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39 Table 23. Climatic zone variations of fracture and healing coefficients for HMA. Climatic Zone Coefficient Wet-Freeze Wet-No Freeze Dry-Freeze Dry-No Freeze g0 -2.09 -1.429 -2.121 -2.024 g1 1.952 1.971 1.677 1.952 g2 -6.108 -6.174 -5.937 -6.107 g3 0.154 0.19 0.192 1.53 g4 -2.111 -2.079 -2.048 -2.113 g5 0.037 0.128 0.071 0.057 g6 0.261 1.075 0.762 0.492 equation for the logarithm of the Paris and Erdogan's Law to 1.0. If the stress wave is a rising and falling shape as is coefficient, A. commonly the case with traffic and thermal stresses, the value of ak is usually considerably less than 1.0. Appendix F shows the patterns of the rise and fall of the stress waves caused by the pas- Healing Coefficients sage of single, tandem, tridem, and quadrem axles. These pat- In addition to the fracture coefficients, the healing coeffi- terns were used in determining the effect during each day of cients obtained in earlier studies (4) are used to account for each set of axle groupings on the growth of reflection cracks. the healing shift function that occurs between the traffic loads on the overlay. The healing shift function is Computational Method for Crack Growth Due to Traffic SFhealing = 1 + g 5 ( t rest ) 6 g (29) Although the SIF for bending and shear occur at the same The rest period in seconds between load applications is cal- time under traffic loads, the crack growth technique adopted culated as the number of seconds in a day (86,400) divided by in this project calculates the growth of cracks due to each of the average daily traffic in vehicles per day. Values for the coef- the two stresses separately. Thus, Paris and Erdogan's Law for ficients g0 through g6 were determined for each of the four cli- bending and shearing are provided by Equations 31 and 32, matic zones; these are listed in Table 23 (4). These coefficients respectively. were used without alteration and the fracture coefficients g0 dc = A [ K1 ( bending )] [ a k ( bending )] n through g4 wereappliedwithout modification to determine both (31) the thermal and traffic fracture properties. The healing coeffi- dN cients were used only with the traffic crack growth equations. dc = A [ 2K11 ( shearing )] [ a k ( shearing )] n (32) dN Stress Wave Pattern Correction for Viscoelastic Crack Growth The wave patterns for ak--the viscoelastic factor--are shown in Appendix F for each of the types of traffic loading: Schapery's theory of crack growth in viscoelastic materials bending and shearing and each of the four axle groupings. takes into account the loading time and the shape of the stress With shearing stresses, there is a peak shearing stress as the pulse during the time that the material is being loaded (30, leading edge of the tire approaches the reflection crack and 31). The normalized wave shape, w(t), has a peak value of 1.0. then another peak shearing stress of a different sign as the The wave shape rises to 1.0 and falls back to zero in a length trailing edge of the tire leaves the location of the reflection of time, t. The correction term for viscoelastic crack growth crack. Thus there are two peak shearing SIF with the passage ak is given by the following equation. of a single tire. Examples of these patterns are shown in Fig- t ures 35 and 36 for bending and shearing, respectively. ak = 0 w ( t ) dt n (30) The time increment, t, for the tridem axle group to pass over a given point on a pavement is given in Equation 33. The exponent, n, is the Paris and Erdogan's Law exponent 18 + L j ft which is given in Equation 25 and is typically between 2 and 6. t ( sec ) = (33) If the applied load is a square wave, the integral is equal V ft sec
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40 4.0 ft 4.0 ft 4.0 ft Overlay Lj Lj Overlay Lj Lj Lj Old Surface Crack or Joint Old Surface Crack or Joint Figure 35a. Bending loading pattern for a tridem axle. Lj Lj The incremental crack growth each day is calculated from the accumulated effects of all of the traffic that have passed over the reflection crack during that day as follows: Lj Lj i =n 1 dc = A ( K Ii ) ( a ki ) dNi n for bending and (34) i =1 SFhealing i=n 1 dc = A ( 2K IIi ) ( a ki ) dNi n Lj Lj for shearing (35) i=1 SF healing Crack length, cn on the nth day of this crack growth process Figure 36a. Shearing loading pattern for a is the sum of all of the n incremental crack growth increments: tandem axle. i =n lengths above this point, bending stresses no longer contribute cn = ci (36) i =1 to the growth of cracks and crack growth is due only to ther- mal and shearing stresses. The number of days that are required When the sum of the bending crack increments reaches the for cracks caused by each type of stress to reach Position 1 are point in the overlay where the bending stresses become com- recorded. Then the number of days required to grow a crack pressive, that defines what is termed "Position 1." At crack from Position 1 to the surface of the overlay because of shear 4.0 ft 4.0 ft 5.0 ft Lj Lj Lj 5.0 ft W(t) 0.84 0.84 Load 0.72 0.76 0.76 0.72 Wave 0.92 0.92 Shape 0.84 0.0 0.0 0.095 0.095 (18 + Lj) ft [W(t)]n (0.92)n (0.92)n (0.84)n (0.84)n (0.84)n (0.76)n (0.76)n (0.72)n (0.72)n 0.0 0.0 (0.095)n (0.095)n t Figure 35b. Normalized SIF and ak wave patterns for tridem bending loading.