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On predicting mode II fracture toughness (KIIc) of hot mix asphalt mixtures using the strain energy density criterion
Accepted Manuscript On Predicting mode II fracture toughness (K IIc) of hot mix asphalt mixtures using the strain energy density criterion MRM Aliha PII: DOI: Reference:
S0167-8442(17)30481-0 https://doi.org/10.1016/j.tafmec.2018.11.001 TAFMEC 2123
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
19 October 2017 7 September 2018 2 November 2018
Please cite this article as: M. Aliha, On Predicting mode II fracture toughness (K IIc) of hot mix asphalt mixtures using the strain energy density criterion, Theoretical and Applied Fracture Mechanics (2018), doi: https://doi.org/ 10.1016/j.tafmec.2018.11.001
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On Predicting mode II fracture toughness (KIIc) of hot mix asphalt mixtures using the strain energy density criterion 1
MRM Aliha Welding and Joining research center, School of Industrial Engineering, Iran University of Science and Technology (IUST), Narmak, 16846-13114, Tehran, Iran. [email protected]
Abstract Mode II or in-plane sliding of crack faces is one of the possible fracture modes for asphalt pavements which can often take place due to traffic loads passing over the crack at low temperature conditions. Hence pavement engineers need to know the value of mode II fracture toughness (KIIc) of asphalt mixtures as an important design parameter. However, despite a large number of mode I fracture toughness (KIc) values available in the literature for different asphalt concrete materials, only limited KIIc values have been reported for such construction materials. In this paper to fill this research gap, a number of pure mode I and pure mode II fracture toughness experiments are conducted at low temperature using symmetric and asymmetric semi-circular bend specimens made of different asphalt mixtures. It was found that the mode II fracture toughness of each mixture is approximately 0.9 to 1.1 times the corresponding value of KIc. It was demonstrated that the experimentally obtained fracture toughness ratio (KIIc/KIc) for the investigated asphalt concretes are in good agreement with theoretical predictions of the minimum strain energy density criterion. Consequently, as an engineering application of this research, mode II fracture toughness of investigated asphalt mixtures was estimated in terms of their more common mechanical parameters such as KIc and Poisson’s ratio values. Keywords: Hot mix asphalt, Additive type, Mode II fracture toughness, Asymmetric semi-circular bend specimen, Minimum strain energy density criterion, KIIc/KIc prediction. 1. Introduction Brittle fracture of asphalt concretes (which usually occurs at low temperatures) is one of the main deterioration and failure modes of pavements in many countries having a cold climate. Due to high rehabilitation cost of roads, it is necessary to understand the mechanism of crack growth in these materials for developing a suitable maintenance program for the network of overlays. Since hot mix asphalt (HMA) concretes are complex materials and are often subjected to diverse environmental and mechanical loads (e.g. induced by traffic or temperature changes), different types of cracks can be initiated and then propagated in the pavements. A number of research studies in this field [e.g. 14] have shown that a cracked asphalt pavement subjected to traffic loading of moving
vehicles experiences complex tensile-shear deformations. In particular, for many practical loading situations, top-down crack extension in the pavement takes place when the crack is subjected to pure or dominantly mode II loading conditions. While pure mode I deformation, refers to the opening of the crack faces, pure mode II deformation takes place for in-plane sliding of the crack faces without any opening. Accordingly, when the wheel of vehicle passes over the top-down crack (as shown schematically in Fig. 1), the crack would experience pure mode II deformation.
Top-down crack
Fig. 1: Top-down crack subjected to pure or dominantly mode II deformation.
Under shear loads, the crack propagates if the mode II stress intensity factor (KII), reaches a critical value that is known as mode II fracture toughness, KIIc. Therefore, it is important to develop appropriate experimental or theoretical methods for determining and predicting mode II fracture toughness KIIc in HMA materials. However, review of published papers indicates that most of the previous research works have only focused on pure mode I fracture toughness (KIc) determination of asphalt mixtures. For example, rectangular edge cracked beam subjected to three or four-point bending [5], disc shape compact-tension (DCT) specimen [6], modified indirect diametral tension (IDT) specimen [7], edge notch disc bend (ENDB) specimen [8-12] and semi-circular bend
(SCB) specimen [13-23] have been used extensively for obtaining KIc of asphalt concrete materials. In any of the above mentioned specimens, the effects of different parameters such as the specification of asphalt mixture (i.e. binder and aggregate type), environmental conditions (e.g. test temperature, loading rate, freeze/thaw cycles, additive type and etc.), have been investigated on the tensile type mode I cracking characteristics of asphalt materials. Recently, the DCT and SCB specimens have been suggested by the American Association of State Highway and Transportation Officials (AASHTO) and American Society for Testing and Materials (ASTM) as standard test configurations for determining a versatile and valid KIc value for the asphalt mixtures [6,24]. But very few test specimens have been used and investigated in the past for obtaining KIIc of asphalt materials and thus no standard testing methods is available for determining mode II fracture toughness of these materials. Double edge cracked four-point bend specimen subjected to lateral confining pressure [25], inclined edge cracked SCB specimen [26] and the SCB specimen containing a vertical edge crack and subjected to asymmetric three-point bending [15,27] are among very limited test configurations employed for mode II fracture toughness study of asphalt concretes. Hence, in comparison with the rich set of pure mode I asphalt fracture toughness (KIC) data reported by a large number of researchers, there is very little experimental KIIc data in the literature for asphalt mixtures. Therefore as the first aim of this research, mode II fracture toughness of different asphalt mixtures with different characteristics and ingredients is obtained experimentally using the asymmetric SCB (ASCB) configuration that has been proposed earlier by the authors and coworkers for testing mode II fracture of other brittle materials [28]. These data can help asphalt fracture researchers and pavement designers in better understanding the fracture behavior of asphalt concretes and provide typical fracture resistance values against shear type (or mode II) crack growth for different HMA mixtures.
On the other hand, the experimental fracture toughness data obtained from different engineering materials are usually predicted by means of theoretical fracture criteria. For example, pure mode II or mixed mode I/II fracture toughness behavior of brittle polymers, metals, rocks, concretes, ceramics and etc. have been already predicted by means of stress/strain or energy based criteria [29-45]. However, this issue has been rarely investigated for asphalt mixtures till now. Thus as the second aim of this paper and to fill this research gap, the applicability of a conventional fracture theory called the minimum strain energy density (MSED) criterion is examined for predicting the obtained mode II asphalt fracture toughness data. It is shown that the KIIc value of investigated asphalt mixtures can be predicted very well in terms of their mode I fracture toughness (i.e. KIc). 2. Mode I and mode II fracture toughness test specimen Fig. 2 shows schematic representation of test specimens used for pure mode I and pure mode II asphalt fracture toughness testing of this research that are semi-circular specimens of radius R and thickness t. A vertical edge crack of length a is emanated from the middle of bottom flat edge of semi-circle and both modes I and II fracture specimens are subjected to three-point bend loading. When the bottom loading supports are symmetric relative to the crack plane (i.e. S1= S2), the symmetric semi-circular bend (SSCB) specimen is subjected to pure mode I (or pure opening mode) loading. But for the asymmetric spans (i.e. moving the right hand support S2 towards the crack) the asymmetric semi-circular bend (ASCB) specimen would be subjected to different mixed mode I/II (i.e. tensile-shear) loading conditions. The contribution of modes I and II deformations is controlled in this specimen by changing the S2 value. Consequently, for any given crack length ratio (a/R), a special S2 value can be found in which the ASCB specimen experiences pure shear deformation or pure mode II condition (in which KI=0
but KII≠0). Thus with a same vertically edge cracked SCB specimen, both KIc and KIIc experiments can be conducted simply by setting S1 and S2 values at suitable positions. From practical view point, this configuration has some other advantages for being used in asphalt fracture toughness studies. simple geometry of specimen, convenience of sample preparation from gyratory compactor machines or field coring devices, ease of introducing an edge crack, simple loading and testing apparatus either for pure mode I or pure mode II (i.e. SSCB and ASCB configurations) and ability of determining KIIc in addition to KIc, are to name a few. Hence, the SSCB and ASCB specimens have been used earlier by several researchers for modes I and II fracture toughness study of asphalt mixtures and other types of quasi-brittle materials, like rocks and cement concretes [14,20,47-51].
