A dimensional analysis on asphalt binder fracture and fatigue cracking

Abstract

Fracture and fatigue cracking in asphalt binder are two of most serious problems for pavement engineers. In this paper, we present a new comprehensive approach, which consists both of dimensional analysis using Buckingham \mathrm{\Pi } Theorem and J-integral analysis based on classic fracture mechanics, to evaluate the fracture and fatigue on asphalt binder. It is discovered that the dimensional analysis could provide a new perspective to analyze the asphalt fracture and fatigue cracking mechanism.

Keywords

Dimensional analysis / asphalt / fracture / fatigue cracking

Introduction

Fracture and fatigue cracking in asphalt binder has always been one of the most serious problems for pavement engineers. When the internal stress caused by fracture or fatigue cracking is equal to or greater than the tensile strength of the pavement, a micro-crack inside asphalt pavement may appear. If external loadings are continuously applied, this crack will propagate through the pavement, which will consequently cause the failure of the asphalt pavement. Fig. 1 shows a typical cracking in asphalt pavements.

Gradually, the low temperature cracking has been formed and can eventually contribute to poor ride quality and shortened service lifetime [¹]. Due to detrimental effects of low-temperature cracking and fatigue cracking, significant research has been made both in laboratory tests and numerical analysis in the areas in recent years. According to a series of compact tension tests of 28 HMA mixtures designed for cold climates, a testing and modeling system based on four parameters (aggregate type, temperature, asphalt content and air void) was conducted to predict fracture behavior of asphalt concrete at low temperatures [²]. Combined with several verification cases, a computer simulation model has been proposed with temperature dependent bulk and fracture properties [³]. Hou et al. [²⁹,³⁰] discovered at asphalt microstructure evolution will occur at a temperature dropping cycle. In order to investigate the low-temperature behavior of asphalt binders, Direct Tension Tests were conducted and compared in terms of trend of stress–strain curve instead of conventional failure stress or failure strain values [⁴]. Asphalt mixture properties influencing thermal stress development were investigated within the Asphalt Concrete Cracking Device (ACCD) [⁵]. The fracture resistance of asphalt concrete at low temperatures was studied under different modes of loading, by performing three-point bend tests [⁶]. Quasi-brittle fracture in bitumen was stimulated by a diffuse interface model to understand the cracking mechanism under thermal cycle [³¹].

The fracture mechanics treats the leading edge of a crack as a line disturbance zone [⁷]. Research results have shown that the fracture mechanics principle can be used appropriately to evaluate or analyze the behavior of asphalt binder at low temperatures [⁸–¹⁰]. Experimental tests such as the Disk-shaped Compact Tension [¹¹], direct tension test, cyclic shear cooling failure test or the Moriyoshi breaking point (MBP) [¹²], Atomic-Force Microscopy (AFM) [³²], and acoustic emission (AE) based laboratory test [¹³] have been usually used for assessing low temperature behavior of asphalt binder.

The main aim of this paper is to present a new comprehensive approach to evaluate the low temperature fracture and fatigue cracking on asphalt binder, which consists both of dimensional analysis using Buckingham \mathrm{\Pi } Theorem and J-integral analysis based on classic fracture mechanics. We herein use a technique that proposed by [¹⁴] in fracture transition analysis and later developed by [¹⁵]in a scratch experiment analysis and now we expand the area to the fatigue life analysis on some daily use materials.

Fracture analysis

Dimensional analysis

The standard method to test the asphalt fracture property at low temperature is the three point bending test which has a small crack as the initial discontinuity. Set the crack cross-section is rectangular-shape, we are interested in the following quantities during the cracking process.

Young’s modulus E, Possion’s ratio v, material tensile strength {{\displaystyle \sigma }}_b and fracture toughness {{\displaystyle K}}_C; initial crack length {{\displaystyle l}}_{\mathrm{0}}, crack width w and crack depth d, specimen length L and cross section area S; tension loading force F=kt as a linearly increasing function function with minimum value zero and maximum value {{\displaystyle F}}_{\mathrm{0}}, note that maximum value is set at the moment when specimen immediately fails, k has the unit of N/s; time t₀ is the time from loading applying to the moment when specimen immediately fails.

In order to understand the fracture mechanism in asphalt low temperature cracking, scaling laws concerning the fracture mechanics is given. For a crack extension, it requires

(1)f\left(E,v,{{\displaystyle \sigma }}_b,{{\displaystyle K}}_C,{{\displaystyle l}}_{\mathrm{0}},w,d,L,S,F,t\right)=\mathrm{0}

We use FLT (force, length, time system) base dimension system to rationalize the dimensional variables as the following Table 1.

