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Stress–strain relation for water-driven particle erosion of quasi-brittle materials
Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
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Stress±strain relation for water-driven particle erosion of quasi-brittle materials A.W. Momber * WOMA Apparatebau GmbH, Werthauser Straûe 77-79, 47226 Duisburg, Germany
Abstract This paper is concerned with the behaviour of quasi-brittle materials eroded by water-driven particles (WDP). Determined are the compressive stress±strain curves for the material. The measured relations are approximated by parabolic regression. The areas under these curves are then used to characterise the speci®c energy absorption capability of the investigated materials. Erosion is then investigated by WDP with varying water pressures and exposure time. The depth of penetration, the material volume loss, and the kerf geometry are measured. Based on these measurements, the speci®c energy of the erosion process is calculated. A characteristic eciency number U is then de®ned and estimated for all materials at dierent erosion conditions. It is shown that this parameter eectively characterises the erodability of the materials. Moreover, relations are found between the characteristic erosion parameters, such as threshold pressure, machinability number, depth of penetration, material removal rate, and the speci®c energy absorption capability. A preliminary phenomenological model based on the relationship between crack propagation processes and energy absorption capability is developed. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Water-driven particle erosion (WDPE) is the basic process of the abrasive waterjet technology ± a newly introduced machining concept that oers some advantages over many other traditional and non-traditional machining technologies, such as no thermal distortion, high machining versatility, low cutting force, and high ¯exibility [1]. On the basis of jet generation, WDPE can be categorised as injection-based or suspension-based. For practical applications, injection-based WDPE is more commonly used.
* Corresponding author. Tel.: +49-206-5304-380; fax: +49206-5304-200. Habilitation Fellow at the RWTH Aachen. E-mail address: [email protected] (A.W. Momber).
An injection jet is formed by accelerating small abrasive particles (usually garnet) through contact with a high-speed waterjet. The mixing between abrasive, water, and air takes place in a mixing chamber, and the acceleration process occurs in an acceleration focus. The abrasive particles leave the focus at velocities of several hundred meters per second. A high number of abrasives generates a high-frequency impingement on the material surface. The high-speed abrasive jets investigated are formed by this method. WDPE is a promising alternative method for the 3D-machining (milling, turning, drilling, polishing) of quasi-brittle materials, such as ceramics, concrete, rocks, and reinforced materials. Although several parameter studies have been identi®ed for ceramics, concrete and rocks, the general mechanism of the material removal process,
0167-8442/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 0 0 ) 0 0 0 4 6 - X
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A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
Nomenclature A; B; C constants b kerf width constants C0±5 focus diameter dF water jet nozzle diameter dw jet energy EA EAbs modulus of fracture absorbed energy per volume during EAbs compressive test dyn dynamically absorbed energy Eabs EM Young's modulus speci®c removal energy Esp h depth of penetration L kerf length abrasive ¯ow rate m_ a m_ w water ¯ow rate
especially in multiphase, quasi-brittle materials, is not well understood. Conducted in [2] is an SEM-study to understand the behaviour of polycrystalline ceramics in WDPE. Through grooving, sweeping and cutting experiments, a mixed material removal mode has been identi®ed that corresponds to brittle-fracture and plastic deformation. For low impact angle, scratching marks by single abrasive grains tend to dominate while intergranular cracking also appears. At perpendicular impact angles, intergranular cracking dominates the erosion process. Transgranular cracking has not been observed. It has been found [3,4] in a combined SEM-/ AE-study for WDPE of refractory ceramics that the certain failure mechanism depends on the kinetic energy delivered to the erosion site. The higher the kinetic energy, the higher the probability of transgranular fracture. It was also found that the type of bond between matrix and inclusion is important. For materials with directly bonded inclusions, such as magnesia chromite, a mixed material removal mode dominates. The inclusions are either exposed due to matrix removal or cut. In contrast, in a material with very pronounced interfaces, such as bauxite, the hard
Nm p pC PH Q v V_M VM a v U q qM r rC e ecr
machinability number pump pressure threshold pressure hydraulic pump power water volume ¯ow rate traverse rate volumetric removal rate volume removal nozzle eciency parameter correction parameter eciency parameter water density material density stress ultimate stress strain ultimate strain
inclusions resist the WDPE and are pulled out almost intact. Recently, the acoustic emission technique (AE) has been applied [5] to monitor the WDPE of quasi-brittle materials. It was found that materials with a low stiness, such as a mortar, fail by matrix erosion and separate inclusion pull-out. In contrast, materials with a high stiness, such as a high-strength concrete, are eroded by intense spalling fracture with cracks running through the matrix as well as the inclusions. Performed in [6] is an investigation of WDPE of marble and calcitic rocks. It was found that the amount and distribution of pre-existing ¯aws such as pores and cracks are essential for the erosion process. Both types of rock were worn by intergranular cracking and cleavage of calcite as well, but intergranular cracking dominates the erosion for calcareous rocks because of the poor bond between calcitic grains and the surrounding matrix. An elasto-plastic erosion model for brittle materials is developed in [2]. The material removal is assumed to be a combination of stresswave induced brittle (intergranular) fracture and plastic deformation. A crack network model relates the speci®c surface energy to the energy of
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
the stress-waves generated during particle impact. This approach oers some possibility to link the material erosion resistance to fracture mechanics parameters. The contribution of plastic ¯ow is considered in the model by employing a microcutting model for solid-particle erosion of ductile materials. Nevertheless, AE-measurements [5] have shown that intergranular fracture occurs mainly in materials with low homogeneity (small inclusions, low-strength matrix, large stiness deviations). Moreover, the probability of transgranular fracture in some directly bonded ceramics increases as the kinetic energy of the WDP increases [4].
2. WDPE resistance parameters For the WDPE of brittle-behaving materials, a number of resistance parameters are introduced. First, there is the so-called `threshold velocity' which has to be exceeded to introduce the macroscopic material removal process [1]. For eroding rocks and concrete by plain waterjets, the threshold velocity can be related to the critical stress intensity factor of the material [7,8]. A similar
21
parameter, a `threshold impact velocity', exists for the air-driven solid-particle erosion of brittle materials [9]. For WDPE, a so-called `machinability number' has been developed to describe the material resistance [1,10]. This material parameter is NM A
DM rF B cM EM rF
1
Due to reference tests, this parameter can also experimentally be estimated Nm
C0 h dw0:618 v0:866 : p1:25 m_ 0:687 m_ 0:343 w a
2
It can be seen from Eq. (2) that the depth of penetration is proportional to the machinability number. The smaller the machinability number, the higher the material resistance in WDPE. The `modulus of fracture' has been used to estimate the WDPE resistance of several materials [11,12]. This modulus of fracture was estimated as EAbs
r2T : 2EM
3
Fig. 1. Relation between modulus of fracture and WDPE parameters: (a) piercing by WDPE [11]; (b) cutting by WDPE [12].
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A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
Fig. 1 shows some of the investigated materials that did not ®t into the relations. If the resistance of conventionally ductile behaving materials and of ceramics is related to the modulus of fracture, then the ceramics fail, Fig. 1(a). If the resistance of ceramics and rocks are related to the modulus of fracture, then the rock materials would fail, Fig. 1(b). Obviously, due to their quasi-brittle characteristics, the simple procedure for estimating the modulus of fracture does not cover the more complex stress±strain behaviour. For both cases, the energy absorption capability of the precracked materials is underestimated in Fig. 1.
Fig. 2. Basic types of fracture, adapted from [13]: (a) brittle; (b) ductile; (c) quasi-brittle.
3. Stress±strain behaviour and failure modes 3.1. Stress±strain characteristics of quasi-brittle materials Generally, material behaviour can be categorised as brittle, ductile, and quasi-brittle. These types of material behaviour can be quanti®ed by the ratio between the non-linear zone and the linear-elastic zone around an existing crack as illustrated in Fig. 2. If the ratio is small, Fig. 2(a), the corresponding material is referred to as brittle such as glass and ceramics. If the ratio is large such that a certain amount of non-linear hardening and yielding occur in the non-linear zone as shown by Fig. 2(b), then the material is referred to as ductile such as metal alloys. If the ratio is large and the major part of the non-linear zone corresponds to microcracking or interface breakage, then the material is referred to as quasibrittle Fig. 2(c). The foregoing subdivisions can also be illustrated by the stress±strain relations. A typical stress±strain curves for a quasi-brittle material is given in Fig. 3. It is characterised by non-linearity, especially at high stress levels. The fracture process must pass through three stages: crack initiation,
Fig. 3. Diagrammatic stress±strain curves for concrete in compression, adapted from [14].
