J.6_Small_Cyclic_Fatigue.pdf

DRAFT PROCEDURE

Determining the Damage Characteristic Curve and Failure Criterion Using Small

Specimens in the Asphalt Mixture Performance Tester (AMPT) Cyclic Fatigue Test

1. SCOPE

1.1. This test method covers procedures for preparing and testing asphalt concrete mixture specimens to determine the damage characteristic curve and fatigue analysis parameters via direct tension cyclic fatigue test using the Asphalt Mixture

Performance Tester (AMPT).

1.2. This standard is intended for dense-graded mixtures with nominal maximum size aggregate less than or equal to 25.0 mm (0.98 in.). Mixtures with a nominal maximum aggregate size greater than 25.0 mm (0.98 in.) should be tested following AASHTO TP 107.

1.3. This standard may involve hazardous material, operations, and equipment. This standard does not purport to address all safety problems associated with its use. It is the responsibility of the user of this procedure to establish appropriate safety and health practices and to determine the applicability of regulatory limitations prior to use.

2. REFERENCED DOCUMENTS

2.1. AASHTO Standards:

PP 61, Developing Dynamic Modulus Master Curves for Asphalt Mixtures

Using the Asphalt Mixture Performance Tester (AMPT)

PP XX, Preparation of 38-mm Diameter Cylindrical Performance Test

Specimens Using the Superpave Gyratory Compactor (SGC)

R 30, Mixture Conditioning of Hot Mix Asphalt (HMA)

R 62, Developing Dynamic Modulus Master Curves for Asphalt Mixtures

TP 79, Determining the Dynamic Modulus and Flow Number for Asphalt

Mixtures Using the Asphalt Mixture Performance Tester (AMPT)

TP XX, Determining the Dynamic Modulus for Asphalt Mixtures Using Small

Specimens in the Asphalt Mixture Performance Tester (AMPT)

2.2. ASTM Standards:

E 4, Standard Practices for Force Verification of Testing Machines

2.3. Other Document:

NCHRP Report 629, Equipment Specifications for the Simple Performance

Test System, Appendix E, October 16, 2007.

FHWA Report, Development of Asphalt Mixture Performance Related

Specifications, Final Report of Project DTFH61-08-H-00005, In Press, 2017.

3. TERMINOLOGY

3.1. alpha term (α)—value corresponding to the slope of the relaxation modulus master curve which is used in the accumulation of damage with time.

3.2. average rate of dissipated pseudo strain energy per cycle (GR)—parameter used in the failure criterion for fatigue performance which is generated from a pseudo strain energy density function relating pseudo strain and damage.

3.3. complex modulus (E*)—a complex number that defines the relationship between stress and strain for a linear viscoelastic material.

3.4. cyclic pseudo secant modulus (C*)—the secant modulus in stress–pseudo strain space for a single cycle. This pseudo modulus differs from C because it is computed using a steady-state assumption and is used only with cycle-based data.

3.5. damage (S)—the internal state variable that quantifies microstructural changes in asphalt concrete.

3.6. damage characteristic curve (C versus S curve)—the curve formed when plotting the damage on the x-axis and the pseudo secant modulus on the y-axis. It defines the unique relationship between the structural integrity and amount of damage in a given mixture.

3.7. dynamic modulus (|E*|)—the norm of the E*, which is calculated by dividing the peak-to-peak stress by the peak-to-peak axial strain measured during the steady-state period.

3.8. dynamic modulus ratio (DMR)—the ratio between the dynamic modulus fingerprint and the dynamic modulus value from a master curve construction, both evaluated at the same temperature and frequency condition. This value is also used to characterize specimen-to-specimen variability.

3.9. failure cycle (Nf)—the cycle in which the measured phase angle drops sharply after a stable increase during cyclic loading.

3.10. fatigue analysis coefficients (K1, K2, K3)—fitting coefficients to describe the classical stress (or strain) versus cycles to failure relationship.

3.11. phase angle (φ)—the angle, expressed in degrees, between an applied sinusoidal stress and the resulting sinusoidal strain measured during the steady-state period.

3.12. pseudo strain (εR)—a quantity that is similar to strain but does not include time effects.

Pseudo strain is calculated by solving the convolution integral of the strain and

E(t).

3.13. pseudo secant modulus (C)—the secant modulus in stress–pseudo strain space.

3.14. relaxation modulus (E(t))—the quotient of the stress response of a material with time to a constant step amplitude of strain.

4. SUMMARY OF METHOD

4.1. An actuator displacement-controlled and repeated cyclic loading is applied to a cylindrical asphalt concrete specimen until failure. The applied stress and on-specimen axial strain response are measured and used to calculate the necessary quantities. The relationship between the damage (S) and the pseudo secant modulus (C) is determined and expressed as the damage characteristic curve.

