A Mechanistic–Empirical Model for Top–Down Cracking of Asphalt Pavements Layers (2018)

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B-1 APPENDIX B. INVERSE APPROACH TO DERIVE COMPLEX MODULUS GRADIENT OF ASPHALT FIELD CORES The objective of this appendix is to determine the complex modulus based on the unique characteristics of field cores. More specifically, there are two main subjects that are addressed: 1) How to determine the parameters of the modulus gradient of a field core specimen; and 2) How to convert the measured data and parameters of modulus gradient to its corresponding viscoelastic property: complex modulus. Inverse Application of Viscoelastic-elastic Correspondence Principle The stress, strain, and modulus of an asphalt field core are nonuniformly distributed. This adds significant difficulty in the viscoelastic analysis of field core specimens. The solution to this problem is to introduce the viscoelastic-elastic correspondence principle (2), so a viscoelastic problem can be inferred from a reference elastic problem. For an undamaged viscoelastic material, there is a linear relationship between the stress and the pseudo strain:   ( )RRt E t  (B-1) where σ(t) is the stress in the undamaged viscoelastic material, or called viscoelastic stress; εR(t) is the pseudo strain; and ER is the reference modulus, which can be assigned as the Young’s modulus (3). The pseudo strain is defined as 0 1 ( )( ) ( ) t R R dt E t d E d        (B-2) where E(t) is the relaxation modulus of the material; ( )  is the strain history;  is a time between 0 and t; t is the loading time. The relaxation modulus for a short loading time like the one in the direct tension test above can be defined by (4): 1( ) t E t E E e     (B-3) where E is the long term relaxation modulus; 1E is the relaxation modulus coefficient; and  is the relaxation time. Once the dynamic modulus of a viscoelastic material is known from the measured load and strain, the relaxation modulus can be calculated from the dynamic modulus-relaxation modulus relationship (4). Then the reference modulus and pseudo strain can be obtained from Eqs. (B-1) to (B-3). The reason why the pseudo strain needs to be determined and used other than the measured strain is that it is not appropriate to use the measured strain which is the viscoelastic strain in the elastic theory (i.e., bending theory) to solve for the modulus gradient parameters n and k. In this study, due to the complexities of stress and strain in the field core specimen, an inverse analysis with an iteration process is proposed to determine the pseudo strain and the gradient parameters. More specifically, it contains the following steps: I. In the first iteration: 1) Use the measured tensile strain of an undamaged field core specimen as the seed value for the pseudo strain. In other words, temporarily, there is an elastic relationship between the measured stress and the measured strain;

B-2 2) Utilize the elastic theory along with the measured load/strains to determine the modulus gradient parameters. The values of n and k are determined and checked for their dependence on loading time and frequency; 3) Convert the functions of the measured load/strains and modulus gradient parameters using the Laplace transform to calculate the corresponding viscoelastic property: complex modulus; 4) Calculate the relaxation modulus then the reference modulus using the calculated complex modulus; and 5) Calculate the pseudo strain using the reference modulus. II. In the second iteration: replace the measured strain with the calculated pseudo strain as the seed value after the first iteration and repeat steps 2 to 5. III. In the following iterations (normally 3 to 5): 1) Replace the pseudo strain in the previous iteration with the newest one and repeat steps 2 to 5; and 2) Stop the iteration when the pseudo strain is stable. Then the modulus gradient parameters converge, the complex modulus and pseudo strain will not change. Determination of Modulus Gradient Parameters Using the pseudo strain to determine the modulus gradient parameters contains three steps discussed below. Step 1. Decomposition of vertical strains in field core specimens As indicated above, there is an eccentricity between the location of the load and neutral axis in the field core specimens. As a result, the vertical pseudo strains at top, center and bottom of the specimen can be decomposed into the tensile portions and bending portions as follows: 0 0 0 0 0 t b aP M Z AE IE       (B-4) ( ) 2 c ct cb c c dM ZbP AE IE         (B-5) ( ) d dt db d d cP M d Z AE IE        (B-6) where 0 , c , d are the vertical pseudo strains at top, center and bottom of the specimen, respectively; 0t , ct and dt are the tensile portions of the vertical pseudo strains at top, center and bottom, respectively; 0b , cb and db are the bending portions of the vertical pseudo strains at top, center and bottom, respectively; P is the magnitude of the load; and a, b, and c are the coefficients to account for the nonuniform distribution of the stress in the field core specimen; A is the loading area; M is the induced moment [ ( )] 2 dM P z  ; Z is the distance from the neutral axis to the top, 2 d Z and d Z are the distances from the neutral axis to the center and from the neutral axis to the bottom, respectively; 0E , cE and dE are the modulus at top, center and bottom, respectively; and I is the moment of inertia. Note that the bending strain