Result
Volume loss / 1=EAbs Volume loss / 1=EAbs General discussion Comminution resistance / n Erosion resistance / EAbs Erosion eciency / EAbs Piercing time / 1=EAbs (except ceramics) Cutting rate / EAbs (except rocks)
Loading type (material)
Solid particle erosion (ductile materials) Cavitation and drop impact (metals) Solid particle erosion (plastics) Impact fragmentation (rocks, minerals) Waterjet erosion (concrete)
AWJ piercing (ceramics, metals) AWJ cutting (ceramics, rocks, glass)
Stress±strain characteristics
Integrated product of stress and strain Ultimate resilience EAbs r2T =2EM Strain energy EAbs rF =2eF n EAbs =EEl (EEl ± elastic energy)R Absorbed fracture energy EAbs r e de
Modulus of fracture EAbs r2T =2EM Modulus of fracture EAbs r2T =2EM
Reference
Bitter [16] Thiruvengadam [17] Kriegel [19] Ocepek [20] Medeot [21] Momber [22] Hunt et al. [11] Matsui et al. [12]
Table 1 Stress±strain in relation to erosion and impact
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
23
slow crack growth, and rapid crack growth. For concrete as a quasi-brittle material, these stages are illustrated in Fig. 3. Toughening and energy absorbing mechanisms such as crack shielding, crack de¯ection and crack arrest on inclusions, and grain bridging, acting in a so-called `fracture process zone' in front of a macrocrack, posses such character [14]; they include concrete, rocks, cemented sands, and some ceramics. Several toughening mechanisms were observed not only under normal fracture, but also during waterjet erosion of concrete [15]. However, Fig. 3 shows that Eq. (3) does not cover the actual conditions as far as quasi-brittle materials are concerned. The area under the stress±strain curves is frequently assumed to represent the energy absorption capability of a certain material volume. This parameter is often termed as absorbed fracture energy, modulus of fracture, ultimate resilience, or strain energy density. 3.2. Stress±strain in relation to erosion and impact Several attempts have been made to associate the stress±strain functions of a material to the resistance against erosive loading, mainly based on Eq. (3). These attempts are summarised in Table 1. For the solid-particle erosion of materials, a material constant has been used in [16] to describe the plastic±elastic behaviour of materials; this constant is identical to the area enveloped by the stress±strain curves. The material removal rate is assumed to be inversely proportional to the energy absorption capacity. Developed in [17] is a concept for the erosion strength of metals for water drop and cavitation erosion. The ultimate resilience was also used, Eq. (3). A unique relation was found between this parameter and the volume loss due to cavitation erosion. Other approaches in the ®eld of liquid-¯ow erosion can be found in [18]. A very similar approach was used in [19]. This work also relates the material resistance against solid-particle erosion to the energy absorbed by the material under tensile test. Discussed in [20] is a comminution parameter which describes the resistance against impact fragmentation of minerals. This parameter is the ratio between the elastic defor-
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
1962 2107 2135 2275 2343
mation energy and the totally absorbed energy. Use of the energy absorbed during the compression of concrete was suggested in [21] for the definition of its resistance against waterjet erosion. This idea was further developed in [22] for describing the eciency of waterjet erosion based on the speci®c energy absorption during compression test. For WDPE, simpli®ed stress±strain curves were used in [11,12] for brittle and ductile materials to calculate the erosion eciency and the piercing time. As already illustrated in Fig. 1(b), the model in [12] fails for rocks which indicates that a simple linear (ideal) brittle approach does not characterise the behaviour of this group of material in the case of WDPE.
b
a
80 70 60 50 40 2.24 2.51 2.13 2.25 2.26 #1 #2 #3 #4 #5
Coarse limestone aggregates (maximum grain size 19 mm) and river sand (maximum grain size 4.7 mm). Portland cement, type I. c River sand (maximum grain size 4.7 mm).
858.1 603.6 516.4 499.5 467.9 55.2 60.4 78.2 80.8 84.7 2.1 11.1 38.9 62.6 72.8 0.85 0.71 0.45 0.38 0.32
10.7 24.1 33.0 34.3 42.3
4.0 12.5 27.1 34.2 41.1
pC (MPa) EM (GPa) Water/cement (%) c
Sand b
Aggregate a /cement Concrete mixture
Table 2 Mix design, mechanical properties and machining characteristics of the used materials
rC (MPa)
dyn Eabs MJ=m3
Nm (±)
qM kg=m3
24
3.3. Energy absorption during compression It is assumed that the compressive stress±strain function of a concrete as shown in Fig. 2 can be approximated by a parabolic equation r e A e ÿ B2 C:
4
This relation holds only for stress±strain functions obtained from a `soft', less rigid testing machine. Eq. (4) attains a physical meaning if B is assumed to be the strain at the ultimate stress and C is assumed to be the ultimate stress. This leads to 2
r e A e ÿ ecr rC :
5
The energy absorbed per volume during the compression test is simply given by the area under the stress±strain curves Z ecr 1 r e de Ae3cr ecr rC : 6 EAbs 3 0 The superscript * stands for quasi-brittle materials to distinguish the relation from Eq. (3). The parameters A, ecr and rC have to be estimated by compression tests. The quantity in Eq. (6) has been referred to in fracture mechanics as the strain energy density function [23] and used successfully for characterising the fracture behaviour of brittle and ductile materials, glass, ceramics, metal alloys etc.
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37 Table 3 Erosion conditions and parameters
dyn _ rC EAbs EAbs v r; :
Parameter
Symbol
Unit
Range
Pressure Traverse rate Abrasive ¯ow rate Nozzle diameter Focus diameter Abrasive size Abrasive type
p v m_ a dw dF ± ±
MPa mm/s g/s mm mm mesh ±
100±350 2.0±12.0 6.12±19.05 0.457 1.270 #36 garnet
The application of Eq. (6) is limited to static loading while WDPE is a highly dynamic process. A correction parameter v may be introduced to account for stress rate eects on the energy absorption capability of concrete. It was observed in [24,25] that energy absorption is signi®cantly higher for impact loading compared to static loading. A relation between the stress rate and the energy absorption was found in [25] for estimating the parameter v which attains a saturation value at very high stress rates. The in¯uence of the compressive strength on v is approximated from the values given in [25] (see Table 3). Introducing this correction parameter, the dynamic absorbed energy can be calculated from
25
7
A dynamic fracture analysis of the Charpy Vnotch specimen has been made in [26] using the strain energy density function or the absorbed energy which applies equally well to the dynamic case.
4. Experimental setup and experimental work In this work, ®ve dierent concrete mixtures were designed. Information on the mix designs is given in Table 2. These mixtures were cured and hardened for 28 days under water. After hardening, the cylindrical specimens were tested. The specimen dimensions were 14 cm in height and 7 cm in diameter for the compression tests, Fig. 4(a); and 30.48 cm in height and 15.24 cm in diameter for the erosion tests, Fig. 4(b). To minimise lateral restraint stresses, the top and bottom of the specimens used for the compression tests were covered with thin rubber plates. The compressive strength was estimated according to ASTM Standard C 39; the modulus of elasticity was measured as the static
Fig. 4. Specimens used for the experiments: (a) estimation of the stress±strain curves; (b) WDPE tests.
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A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
the generated kerfs were measured, including depth of penetration, kerf width, and kerf length, Fig. 4(b). These values were used to calculate the material removal rate. Based on measurements of 18 erosion tests under dierent erosion conditions, the machinability number of each material was calculated using Eq. (1). The results are given in Table 2. Optical microscopy and SEM records magni®ed up to 25 were used for erosion site inspection. 5. Compressive test results and discussion 5.1. General failure modes
Fig. 5. Compression stress±strain curves of the investigated concrete mixtures.
chord modulus according to ASTM Standard C 469. The stress±strain curves of each mixture was estimated by loading and unloading the specimen with strain-gauge equipped 11 times. The results of these measurements are given in Table 2. After the tests, the fracture debris were collected and analysed by sieve analysis using a conventional sieve shaker. The test device used to perform WDPE consists of an intensi®er pump, abrasive mixing and acceleration head, abrasive storage and metering system, catcher, and CNC-controlled positioning system. The position of the mixing and acceleration head was controlled using an x-y-z-positioning table. Erosion parameters and conditions are listed in Table 3. After erosion, the dimensions of
Fig. 5 shows the stress±strain test results that are approximated by a parabolic equation as given by Eq. (4). The regression parameters are listed in Table 4. The absorbed energy per volume was calculated via Eq. (6). The estimated static values are between 1:5 and 47:6 MJ=m3 . This is of the same order of magnitude as values published in [21,22]. The curves can be divided into several stages 1±4, as shown in Fig. 3. Up to point 1, the stress increases linearly with increasing strain. After that, pre-existing micro¯aws (mainly interfacial ¯aws at the aggregate-cement paste-interface) start to open. They propagate into the matrix and coalesce into cracks. Detected in [27,28] is a strain localisation at this stage that accelerates the failure process. The major cracks propagate stably and one of them reaches the critical length. This stage is marked as point 4. At that point, the peak stress is reached and failure occurs. Dependent on the ®nal failure mode, the specimen is separated at least into two major
Table 4 Regression parameters of the stress±strain curves Concrete mixture
A
ecr
rC [MPa]
Eabs MJ=m3
va
#1 #2 #3 #4 #5
)12.5 )13.9 )12.1 )10.6 )13.6
0.57 0.95 1.50 1.80 1.73
4.0 12.5 27.1 34.2 41.1
1.5 7.9 27.0 40.9 47.6
1.40 1.40 1.44 1.53 1.53
a
Taken from [25].
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
parts: A low number of primary debris is ®rst obtained. (For non-covered or slender specimens this number is typically two.) Fine-grained secondary debris are then observed. Fig. 6 shows primary debris from mixtures 1 and 5, respectively. Both mixtures exhibit dierent failure behaviour. Concrete 1 in Fig. 6(a) is characterised by the cone-type failure as a result of shear failure. In contrast, mixture 5 in Fig. 6(b) corresponds to the splitting mode characterised by a very brittle fracture. It is also very signi®cant that the number of broken aggregates on the surface of these cones depends on the concrete mixture. It is about zero for specimen 1 and about 100% for mixture 5. Additional results of these experiments including the discussion of secondary debris, can be found in [29,30].
