Fracture Toughness Effects in Geomaterial Solid Particle Erosion

A. W. Momber

doi:10.1007/s00603-014-0658-x

Rock Mechanics and Rock Engineering

July 2015, Volume 48, Issue 4, pp 1573–1588 | Cite as

Fracture Toughness Effects in Geomaterial Solid Particle Erosion

Authors
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A. W. MomberEmail author

Original Paper

First Online: 11 October 2014

Received: 11 May 2014

Accepted: 02 October 2014

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Abstract

Effects of fracture toughness on the impingement of geomaterials (rocks and cementitious composites) by quartz particles at velocities between 40 and 140 m/s are investigated experimentally and analytically. If schist is excluded, relative erosion (in g/g) reduces according to a reverse power function if fracture toughness increases. The power exponent depends on impingement velocity, and it varies between −0.64 and −1.33. Lateral cracking erosion models, developed for brittle materials, deliver too high values for relative material erosion. This discrepancy is partly attributed to stress rate effects. Effects of R-curve behavior seem to be marginal. An integral approach E _R = K ₁ · E _R ^P + (1 − K ₁) · E _R ^L is introduced, which considers erosion due to plastic deformation and lateral cracking. A transition function $K_{1} = f\left( {K_{\text{Ic}}^{12/4} /\sigma_{\text{C}}^{23/4} } \right)$ is suggested in order to classify geomaterials according to their response against solid particle impingement.

Keywords

Erosion Fracture toughness Geomaterials Impact

List of symbols

b: Distribution shape parameter
c: Crack length
c_I: Material parameter
D: Fracture toughness exponent
d_P: Erodent particle diameter
E_K: Kinetic energy erodent particle
E_M: Young’s modulus target material
E_P: Young’s modulus erodent material
E_R: Relative erosion
H_M: Hardness target material
k: Elastic parameter
K₁: Erosion parameter
K_Ic: Fracture toughness target material
m: R-curve parameter
M_M: Eroded target mass
M_P: Erodent particle mass
$\dot{M}_{\text{P}}$: Erodent mass flow rate
n: Stress rate parameter
P_C: Contact force
r_B: Contact radius
r_P: Particle radius
t_E: Exposure time
t_P: Contact time
v_P: Erodent particle velocity
β: Indenter angle
χ: Transition parameter
Γ_Ic: Critical energy release rate
λ: Distribution scale parameter
ν_M: Poisson’s ratio target material
ν_P: Poisson’s ratio erodent material
ρ_P: Density erodent material
ρ_M: Density target material
$\dot{\sigma }$: Stress rate
σ_C: Compressive strength target material
σ_P: Contact stress
σ_Y: Yield stress

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Notes

Acknowledgments

The author is thankful to the German Academic Exchange Service (DAAD), Bonn, Germany, for providing an Exchange Lecturer Fellowship for a stay at the University of Cambridge, UK. Special thanks is addressed to the Fracture Group, Physics and Chemistry of Solids, Cavendish Laboratory, for its kind hospitality and the permission to use experimental facilities.

Appendix A: Calculation of Eroded Volume According to the Model of Chen et al. (2005)

The volume eroded by an impinging solid particle is estimated as follows (Chen et al. 2005):

$$V_{\text{E}} \approx 0.41 \cdot \left[ {\frac{1}{{\pi^{2} \cdot c_{\text{I}}^{ 2} \cdot \tan^{2} \beta }} \cdot \left( {\frac{{(M_{\text{P}} \cdot v_{\text{P}}^{ 2} )^{14} }}{{E_{\text{M}}^{ 6} \cdot \varGamma_{\text{Ic}}^{6} \cdot \sigma_{\text{Y}}^{2} }}} \right)} \right]^{1/12}$$

(14)

Yield strength σ _Y can be related to material hardness as follows (Tabor 1951):

$$\sigma_{\text{Y}} = \frac{{H_{\text{M}} }}{{c_{\text{I}} }}$$

(15)

The parameter c _I is a material-dependent parameter; it can be estimated from a function f(E _M/H _M) provided by Bushby and Swain (1995). Hardness is linearly related to the compressive strength for many rocks and cement-based materials (Szwedzicki 1998; Winslow 1984; Igrashi et al. 1996). For the relationship between indentation hardness and uniaxial compressive strength of rock materials, the following correlation (R ² = 0.974) can be applied (Jung et al. 1994):

$$H_{\text{M}} = 20.2 \cdot \sigma_{\text{C}} + 277$$

(16)

