AP08_4.vp 1 Introduction Scientific and technical developments (in all areas of world-wide industry) are affected by the growing demand for basic raw materials and energy. The provision of sufficient quantities of raw materials and energy for the processing industry is the main limiting factor of further development. It is therefore very important to understand the ore disintegration process, including an analysis of the bit (i.e. excavation tool) used in mining operations. The main focus is on modeling the mechanical contact between the bit and the ore, see Fig. 1. 2 Finite element model of the ore disintegration process FEM (i.e. MSC.Marc/Mentat 2005r3 and 2008r1 software) was used in modeling the ore disintegration process. Figure 2 shows the basic scheme (plain strain formulation, mechani- © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 51 Acta Polytechnica Vol. 48 No. 4/2008 Probabilistic Analysis of the Hard Rock Disintegration Process K. Frydrýšek This paper focuses on a numerical analysis of the hard rock (ore) disintegration process. The bit moves and sinks into the hard rock (mechanical contact with friction between the ore and the cutting bit) and subsequently disintegrates it. The disintegration (i.e. the stress-strain relationship, contact forces, reaction forces and fracture of the ore) is solved via the FEM (MSC.Marc/Mentat software) and SBRA (Simulation-Based Reliability Assessment) method (Monte Carlo simulations, Anthill and Mathcad software). The ore is disintegrated by deactivating the finite elements which satisfy the fracture condition. The material of the ore (i.e. yield stress, fracture limit, Young’s modulus and Poisson’s ratio), is given by bounded histograms (i.e. stochastic inputs which better describe reality). The results (reaction forces in the cutting bit) are also of stochastic quantity and they are compared with experimental measurements. Application of the SBRA method in this area is a modern and innovative trend in mechanics. However, it takes a long time to solve this problem (due to material and structural nonlinearities, the large number of elements, many iteration steps and many Monte Carlo simulations). Parallel computers were therefore used to handle the large computational needs of this problem. Keywords: Hard rock (ore), cutting bit, disintegration process, FEM, probability, SBRA method, parallel computing. Fig. 1: A typical example of mechanical interaction between bits and hard rock (example of the ore disintegration process) Fig. 2: Geometry of 2D FE model, boundary conditions and details cal contact with friction between the bit and platinum ore, boundary conditions, etc.). Fig. 2 shows that the bit moves into the ore with the pre- scribed time dependent function u f t� ( ), and subsequently disintegrates it. When the bit moves into the ore (i.e. a me- chanical contact occurs between the bit and the ore) the stresses �HMH (i.e. the equivalent von Mises stresses) in the ore increase. When the situation �HMH m� R occurs (i.e. the equivalent stress is greater than the fracture limit) in some ele- ments of the ore, then these elements break off (i.e. these ele- ments are dead). Hence, a part of the ore disintegrates. In MSC.Marc/Mentat software, this is done by deactivating the elements that satisfy the condition �HMH m� R . This deacti- vation of the elements was performed in every 5th step of the solution. For further information see references [1] and [2]. 3 Probabilistic inputs – SBRA (Simulation-Based Reliability Assessment) method A deterministic approach (i.e. all types of loading, dimen- sions and material parameters etc. are constant) provides an older but simple way to simulate mechanical systems. However, a deterministic approach cannot truly include the variability of all inputs, because nature and the world are sto- chastic. Simulations of the ore disintegration process via a 52 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 4/2008 Fig. 3: Material properties – whole model and material of the ore (stress � vs. plastic strain �p) Fig. 4: Stochastic inputs for the material of the ore (histograms for yield stress and fracture stress, results of Anthill software) deterministic approach are shown in [1] and [2]. However, this problem is also solved via probabilistic approaches which are based on statistics. Let us consider the “Simulation-Based Reliability Assess- ment” (SBRA) Method, a probabilistic approach, in which all inputs are given by bounded histograms. Bounded histo- grams include the real variability of the inputs. Application of the SBRA method (based on Monte Carlo simulations) is a modern and innovative trend in mechanics, see for example [3] to [5]. The material properties (i.e. isotropic and homogeneous materials) of the whole system are described in Fig. 3, where E is Young’s modulus of elasticity and � is Poisson’s ratio. The bit is made of sintered carbide (sharp edge) and