27
5.2. Energy absorption eciency Secondary debris have not attracted much attention in the past although they may contain information about the fracture process, especially about the energy balance during failure. This information was acquired in [30] with reference to sieve analysis and debris image analysis. The speci®c crack energy and the speci®c crack length were calculated. They are plotted in Fig. 7. It can be seen that the energy required to generate a crack network with a given length increases linearly with the absorbed fracture energy, Fig. 7(a). This result is not unexpected and can be used to relate the energy absorption capability of the materials to the microcracking processes in the compression test. An interesting result is that the dierence in the absorbed energy during com-
Fig. 6. Primary fracture debris from the compressive failure of cylindrical concrete specimens: (a) concrete 1 (conical); (b) concrete 5 (splitting).
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A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
Fig. 7. Fracture parameters obtained from the compression tests, adapted from [30]: (a) speci®c crack energy; (b) speci®c crack length.
pression is not caused by the dierence in crack length which is approximately identical, Fig. 7(b). Rather, it is due to dierent crack path. The cracks run preferably through the interface between cement matrix and aggregate in samples 1 and 2; they run through the aggregates in samples 4 and 5 giving rise to transgranular fracture. This may be a result of the coarser aggregates as well as of the higher matrix strength of these samples; both effects increase the material stiness which is further enhanced by the higher values of the Young's modulus for the stronger mixture. It was observed in concrete that the work of fracture increases with the aggregate size [31]. The probability of transgranular fracture also increases with cement matrix strength [32]. Due to the higher speci®c surface energy of the aggregate material, which is more than two orders of magnitude higher than those of the aggregate±matrix interface, more energy is absorbed during transgranular fracture at comparable crack length. Assuming that the WDPE of quasi-brittle materials is characterised by the generation and intersection of a microcrack network [2], there should be a signi®cant dependence of the behaviour of the dierent material mixtures on the resistance against WDPE.
6. Erosion test results and discussion 6.1. Threshold pressure The results of the erosion depth measurements are presented elsewhere, but not discussed in terms of erosion resistance [33]. Fig. 8 shows a relation between operating pressure and depth of penetration. The depth of penetration increases with the pressure. At high pressure levels, the progress of the function drops indicating a reduced energy eciency. The relation for each mixture was approximated by a second-order polynomial, and the intersection of the polynomial and the pressure axis is de®ned as the `threshold pressure'. The estimated values are given in Table 2. It is an experimentally de®ned parameter and describes the pressure required to introduce the visible material removal process. Because the abrasive particles are accelerated by a high-speed waterjet, the threshold pressures describes a critical particle velocity. A simpli®ed relation between water pressure and particle impact velocity is [1] given by p 8 vPC / pC : As shown in [9] for brittle, pre-cracked materials, the threshold impact velocity depends on the
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
Fig. 8. Relation between pump pressure and erosion depth (concrete 1, v 4:0 mm/s, m_ a 7:41 g/s).
wave velocity and the fracture toughness of the material. This ®nding is veri®ed for WDPE in [33]. Alternatively, it may be assumed that the visible material removal process starts if a certain (small) volume of material is removed. This `threshold volume' is a constant geometrical parameter for any material. If the pressure is lower than that of threshold, then the generated crack network is not dense enough and no intersection of penetrating cracks occurs aside from those occurred spontaneously. This period is assumed to be that of incubation. To erode this threshold volume, a certain amount of energy is required, i.e., dyn C pC3=2 f pC EC VC EAbs
9
with E / p3=2 [1]. The function f pC considers the relationship between operating pressure and energy transformation during waterjet formation, abrasive mixing and acceleration, and material erosion that in¯uences the kinetic energy of the eroding jet [1]. In the narrow range of threshold pressures, f pC can be neglected, and hence 0:67 dyn : 10 pC C1 EAbs
29
Fig. 9. Relation between absorbed fracture energy and threshold pressure (v 4:0 mm/s, m_ a 7:41 g/s).
Linearisation yields dyn ln pC 0:67 ln EAbs C2 :
11
In Fig. 9, the threshold pressure is plotted against the absorbed energy in a double-logarithmic diagram. The data trend ®ts qualitatively using Eq. (11). This indicates a relationship between the material resistance against WDPE and the failure under compression. The threshold pressure shows the dierent energy absorption capability of the concrete during crack propagation in the incubation pressure range. For a pressure of p 80 MPa, material removal has already occurred in the materials 1±4, whereas no visible erosion could be seen in concrete 5. Fig. 7(a) shows that a higher amount of energy is required for concrete 5 to produce a network of cracks of a given length. The analytical curve is about ®ve times higher than that of the experimental curve. The absolute in¯uence of the absorbed fracture energy is overestimated by Eq. (10). This may be contributed by the localised material attack for WDPE. The jet starts to remove material at the points of the lowest resistance in concrete. A certain range of
30
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
these weak zones is distributed over the surface. Tracking a 15 cm long erosion path, the jet hits one of the weak zones with a certain probability. This probability is about the same for the two lowstrength concretes (the average deviation in threshold pressure in about 5%), and it is about the same for the three stronger concretes (the average deviation in threshold pressure is about 4%). Under compression, a few weak zones do not lead to specimen fracture as far as the stresses are not forced in a general direction, even if a failure stress is generated. Whereas, material is immediately removed in WDPE. 6.2. Machinability number Since the absorbed energy characterises the behaviour of quasi-brittle materials in WDPE, it should be related to the machinability number being estimated between Nm 470 and Nm 860 as given in Table 2. They correspond to those of asphalt concrete, marble and glass [1]. The results plotted in Fig. 10 suggest a relationship that can be ®tted by ÿC4 dyn : 12 Nm C3 EAbs
Fig. 10. Relation between absorbed fracture energy and machinability number.
Note that the higher the energy absorbed during the compressive test, the lower is the machinability number. This implies a higher material resistance against WDPE.
Fig. 11. Relation between absorbed fracture energy and eciency for dierent erosion conditions (m_ a 7:41 g/s).
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
31
Fig. 12. Surfaces generated by WDPE (p 200 MPa, v 8:0 mm/s, m_ a 7:41 g/s). Left: Concrete 1; Right: Concrete 5.
6.3. Erosion eciency The eciency of WDPE can be evaluated by relating the energy absorbed for the compressive test to the speci®c energy involved in material removal. Attempts were made in [34,35] to relate the `machinability' of rocks by conventional tools to the behaviour during compression test. For the present case, a non-dimensional eciency parameter U is de®ned: U
dyn EAbs
Esp
:
The energy portion 1 ÿ UEA is partly dissipated at the erosion site due to damping and friction, and is carried away partly by the exiting abrasivewater-debris suspension after erosion [36]. The estimated eciency parameters are plotted against the absorbed fracture energy in Fig. 11. The eciency values are between U 2:9 10ÿ5 and U 1:2 10ÿ3 ; they increase linearly with the energy absorption capability: dyn b1 : U a1 EAbs
17
13
The energy required to remove a certain volume of material by WDPE is Esp
EA : VM
14
Here, EA is the kinetic energy of the eroding jet given by [1] s Z xL a2 u2 pdw2 2 3=2 p dx: 15 EA q 4v x0 The removed volume VM has to be estimated by erosion tests. Using Eqs. (7), (8) and Eqs. (13)± (15), the eciency of WDPE can be calculated: 3 ecr rC VM vp 4v 0:3Ae q p cr : 16 U 2a2 u2 pdw2 p3=2 L
Fig. 13. Relation between jet energy, machinability number, and erosion eciency.
32
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
With a1 1=Esp and b1 0 this is identical to Eq. (13). This trend, which was also observed for concrete erosion by waterjets [22], expresses the dierent crack propagation modes in the material. As pointed out earlier, the concrete mixtures with the higher energy absorption capability show
a more brittle failure with large fracture debris and transcrystalline fracture planes. This character prevails also for the WDPE as shown in Fig. 12 that displays photographs of eroded cross-sections from two concrete samples. Note that the crosssection of concrete 1 is very rough and shows
Fig. 14. Micrographs from the erosion sites: (a) concrete 1 (optical 1:6); (b) concrete 2 (SEM, 1:25); (c) concrete 4 (SEM 1:25); (d) concrete 5 (optical 1:25).
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
predominatly intergranular fracture, while the cross-section of mixture 5 is smooth and exhibits predominatly transgranular fracture. One can assume that this latter behaviour allows the generation of larger fracture debris in the concrete with the high energy absorption capability during
33
WDPE. More evidence is given in Fig. 7(a). It shows that for comparable speci®c crack length the energy required for crack generation increases with the concrete strength. Even if the transgranular fracture in concrete 5 reduces the crack length, this process absorbs more energy because
Fig. 14. (continued).