(a)
(b)
Fig. 2: Schematic representation of (a) symmetric semi-circular bend (SSCB) and (b) asymmetric semi-circular bend (ASCB) specimens for conducting pure mode I and pure mode II fracture toughness experiments on HMA mixtures.
Based on the principals of fracture mechanics the crack tip stress/strain field in cracked components subjected to any arbitrary loading conditions is defined by a key parameter
called the stress intensity factor. This parameter describes the severity or singularity of stress/strain at crack tip outlined by William’s series expansion as [52]:
ij An r
n2 n
fij ( n ) ( )
n 1
(1)
where r, are the polar coordinates, An are constant factors and fij (n) are the angular functions of nth term and ij is the stress tensor. For example, the tangential stress component σθθ (as shown schematically in Fig. 3) is written as following equations for pure mode I and Pure mode II loading condition [53-55]:
KI
(@ pure mode I)
(@ pure mode II)
2r
cos 3
2
H .O.T
cos sin H .O.T 2 2 2r
(2)
3K II
(3)
in which KI and KII are the mode I and mode II stress intensity factors that are related to the singular terms of Williams’s infinite series expansion and “H.O.T” presents the remaining non-singular and higher order stress terms that are often ignored in comparison with the singular terms.
Fig. 3: Tangential stress component in the vicinity of crack tip under pure mode I and pure mode II conditions.
The mode I and mode II stress intensity factors (KI and KII) in the SSCB and ASCB specimens are functions of crack length over the radius ratio (a/R), bottom support distances over the radius ratios (S1/R and S2/R) and the applied load (P) and can be written as:
K I ( SSCB)
K II ( ASCB )
P
a YI ( SSCB)
(4)
P a YII ( ASCB ) 2R t
(5)
2R t
where YI(SSCB) and YII (ASCB) are the geometry factors corresponding to the pure modes I and II in the semi-circular bend specimen, respectively. Ayatollahi et al. [55] computed numerically these geometry factors under mixed mode I/II loading condition but only for limited values of a/R, S1/R and S2/R. Using the same procedure described in Ref. [55], we performed here a number of finite element analyses to obtain YI (SSCB) and YII (ASCB) for a wider range of a/R, S1/R, and S2/R ratios. Fig. 4 shows the finite element model of SSCB and ASCB specimens created in the ABAQUS code and Fig. 5 presents the corresponding values of pure mode I and pure mode II geometry factors (i.e. YI (SSCB) and YII (ASCB)), for different crack lengths and bottom loading spans. In the following sections, fracture toughness of some asphalt mixtures are determined experimentally using the SSCB and ASCB specimens.
SSCB model (pure mode I)
ASCB model (pure mode II)
Fig. 4: Finite element model of SSCB and ASCB specimens created in the ABAQUS software.
Fig. 5: Variation of YI (SSCB) and YII (ASCB) for different geometry and loading conditions of SSCB and ASCB specimens.
3. Asphalt fracture toughness experiments In the experimental part of this research, a number of hot mix asphalt concretes with different aggregates and binder modifiers were manufactured. For preparation of asphalt
mixtures, two aggregate types (i.e. limestone and siliceous) with nominal maximum aggregate size (NMAS) of 12.5 mm (which is known as aggregate size No. 4 according to Iranian paving standard code-234) and a binder with performance grade of (PG 64-22) was used. This bitumen is the most frequently used binder for paving the roads and highways in Iran that is a moderately soft binder with penetration grade of 60/70. Fig. 6 and Table 1 present the gradation of aggregate No. 4 and the specifications of 60/70 binder. Moreover, a number of frequently used additives (including Styrene Butadiene Styrene (SBS), Polyphosphoric Acid (PPA), Crumb Rubber (CR), FT-Paraffin Wax (Sasobit) and Anti stripping agent) were also used in the composition of manufactured HMA materials. Details of mixing procedure of the additives with the base bitumen and physical specifications of the modified binders have been summarized in Tables 2 and 3, respectively. In order to manufacture the HMA mixtures, the optimum percentage of binder for each modified and unmodified mixtures was determined using the Marshall Mix design method. For manufacturing the HMA mixtures, binders modified with the crumb rubber and SBS were mixed with aggregate at 160oC but the rest of modified binders (i.e. Sasobit, PPA, Anti stripping) were mixed at 135oC. Then several cylindrical specimens with diameter of 150 mm and height of 130 mm were manufactured using superpave gyratory compactor machine (SGC). Two gyratory rotations (i.e. 90 and 30 rotations) were chosen for compacting the HMA mixtures. Consequently, the air void content of manufactured HMA samples were approximately equal to 3 and 7%, respectively.
Fig. 6: Gradation of aggregate used for preparation of HMA.
Table 1. Specifications of 60/70 binder with PG of (64-22). Test
Standard
Specific Gravity (25 °C) Flash Point (Cleveland) Penetration (25 °C) Ductility (25 °C) Softening point Kinematic Viscosity @ 120 °C Kinematic Viscosity @ 135 °C Kinematic Viscosity @ 150 °C Penetration index (PI)a
ASTM D70 ASTM D92 ASTM D5 ASTM D113 ASTM D36 ASTM D2170 ASTM D2170 ASTM D2170 –
Unit
value 3
gr/cm °C °C cm °C mm2/s mm2/s mm2/s –
1.03 308 62 100 49 810 420 232 -1.12
Table 2. Weight percentage and mixing method of different additives with the base binder for manufacturing the HMA samples of this research. Type of Binder BCR BSb BSb BP BA
Additive type Crumb rubber SBS Sasobit PPA Anti Stripping
Weight percentage of Binder (%) 15 5 2.5 1 0.4
High shear (rpm) 6000 6000 -
Mixing temperat ure (⁰C) 170 170 -
Mix Method Mixing low shear time (rpm) (min) 90 1000 90 1000 300 1000 1000
Mixing temperat ure (⁰C) 130 130 130 160 150
Mixing time (min) 120 120 10 30 30
Table 3. Physical properties of binders used for HMA manufacturing. Test B1 BA2 BP3 BSa4 BSb5 Penetration (25 °C) (°C) 62 56 48 45 43 Ductility (25 °C) ( cm) >100 >100 >100 >100 >100 Softening point (°C) 49 53 61 63 66 Performance Grade 64-22 64-22 64-22 64-22 64-28 (1) Base Bitumen, (2) Anti stripping modified bitumen, (3)Polyphosphoric modified modified bitumen, (5) SBS modified bitumen, (6) Crumb rubber modified bitumen.