Based on Buckingham \mathrm{\pi } Theorem, we pick {{\displaystyle \sigma }}_b, d and t as repeating variables, rationalize the other quantities as dimensionless ones and get

(2){\displaystyle \frac{F}{{{\displaystyle \sigma }}_b{d}^{\mathrm{2}}}}=\mathrm{\pi }\left({\displaystyle \frac{E}{{{\displaystyle \sigma }}_b}},\upsilon ,{\displaystyle \frac{{{\displaystyle K}}_C}{\sqrt{d}{{\displaystyle \sigma }}_b}},{\displaystyle \frac{w}{d}},{\displaystyle \frac{{{\displaystyle l}}_{\mathrm{0}}}{d}},{\displaystyle \frac{L}{d}},{\displaystyle \frac{S}{d^{\mathrm{2}}}}\right).

Note that the third dimensionless term on the right side of Eq. (2) could be rewrote as {\displaystyle \frac{{{\displaystyle K}}_C}{\sqrt{d}{{\displaystyle \sigma }}_b}}={\displaystyle \frac{\mathrm{1}}{\mathrm{I}\mathrm{I}}}, where II is Irwin Number that used to determine whether the cracking is ductile () or not (\mathrm{I}\mathrm{I}>>\mathrm{1}). It is easy to understand that under same conditions, fatigue cracking in a ductile material will be slower compared fatigue cracking in a brittle material due to the existence of plastic deformation at crack tip.

J-integral analysis

It is believed that at low temperature (−20°C or even lower), asphalt binder will behave linearly elasticity, which satisfies the fundamental assumption of Linear Fracture Mechanics Theory first proposed by Griffith. J-integral is the key concept in Griffith’s theory since it represents the elastic strain energy release rate and the elastic strain energy will be transformed to the surface free energy to form new crack surfaces during crack propagation process.

We start our analysis from the expression of the classic J integral first proposed by [¹⁶], shown as

(3)J={\int }_{\mathrm{\Gamma }}(Wdy-\mathbf{T}\cdot \frac{\partial u}{\partial x}\mathrm{d}S),

where W is the elastic strain energy, \mathbf{T}=\mathbf{\sigma }·\widehat{n} is the traction vector, where \widehat{n} is the unit vector normal to \partial \mathrm{\Omega }, dS is the length increment along the given path \partial \mathrm{\Omega }..

Note that there is no traction force on the cracking surface, which means T=0 [¹⁵]. suggested the specific J-integral expression for a rectangular crack propagation as

(4)J={\frac{1}{p}\int }_A(W{n}_x-t_i\frac{{\partial u}_i}{\partial x})\mathrm{d}A,

where p= w+2d is the perimeter edge of the crack. Note that in our research {{\displaystyle \sigma }}_{xx}=\mathrm{0}, and all of the elastic strain energy are contributed by {{\displaystyle \sigma }}_{yy} which is different from Akon’s previous work.

The stress along y direction could be calculated based on the tension loading F and the actual loading area S^* as:

(5){{\displaystyle \sigma }}_{yy}={\displaystyle \frac{F}{S^{*}}}={\displaystyle \frac{F}{S-\left({{\displaystyle l}}_{\mathrm{0}}+\dot{l}t\right)d}}

where the effective actual loading area S^* could be calculated as:

(6)S^{*}=S-ld

and l is the crack length and could be expressed as:

(7)l={{\displaystyle l}}_{\mathrm{0}}+\dot{l}t

where \dot{l} is the crack propagation speed.

It should be pointed out that Eq. (5) could accurately calculate the stress along y direction. However, by conducting a Direct Tension Test and Three Point Bending Test, it is observed that for a laboratory experiment on asphalt specimen at low temperature, it fails immediately at the moment crack begins to propagate when the tension loading reaches critical value. And that indicate that actually remains the same during the cracking process since stress will fast decrease to zero when crack initiates and specimen fails almost at the same time. And thus we have

(8){{\displaystyle \sigma }}_{yy}={\displaystyle \frac{F}{S^{*}}}={\displaystyle \frac{F}{S-{{\displaystyle l}}_{\mathrm{0}}d}}

Based on Hooke’s law, we further have elastic strain energy density as W={\displaystyle \frac{\mathit{\kappa }}{\mathrm{2}E}}{{\displaystyle \sigma }}_{yy}^{\mathrm{2}}; dA= (w+2d)ds, ds=wdz. Since the J-integral only has physical meaning when crack begins to propagate, we substitute the critical value F={{\displaystyle F}}_{\mathrm{0}} into Eq. (8) and we have {{\displaystyle \sigma }}_{yy}={\displaystyle \frac{{{\displaystyle F}}_{\mathrm{0}}}{S-{{\displaystyle l}}_{\mathrm{0}}d}}.