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A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
of the high speci®c surface energy of the aggregate material. The regression parameters a1 and b1 in Eq. (17) depend on pump pressure, traverse rate and hence on the jet energy. From Fig. 11, the eciency decreases with increasing pressure and decreasing traverse rate. Higher jet energy probably leads to worse energy transfer in the given parameter range. This trend agrees with the results in [37]. On the average, just 0.1% of the jet energy is required to separate material. This value is one order of magnitude lower than those estimated in [37]: Based on wear debris analysis for cast iron, about 2% of the jet energy was found to be absorbed by the generation of fracture surfaces. This could be caused by the lower machinability number of cast iron (Nm 121). As already shown in Fig. 10, the eciency increases as the machinability number drops. It can thus be concluded that the eciency increases as the ratio between machinability number and jet energy drops. This is illustrated in Fig. 13. By using waterjets for eroding concrete, [22] estimated an eciency of about U 0:05 (5%), which is relatively high. This may be attributed to the fact that a plain waterjet is able to break large pieces from brittle specimens. The eciency of tools for the removal of rocks and rocklike materials
drops signi®cantly as the tool size decreases [34]. For the WDPE with extremely small tool dimensions (abrasive particle diameter dP 300 lm, jet diameter dJet 1:27 mm), the removal eciency may be comparatively low. 6.4. Microscopic investigations and AE-measurements An inspection of the surfaces arising from WDPE gives results similar to those obtained from the compression tests. Fig. 14(a) shows that almost all inclusions are unbroken in concrete 1 after WDPE. Some broken grains are found on top of the cut surface. Also, striations are found running from top to the bottom due to intergranular matrix removal. The situation is very similar in concrete 2 shown in Fig. 14(b). Again, intergranular fracture can be noticed accompanied by grain pullout (upper center and right section). A large striation can also be noticed (left from the center). In contrast, sample 5 shown in Fig. 14(d) shows broken aggregate grains. A grain, cut with the matrix, can be seen at the top right. This grain also shows several striation marks. The eroded surface is very smooth; the grain boundaries are intact and adopt a sharp appearance. These results suggest
Fig. 15. Relation between absorbed fracture energy and AE-signal parameters (AE-signal parameters taken from [5], Table 3): (a) signal amplitude; (b) signal frequency.
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
dierent mechanisms of material removal for the mixture. For mixture 1, the matrix is removed only without cutting the aggregate. Matrix and aggregate are cut simultaneously in mixture 5. Mixture 4, shown in Fig. 14(c) exhibits a mixed material removal mode where both grain pull-out and brittle fracture can be found. These ®ndings are supported by Fig. 15 that shows the in¯uence of absorbed energy on AE signals acquired on-line during WDPE [5]. Two trends can be noticed from the graphs. First, the signal amplitude tends to reduce with an increase in the absorbed fracture energy for mixtures 1±4. This is independent of the exposure time (which is the reverse of the traverse rate). Next, the AE-signal energy increases for concrete 5, Fig. 15(a). The same trend prevails between the absorbed energy and the peak frequency of the AE-signal. Here, the frequency rises with an increase in the absorbed energy for mixtures 1±4. For concrete 5, the frequency drops, Fig. 15 (b). Switching in the trends for the two AE-signal parameters is due to a change in the erosion mode. Mixtures 1±3 are eroded by intergranular grain pullout and independent matrix removal. Mixture 5 shows a transgranular fracture mode during WDPE. Mixture 4 shows a mixed mode which is probably responsible for the high frequency value. Most probably, there is a transition range in the absorbed energy between 50 and 70 MJ=m3 where a change in the fracture mode from intergranular erosion to transgranular fracture has occurred.
dyn EAbs U
PH : hbv
35
20
Assuming that the width is independent of the erosion depth, and using Eq. (17), Eq. (20) solves for the erosion depth as given by dyn a1 EAbs b1 pQ_ : 21 h dyn bv EAbs The experimentally estimated depths are plotted against the theoretical values in Fig. 16. Despite a few exceptions, the results ®t reasonably by Eq. (21). The strayed points away from the line belong to the low-strength concrete 1. The reason is that WDPE did not produce straight kerfs as assumed in Eq. (21). Hence, the parameter b could be replaced by a function b h. The general in¯uence of pump pressure and traverse rate are also included in Eq. (21). 8. Summary and conclusions A systematic study has been made for the energy absorbed during compression test of quasibrittle and multiphase materials subjected to
7. Volume removal rate and depth of cut Volume removal rate and erosion depth can be estimated from a known stress±strain curves. An energy balance and Eq. (17) give dyn U EAbs
and V_M
PH V_M
dyn PH a1 EAbs b1 dyn EAbs
18
:
The assumption V_M hbv leads to
19 Fig. 16. Comparative plot between experimental results and Eq. (21).
36
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
WDPE. The absorbed energy is estimated by calculating the area under the stress±strain curves of ®ve concrete materials. Moreover, fracture tests for compression and erosion tests with WDP are performed to estimate the erodability. The following results can be concluded: · The amount of energy absorbed during compression can be related to failure by crack propagation. In a material with high energy absorption capacity, the cracks run preferably through the aggregate. This process is more energy consuming because of the higher speci®c surface energy of the inclusions. · The absorbed energy can be related to WDPE threshold pressure that may describe the establishment of a microcrack network at the ®nal stage. · There exists a relationship between the energy absorption capability during compression and erodability by WDP. · Material removal rate and depth of penetration are related to the energy absorbed during compression. · The eciency of WDPE of quasi-brittle and multiphase materials is about U 0:001. However, this is estimated from volume removal, which is not a typical process for commercial WDPE applications. · The eciency increases with absorbed fracture energy and decreases with machinability number. This suggests that WDPE is more eective for machining highly resistant materials. · Material removal mode for WDPE depends on energy absorption capability. Materials with low energy absorption capability will be eroded by intergranular fracture. In contrast, materials with high energy absorption capability will be eroded by transgranular fracture.
Acknowledgements The author is thankful to the German Research Foundation, Bonn, for ®nancial support. Thank is addressed to WOMA Apparatebau GmbH, Duisburg, Germany, and RWTH Aachen, Aachen, Germany, for administrative support.
References [1] A.W. Momber, R. Kovacevic, Principles of Abrasive Water Jet Machining, Springer, London, 1998. [2] J. Zeng, T.J. Kim, Development of an abrasive waterjet kerf cutting model for brittle materials, in: A. Lichtarowicz (Ed.), Jet Cutting Technology, Kluwer Academic Publishers, Dordrecht, MA, 1992, pp. 483±501. [3] A.W. Momber, I. Eusch, R. Kovacevic, Machining refractory ceramics with abrasive water jet, J. Mater. Sci. 31 (1996) 6485±6493. [4] A.W. Momber, R.S. Mohan, R. Kovacevic, An acoustic emission study of cutting bauxite refractory ceramics by abrasive water jets, ASM J. Mater. Eng. Perform. 8 (1999) 450±454. [5] A.W. Momber, R.S. Mohan, R. Kovacevic, On-line analysis of hydro-abrasive erosion of pre-cracked materials by acoustic emission, Theoret. Appl. Fracture Mech. 31 (1999) 1±17. [6] R.M. Miranda, T.J. Kim, Abrasive waterjet cutting of marble and calcareous stones ± a phenomenological study, in: C. Gee (Ed.), Jetting Technology, Mechanical Engineering Publishing, Bury St. Edmunds, 1996, pp. 415±424. [7] J. Wiedemeier, Fl ussigkeitsfreistrahlen hoher Relativgeschwindigkeit und Bruchkinetik spr oder Werkstoe, Ph.D. Thesis, Univ. Hannover, 1981. [8] A.W. Momber, R. Kovacevic, Statistical character of the failure of rock-like materials due to high energy water jet impingement, Int. J. Fract. 71 (1995) 1±14. [9] A.G. Evans, M.E. Gulden, M. Rosenblatt, Impact damage in brittle materials in the elastic-plastic response regime, Proc. Roy. Soc. London, Ser. A 361 (1978) 343±356. [10] J. Zeng, T.J. Kim, R.J. Wallace, Quantitative evaluation of machinability in abrasive water jet machining, ASME-PED 58 (1992) 169±179. [11] D.C. Hunt, T.J. Kim, J.G. Sylvia, A parametric study of abrasive waterjet by piercing experiments, in: D. Saunders (Ed.), Proceedings of the Eighth International Symposium Jet Cutting Technology, BHRA Fluid Engineering, Cran®eld, 1986, pp. 287±295. [12] S. Matsui, Y. Matsumura, H. Ikemoto, H. Shimizu, I. Takada, Prediction equations for depth of cut made by abrasive water jet, in: T.J. Labus (Ed.), Proceedings of the Sixth American Water Jet Conference, Water Jet Technical Association, St. Louis, 1991, pp. 31±41. [13] Z.P. Bazant, J. Planas, Fracture and Size Eect in Concrete and Other Quasibrittle Materials, CRC Press, Boca Raton, 1997. [14] S.P. Shah, S.E. Swartz, C. Ouyang, Mechanics of Concrete, Wiley, New York, 1995. [15] A.W. Momber, R. Kovacevic, Fundamental investigations on concrete wear by high velocity water ¯ow, Wear 117 (1994) 55±62. [16] J.G. Bitter, A study of erosion phenomena, part I+II Wear 6 (1963) 5±21, 169±190. [17] A. Thiruvengadam, The concept of erosion strength, ASTM STP 408 (1967) 22±35.
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37 [18] I.M. Hutchings, The Erosion of Materials by Liquid Flow, MTI Publication No. 25, MIT, Columbus, 1986. [19] E. Kriegel, Der Strahlverschleiû von werkstoen, Chemie Inorg. Techn. 40 (1968) 31±36. [20] D. Ocepek, Uber die Methoden der Bestimmung des Zerkleinrungswiderstandes, Aufbereitungs-Technik (1969) 290±293. [21] R. Medeot, History, theory and practice of hydrodemolition, in: M.M. Vijay, G.A. Savanick (Eds.), Proceedings of the Fifth American Water Jet Conference, Water Jet Technical Association, Toronto, 1989, pp. 99±110. [22] A.W. Momber, Untersuchungen zum Verhalten von Beton unter der Belastung durch Druckwasserstrahlen, VDI Fortschrittsberichte, Nr. 109, VDI-Verlag, D usseldorf, 1992. [23] G.C. Sih, Mechanics of Fracture Initiation and Propagation, Kluwer Academic Publishers, Boston, 1991. [24] A.J. Zielinski, Model for tensile fracture of concrete at high rates of loading, Cement Concr. Res. 14 (1984) 215±224. [25] B.L. Atchley, H.L. Furr, Strength and energy absorption capability of plain concrete under dynamic and static loading, ACI J. (1967) 745±756. [26] G.C. Sih, D.Y. Tzou, Dynamic fracture rate of Charpy V-notch specimens, Theoret. Appl. Fract. Mech. 5 (1986) 189±203. [27] A.K. Maji, S.P. Shah, Application of acoustic emission and laser holography to study microfracture in concrete, in: H.S. Lew (Ed.), ACI-SP 112, American Concrete Institute, Detroit, 1988, pp. 83±109.