BCR6 44 >100 67 64-28 bitumen,
(4)
Sosobit
Each manufactured cylinder was sliced using a rotary diamond saw machine to obtain circular discs with thickness of 30 mm. Then each disc was splited along the diameter direction to obtain two semi-circular discs. A narrow artificial crack with length of 22.5 mm and width of 0.4 mm was then introduced at the middle of flat edge by a very thin rotary high speed diamond saw blade. Consequently, several semi-circular specimens containing vertical edge cracks were manufactured. In order to ensure the brittle fracture behavior of the manufactured asphalt samples, the cracked SCB specimens were maintained inside a freezer at -15oC for 8h before conducting the modes I and II fracture toughness experiments. Using a compression test machine the SCB specimens were tested under displacement control conditions with a constant cross head speed of 0.05 mm/s. The tests were carried out under. For mode I testing, the span of bottom loading fixture relative to the crack line was set at S1=S2=50 mm. But for pure mode II fracture toughness testing, the S1 and S2 values were considered as 50 and 9 mm, respectively. Fig. 7a and 7b shows the test setup used for the SSCB (pure mode I) and ASCB (pure mode II) fracture toughness tests.
(a)
(b)
(d) (c)
Fig. 7: Test setup used for (a) the SSCB (pure mode I) and (b) ASCB (pure mode II) fracture toughness tests; and typical fracture trajectories observed of (c) mode I and (d) mode II specimens. By increasing the applied monotonic load, failure of both SSCB and ASCB specimens was initiated suddenly from the tip of crack at a critical load level and then the path of fracture was extended towards the upper loading roller. While the mode I fracture path was straight along the crack line (and even passing through the aggregates), pure mode II fracture trajectory was propagated along a curvilinear path. Figs. 7c and 7d show typical SSCB and ASCB specimens fractured under pure modes I and II conditions, respectively. As stated in previous works of authors [1,27], depending on the position of wheels of moving vehicles relative to a top down crack initiated in the surface of pavement, different combinations of tensile and shear modes can be produced at the tip of top-down crack. The configuration of loading and the locations of bottom rollers in the proposed SSCB and ASCB specimens can simulate nearly the same condition that a top-down crack would experience in practice under both pure modes I and II loading conditions. The load-displacement curves of tested samples were linear for both modes I and II showing the validity of linear elastic fracture mechanics approach in determining low temperature fracture toughness of investigated HMA mixtures. For the tested SSCB and ASCB samples of this research (with a/R = 0.3, S1/R = 0.7, S2/R = 0.105), the corresponding geometry factors YI and YII were obtained as 4.01 and 1.805, respectively from Fig. 5. Hence, by replacing the critical peak load of each specimen (Pcr) obtained from the experiments into Eqs. (4) and (5) the corresponding KIc and KIIc values of tested HMA mixtures were determined from the following relations:
K Ic ( SSCB)
1.805 Pcr a 2R t
(6)
K IIc ( ASCB )
4.01 Pcr a 2R t
(7)
The specimens were labeled and designated as X-Y-Z in which X indicates the aggregate type (‘S’ for Siliceous and ‘L’ for limestone), Y refers to the air void content (3% and 7%) and Z indicates the additive type (‘B’ for neat binder with no additive, ‘A’ for AntiStripping agent, ‘Sa’ for Sasobit, ‘CR’ for crumb rubber, ‘P’ for Polyphosphoric acid and ‘Sb’ for Styrene Butadiene Styrene). Three replicates were also tested for each X-Y-Z code. The standard devotions obtained from the modes I and II fracture toughness experiments were varied from 0.25 to 0.40 demonstrating reasonably low scatter of KIc, KIIc results for the investigated HMA mixtures.
4. Results and discussions Fig. 8 compares the average of KIc and KIIc values obtained for the investigated modified and unmodified asphalt mixtures. Based on this figure, the whole modifiers investigated in this research enhance both mode I and mode II cracking resistance of control HMA mixture made of base (i.e. unmodified) bitumen. This finding is in agreement with the previous data reported for mode I fracture toughness of different HMA modified mixtures [56-60] that demonstrates the well-known and commonly used binder modifiers increase in general the low temperature performance of asphalt concretes against fracturing. Also, noticeable influence of HMA composition (i.e. type of binder, aggregate and air void content) on the tensile and shear fracture resistance of tested mixtures is obvious from Fig. 8. The mixture made of FT-Paraffin wax (Sasobit) and crumb rubber showed the highest resistance values against mode I fracture. The crumb rubber additive provided also excellent performance under pure shear (or mode II) load. In contrary, adding the anti-stripping agent and Polyphosphoric acid additives had no sound influence on KIc and KIIc values of control HMA material, respectively. Furthermore, the influence of air void
content on modes I and II fracture toughness data has been shown in Fig. 9. Based on this figure, by increasing the air void percentage from 3% to 7% both KIc and KIIc values are reduced for all tested mixtures. It is worth mentioning that based on Fig. 8, KIIc of each mixture is lower than the corresponding KIc value except for the mixture modified with crumb rubber in which its KIIc value is slightly greater than KIc. In other words, the risk of shear type cracking in most of the investigated HMA concretes is higher than the tensile type fracture. It is shown in the following section that the experimental mode II data of each mixture can be predicted theoretically in terms of its KIc value. 3% air void content
7% void content 1.3
1.5
KIc KIIc
1.2
0.5
0.5
Fracture toughness (MPa m )
Fracture toughness (MPa m )
KIc KIIc
1.4
1.3
1.2
1.1
1.1
1.0
0.9
1.0
0.9
0.8
S-3-B
L-3-B
L-3-A
L-3-Sa L-3-Cr
L-3-P
S-7-B
L-3-Sb
L-7-B
L-7-A
L-7-Sa
L-7-Cr
L-7-P
L-7-Sb
Fig. 8: Comparison of pure modes I and II fracture toughness results (KIc and KIIc) obtained for different HMA mixtures using the SSCB and ASCB specimens. 1.4
1.3
1.5
3% air void 7% air void
1.4
3% air voids 7% air voids
0.5
K IIc (MPa m )
0.5
K Ic (MPa m )
1.3
1.2
1.1
1.2
1.1
1.0
1.0 0.9
0.9
0.8
S-Y-B
L-Y-B
L-Y-A
L-Y-Sa
L-Y-Cr
L-Y-P
L-Y-Sb
S-Y-B
L-Y-B
L-Y-A
L-Y-Sa
L-Y-Cr
L-Y-P
L-Y-Sb
Fig. 9: The influence of air void content on KIc and KIIc values of investigated HMA mixtures. As stated earlier, there are some theoretical fracture theories for predicting the onset of fracture growth under different combinations of tensile-shear loads [e.g. 29-33]. It has been demonstrated that by accurate and precise description of crack tip stress/strain field, reasonably good theoretical predictions can be obtained for the mixed mode I/II fracture behavior of brittle and quasi-brittle materials including rocks, ceramics, concrete, brittle polymers and etc. [33,34,37,55,61-67]. However at the best knowledge of authors, the practical ability of theoretical fracture criteria has rarely been investigated in the past for predicting the asphalt fracture toughness data. Here, a conventional fracture theory namely the minimum strain energy density (MSED) criterion is described briefly and it will be shown that this criterion can be used successfully for predicting mode II brittle fracture toughness of asphalts mixtures obtained from the ASCB specimens. Consider a volume element dV at a distance r from the crack tip. In the vicinity of crack tip and for the general state of plane strain condition, the strain energy function dW/dV can be written as [30]:
dW 1 2 2 1 K I 2 2 K I K II 3 K II dV r
(8)
where the coefficients α1, α2 and α3 are: 1 (3 4 cos )(1 cos ) 16G 2 2 sin cos (1 2 ) 16G 1 4(1 )(1 cos ) (1 cos )(3 cos 1) 3 16G
1
(9) (10) (11)
where ν is the Poisson’s ratio, G is the shear modulus and is the tangential coordinate measured from the crack line. The intensity of strain energy function dW/dV is shown here by S and is called strain energy factor. This parameter is written as:
S ( K I , K II , , , G) K I 2 2 K I K II 3 K II 2
2
)12(
According to the minimum strain energy density (MSED) criterion, the brittle fracture initiates from the crack tip in a radial direction 0 along which the strain energy factor is minimum. This can be shown mathematically as follows:
S 2S 0 , 0 along o
)13(
Once the angle of fracture initiation 0 is determined from Eq. (8), the critical load of mode II brittle fracture can be predicted using the second hypothesis of the minimum strain energy density criterion. This hypothesis suggests that the onset of pure mode II fracture takes place when the strain energy density factor (i.e. S) along the angle 0 reaches a critical value Sc:
S ( K I , K II , , , G) Sc
(14)
Thus: Sc
1 8G
K Ιc2
(15)
where 3 4 for plane strain and (3 ) /(1 ) for plane stress conditions. Consequently, based on the MSED criterion pure mode II fracture toughness over pure mode I fracture toughness ratio (i.e. KIIc/KIc) can be determined theoretically from:
K IIc 8 G 3 K Ic 1
(16)
According to Eq. 16 KIIc/KIc depends on the Poisson’s ratio of material. The value of ν for the tested asphalt mixtures was determined experimentally in the range of 0.25 and 0.4. Fig. 10 shows the KIIc/KIc results obtained for the tested modified and unmodified HMA mixtures. Shown in this figure is also the prediction of Eq. 16 (i.e. the MSED criterion). It is seen that the experimental results lie in the theoretical bound that is predicted by the MSED criterion. Thus, it can be concluded that the MSED criterion provides good
estimations for the fracture toughness ratio (KIIc/KIc) of asphalt mixtures under low temperature conditions.
1.2
3% air void content 7% air void content MSED prediction (Poisson's ratio = 0.40) MSED prediction (Poisson's ratio = 0.25)
K IIc / K Ic
1.1
1.0
0.9
0.8 S-Y-B
L-Y-B
L-Y-A
L-Y-Sa
L-Y-CR
L-Y-P
L-Y-Sb
Asphalt mixture type
Fig. 10: Predictions of the MSED criterion for fracture toughness ratio (KIIc/KIc) of tested HMA mixtures. Indeed, this criterion provides a suitable framework for predicting the mode II fracture toughness of tested asphalt mixtures in terms of their KIc. As stated earlier, despite comprehensive mode I fracture toughness data that are available for various asphalt concretes, only limited KIIc data have been reported in the literature for such materials. Thus as an engineering estimation of HMA resistance against pure shear type deformations, one can employ theoretical relation of MSED criterion presented by Eq.16. Accordingly, the KIIc value of asphalt concrete can be estimated by knowing two commonly available properties of HMA mixtures (i.e. Poisson’s ratio and mode I fracture toughness value). Based on the findings of this research, KIIc value of HMA mixtures tested by the ASCB specimen is typically about 0.9 -1.05 times the corresponding value of KIc. Furthermore, it is seen from Fig. 10 that the fracture toughness ratio (KIIc/KIc) reduces by decreasing the air void content. Among the investigated asphalt concretes, the
fracture toughness ratio of mixture modified with the anti-striping agent was more sensitive to the change of air void content. This reveals that when an asphalt pavement is compacted during its service life (by passing the vehicles), its load bearing capacity against shear type cracking might be reduced.
4. Conclusions 1- A series of pure mode I and pure mode II fracture toughness data (KIc and KIIc) were obtained experimentally at low temperature conditions using the SSCB and ASCB specimens for different modified and unmodified HMA mixtures. 2- Adding crumb rubber, SBS, sasobit, PPA and anti-striping modifiers enhanced in general both KIc and KIIc values of tested mixtures relative to the unmodified and control asphalt concrete. 3- The KIIc values of the whole modified HMA mixtures (except the mixture modified with the crumb rubber) were slightly lower than the corresponding value of KIc, showing the higher risk of HMA cracking against shear type loads with respect to the tensile loads at low temperature conditions. 4- Fracture toughness ratio (KIIc/KIc) of investigated HMA mixtures was decreased by reducing the air void content of mixture. In other words, by aging and compacting the pavement during its service life, the probability of mode II crack growth in the overlay becomes more. 5- The KIIc values of modified and unmodified asphalt mixtures were predicted theoretically by means of minimum strain energy density (MSED) criterion, and in terms of two well–known and common available properties of HMA mixture (i.e. KIC and ν). 6- The mixtures made of crumb rubber showed the highest resistance values against
both modes I and II loading conditions. However, adding the anti-stripping agent and Polyphosphoric acid had no sound influence on KIc and KIIc values of control HMA material, respectively.
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[53] Ayatollahi MR, Aliha MRM (2006) On determination of mode II fracture toughness using semi-circular bend specimen. Int J Solid Struct, 43: 5217-5227. [54] Aliha MRM, Ayatollahi MR (2014) Rock fracture toughness study using cracked chevron notched Brazilian disc specimen under pure modes I and II loading–A statistical approach. Theoretical Applied Fract Mech 69:17-25. [55] Ayatollahi MR, Aliha MRM, Saghafi H (2011). An improved semi-circular bend specimen for investigating mixed mode brittle fracture. Eng Fract Mech, 78:110-123. [56] Kim KW, Kweon SJ, Doh YS, Park TS (2003) Fracture toughness of polymermodified asphalt concrete at low temperatures. Canadian J Civil Eng 30: 406-413. [57] Kim KW, El Hussein M (1997) Variation of fracture toughness of asphalt concrete under low temperatures. Const build Mater, 11:403-411. [58] Behnia BE, Dave V, Ahmed S, Buttlar WG, Reis H (2011) Effects of recycled asphalt pavement amounts on low-temperature cracking performance of asphalt mixtures using acoustic emissions. Transp Res Rec 2208:64–71. [59] Mubaraki MA, Abd-Elhady AA, Sallam HEM (2013) Mixed mode fracture toughness of recycled tire rubber-filled concrete for airfield rigid pavements. Int J Pave Res Tech 6:8–14. [60] Zegeye, K. Moon, M. Turos, T. Clyne, M.O. Marasteanu, Low temperature fracture properties of polyphosphoric acid modified asphalt mixtures, J. Mater. Civ. Eng. 24 (2012) 1089–1096.