Since there is no traction on the fracture surface, we thus have

(9)J={\displaystyle \frac{\mathrm{1}}{w+\mathrm{2}d}}{\displaystyle {\int }_{\mathrm{0}}^{d}Ww\mathrm{d}z}={\displaystyle \frac{\mathit{\kappa }{{\displaystyle F}}_{\mathrm{0}}^{\mathrm{2}}wd}{\mathrm{2}E{\left(S-{{\displaystyle l}}_{\mathrm{0}}d\right)}^{\mathrm{2}}\left(w+\mathrm{2}d\right)}}

where \mathit{\kappa }=\mathrm{1}-v^{\mathrm{2}} for plane strain and \mathit{\kappa }=\mathrm{1} for plane strain.

Also noting during each loading cycle, the elastic strain energy will be totally transformed to the fracture energy, i.e.,

(10)J=G={\displaystyle \frac{\mathit{\kappa }{{\displaystyle K}}_c^{\mathrm{2}}}{E}}

Compare equation (9) and (10), we now have

(11){\displaystyle \frac{{{\displaystyle F}}_{\mathrm{0}}}{{{\displaystyle K}}_{c}S}}=\sqrt{{\displaystyle \frac{\mathrm{2}\left(w+\mathrm{2}d\right)}{wd}}}

Eq. (11) can be further used to determine the material fracture toughness that represents material fatigue-resistance capability in Griffith’s theory. Write Eq. (11) in the dimensionless expressed we derived in Eq. (2) and we further get

(12)\mathrm{I}\mathrm{I}={\displaystyle \frac{{\displaystyle \frac{S}{d^{\mathrm{2}}}}}{{\displaystyle \frac{{{\displaystyle F}}_{\mathrm{0}}}{{{\displaystyle \sigma }}_b{d}^{\mathrm{2}}}}}}\sqrt{\mathrm{2}\left(\mathrm{1}+\mathrm{2}{\displaystyle \frac{d}{w}}\right)} .

Eq. (12) implies that if the crack depth d is sufficient large, \mathrm{I}\mathrm{I}>>\mathrm{1} and implies a brittle fatigue crack is growing while a sufficient large crack width w implies , which implies a ductile fatigue crack is growing.

Fatigue analysis

Most of the current researches on asphalt materials fatigue are using four point bending test, as shown in Fig. 2. The typical fatigue life of asphalt materials could be summarized into three stages: stage 1, crack initiation; stage 2, stable fatigue cracking; stage 3, failure due to unstable instant fracture. And since the most common situation it will occur in reality is stage 2, we are focusing on how the initial crack will propagate during fatigue cracking process.

Assume the crack cross-section is rectangular-shape, we are interested in the following controlling variables during fatigue cracking process:

Young’s modulus E, Possion’s ratio v, material tensile strength (for brittle material), yield stress (for ductile material) {{\displaystyle \sigma }}_b and fracture toughness {{\displaystyle K}}_C; initial crack length {{\displaystyle l}}_{\mathrm{0}}, crack width w and crack depth d, specimen length L and cross section area S; tension loading force F as a sinewave function with minimum value zero and maximum value {{\displaystyle F}}_{\mathrm{0}} within a loading cycle, and loading frequency H.

In order to understand the fatigue mechanism that is useful in everyday life, scaling laws concerning the fatigue mechanics is given. For a fatigue crack extension, it requires:

(13)f\left(E,v,{{\displaystyle \sigma }}_b,{{\displaystyle K}}_C,{{\displaystyle l}}_{\mathrm{0}},w,d,L,S,F\left(t\right),H\right)=\mathrm{0}

We use FLT (force, length, time system) base dimension system to rationalize the dimensional variables as the following Table 2.

Eq. (2) is also employed based on Buckingham \mathrm{\Pi } Theorem.

For fatigue cracking, substitute the J expression into the Paris’ law as follows:

(14){\displaystyle \frac{\mathrm{d}l}{\mathrm{d}N}}=C{\left(\mathrm{\Delta }K\right)}^m

where l is crack length and N is loading cycles; C and m are material constants; \mathrm{\Delta }K is the stress intensity factor amplitude.

