Numerical modelling of induced tensile stresses in rock in response to impact loading

Transcription

1 Numerical modelling of induced tensile stresses in rock in response to impact loading M.T. Mnisi, D.P. Roberts and J.S. Kuijpers Council for Scientific and Industrial Research (CSIR): Natural Resources and Environment (NRE) P.O. Box 91230, Auckland Park, Johannesburg 2006 Key words: Rock breaking, impact loading, numerical modelling, discrete element Abstract: Computer simulations have shown that a compressive stress wave imposed on the surface of a half-space can induce a corresponding tensile stress wave. The absence of a reflecting boundary is of particular interest in this case. Optimization of the stress wave with the aim of breaking rock in direct tension may have significant implications for mining in terms of the production cycle and in terms of minimizing energy input. Computer simulations have been used to confirm the presence of the tensile stress wave, as analytical solutions are not readily available. This paper presents an investigation of the parameters affecting the propagation and characteristics of the tensile stress wave through finite/discrete element modelling. The rate of (un)loading the rock surface has been found to be key in the parameter sensitivity analysis. Introduction Li-Lih Wang (2003) presented the failures due to unloading effect of waves during impact. He illustrated in an example showing surface damage caused by a 14.3 mm diameter ice ball moving at 192m/s on a polymethylmethacrylate (PMMA) plate. This damage is caused by the introduction significant tensile stresses into the surface by Rayleigh surface waves. From this study it is deduced that effect of unloading corresponds to the frequency of an impact. This idea is applicable to a study of induced tensile stresses that are observed to follow closely on a P wave along a rock medium during an imposed compressive front. These tensile stresses do not result from surface waves nor boundary reflections. It is part of the study to interpret their origin and characteristics. A possible way to induce tension in a rock medium is by means of dynamic loading effects. Such effects can only occur if the (un)loading rate is sufficiently fast, so that the inertial momentum of the rock mass can be exploited. For the investigation, there are two techniques used as sources for the compressive wave front in the rock block, i.e. an impact from an accelerated mass and a single sinusoidal pulse imposed on the rock surface. Kuijpers et al (2002) demonstrated using a numerical model from WAVE and FLAC that there is a potential for macro fracturing a distance away from the surface with the technique of inducing direct tension. From these results, it was established that the speed of unloading needs to be sufficiently fast in order to allow for a natural system response, which would include an overshoot due to the momentum of that volume of rock which is effectively unloaded. In practice it is hardly possible to apply direct tension to rock and the actual fracturing of rock commonly results from tensile stresses, which are induced by primary failure mechanisms.

2 The importance of the tensile wave being generated during the introduction of a compressive wave is that rocks fail in less the magnitude of tensile than compressive strength. So if this tensile wave can be optimised, a new technique of rock breaking can be formulated. Such a technique may prove economical especially to hard rock breaking. Different parameters are investigated for optimizing the tensile wave. These include impactor geometry and density. For the sinusoidal pulse problem, frequency and loading area are investigated. The study however, limits itself to the fundamentals of the process; therefore only elastic waves are applied to the rock block. After the tensile wave parameters have been optimized, the applied strengths could be upscaled to allow for fracture. FLAC and ELFEN are used as modelling tools for this investigation. Preliminary work has shown significant differences in results from half space models and plane strain models. The tensile stresses are more pronounced on former than the latter model. The axi-symmetric option was therefore chosen for the research. Experimental investigations for the numerical modelling will be covered in ongoing research.future work for further modelling may cover parameters such as pre-existing stress state, the rock properties, sample geometry and the presence and characteristics of discontinuities within the rock sample. The results from the sensitivity analyses provide a range of parameters that can be used in the design of the impactor machinery. Various impactor machine configurations, including radial and sequential impact designs, are to be investigated by further modelling. Simulation Geometry As indicated in the previous section, there were two techniques used for inducing the tensile stresses. This translates to two geometrical models. The first model is axisymmetric in geometry, which consists of an impactor and the rock medium. About half the rock radius is finer meshed for improved solutions around the applied loading area. The gap in between the impactor and the rock was set to be 0.01m.The impactor size is altered as necessary during the course of the investigation. The second geometry is also half space with a rock medium and a pulse area on the rock surface. The rock medium has been extended to large enough a distance that would avoid any boundary interactions of the imposed compressive wave. As indicated earlier, a free surface may cause reflections that could result in interferences to the induced tensile wave. Constraints The rock block and the impactor were constrained along their centre line to have no displacement in the x direction. In addition, the rock was constrained in its boundaries to have no displacement in the y directions. This applies to both the single pulse and the impactor models. Mesh The mesh density was set at to within the four times radius area of the rock and within the impactor to improve the quality of results. Anywhere else in the rock, it

3 was set to the default one. For all cases the mesh geometry was unstructured triangular elements. Types of Loading Sources The first numerical modelling case used to proof the presence of tensile stresses along the centre-line of the rock was FLAC. In this case, a hollow cylindrical impact was used as a loading source (see Figure 1 below). Later, ELFEN was used to prove the presence of the tensile wave in response to impact loading by an accelerated solid mass (see Figure 2). The impactor was accelerated at an applied velocity of 10m/s and just before impact; the load was removed to allow for natural frequency wave propagation. Figure 1: Induced tensile stresses from a hollow impactor using FLAC Figure 2: Tensile stresses induced through a solid impact using ELFEN As part of the investigations regarding the maximising of the induced tensile wave, a third loading source was introduced, i.e. the single sinusoidal pulse. This was done with both FLAC and ELFEN and it showed similar results to the other two loading sources. Figure 3 shows the ELFEN model of the single sinusoidal pulse.

