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The efficiency of mechanical concrete comminution
Engineering Fracture Mechanics 70 (2003) 81–91 www.elsevier.com/locate/engfracmech
The efficiency of mechanical concrete comminution Andreas W. Momber
*,1
WOMA Apparatebau GmbH, P.O. Box 141820, D-47226 Duisburg, Germany Received 8 August 2001; received in revised form 22 January 2002; accepted 29 January 2002
Abstract The paper reports about the comminution of cementitious composites in a laboratory jaw breaker. Six different concrete types, two mixtures of hardened cement paste, and one mortar sample are investigated. The efficiency of the breaking process is calculated based on a non-linear fracture approach. The efficiency g is the ratio between net energy delivered by the crusher and the fracture energy dissipated in the materials during the fragmentation process. Typical efficiency values between 0:03 < g < 0:19 are estimated. However, efficiency increases as the fracture energy of the material increases and brittleness decreases. Jaw breakers are efficient for low-brittle materials. It is suggested to use the approach developed in the paper for estimating the efficiency of rock comminution processes. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction In the countries of the European community, about 2 billion tons of aggregates are consumed each year, mainly for concrete manufacturing [1]. For that reason, recycling and reuse of construction demolition waste is becoming a notable aspect in national economics and in environmental protection. Again in the countries of the European community, the annual amount of construction demolition waste is about 300 million tons. About 60% of this material (180 million tons) is being recycled currently [1]. In industry, recycled concrete will be used only if quality and price are comparable to those of newly manufactured aggregate materials. However, about 70% of the recycling costs is covered by the mechanical processing of the construction demolition waste, primarily by comminution and grinding. For that reason, knowledge about how demolition waste, in particular concrete, behaves in industrial fragmentation devices is an important contribution to a more efficient recycling of aggregate materials. Although the comminution and grinding of minerals, rock and ore is a research issue since decades, not much is known about the behavior of cementitious materials. A review about recent work is given in [2]. A major concern of these studies was the selective fragmentation of concrete: attempts were made to expose the aggregate in the concrete completely and to break the surrounding cement paste away. However, no
*
Tel.: +49-206-530-4380; fax: +49-206-530-4200. E-mail address: [email protected] (A.W. Momber). 1 Habilitation-fellow of the German Research Association, Bonn, Germany, with the Aachen University, Department of Mining, Metallurgy and Geosciences. 0013-7944/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 2 ) 0 0 0 2 0 - 6
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A.W. Momber / Engineering Fracture Mechanics 70 (2003) 81–91
Nomenclature A0 ASP BM Ci d dF dmax EEL EN EM GF LCH mM n PS R tE w / cM g qM r rC rT
grain sample surface specific grain sample surface brittleness regression parameters RRSB-distribution parameter fragment diameter maximum aggregate size elastic strain energy density crusher net energy material Young’s modulus fracture energy characteristic length specimen weight RRSB-distribution parameter measured crusher power sieve residue net exposure time displacement grain shape parameter specific surface energy comminution efficiency material density stress compressive strength tensile strength
investigations were performed in these studies using different materials or concrete mixtures. Neither was the efficiency of the comminution process investigated. It will be the objective of this paper to address this important problem.
2. Tension-softening materials 2.1. Fracture behaviour of tension-softening materials The materials discussed in this paper are characterised by a ‘fracture process zone’ (FPZ). The FPZ is a zone characterised by progressive softening, for which the stress decreases at increasing deformation. This is illustrated in Fig. 1. Assuming a general material, the failure zone consists of a linear range and a non-linear range. The non-linear range is formed by the FPZ and a non-softening zone (yielding, as known from ductile metals). Depending on the relative size of these zones to the structural size, three basic types of fracture behaviour can be distinguished. In the first fracture type, the non-linear range is small compared to the structural size, and the entire fracture process takes place in a small linear range at the crack tip. This type of fracture approximates materials that usually are called brittle, like PMMA, glass, brittle ceramics,
A.W. Momber / Engineering Fracture Mechanics 70 (2003) 81–91
83
Fig. 1. General structure of a fracture process zone in concrete.
and, to a certain amount, hardened cement paste. In the second fracture type, most of the non-linear range consists of elasto-plastic hardening or yielding. The actual softening zone (FPZ) is comparatively small compared with the zone occupied by hardening or yielding. Materials that are usually called ductile, such as many metals, especially if they are alloyed, fall into this category. The third fracture type, considered in this paper, is characterised by a comparatively large FPZ; yielding can be neglected. A typical FPZ for a brittle, tension-softening material is shown in Fig. 1. Typical parameters characterising the fracture behaviour of quasi-brittle materials are the crack tip opening displacement (CTOD, or w in Fig. 1), the fracture energy GF , and the shape of the softening curve B–C–D as illustrated in Fig. 1. For more brittle-behaving materials, the FPZ is short. The major mechanisms responsible for the formation of an FPZ are assumed to be grain bridging and crack-rim friction behind the actual crack tip and microcracking in front of the crack tip as illustrated in Fig. 1. Several of these features have been observed through an SEM-study on concrete debris formed in a mechanical jaw breaker [2]. The following phenomenological approaches for modelling the non-linear behaviour of brittle materials are known: 1. 2. 3. 4.