The indenter angle β can be related to the ratio r _B/r _P for spherical indenters (Chen et al. 2005). The graph in Chen et al. (2005; Fig. 5, p. 1236), can be approximated with a second-power regression (R ² = 0.991):

$$\beta = 2 8. 3 2\cdot (r_{\text{B}} /r_{\text{P}} )^{ 2} + 2 6. 4 1\cdot (r_{\text{B}} /r_{\text{P}} )+ 1. 6 1$$

(17)

The contact radius can be calculated as follows (Fischer-Cipps 2007):

$$r_{\text{B}} = \left( {\frac{{3 \cdot P_{\text{C}} \cdot d_{\text{P}} \cdot k}}{8}} \right)^{1/3}$$

(18)

The ratio r _B/r _P can be expressed as follows:

$$\frac{{r_{\text{B}} }}{{r_{\text{P}} }} = \left( {\frac{{5 \cdot P_{\text{C}} \cdot k}}{8}} \right)^{1/3} \cdot r_{\text{P}}^{ - 2/3}$$

(19)

The contact force generated by an impacting spherical solid particle can be calculated as follows (Knight et al. 1977):

$${P_{\text{C}} = \left( {\frac{5}{3} \cdot \pi \cdot \rho_{\text{P}} } \right)^{3/5} \cdot \left( {\frac{3}{4} \cdot k} \right)^{ - 2/5} \cdot v_{\text{P}}^{6/5} \cdot \left( {\frac{{d_{\text{P}} }}{2}} \right)^{2} }$$

(20)

The parameter k balances the elastic properties of erodent and target material as follows (Fischer-Cipps 2007):

$${k = \frac{{1 - \nu_{\text{P}}^{2} }}{{E_{\text{P}} }} + \frac{{1 - \nu_{\text{M}}^{2} }}{{E_{\text{M}} }}} .$$

(21)

From linear-elastic fracture mechanics, E _M⋅Γ _Ic can be expressed as follows:

$$K_{\text{Ic}}^{2} = E_{\text{M}} \cdot \varGamma_{\text{Ic}}$$

(22)

Appendix B: Calculation of Stress Rates

Loading rate in terms of stress rate can be expressed as follows:

$$\dot{\sigma } = \frac{{\sigma_{\text{P}} }}{{t_{\text{P}} }}$$

(23)

Here, σ _P is the contact tensile stress, and t _P is the period of contact. The tensile stress generated by an impinging spherical particle can be calculated as follows (Fischer-Cipps 2007):

$$\sigma_{\text{P}} = \frac{{(1 - 2 \cdot \nu_{\text{M}} ) \cdot P_{\text{C}} }}{{2 \cdot \pi \cdot r_{\text{B}}^{2} }}$$

(24)

Here, r _B is the contact radius, and P _C is the contact force. They are given in Appendix A. A solution for the contact time for elastic contact is provided by Timoshenko und Goodier (1970); it can be expressed as follows:

$$t_{\text{P}}^{\text{E}} = 2.94 \cdot \left[ {\frac{{5 \cdot \pi \cdot \rho_{\text{P}} \cdot k}}{4}} \right]^{2/5} \cdot \frac{{d_{\text{P}} }}{2} \cdot v_{\text{P}}^{ - 1/5}$$

(25)

The parameter k is given in Appendix A. This period can increase notably, if plastic deformation, toughening or erodent fragmentation occurs. Thus, a plastic contact time shall be added. A solution for the contact time for plastic contact is provided by Chaudri und Walley (1978); it reads as follows:

$$t_{\text{P}}^{\text{PL}} = \frac{\pi }{2} \cdot \left( {\frac{{M_{\text{P}} }}{{\pi \cdot d_{\text{P}} \cdot H_{\text{M}} }}} \right)^{1/2}$$

(26)

It can be seen that this time period does not depend on impingement velocity. The equations deliver the following relationship:

$$\dot{\sigma } \propto v_{\text{P}}^{3/5}$$

(27)