5. SIGNIFICANCE AND USE

5.1. The damage characteristic curve represents the fundamental relationship between damage and material integrity for asphalt concrete mixtures. This property is independent of temperature, frequency, and mode of loading. Combined with the linear viscoelastic properties of asphalt concrete, the damage characteristic curve can be used to analyze the fatigue characteristics of asphalt concrete mixtures.

5.2. Damage characteristic curves can also be combined with additional fatigue parameters and pavement response models to predict the fatigue behavior of in-service asphalt concrete mixtures.

6. APPARATUS

6.1. Asphalt Mixture Performance Tester—An AMPT or other system meeting or exceeding the requirements of Equipment Specifications for the Simple Performance Test

System, NCHRP Report 629, Appendix E, with the additional capability to conduct direct tension testing, as shown in Figure 1.

Figure 1—General Schematic of Direct Tension Test Setup

6.2. External Conditioning Chamber (optional)— An environmental chamber for conditioning the test specimens to the desired testing temperature. The chamber shall be capable of controlling the temperature of the specimen over a temperature range of 5 to 25°C (41 to 77°F) to within ±0.5°C (±1°F). The chamber shall be large enough to accommodate at least a single test specimen and a “dummy” specimen with a thermocouple or other calibrated temperature-measuring device mounted at the center for temperature verification.

6.3. Loading Platens—Are required above and below the specimen to transfer the load from the testing machine to the specimen. The diameter of the loading platens shall be

38±1 mm. These platens should be made of hardened or plated steel, or anodized high strength aluminum. Materials that have linear elastic modulus properties and hardness properties lower than that of 6061-T6 aluminum shall not be used. The face of each load platen shall be grooved to provide better adhesion between the glue and plate. The top loading platen shall be designed so that it can be mated to the test machine without inducing loading eccentricity.

6.4. Ball Bearing (optional) —Users may place a ball bearing between the platens and the top loading platform in an attempt to account for loading eccentricity. Extra care should be taken using the ball bearing because if the screws are not tightened evenly around then excessive tensile stresses will develop on one side of the specimen.

6.5. End Plate Gluing Apparatus—Should be available for gluing the end plates to the asphalt concrete specimen. The device should ensure that the end plates and specimen are all centered, that the two platens are held parallel, and that the specimen is standing perpendicular to the plates. The weight resting on the specimen during curing of the adhesive shall not exceed 0.02 kN (4.5 lb), otherwise it shall be possible to clamp or otherwise hold the gluing apparatus at a fixed height for a period of at least 4 h.

7. HAZARDS

7.1. Standard laboratory safety precautions must be observed when preparing and testing asphalt concrete specimens.

8. TESTING EQUIPMENT CALIBRATION

8.1. The guidelines provided in TP 79 shall be followed to ensure that the test equipment and on-specimen measurement devices are properly calibrated.

8.2. If any of the verifications yield data that do not comply with the accuracy specified, the problem shall be corrected prior to further testing.

8.3. The hydraulic machine shall be properly tuned in displacement control mode, to enable use of the strain selection guidance in this standard. In displacement control mode, the tuning shall be such that there is a sinusoidal actuator deformation shape and the actuator displacement returns close to the initial position on the first cycle, as this will ensure the cycles are uniform and the input strain closely matches the output strain. Consult the equipment manufacturer for guidance on the specific equipment. If the machine is not tuned properly, the strain selection guidance may need to be adjusted for the specific machine.

9. TEST SPECIMEN INSTRUMENTATION PROCEDURE

9.1. Test specimens, 38-mm diameter by 110-mm height, shall be fabricated in accordance with PP XX.

9.2. Attach the mounting studs for the axial sensors to the sides of the test specimen using epoxy cement. Take care to avoid placing mounting studs directly in-line with screw holes in the end plates, which will be used to attach the specimen to the testing machine. A spacing fixture may be used to facilitate the mounting of the axial deformation measuring hardware.

Note 1—Quick-setting steel putty, such as Devcon 10240, has been found to be satisfactory for attaching the studs.

9.3. Verify that the gauge length is 70 mm ± 1 mm, measured center to center of the gauge points.

9.4. Thoroughly clean all end plates by first heavily brushing the face of each platen using either a hand operated wire brush, sandpaper, or a wire brush attached to a standard electric drill. After cleaning the platen with the wire brush, wipe the surface clean of any dust by using a towel dipped in acetone or similar solvent.

9.5. Using the same towel, but with only a small amount of solvent, wipe the ends of the specimen clean of any residual dust.

9.6. Weigh out an appropriate amount of adhesive to adhere the end plates and specimen to one another. The gluing process will require approximately 5 minutes, so prepare an adhesive that is appropriate for this length of working time.