B-3 at the top is negative (compressive), so it is subtracted from the strain at the top as in Eq. (B-4). At the other two locations, the bending strains are positive (tensile). Step 2. Formulation of value and location of the load in field core specimens The linear distribution of the tensile portion of the pseudo strain is:   00 dt tt tz zd      (B-7) The modulus has a distribution defined in Eq. (A-1). Then the magnitude of the load is calculated by the integral of the tensile stress as follows:     00 1 1 1 1{ ( ) [ ]} 2 2 2 ( 1)( 2) z d t t dt dz k kP m z E z dz A E n n n                 (B-8) where m and d are the width and thickness of the field core specimen, respectively; and A is the cross-sectional area (A = md). The location of the neutral axis relative to the top of the specimen is determined by Eq. (B-9):     0 0 0 1 1 ( 1)( 5) 1 2( 1){ [ ]} 6 ( 1)( 2)( 3) 3 ( 1)( 2)( 3) 1 1 1 1[ ( )] { [ ] 2 2 2 ( 1)( 2) z d t z t dt t dt Z mz z E z dz P k n kd n n n n n n k k n n n                              (B-9) For the case of the LMLC specimen, the pseudo strains are the same at different locations and k equals to 1. Thus Z reduces in 2 d in Eq. (B-9), which is the centerline of the specimen. However, for a field specimen, Z is always smaller than 2 d given that k is larger than 1. Step 3. Solve for n and k in the modulus gradient model Select the values of the pseudo strain and load at different loading times (in this case from 5s to 35s) of the direct tension test, which are given in Table B-1. For every second of the loading time, substitute the measured values into Eq. (B-4) to (B-9) and solve for a, b, c, n, and k. The calculated results of a field core specimen are also given in Table B-1. For the first iteration, the values of the pseudo strains are equal to the tensile strains measured from the direct tension test, which are used as the seed values. In the following iterations, these strains are the pseudo strains calculated from the previous iteration after determining the complex modulus and relaxation modulus detailed in the following subsection. The changes of the values of a, b, c, n, and k become small from the second iteration to the third one. Therefore, it is regarded that the results converge at the third iteration. The details regarding the determinations of iterations and pseudo strains will be discussed in the next section. It should be noted that the exponent n and the ratio k are material properties since they are the two parameters in the modulus gradient equation and must be included in the application of the correspondence principle which transforms an elastic equation into the Laplace transform of a viscoelastic equation. With each iteration, both n and k are determined to be time-dependent, as seen in Table B-1, which shows their final converged values. They both increase slightly with loading time and decrease slightly with frequency.

