A transition function for the solid particle erosion of rocks

Wear 328-329 (2015) 348–355

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A transition function for the solid particle erosion of rocks A.W. Momber n Aachen University of Technology (RWTH Aachen), Faculty of Georesources and Materials Technology, Wolfgangsweg 12, D-20459 Hamburg, Germany

art ic l e i nf o

a b s t r a c t

Article history: Received 13 November 2014 Received in revised form 3 March 2015 Accepted 4 March 2015 Available online 13 March 2015

A simple two-parameter model K 1 ¼ 1  exp ½  ðλ U χÞb  is derived, which can describe the response of rock materials during the erosion through rounded solid particles at normal incidence. A procedure for estimating the distribution parameters λ and b is proposed, and the procedure is applied to four rock materials (rhyolite, granite, limestone, and schist). Erosion experiments with quartz particles are performed in order to estimate K1 numbers for the rock materials. It is shown that the model covers different types of material response. The parameter K1 can be linked to general rock classification schemes. & 2015 Elsevier B.V. All rights reserved.

Keywords: Erosion Impact Rock mechanics Weibull distribution

response to lateral cracking for geomaterials is presented earlier [2]; it reads like follows:

1. Introduction and procedure The impingement of solid particles on brittle materials is associated with a number of failure modes, namely plastic deformation, ring or cone crack formation, and radial or lateral crack formation. The former mode is usually referred to as ductile failure, whereas the latter modes are referred to as brittle failure (elastic failure or elastic– plastic failure). Additionally, erodent particle fragmentation occurs. The occurrence of an individual failure mode is governed by threshold, or transition, conditions [1]. In an earlier issue of this journal [2], the author applied an erosion model [3], which combines erosion due to plastic deformation (EPR) and due to lateral cracking (ELR) to geomaterials ER ¼ K 1 U EPR þ K 2 UELR

ð1Þ

A procedure for the estimation of the two parameters K1 and K2 does not exist yet. This communication deals with a calculation procedure for rock materials. The first step is the reduction to a single-parameter model. This can simply be done for the condition K2 ¼1  K1, which delivers ER ¼ K 1 UEPR þ ð1  K 1 Þ U ELR

ð2Þ

The constant K1 balances the amount of either erosion mode. The following conditions apply: For K1 ¼ 0, lateral cracking dominates the erosion process, and elastic–plastic erosion models shall be valid. For K1 ¼ 1, ductile erosion modes (ploughing, lip formation, platelet formation) dominate the erosion process, and erosion models for ductile materials (like soft metals) shall be valid. For values 0oK1 o1, mixed mode erosion occurs. A criterion for the transition from plastic n

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http://dx.doi.org/10.1016/j.wear.2015.03.005 0043-1648/& 2015 Elsevier B.V. All rights reserved.

23=4 χ ¼ K 12=4 =σ C Ic

ð3Þ

In the equation, χ is a transition number, KIc is target material fracture toughness, and σC is target material compressive strength. Numbers 23/4 for the ratio K12/4 are provided in Table 1 for relevant materials. Ic /σC Low numbers characterize a preference for lateral cracking [K1-0 in Eq. (2)], and vice versa. Fracture toughness becomes the dominating erosion resistance parameter if χ decreases. For high χ-numbers, other resistance parameters become important, namely hardness, respectively compressive strength. Compressive strength is also the governing parameter in Eq. (3) because of its higher power exponent. For the three frame conditions: (i) K1 ¼0 for χ ¼0; (ii) K1 ¼1 for very high χ-numbers; (iii) inflection point between erosion modes, the relationship between χ and K1 can be approximated with a two-parameter Weibull distribution function [8] K 1 ¼ 1  exp½ ðλ U χ Þb 

ð4Þ

In that equation, λ is a scale parameter, and b is a shape parameter. A method for the estimation of λ and b is provided in [9]

 1 1 ln ln ; ð5:1Þ ¼ b U ln χ  b U ln 1 K1 λ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} A