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Highlights -
Modes I and II fracture toughness values were determined for different HMA mixtures using SSCB and ASCB specimens The fracture toughness ratio KIIc/KIc was predicted for the tested HMA using the SED criterion. Low temperature mode II fracture toughness of asphalt mixtures can be estimated in terms of KIc.
S0167-8442(17)30481-0 https://doi.org/10.1016/j.tafmec.2018.11.001 TAFMEC 2123
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
19 October 2017 7 September 2018 2 November 2018
Please cite this article as: M. Aliha, On Predicting mode II fracture toughness (K IIc) of hot mix asphalt mixtures using the strain energy density criterion, Theoretical and Applied Fracture Mechanics (2018), doi: https://doi.org/ 10.1016/j.tafmec.2018.11.001
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On Predicting mode II fracture toughness (KIIc) of hot mix asphalt mixtures using the strain energy density criterion 1
MRM Aliha Welding and Joining research center, School of Industrial Engineering, Iran University of Science and Technology (IUST), Narmak, 16846-13114, Tehran, Iran. [email protected]
Abstract Mode II or in-plane sliding of crack faces is one of the possible fracture modes for asphalt pavements which can often take place due to traffic loads passing over the crack at low temperature conditions. Hence pavement engineers need to know the value of mode II fracture toughness (KIIc) of asphalt mixtures as an important design parameter. However, despite a large number of mode I fracture toughness (KIc) values available in the literature for different asphalt concrete materials, only limited KIIc values have been reported for such construction materials. In this paper to fill this research gap, a number of pure mode I and pure mode II fracture toughness experiments are conducted at low temperature using symmetric and asymmetric semi-circular bend specimens made of different asphalt mixtures. It was found that the mode II fracture toughness of each mixture is approximately 0.9 to 1.1 times the corresponding value of KIc. It was demonstrated that the experimentally obtained fracture toughness ratio (KIIc/KIc) for the investigated asphalt concretes are in good agreement with theoretical predictions of the minimum strain energy density criterion. Consequently, as an engineering application of this research, mode II fracture toughness of investigated asphalt mixtures was estimated in terms of their more common mechanical parameters such as KIc and Poisson’s ratio values. Keywords: Hot mix asphalt, Additive type, Mode II fracture toughness, Asymmetric semi-circular bend specimen, Minimum strain energy density criterion, KIIc/KIc prediction. 1. Introduction Brittle fracture of asphalt concretes (which usually occurs at low temperatures) is one of the main deterioration and failure modes of pavements in many countries having a cold climate. Due to high rehabilitation cost of roads, it is necessary to understand the mechanism of crack growth in these materials for developing a suitable maintenance program for the network of overlays. Since hot mix asphalt (HMA) concretes are complex materials and are often subjected to diverse environmental and mechanical loads (e.g. induced by traffic or temperature changes), different types of cracks can be initiated and then propagated in the pavements. A number of research studies in this field [e.g. 14] have shown that a cracked asphalt pavement subjected to traffic loading of moving
vehicles experiences complex tensile-shear deformations. In particular, for many practical loading situations, top-down crack extension in the pavement takes place when the crack is subjected to pure or dominantly mode II loading conditions. While pure mode I deformation, refers to the opening of the crack faces, pure mode II deformation takes place for in-plane sliding of the crack faces without any opening. Accordingly, when the wheel of vehicle passes over the top-down crack (as shown schematically in Fig. 1), the crack would experience pure mode II deformation.
Top-down crack
Fig. 1: Top-down crack subjected to pure or dominantly mode II deformation.
Under shear loads, the crack propagates if the mode II stress intensity factor (KII), reaches a critical value that is known as mode II fracture toughness, KIIc. Therefore, it is important to develop appropriate experimental or theoretical methods for determining and predicting mode II fracture toughness KIIc in HMA materials. However, review of published papers indicates that most of the previous research works have only focused on pure mode I fracture toughness (KIc) determination of asphalt mixtures. For example, rectangular edge cracked beam subjected to three or four-point bending [5], disc shape compact-tension (DCT) specimen [6], modified indirect diametral tension (IDT) specimen [7], edge notch disc bend (ENDB) specimen [8-12] and semi-circular bend
(SCB) specimen [13-23] have been used extensively for obtaining KIc of asphalt concrete materials. In any of the above mentioned specimens, the effects of different parameters such as the specification of asphalt mixture (i.e. binder and aggregate type), environmental conditions (e.g. test temperature, loading rate, freeze/thaw cycles, additive type and etc.), have been investigated on the tensile type mode I cracking characteristics of asphalt materials. Recently, the DCT and SCB specimens have been suggested by the American Association of State Highway and Transportation Officials (AASHTO) and American Society for Testing and Materials (ASTM) as standard test configurations for determining a versatile and valid KIc value for the asphalt mixtures [6,24]. But very few test specimens have been used and investigated in the past for obtaining KIIc of asphalt materials and thus no standard testing methods is available for determining mode II fracture toughness of these materials. Double edge cracked four-point bend specimen subjected to lateral confining pressure [25], inclined edge cracked SCB specimen [26] and the SCB specimen containing a vertical edge crack and subjected to asymmetric three-point bending [15,27] are among very limited test configurations employed for mode II fracture toughness study of asphalt concretes. Hence, in comparison with the rich set of pure mode I asphalt fracture toughness (KIC) data reported by a large number of researchers, there is very little experimental KIIc data in the literature for asphalt mixtures. Therefore as the first aim of this research, mode II fracture toughness of different asphalt mixtures with different characteristics and ingredients is obtained experimentally using the asymmetric SCB (ASCB) configuration that has been proposed earlier by the authors and coworkers for testing mode II fracture of other brittle materials [28]. These data can help asphalt fracture researchers and pavement designers in better understanding the fracture behavior of asphalt concretes and provide typical fracture resistance values against shear type (or mode II) crack growth for different HMA mixtures.