B-4 Table B-1. Results of Calculations of Modulus Gradient of a Field Core Specimen (8 Months Aged at 30˚C) from Direct Tension Test Iteration Loading Time (s) Pseudo Strain (με) Load (N) a b c n k n0 k0 1st 5-15 6.12 30.41 1.26 0.73 0.85 2.96 2.44 2.54 2.16 16-25 20.42 165.6 1.28 0.72 0.86 3.73 2.82 26-35 34.71 268.34 1.30 0.72 0.87 3.89 3.09 2nd 5-15 3.67 30.41 1.23 0.70 0.84 2.91 2.40 2.53 2.12 16-25 12.15 165.6 1.26 0.71 0.85 3.68 2.79 26-35 20.3 268.34 1.29 0.71 0.85 3.86 3.06 3rd 5-15 3.53 30.41 1.22 0.68 0.82 2.90 2.39 2.52 2.11 16-25 11.79 165.6 1.25 0.70 0.84 3.67 2.76 26-35 19.15 268.34 1.27 0.70 0.84 3.85 3.05 Determination of Complex Modulus Using Approximated n and k After obtaining the modulus gradient parameters, the next step is to convert the elastic property to the corresponding viscoelastic property using the Laplace transform. The procedure is given below in sequence. In the direct tension test, the measured load and tensile portions of the strains versus time of a field core specimen are modeled as follows:  Monotonic tensile load ( )P t : ( ) (1 )Pb tPP t a e   (B-10)  Tensile portions of the strains at the top and bottom of the field core specimen: 00 0( ) (1 ) b t t t a e   (B-11)    1 db tdt dt a e   (B-12)  Modulus gradient parameters n and k: 0 nb tn n e (B-13) 0 k b tk k e (B-14) where Pa and Pb are the fitting parameters for the load; 0a and 0b are the fitting parameters for the tensile portion of strain at the top; da and db are the fitting parameters for the tensile portion of the strain at the bottom; and 0n , 0k , nb and kb are the fitting parameters for the modulus gradient parameters n and k.

B-5 Using the Laplace transform, the elastic forms in Eqs. (B-10) to (B-14) can be rewritten as viscoelastic solutions in the Laplace domain by an s-multiplied Laplace transform (Carson-transform), which are shown in Eqs. (B-15) to (B-19): ( ) ( ) P P P a bP s s s b   (B-15) 0 00 0 ( ) ( )t a bs s s b    (B-16) ( ) ( ) d d dt d a bs s s b    (B-17) 0( ) n n ssn s s b   (B-18) 0( ) k k ssk s s b   (B-19) where s is the variable in the Laplace domain; ( )P s , 0 ( )t s , ( )dt s , ( )n s , and ( )k s are the corresponding load, strains, n and k in the Laplace domain. The viscoelastic forms of n and k are shown in Eqs. (B-20) and (B-21). For small values of nb and kb , the values of ( )sn s and ( )sk s are closely approximated by the constants 0n and 0k , as shown in Eqs. (B-20) and (B-21). 2 0 0 0 02 2( ) n s i n ns i n s n n bsn s n s b b                  (B-20) 2 0 0 0 02 2( ) k s i k ks i k s k k bsk s k s b b                  (B-21) To obtain the modulus in the Laplace domain, the Laplace transform is taken on both sides of Eq. (B-22) and used to solve for the modulus at the bottom, which gives: 0 0 0 0 0 0 ( )( ) 1 11 1( ) ( ) 2 2 2 ( 1)( 2) d t dt P sE s k ksA s s n n n                        (B-22) where ( )dE s is the bottom modulus in the Laplace domain. The relationship between the complex modulus and relaxation modulus is shown in Eq. (B-23) (5): *( ) { ( )} [ ( )]s i s iE i L E t sE s      (B-23) Therefore, the complex modulus at the bottom of the field core specimen can be obtained by substituting Eq. (B-22) into Eq. (B-23), which is shown by: * 0 0 0 0 0 0 ( ) [ ( )] ( ) 1 11 1( ) ( ) 2 2 2 ( 1)( 2) d d s i t dt s i E sE s P s k kA s s n n n                              (B-24)