λ ¼ exp




 A  b

ð5:2Þ

Shape and scale parameter can be estimated if Eq. (5.1) is fitted to experimentally estimated K1-values. Values for K1 for particular materials and erosion conditions can be estimated through microscopic inspections of eroded surfaces. The following simple

A.W. Momber / Wear 328-329 (2015) 348–355

Table 1 Numbers for the transition criterion χ ¼K12/4 /σ23/4 for different geomaterials. Values Ic C for σC and KIc are taken from [2,4–7]. Material

σC (MPa)

KIc (MN/m3/2)

χ (MPa  11/4 m3/2)

Arenaceous shale Argillaceous schist Äspö diorite Aue granite Basalt Basalt Basalt Carrara marble Coral limestone Dolomite Dolostone (Falkirk) Dolostone (Kankakee) Dolostone (Markgraf) Dolostone (Oatka) Dolostone (Remeo) Granite (Newhurst) Granite (Portuguese) Flechtinger sandstone Jurassic limestone Marble (coarse) Mizunami granite Ogino tuff Porphyric rhyolite Rüdersdorf limestone Ruhr sandstone Ryefield sandstone Sandstone (coarse) Siltstone Syenite White limestone

143 50 219 166 274 120 187 101 59 95 172 152 177 142 264 175 160 96 55 40 134 55 240 40 95 35 33 51 233 50

2.12 2.70 3.83 2.38 2.27 1.80 3.01 2,44 1.32 1.47 1.66 1.66 1.80 1.78 2.47 1.75 0.80 1.15 1.21 1.12 1.60 0.60 1.17 1.11 1.28 1.04 0.27 0.80 1.73 1.38

3.85  10–12 3.35  10–9 1.96  10–12 2.31  10–12 1.12  10–13 6.46  10–12 2.36  10–12 4.34  10–11 1.51  10–10 1.35  10–11 6.40  10–13 1.30  10–12 6.92  10–13 2.37  10–12 1.70  10–13 4.90  10–13 1.09  10–13 6.08  10–12 1.74  10–10 8.63  10–10 2.41  10–12 3.83  10–11 3.30  10–14 8.63  10–10 8.91  10–12 1.49  10–9 4.07  10–11 7.78  10–11 1.68  10–13 4.47  10–10

approaches can be utilized: NP K 1 ¼ Pn

i¼1

NL K 1 ¼ 1  Pn

i¼1

Ni

was argillaceous schist with a layered structure. The layers could be identified as a white, quartz-rich pale-band, and a dark band containing a high amount of mud and organic substances. Pronounced cleavage could be noticed as the specimens were loaded parallel to the layers. The properties listed in Table 2 were estimated perpendicular to the layer structure. Erosion conditions and erodent properties are listed in Table 3. The feeder mechanism of the erosion device allowed steady and reproducible particle delivery. Erodent particles entered into the stream of air under the suction provided by a constriction in the inlet section. The particles were then accelerated by the air flow up to an appreciable fraction of the air speed. The particle velocity was calibrated through the pressure of the air delivery system. The targets were mounted a fixed distance from the end of the blasting nozzle, allowing precise control over the exposure position and the impact angle. An impact angle of 901 was applied for all experiments. The exposure time was 30 s, which was in the range of steady-state erosion. Three erosion spots with a cross section of 0.78 cm², generated at an impingement velocity of 140 m/s, were inspected under scanning electron microscopes at magnifications between 10  and 40,000  . Two types of SEM were used. The first microscope was type “JEOL 840 SEM” with the following parameters: scanning distance 3.9 mm; voltage 15 kV; electric current 6  10–9 A. The second microscope was type “Zeiss Supra VP55” with the following parameters: scanning distance 4–9.8 mm; voltage 10 kV; electric current 10–9 A. The samples were sputtered with gold (5–8 nm). Each spot was separated into four sections, and each section was inspected in detail in terms of erosion modes (see Figs. 1a, 2a, 3a and 4a). A total of 250 individual impact sites were

Table 2 Properties of the investigated rock materials.

