm]

D [mm]

CSx

CSy

CBx

CBy

38

150

0.973295

-0.978908

0.963675

0.955864

Correction coefficients for different variations of the ZDN procedure were calculated using the methodology outlined earlier in Section 3. Correction coefficients and corresponding A, B, C, and D coefficients for Equation [17] are presented in Table 2 and 3, respectively. For 3-point bending simulations, virtual beam deflections were recorded from the neutral axis in the middle of the span and Bernoulli-Euler beam theory (Equation [21]) was used to determine creep compliance D(t).

426



Table 3. Coefficients for ZDN method (for Equation [17]) Model

A

B

C

D

ZDN original,

1.664076

-1.52897

-5.65888

-3.68501

ZDN with CSx CSy

1.702275

-1.56409

-5.76878

-3.75112

ZDN with CBx CBy CSx CSy

1.640441

-1.49506

-5.55923

-3.58557

Figure 6. Comparison of interpretation procedures based on FEA simulations 4.3. Results of FEA comparison The FEA comparison is presented in Figure 6 in terms of the ratios between calculated D(t) by five IDT interpretations procedures and 3-point bending versus D(t)INPUT function which is a “true” material response in this case. The closer a given ratio to 1, the better the data interpretation procedure. Several observations can be made based on these results: – All IDT interpretation procedures are within 5% error range from the D(t)INPUT. – Original ZDN procedure (without any coefficient C) produces good results within 3% from the D(t)INPUT. – Adding correction factors CSx CSy to the ZDN procedure significantly improve accuracy of this method and make it comparable with AASHTO.


427


– Alternative AASHTO without CBx and CBy and ZDN with only CSx and CSy produce excellent results (less than 1% apart from the D(t)INPUT). It is believed that at least part of the error for both methods is due to fitting process as well as other numerical approximations in calculations. – As expected, AASHTO and ZDN procedures that included CBx and CBy factors give less accurate predictions due to misusage of bulging coefficients for the FEA results. – Bernoulli-Euler beam theory produces very good creep compliance D(t) and it is believed that the error is due to shear effect as explained earlier. 5. Comparison of IDT interpretation procedures – laboratory experiment on reference materials The most accurate interpretation procedures from the FEA comparison were applied next to the laboratory results on two reference materials: High-Density Polyethylene (HDPE) and Ultra-High Molecular Weight polyethylene (UHMW). The idea to use plastic reference materials that exhibit viscoelastic behavior is not new in the asphalt-related testing. At least two studies – Christensen et al. (1998) (IDT) and Buttlar et al. (2004) (Hollow Cylinder Tensile, HCT, test) incorporated specimens prepared from various reference materials. Christensen et al. (1998) used HDPE specimens to develop calibration standard procedure for the IDT setup whereas Buttlar et al. (2004) used Delrin plastic to evaluate a new method of interpreting HCT experimental results. Using specimens prepared from the reference material allows to focus on evaluating testing protocol, testing equipment and, like in this study, on the data interpretation procedure.

5.1. Details on laboratory testing Two different plastic materials were used in this study as reference materials: High-Density Polyethylene (HDPE) and Ultra-High Molecular Weight polyethylene (UHMW). Table 4 presents general physical and mechanical properties from the manufacturer product specifications. Generally speaking, UHMW material is slightly lighter and softer than HDPE which suggest it will have higher compliance in the creep test. Both reference materials were received as blocks: 38x305x305 mm for UHMW and 36x305x305mm for HDPE. Each block was cut into four IDT specimens of 142mm in diameter. Two IDT specimens from each material were used for the IDT testing and after IDT testing was finalized, each IDT specimen was further cut into beams for 3-point bending test. All testing was done in creep mode at room temperature for 240 s. The design of the experiment comprised three factors: specimen, load level and direction of load application which are explained next separately for the IDT and 3-point bending testing.