B-6 The final expression of the complex modulus at the bottom is shown in Eq. (B-25) by substituting Eqs. (B-15) to (B-17), (B-20) and (B-21) into Eq. (B-25): * 2 2 ( ) ( )( )d AC BD AD BC iE A B      (B-25) in which 20 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 11 1 1 1{ [( ) ( ) ] ( ) ( ) } 2 2 2 ( 1)( 2) 2 2 2 ( 1)( 2)d d p d d d p d k k k kA a b a b a b b b a b b b md n n n n n n                   0 0 0 0 0 0 0 0 1 11 1[( ) ( ) ( ) ( )] 2 2 2 ( 1)( 2)p d d d p k kB a b b b a b b b md n n n           2 0( )p p dC a b b b   0( )p p dD a b b b   When the complex modulus at the bottom is determined, the complex modulus at the top in the Laplace domain can be determined as shown in Eq. (B-26): 0 0( ) ( )dsE s sk E s (B-26) where 0( )E s is the corresponding modulus at the top of a field core in the Laplace domain. The complex modulus at the center of the field core in the Laplace domain is determined by: 0 0 1( ) ( )[1 ] 2c d n ksE s sE s   (B-27) Similarly, the complex modulus at the top and that at the center can be determined as follows: * 0 0 0 2 2 ( ) ( )( ) [ ( )]d s i k AC BD k AD BC iE sk E s A B         (B-28) 0 0 0 0 * 2 2 1 1[1 ]( ) [1 ]( ) 2 2( ) [ ( )] n n c c s i k kAC BD AD BC i E sE s A B             (B-29) The complex modulus includes a real part and an imaginary part, and the dynamic modulus is defined as * '2 ''2( )E E E   (B-30) where 'E is real or storage modulus component; ''E is imaginary or loss modulus component; *( )E  is the magnitude of the complex modulus, or dynamic modulus. The phase angle of the complex modulus is calculated using Eq. (B-31), which is also frequency dependent. '' 'arctanE E E         (B-31) Note that the range of the frequency for the dynamic modulus depends on the duration of the loading time of the direct tension test. A time-frequency relationship is needed to convert the ranges in the time domain to frequency domain. In this study, Eq. (B-32) is used to make the approximate inverse Laplace transform based on the ref. (2): 1 2 ( ) [ ( )] s t f t s f s   (B-32)

B-7 Determination of complex modulus using complex n and k It should be mentioned that the calculations of the complex modulus above are based on the approximated results of the Laplace transform of n and k by Eqs. (B-20) and (B-21). This generates a dynamic modulus gradient, but results in an issue that the phase angles at the top, center, and bottom are the same according to Eqs. (B-25), (B-28), (B-29), and (B-31). As a matter of fact, the phase angle should also have a gradient along the pavement depth. However, the derivations and computations become too complicated when using the accurate results of Laplace transform of n and k. In this study, the approximation method to calculate the complex moduli is adopted. The derivations and expressions of the complex moduli with complex n and k are presented in the Appendix, which also provides the phase angle gradient accurately. Determination of relaxation modulus, reference modulus and modulus gradient After obtaining the initial complex modulus, the corresponding relaxation modulus and reference modulus can be computed. First, the master curve of the dynamic modulus is constructed at a reference temperature of 20°C using the sigmoidal model shown in Eq. (B-33). * log( )log ( ) 1 Ta E e          (B-33) where  is the value of the lower asymptote,  is the difference between the upper and lower asymptotes,  and  are shape coefficients, and Ta is the time-temperature shift factor. The Williams-Landel-Ferry (WLF) equation is employed as the shift factor equation: 1 2 ( )log ( ) r T r C T Ta C T T      (B-34) where T is the test temperature, rT is the reference temperature, 1C and 2C are the positive fitting parameters. Figure B-1 shows the master curve constructed by Eqs. (B-33) and (B-34) for the bottom modulus of a field core specimen. Figure B-1. Dynamic Modulus Master Curve of a Field Core Specimen

B-8 Once the dynamic modulus master curve is determined, the relaxation modulus can also be constructed according to their relationships shown in Eqs. (B-35) and (B-36). When the relaxation modulus is fitted by the Prony series model:   1 e j tM j j E t E E       (B-35) where E is the long term relaxation modulus; jE are the relaxation modulus coefficients; and j are the relaxation times. The dynamic modulus is given by: 2 22 2 * 2 2 2 2 1 1 ( ) 1 1 M M j j j j j jj j E E E E                              (B-36) As a result, the fitting parameters for the relaxation modulus can be computed by Eq. (B- 36) based on the dynamic modulus master curve determined above. The calculation result is given in Figure B-2. Figure B-2. Relaxation Modulus Determined from the Dynamic Modulus Master Curve To faciliate the calcualtion of the pseudo strain, fit the relaxation modulus determimed above by a simpler model like that in Eq. (B-3). Substitute Eq. (B-3) and the strain history formulated by Eq. (B-11) into Eq. (B-2), which gives: 11( ) [( (1 ) ( )]1 d d t b t b td d d R d E at E a e e e E b          (B-37) where ( )d t is the pseudo strain at the bottom of the field core specimen. Since the Young’s modulus of asphalt materials is not easy to determine using (3), the representative elastic modulus formulated by Eq. (B-38) is used to estimate the reference modulus (6). * 1 1 2 2 p p R re f t t E E E E t               (B-38) where Ere is the representative elastic modulus; *E is the dynamic modulus; f is the frequency of a load pulse; and tp is the pulse time of a load. The pulse time of 0.1 s is chosen in