The The The The
fictitious crack model [3]. crack-band model [4]. two-parameter model [5]. effective crack model [6].
Detailed information about these models is available in the original papers. In this paper, the fictitious crack model will be considered.
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A.W. Momber / Engineering Fracture Mechanics 70 (2003) 81–91
2.2. Fracture energy of cementitious materials The fracture energy, GF , is usually considered to be the energy absorbed in the FPZ during fracture. In the fictitious crack model [3], it is given by the area enveloped by the load-displacement-curve shown in Fig. 1: Z w1 rðwÞ dw: ð1Þ GF ¼ 0
Methods for the experimental estimation of the fracture energy are recommended in [7]. For concrete materials, fracture energy mainly depends on compressive strength and maximum aggregate size. Some relations are discussed, among others, in [8]. According to the CEB-FIP Model Code [9], the fracture energy (given in J/m2 ) for concrete can be approximated by the following equation: GF ¼ aðdmax Þr0:7 C :
ð2Þ
Here, rC is the compressive strength in MPa, and dmax is the maximum aggregate diameter in mm. The empirical coefficient a is tabulated in Table 1. A regression delivers the relationship: a ¼ 0:25dmax þ 2:
ð3Þ
These values are used along with Eq. (2) to calculate the fracture energies of the mixtures used in this study. The results of these calculations are listed in Table 4.
3. Estimation of the efficiency The efficiency of mechanical size reduction devices can be estimated as follows [11]: g¼
c M A0 : EN
ð4Þ
Here, cM is the thermodynamic specific surface energy required to create a defined new surface in the material, A0 is the new surface generated during the size reduction process, and EN is the measured net energy of the breaker. From this equation, the efficiency for minerals is always as low as g < 0:01 [12]. However, Eq. (4) is valid only for perfectly brittle (elastically) behaving materials. Concrete, however, dissipates additional energy GF required to separate the fracture surfaces. This is valid also for the comminution in mechanical breakers [2]. Assuming cM GF , Eq. (4) reads: g¼
GF A0 : EN
ð5Þ
The values for EN are estimated from wattmeter readings (see Section 4) and listed in Table 4. It is found in a previous study [2] that the fragments of cementitious materials formed in a mechanical jaw breaker follow a Rosin–Rammler–Sperling–Bennett-distribution (RRSB-distribution). Interestingly, this distribution also applies to debris formed during the uncontrolled post-peak failure in a force-controlled compressive tests [13]. The RRSB-distribution has the following form: Table 1 Typical values for a [10] dmax (mm)
a
8 16 32
4 6 10
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85
Table 2 Parameters of the RRSB-distributions [2] Mixture
n
d (lm)
Corr2 (RRSB)
SD (RRSB)
1 2 3 4 5 6 7 8
1.14 1.24 1.11 1.67 1.05 1.09 1.05 1.12
807 820 822 863 778 872 960 870
0.997 0.997 0.997 0.972 0.994 0.995 0.992 0.997
0.02 0.02 0.02 0.51 0.04 0.03 0.05 0.02
n R dF ¼ exp : 100 d
ð6Þ
In the equation, dF is the fragment size. The correlation between Eq. (6) and the experimental results is always Corr2 > 0:97 for the materials used in this study (see Table 2). The distribution parameter n can be calculated from the slope of a LogLog–Log-plot of Eq. (6). The size parameter d is defined as the fragment diameter at R ¼ 36:8%. More details about the RRSB-distribution are provided, among others, in [14,15]. All RRSB-parameters of the analysed fragments are listed in Table 2. The specific surface of an RRSBdistributed sample can be estimated using an approximation developed in [16]: ASP ¼
6:39 expð1:795=n2 Þ: d qM
ð7Þ
In the equation, ASP is the specific surface given in m2 /kg. Thus, the absolute surface can be calculated as follows: A0 ¼ ASP mM :
ð8Þ
Here, mM is given in kg. However, as shown in Fig. 2 the individual debris are not of an ideally spherical shape as assumed for Eq. (7). Therefore, a shape factor / is included in order to estimate the real surface: A0 ¼ /A0 :
ð9Þ
The following empirical values were used according to [17]: / ¼ 1:1 for mortar, / ¼ 1:4 for concrete, / ¼ 2 for cement paste. The results of these calculations are listed in Table 4.
Fig. 2. Debris formed during comminution (scale: 1 mm). (a) Concrete 2; (b) mortar 4.
