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

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N-1 APPENDIX N. LIST OF ALL VARIABLES The Appendix N lists all variables included in this report with the equation number where each of them is used. Name Description Equation Number ( )E z dynamic modulus in pavement depth z (4.1) dE dynamic moduli at the bottom (4.1) 0E dynamic moduli at the top (4.1) d thickness of the field core specimen (4.1) z pavement depth (4.1) n model parameter (4.1) k ratio of the modulus at the top to the modulus at the base-line 1.5 inches (38 mm) below the surface (4.2) ( )E  dynamic modulus (4.3) eE equilibrium modulus (4.3) gE glassy modulus (4.3) cE crossover frequency (4.3)  frequency (4.3) Ta time-temperature shift factor (4.3) ER rheological index of the asphalt mixtures (4.3) 1C fitting parameter of the WLF equation (4.4) 2C fitting parameter of the WLF equation (4.4) T test temperature in Kelvin (4.4) RT reference temperature in Kelvin (4.4) gxE glassy modulus at aging time of x months (4.5) 0gE glassy modulus at the reference aging time (4.5) xa aging time of interest aging time (4.5) 0a aging time of reference aging time (4.5) En model parameter (4.5) ExR rheological index at x months aging time (4.6) 0ER rheological index at the reference aging time (4.6) Rn model parameter (4.6) Aa horizontal aging shift factor (4.7) a aging time (4.7) T aging temperature in Kelvin (4.7) b Horizontal aging shift factor parameter (4.7)

N-2 0a initial aging time or unaged time (4.7) A acceleration factor to account for the laboratory accelerated aging process (4.7) aE activation energy (4.7) R universal gas constant (4.7) 0T initial average monthly temperature in that field location or the reference aging temperature in the laboratory (4.7) T average monthly temperature in a field location or the laboratory aging temperature for an aging period (4.7) 'f reduced frequency (4.8) f previous frequency after the time-temperature shift (4.8) da depth shift factor (4.9) db fitting parameter for the depth shift function (4.9) 0d thickness of the layer down to the baseline depth (4.9) d depth of interest (4.9) * b E base-line modulus (4.10) * bi E initial base-line modulus (4.10) * 0b E intercept of the constant-rate line of base-line modulus (4.10) fbk fast-rate reaction constants for the base-line modulus (4.10) t aging time in days (4.10) cbk constant-rate reaction constants for the base-line modulus (4.10) fbA fast-rate pre-exponential factors for the base-line modulus (4.11) afbE fast-rate aging activation energies for the base-line modulus (4.11) fieldT field aging absolute temperature (4.11) cbA constant-rate pre-exponential factors for the base- line modulus (4.12) acbE constant-rate aging activation energies for the base- line modulus (4.12) * s E surface modulus (4.13) * si E initial surface modulus (4.13) * 0s E intercept of the constant-rate line of surface modulus (4.13) fsk fast-rate reaction constants for the surface modulus (4.13) csk constant-rate reaction constants for the surface modulus (4.13)

N-3 fsA fast-rate pre-exponential factors for the surface modulus (4.14) afsE fast-rate aging activation energies for the surface modulus (4.14) csA constant-rate pre-exponential factors for the surface modulus (4.15) acsE constant-rate aging activation energies for the surface modulus (4.15) n aging exponent (4.16) in initial aging exponent (4.16) 0n intercept of the constant-rate line of the aging exponent (4.16) fnk fast-rate reaction constants for aging exponent (4.16) cnk constant-rate reaction constants for aging exponent (4.16) fnA fast-rate pre-exponential factors for aging exponent (4.17) afnE fast-rate aging activation energies for aging exponent (4.17) cnA constant-rate pre-exponential factors for aging exponent (4.18) acnE constant-rate aging activation energies for aging exponent (4.18)  bitumen viscosity of field cores (4.19) rA rheological pre-exponential factor (4.19) arE rheological activation energy of an asphalt binder (4.19) labT absolute temperature at which field cores are tested in the laboratory (4.19) E asphalt mixture modulus (4.20) pT pavement temperature (°F) (4.20) C volumetric heat capacity of the pavement (4.21) sT pavement surface temperature (4.21) sQ heat flux due to solar radiation (4.21)  albedo of the pavement surface (4.21) aQ downwelling long-wave radiation heat flux from the atmosphere (4.21) rQ outgoing long-wave radiation heat flux from the pavement surface (4.21) cQ convective heat flux between the surface and the air (4.21) fQ conduction from the surface into the pavement (4.21) c crack length (4.22) N number of load repetitions (4.22)