B-9 this report, so using Eq. (B-38), the reference modulus is calculated with the dynamic modulus master curve and relaxation modulus determined previously. The pseudo strains at 30°C at different iterations and the strain measured from the direct tension test are shown in Figure B-3. It can be seen that the pseudo strain is smaller than the measured strain, especially for the longer loading time. This phenomenon matches the understanding that the viscous effect is more active when the temperature is higher, which corresponds to a lower loading frequency or a higher loading time. Once the relationships of the pseudo strains and time are determined, the measured strains used in the first iteration are replaced by the pseudo strains to recalculate the values of n and k using Eqs. (B-4) to (B-9). Then the updated values of n and k are inserted into Eqs. (B-24) to (B-38) to obtain the new dynamic modulus master curve and relaxation modulus again. This procedure is repeated until the convergence requirement of the values of n and k are met. In general, the values of n and k become stable within 5 iterations. For instance, in Figure B-3, the change of the pseudo strain at 30°C is minimal after 3 iterations. Once the convergence is reached, the complex modulus and the modulus gradient parameters are the actual material properties. The three complex moduli are determined with the updated n and k using Eqs. (B-25), (B-28) and (B-29). The modulus gradient is then extracted from the dynamic modulus curves at the three depths and three temperatures for 8 and 22 months aged field core specimens when the loading frequency is 0.1 Hz, which is shown in Figure B-4. Figure B-3. Measured Strain at the Bottom of a Field Core Specimen and Associated Pseudo Strains at Different Iterations

B-10 Figure B-4. Modulus gradient of 8 months and 22 months field core specimens at three temperatures References: 1. Glover C.J., Liu, G., Rose, A.A., Tong, Y., Gu, F., Ling, M., Arambula, E., Estakhri, C., Lytton, R.L. (2014). “Evaluation of binder aging and its influence in aging of hot mix asphalt concrete: Technical Report.” Publication FHWA/TX-14/0-6613-1. Texas A&M Transportation Institute, College Station, TX 2. Schapery R.A. (1984). “Correspondence principle and a generated J integral for large deformation and fracture analysis of viscoelastic media.” International Journal of Fracture, 25(3), pp. 195-223 3. Zhang Y., Luo R., Lytton R.L. (2012). “Characterizing permanent deformation and fracture of asphalt mixtures by using compressive dynamic modulus tests.” Journal of Materials in Civil Engineering, 24(7), pp. 898-906 4. Luo X., Luo R., Lytton R.L. (2015). “Mechanistic modeling of healing in asphalt mixtures using internal stress.” International Journal of Solids and Structure, 60, pp. 35-47 5. Findley, W.N., Lai, J.S., Onaran, K. (1989). “Creep and relaxation of nonlinear viscoelastic materials: with an introduction to linear viscoelasticity.” Dover Publications, New York 6. Luo, X., Y. Zhang, and R.L. Lytton (2016). “Implementation of Pseudo J-Integral Based Paris’ Law for Fatigue Cracking in Asphalt Mixtures and Pavements.” Materials and Structures, Vol. 49, No. 9, pp. 3713-3732. 0 1 2 3 4 5 6 0 1000 2000 3000 4000 5000 6000 De pt h  (c m ) Dynamic Modulus (MPa) 10C (8 Months) 20C (8 Months) 30C (8 Months) 10C (22 Months) 20C (22 Months) 30C (22 Months)