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

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A. W. Momber

Original Paper

First Online: 11 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)

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A. W. Momber
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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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