1 Introduction
2 Experimental procedure of the fragmentation test Equation Section (Next)
3 Methodology for numerical simulation of the fragmentation test
3.1 Numerical integration procedure
3.2 SPH method
4 Modelling and simulation of the fragmentation testEquation Section (Next)
4.1 Model definition
4.2 Constitutive modelling
4.2.1 Rock dolomite
Parameter | Value | Unit |
---|---|---|
Density, ρ | 2840.0 | kg/m3 |
Poisson’s ratio, v | 0.24 | – |
Bulk modulus, K1 | 30,834.5 | MPa |
Shear modulus, G | 19,059.6 | MPa |
Elastic modulus, E | 47,410.3 | MPa |
Hugoniot elastic limit, HEL | 2750.0 | MPa |
HEL pressure, PHEL | 1945.0 | MPa |
Maximum tensile strength*, T | 29.2 | MPa |
Intact strength coefficient, A | 0.78 | – |
Fractured strength coefficient, B | 0.65 | – |
Strain rate coefficient, C | 0.02 | – |
Intact strength exponent, N | 0.45 | – |
Fractured strength exponent, M | 0.45 | – |
Bulk factor, β | 1.0 | – |
Damage coefficient, D1 | 0.001 | – |
Damage coefficient, D2 | 1.15 | – |
Pressure coefficient 2, K2 | 700,000.0 | MPa |
Pressure coefficient 3, K3 | 5,650,000.0 | MPa |
Maximum normalized fracture strength, σ*max | 0.35 | – |
4.2.2 Stemming and high-explosive material models
Parameter | Value | Unit |
---|---|---|
Initial density, ρ0 | 1130.0 | kg/m3 |
Detonation velocity, D | 4805.0 | m/s |
Chapman-Jouguet pressure, pCJ | 7400.0 | MPa |
AHE | 252,000.0 | MPa |
BHE | 15,570.0 | MPa |
R1 | 6.08 | – |
R2 | 2.05 | – |
ω | 0.25 | – |
Detonation energy per unit volume, E0 | 3700.0 | MPa |
Parameter | Value | Unit |
---|---|---|
Density, ρst | 2350.0 | kg/m3 |
Specific gravity, SPGRAV | 2.65 | – |
Density of water, ρwat | 1000.0 | kg/m3 |
Viscoplasticity parameter, Vn | 1.1 | – |
Viscoplasticity parameter, γr | 0.0 | – |
Max. number of plasticity iterations, ITERmax | 10.0 | – |
Shear modulus, Gst | 19.5 | MPa |
Bulk modulus, Kst | 15.3 | MPa |
Peak shear strength angle, φmax | 0.42 | radians |
Cohesion, COH | 0.011 | MPa |
Eccentricity parameter, ECCEN | 0.7 | – |
Strain hardening % of φmax where nonlinear effects start, An | 0.0 | – |
Strain hardening amount of non-linear effects, Et | 10.0 | – |
Moisture content, MCONT | 6.2 | % |
Parameter for pore water effects on bulk modulus, PWD1 | 0.0 | – |
Skeleton bulk modulus, PWKSK | 0.153 | MPa |
Parameter for pore water effects on confinement pressure, PWD2 | 0.0 | – |
Residual friction angle, φres | 0.063 | radians |
Volumetric strain at initial damage threshold, Dint | 0.001 | – |
Void formation (fracture) energy, Vdfm | 10.0 | – |
Level of damage causing element deletion, DAMlev | 0.0 | – |
Maximum principal failure strain, EPSmax | 0.0 | – |
5 Results and discussion
5.1 Rock sample no. 1
5.1.1 Calibration of damping properties
5.2 Rock sample no. 2
Fragment | FEA/Exp. | Volume (mm3) | % of the experiment | % of whole volume |
---|---|---|---|---|
No. 1 | FEA | 0.171 | 98.85 | 36.23 |
Experiment | 0.173 | 100.00 | 36.66 | |
No. 2 | FEA | 0.286 | 109.16 | 60.59 |
Experiment | 0.262 | 100.00 | 55.51 | |
Whole | FEA/Experiment | 0.472 | 100.00 | 100.00 |
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the smaller fragments visible in Fig. 14 were not measured.
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The fully damaged finite elements, which were blanked for volume calculations, were not taken into consideration.
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The fragments within the area under the HE were treated as parts of fragments no. 1 and no. 2.
6 Conclusions
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Knowledge of the damping mechanism is crucial in order to obtain reliable results from numerical simulations. In this study, the Rayleigh classical approach, which is commonly used in FE dynamic analyses, was considered. Due to the difficulty of determining damping properties for rock-like materials, several values of coefficients of stiffness-weighted damping were tested. β significantly influenced the acceleration oscillations and energy loss caused by damping. By contrast, the mass-weighted damping α only influenced lower frequencies and thus did not significantly impact the results in the present analysis. Consequently, α was set to 1.0 in all FEA.
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The closest agreement between the acceleration history measured in FEA and the maximum FFT magnitude was obtained at β equal to 0.20. The numerical simulations satisfactorily reproduced the cracking characteristics and fragmentation of both rock samples. Furthermore, the fragments from rock sample no. 2 were scanned using a 3D laser scanner to compare the volumes and geometries of the real-world and simulated fragments.
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The implemented JH-2 constitutive model with parameters determined for dolomite proved to be efficient and accurate for simulating the blasting and subsequent fracture and fragmentation of the rock. Nevertheless, further studies are planned to validate the model in penetration and small-scale tests. The influence of the mesh will be also analyzed in future work.
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The FEM-SPH coupled method of modelling was sufficiently effective, and the results of FEA were very similar to those of the actual tests. The main advantage of the adopted meshless approach for simulating the HE and stemming is the significantly shorter computational time compared with the ALE approach.