Note 2—Quick setting steel putty, such as Devcon 10240, has been found to be satisfactory for attaching the end plates, provided gluing can be completed within

5 minutes of blending.

Note 3—Approximately 6 grams of the Devcon 10240 has been found to be suitable for 38-mm diameter specimens.

9.7. Fill in any surface voids and pores in the top and bottom surfaces of the specimen with the adhesive.

9.8. Divide the remaining adhesive in half and spread evenly between the end plates, ensuring grooves are filled. Insert and secure the end plates into the gluing jig and gently place the specimen on top of the bottom end plate, as close as possible to the center. Engage the centering mechanism to center the specimen on the end plate.

Lower the top plate into position, and secure if necessary. The final glue thickness should be approximately 1 mm (0.04 in.) thick. Allow the adhesive to reach its initial set before moving the specimen from the jig.

9.9. Remove the specimen from the gluing jig, taking special care to support the specimen from the bottom. Do not lift the specimen by the top end plate, so as not to put any tension stress on the specimen and adhesive.

Note 4— Users may attach mounting studs before or after the end plates, however the gluing jigs may require that the end plates and mounting studs are attached in a specific order.

9.10. Allow all adhesives to reach full cure before testing. Follow the manufacturer’s recommendation to determine the time needed to reach full cure.

10. TEST INFORMATION

10.1. This test procedure is designed to first test the specimens, at a specific temperature and frequency in oscillation (cyclic) mode to obtain linear viscoelastic fingerprints and then at the desired frequency and temperature until failure.

10.2. A total of three tests need to be done at three different strain levels. This is sufficient for ranking of materials’ sensitivity to fatigue damage. However, if pavement performance predictions are being done, a fourth test at a fourth strain level is strongly recommended. Guidance on selecting strain levels can be found in

Appendix X1.

10.3. An appropriate testing temperature can be selected using Equation 1. The climatic

Performance Grade (PG) for the location of interest at 98 percent reliability should be determined based on LTPPBind Version 3.1. The input depth in

LTPPBind should be 20 mm and no traffic adjustment shall be used. For example, the testing temperature for the LTPPBind high temperature of 64°C and low temperature of –22°C is 18°C.

𝑇𝑒𝑠𝑡 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 (℃) = { 𝑖𝑓 (

𝑇𝐻+𝑇𝐿

− 3) ≤ 21℃ , (

𝑇𝐻+𝑇𝐿

− 3) 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, 21℃ (1) where:

TH = High-temperature PG Grade from LTPPBind (ºC); and

TL = Low-temperature PG Grade from LTPPBind (ºC), generally a negative number.

Note 5—This general temperature selection guidance may not be appropriate for all mixtures, such as mixtures from cold climates with recycled materials, which may need to be tested at a higher temperature than prescribed to account for the more brittle nature of mixtures with high recycled asphalt binder ratios. This brittleness is typically manifested through repeated end failures or a very low number of cycles to failure at a particular strain input.

11. PROCEDURE

11.1. Test Setup:

11.1.1. Spacers may be needed to compensate for the reduced size of the specimens. If so, attach the spacers to the machine according to the manufacturer’s instructions.

11.1.2. If using a separate chamber to temperature-condition the specimens, continue to Section

11.1.3. If using the testing device to temperature-condition the specimens, skip to

Section 11.1.5.

Note 6—Use of a separate chamber to temperature-condition specimens is advantageous only if the specimen can be secured to the testing machine quickly.

Otherwise temperature conditioning in the machine may be more efficient.

11.1.3. Insert all specimens to be tested and the “dummy” specimen with the center-mounted temperature monitoring device into the external conditioning chamber, and start the temperature control of the testing machine.

11.1.4. Allow the test specimens to reach the specified testing temperature ±0.5°C (±1°F) by monitoring the temperature of the “dummy” specimen.

11.1.5. Insert the specimen into the testing machine, and tighten the specimen securely to the bottom support.

11.1.6. Bring the actuator into position and apply approximately 0.01 kN (2.2 lbs) of seating load. Quickly secure the upper loading platen, making sure not to overtighten any screws, which can shear the specimen.

Note 7—Feeler gauges may be placed between the specimen end plate and the top support of the testing machine to account for any small gaps. This is done at the user’s discretion. Users should use the ball bearing for specimens which exhibit

(after the seating load is applied) any gap between the specimen end plates and top support of the testing machine in excess of 1 mm (0.04 in.).

11.1.7. Reduce the load on the specimen to 0 kN ± 0.01 kN (0 lb ± 2 lb).

11.1.8. Attach the sensors to the specimen. Position the sensors in a location along its travel range where the elongation of the specimen as it undergoes damage will not exceed the range of the sensors, but the compression on the fingerprint can also be measured. This may not be the zero position; the exact position depends on the sensors.