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Table 4. General properties of HDPE and UHMW Property

HDPE

UHMW

Density [g/cm3]

0.95

0.93

Tensile Strength [MPa]

32.0

21.4

Flexural Modulus [MPa]

1380.0

860.0

Hardness, Shore D

D69

D62-D66

IDT testing was performed according to AASHTO T 322-03 with some modifications regarding the experimental procedure. The creep compliance D(t) was calculated using AASHTO simple mean approach and ZDN method with CSx, CSy, CBx and CBy coefficients. Two specimens from each reference material were used. After cutting from the block each IDT specimen was marked and two perpendicular directions (N and W) were set. The testing was performed at three different load levels in each direction with three repetitions for each load-direction combination. Sufficient time period was left between each test to allow the specimens to ‘recover’ and reasonable small load levels were used to keep the material within linear range. This resulted in a total of 72 tests: 2 materials * 2 specimens * 2 directions * 3 loads * 3 repetitions. The details on factor levels are presented in Table 5. Figure 7a presents the samples before the IDT testing and Figure 7b shows the sample in the IDT setup with strain gages attached.

Table 5. Factors in IDT testing on reference materials Factors

No. of Levels

specimen

2 (#1 and #2)

load

3 (High Medium Low)

direction

2 (North West)

The experiment design allowed for comprehensive evaluation of the IDT interpretation procedures considering material-related factors like homogeneity, isotropy, and linearity. Both materials could be evaluated simultaneously and in this case more complex statistical designs such as Randomized Complete Block Design or Split-Plot design would have to be used. If each material was analyzed separately then Completely Randomized Design could be used with partially nested factorial


429


treatment structure and mixed effects. Full details on the experimental design, testing and Analysis of Variance (ANOVA) results can be found in Zofka (2007). In this paper, two-sample t-test is used to compare two IDT interpretation procedures and some observations are supported by the ANOVA results.

(a)

(b)

Figure 7. Samples before the testing (a) and sample in the IDT setup (b)

3-point bending creep test was performed at room temperature according to AASHTO T 313-05 (BBR). After testing each IDT specimen was cut into two round slices 12 mm thick each – one slice per each load direction (North and West). Then, at least 6 beams were cut from each slice along the load directions from the IDT testing. Therefore a total of 12 beams were cut from one IDT specimen: with 6 beams cut in N direction and another 6 beams cut in W direction. The final beam dimensions were 127x5x12mm (LxHxD). Two load levels were used during testing and there were three replicates for each specimen-direction-load combination which resulted in 48 tests in total: 2 materials * 2 specimens * 2 directions * 2 loads * 3 replicates. The factor details are shown in Table 6. Figure 8a illustrates round slice from the IDT specimens with cut beams and Figure 8b shows a beam during the testing.

Table 6. Factors in 3-point bending tests on reference materials Factors

No. of Levels

specimen

2 (#1 and #2)

load

2 (High Low)

direction

2 (North West)


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(a) (b) Figure 8. Round slice from IDT specimen with cut beams (a) and sample in 3-point bending (b)

5.2. Interpretation procedures for laboratory comparison Creep compliance function D(t) was calculated from IDT displacement measurements using two procedures: 1) Full AASHTO procedure (simple mean approach), and 2) ZDN method with coefficients CSx, CSy, CBx and CBy. Based on the FEA comparison it was determined that this variation of ZDN procedure should yield the most accurate results. Conceptually, coefficients CBx and CBy should be added to the ZDN method since in this comparison real measurements are considered and they require correction for specimen bulging effect.

Table 7. Coefficients for ZDN procedure with CSx, CSy, CBx and CBy for plastic reference materials Coefficient*

HDPE (n=36)

UHMW (n=36)

A

1.5278** 0.2603***

1.4997 0.7770

B

-1.3687 0.3417

-1.3406 1.0201

C

-5.1693 0.2407

-5.1112 0.6552

D

-3.2784 0.3136

-3.2443 0.8468

* For Equation [17] ** Mean value *** Coefficient of Variation, CV, in %


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Correction coefficients CSx, CSy, CBx and CBy for ZDN method were determined separately for each IDT test using the methodology outlined in Section 3. Average value of ûU/ûV was obtained from the displacement measurement at 15, 60 and 120 s. These values were chosen somehow arbitrary but they cover the first and middle part of 240 s creep test. Such approach yielded a unique set of parameters A, B, C and D (see Equation [17]) for each IDT test, i.e. 36 sets of coefficients were derived for the HDPE and 36 sets were obtained for the UHMW. The summary of all coefficients is presented in Table 7 in terms of their mean values and Coefficients of Variation (CV). Observation of the CV values shown in Table 7 implies that coefficients for the UHMW material have wider spread than coefficients for the HDPE. In fact closer look at the distributions of coefficients revealed that direction of testing (North or West) played important role for the UHMW material. Each direction produced a distinct set of values. Figure 9 presents cumulative distribution functions (CDF) for A coefficient for UHMW and HDPE materials together with Weibull distribution fits to each CDF. It can be observed that indeed for North and West directions for the UHMW, A coefficient clearly follow unique pattern which resulted in higher CV in Table 7. This observation can be also a first signal of material anisotropy which was definitely confirmed by the ANOVA results (Zofka, 2007). 3-point bending tests were conducted in the BBR equipment which record the mid-span beam defections during the creep test. The deflections were processed using Bernoulli-Euler beam theory (Equation [21]) and creep compliance D(t) function was found for each factor-level combination.