ð6:1Þ

Ni

349

Material

Splitting tensile strengtha (MPa)

Density (kg/m³)

Fracture toughnessb (MN/m3/2)

Compressive strengthc (MPa)

Young's Modulusd (GPa)

Granite Limestone Rhyolite Schist

10.7 10.7 12.2 –

2,500 2,500 2,700 2,600

0.80 1.21 1.17 2.70

160 55 240 50

52 82 45 –

ð6:2Þ

In the equation, NL is the number of impact sites showing lateral cracking, NP is the number of impact sites showing ductile erosion, i is the number of inspected impact sites. With known numbers for K1 (from the counting procedure) and for χ (from materials testing; see Table 1), values for λ and b can be estimated, and Eq. (2) can be quantified for different erosion conditions.

a

Brazilian disc splitting tensile test. Notch bend three-point bending test. c Uniaxial cylinder test. d Slow-deformation three-point bending test. b

2. Experimental procedure Experiments were performed on four rock materials, namely rhyolite, granite, limestone, and schist. Their mechanical properties are listed in Table 2. The granite was Portuguese granite with a crystalline structure. The rock forming minerals were mica, quartz and feldspar. The structure was dense. The fracture behaviour was dominated by the cleavage of the minerals. Due to tectonic loading, a pronounced pre-existing micro-crack net was formed in the material. The porphyry was a porphyric rhyolite, consisting of a matrix (approximately 50 vol%) and embedded coarse particles. Major mineral components were potassium feldspar, sodium feldspar and quartz. The non-crystalline matrix was dense and finegrained (average matrix particle size about 0.1 mm). The inclusions had a maximum grain size of about 13 mm. The limestone was a sedimentary Jurassic limestone consisting of a fine-grained matrix with an average grain size in the 1/10-mm-range and embedded broken shells. These organic inclusions may lead to local strength reduction. However, the calcitic matrix was very dense. The schist

Table 3 Experimental conditions and erodent properties. Parameter

Value

Nozzle diameter Stand-off distance Erodent mass flow rate Angle of impingement Impingement velocity Erodent type Particle size Particle roundnessa Erodent density Erodent indentation hardnessb Erodent fracture toughnessb Erodent Poisson's ratioc Erodent Young's modulusc

10 mm 80 mm 0.45 g/s 901 140 m/s Quartz sand 0.3–0.6 mm 1.32 2,650 kg/m³ 12 GPa 1.6 MPa m1/2 0.17 87 GPa

a b c

Elongation ratio. Ref. [10]. Ref. [11].

350

A.W. Momber / Wear 328-329 (2015) 348–355

L

L L

L

L

L

L

L

L

L L

L

L

L

L

L L

L

L L

L

L

L

L L

L

L

L

L

1 2

Fig. 1. SEM images of eroded rhyolite sections. (a) General sectional view with lateral cracking (denoted “L”). (b) Overlap of individual impact sites (circled). (c) Magnification of a lateral cracking site, illustrating modified cracking processes described in [12,13]; “L” – Laterally cracked sections. (d) Fracture marks in quartz. “1” – Fractal (Cantor set) crack lance front (see [13]); “2” – Coalescence of fracture marks (see [13]).

categorized for each material according to brittle or ductile removal mode, and K1 values were calculated with Eqs. (6.1) and (6.2).

3. Results and discussion SEM images of erosion sites in rhyolite are provided in Fig. 1. Fig. 1a illustrates the dominance of lateral fractures (denoted “L”). Fig. 1b shows intersecting lateral fractures leading to material removal. An individual impact site is provided in Fig. 1c. The plastically deformed cavity in the centre can clearly be distinguished. The lateral vent cracks, grouped around the centre, are denoted “L”. They differ from lateral cracks formed in isotropic brittle materials (see Table 4, upper row). This particular feature agrees with observations reported in [12] for a number of rock materials; this author attributed it to the existence of pre-existing cracks. Lateral vent cracks do not grow to the free surface immediately, but follow either weak mineral interfaces or pre-existing cracks. Fig. 1d illustrates fracture marks in quartz in more detail. The right section features conchoidal fractures, whereas the left section is characterized through the formation of stepwise fractures. All fracture surfaces feature fracture marks, namely fracture lance fronts. Two very typical examples are marked. Arrow “1” highlights the formation of a fractal (Cantor-set) fracture front as reported in [13] for glass indented with tungsten carbide balls. Arrow “2” marks the coalescence of fracture marks into crack growth direction, or lance coarsening [13]. Because lance formation occurs under localized mixed loading regimes, these particular features support an approach of a mode-II crack propagation mode being active during lateral crack formation due to solid particle impingement [14]. The results also support the conclusion drawn in [8], that erosion of rocks due to solid particle impingement is insensitive to mode-I R-curve behaviour. In