References

Amaral PM, Fernandes JC, Rosa LG (2009) Wear mechanisms in materials with granitic textures—applicability of a lateral crack system model. Wear 266:753–764CrossRefGoogle Scholar
Bertram B (2008) Bruchenergie laufender Risse in Gestein. Dissertation, Bochum: Ruhr-Universität: 1988Google Scholar
Beste U, Lundvall A, Jacobson S (2004) Micro-scratch evaluation of rock types—a means to comprehend rock drill wear. Tribol Int 37:203–210CrossRefGoogle Scholar
Blange JJ, van Nieuwkoop P (2011) Method of drilling and jet drilling system. Interns Patent Application, WO 2011/076845 A1, 30 June 2011Google Scholar
Breder K, Giannakopoulos AE (1990) Erosive wear in Al₂O₃ exhibiting Mode-I R-curve behavior. Ceram Eng Sci Proc 11:1046–1060CrossRefGoogle Scholar
Breder K, De Portu G, Ritter JE, Fabbriche DD (1988) Erosion damage and strength degradation of zirconia-toughened alumina. J Amer Ceram Soc 71:770–775CrossRefGoogle Scholar
Bujis M (1994) Erosion of glass as modeled by indentation theory. J Amer Ceram Soc 77:1676–1678CrossRefGoogle Scholar
Bushby AJ, Swain MV (1995) Spherical indentations as a means for investigating the plastic deformation of ceramics. In: Bradt RC et al (eds) Plastic Deformation of Ceramics. Plenum Press, New York, pp 161–172CrossRefGoogle Scholar
Chaudri MM, Walley SM (1978) A high-speed photographic investigation of the impact damage in sodalime and borosilicate glasses by small glass and steel spheres. In: Bradt RC et al (eds) Fracture Mechanics of Ceramics, vol 3. Plenum Press, New York, pp 349–364Google Scholar
Chen X, Hutchinson JW, Evans AG (2005) The mechanics of indentation induced lateral cracking. J Amer Ceram Soc 88(5):1233–1238CrossRefGoogle Scholar
Chen P, Cui J, Zhou X (2013) Research on effects of pressure loading on rock lateral cracks under confining pressure. Adv Mater Res 838–841:850–853Google Scholar
Cui M, Zhay Y, Ji G (2011) Experimental study of rock breaking effect of steel particles. J Hydrodyn 23:241–246CrossRefGoogle Scholar
Dai F, Chen R, Iqbal MJ, Xia K (2010) Dynamic cracked chevron notched Brazilian disc method for measuring rock fracture parameters. Int J Rock Mech Min Sci 47:606–613CrossRefGoogle Scholar
Du Y, Wang R, Ni H, Ma L, Zhang S, Han Z (2011) Study on the rock-breaking performance of particles jet drilling technology. Electr J Geomech Engng 16:431–440Google Scholar
Du Y, Wang R, Ni H (2012) Feasibility of particle jet as a drilling medium for the development of deep complicated oil-gas reservoir. Adv Mater Res 361–363:465–468Google Scholar
Erarslan N, Williams DJ (2012) The damage mechanism of rock fatigue and its relationship to the fracture toughness of rocks. Int J Rock Mech Min Sci 56:15–26Google Scholar
Evans AG, Gulden ME, Rosenblatt ME (1978) Impact damage in brittle materials in the elastic–plastic response regime. Proc Roy Soc Lond A 361:343–356CrossRefGoogle Scholar
Fischer-Cipps AC (2007) Introduction to Contact Mechanics. Springer, New-YorkCrossRefGoogle Scholar
Gordon AT, Greg GG (2008) Particle drilling alters standard rock-cutting approach. World Oil 229(6):37–44Google Scholar
Gotzmann J (1989) Modellierung des Strahlverschleißes an keramischen Werkstoffen. Schmierungstechnik 20:324–329Google Scholar
Harder NJ, Curlett HB, Padgett O, Hazel B (2010) Impact excavation system and method with injection system. United States Patent, US 7,793,741 B2, 14 Sep 2010Google Scholar
Heßling M (1988) Grundlagenuntersuchungen über das Schneiden von Gestein mit abrasiven Höchstdruckwasserstrahlen. Dissertation, Aachen, RWTH AachenGoogle Scholar