Rewrite it in energy form as:

(15){\displaystyle \frac{\mathrm{d}l}{\mathrm{d}N}}=C{\left(\mathrm{\Delta }J\right)}^m

where \mathrm{\Delta }J is the energy release in each loading cycle. And we further get:

(16){\displaystyle \frac{\mathrm{d}l}{\mathrm{d}N}}=C{{\displaystyle \left[{\displaystyle \frac{\mathit{\kappa }{{\displaystyle F}}_{\mathrm{0}}^{\mathrm{2}}wd}{\mathrm{2}E{\left(S-{{\displaystyle l}}_{\mathrm{0}}d\right)}^{\mathrm{2}}\left(w+\mathrm{2}d\right)}}\right]}}^m

Now consider a situation that we know the initial crack length {{\displaystyle l}}_{\mathrm{0}} and the critical crack length {{\displaystyle l}}_{crit} that would lead to a sudden failure of the material specimen. Denote N as the accumulated loading cycles that would lead to the fatigue failure, we have

(17){\displaystyle {\int }_{\mathrm{0}}^{N}dN}={\displaystyle {\int }_{{{\displaystyle l}}_{\mathrm{0}}}^{{{\displaystyle l}}_{crit}}{\displaystyle \frac{\mathrm{d}l}{C{{\displaystyle \left[{\displaystyle \frac{k{{\displaystyle F}}_{\mathrm{0}}^{\mathrm{2}}wd}{\mathrm{2}E{\left(S-{{\displaystyle l}}_{\mathrm{0}}d\right)}^{\mathrm{2}}\left(w+\mathrm{2}d\right)}}\right]}}^m}}}

We finally get

(18)N={\displaystyle \frac{{{\displaystyle l}}_{crit}-{{\displaystyle l}}_{\mathrm{0}}}{C{{\displaystyle \left[{\displaystyle \frac{\mathit{\kappa }{{\displaystyle F}}_{\mathrm{0}}^{\mathrm{2}}wd}{\mathrm{2}E{\left(S-{{\displaystyle l}}_{\mathrm{0}}d\right)}^{\mathrm{2}}\left(w+\mathrm{2}d\right)}}\right]}}^m}}

Eq. (14) simply shows the remaining loading cycles of the sample before it fails. A more intuitive equation to reveal the fatigue life is T={\displaystyle \frac{\mathrm{1}}{H}}N={\displaystyle \frac{{{\displaystyle l}}_{crit}-{{\displaystyle l}}_{\mathrm{0}}}{CH{{\displaystyle \left[{\displaystyle \frac{\mathit{\kappa }{{\displaystyle F}}_{\mathrm{0}}^{\mathrm{2}}wd}{\mathrm{2}E{\left(S-{{\displaystyle l}}_{\mathrm{0}}d\right)}^{\mathrm{2}}\left(w+\mathrm{2}d\right)}}\right]}}^m}}

Rewrite it in dimensionless expression as followings:

(19)T=\mathrm{2}\mathit{\kappa }C\left({{\displaystyle l}}_{cit}-{{\displaystyle l}}_{\mathrm{0}}\right){\displaystyle \frac{\mathrm{1}}{H}}\left({\displaystyle \frac{E}{{{\displaystyle \sigma }}_b}}\right){{\displaystyle \left({\displaystyle \frac{S-{{\displaystyle l}}_{\mathrm{0}}d}{d}}\right)}}^{\mathrm{2}}{\displaystyle \frac{\mathrm{1}}{{{\displaystyle \left({\displaystyle \frac{{{\displaystyle F}}_{\mathrm{0}}}{{{\displaystyle \sigma }}_b{d}^{\mathrm{2}}}}\right)}}^{\mathrm{2}}}}{\displaystyle \frac{\mathrm{1}+\mathrm{2}{\displaystyle \frac{d}{w}}}{d^{\mathrm{3}}}}

Based on Eq. (19), it is interesting to find that a deeper crack indicates a shorter fatigue life while a wider crack has a longer fatigue life, a smaller meanwhile some obvious conclusion can be obtained such as a high frequency and a high loading force both imply a short fatigue life.

Also note that the critical crack length is beyond the plastic zone and as [¹⁷,¹⁸]

(20){{\displaystyle l}}_{crit}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }}}{{\displaystyle \left({\displaystyle \frac{{{\displaystyle K}}_C}{{{\displaystyle \sigma }}_b}}\right)}}^{\mathrm{2}}

Summary

Fracture and fatigue cracking in asphalt binder are two of most serious problems for pavement engineers. In this paper, we present a new comprehensive approach, which consists both of dimensional analysis using Buckingham \mathrm{\Pi } Theorem and J-integral analysis based on classic fracture mechanics, to evaluate the fracture and fatigue on asphalt binder. It is discovered that the dimensional analysis could provide a new perspective to analyze the asphalt fracture and fatigue cracking mechanism.

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