11.1.9. Allow the specimen to reach the specified testing temperature ± 0.5°C (± 1°F). If the specimen is conditioned in a separate temperature-conditioning chamber and the steps presented in Sections 11.1.5 through 11.1.8 can be completed within 5 minutes, 30 minutes is enough to bring the specimen temperature back to the test temperature. If more than 5 minutes are needed, then calibrate the time for specific equipment. If temperature conditioning is done in the machine, calibrate this time by using a monitoring specimen alone in the machine chamber. To calibrate the conditioning time for testing without a monitoring specimen, record the temperature using a logger or other device with time to capture the equilibrium time needed for specimen to reach the appropriate temperature both before (i.e., conditioning) and after sensors (i.e., regaining temperature equilibrium before testing commences) are attached.

11.1.10. If the system does not automatically, adjust, balance, and zero the electronic measuring system.

11.2. Dynamic Modulus Fingerprint Test:

11.2.1. Input the required information for the dynamic modulus fingerprint test into the equipment control software. The fingerprint test shall be performed at the frequency of 10 Hz, at a target strain range of 50-75 microstrain, at the target test temperature, and in the tension-compression mode of loading.

Note 8—Some software may require input of an estimated dynamic modulus value to estimate the starting load amplitude. In this case, enter a value similar to the modulus obtained during frequency sweep testing (using TP XX) if available.

If not available, enter a conservative value of dynamic modulus so that the initial load does not damage the specimen.

11.2.2. Start the fingerprint test. The machine shall calculate the load level necessary to achieve

50 to 75 microstrain using the results of these first few cycles and then apply this load level for 50 cycles. A minimum 50 data points per cycle shall be recorded using equipment control software.

11.2.3. Compute the dynamic modulus for the last five cycles according to the method recommended in T 342 or TP 79. If the peak-to-peak strain exceeds 150 microstrain, discard the specimen.

11.2.4. The specimen shall rest at a load of 0 kN ± 0.01 kN (0 lb ± 2 lb) for a minimum of 20 minutes following the fingerprint testing.

11.3. Cyclic Fatigue Test:

11.3.1. Perform a constant positive movement actuator oscillation (cyclic) fatigue experiment at a frequency of 10 Hz (e.g., a pull-pull actuator displacement test). Start the first cyclic fatigue test with the target peak-to-peak on-specimen strain amplitude of

300, 500, or 800 microstrain (εos1) based on the |E*|fingerprint ranges in Table 3. For the first specimen, the target Nf shall be greater than 500 or the data shall be discarded.

Table 3—Target On-Specimen Strain Levels for the First Specimen

Case (units in MPa) εos1

|E*|fingerprint > 8,800 300

4,400 < |E*|fingerprint < 8,800 500

|E*|fingerprint < 4,400 800

11.3.2. Stop the test when propagated micro-cracks form one clear macrocrack. This macrocrack can be visually seen on the specimen surface and will cause the specimen to break into two completely separate parts. If the macrocracks occur outside the on-specimen deformation sensors, the test data can only be used to construct the damage characteristic (C versus S) curve. However, for calculating other parameters such as GR and Nf, a new specimen must be tested.

11.4. If the resultant number of cycles to failure of the first test (Nf1) is less than 500, the first test has to be discarded and redone with the target on-specimen peak-to-peak strain level (εos1) at the next lowest microstrain level in Table 3 (e.g., 300 microstrain instead of 500 microstrain). If the first specimen was done at 300 microstrain and failed before 500 cycles, discard the specimen and redo the testing at 250 microstrain. .

11.5. Repeat steps in Section 11.1 through 11.3 on the remaining two or more specimens.

Recommended target on-specimen peak-to-peak strain levels in microstrain for the remaining tests can be found from Table X1.1 based on the resultant number of cycles to failure of the first test (Nf1).

12. CALCULATIONS

12.1. This section presents the equations used to calculate the pseudo strain, pseudo secant modulus, and damage parameter for the fatigue tests. All the calculations in this section can be automatically performed using a combination of the AMPT control software and the ALPHA-Fatigue software or the FlexMAT spreadsheet described in the FHWA PRS Report.

12.2. Determine the E(t) Prony coefficients from the dynamic modulus and phase angle measured using TP XX and PP 61. It is assumed that the relaxation modulus can be represented by Equation 2.

m

N t m m

E t E E e

(2) where:

E(t) = relaxation modulus as a function of time, t, (kPa or psi);

E∞ = long-time equilibrium modulus, (kPa or psi);

Em = modulus of Prony term number m, (kPa or psi);

ρm = relaxation time of Prony term m (s); and

N = number of Prony terms used.