37
[28] J.M. Torrenti, J. Desrues, E.H. Benaija, C. Boulay, Stereophotogrammetry and localization in concrete under compression, ASCE J. Eng. Mech. 117 (1991) 1455± 1465. [29] A.W. Momber, The fragmentation of standard concrete cylinders during the compressive testing, Int. J. Fract. 92 (1999) L29±L34. [30] A.W. Momber, The fragmentation of standard concrete cylinders under compression: The role of secondary fracture debris, Eng. Fract. Mech. 67 (2000) 445±459. [31] F.H. Wittmann, K. Rokugo, E. Br uhwiler, H. Mihashi, P. Simonin, Fracture energy and strain softening of concrete as determined by means of compact tension specimens, Mater. Struct. 21 (1988) 21±32. [32] H.W. Reinhardt, Concrete under impact loading: tensile strength and bond, Heron 27 (1982) 5±41. [33] A.W. Momber, R. Kovacevic, Test parameter analysis in abrasive water jet cutting of rocklike materials, Int. J. Rock Mech. Min. Sci. 34 (1997) 17±25. [34] H.M. Hughes, Some aspects of rock machining, Int. J. Rock Mech. Min. Sci. 9 (1972) 205±211. [35] M. Mellor, Normalization of speci®c energy values, Int. J. Rock Mech. Min. Sci. 9 (1972) 661±663. [36] A.W. Momber, R. Kovacevic, Energy balance of highspeed abrasive water jet erosion, Inst. Mech. Engrs., J. Eng. Tribol. 213 (1999) 463±472. [37] A.W. Momber, H. Kwak, R. Kovacevic, Investigation in abrasive water jet erosion based on wear particle analysis, ASME J. Tribol. 118 (1996) 759±766.
www.elsevier.com/locate/tafmec
Stress±strain relation for water-driven particle erosion of quasi-brittle materials A.W. Momber * WOMA Apparatebau GmbH, Werthauser Straûe 77-79, 47226 Duisburg, Germany
Abstract This paper is concerned with the behaviour of quasi-brittle materials eroded by water-driven particles (WDP). Determined are the compressive stress±strain curves for the material. The measured relations are approximated by parabolic regression. The areas under these curves are then used to characterise the speci®c energy absorption capability of the investigated materials. Erosion is then investigated by WDP with varying water pressures and exposure time. The depth of penetration, the material volume loss, and the kerf geometry are measured. Based on these measurements, the speci®c energy of the erosion process is calculated. A characteristic eciency number U is then de®ned and estimated for all materials at dierent erosion conditions. It is shown that this parameter eectively characterises the erodability of the materials. Moreover, relations are found between the characteristic erosion parameters, such as threshold pressure, machinability number, depth of penetration, material removal rate, and the speci®c energy absorption capability. A preliminary phenomenological model based on the relationship between crack propagation processes and energy absorption capability is developed. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Water-driven particle erosion (WDPE) is the basic process of the abrasive waterjet technology ± a newly introduced machining concept that oers some advantages over many other traditional and non-traditional machining technologies, such as no thermal distortion, high machining versatility, low cutting force, and high ¯exibility [1]. On the basis of jet generation, WDPE can be categorised as injection-based or suspension-based. For practical applications, injection-based WDPE is more commonly used.
* Corresponding author. Tel.: +49-206-5304-380; fax: +49206-5304-200. Habilitation Fellow at the RWTH Aachen. E-mail address: [email protected] (A.W. Momber).
An injection jet is formed by accelerating small abrasive particles (usually garnet) through contact with a high-speed waterjet. The mixing between abrasive, water, and air takes place in a mixing chamber, and the acceleration process occurs in an acceleration focus. The abrasive particles leave the focus at velocities of several hundred meters per second. A high number of abrasives generates a high-frequency impingement on the material surface. The high-speed abrasive jets investigated are formed by this method. WDPE is a promising alternative method for the 3D-machining (milling, turning, drilling, polishing) of quasi-brittle materials, such as ceramics, concrete, rocks, and reinforced materials. Although several parameter studies have been identi®ed for ceramics, concrete and rocks, the general mechanism of the material removal process,
0167-8442/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 0 0 ) 0 0 0 4 6 - X
20
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
Nomenclature A; B; C constants b kerf width constants C0±5 focus diameter dF water jet nozzle diameter dw jet energy EA EAbs modulus of fracture absorbed energy per volume during EAbs compressive test dyn dynamically absorbed energy Eabs EM Young's modulus speci®c removal energy Esp h depth of penetration L kerf length abrasive ¯ow rate m_ a m_ w water ¯ow rate
especially in multiphase, quasi-brittle materials, is not well understood. Conducted in [2] is an SEM-study to understand the behaviour of polycrystalline ceramics in WDPE. Through grooving, sweeping and cutting experiments, a mixed material removal mode has been identi®ed that corresponds to brittle-fracture and plastic deformation. For low impact angle, scratching marks by single abrasive grains tend to dominate while intergranular cracking also appears. At perpendicular impact angles, intergranular cracking dominates the erosion process. Transgranular cracking has not been observed. It has been found [3,4] in a combined SEM-/ AE-study for WDPE of refractory ceramics that the certain failure mechanism depends on the kinetic energy delivered to the erosion site. The higher the kinetic energy, the higher the probability of transgranular fracture. It was also found that the type of bond between matrix and inclusion is important. For materials with directly bonded inclusions, such as magnesia chromite, a mixed material removal mode dominates. The inclusions are either exposed due to matrix removal or cut. In contrast, in a material with very pronounced interfaces, such as bauxite, the hard
Nm p pC PH Q v V_M VM a v U q qM r rC e ecr
machinability number pump pressure threshold pressure hydraulic pump power water volume ¯ow rate traverse rate volumetric removal rate volume removal nozzle eciency parameter correction parameter eciency parameter water density material density stress ultimate stress strain ultimate strain
inclusions resist the WDPE and are pulled out almost intact. Recently, the acoustic emission technique (AE) has been applied [5] to monitor the WDPE of quasi-brittle materials. It was found that materials with a low stiness, such as a mortar, fail by matrix erosion and separate inclusion pull-out. In contrast, materials with a high stiness, such as a high-strength concrete, are eroded by intense spalling fracture with cracks running through the matrix as well as the inclusions. Performed in [6] is an investigation of WDPE of marble and calcitic rocks. It was found that the amount and distribution of pre-existing ¯aws such as pores and cracks are essential for the erosion process. Both types of rock were worn by intergranular cracking and cleavage of calcite as well, but intergranular cracking dominates the erosion for calcareous rocks because of the poor bond between calcitic grains and the surrounding matrix. An elasto-plastic erosion model for brittle materials is developed in [2]. The material removal is assumed to be a combination of stresswave induced brittle (intergranular) fracture and plastic deformation. A crack network model relates the speci®c surface energy to the energy of
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
the stress-waves generated during particle impact. This approach oers some possibility to link the material erosion resistance to fracture mechanics parameters. The contribution of plastic ¯ow is considered in the model by employing a microcutting model for solid-particle erosion of ductile materials. Nevertheless, AE-measurements [5] have shown that intergranular fracture occurs mainly in materials with low homogeneity (small inclusions, low-strength matrix, large stiness deviations). Moreover, the probability of transgranular fracture in some directly bonded ceramics increases as the kinetic energy of the WDP increases [4].
2. WDPE resistance parameters For the WDPE of brittle-behaving materials, a number of resistance parameters are introduced. First, there is the so-called `threshold velocity' which has to be exceeded to introduce the macroscopic material removal process [1]. For eroding rocks and concrete by plain waterjets, the threshold velocity can be related to the critical stress intensity factor of the material [7,8]. A similar
21
parameter, a `threshold impact velocity', exists for the air-driven solid-particle erosion of brittle materials [9]. For WDPE, a so-called `machinability number' has been developed to describe the material resistance [1,10]. This material parameter is NM A
DM rF B cM EM rF
1
Due to reference tests, this parameter can also experimentally be estimated Nm
C0 h dw0:618 v0:866 : p1:25 m_ 0:687 m_ 0:343 w a
2
It can be seen from Eq. (2) that the depth of penetration is proportional to the machinability number. The smaller the machinability number, the higher the material resistance in WDPE. The `modulus of fracture' has been used to estimate the WDPE resistance of several materials [11,12]. This modulus of fracture was estimated as EAbs
r2T : 2EM
3
Fig. 1. Relation between modulus of fracture and WDPE parameters: (a) piercing by WDPE [11]; (b) cutting by WDPE [12].