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4. Materials and experimental set-up Eight different mixtures were manufactured for this investigation: five concrete mixtures, one mortar mixture, and two hardened cement paste samples. The cement was a general purpose cement. As fine aggregate quartz sand was used, and as coarse aggregate broken basalt was used. The maximum sand grain size was 2.8 mm, and the coarse aggregate grain size was between 5 and 20 mm. The grading curves of the aggregate materials followed the grading curves recommended by the ASTM Standard C 33. Basically, water–cement-ratio and aggregate content were varied. The mix compositions are listed in Table 3. The mixtures were placed in conventional cylinder forms (15 cm in diameter and 30 cm in height). After 24 h, the samples were released and placed under water for hardening. After 28 days, the compressive strength and the density as given in Table 4 were estimated on three specimens of each mixture. The compressive strength was measured with a force-controlled testing machine Type ‘Avery 7112 CCG’ following the standard specification no. AS 1012, part 8. The density was estimated as the ratio between specimen mass and specimen volume measured on three specimen of each mixture prior to the compressive testing. For the comminution experiments, five half-circular specimens, each 7 cm in height, were prepared for each mixture; they were separated by a mechanical diamond saw. The weight of each specimen, mM , was estimated before and after the comminution experiments as the average of five measurements. The comminution tests were performed in a laboratory jaw breaker Type ‘N.V. Tema’. This breaker is illustrated in Fig. 3. The entry width of the breaker was DIN ¼ 80 mm (140 mm), and the exit width was DEX ¼ 3:5 mm. The drive was a three-phase current E-motor (current: 415 V) with a maximum power of 4 kW. However, the average basic power required to drive the unit without any material was PB ¼ 1:6 kW. The driving motor was coupled to a commercial wattmeter in order to measure the consumed power onTable 3 Compositions of the used materials (all values in kg) Mixture 1 2 3 4 5 6 7 8
Water 190 175 137 202 171 180 750 485
Cement 320 400 340 520 280 560 1250 1515
Coarse aggregates (mm)
Sand 840 756 703 1459 841 609 – –
Summary
5
10
20
– – – – 481 481 – –
808 303 – – –
– 707 1212 – 538 538 – –
– –
2158 2341 2392 2181 2311 2368 2000 2000
Table 4 Results of the efficiency calculations Mixture 1 2 3 4 5 6 7 8 a
Parameter rC (MPa)
qM (kg/m3 )
GF (J/m2 )
BM (mm1 )
A0 (m2 )
EN (kJ)
g (%)
11.3 30.3 5.2 9.1 17.2 19.8 3.8 6.7
2026 2193 1660 1871 2016 2168 1680 1650
24.6 49.0 22.2 12.7 51.3 56.6 10a 5a
0.00038 0.00033 0.00019 0.00250 0.00028 0.00029 0.05 0.10
21.64 17.48 21.53 8.21 29.24 22.22 24.34 20.03
0.9 1.0 1.1 5.0 11.2 11.7 8.4 10.5
8.3 12.0 6.0 2.31 18.8 14.9 5.2 3.4
GF replaced by GIc ; GIc ––values from [21].
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87
Fig. 3. Experimental set-up.
line. The wattmeter measured the entire power, PS , consumed during the breaking process. Based on the power plots shown on the wattmeter screen, the net exposure time, tE , could be estimated, and the net energy was calculated: EN ¼ P S t E :
ð10Þ
The collected fragment samples were sieved in a commercial mechanical sieve shaker for five minutes, and the fragment size distributions was estimated. Two generated debris samples are shown in Fig. 2. The fragment size distributions are discussed in detail in [2].
5. Experimental results and discussion The results of the calculations are listed in Table 4. The fracture energy ranges between 22:2 J=m2 < GF < 56:6 J/m2 for the concrete mixtures. Mixture 3, which is a material with a high porosity caused by the absence of coarse aggregates, exhibits the lowest fracture energy among the concrete samples. This agrees with results from [18] who found that a lightweight concrete has always a lower fracture energy than a normal concrete in a comparative range of comressive stengths. The mortar owns the lowest fracture energy as far as all composite materials are considered. However, this could be expected since a mortar is generally considered to behave more brittle than a concrete. For the cement paste samples, as a plain matrix material, the crack extension force GIc is used instead of the fracture energy. In a cement paste no coarse inclusions are involved; therefore, no aggregate bridging occurs. The efficiency for the materials used in this study is between 2.3% and 18.8%; thus, it varies over an order of magnitude. It is shown in Fig. 4 that the efficiency does not depend on the compressive strength; no distinct general relationship exists between these parameters. This was also found for the relationship between compressive strength and specific breaking energy [2]. However, if mixtures with equal maximum aggregate size or, respectively, the cement paste samples are compared, compressive strength balances the efficiency: the higher the compressive strength the lower the efficiency. This trend is not valid for the high-porous mixture 3. If materials with comparable compressive strengths are compared (mixtures 1 and 4) the mixture with the larger aggregates has a higher
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A.W. Momber / Engineering Fracture Mechanics 70 (2003) 81–91
Fig. 4. Relationship between compressive strength and efficiency; numbers correspond to the mixtures.