12.3. Compute the storage modulus, E′, for each temperature and frequency combination measured via Equation 3.

* cos

E E

(3)

E′ = storage modulus (kPa or psi);

|E*| = dynamic modulus determined via experiment (kPa or psi); and θ = phase angle determined via experiment (degrees).

12.4. Because measured data contain some variability, a smoothing process is needed to obtain reliable coefficients. Optimizing the coefficients in Equation 4 and Equation 8 simultaneously.

log log(max E') log , log

1 R

RE T E

e

(4)

E′(ω,T) = storage modulus at a particular temperature and angular frequency

(kPa or psi);

E′(ωR) = storage modulus at a particular reduced angular frequency (kPa or psi);

ωR = reduced angular frequency, Equation 7 (rad/s);

max E′ = defined by Equation 5; and κ, δ, γ = fitting coefficients.

max E' 1

100 10,000 c c

PVMA VFA xVMA P A B

VMA

VMA

A B VFA

(5)

Pc = defined in Equation 6 (kPa or psi);

VMA = voids in mineral aggregate, %;

VFA = voids filled with asphalt, %;

A = 4,200,000 for prediction in psi or 29,000,000 for prediction in kPa; and

B = 435,000 for prediction in psi or 3,000,000 for prediction in kPa.

0 58

0 58

435 000

435 000 c .

, (VFA )

VMA

P

, (VFA )

VMA

(6)

*R Ta (7) where:

aT = time–temperature shift factor at a given temperature, as shown in Equation

8.

2 2 1 2 ref refT T T T

Ta

(8)

T = temperature (°C or °F);

Tref = reference temperature (°C or °F); and α1, α2 = fitting coefficients.

12.5. Compute the total Storage modulus according to Equation 9.

2 2

2 2

N m R m

R mm

E

(9)

12.6. Compute the total Loss modulus according to Equation 10.

2 2

N m R m

R mm

(10)

12.7. Compute the specimen-to-specimen normalization parameter using Equation 11 and denote this parameter as the DMR (Dynamic Modulus Ratio).

fingerprint

DMR

LVE

E

E (11)

|E*|fingerprint = dynamic modulus determined from Section 11.16 (kPa or psi);

|E*|LVE = average representative dynamic modulus for the mixture of interest at the temperature and frequency of interest (kPa or psi), and computed by Equation 12; and

DMR = dynamic modulus ratio, which is the specimen variability compensation parameter.

2 2 2 2

2 2 2 2 1 11 1

N N m R m m R m

LVE

R m R mm m

E* E

(12) ω = angular frequency used in the fingerprint experiment;

aT = time–temperature shift factor for the fingerprint test temperature;

ωR = reduced angular frequency, Equation 7, used in the fingerprint experiment; and

E∞, Em, ρm = Prony coefficient terms.

12.8. Separate the data into two parts. The first part, Dataset 1, comprises the data for the first half of the first loading path (from zero to first peak stress). The second part, Dataset 2, comprises the rest of the data.

12.9. For Dataset 1, average all sensor readings and compute the average strain for all data points.

12.10. Calculate the axial stress for each data point in Dataset 1.

12.11. Compute the reduced time for each data point in Dataset 1 using Equation 13.

R T t t a (13) where:

aT = time–temperature shift factor at a given temperature;

t = time measured from the experiment (s), and tR = reduced time (s).

12.12. Compute the pseudo strain for each data point in Dataset 1 using the state variable formulation shown in Equation 13.

1 1 1

N

R n n n o m

R m

(14) εR(n+1) = pseudo strain at the next time step;

ER = reference modulus, a value of 1 should be chosen;

η = elastic component of the pseudo strain (Equation 15);

ηm = pseudo strain contribution of Prony element m (Equation 16);

n = time step used in the calculation;

ε = strain calculated for the current or subsequent time step;

ΔtR = duration of the reduced time step, tR n+1–tR n; and tR = reduced time.

1 1 n nE

(15)

1 1R m R m n n t tn n m m m m R e E e t n nE

(16)

12.13. Compute the normalized pseudo secant modulus for each data point in Dataset 1 using

Equation 17.

*R C

DMR

(17)

12.14. The continuum damage model power term, α, is related to the log-log slope of the relaxation modulus, E(t). Its value is found numerically using the Prony series representation of the E(t) in Equation 1. Determine m, the maximum value of the tangential slope of the relaxation modulus versus time relationship in log-log scale, and determine the α value using Equation 18.

m (18)

12.15. Compute the change in damage, ΔS, for each time step using Equation 19. Due to inherent electronic interference (data noise) during data acquisition, a few sequential data points may have positive ΔC values. A few of these spurious data points do not negatively affect the overall value of damage (S), but they do complicate the calculation. An efficient method that accounts for these spurious data points is the piecewise function shown in Equation 19.