general, features of brittle fracture dominate the material removal process on the microscopic scale. SEM images of erosion sites in granite are provided in Fig. 2. Again, Fig. 2a illustrates the dominance of lateral fractures (denoted “L”). Fig. 2b shows intersecting lateral fractures leading to material removal. The typical damage structure, consisting of a plastically deformed cavity and lateral vent cracks, can be recognized. An individual lateral fracture section is provided in Fig. 2c. A striking feature is the formation of fracture lances (arrowed) in the outer section of the fracture plane. A further magnification (Fig. 2d) reveals the formation of brittle fracture marks, namely fracture steps and very regular fracture facets in the central region. A further magnification in Fig. 2e very well illustrates the partial adjustment of individual crack planes along the crack front (crack front formation from bottom to top). Similar to Fig. 1d, these features confirm a mixed loading regime as well as the insensitivity of the erosion process to mode-I Rcurve behaviour [8]. In general, brittle fracture features dominate the material removal process on the microscopic scale. SEM images of erosion sites in limestone are provided in Fig. 3. In contrast to Figs. 1 and 2a, a mixed material removal mode can be recognized in Fig. 3a and b. Fig. 3a shows a large number of plastically deformed impact craters (“P”), along with features of material pile-up and lip formation as suggested for the erosion of ductile metals at high angles of impingement [15,16]. These material removal modes dominate the erosion process. Features of lateral fracture (“L”) can be seen at places. The formation of extruded lips can clearly be recognized in Fig. 3c. This figure also shows the formation of lateral fractures (arrowed). The fractures, however, are shallow and seem to be restricted to a rather thin material layer. As can be seen in Fig. 3d, the calcitic matrix between impact sites is exposed, and the material is removed

A.W. Momber / Wear 328-329 (2015) 348–355

L

L

L

351

L L

L L

L L L

L

L

L

L L

L

L

L L

Fig. 2. SEM images of eroded granite sections. (a) General sectional view with lateral cracking (denoted “L”). (b) Overlap of individual impact sites (circled); scale bar: 200 mm. (c) Magnification of a lateral cracking site, illustrating modified cracking process described in [12] as well as fracture lance formation (arrowed) described in [13]. Fracture direction from lower left to upper right. (d) Fracture marks on brittle fracture surfaces in quartz. (e) Magnified section of Fig. 2d: Partial adjustment of individual crack planes along the crack front (mixed loading; see [13]).

through intergranular separation of calcite crystals. This situation distinguishes the lateral fracture formation process from those observed in rhyolite and granite. SEM images of erosion sites in schist are provided in Fig. 4. Fig. 4a illustrates the pre-dominance of ductile erosion features, namely platelet formation; denoted “P”. Fig. 4b illustrates the formation of elevated walls at the rim of permanently deformed impact craters. The appearance of the eroded surface sections is in agreement with results reported in [17] for the impingement of shale with high-speed hemispherical indenters. Figs. 4c and d illustrate the formation and detachment of platelets, extruded off the surface, at higher magnifications. Droplet-shaped features in Fig. 4e (arrowed) supports results in [18], where local melting of schist due to impinging solid particles is proposed. The erosion features provided in Fig. 4 are, however, partly in contradiction to results reported earlier [18], where a mixed erosion mode in