Figure 9. CDF with Weibull fit for coefficient A for UHMW and HDPE

432


5.3. Results of laboratory comparison The following sections present the comparison between D(t) functions determined from the laboratory experiments. First, the visual comparison is made on the entire D(t) curves for both IDT and 3-point bending tests. Next, AASHTO and ZDN methods are compared in terms of D(t) values obtained at specific time points (24, 60, 120 and 240 s) Finally, two-sample t-test evaluates statistical difference between AASHTO and ZDN procedures at the same time points.


5.3.1. Visual comparison Figure 10 shows the direct comparison of example D(t) curves separately for HDPE and UHMW materials. Each plot shows three D(t) curves: one for the IDT interpreted with AASHTO method, one for IDT interpreted with ZDN procedure (with CSx, CSy, CBx and CBy coefficients) and one D(t) curve from 3-pont bending (BBR). For both materials, the IDT curves represent averages from all load levels in West direction from one specimen which totals to nine tests per material. The bars represent +/- one standard deviation. In case of 3-point bending, the curve represents the average of 6 tests since beams were tested only at two load levels. Figure 10 presents an excellent match of all three D(t) curves for considered factor-level combination. It should be mentioned here that data from all tests was intentionally not presented in the form of the average D(t) curves. As explained briefly in the next section, ANOVA results indicted isotropy of both materials and pooling all results together in the form of single curve could lead to false conclusions on the comparison of IDT interpretation procedures.

Figure 10. Creep compliance D(t) curves


433

Although not shown in this paper, ZDN method with coefficients CSx, CSy, CBx and CBy coefficients performed better, i.e. produced D(t) values closer to AASHTO than original ZDN procedure and ZDN with only CSx and CSy.


Figure 11 shows the direct comparison of two IDT interpretation procedures for four arbitrarily chosen time points: 24, 60, 120 and 240 s. Each point in this figure corresponds to exactly the same factor-level-time combination for both interpretation procedures. UHMW material produces slightly better coefficient of determination R2 while HDPE shows more variability with still very high correlation between both interpretation procedures.

Figure 11. Direct comparison of AASHTO and ZDN at four time points

Figure 12. Dd(t) and Dv(t) for HDPE and UHMW from single tests

434


One of the advantages of the ZDN procedure is to provide the separate equations for deviatoric Dd(t) and volumetric Dv(t) parts of the creep compliance function (see Equation [17]). Figure 12 show the examples of Dd(t) and Dv(t) curves for both reference materials. As expected, volumetric parts of creep compliance are significantly smaller than corresponding isotropic parts and both materials creep primarily due to shear.


5.3.2. Statistical comparison Two-sample t-test was use to evaluate statistical difference between AASHTO and ZDN procedures applied to the IDT measurements at the four time points: 24, 60, 120 and 240 s. This statistical test was chosen due to its robustness against nonnormality of the data especially for larger sample sizes. The results of t-test in terms of p-values are shown in Table 8. If p-values are higher than the significance level (for example 0.05), this implies the null hypothesis that the average difference between two procedures is zero, cannot be rejected. Following this reasoning, Table 8 suggests that AASHTO and ZDN procedures yield different D values at 120 s and 240 s for HDPE material. However closer look at the individual values in Figure 13 shows that the difference between AASHTO and ZDN is coming primarily from testing the second specimen. Thus the difference is rather due to the anisotropy and inhomogeneity of the material than to interpretation method and similar conclusions can be drawn from the ANOVA results (Zofka, 2007). For practical purposes it can be assumed that both interpretation methods produce very close results.

a)

b)