12 1

1 1

R i i R i i i i i

DMR

C C t C C

S

C C

(19)

Ci = pseudo secant modulus at the current time step;

Ci – 1 = pseudo secant modulus at the previous time step;

ΔtR = change in the reduced time step; and α = continuum damage power term related to material time dependence, Equation 18.

12.16. Determine the damage at each time step using Equation 20.

N i i i

S S

(20)

Si = cumulative damage at the current time step; and

ΔSi = incremental damage for all time steps to be summed from the initial time step, i = 1, to the current time step, N.

12.17. Define the damage at the final point in Dataset 1 as SDataset 1.

12.18. Compute the peak-to-peak strain for each sensor and each cycle in Dataset 2.

12.19. For each cycle in Dataset 2, average all sensor strains and denote this strain as the test peak-to-peak strain amplitude, εpp.

12.20. Compute the peak-to-peak stress for each cycle in Dataset 2.

12.21. Compute the phase angle for each sensor and average the values together for each cycle.

Depending on the test equipment (e.g., an AMPT), phase angle may be automatically calculated per sensor and averaged.

12.22. Compute the peak-to-peak pseudo strain for each cycle in Dataset 2 using Equation 21.

* *R pp pp LVE

E (21) where:

εpp = average peak-to-peak axial strain; and εR pp = peak-to-peak pseudo strain.

12.23. Compute the cyclic pseudo secant modulus for each cycle in Dataset 2 using Equation 22.

pp

R pp

C

DMR

(22)

12.24. Compute the functional form factor, β, for each cycle in Dataset 2 using Equation 23.

peak valley peak valley

F F

F F

(23)

Fpeak = peak axial force measured by the load transducer (kN or lb); and

Fvalley = valley axial force measured by the load transducer (kN or lb).

12.25. Compute the tension amplitude pseudo strain for each cycle in Dataset 2 using Equation

24.

R R ta pp

(24) εR ta = tension amplitude pseudo strain.

12.26. Compute the time within a cycle when tensile loading begins, tb, for each cycle in Dataset

2 by using Equation 25.

1cos

62.83 bt

(25)

12.27. Compute the time within a cycle when tensile loading ends, te, for each cycle in Dataset 2 by using Equation 26.

12 cos

62.83 et

(26)

12.28. Compute the form adjustment factor for each cycle in Dataset 2 using Equation 27.

Equation 27 should be solved for each cycle, but generally β does not change significantly after the first few cycles, and a constant value may be applied after this transient period. Values of K1 have been tabulated for typical values of β and α in Table 4.

Table 4—Compiled K1 Values for Typical Material and Test Conditions

Beta

Alpha

4.333 4.077 3.857 3.667 3.500 3.353 3.222

–0.5 0.277 0.285 0.293 0.300 0.306 0.312 0.318

0.0 0.263 0.271 0.278 0.285 0.291 0.297 0.302

0.2 0.256 0.264 0.271 0.277 0.284 0.289 0.295

0.4 0.248 0.256 0.262 0.269 0.275 0.280 0.286

0.6 0.238 0.245 0.252 0.258 0.264 0.269 0.274

0.8 0.225 0.231 0.238 0.243 0.249 0.254 0.259

1.0 0.189 0.195 0.200 0.205 0.209 0.214 0.218

1 1 cos 62.83* e b t e b t

K t dt t t

(27)

Note 26—Equation 27 can be solved with sufficient accuracy using a suitable numerical technique with the number of time steps equal to 100.

12.29. Compute the average reduced time for each cycle in Dataset 2 using Equation 28.

R

T

N t a

(28)

N = cycle number.

12.30. Compute the change in damage, ΔS, for each cycle in the Dataset 2 using Equation 29.

Note 27—Even with data reduction, a few sequential data points may have positive ΔC values. A few of these spurious data points do not negatively affect the overall value of S, but they do complicate the calculation. An efficient method that accounts for these spurious data points is to use the piecewise function shown in Equation 23.

12 1 1

1 1 1

0 * *

R ta n n R n n n n n

DMR

C C t K C C

S

C C

(29)

C*n = the cyclic pseudo secant modulus at the current analysis cycle;

C*n – 1 = the cyclic pseudo secant modulus at the previous analysis cycle; and

ΔtR = the change in the average reduced time between analysis cycles.

12.31. Determine the damage at each analysis cycle using Equation 30.

dataset1

N n n n

S S S

(30)

SDataset1 = cumulative damage value at the end of Dataset 1;

Sn = cumulative damage at the current analysis cycle; and

ΔSn = incremental damage for all analysis cycles to be summed from the initial analysis cycle step, n = 1, to the current time step, N.