schist, eroded by glass spheres at normal impingement, was observed, consisting of plastic deformation and lateral cracking. This author also found that an elastic–plastic model could describe the relationship between relative erosion (by rounded quartz) and particle impingement speed (ER p v2.44 ). However, ductile erosion P models can cover velocity exponents between 2 and 3 and may be applicable as well to the erosion by spherical particles at normal incidence [19–21]. Neither plainly “ductile” nor plainly “brittle” response can be expected for rock material due to their more complex anisotropic structure and morphology. Results of the calculation procedure for K1 are listed in Table 4. Two materials that represent theoretical limits are added to the results of this study. The lower theoretical limit (K1 ¼ 0) is represented through magnesium oxide. This isotropic brittle material shows an ideal lateral cracking mode. The upper theoretical limit (K1 ¼1) is represented through aluminium. This isotropic ductile material is characterized by

352

A.W. Momber / Wear 328-329 (2015) 348–355

P

P

L

P

P

P P

P

P

P

P

P P

L L

L L L

P

P P

P

L

P

L

P

Fig. 3. SEM images of eroded limestone sections. (a, b) General sectional view with mixed erosion modes (lateral cracking “L”, and lip and platelet formation “P”). (c) Magnification of individual erosion sites, illustrating lip formation, described in [15,16], and the formation of shallow lateral fractures (arrowed) between plastically deformed sections. (d) Magnification of a shallow radial fracture, illustrating intergranular crystal separation in the calcite matrix.

an ideal plastic material response. The rock materials investigated in this study are situated between these two limits. The regression between measurement and Eqs. (5.1) and (5.2) delivered the following results: The scale parameter was estimated to λ ¼5.1  109, and the shape parameter was estimated to b ¼0.26. The accuracy of the regression to Eq. (5.1) was R² ¼0.98. The twoparameter transition function K1 ¼f(χ) according to Eq. (4) is plotted in Fig. 5. The transition parameter χ it is plotted in a logarithmic style. The data points are grouped according to general rock types. Numbers for basalt, diorite, dolostone, granite, syenite range between K1 ¼0.13 and 0.4; numbers for sandstone vary between K1 ¼ 0.3 and 0.8; and numbers for limestone and marble cover the range between K1 ¼ 0.55 and 0.8. Schist has a high number of K1 ¼0.88. The number for mortar (as a cementitious composite) is moderate: K1 ¼ 0.32. The graph in Fig. 5 is subdivided into three ranges. The designations of these ranges refer to a rock classification scheme defined in [24] according to the shape of stress–strain curves of rocks in uniaxial compression. Types I and II exhibit either elastic or elastic–plastic stress–strain curves; these materials (basalt, diorite, dolostone, and granite), cover the lower section in Fig. 5 up to K1 E0.3. Types III and IV exhibit either plastic–elastic or plastic–elastic–plastic (high Young's modulus) stress–strain curves; these materials (sandstone, marble), cover the range 0.34K1 40.8. Type V exhibits plastic–elastic–plastic (low Young's modulus) stress–strain curves; these materials (schist, some sandstones), cover the upper region K1 40.8. Type I and II rocks will basically respond due to lateral cracking (although not ideally) to solid particle impingement. Type III and IV rocks will respond in a mixed mode, and small modification in loading regime or in rock structure (e.g. grain size) may shift their response either to lateral cracking or to ductile erosion. Type V rocks will basically respond due to ductile erosion to solid particle impingement. The low K1 numbers for granite and rhyolite are confirmed through the photographs

provided in Figs. 1 and 2. The low transition number for granite is also supported by results in [25], where a good applicability of a lateral crack model to the abrasion of rocks with granitic textures (granite, gabbro) was reported. A material with a layered structure similar to schist could, in contrast, not be linked to the lateral crack model [25]. The high K1 numbers for limestone and particularly for schist are confirmed through the photographs provided in Figs. 3 and 4.