Figure 13. Individual data points from AASHTO and ZDN for HDPE at a) 120sec, b) 240 sec


435

Table 8. p-values for two-sample t-test p-value


D time [s] HDPE

UHMW

24

0.449

0.671

60

0.355

0.620

120

0.013

0.726

240

0.005

0.914

6. Example application of improved ZDN procedure to asphalt mixtures Specimen dimensions and Poisson ratio value change with each asphalt mixture specimen. Thus there is no single set of ZDN equation coefficients that can be universally applied to all IDT results. To present possible ranges and distributions of these coefficients for asphalt mixtures, the results from 178 IDT tests were analyzed. The IDT experiments were conducted on temperatures ranges from -0°C to -42°C. Average values of ûU/ûV for each test were obtained from the displacement measurements at 16, 60, 120, 240, and 500 s which resulted in a unique set of equation coefficients A, B, C and D for each IDT test. During calculations, there was an upper limit for Poisson ratio values set to 0.5. More details on the mixtures and laboratory campaign can be found in Marasteanu et al. (2007). Figure 14 presents the CDFs curves together with Weibull fits for each ZDN coefficient. The sharp drop at lower end of each distribution is the result of setting the top limit for Poisson ratio. Example plots of volumetric Dv(t) and deviatoric Dd(t) parts of the creep compliance function are shown also in Figure 15. Similar to plastic references materials, volumetric part of creep compliance is significantly smaller than corresponding isotropic part which suggests that asphalt mixture creeps primarily due to shear.


436


Figure 14. CDF curves with Weibull fits for ZDN coefficients (absolute values) for asphalt mixtures

Figure 15. Example of Dd(t) and Dv(t) curves for asphalt mixture

7. Summary This paper presents numerical and experimental comparison of two interpretation procedures for the IDT creep test. The current AASHTO method is compared with the Zhang-Drescher-Newcomb (ZDN) method. ZDN method has been successfully improved with two sets of correction coefficients to account for 3D stress solution and bulging of the specimen during the test. The results and analysis presented in this paper can be summarized as follows: 1) ZDN interpretation procedure and current AASHTO standard represent comparable methods to determine creep compliance function D(t).



437

2) FEA isothermal viscoelastic simulations showed the following: a. ZDN procedure even without any correction coefficients produces decent D(t) function within 3% from the true D(t) function. b. ZDN procedure can be improved by adding correction coefficients. For the FEA, CSx and CSy coefficients improved this procedure and placed it less than 1% from the true D(t) function. c. Conceptually inappropriate coefficients worsen the results from both interpretation procedures - AASHTO and ZDN. d. 3-point bending test produces very accurate creep compliance function D(t) and it is believed that 1% error is due to non-accounted shear effect. 3) Experimental testing on two reference materials lead to the following conclusions: a. Average D(t) functions determined from the IDT and the 3-point bending testing are practically the same. b. ZDN procedure improved with CSx, CSy, CBx and CBy coefficients produces very similar individual D(t) values as compared to AASHTO standard. Some statistical differences were observed at later times but they were contributed to the anisotropy and inhomogeneity of the reference material rather than to interpretation procedures. 4) Reference materials, such as HDPE and UHMW, provide very convenient and practical ways to evaluate testing protocol, testing equipment and, like in this study, the data interpretation procedures. However, the experimental design must be prepared very carefully since even apparent homogenous material can exhibit anisotropic behavior. 5) Main advantages of the ZDN procedure is to provide the separate equations for deviatoric Dd(t) and volumetric Dv(t) parts of the creep compliance function which could be useful for researching and modeling of asphalt mixtures. Appropriate step-by-step procedure for the improved ZDN procedure is proposed in Section 3.3. and typical ranges of A, B, C, and D coefficients for asphalt mixtures are presented in Section 6 (determined from 178 IDT experimental results) 8. Bibliography Abaqus online documentation, ver. 6.6, 2006 American Association of State Highway and Transportation Officials (AASHTO) Standard T 322-03, “Determining the Creep Compliance and Strength of Hot-Mix Asphalt (HMA) Using the Indirect Tensile Test Device”, Standard Specifications for Transportation Materials and Methods of Sampling and Testing, 25th Edition, 2005. American Association of State Highway and Transportation Officials (AASHTO) Standard T 313-05, “Standard method of test for determining the flexural creep stiffness of asphalt

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