On the other hand, the experimental fracture toughness data obtained from different engineering materials are usually predicted by means of theoretical fracture criteria. For example, pure mode II or mixed mode I/II fracture toughness behavior of brittle polymers, metals, rocks, concretes, ceramics and etc. have been already predicted by means of stress/strain or energy based criteria [29-45]. However, this issue has been rarely investigated for asphalt mixtures till now. Thus as the second aim of this paper and to fill this research gap, the applicability of a conventional fracture theory called the minimum strain energy density (MSED) criterion is examined for predicting the obtained mode II asphalt fracture toughness data. It is shown that the KIIc value of investigated asphalt mixtures can be predicted very well in terms of their mode I fracture toughness (i.e. KIc). 2. Mode I and mode II fracture toughness test specimen Fig. 2 shows schematic representation of test specimens used for pure mode I and pure mode II asphalt fracture toughness testing of this research that are semi-circular specimens of radius R and thickness t. A vertical edge crack of length a is emanated from the middle of bottom flat edge of semi-circle and both modes I and II fracture specimens are subjected to three-point bend loading. When the bottom loading supports are symmetric relative to the crack plane (i.e. S1= S2), the symmetric semi-circular bend (SSCB) specimen is subjected to pure mode I (or pure opening mode) loading. But for the asymmetric spans (i.e. moving the right hand support S2 towards the crack) the asymmetric semi-circular bend (ASCB) specimen would be subjected to different mixed mode I/II (i.e. tensile-shear) loading conditions. The contribution of modes I and II deformations is controlled in this specimen by changing the S2 value. Consequently, for any given crack length ratio (a/R), a special S2 value can be found in which the ASCB specimen experiences pure shear deformation or pure mode II condition (in which KI=0
but KII≠0). Thus with a same vertically edge cracked SCB specimen, both KIc and KIIc experiments can be conducted simply by setting S1 and S2 values at suitable positions. From practical view point, this configuration has some other advantages for being used in asphalt fracture toughness studies. simple geometry of specimen, convenience of sample preparation from gyratory compactor machines or field coring devices, ease of introducing an edge crack, simple loading and testing apparatus either for pure mode I or pure mode II (i.e. SSCB and ASCB configurations) and ability of determining KIIc in addition to KIc, are to name a few. Hence, the SSCB and ASCB specimens have been used earlier by several researchers for modes I and II fracture toughness study of asphalt mixtures and other types of quasi-brittle materials, like rocks and cement concretes [14,20,47-51].
(a)
(b)
Fig. 2: Schematic representation of (a) symmetric semi-circular bend (SSCB) and (b) asymmetric semi-circular bend (ASCB) specimens for conducting pure mode I and pure mode II fracture toughness experiments on HMA mixtures.
Based on the principals of fracture mechanics the crack tip stress/strain field in cracked components subjected to any arbitrary loading conditions is defined by a key parameter
called the stress intensity factor. This parameter describes the severity or singularity of stress/strain at crack tip outlined by William’s series expansion as [52]:
ij An r
n2 n
fij ( n ) ( )
n 1
(1)
where r, are the polar coordinates, An are constant factors and fij (n) are the angular functions of nth term and ij is the stress tensor. For example, the tangential stress component σθθ (as shown schematically in Fig. 3) is written as following equations for pure mode I and Pure mode II loading condition [53-55]:
KI
(@ pure mode I)
(@ pure mode II)
2r
cos 3
2
H .O.T
cos sin H .O.T 2 2 2r
(2)
3K II
(3)
in which KI and KII are the mode I and mode II stress intensity factors that are related to the singular terms of Williams’s infinite series expansion and “H.O.T” presents the remaining non-singular and higher order stress terms that are often ignored in comparison with the singular terms.
Fig. 3: Tangential stress component in the vicinity of crack tip under pure mode I and pure mode II conditions.
The mode I and mode II stress intensity factors (KI and KII) in the SSCB and ASCB specimens are functions of crack length over the radius ratio (a/R), bottom support distances over the radius ratios (S1/R and S2/R) and the applied load (P) and can be written as:
K I ( SSCB)
K II ( ASCB )
P
a YI ( SSCB)
(4)
P a YII ( ASCB ) 2R t
(5)
2R t
where YI(SSCB) and YII (ASCB) are the geometry factors corresponding to the pure modes I and II in the semi-circular bend specimen, respectively. Ayatollahi et al. [55] computed numerically these geometry factors under mixed mode I/II loading condition but only for limited values of a/R, S1/R and S2/R. Using the same procedure described in Ref. [55], we performed here a number of finite element analyses to obtain YI (SSCB) and YII (ASCB) for a wider range of a/R, S1/R, and S2/R ratios. Fig. 4 shows the finite element model of SSCB and ASCB specimens created in the ABAQUS code and Fig. 5 presents the corresponding values of pure mode I and pure mode II geometry factors (i.e. YI (SSCB) and YII (ASCB)), for different crack lengths and bottom loading spans. In the following sections, fracture toughness of some asphalt mixtures are determined experimentally using the SSCB and ASCB specimens.
SSCB model (pure mode I)
ASCB model (pure mode II)
Fig. 4: Finite element model of SSCB and ASCB specimens created in the ABAQUS software.
Fig. 5: Variation of YI (SSCB) and YII (ASCB) for different geometry and loading conditions of SSCB and ASCB specimens.
3. Asphalt fracture toughness experiments In the experimental part of this research, a number of hot mix asphalt concretes with different aggregates and binder modifiers were manufactured. For preparation of asphalt
mixtures, two aggregate types (i.e. limestone and siliceous) with nominal maximum aggregate size (NMAS) of 12.5 mm (which is known as aggregate size No. 4 according to Iranian paving standard code-234) and a binder with performance grade of (PG 64-22) was used. This bitumen is the most frequently used binder for paving the roads and highways in Iran that is a moderately soft binder with penetration grade of 60/70. Fig. 6 and Table 1 present the gradation of aggregate No. 4 and the specifications of 60/70 binder. Moreover, a number of frequently used additives (including Styrene Butadiene Styrene (SBS), Polyphosphoric Acid (PPA), Crumb Rubber (CR), FT-Paraffin Wax (Sasobit) and Anti stripping agent) were also used in the composition of manufactured HMA materials. Details of mixing procedure of the additives with the base bitumen and physical specifications of the modified binders have been summarized in Tables 2 and 3, respectively. In order to manufacture the HMA mixtures, the optimum percentage of binder for each modified and unmodified mixtures was determined using the Marshall Mix design method. For manufacturing the HMA mixtures, binders modified with the crumb rubber and SBS were mixed with aggregate at 160oC but the rest of modified binders (i.e. Sasobit, PPA, Anti stripping) were mixed at 135oC. Then several cylindrical specimens with diameter of 150 mm and height of 130 mm were manufactured using superpave gyratory compactor machine (SGC). Two gyratory rotations (i.e. 90 and 30 rotations) were chosen for compacting the HMA mixtures. Consequently, the air void content of manufactured HMA samples were approximately equal to 3 and 7%, respectively.
Fig. 6: Gradation of aggregate used for preparation of HMA.