4. Additional transition conditions Although the qualitative trend and the order of the rock materials in Fig. 5 are generally valid, the precise numbers for K1 are expected to vary for changing erosion conditions. In terms of process parameters, governing transition parameters are impact velocity, erodent particle size, erodent particle shape, and angle of impingement. In terms of materials parameters, the most important transition parameters are fracture toughness and hardness of erodent and target materials. Hardness is often discussed in terms of hardness ratio. Solid particle impact transition conditions experienced quite some interest in the past, and a number of references have investigated the corresponding relationships in detail. Hutchings [1], Geltnik-Verspui [26] and Morozov et al. [27] designed transition nomograms, where different failure modes were situated in dependence of impact velocity, erodent particle size and particle shape. Wada [28] and Akimune et al. [29] followed a similar approach, but failure modes are situated according to hardness ratio and particle kinetic energy (resp. impact velocity) in their nomograms. Wensink and Elwenspoek [30] derived a transition model, based on changes in kinetic erodent particle energy. Plastic response can be expected at moderate particle velocities and small erodent particles [1,26,27]. Hertzian fracture (ring or cone cracks) are expected to occur at rather low impact velocities and lager particles [1,26], at hardness ratios

A.W. Momber / Wear 328-329 (2015) 348–355

353

P P P

P P

P

P

P L

P P

P

P

P

P

P

P

P

P

P

P P

P P P

P

P

P P

P P

P

Figure c

Figure d

Fig. 4. SEM images of eroded schist sections. (a) General sectional view with platelet formation (denoted “P”). (b) Overlap of individual impact sites. (c, d) Magnification of individual erosion sites from Fig. 4b, illustrating formation and detachment of platelets. (e) Formation of platelets at different dimensional levels and features of local melting (arrowed).

target/erodent 41 [28], and with spherical particles. The formation of lateral cracks (elastic–plastic mode) is associated with rather high impact velocities and large particles [1,26], with high impact energies [30], and with low hardness ratios target/erodent [28,29,31,32]. The probability of erodent particle fragmentation is high, if impact velocity and particle size are high [1], if the hardness ratio erodent/target is low [31], and if the fracture toughness ratio erodent/target is low [33]. Based on a review of these references, the restrictions and limiting conditions of this investigation can be evaluated. The experimental results of this study are characterized through the following conditions: (i). Hardness: HP 4HM. This condition would not allow for a notable fragmentation of erodent particles, and it promotes indentationtype failure. Lower HP-values would promote ductile failure and would lead to crushing and fragmentation of erodent particles. (ii). Fracture toughness: KIc/P 4KIc/M (except schist). This condition would reduce the probability of erodent particle fragmentation.

Lower KIc/P-values would lead to crushing and fragmentation of erodent particles. (iii). Particle shape: rounded particles. This condition promotes the formation of lateral cracks (elastic–plastic) compared to spherical particles. Angular particles would further increase the probability of lateral cracking. Spherical particles would promote the formation of Hertzian (elastic) crack configurations. Effects of particle shape would become particularly important for a combination of high impact velocity and small particles. (iv). Particle size: dP ¼300–600 mm. Smaller particles (equal impact velocities provided) would promote ductile failure in the harder rocks. Very small erodent particles would induce ductile scratching even in the harder rocks. (v). Impact velocity: vP ¼90 m/s (respectively vP ¼ 40–140 m/s in [2]). Notably higher velocities (equal erodent particle sizes provided) would induce notably higher amounts of lateral cracking in the softer rocks.

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A.W. Momber / Wear 328-329 (2015) 348–355

Table 4 Lower (elastic–plastic) and upper (plastic) limits for χ for orthogonal impact of spheres and situation of the investigated rock materials (image widths for rocks: 150 mm). Material

Failure mode and K1 number

Reference

Magnesium oxide (MgO)