12.32. Combine the damage and pseudo secant modulus from for each time step in the first cycle, Sections 12.13 and 12.16, with the cyclic pseudo secant moduli and damage values from Sections 12.23 and 12.31, into a single dataset.

12.33. Determine the damage characteristic relationship by fitting one of the following equations to the plot of the pseudo secant modulus and damage from all of the fatigue tests.

baSC e or (31)

1 zC yS (32) where:

a, b = the fitting coefficients for the exponential model; and y, z = the fitting coefficients for the power model.

Note 28—The coefficients K1, K2, and K3 can be fit using the Nf and εt for use in the AASHTOWare Pavement ME Design software with additional localized calibration coefficients. This can be calculated in the FlexMAT spreadsheet.

12.34. Determine the average released pseudo strain energy per cycle to aid in the fatigue assessment of the mixture.

2 *1 ( ) (1 )

R R

C ta n nW C (33)

2 ( ) fN R

CR

f

W dN G

N

(34)

R

CW WC

R = released pseudo strain energy per cycle;

RG = average released pseudo strain energy per cycle; and fN = cycles to failure.

12.35. Using the data from all test specimens for the mixture, fit the GR and Nf using the power law shown in Equation 35.

R fG N (35) where:

γ, δ = fitting parameters; and fN = cycles to failure.

13. REPORT

13.1. Report the following for each specimen tested:

13.1.1. Test temperature;

13.1.2. The fingerprint dynamic modulus, |E*|fingerprint;

13.1.3. DMR value;

13.1.4. The model term related to the log-log slope of the relaxation modulus master curve, α;

13.1.5. C versus S curve coefficients (a and b in Equation 31 or y and z in Equation 32); and

13.1.6. GR and Nf for each test specimen.

14. KEYWORDS

14.1. AMPT; axial deformation; complex modulus; cyclic fatigue; damage characteristic curve;

direct tension; DMR; dynamic modulus; failure cycle; fingerprint; modulus; phase angle; Prony coefficients; relaxation modulus; pseudo secant modulus; pseudo strain; specimen deformation; strain.

APPENDIXES (NONMANDATORY INFORMATION)

X1. PROCEDURE FOR ESTIMATING ON-SPECIMEN STRAIN LEVELS IN THE

AMPT

X1.1. This appendix elaborates on the procedure for estimating strain levels for fatigue tests based upon the fingerprint dynamic modulus of the mixture, which is presented in Table 3. The approach is based on test performance from a large database of mixtures. This section is intended to provide guidance for selecting a programmed target on-specimen strain value for testing materials with unknown fatigue characteristics in the Asphalt Mixture Performance Tester (AMPT).

X1.2. Perform a fingerprint dynamic modulus test as prescribed in Section 11.3.

X1.3. Using the |E*|fingerprint, select the appropriate programmed target on-specimen strain level (εos1) for

Specimen 1. Record the cycles to failure, Nf. Discard any specimens that do not fail between the gauge points.

X1.4. Based on the Nf for the first test specimen, use Table X1.1 to select a target on-specimen strain level for

Specimens 2-4. Based on the database used to develop this approach, the mixture should lie on or between one of the families of curves in Table X1.1. Recall the intent of selecting different strain levels is to obtain a range of cycles to failure which are adequately spaced in log-log space for subsequent fatigue analyses. Due to the difficulty of obtaining specific Nf values, it is not possible to provide rigid guidance on strain selection processes. If this approach does not yield a sufficient range of Nf values, users are encouraged to adjust the strain inputs in increments of about 50 microstrain to achieve values in an acceptable range.

X1.5. As an example, Specimen A1 has a |E*|fingerprint of 7,500 MPa. Using Table 3, the first specimen programmed actuator microstrain is 500 microstrain, which results in a Nf value of 4,900 cycles.

The user could then use Table X6.1 and decide to test Specimen A2 at 450 microstrain to obtain an Nf around 10,000 cycles, Specimen A3 at about 550 microstrain to obtain an Nf of around 2,500 cycles, and Specimen A4 at around 600 microstrain to obtain an Nf of around 1,400 cycles. The values used in this example are underlined and italicized in Table X1.1.

X1.6. Table X6.1 was originally calibrated using |E*| values constructed from the master curve, such as in the PP

61 procedure. To simplify the strain selection process shown here, the values have been scaled to the fingerprint condition by using a DMR of 0.80. Users may choose to calibrate their own families of curves in a different fashion, but it is believed the |E*|fingerprint is an easier approach for a testing laboratory.

X1.7. A family of curves is developed by plotting test data in the fashion shown in Figure X1.1. A power function is then fit to the data, which can be done in a spreadsheet application such as Microsoft

Excel. The Nf can now be predicted based on the programmed actuator strain level.