22
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
Fig. 1 shows some of the investigated materials that did not ®t into the relations. If the resistance of conventionally ductile behaving materials and of ceramics is related to the modulus of fracture, then the ceramics fail, Fig. 1(a). If the resistance of ceramics and rocks are related to the modulus of fracture, then the rock materials would fail, Fig. 1(b). Obviously, due to their quasi-brittle characteristics, the simple procedure for estimating the modulus of fracture does not cover the more complex stress±strain behaviour. For both cases, the energy absorption capability of the precracked materials is underestimated in Fig. 1.
Fig. 2. Basic types of fracture, adapted from [13]: (a) brittle; (b) ductile; (c) quasi-brittle.
3. Stress±strain behaviour and failure modes 3.1. Stress±strain characteristics of quasi-brittle materials Generally, material behaviour can be categorised as brittle, ductile, and quasi-brittle. These types of material behaviour can be quanti®ed by the ratio between the non-linear zone and the linear-elastic zone around an existing crack as illustrated in Fig. 2. If the ratio is small, Fig. 2(a), the corresponding material is referred to as brittle such as glass and ceramics. If the ratio is large such that a certain amount of non-linear hardening and yielding occur in the non-linear zone as shown by Fig. 2(b), then the material is referred to as ductile such as metal alloys. If the ratio is large and the major part of the non-linear zone corresponds to microcracking or interface breakage, then the material is referred to as quasibrittle Fig. 2(c). The foregoing subdivisions can also be illustrated by the stress±strain relations. A typical stress±strain curves for a quasi-brittle material is given in Fig. 3. It is characterised by non-linearity, especially at high stress levels. The fracture process must pass through three stages: crack initiation,
Fig. 3. Diagrammatic stress±strain curves for concrete in compression, adapted from [14].
Result
Volume loss / 1=EAbs Volume loss / 1=EAbs General discussion Comminution resistance / n Erosion resistance / EAbs Erosion eciency / EAbs Piercing time / 1=EAbs (except ceramics) Cutting rate / EAbs (except rocks)
Loading type (material)
Solid particle erosion (ductile materials) Cavitation and drop impact (metals) Solid particle erosion (plastics) Impact fragmentation (rocks, minerals) Waterjet erosion (concrete)
AWJ piercing (ceramics, metals) AWJ cutting (ceramics, rocks, glass)
Stress±strain characteristics
Integrated product of stress and strain Ultimate resilience EAbs r2T =2EM Strain energy EAbs rF =2eF n EAbs =EEl (EEl ± elastic energy)R Absorbed fracture energy EAbs r e de
Modulus of fracture EAbs r2T =2EM Modulus of fracture EAbs r2T =2EM
Reference
Bitter [16] Thiruvengadam [17] Kriegel [19] Ocepek [20] Medeot [21] Momber [22] Hunt et al. [11] Matsui et al. [12]
Table 1 Stress±strain in relation to erosion and impact
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
23
slow crack growth, and rapid crack growth. For concrete as a quasi-brittle material, these stages are illustrated in Fig. 3. Toughening and energy absorbing mechanisms such as crack shielding, crack de¯ection and crack arrest on inclusions, and grain bridging, acting in a so-called `fracture process zone' in front of a macrocrack, posses such character [14]; they include concrete, rocks, cemented sands, and some ceramics. Several toughening mechanisms were observed not only under normal fracture, but also during waterjet erosion of concrete [15]. However, Fig. 3 shows that Eq. (3) does not cover the actual conditions as far as quasi-brittle materials are concerned. The area under the stress±strain curves is frequently assumed to represent the energy absorption capability of a certain material volume. This parameter is often termed as absorbed fracture energy, modulus of fracture, ultimate resilience, or strain energy density. 3.2. Stress±strain in relation to erosion and impact Several attempts have been made to associate the stress±strain functions of a material to the resistance against erosive loading, mainly based on Eq. (3). These attempts are summarised in Table 1. For the solid-particle erosion of materials, a material constant has been used in [16] to describe the plastic±elastic behaviour of materials; this constant is identical to the area enveloped by the stress±strain curves. The material removal rate is assumed to be inversely proportional to the energy absorption capacity. Developed in [17] is a concept for the erosion strength of metals for water drop and cavitation erosion. The ultimate resilience was also used, Eq. (3). A unique relation was found between this parameter and the volume loss due to cavitation erosion. Other approaches in the ®eld of liquid-¯ow erosion can be found in [18]. A very similar approach was used in [19]. This work also relates the material resistance against solid-particle erosion to the energy absorbed by the material under tensile test. Discussed in [20] is a comminution parameter which describes the resistance against impact fragmentation of minerals. This parameter is the ratio between the elastic defor-
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
1962 2107 2135 2275 2343
mation energy and the totally absorbed energy. Use of the energy absorbed during the compression of concrete was suggested in [21] for the definition of its resistance against waterjet erosion. This idea was further developed in [22] for describing the eciency of waterjet erosion based on the speci®c energy absorption during compression test. For WDPE, simpli®ed stress±strain curves were used in [11,12] for brittle and ductile materials to calculate the erosion eciency and the piercing time. As already illustrated in Fig. 1(b), the model in [12] fails for rocks which indicates that a simple linear (ideal) brittle approach does not characterise the behaviour of this group of material in the case of WDPE.
b
a
80 70 60 50 40 2.24 2.51 2.13 2.25 2.26 #1 #2 #3 #4 #5
Coarse limestone aggregates (maximum grain size 19 mm) and river sand (maximum grain size 4.7 mm). Portland cement, type I. c River sand (maximum grain size 4.7 mm).
858.1 603.6 516.4 499.5 467.9 55.2 60.4 78.2 80.8 84.7 2.1 11.1 38.9 62.6 72.8 0.85 0.71 0.45 0.38 0.32
10.7 24.1 33.0 34.3 42.3
4.0 12.5 27.1 34.2 41.1
pC (MPa) EM (GPa) Water/cement (%) c
Sand b
Aggregate a /cement Concrete mixture
Table 2 Mix design, mechanical properties and machining characteristics of the used materials
rC (MPa)
dyn Eabs MJ=m3
Nm (±)
qM kg=m3
24
3.3. Energy absorption during compression It is assumed that the compressive stress±strain function of a concrete as shown in Fig. 2 can be approximated by a parabolic equation r e A e ÿ B2 C:
4
This relation holds only for stress±strain functions obtained from a `soft', less rigid testing machine. Eq. (4) attains a physical meaning if B is assumed to be the strain at the ultimate stress and C is assumed to be the ultimate stress. This leads to 2
r e A e ÿ ecr rC :
5
The energy absorbed per volume during the compression test is simply given by the area under the stress±strain curves Z ecr 1 r e de Ae3cr ecr rC : 6 EAbs 3 0 The superscript * stands for quasi-brittle materials to distinguish the relation from Eq. (3). The parameters A, ecr and rC have to be estimated by compression tests. The quantity in Eq. (6) has been referred to in fracture mechanics as the strain energy density function [23] and used successfully for characterising the fracture behaviour of brittle and ductile materials, glass, ceramics, metal alloys etc.
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37 Table 3 Erosion conditions and parameters
dyn _ rC EAbs EAbs v r; :
Parameter
Symbol
Unit
Range
Pressure Traverse rate Abrasive ¯ow rate Nozzle diameter Focus diameter Abrasive size Abrasive type
p v m_ a dw dF ± ±
MPa mm/s g/s mm mm mesh ±
100±350 2.0±12.0 6.12±19.05 0.457 1.270 #36 garnet
The application of Eq. (6) is limited to static loading while WDPE is a highly dynamic process. A correction parameter v may be introduced to account for stress rate eects on the energy absorption capability of concrete. It was observed in [24,25] that energy absorption is signi®cantly higher for impact loading compared to static loading. A relation between the stress rate and the energy absorption was found in [25] for estimating the parameter v which attains a saturation value at very high stress rates. The in¯uence of the compressive strength on v is approximated from the values given in [25] (see Table 3). Introducing this correction parameter, the dynamic absorbed energy can be calculated from
25
7
A dynamic fracture analysis of the Charpy Vnotch specimen has been made in [26] using the strain energy density function or the absorbed energy which applies equally well to the dynamic case.
4. Experimental setup and experimental work In this work, ®ve dierent concrete mixtures were designed. Information on the mix designs is given in Table 2. These mixtures were cured and hardened for 28 days under water. After hardening, the cylindrical specimens were tested. The specimen dimensions were 14 cm in height and 7 cm in diameter for the compression tests, Fig. 4(a); and 30.48 cm in height and 15.24 cm in diameter for the erosion tests, Fig. 4(b). To minimise lateral restraint stresses, the top and bottom of the specimens used for the compression tests were covered with thin rubber plates. The compressive strength was estimated according to ASTM Standard C 39; the modulus of elasticity was measured as the static
Fig. 4. Specimens used for the experiments: (a) estimation of the stress±strain curves; (b) WDPE tests.