Fig. 5. Relationship between fracture energy and efficiency; numbers correspond to the mixtures.
efficiency. Fig. 5 shows the relationship between efficiency and fracture energy. Here, a distinct relationship can be noticed. A regression with a high correlation is: gðGF Þ ¼ C1 C2GF :
ð11Þ
For the conditions in this study the constants are C1 ¼ 2:81 and C2 ¼ 1:03; the coefficient of regression is Corr2 ¼ 0:91, and the standard deviation is SD ¼ 0:11. For the materials usually considered to behave in a more brittle fashion (mortar and hardened cement paste), the efficiency is comparatively low.
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89
However, fracture energy alone can not distinguish ductility from brittleness. Additional parameters are required to make this distinction. The brittleness as the inverse of the characteristic length as suggested in [19] is such a parameter: BM ¼
1 elastic energy : ¼ LCH crack energy
ð12Þ
The higher the value for BM , the more brittle the material behaves. The parameter LCH is the characteristic length according to the fictitious crack model [3]: LCH ¼
EM GF : r2T
ð13Þ
Assuming EEL ¼ r2T =2EM is the elastic strain energy density according to the lower left part of Fig. 1, brittleness can be expressed by the following equation: BM ¼
1 EEL ¼ : 2LCH GF
ð14Þ
Because EEL is given in J/m3 (volume related) and GF is given in J/m2 (surface related), the unit of the brittleness is that of an inverse length. Therefore, Eqs. (12) and (14) can be used only in comparative test series adopting a given specimen size, as done in this study. The relationship between brittleness and efficiency is shown in Fig. 6. The efficiency notably drops as brittleness increases. However, mixture 3 again does not fit into the relationship; its efficiency is rather low. From the arguments used above, the efficiency of this material must be reduced by other events than crack branching, aggregate bridging, or microcracking. In fact, this material has an extraordinarily high amount of large voids caused by the absence of fine aggregate material. This is also the reason for the low density of this material (see Table 4). In a previous study [2] it was observed that this material requires the highest specific breaking energy (in kJ/g) among all tested materials; it is most difficult to crush. The porous regions in the material may interrupt
Fig. 6. Relationship between brittleness and efficiency; numbers correspond to the mixtures.
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A.W. Momber / Engineering Fracture Mechanics 70 (2003) 81–91
stress transfer performed by stress waves originating from the impact of the breakage plate, thus reducing the loading intensity.
6. Conclusions As far as process efficiency is considered, a jaw breaker is efficient for breaking concrete with a high fracture energy and a low brittlenes. If this type of breaker is used for more brittle behaving cementitious materials, such as mortar or hardened cement paste, the efficiency is one order of magnitude lower. For materials with equal maximum aggregate sizes, compressive strength determines the efficiency. The efficiency for porous concrete is rather low. As many rocks exhibit tension-softening behaviour if exposed to load, it may be interesting to perform a similar study with rock materials. Among many rocks basalt has the highest bond-index [15], which points to a high fragmentation resistance. Basalt has also a very high brittleness (low characteristic length) compared to other rocks [20]. This very crude argument may, however, support the results obtained in this study.
Acknowledgements This investigation was funded by the German Research Association, Bonn, Germany. The author is also thankful to the Industrial Research Institute Swinburne (IRIS), and the Department of Chemical Engineering of the Swinburne University of Technology, Melbourne, Australia, for technical and administrative support. Thanks is also addressed to WOMA Apparatebau GmbH, Duisburg, Germany.
References [1] Braus H-P. Recycling-Baustoffe––Chancen und Risiken. In: Rese F, editor. Abbruch and Recycling. Baden––Baden: Stein-Verlag; 2000. p. 8–10. [2] Momber AW. The fragmentation of cementitious composites in a jaw breaker, in review. [3] Hillerborg A, Modeer M, Petersson PE. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement Concrete Res 1976;6:773–81. [4] Bazant ZP, Oh BH. Crack band theory for fracture of concrete. Mater Struct 1983;16:155–77. [5] Jenq Y, Shah SP. Two parameter fracture model for concrete. ASCE J Eng Mech 1985;111(10):1227–41. [6] Nallathambi P, Karihaloo BL. Determination of the specimen-size independent fracture toughness of plain concrete. Magaz Concrete Res 1986;38:67–76. [7] RILEM, Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams. RILEM Draft Recommendation 50-FMC. RILEM, 1985. [8] Wittmann FH, Roelfstra PE, Mihashi H, Huang YY, Zhang XH, Nomura N. Influence of age of loading, water–cement-ratio and rate of loading on fracture energy of concrete. Mater Struct 1987;20:103–10. [9] Comite Europe International du Beton, CEB FIP Model Code 1990, Lausanne, 1993. [10] Hilsdorf HK, Brameshuber W. Code-type formulation of fracture mechanics concepts for concrete. Int J Fract 1991;51:61–72. [11] Rumpf H. Die Einzelkornzerkleinerung als Grundlage einer technischen Zerkleinerungswissenschaft. Chemie-Ing-Techn 1965; 37:187–202. [12] H€ offl K. Zerkleinerungs- und Klassiermaschinen. Berlin, Heidelberg: Springer Verlag; 1986. [13] Momber AW. The fragmentation of standard concrete cylinders under compression: the role of secondary fracture debris. Eng Fract Mech 2000;67:445–59. [14] Kelly FG, Spottiswood DJ. Introduction to Mineral Processing. New York: Wiley & Sons; 1982. [15] Schubert H. Aufbereitung fester mineralischer Rohstoffe. Band 1, VEB Deutscher Verlag f€ ur Grundstoffindustrie, 1988. [16] Kiesskalt S, Matz G. Zur Ermittlung der spezifischen Oberfl€ache von Kornverteilungen. VDI-Zeitschrift 1951;(3):58–60.