4 Single sinusoidal pulse Figure 4: Single sinusoidal pulse induced tensile stress wave The rock block was free from any external stresses. Material properties The rock medium was set at a density of 2800 kg/m 3, poisson s ratio of 0.15 and an elastic modulus of 70 Gpa for both FLAC and ELFEN. The impactor properties were altered as required by the investigation. Results and Discussions In the process of this investigation, the mechanical response concurrent with the tensile wave front was analysed, including the principal, cartesian and deviatoric stress components. The displacements and velocities of particular nodes were also monitored to determine the response of the rock medium. The most important aspect of the result was expressed in terms of the response efficiency, which is simply the ratio of the greatest vertical tensile stress amplitude to the greatest vertical compressive stress amplitude. To begin with, the ELFEN accelerated solid mass problem is considered. The first parameter to investigate on the sensitivity of the induced tensile wave was impact frequency. Assuming the spring mass analogy: 1 f k / m [1] 2 where, f is frequency, k is spring stiffness, and m is the mass

5 Efficiency (%) Key to equation 1 is the fact that a decrease in mass (material density) results in an increase in the frequency response of the material. Density was then used as a parameter that controls material frequency. With the impactor geometry remaining unchanged at radius of 0.1m and length of 02m, poisson s ratio at 0.3 and the elastic modulus at 200 Gpa the density was changed between 10 and kg/m 3. Figure 5 shows the results of the investigation. Efficiency vs Density Curve Density (kg/m 3 ) Figure 5: Tensile wave efficiency as a function of impactor frequency From the curve above, it is clear that decreasing the frequency increases the tensile wave efficiency. It is understood though, that the lower densities have practical limitations. A similar result was obtained from increasing the radius of impactor and leaving the other properties unchanged using a density of 1000 m 3. This effectively increases the mass, thus decreasing the frequency of the impactor. It was also observed that doubling the impact area with all other parameters held constant, doubles the period thus decreases the frequency. This analogy is used in the next sections in order to overcome numerical instabilities for high frequency analysis. The occurrence of the maximum tensile strength from the impactor model was only a short duration as shown in Figure 6 below. Figure 6: Centreline stresses for the impactor model

6 tensile stress efficiency (%) Optimization of this wave would be difficult especially in experimental applications. Due to this reason, the single sinusoidal pulse was also considered as an option to investigate. The load is defined by equation 2. Where, P is resultant vertical stress P A[ 1 cos(2 ft)] [2] A is the maximum compressive stress applied f is the frequency t is time The type of distribution of the stresses along the rock centre line due to this load is shown in Figure 7. Figure 7: Stress distribution along rock centreline elements in response to single pulse loading It has been shown from the ELFEN model that an increase in impact frequency results in increased efficiency of the induced tensile wave. In applying the model in ELFEN, a range of frequencies between 1 khz and 320 khz was applied to the function. For FLAC a model 200 times in size was used with frequencies 200 times less to those of ELFEN. In addition, a constant frequency of 80 khz was used in ELFEN and the loading area was altered to match the lower and higher frequencies of the range. Similar adjustments were done with FLAC and the normalised results are shown in Figure 9 below. Effect of Pulse Frequency frequency (Hz) FLAC frequency FLAC scale ELFEN frequency ELFEN scale

7 Figure 9: The effect of frequency and scaling on tensile wave efficiency Consistency is observed between the FLAC and ELFEN models from the plots. A numerical instability at around 700 Hz is noticed on the higher frequencies; there is a decrease in tensile wave efficiency and a steady efficiency of close to 100% for the scaled points. This implies that once the maximum efficiency is reached, no matter how much the frequency is increased, the there will be little difference on the efficiency value. Conclusion The impact frequency on a rock surface is critical in maximising production of the tensile wave during the induction of a compressive wave. Additional to this is the scaling effect of the area of impact. As the frequency is held constant, and the loading area is decreased or increased, the efficiency follows respectively. References Kuijpers JS, Güler G and Ilgner H Literature Survey on Mechanised Mining Including Electric Shock Wave Comminution and Simple Laboratory Tests, CSIR : Division of Mining Technology, STEP project number Y5265 Li-Lih Wang Unloading Waves and Unloading Failures in Structures Under Impact Loading, Mechanics and Materials Science Research Centre, Ningbo University, Ningbo, Zhejiang , China, International Journal of Impact Engineering, Elsevier.