Table 1. Specifications of 60/70 binder with PG of (64-22). Test
Standard
Specific Gravity (25 °C) Flash Point (Cleveland) Penetration (25 °C) Ductility (25 °C) Softening point Kinematic Viscosity @ 120 °C Kinematic Viscosity @ 135 °C Kinematic Viscosity @ 150 °C Penetration index (PI)a
ASTM D70 ASTM D92 ASTM D5 ASTM D113 ASTM D36 ASTM D2170 ASTM D2170 ASTM D2170 –
Unit
value 3
gr/cm °C °C cm °C mm2/s mm2/s mm2/s –
1.03 308 62 100 49 810 420 232 -1.12
Table 2. Weight percentage and mixing method of different additives with the base binder for manufacturing the HMA samples of this research. Type of Binder BCR BSb BSb BP BA
Additive type Crumb rubber SBS Sasobit PPA Anti Stripping
Weight percentage of Binder (%) 15 5 2.5 1 0.4
High shear (rpm) 6000 6000 -
Mixing temperat ure (⁰C) 170 170 -
Mix Method Mixing low shear time (rpm) (min) 90 1000 90 1000 300 1000 1000
Mixing temperat ure (⁰C) 130 130 130 160 150
Mixing time (min) 120 120 10 30 30
Table 3. Physical properties of binders used for HMA manufacturing. Test B1 BA2 BP3 BSa4 BSb5 Penetration (25 °C) (°C) 62 56 48 45 43 Ductility (25 °C) ( cm) >100 >100 >100 >100 >100 Softening point (°C) 49 53 61 63 66 Performance Grade 64-22 64-22 64-22 64-22 64-28 (1) Base Bitumen, (2) Anti stripping modified bitumen, (3)Polyphosphoric modified modified bitumen, (5) SBS modified bitumen, (6) Crumb rubber modified bitumen.
BCR6 44 >100 67 64-28 bitumen,
(4)
Sosobit
Each manufactured cylinder was sliced using a rotary diamond saw machine to obtain circular discs with thickness of 30 mm. Then each disc was splited along the diameter direction to obtain two semi-circular discs. A narrow artificial crack with length of 22.5 mm and width of 0.4 mm was then introduced at the middle of flat edge by a very thin rotary high speed diamond saw blade. Consequently, several semi-circular specimens containing vertical edge cracks were manufactured. In order to ensure the brittle fracture behavior of the manufactured asphalt samples, the cracked SCB specimens were maintained inside a freezer at -15oC for 8h before conducting the modes I and II fracture toughness experiments. Using a compression test machine the SCB specimens were tested under displacement control conditions with a constant cross head speed of 0.05 mm/s. The tests were carried out under. For mode I testing, the span of bottom loading fixture relative to the crack line was set at S1=S2=50 mm. But for pure mode II fracture toughness testing, the S1 and S2 values were considered as 50 and 9 mm, respectively. Fig. 7a and 7b shows the test setup used for the SSCB (pure mode I) and ASCB (pure mode II) fracture toughness tests.
(a)
(b)
(d) (c)
Fig. 7: Test setup used for (a) the SSCB (pure mode I) and (b) ASCB (pure mode II) fracture toughness tests; and typical fracture trajectories observed of (c) mode I and (d) mode II specimens. By increasing the applied monotonic load, failure of both SSCB and ASCB specimens was initiated suddenly from the tip of crack at a critical load level and then the path of fracture was extended towards the upper loading roller. While the mode I fracture path was straight along the crack line (and even passing through the aggregates), pure mode II fracture trajectory was propagated along a curvilinear path. Figs. 7c and 7d show typical SSCB and ASCB specimens fractured under pure modes I and II conditions, respectively. As stated in previous works of authors [1,27], depending on the position of wheels of moving vehicles relative to a top down crack initiated in the surface of pavement, different combinations of tensile and shear modes can be produced at the tip of top-down crack. The configuration of loading and the locations of bottom rollers in the proposed SSCB and ASCB specimens can simulate nearly the same condition that a top-down crack would experience in practice under both pure modes I and II loading conditions. The load-displacement curves of tested samples were linear for both modes I and II showing the validity of linear elastic fracture mechanics approach in determining low temperature fracture toughness of investigated HMA mixtures. For the tested SSCB and ASCB samples of this research (with a/R = 0.3, S1/R = 0.7, S2/R = 0.105), the corresponding geometry factors YI and YII were obtained as 4.01 and 1.805, respectively from Fig. 5. Hence, by replacing the critical peak load of each specimen (Pcr) obtained from the experiments into Eqs. (4) and (5) the corresponding KIc and KIIc values of tested HMA mixtures were determined from the following relations:
K Ic ( SSCB)
1.805 Pcr a 2R t
(6)
K IIc ( ASCB )
4.01 Pcr a 2R t
(7)
The specimens were labeled and designated as X-Y-Z in which X indicates the aggregate type (‘S’ for Siliceous and ‘L’ for limestone), Y refers to the air void content (3% and 7%) and Z indicates the additive type (‘B’ for neat binder with no additive, ‘A’ for AntiStripping agent, ‘Sa’ for Sasobit, ‘CR’ for crumb rubber, ‘P’ for Polyphosphoric acid and ‘Sb’ for Styrene Butadiene Styrene). Three replicates were also tested for each X-Y-Z code. The standard devotions obtained from the modes I and II fracture toughness experiments were varied from 0.25 to 0.40 demonstrating reasonably low scatter of KIc, KIIc results for the investigated HMA mixtures.
4. Results and discussions Fig. 8 compares the average of KIc and KIIc values obtained for the investigated modified and unmodified asphalt mixtures. Based on this figure, the whole modifiers investigated in this research enhance both mode I and mode II cracking resistance of control HMA mixture made of base (i.e. unmodified) bitumen. This finding is in agreement with the previous data reported for mode I fracture toughness of different HMA modified mixtures [56-60] that demonstrates the well-known and commonly used binder modifiers increase in general the low temperature performance of asphalt concretes against fracturing. Also, noticeable influence of HMA composition (i.e. type of binder, aggregate and air void content) on the tensile and shear fracture resistance of tested mixtures is obvious from Fig. 8. The mixture made of FT-Paraffin wax (Sasobit) and crumb rubber showed the highest resistance values against mode I fracture. The crumb rubber additive provided also excellent performance under pure shear (or mode II) load. In contrary, adding the anti-stripping agent and Polyphosphoric acid additives had no sound influence on KIc and KIIc values of control HMA material, respectively. Furthermore, the influence of air void
content on modes I and II fracture toughness data has been shown in Fig. 9. Based on this figure, by increasing the air void percentage from 3% to 7% both KIc and KIIc values are reduced for all tested mixtures. It is worth mentioning that based on Fig. 8, KIIc of each mixture is lower than the corresponding KIc value except for the mixture modified with crumb rubber in which its KIIc value is slightly greater than KIc. In other words, the risk of shear type cracking in most of the investigated HMA concretes is higher than the tensile type fracture. It is shown in the following section that the experimental mode II data of each mixture can be predicted theoretically in terms of its KIc value. 3% air void content
7% void content 1.3
1.5
KIc KIIc
1.2
0.5
0.5
Fracture toughness (MPa m )
Fracture toughness (MPa m )
KIc KIIc
1.4
1.3
1.2
1.1
1.1
1.0
0.9
1.0
0.9
0.8