Lower theoretical limit: Elastic–plastic (lateral cracking) “L” K1 ¼0

[22]a

Rhyolite σC ¼ 240 MPa KIc ¼ 1.17 MPa m1/2 χ ¼ 3.3  10–14 MPa-11/4 m3/2

Elastic–plastic (non-ideal lateral cracking) “L” K1 ¼0.10

This paper

Granite σC ¼ 160 MPa KIc ¼ 0.80 MPa m1/2 χ ¼ 1.09  10–13 MPa-11/4 m3/2

Elastic–plastic (non-ideal lateral cracking) “L” K1 ¼0.15

This paper

Limestone σC ¼ 55 MPa KIc ¼ 1.21 MPa m1/2 χ ¼ 1.74  10–10 MPa-11/4 m3/2

Plastic (wall and lip formation) “P”þ elastic–plastic (lateral cracking) K1 ¼0.60

This paper

Schist σC ¼ 50 MPa KIc ¼ 2.70 MPa m1/2 χ ¼ 3.35  10–9 MPa-11/4 m3/2

Plastic (platelet and lip formation) “P”þ strictly limited elastic–plastic (lateral cracking) K1 ¼0.90

This paper

Aluminium

Upper theoretical limit: Plastic “P” (lip formation; elevated walls) K1 ¼1

[23]b

a b

Tungsten carbide spheres, dP ¼1.58 mm, vP ¼ 90 m/s, orthogonal impact. Tungsten carbide spheres, dP ¼ 1.58 mm, vP ¼ 50 m/s, almost orthogonal impact.

A.W. Momber / Wear 328-329 (2015) 348–355

Fig. 5. Proposed relationship between transition parameter χ and erosion parameter K1 according to Eq. (4). Numbers correspond to rocks in Table 2 as follows: 1¼ rhyolite; 2¼ granite; 3¼ limestone; 4¼ schist.

(vi). Impact angle: orthogonal impact (901). This condition promotes lip and platelet formation for plastically responding materials. Multiple impacts are required in order to remove material. Smaller impact angles would favour microcutting or microploughing, and they would reduce the amount of brittle cracking. Further research is needed in order to investigate the effects of these additional transition conditions on the solid particle erosion of geomaterials. 5. Summary A procedure for the quantification of a two-parameter model, which can describe the response of rock materials against solid particle impingement, has been derived. The model covers different types of material response, and the model parameters can be linked to general rock classification schemes. Acknowledgements This investigation was funded through an Exchange Lecturer Fellowship provided by the DAAD, Bonn, Germany. Thank is addressed to the Fracture Group, Cavendish Laboratory, University of Cambridge, Cambridge, UK for its hospitality and the permission to use experimental facilities. References [1] I.M. Hutchings, Ductile-brittle transitions and wear maps for the erosion and abrasion of brittle materials, J. Phys. D: Appl. Phys. 25 (1992) A212–A221.