26
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
the generated kerfs were measured, including depth of penetration, kerf width, and kerf length, Fig. 4(b). These values were used to calculate the material removal rate. Based on measurements of 18 erosion tests under dierent erosion conditions, the machinability number of each material was calculated using Eq. (1). The results are given in Table 2. Optical microscopy and SEM records magni®ed up to 25 were used for erosion site inspection. 5. Compressive test results and discussion 5.1. General failure modes
Fig. 5. Compression stress±strain curves of the investigated concrete mixtures.
chord modulus according to ASTM Standard C 469. The stress±strain curves of each mixture was estimated by loading and unloading the specimen with strain-gauge equipped 11 times. The results of these measurements are given in Table 2. After the tests, the fracture debris were collected and analysed by sieve analysis using a conventional sieve shaker. The test device used to perform WDPE consists of an intensi®er pump, abrasive mixing and acceleration head, abrasive storage and metering system, catcher, and CNC-controlled positioning system. The position of the mixing and acceleration head was controlled using an x-y-z-positioning table. Erosion parameters and conditions are listed in Table 3. After erosion, the dimensions of
Fig. 5 shows the stress±strain test results that are approximated by a parabolic equation as given by Eq. (4). The regression parameters are listed in Table 4. The absorbed energy per volume was calculated via Eq. (6). The estimated static values are between 1:5 and 47:6 MJ=m3 . This is of the same order of magnitude as values published in [21,22]. The curves can be divided into several stages 1±4, as shown in Fig. 3. Up to point 1, the stress increases linearly with increasing strain. After that, pre-existing micro¯aws (mainly interfacial ¯aws at the aggregate-cement paste-interface) start to open. They propagate into the matrix and coalesce into cracks. Detected in [27,28] is a strain localisation at this stage that accelerates the failure process. The major cracks propagate stably and one of them reaches the critical length. This stage is marked as point 4. At that point, the peak stress is reached and failure occurs. Dependent on the ®nal failure mode, the specimen is separated at least into two major
Table 4 Regression parameters of the stress±strain curves Concrete mixture
A
ecr
rC [MPa]
Eabs MJ=m3
va
#1 #2 #3 #4 #5
)12.5 )13.9 )12.1 )10.6 )13.6
0.57 0.95 1.50 1.80 1.73
4.0 12.5 27.1 34.2 41.1
1.5 7.9 27.0 40.9 47.6
1.40 1.40 1.44 1.53 1.53
a
Taken from [25].
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
parts: A low number of primary debris is ®rst obtained. (For non-covered or slender specimens this number is typically two.) Fine-grained secondary debris are then observed. Fig. 6 shows primary debris from mixtures 1 and 5, respectively. Both mixtures exhibit dierent failure behaviour. Concrete 1 in Fig. 6(a) is characterised by the cone-type failure as a result of shear failure. In contrast, mixture 5 in Fig. 6(b) corresponds to the splitting mode characterised by a very brittle fracture. It is also very signi®cant that the number of broken aggregates on the surface of these cones depends on the concrete mixture. It is about zero for specimen 1 and about 100% for mixture 5. Additional results of these experiments including the discussion of secondary debris, can be found in [29,30].
27
5.2. Energy absorption eciency Secondary debris have not attracted much attention in the past although they may contain information about the fracture process, especially about the energy balance during failure. This information was acquired in [30] with reference to sieve analysis and debris image analysis. The speci®c crack energy and the speci®c crack length were calculated. They are plotted in Fig. 7. It can be seen that the energy required to generate a crack network with a given length increases linearly with the absorbed fracture energy, Fig. 7(a). This result is not unexpected and can be used to relate the energy absorption capability of the materials to the microcracking processes in the compression test. An interesting result is that the dierence in the absorbed energy during com-
Fig. 6. Primary fracture debris from the compressive failure of cylindrical concrete specimens: (a) concrete 1 (conical); (b) concrete 5 (splitting).
28
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
Fig. 7. Fracture parameters obtained from the compression tests, adapted from [30]: (a) speci®c crack energy; (b) speci®c crack length.
pression is not caused by the dierence in crack length which is approximately identical, Fig. 7(b). Rather, it is due to dierent crack path. The cracks run preferably through the interface between cement matrix and aggregate in samples 1 and 2; they run through the aggregates in samples 4 and 5 giving rise to transgranular fracture. This may be a result of the coarser aggregates as well as of the higher matrix strength of these samples; both effects increase the material stiness which is further enhanced by the higher values of the Young's modulus for the stronger mixture. It was observed in concrete that the work of fracture increases with the aggregate size [31]. The probability of transgranular fracture also increases with cement matrix strength [32]. Due to the higher speci®c surface energy of the aggregate material, which is more than two orders of magnitude higher than those of the aggregate±matrix interface, more energy is absorbed during transgranular fracture at comparable crack length. Assuming that the WDPE of quasi-brittle materials is characterised by the generation and intersection of a microcrack network [2], there should be a signi®cant dependence of the behaviour of the dierent material mixtures on the resistance against WDPE.
6. Erosion test results and discussion 6.1. Threshold pressure The results of the erosion depth measurements are presented elsewhere, but not discussed in terms of erosion resistance [33]. Fig. 8 shows a relation between operating pressure and depth of penetration. The depth of penetration increases with the pressure. At high pressure levels, the progress of the function drops indicating a reduced energy eciency. The relation for each mixture was approximated by a second-order polynomial, and the intersection of the polynomial and the pressure axis is de®ned as the `threshold pressure'. The estimated values are given in Table 2. It is an experimentally de®ned parameter and describes the pressure required to introduce the visible material removal process. Because the abrasive particles are accelerated by a high-speed waterjet, the threshold pressures describes a critical particle velocity. A simpli®ed relation between water pressure and particle impact velocity is [1] given by p 8 vPC / pC : As shown in [9] for brittle, pre-cracked materials, the threshold impact velocity depends on the
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
Fig. 8. Relation between pump pressure and erosion depth (concrete 1, v 4:0 mm/s, m_ a 7:41 g/s).
wave velocity and the fracture toughness of the material. This ®nding is veri®ed for WDPE in [33]. Alternatively, it may be assumed that the visible material removal process starts if a certain (small) volume of material is removed. This `threshold volume' is a constant geometrical parameter for any material. If the pressure is lower than that of threshold, then the generated crack network is not dense enough and no intersection of penetrating cracks occurs aside from those occurred spontaneously. This period is assumed to be that of incubation. To erode this threshold volume, a certain amount of energy is required, i.e., dyn C pC3=2 f pC EC VC EAbs
9
with E / p3=2 [1]. The function f pC considers the relationship between operating pressure and energy transformation during waterjet formation, abrasive mixing and acceleration, and material erosion that in¯uences the kinetic energy of the eroding jet [1]. In the narrow range of threshold pressures, f pC can be neglected, and hence 0:67 dyn : 10 pC C1 EAbs
29
Fig. 9. Relation between absorbed fracture energy and threshold pressure (v 4:0 mm/s, m_ a 7:41 g/s).
Linearisation yields dyn ln pC 0:67 ln EAbs C2 :
11
In Fig. 9, the threshold pressure is plotted against the absorbed energy in a double-logarithmic diagram. The data trend ®ts qualitatively using Eq. (11). This indicates a relationship between the material resistance against WDPE and the failure under compression. The threshold pressure shows the dierent energy absorption capability of the concrete during crack propagation in the incubation pressure range. For a pressure of p 80 MPa, material removal has already occurred in the materials 1±4, whereas no visible erosion could be seen in concrete 5. Fig. 7(a) shows that a higher amount of energy is required for concrete 5 to produce a network of cracks of a given length. The analytical curve is about ®ve times higher than that of the experimental curve. The absolute in¯uence of the absorbed fracture energy is overestimated by Eq. (10). This may be contributed by the localised material attack for WDPE. The jet starts to remove material at the points of the lowest resistance in concrete. A certain range of
30
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
these weak zones is distributed over the surface. Tracking a 15 cm long erosion path, the jet hits one of the weak zones with a certain probability. This probability is about the same for the two lowstrength concretes (the average deviation in threshold pressure in about 5%), and it is about the same for the three stronger concretes (the average deviation in threshold pressure is about 4%). Under compression, a few weak zones do not lead to specimen fracture as far as the stresses are not forced in a general direction, even if a failure stress is generated. Whereas, material is immediately removed in WDPE. 6.2. Machinability number Since the absorbed energy characterises the behaviour of quasi-brittle materials in WDPE, it should be related to the machinability number being estimated between Nm 470 and Nm 860 as given in Table 2. They correspond to those of asphalt concrete, marble and glass [1]. The results plotted in Fig. 10 suggest a relationship that can be ®tted by ÿC4 dyn : 12 Nm C3 EAbs
Fig. 10. Relation between absorbed fracture energy and machinability number.
Note that the higher the energy absorbed during the compressive test, the lower is the machinability number. This implies a higher material resistance against WDPE.
Fig. 11. Relation between absorbed fracture energy and eciency for dierent erosion conditions (m_ a 7:41 g/s).
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
31
Fig. 12. Surfaces generated by WDPE (p 200 MPa, v 8:0 mm/s, m_ a 7:41 g/s). Left: Concrete 1; Right: Concrete 5.
6.3. Erosion eciency The eciency of WDPE can be evaluated by relating the energy absorbed for the compressive test to the speci®c energy involved in material removal. Attempts were made in [34,35] to relate the `machinability' of rocks by conventional tools to the behaviour during compression test. For the present case, a non-dimensional eciency parameter U is de®ned: U
dyn EAbs
Esp
:
The energy portion 1 ÿ UEA is partly dissipated at the erosion site due to damping and friction, and is carried away partly by the exiting abrasivewater-debris suspension after erosion [36]. The estimated eciency parameters are plotted against the absorbed fracture energy in Fig. 11. The eciency values are between U 2:9 10ÿ5 and U 1:2 10ÿ3 ; they increase linearly with the energy absorption capability: dyn b1 : U a1 EAbs
17
13
The energy required to remove a certain volume of material by WDPE is Esp
EA : VM
14
Here, EA is the kinetic energy of the eroding jet given by [1] s Z xL a2 u2 pdw2 2 3=2 p dx: 15 EA q 4v x0 The removed volume VM has to be estimated by erosion tests. Using Eqs. (7), (8) and Eqs. (13)± (15), the eciency of WDPE can be calculated: 3 ecr rC VM vp 4v 0:3Ae q p cr : 16 U 2a2 u2 pdw2 p3=2 L
Fig. 13. Relation between jet energy, machinability number, and erosion eciency.
