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 PR + (1 − K 1) · E LR 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.
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Abbreviations
- 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
- K 1 :
-
Erosion parameter
- KIc :
-
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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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.
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Appendices
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):
Yield strength σ Y can be related to material hardness as follows (Tabor 1951):
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):
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):
The contact radius can be calculated as follows (Fischer-Cipps 2007):
The ratio r B/r P can be expressed as follows:
The contact force generated by an impacting spherical solid particle can be calculated as follows (Knight et al. 1977):
The parameter k balances the elastic properties of erodent and target material as follows (Fischer-Cipps 2007):
From linear-elastic fracture mechanics, E M⋅Γ Ic can be expressed as follows:
Appendix B: Calculation of Stress Rates
Loading rate in terms of stress rate can be expressed as follows:
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):
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:
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:
It can be seen that this time period does not depend on impingement velocity. The equations deliver the following relationship:
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Momber, A.W. Fracture Toughness Effects in Geomaterial Solid Particle Erosion. Rock Mech Rock Eng 48, 1573–1588 (2015). https://doi.org/10.1007/s00603-014-0658-x
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DOI: https://doi.org/10.1007/s00603-014-0658-x