Huang H, Damjanac B, Detournay E (1998) Normal wedge indentation in rocks with lateral confinement. Rock Mech Rock Engng 31(2):81–94CrossRefGoogle Scholar
Hutchings IM (1977) Strain rate effects in microparticle impact. J Phys D Appl Phys 10:L179–L184CrossRefGoogle Scholar
Igrashi S, Bentur A, Mindess S (1996) Microhardness testing of cementitious materials. Advn Cem Bas Mat 4:48–57CrossRefGoogle Scholar
Iqbal MJ, Mohanty B, Xia K (2008) Dynamic tensile strength and mode-I fracture toughness in granitic rocks. In: Proceedings 11th International Congress. Exposition, Orlando; Society for Experimental Mechanics 2–5 June 2008Google Scholar
Jung SJ, Prisbrey K, Wu G (1994) Prediction of rock hardness and drillability using acoustic signatures during indentation. Int J Rock Mech Min Sci Geomech Abstr 31:561–567CrossRefGoogle Scholar
Kleis I, Kulu P (2008) Solid Particle Erosion. Springer, LondonGoogle Scholar
Knight CG, Swain MV, Chaudri MM (1977) Impact of small steel spheres on glass surfaces. J Mater Sci 12:1573–1586CrossRefGoogle Scholar
Lambe TW (1969) Soil Mechanics. John Wiley, New YorkGoogle Scholar
Lambert DE, Ross CA (2000) Strain rate effects on dynamic fracture and strength. Int J Impact Eng 24:985–998CrossRefGoogle Scholar
Marshall DB, Lawn BR, Evans AG (1982) Elastic/plastic indentation damage in ceramics: the lateral crack system. J Amer Ceram Soc 65:561–566CrossRefGoogle Scholar
Maurer WCJ, Rinehart JS (1960) Impact crater formation in rock. J Appl Phys 31:1247–1252CrossRefGoogle Scholar
Momber AW (2003a) Cavitation damage to geomaterials in a flowing system. J Mater Sci 38:747–757CrossRefGoogle Scholar
Momber AW (2003b) An SEM-study of high-speed hydrodynamic erosion of cementitious composites. Compos B 34:135–142CrossRefGoogle Scholar
Momber AW (2004) Damage to rocks and cementitious materials from solid impact. Rock Mech Rock Eng 37:57–82CrossRefGoogle Scholar
Momber AW (2005) Hydrodemolition of concrete surfaces and reinforced concrete. Elsevier, LondonGoogle Scholar
Momber AW (2008) Blast Cleaning Technology. Springer, LondonCrossRefGoogle Scholar
Momber AW (2013) Erosion of schist due to solid particle impingement. Rock Mech Rock Eng 46:849–857CrossRefGoogle Scholar
Momber AW (2014) Effects of target material properties on solid particle erosion of geomaterials at different impingement velocities. Wear 319:69–83CrossRefGoogle Scholar
Momber AW, Kovacevic R (1998) Principles of Abrasive Water Jet Machining. Springer, LondonCrossRefGoogle Scholar
Momber AW, Budidharma E, Tjo R (2011) The separation of reinforced cementitious composites with a stream-line cutting tool. Res Conserv Recycl 55:507–514CrossRefGoogle Scholar
Murakami Y (1987) Stress Intensity Handbook, vol 1. Pergamon Press, Oxford, pp 13–15Google Scholar
Murugesh L, Srinivasan S, Scattergood RO (1991) Models and material properties for erosion of ceramics. J Mater Eng 13:55–61CrossRefGoogle Scholar
Ouchterlony F (1982) Review on fracture toughness testing of rock. SM Archives 71:131–211Google Scholar
Ouchterlony F (1990) Fracture toughness testing of rock with core based specimens. Eng Fract Mech 35:351–366CrossRefGoogle Scholar
Pickering EG, Deshpande VS (2014) The ballistic impact of ceramics—an investigation into the effects of confinement and material properties. In: 4th International Conference on Impact Loading of Lightweight Structures. Cape Town, SA; 12–16 Jan 2014Google Scholar
Ritter JE (1985) Erosion damage in structural ceramics. Mater Sci Eng 71:194–201CrossRefGoogle Scholar
Ritter JE, Strzepa P, Jakus K, Rosenfeld L, Buckman KJ (1984) Erosion damage in glass and alumina. J Amer Ceram Soc 67:769–774CrossRefGoogle Scholar