32
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
With a1 1=Esp and b1 0 this is identical to Eq. (13). This trend, which was also observed for concrete erosion by waterjets [22], expresses the dierent crack propagation modes in the material. As pointed out earlier, the concrete mixtures with the higher energy absorption capability show
a more brittle failure with large fracture debris and transcrystalline fracture planes. This character prevails also for the WDPE as shown in Fig. 12 that displays photographs of eroded cross-sections from two concrete samples. Note that the crosssection of concrete 1 is very rough and shows
Fig. 14. Micrographs from the erosion sites: (a) concrete 1 (optical 1:6); (b) concrete 2 (SEM, 1:25); (c) concrete 4 (SEM 1:25); (d) concrete 5 (optical 1:25).
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
predominatly intergranular fracture, while the cross-section of mixture 5 is smooth and exhibits predominatly transgranular fracture. One can assume that this latter behaviour allows the generation of larger fracture debris in the concrete with the high energy absorption capability during
33
WDPE. More evidence is given in Fig. 7(a). It shows that for comparable speci®c crack length the energy required for crack generation increases with the concrete strength. Even if the transgranular fracture in concrete 5 reduces the crack length, this process absorbs more energy because
Fig. 14. (continued).
34
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
of the high speci®c surface energy of the aggregate material. The regression parameters a1 and b1 in Eq. (17) depend on pump pressure, traverse rate and hence on the jet energy. From Fig. 11, the eciency decreases with increasing pressure and decreasing traverse rate. Higher jet energy probably leads to worse energy transfer in the given parameter range. This trend agrees with the results in [37]. On the average, just 0.1% of the jet energy is required to separate material. This value is one order of magnitude lower than those estimated in [37]: Based on wear debris analysis for cast iron, about 2% of the jet energy was found to be absorbed by the generation of fracture surfaces. This could be caused by the lower machinability number of cast iron (Nm 121). As already shown in Fig. 10, the eciency increases as the machinability number drops. It can thus be concluded that the eciency increases as the ratio between machinability number and jet energy drops. This is illustrated in Fig. 13. By using waterjets for eroding concrete, [22] estimated an eciency of about U 0:05 (5%), which is relatively high. This may be attributed to the fact that a plain waterjet is able to break large pieces from brittle specimens. The eciency of tools for the removal of rocks and rocklike materials
drops signi®cantly as the tool size decreases [34]. For the WDPE with extremely small tool dimensions (abrasive particle diameter dP 300 lm, jet diameter dJet 1:27 mm), the removal eciency may be comparatively low. 6.4. Microscopic investigations and AE-measurements An inspection of the surfaces arising from WDPE gives results similar to those obtained from the compression tests. Fig. 14(a) shows that almost all inclusions are unbroken in concrete 1 after WDPE. Some broken grains are found on top of the cut surface. Also, striations are found running from top to the bottom due to intergranular matrix removal. The situation is very similar in concrete 2 shown in Fig. 14(b). Again, intergranular fracture can be noticed accompanied by grain pullout (upper center and right section). A large striation can also be noticed (left from the center). In contrast, sample 5 shown in Fig. 14(d) shows broken aggregate grains. A grain, cut with the matrix, can be seen at the top right. This grain also shows several striation marks. The eroded surface is very smooth; the grain boundaries are intact and adopt a sharp appearance. These results suggest
Fig. 15. Relation between absorbed fracture energy and AE-signal parameters (AE-signal parameters taken from [5], Table 3): (a) signal amplitude; (b) signal frequency.
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
dierent mechanisms of material removal for the mixture. For mixture 1, the matrix is removed only without cutting the aggregate. Matrix and aggregate are cut simultaneously in mixture 5. Mixture 4, shown in Fig. 14(c) exhibits a mixed material removal mode where both grain pull-out and brittle fracture can be found. These ®ndings are supported by Fig. 15 that shows the in¯uence of absorbed energy on AE signals acquired on-line during WDPE [5]. Two trends can be noticed from the graphs. First, the signal amplitude tends to reduce with an increase in the absorbed fracture energy for mixtures 1±4. This is independent of the exposure time (which is the reverse of the traverse rate). Next, the AE-signal energy increases for concrete 5, Fig. 15(a). The same trend prevails between the absorbed energy and the peak frequency of the AE-signal. Here, the frequency rises with an increase in the absorbed energy for mixtures 1±4. For concrete 5, the frequency drops, Fig. 15 (b). Switching in the trends for the two AE-signal parameters is due to a change in the erosion mode. Mixtures 1±3 are eroded by intergranular grain pullout and independent matrix removal. Mixture 5 shows a transgranular fracture mode during WDPE. Mixture 4 shows a mixed mode which is probably responsible for the high frequency value. Most probably, there is a transition range in the absorbed energy between 50 and 70 MJ=m3 where a change in the fracture mode from intergranular erosion to transgranular fracture has occurred.
dyn EAbs U
PH : hbv
35
20
Assuming that the width is independent of the erosion depth, and using Eq. (17), Eq. (20) solves for the erosion depth as given by dyn a1 EAbs b1 pQ_ : 21 h dyn bv EAbs The experimentally estimated depths are plotted against the theoretical values in Fig. 16. Despite a few exceptions, the results ®t reasonably by Eq. (21). The strayed points away from the line belong to the low-strength concrete 1. The reason is that WDPE did not produce straight kerfs as assumed in Eq. (21). Hence, the parameter b could be replaced by a function b h. The general in¯uence of pump pressure and traverse rate are also included in Eq. (21). 8. Summary and conclusions A systematic study has been made for the energy absorbed during compression test of quasibrittle and multiphase materials subjected to
7. Volume removal rate and depth of cut Volume removal rate and erosion depth can be estimated from a known stress±strain curves. An energy balance and Eq. (17) give dyn U EAbs
and V_M
PH V_M
dyn PH a1 EAbs b1 dyn EAbs
18
:
The assumption V_M hbv leads to
19 Fig. 16. Comparative plot between experimental results and Eq. (21).
36
A.W. Momber / Theoretical and Applied Fracture Mechanics 35 (2001) 19±37
WDPE. The absorbed energy is estimated by calculating the area under the stress±strain curves of ®ve concrete materials. Moreover, fracture tests for compression and erosion tests with WDP are performed to estimate the erodability. The following results can be concluded: · The amount of energy absorbed during compression can be related to failure by crack propagation. In a material with high energy absorption capacity, the cracks run preferably through the aggregate. This process is more energy consuming because of the higher speci®c surface energy of the inclusions. · The absorbed energy can be related to WDPE threshold pressure that may describe the establishment of a microcrack network at the ®nal stage. · There exists a relationship between the energy absorption capability during compression and erodability by WDP. · Material removal rate and depth of penetration are related to the energy absorbed during compression. · The eciency of WDPE of quasi-brittle and multiphase materials is about U 0:001. However, this is estimated from volume removal, which is not a typical process for commercial WDPE applications. · The eciency increases with absorbed fracture energy and decreases with machinability number. This suggests that WDPE is more eective for machining highly resistant materials. · Material removal mode for WDPE depends on energy absorption capability. Materials with low energy absorption capability will be eroded by intergranular fracture. In contrast, materials with high energy absorption capability will be eroded by transgranular fracture.
Acknowledgements The author is thankful to the German Research Foundation, Bonn, for ®nancial support. Thank is addressed to WOMA Apparatebau GmbH, Duisburg, Germany, and RWTH Aachen, Aachen, Germany, for administrative support.
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[28] J.M. Torrenti, J. Desrues, E.H. Benaija, C. Boulay, Stereophotogrammetry and localization in concrete under compression, ASCE J. Eng. Mech. 117 (1991) 1455± 1465. [29] A.W. Momber, The fragmentation of standard concrete cylinders during the compressive testing, Int. J. Fract. 92 (1999) L29±L34. [30] A.W. Momber, The fragmentation of standard concrete cylinders under compression: The role of secondary fracture debris, Eng. Fract. Mech. 67 (2000) 445±459. [31] F.H. Wittmann, K. Rokugo, E. Br uhwiler, H. Mihashi, P. Simonin, Fracture energy and strain softening of concrete as determined by means of compact tension specimens, Mater. Struct. 21 (1988) 21±32. [32] H.W. Reinhardt, Concrete under impact loading: tensile strength and bond, Heron 27 (1982) 5±41. [33] A.W. Momber, R. Kovacevic, Test parameter analysis in abrasive water jet cutting of rocklike materials, Int. J. Rock Mech. Min. Sci. 34 (1997) 17±25. [34] H.M. Hughes, Some aspects of rock machining, Int. J. Rock Mech. Min. Sci. 9 (1972) 205±211. [35] M. Mellor, Normalization of speci®c energy values, Int. J. Rock Mech. Min. Sci. 9 (1972) 661±663. [36] A.W. Momber, R. Kovacevic, Energy balance of highspeed abrasive water jet erosion, Inst. Mech. Engrs., J. Eng. Tribol. 213 (1999) 463±472. [37] A.W. Momber, H. Kwak, R. Kovacevic, Investigation in abrasive water jet erosion based on wear particle analysis, ASME J. Tribol. 118 (1996) 759±766.