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[17] Vauck WR, M€ uller HA. Grundoperationen chemischer Verfahrenstechnik. Leipzig: Deutscher Verlag f€ ur Grundstoffindustrie; 1992. [18] Wu K, Zhang B. Fracture energy of lightweight concrete. In: Mihashi H et al., editors. Fracture Toughness and Fracture Energy. Rotterdam: A.A. Balkema; 1989, p. 117–24. [19] Elfgren L. Applications of fracture mechanics to concrete structures. In: Mihashi H et al., editors. Fracture Toughness and Fracture Energy. Rotterdam: A.A. Balkema; 1989. p. 575–90. [20] Basham KD, Chong KP, Boresi AP. A new development in devising tension-softening curves for brittle materials. In: Proc 9th Int Conf on Exper Mechanics Copenhagen, 1990. p. 1423–32. [21] Mindess S. Fracture toughness testing of cement and concrete. In: Carpintieri A, Ingraffea AR, editors. Fracture Mechanics of Concrete: Material Characterization and Testing. The Hague: Martinus Nijhoff Publishers; 1991. p. 85–93.
The efficiency of mechanical concrete comminution Andreas W. Momber
*,1
WOMA Apparatebau GmbH, P.O. Box 141820, D-47226 Duisburg, Germany Received 8 August 2001; received in revised form 22 January 2002; accepted 29 January 2002
Abstract The paper reports about the comminution of cementitious composites in a laboratory jaw breaker. Six different concrete types, two mixtures of hardened cement paste, and one mortar sample are investigated. The efficiency of the breaking process is calculated based on a non-linear fracture approach. The efficiency g is the ratio between net energy delivered by the crusher and the fracture energy dissipated in the materials during the fragmentation process. Typical efficiency values between 0:03 < g < 0:19 are estimated. However, efficiency increases as the fracture energy of the material increases and brittleness decreases. Jaw breakers are efficient for low-brittle materials. It is suggested to use the approach developed in the paper for estimating the efficiency of rock comminution processes. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction In the countries of the European community, about 2 billion tons of aggregates are consumed each year, mainly for concrete manufacturing [1]. For that reason, recycling and reuse of construction demolition waste is becoming a notable aspect in national economics and in environmental protection. Again in the countries of the European community, the annual amount of construction demolition waste is about 300 million tons. About 60% of this material (180 million tons) is being recycled currently [1]. In industry, recycled concrete will be used only if quality and price are comparable to those of newly manufactured aggregate materials. However, about 70% of the recycling costs is covered by the mechanical processing of the construction demolition waste, primarily by comminution and grinding. For that reason, knowledge about how demolition waste, in particular concrete, behaves in industrial fragmentation devices is an important contribution to a more efficient recycling of aggregate materials. Although the comminution and grinding of minerals, rock and ore is a research issue since decades, not much is known about the behavior of cementitious materials. A review about recent work is given in [2]. A major concern of these studies was the selective fragmentation of concrete: attempts were made to expose the aggregate in the concrete completely and to break the surrounding cement paste away. However, no
*
Tel.: +49-206-530-4380; fax: +49-206-530-4200. E-mail address: [email protected] (A.W. Momber). 1 Habilitation-fellow of the German Research Association, Bonn, Germany, with the Aachen University, Department of Mining, Metallurgy and Geosciences. 0013-7944/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 2 ) 0 0 0 2 0 - 6
82
A.W. Momber / Engineering Fracture Mechanics 70 (2003) 81–91
Nomenclature A0 ASP BM Ci d dF dmax EEL EN EM GF LCH mM n PS R tE w / cM g qM r rC rT
grain sample surface specific grain sample surface brittleness regression parameters RRSB-distribution parameter fragment diameter maximum aggregate size elastic strain energy density crusher net energy material Young’s modulus fracture energy characteristic length specimen weight RRSB-distribution parameter measured crusher power sieve residue net exposure time displacement grain shape parameter specific surface energy comminution efficiency material density stress compressive strength tensile strength
investigations were performed in these studies using different materials or concrete mixtures. Neither was the efficiency of the comminution process investigated. It will be the objective of this paper to address this important problem.