N-4 A fracture coefficient (4.22) K range of stress intensity factor (4.22) J J-integral range (4.22) n fracture coefficients (4.22) reE representative elastic modulus (4.23) f frequency of a load pulse (4.23) t loading time (4.23) pt pulse time of a load (4.23) *E dynamic modulus (4.23)  damage density (4.24) A modified Paris’ law parameter associated with the evolution of the damage density (4.24) RJ pseudo J-integral (4.24) n modified Paris’ law parameter associated with the evolution of the damage density (4.24) All parameters are defined in (4.22) and (4.24) (4.25) DPW dissipated pseudo work (4.26) 1t start time of a loading period (4.27) 2t end time of a loading period (4.27) V volume of the material (4.27) R pseudo strain (4.27)  stress applied to the material; (4.27) S cross-sectional area of the material (4.28) w thickness of the material (4.28)  Poisson’s ratio (4.29) RE representative elastic modulus (4.29) IK Mode I (opening) stress intensity factor (4.29) IIK Mode II (in-plane shear) stress intensity factor (4.29) IIIK Mode III (out of plane shear) stress intensity factor (4.29) a regression coefficient of the curve of the DPW versus N (4.30) b regression coefficient of the curve of the DPW versus N (4.30) 0 initial damage density (4.31) 0c initial crack size (4.32) All parameters are defined (4.33)  E t relaxation modulus of asphalt mixtures (4.34) 1E relaxation modulus parameter (4.34)

N-5 m relaxation modulus parameter (4.34) ( )f x cumulative percent passing size x (4.35)  gradation aggregate scale parameter (4.35) x sieve size (4.35)  gradation aggregate shape parameter (4.35) AV air void content (4.36) AB asphalt binder content by weight of mixture (4.36)  damage density of the cross-sectional area (4.37) RJ pseudo J-integral (4.37) N number of loading cycles for a load level (4.37) RW load induced pseudo displacement work (4.38) CSA cracked surface area (4.38) RiW loading cycle N of the load level i (4.39) ia energy parameter of the pseudo displacement work (4.39) ib energy parameter of the pseudo displacement work (4.39) 0a energy parameter of a 3000 lb of single axle (4.40) iP load level i (4.40) 0P minimum load level defined in the load spectra, 3000 lb (4.40) 0A critical cross-sectional area (4.41) All parameters are defined (4.42) iN loading cycles for crack initiation of load level i (4.43) c critical damage density corresponding to the critical cracked percentage at which multiple distributed cracks coalesce into a single crack and the single crack then propagates (4.43) 0 initial damage density equal to the percentage of the air voids at the pavement surface (4.43) mV volume of asphalt mastic (4.44) bV volume of asphalt binder (4.44) 200V volume of aggregate passing # 200 sieve (4.44) a mean air void content (4.44)  a z air void distribution at pavement depth z (4.45) mina minimum air void content (4.45) maxa maximum air void content (4.45) h thickness of the asphalt layer (4.45) All parameters are defined (4.46) All parameters are defined (4.47)

N-6 ijn annual number of axle loads at load level i of axle category j (4.48) ijN number of load at load level i of axle type j (4.48) 0t TDC initiation time (year) (4.48)  ic P cumulative axle load distribution factor at the load level iP (4.49)  scale parameter of the cumulative distribution curve (4.49)  shape parameter of the cumulative distribution curve (4.49) ijP load level i at axle category j (4.49) ( )l t longitudinal crack length (4.50) 0l maximum longitudinal crack length (4.50)  scale parameter of the distress curve (4.50) t pavement service time (days) (4.50) 0t crack initiation time (days) (4.50)  shape parameter of the distress curve (4.50) The whole equation was defined before (4.51)  condition number (4.52) max largest eigenvalue (4.52) min smallest eigenvalue (4.52) aH total thickness of the asphalt layer (inch) (4.53) 0a released energy at smallest load level (4.53) 0A cross-sectional area (4.53) A calibration coefficient (4.54) B calibration coefficient (4.54) C calibration coefficient (4.54) D calibration coefficient (4.54) HT annual number of days above 32°C (4.54) LT annual number of days above 0°C (4.54) AADTT annual average daily truck traffic (4.54) CP maximum contact stress in kPa (4.55) 1k regression coefficient (4.55) 2k regression coefficient (4.55) 3k regression coefficient (4.55) IP inflation pressure in kPa (4.55) LP tire load in kN (4.55) l tire length (4.56) w tire width (4.56)