Rogers CO, Pang SS, Kumano A, Goldsmith W (1986) Response of dry- and liquid-filled porous rocks to static and dynamic loading by variously-shaped projectiles. Rock Mech 19:235–260CrossRefGoogle Scholar
Routbort JL, Matzke HJ (1983) On the correlation between solid-particle erosion and fracture parameters in SiC. J Mater Sci 18:1491–1496CrossRefGoogle Scholar
Ruff AW, Wiederhorn SM (1979) Erosion by solid particle impact. In: Preece CM (ed) Erosion. Academic Press, New York, pp 69–126Google Scholar
Schmidt RA, Lutz TJ (1979) KIc and JIc of Westerley granite—effects of thickness and in-plane dimensions. In: Freiman SW (ed) Fracture Mechanics Applied to Brittle Materials. ASTM, New York, pp 166–182CrossRefGoogle Scholar
Slikkerveer PJ, Verspui M, Skerka E (1999) Erosion and damage by hard spherical particles on glass. J Amer Ceram Soc 82:3173–3178CrossRefGoogle Scholar
Szwedzicki T (1998) Indentation hardness testing of rock. Int J Rock Mech Min Sci 35:825–829CrossRefGoogle Scholar
Tabor D (1951) The hardness of metals. Clarendon Press, OxfordGoogle Scholar
Tandon S, Faber KT (1999) Effects of loading rate on the fracture of cementitious materials. Cem Concr Res 29:397–406CrossRefGoogle Scholar
Tibbitts GA, Galloway G, Vuyk A, Terry J (2009) Methods of using a particle impact drilling system for removing near-borehole damage, milling objects in a wellbore, under reaming, coring, perforating, assisting, annular flow, and associated methods. United States Patent Application Publication, US 2009/0218098 A1, 3 Sep 2009Google Scholar
Timoshenko S, Goodier JN (1970) Theory of Elasticity. McGraw-Hill, New YorkGoogle Scholar
Verhoef PN (1987) Sandblast testing of rock. Int J Rock Mech Min Sci 24:185–192CrossRefGoogle Scholar
Verhoef PN, Kuipers TJ, Verwaal W (1984) The use of the sand-blast test to determine rock durability. Bull Int Assoc Engng Geol 29:457–461CrossRefGoogle Scholar
Wallin K (2013) A simple fracture mechanical interpretation of size effects in concrete fracture toughness tests. Eng Fract Mech 99:18–29CrossRefGoogle Scholar
Wang QZ, Yang JR, Zhang CG, Zhou Y (2014) Rock dynamic fracture toughness measured by using crack propagation gauge and universal function. In: 16th International Conference on Experimental Mechanics, Cambridge, UK; European Society Experimental Mechanics: 7–8 July 2014Google Scholar
Wiederhorn SM, Hockey BJ (1983) Effect of material parameters on the erosion resistance of brittle materials. J Mater Sci 18:766–780CrossRefGoogle Scholar
Winslow DN (1984) A Rockwell hardness test for concrete. Cement Concr Aggreg 6:137–141CrossRefGoogle Scholar
Zhang ZX, Kou SQ, Yu J, Yu Y, Jiang LG, Lindqvist PA (1999) Effects of loading rate on rock fracture. Int J Rock Mech Min Sci 36:597–611CrossRefGoogle Scholar
Zhang ZX, Kou SQ, Jiang LG, Lindqvist PA (2000) Effects of loading rate on rock fracture: fracture characteristics and energy partitioning. Int J Rock Mech Min Sci 37:745–762CrossRefGoogle Scholar
Zhang H, Song H, Kang Y, Huabng G, Qu C (2013) Experimental analysis on deformation evolution and crack propagation of rock under cyclic indentation. Rock Mech Rock Eng 46:1053–1059CrossRefGoogle Scholar

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A. W. Momber
- 1
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1.Faculty of Georesources and Materials TechnologyAachen University of Technology (RWTH Aachen)HamburgGermany

Fracture Toughness Effects in Geomaterial Solid Particle Erosion

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