2. Tension-softening materials 2.1. Fracture behaviour of tension-softening materials The materials discussed in this paper are characterised by a ‘fracture process zone’ (FPZ). The FPZ is a zone characterised by progressive softening, for which the stress decreases at increasing deformation. This is illustrated in Fig. 1. Assuming a general material, the failure zone consists of a linear range and a non-linear range. The non-linear range is formed by the FPZ and a non-softening zone (yielding, as known from ductile metals). Depending on the relative size of these zones to the structural size, three basic types of fracture behaviour can be distinguished. In the first fracture type, the non-linear range is small compared to the structural size, and the entire fracture process takes place in a small linear range at the crack tip. This type of fracture approximates materials that usually are called brittle, like PMMA, glass, brittle ceramics,
A.W. Momber / Engineering Fracture Mechanics 70 (2003) 81–91
83
Fig. 1. General structure of a fracture process zone in concrete.
and, to a certain amount, hardened cement paste. In the second fracture type, most of the non-linear range consists of elasto-plastic hardening or yielding. The actual softening zone (FPZ) is comparatively small compared with the zone occupied by hardening or yielding. Materials that are usually called ductile, such as many metals, especially if they are alloyed, fall into this category. The third fracture type, considered in this paper, is characterised by a comparatively large FPZ; yielding can be neglected. A typical FPZ for a brittle, tension-softening material is shown in Fig. 1. Typical parameters characterising the fracture behaviour of quasi-brittle materials are the crack tip opening displacement (CTOD, or w in Fig. 1), the fracture energy GF , and the shape of the softening curve B–C–D as illustrated in Fig. 1. For more brittle-behaving materials, the FPZ is short. The major mechanisms responsible for the formation of an FPZ are assumed to be grain bridging and crack-rim friction behind the actual crack tip and microcracking in front of the crack tip as illustrated in Fig. 1. Several of these features have been observed through an SEM-study on concrete debris formed in a mechanical jaw breaker [2]. The following phenomenological approaches for modelling the non-linear behaviour of brittle materials are known: 1. 2. 3. 4.
The The The The
fictitious crack model [3]. crack-band model [4]. two-parameter model [5]. effective crack model [6].
Detailed information about these models is available in the original papers. In this paper, the fictitious crack model will be considered.
84
A.W. Momber / Engineering Fracture Mechanics 70 (2003) 81–91
2.2. Fracture energy of cementitious materials The fracture energy, GF , is usually considered to be the energy absorbed in the FPZ during fracture. In the fictitious crack model [3], it is given by the area enveloped by the load-displacement-curve shown in Fig. 1: Z w1 rðwÞ dw: ð1Þ GF ¼ 0
Methods for the experimental estimation of the fracture energy are recommended in [7]. For concrete materials, fracture energy mainly depends on compressive strength and maximum aggregate size. Some relations are discussed, among others, in [8]. According to the CEB-FIP Model Code [9], the fracture energy (given in J/m2 ) for concrete can be approximated by the following equation: GF ¼ aðdmax Þr0:7 C :
ð2Þ
Here, rC is the compressive strength in MPa, and dmax is the maximum aggregate diameter in mm. The empirical coefficient a is tabulated in Table 1. A regression delivers the relationship: a ¼ 0:25dmax þ 2:
ð3Þ
These values are used along with Eq. (2) to calculate the fracture energies of the mixtures used in this study. The results of these calculations are listed in Table 4.
3. Estimation of the efficiency The efficiency of mechanical size reduction devices can be estimated as follows [11]: g¼
c M A0 : EN
ð4Þ
Here, cM is the thermodynamic specific surface energy required to create a defined new surface in the material, A0 is the new surface generated during the size reduction process, and EN is the measured net energy of the breaker. From this equation, the efficiency for minerals is always as low as g < 0:01 [12]. However, Eq. (4) is valid only for perfectly brittle (elastically) behaving materials. Concrete, however, dissipates additional energy GF required to separate the fracture surfaces. This is valid also for the comminution in mechanical breakers [2]. Assuming cM GF , Eq. (4) reads: g¼
GF A0 : EN
ð5Þ
The values for EN are estimated from wattmeter readings (see Section 4) and listed in Table 4. It is found in a previous study [2] that the fragments of cementitious materials formed in a mechanical jaw breaker follow a Rosin–Rammler–Sperling–Bennett-distribution (RRSB-distribution). Interestingly, this distribution also applies to debris formed during the uncontrolled post-peak failure in a force-controlled compressive tests [13]. The RRSB-distribution has the following form: Table 1 Typical values for a [10] dmax (mm)
a
8 16 32
4 6 10
A.W. Momber / Engineering Fracture Mechanics 70 (2003) 81–91
85
Table 2 Parameters of the RRSB-distributions [2] Mixture
n
d (lm)
Corr2 (RRSB)
SD (RRSB)
1 2 3 4 5 6 7 8
1.14 1.24 1.11 1.67 1.05 1.09 1.05 1.12
807 820 822 863 778 872 960 870
0.997 0.997 0.997 0.972 0.994 0.995 0.992 0.997
0.02 0.02 0.02 0.51 0.04 0.03 0.05 0.02
n R dF ¼ exp : 100 d
ð6Þ
In the equation, dF is the fragment size. The correlation between Eq. (6) and the experimental results is always Corr2 > 0:97 for the materials used in this study (see Table 2). The distribution parameter n can be calculated from the slope of a LogLog–Log-plot of Eq. (6). The size parameter d is defined as the fragment diameter at R ¼ 36:8%. More details about the RRSB-distribution are provided, among others, in [14,15]. All RRSB-parameters of the analysed fragments are listed in Table 2. The specific surface of an RRSBdistributed sample can be estimated using an approximation developed in [16]: ASP ¼
6:39 expð1:795=n2 Þ: d qM
ð7Þ
In the equation, ASP is the specific surface given in m2 /kg. Thus, the absolute surface can be calculated as follows: A0 ¼ ASP mM :
ð8Þ
Here, mM is given in kg. However, as shown in Fig. 2 the individual debris are not of an ideally spherical shape as assumed for Eq. (7). Therefore, a shape factor / is included in order to estimate the real surface: A0 ¼ /A0 :
ð9Þ
The following empirical values were used according to [17]: / ¼ 1:1 for mortar, / ¼ 1:4 for concrete, / ¼ 2 for cement paste. The results of these calculations are listed in Table 4.