S-3-B
L-3-B
L-3-A
L-3-Sa L-3-Cr
L-3-P
S-7-B
L-3-Sb
L-7-B
L-7-A
L-7-Sa
L-7-Cr
L-7-P
L-7-Sb
Fig. 8: Comparison of pure modes I and II fracture toughness results (KIc and KIIc) obtained for different HMA mixtures using the SSCB and ASCB specimens. 1.4
1.3
1.5
3% air void 7% air void
1.4
3% air voids 7% air voids
0.5
K IIc (MPa m )
0.5
K Ic (MPa m )
1.3
1.2
1.1
1.2
1.1
1.0
1.0 0.9
0.9
0.8
S-Y-B
L-Y-B
L-Y-A
L-Y-Sa
L-Y-Cr
L-Y-P
L-Y-Sb
S-Y-B
L-Y-B
L-Y-A
L-Y-Sa
L-Y-Cr
L-Y-P
L-Y-Sb
Fig. 9: The influence of air void content on KIc and KIIc values of investigated HMA mixtures. As stated earlier, there are some theoretical fracture theories for predicting the onset of fracture growth under different combinations of tensile-shear loads [e.g. 29-33]. It has been demonstrated that by accurate and precise description of crack tip stress/strain field, reasonably good theoretical predictions can be obtained for the mixed mode I/II fracture behavior of brittle and quasi-brittle materials including rocks, ceramics, concrete, brittle polymers and etc. [33,34,37,55,61-67]. However at the best knowledge of authors, the practical ability of theoretical fracture criteria has rarely been investigated in the past for predicting the asphalt fracture toughness data. Here, a conventional fracture theory namely the minimum strain energy density (MSED) criterion is described briefly and it will be shown that this criterion can be used successfully for predicting mode II brittle fracture toughness of asphalts mixtures obtained from the ASCB specimens. Consider a volume element dV at a distance r from the crack tip. In the vicinity of crack tip and for the general state of plane strain condition, the strain energy function dW/dV can be written as [30]:
dW 1 2 2 1 K I 2 2 K I K II 3 K II dV r
(8)
where the coefficients α1, α2 and α3 are: 1 (3 4 cos )(1 cos ) 16G 2 2 sin cos (1 2 ) 16G 1 4(1 )(1 cos ) (1 cos )(3 cos 1) 3 16G
1
(9) (10) (11)
where ν is the Poisson’s ratio, G is the shear modulus and is the tangential coordinate measured from the crack line. The intensity of strain energy function dW/dV is shown here by S and is called strain energy factor. This parameter is written as:
S ( K I , K II , , , G) K I 2 2 K I K II 3 K II 2
2
)12(
According to the minimum strain energy density (MSED) criterion, the brittle fracture initiates from the crack tip in a radial direction 0 along which the strain energy factor is minimum. This can be shown mathematically as follows:
S 2S 0 , 0 along o
)13(
Once the angle of fracture initiation 0 is determined from Eq. (8), the critical load of mode II brittle fracture can be predicted using the second hypothesis of the minimum strain energy density criterion. This hypothesis suggests that the onset of pure mode II fracture takes place when the strain energy density factor (i.e. S) along the angle 0 reaches a critical value Sc:
S ( K I , K II , , , G) Sc
(14)
Thus: Sc
1 8G
K Ιc2
(15)
where 3 4 for plane strain and (3 ) /(1 ) for plane stress conditions. Consequently, based on the MSED criterion pure mode II fracture toughness over pure mode I fracture toughness ratio (i.e. KIIc/KIc) can be determined theoretically from:
K IIc 8 G 3 K Ic 1
(16)
According to Eq. 16 KIIc/KIc depends on the Poisson’s ratio of material. The value of ν for the tested asphalt mixtures was determined experimentally in the range of 0.25 and 0.4. Fig. 10 shows the KIIc/KIc results obtained for the tested modified and unmodified HMA mixtures. Shown in this figure is also the prediction of Eq. 16 (i.e. the MSED criterion). It is seen that the experimental results lie in the theoretical bound that is predicted by the MSED criterion. Thus, it can be concluded that the MSED criterion provides good
estimations for the fracture toughness ratio (KIIc/KIc) of asphalt mixtures under low temperature conditions.
1.2
3% air void content 7% air void content MSED prediction (Poisson's ratio = 0.40) MSED prediction (Poisson's ratio = 0.25)
K IIc / K Ic
1.1
1.0
0.9
0.8 S-Y-B
L-Y-B
L-Y-A
L-Y-Sa
L-Y-CR
L-Y-P
L-Y-Sb
Asphalt mixture type
Fig. 10: Predictions of the MSED criterion for fracture toughness ratio (KIIc/KIc) of tested HMA mixtures. Indeed, this criterion provides a suitable framework for predicting the mode II fracture toughness of tested asphalt mixtures in terms of their KIc. As stated earlier, despite comprehensive mode I fracture toughness data that are available for various asphalt concretes, only limited KIIc data have been reported in the literature for such materials. Thus as an engineering estimation of HMA resistance against pure shear type deformations, one can employ theoretical relation of MSED criterion presented by Eq.16. Accordingly, the KIIc value of asphalt concrete can be estimated by knowing two commonly available properties of HMA mixtures (i.e. Poisson’s ratio and mode I fracture toughness value). Based on the findings of this research, KIIc value of HMA mixtures tested by the ASCB specimen is typically about 0.9 -1.05 times the corresponding value of KIc. Furthermore, it is seen from Fig. 10 that the fracture toughness ratio (KIIc/KIc) reduces by decreasing the air void content. Among the investigated asphalt concretes, the
fracture toughness ratio of mixture modified with the anti-striping agent was more sensitive to the change of air void content. This reveals that when an asphalt pavement is compacted during its service life (by passing the vehicles), its load bearing capacity against shear type cracking might be reduced.
4. Conclusions 1- A series of pure mode I and pure mode II fracture toughness data (KIc and KIIc) were obtained experimentally at low temperature conditions using the SSCB and ASCB specimens for different modified and unmodified HMA mixtures. 2- Adding crumb rubber, SBS, sasobit, PPA and anti-striping modifiers enhanced in general both KIc and KIIc values of tested mixtures relative to the unmodified and control asphalt concrete. 3- The KIIc values of the whole modified HMA mixtures (except the mixture modified with the crumb rubber) were slightly lower than the corresponding value of KIc, showing the higher risk of HMA cracking against shear type loads with respect to the tensile loads at low temperature conditions. 4- Fracture toughness ratio (KIIc/KIc) of investigated HMA mixtures was decreased by reducing the air void content of mixture. In other words, by aging and compacting the pavement during its service life, the probability of mode II crack growth in the overlay becomes more. 5- The KIIc values of modified and unmodified asphalt mixtures were predicted theoretically by means of minimum strain energy density (MSED) criterion, and in terms of two well–known and common available properties of HMA mixture (i.e. KIC and ν). 6- The mixtures made of crumb rubber showed the highest resistance values against
both modes I and II loading conditions. However, adding the anti-stripping agent and Polyphosphoric acid had no sound influence on KIc and KIIc values of control HMA material, respectively.
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Highlights -
Modes I and II fracture toughness values were determined for different HMA mixtures using SSCB and ASCB specimens The fracture toughness ratio KIIc/KIc was predicted for the tested HMA using the SED criterion. Low temperature mode II fracture toughness of asphalt mixtures can be estimated in terms of KIc.