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[2] A.W. Momber, Effects of target material properties on solid particle erosion of geomaterials at different impingement velocities, Wear 319 (2014) 69–83. [3] J. Gotzmann, Modellierung des Strahlverschleißes an keramischen Werkstoffen, Schmierungstechnik 20 (1989) 324–329. [4] T. Backers, Fracture toughness determination and micromechanics of rock under mode I and mode II loading (Dissertation), Universität Potsdam, 2004. [5] M. Heßling, Grundlagenuntersuchungen über das Schneiden von Gestein mit abrasiven Höchstdruckwasserstrahlen, Dissertation, RWTH Aachen (1988). [6] R. Kobayashi, K. Matsuki, N. Otsuka, Size effect in the fracture toughness of Ogino tuff, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 23 (1986) 13–18. [7] B.N. Whittacker, R.N. Singh, G. Sun, Rock Fracture Mechanics: Principles, Design and Applications, Elsevier, Amsterdam, 1992. [8] A.W. Momber, Fracture toughness effects in geomaterial solid particle erosion, Rock Mech. Rock Eng. (2014), http://dx.doi.org/10.1007/s00603-014-0658-x. [9] B. Dodson, The Weibull Analysis Handbook, 2nd ed., American Soc. for Quality, Milwaukee, USA, 2006. [10] D.L. Whitney, M. Broz, R.F. Cook, Hardness, toughness, and modulus of some common metamorphic minerals, Am. Mineral. 92 (2007) 281–288. [11] R.H. Telling, J.E. Field, The erosion of diamond, sapphire and zinc sulphide by quartz particles, Wear 233–235 (1999) 666–673. [12] P.N.W. Verhoef, Sandblast testing of rock, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 24 (1987) 185–192. [13] A.W. Momber, Fracture features in soda-lime glass after testing with a spherical indenter, J. Mater. Sci. 46 (2011) 4494–4508. [14] K. Breder, A.E. Giannakopoulos, Erosive wear in Al2O3 exhibiting Mode-I R-curve behavior, Ceram. Eng. Sci. Proc. 11 (1990) 1046–1060. [15] D.J. O'Flynn, M.S. Bingley, M.S.A. Bradley, A.J. Burnett, A model to predict the solid particle erosion rate of metals and its assessment using heat-treated steels, Wear 248 (2001) 162–177. [16] G. Sundararajan, A comprehensive model for the solid particle erosion of ductile materials, Wear 149 (1991) 111–127. [17] C.O. Rogers, S.S. Pang, A. Kumano, W. Goldsmith, Response of dry- and liquidfilled porous rocks to static and dynamic loading by variously-shaped projectiles, Rock Mech. 19 (1986) 235–260. [18] A.W. Momber, Erosion of schist due to solid particle impingement, Rock Mech. Rock Eng. 46 (2013) 849–857. [19] R. Brown, E.J. Jun, J.W. Edington, Erosion of α-Fe by spherical glass particles, Wear 70 (1981) 347–363. [20] I.M. Hutchings, A model for the erosion of metals by spherical particles at normal incidence, Wear 70 (1981) 269–281. [21] G. Sundararajan, P. Shewmon, A new model for the erosion of metals at normal incidence, Wear 84 (1983) 237–258. [22] D.G. Rickerby, N.H. Macmillan, Erosion of MgO by solid particle impingement at normal incidence, J. Mater. Sci. 16 (1981) 1579–1591. [23] D.G. Rickerby, N.H. Macmillan, The erosion of aluminum by solid particle impingement at oblique incidence, Wear 79 (1982) 171–190. [24] D.U. Deere, R.P. Miller, Engineering Classification and Index Properties for Intact Rocks, Technical Report No. AFWL-TR-65-116, Air Force Weapons Laboratory, Kirtland, NM, USA, December 1966. [25] P.M. Amaral, J.C. Fernandes, L.G. Rosa, Wear mechanisms in materials with granitic textures - applicability of a lateral crack system model, Wear 266 (2009) 753–764. [26] M.A. Geltink-Verspui, Modelling Abrasive Processes of Glass, Proefschrift, Technische Universteit Einhhoven, Netherlands, 1989. [27] N.F. Morozov, Y.V. Petrov, V.I. Smirnov, Transition between brittle and ductile erosion fracture, Dokl. Phys. 47 (2002) 525–527. [28] S. Wada, Effects of hardness and fracture toughness of target materials and impact particles on erosion of ceramic materials, Key Eng. Mater. 71 (1992) 51–74. [29] Y. Akimune, T. Akiba, T. Ogasawara, Damage behavior of silicon nitride for automotive gas turbine use when impacted by several types of spherical particles, J. Mater. Sci. 30 (1995) 1000–1004. [30] H. Wensink, M.C. Elwenspoek, A closer look at the ductile-brittle transition in solid particle erosion, Wear 253 (2002) 1035–1043. [31] R.A. Vaughan, A. Ball, The effect of hardness and toughness on the erosion of ceramics and ultrahard materials, in: K.C. Ludema, R.G. Bayer (Eds.), Wear Mater., Vol. 1, ASME, New York, 1991, pp. 171–175. [32] P.H. Shipway, I.M. Hutchings, The role of particle properties in the erosion of brittle materials, Wear 193 (1996) 105–113. [33] L. Murugesh, S. Srinivasan, R.O. Scattergood, Models and material properties for erosion of ceramics, J. Mater. Eng 13 (1991) 55–61.