Fig. 2. Debris formed during comminution (scale: 1 mm). (a) Concrete 2; (b) mortar 4.
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4. Materials and experimental set-up Eight different mixtures were manufactured for this investigation: five concrete mixtures, one mortar mixture, and two hardened cement paste samples. The cement was a general purpose cement. As fine aggregate quartz sand was used, and as coarse aggregate broken basalt was used. The maximum sand grain size was 2.8 mm, and the coarse aggregate grain size was between 5 and 20 mm. The grading curves of the aggregate materials followed the grading curves recommended by the ASTM Standard C 33. Basically, water–cement-ratio and aggregate content were varied. The mix compositions are listed in Table 3. The mixtures were placed in conventional cylinder forms (15 cm in diameter and 30 cm in height). After 24 h, the samples were released and placed under water for hardening. After 28 days, the compressive strength and the density as given in Table 4 were estimated on three specimens of each mixture. The compressive strength was measured with a force-controlled testing machine Type ‘Avery 7112 CCG’ following the standard specification no. AS 1012, part 8. The density was estimated as the ratio between specimen mass and specimen volume measured on three specimen of each mixture prior to the compressive testing. For the comminution experiments, five half-circular specimens, each 7 cm in height, were prepared for each mixture; they were separated by a mechanical diamond saw. The weight of each specimen, mM , was estimated before and after the comminution experiments as the average of five measurements. The comminution tests were performed in a laboratory jaw breaker Type ‘N.V. Tema’. This breaker is illustrated in Fig. 3. The entry width of the breaker was DIN ¼ 80 mm (140 mm), and the exit width was DEX ¼ 3:5 mm. The drive was a three-phase current E-motor (current: 415 V) with a maximum power of 4 kW. However, the average basic power required to drive the unit without any material was PB ¼ 1:6 kW. The driving motor was coupled to a commercial wattmeter in order to measure the consumed power onTable 3 Compositions of the used materials (all values in kg) Mixture 1 2 3 4 5 6 7 8
Water 190 175 137 202 171 180 750 485
Cement 320 400 340 520 280 560 1250 1515
Coarse aggregates (mm)
Sand 840 756 703 1459 841 609 – –
Summary
5
10
20
– – – – 481 481 – –
808 303 – – –
– 707 1212 – 538 538 – –
– –
2158 2341 2392 2181 2311 2368 2000 2000
Table 4 Results of the efficiency calculations Mixture 1 2 3 4 5 6 7 8 a
Parameter rC (MPa)
qM (kg/m3 )
GF (J/m2 )
BM (mm1 )
A0 (m2 )
EN (kJ)
g (%)
11.3 30.3 5.2 9.1 17.2 19.8 3.8 6.7
2026 2193 1660 1871 2016 2168 1680 1650
24.6 49.0 22.2 12.7 51.3 56.6 10a 5a
0.00038 0.00033 0.00019 0.00250 0.00028 0.00029 0.05 0.10
21.64 17.48 21.53 8.21 29.24 22.22 24.34 20.03
0.9 1.0 1.1 5.0 11.2 11.7 8.4 10.5
8.3 12.0 6.0 2.31 18.8 14.9 5.2 3.4
GF replaced by GIc ; GIc ––values from [21].
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Fig. 3. Experimental set-up.
line. The wattmeter measured the entire power, PS , consumed during the breaking process. Based on the power plots shown on the wattmeter screen, the net exposure time, tE , could be estimated, and the net energy was calculated: EN ¼ P S t E :
ð10Þ
The collected fragment samples were sieved in a commercial mechanical sieve shaker for five minutes, and the fragment size distributions was estimated. Two generated debris samples are shown in Fig. 2. The fragment size distributions are discussed in detail in [2].