N-7 iP input variable (4.57) J output of the J-integral (4.57) 0A activation functions for the output layer (4.57) hjA activation functions for the first hidden layer (4.57) hkA activation functions for second hidden layer (4.57) i subscript for the input layer (4.57) j subscript for the input layer (4.57) k subscript for the input layer (4.57) m number of inputs (4.57) n number of nodes in the first hidden layer (4.57) q number of nodes in the second hidden layer (4.57) 0 kW weight factors for the output layer (4.57) 2h jkW weight factors for the second hidden layer (4.57) 1h ijW weight factors for the first hidden layer (4.57) 1h jb bias factors for the first hidden layer (4.57) 2h kb bias factors for the second hidden layer (4.57) 0b bias factors for the output layer (4.57) (4.61) was defined in (4.21) pC volumetric heat capacity of the pavement (4.61) / 2x differential pavement thickness for the energy balance (4.61) sT pavement surface temperature (4.61) sQ heat flux due to solar radiation (4.61)  albedo of pavement surface, the fraction of reflected solar radiation (4.61) aQ down-welling long-wave radiation heat flux from the atmosphere (4.61) rQ outgoing long-wave radiation heat flux from the pavement surface (4.61) cQ convective heat flux between the surface and the air (4.61) fQ heat flux within the pavement at the pavement surface (4.61) ( )rE t dynamic modulus (4.62) 1c minimum value of the logarithm of the dynamic modulus (4.62) 2c difference between maximum and minimum values of the logarithm of the dynamic modulus (you defined ‘c1+c2’) (4.62)

N-8 3c parameter describing the shape of the sigmoidal function (4.62) 4c parameter describing the shape of the sigmoidal function (4.62) rt reduced time (4.62) iE Prony series parameter for the relaxation modulus curve (4.63) i Prony series parameter for the relaxation modulus curve (4.63) *E dynamic modulus at the angular frequency ω (4.64)  frequency (4.64)  thermal stress (4.65)  reduced time (4.65) ' reduced time (variable of integration) (4.65)  change in strain (4.66)  change in reduced time (4.66) c change in crack depth due a cooling cycle (4.67) A fracture parameter for the asphalt mixture (4.67) n fracture parameter for the asphalt mixture (4.67) RJ pseudo J-integral (4.67) reE reference modulus (representative elastic modulus) (4.68) *E dynamic modulus (4.68) f frequency (4.68) t time (4.68) pt pulse time (4.68) iE initial modulus (4.69) oE intercept of the constant-rate line of modulus (4.69) fk fast-rate reaction constant for modulus (4.69) ck constant-rate reaction constant for modulus (4.69) afE fast-rate aging activation energy for modulus (4.70) fk fast-rate reaction constant for modulus (4.70) fA fast-rate pre-exponential factor for modulus (4.70) aT aging absolute temperature, K (4.70) ck constant-rate reaction constant for modulus (4.71) cA constant-rate pre-exponential factor for modulus (4.71) acE constant-rate aging activation energy for modulus (4.71) R universal gas constant (4.71)

N-9 i distress scale parameter (4.72) TRiN numbers of days for traffic load to reach the standard depth (4.72) THiN numbers of days for thermal load to reach the standard depth (4.72) 0 calibration coefficient (4.72) 1 calibration coefficient (4.72) i distress shape parameter (4.73) 0 calibration coefficient (4.73) 1 calibration coefficient (4.73) 1 calibration coefficient (4.74) 2 calibration coefficient (4.74) 1 calibration coefficient (4.75) 2 calibration coefficient (4.75)