X1.8. For users who wish to construct their own values for the first specimen target on-specimen strain level, a relationship between |E*|fingerprint and Nf is developed. Typically, the mixtures with a higher

|E*|fingerprint will fail at a lower Nf value than mixtures with a lower |E*|fingerprint. Based on the correlation between |E*|fingerprint and Nf, the user can identify ranges or categories of |E*|fingerprint which best align with satisfactory performing and poorly performing materials.

Figure X1.1-Example of Family of Curves Plot with Power Fit Trendlines to Predict Cycles to

Failure

1,000

10,000

100,000

0 500 1000 1500 2000

C yc le s to P e ak

P h as e

A n gl e

Programmed Target On-Specimen Strain Level [microstrain]

Table X1.1-Tool for Identifying Target On-Specimen Strain Levels for Second, Third, and

Fourth Specimen of a Mixture Set.

Target On-

Specimen

Microstrain

|E*|fingerprint ≥ 8,800 MPa

(εos1 = 300 microstrain)

8,800 > |E*|fingerprint > 4,400 MPa

(εos1 = 500 microstrain)

|E*|fingerprint < 4,400 MPa

(εos1 = 800 microstrain)

Estimated cycles to failure for test specimen

200 62,500 129,000 258,700 503,800 - - - - - - - - -

250 4,200 11,000 27,200 64,500 145,900 315,000 - - - - - - -

300 500 1,500 4,300 12,000 31,500 77,500 180,000 - - - - - -

350 - - 900 2,900 8,600 23,700 60,800 144,800 - - - - -

400 - - - 850 2,800 8,500 23,700 61,000 144,500 - - - -

450 - - - - 1,000 3,400 10,500 28,500 71,400 163,500 - - -

500 - - - - 500 1,500 4,900 14,400 38,000 91,000 - - -

550 - - - - - 750 2,500 7,800 21,500 53,500 119,600 - -

600 - - - - - - 1,400 4,400 12,800 33,000 75,850 155,700 -

650 - - - - - - 800 2,600 7,900 21,000 49,900 104,600 -

700 - - - - - - - 1,600 5,100 14,000 33,900 72,400 136,400

750 - - - - - - - 1,000 3,400 9,500 23,600 51,400 98,000

800 - - - - - - - 700 2,300 6,700 16,900 37,300 72,000

850 - - - - - - - - 1,600 4,800 12,300 27,600 53,900

900 - - - - - - - - 1,200 3,500 9,100 20,800 41,000

950 - - - - - - - - 800 2,600 6,900 15,900 31,600

1,000 - - - - - - - - - 1,900 5,300 12,300 24,700

1,050 - - - - - - - - - 1,500 4,100 9,700 19,600

1,100 - - - - - - - - - 1,100 3,200 7,700 15,700

1,150 - - - - - - - - - 900 2,500 6,200 12,700

1,200 - - - - - - - - - - 2,000 5,000 10,300

1,250 - - - - - - - - - - 1,600 4,100 8,500

1,300 - - - - - - - - - - 1,300 3,300 7,000

1,350 - - - - - - - - - - 1,100 2,800 5,900

1,400 - - - - - - - - - - 900 2,300 4,900

1,450 - - - - - - - - - - 750 2,000 4,200

1,500 - - - - - - - - - - - 1,700 3,600

1,550 - - - - - - - - - - - 1,400 3,000

1,600 - - - - - - - - - - - 1,200 2,600

1,650 - - - - - - - - - - - 1,000 2,300

1,700 - - - - - - - - - - - 900 2,000

1,750 - - - - - - - - - - - 750 1,700

1,800 - - - - - - - - - - - - 1,500

1,850 - - - - - - - - - - - - 1,300

1,900 - - - - - - - - - - - - 1,150

1,950 - - - - - - - - - - - - 1,000

2,000 - - - - - - - - - - - - 900

X2. USE OF ALTERNATIVE SMALL SPECIMEN GEOMETRIES

X2.1. Alternative small specimen geometries—Test specimens of geometries other than the one specified can be obtained from constructed pavement layers to measure the dynamic modulus for use in applications such as forensic investigations and field monitoring of test sections. Specimens with a diameter of 50 mm can be used where there are concerns about the use of 38-mm specimens.

Prismatic specimens 25 mm by 50 mm by 110 mm have also been evaluated for thinner construction lifts; if diameter is needed for prismatic specimens, calculate the cross-sectional area and then calculate the effective circular diameter that yields the same cross-sectional area.

X2.2. Alternative geometry test equipment—The same gauge points, same gauge length, and same on-specimen deformation sensors are used. Alternate end plates should be designed or procured with a gluing surface to match the specimen geometry. This may require adapters for an AMPT. Any measurement given directly in units of force (i.e., seating load for securing the specimen) should be scaled to an equal-stress condition with the 38-mm diameter specimens.