5. Experimental results and discussion The results of the calculations are listed in Table 4. The fracture energy ranges between 22:2 J=m2 < GF < 56:6 J/m2 for the concrete mixtures. Mixture 3, which is a material with a high porosity caused by the absence of coarse aggregates, exhibits the lowest fracture energy among the concrete samples. This agrees with results from [18] who found that a lightweight concrete has always a lower fracture energy than a normal concrete in a comparative range of comressive stengths. The mortar owns the lowest fracture energy as far as all composite materials are considered. However, this could be expected since a mortar is generally considered to behave more brittle than a concrete. For the cement paste samples, as a plain matrix material, the crack extension force GIc is used instead of the fracture energy. In a cement paste no coarse inclusions are involved; therefore, no aggregate bridging occurs. The efficiency for the materials used in this study is between 2.3% and 18.8%; thus, it varies over an order of magnitude. It is shown in Fig. 4 that the efficiency does not depend on the compressive strength; no distinct general relationship exists between these parameters. This was also found for the relationship between compressive strength and specific breaking energy [2]. However, if mixtures with equal maximum aggregate size or, respectively, the cement paste samples are compared, compressive strength balances the efficiency: the higher the compressive strength the lower the efficiency. This trend is not valid for the high-porous mixture 3. If materials with comparable compressive strengths are compared (mixtures 1 and 4) the mixture with the larger aggregates has a higher
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Fig. 4. Relationship between compressive strength and efficiency; numbers correspond to the mixtures.
Fig. 5. Relationship between fracture energy and efficiency; numbers correspond to the mixtures.
efficiency. Fig. 5 shows the relationship between efficiency and fracture energy. Here, a distinct relationship can be noticed. A regression with a high correlation is: gðGF Þ ¼ C1 C2GF :
ð11Þ
For the conditions in this study the constants are C1 ¼ 2:81 and C2 ¼ 1:03; the coefficient of regression is Corr2 ¼ 0:91, and the standard deviation is SD ¼ 0:11. For the materials usually considered to behave in a more brittle fashion (mortar and hardened cement paste), the efficiency is comparatively low.
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However, fracture energy alone can not distinguish ductility from brittleness. Additional parameters are required to make this distinction. The brittleness as the inverse of the characteristic length as suggested in [19] is such a parameter: BM ¼
1 elastic energy : ¼ LCH crack energy
ð12Þ
The higher the value for BM , the more brittle the material behaves. The parameter LCH is the characteristic length according to the fictitious crack model [3]: LCH ¼
EM GF : r2T
ð13Þ
Assuming EEL ¼ r2T =2EM is the elastic strain energy density according to the lower left part of Fig. 1, brittleness can be expressed by the following equation: BM ¼
1 EEL ¼ : 2LCH GF
ð14Þ
Because EEL is given in J/m3 (volume related) and GF is given in J/m2 (surface related), the unit of the brittleness is that of an inverse length. Therefore, Eqs. (12) and (14) can be used only in comparative test series adopting a given specimen size, as done in this study. The relationship between brittleness and efficiency is shown in Fig. 6. The efficiency notably drops as brittleness increases. However, mixture 3 again does not fit into the relationship; its efficiency is rather low. From the arguments used above, the efficiency of this material must be reduced by other events than crack branching, aggregate bridging, or microcracking. In fact, this material has an extraordinarily high amount of large voids caused by the absence of fine aggregate material. This is also the reason for the low density of this material (see Table 4). In a previous study [2] it was observed that this material requires the highest specific breaking energy (in kJ/g) among all tested materials; it is most difficult to crush. The porous regions in the material may interrupt
Fig. 6. Relationship between brittleness and efficiency; numbers correspond to the mixtures.
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stress transfer performed by stress waves originating from the impact of the breakage plate, thus reducing the loading intensity.
6. Conclusions As far as process efficiency is considered, a jaw breaker is efficient for breaking concrete with a high fracture energy and a low brittlenes. If this type of breaker is used for more brittle behaving cementitious materials, such as mortar or hardened cement paste, the efficiency is one order of magnitude lower. For materials with equal maximum aggregate sizes, compressive strength determines the efficiency. The efficiency for porous concrete is rather low. As many rocks exhibit tension-softening behaviour if exposed to load, it may be interesting to perform a similar study with rock materials. Among many rocks basalt has the highest bond-index [15], which points to a high fragmentation resistance. Basalt has also a very high brittleness (low characteristic length) compared to other rocks [20]. This very crude argument may, however, support the results obtained in this study.
Acknowledgements This investigation was funded by the German Research Association, Bonn, Germany. The author is also thankful to the Industrial Research Institute Swinburne (IRIS), and the Department of Chemical Engineering of the Swinburne University of Technology, Melbourne, Australia, for technical and administrative support. Thanks is also addressed to WOMA Apparatebau GmbH, Duisburg, Germany.
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