steel. The ore material is elasto-plastic with yield limit Rp � � �9 946 0 911 1722. . . and fracture limit Rm � � �12 661 0 650 0 925. . . , which are given by bounded histograms see Figs. 3 and 4. The elastic properties of the ore are described by Hooke’s law in the histograms E � � �18513 8 2418 8 2608 8. . . MP and � � � �0199 0 019 0 021. . . ), see Figs. 5 and 6. Applications of the SBRA method in combination with FEM and subsequent evaluation of the results are shown © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 53 Acta Polytechnica Vol. 48 No. 4/2008 Fig. 5: Stochastic inputs for the material of the ore (histogram of Young’s modulus, results of Anthill software) Fig.6. Stochastic inputs for the material of the ore (histogram of Poisson’s ratio, results of Anthill software) in Fig. 7. Anthill, MSC.Marc/Mentat and Mathcad software were used. 4 Solution – SBRA method in combination with FEM Because of the material non-linearities, the mechanical contacts with friction, the large number of elements many iteration steps, and the choice of 500 Monte Carlo simula- tions, four parallel computers were used to handle the large computational requirements for this problem, see Table 1. The Domain Decomposition Method (i.e. application of parallel computers) was used, see Fig. 8. 54 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 4/2008 Computer name Computer description Software No. of CPU No. of MC simulations Wall time (hours) Alfa: Linux OS, 8 nodes. Node configuration: 2x CPU AMD Opteron 250 (frequency 2.4 GHz, 1MB l2 cache) with 4 GB RAM (400MHz DDR) MSC.Marc/Mentat 16 312 70.395 Opteron: Linux OS, 1 nodes. Node configuration: 2x CPU AMD Opteron 248 (frequency 2.2 GHz) with 8 GB RAM MSC.Marc/Mentat 2 28 54.6 Quad: Linux OS, 4 nodes. Node configuration: 1x CPU AMD Opteron 848 (frequency 2.2 GHz) with 4 GB RAM MSC.Marc/Mentat 4 86 69.015 Pca632d: MS Windows XP professional 64 bit OS, 4 nodes. Configuration: Intel core 2 quad CPU q9300 (frequency 2.5 GHz) with 8 GB RAM MSC.Marc/Mentat, Anthill, Mathcad 4 74 68.82 � 26 � 500 � cca 70.4 Table 1: Parallel computers used in this study (date: August-September 2008) Fig. 7: Computational procedure – application of the SBRA method (solution of the ore disintegration process) Fig. 8. Domain Decomposition Method Used for Application of 2 CPU and 4 CPU (i.e. Ways of Performing one Monte Carlo Simulation) The whole solution time for the non-linear solution (i.e. 1.04 s) was divided into 370 steps of variable length. The Full Newton-Raphson method was used for solving the non-linear problem. Table 1 shows that the solution of 500 Monte Carlo simu- lations (calculated simultaneously on four different parallel computers) takes cca 70.4 hours. 5 Results – stochastic evaluation Figs. 9 to 14 show the equivalent stress (i.e. �HMH distribu- tions) at some selected time t of the solution calculated for one of 500 Monte Carlo simulations (i.e. for one situation when the material of the ore is described by values Rp � 12 MPa, Rm � 13 5. MPa, E � 20000 MPa and � � 0 2. ). The movement of the bit and also the subsequent disintegration of the ore caused by the cutting are shown. From the FEM results, we can calculate the reaction forces RX, RY and the total reaction force R R RX Y� � 2 2 which acts in the bit, see Figs. 15 and 16. Figure 16 is calculated for one simulation (i.e. for the situation when the material of the ore is described by values Rp � 12 MPa, Rm � 13 5. MPa, E � 20000 MPa and � � 0 2. ). © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 55 Acta Polytechnica Vol. 48 No. 4/2008 Fig. 9: t � 0 s (FEM results, start of the solution) Fig. 10: t � � �3 37 10 2. s (FEM results) Fig. 11: t � � �3 714 10 1. s (FEM results) Fig. 12: t � � �8 335 10 1. s (FEM results) Fig. 13: t � 0 8511. s (FEM results) A distribution of the total reaction forces acquired from 500 simulations is shown in Fig.17. The maximum total reaction force (acquired from 500 Monte Carlo simulation) is given by the histogram RMAXSBRA, FEM � � �5068 984 1098 N, see Fig. 18. 56 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 4/2008 Fig. 14: t � 1 026. s (FEM results) Fig.15. Reaction forces in the bit Fig.16. Reaction forces in the bit (FEM results) Fig. 17: Total reaction forces in the bit (SBRA-FEM results) Fig. 18: Maximum total reaction forces in the bit (SBRA-FEM results of 500 Monte Carlo simulations), and an evaluation 6 Comparison between stochastic results and experimental measurements The calculated maximum forces (i.e. SBRA-FEM solu- tions, see Fig. 18) can be compared with the experimental measurements (i.e. compared with a part of Fig. 19), see also [1] and [2]. The evaluation of one force measurement (Fig. 19) shows that the maximum force is RMAXEXP � 5280 N. Hence, the relative error calculated for the acquired median value RMAXSBRA, FEM MED� � 5068 N, see Fig. 18, is: �R R R RMAX � � � � � MAX MAX MAX EXP SBRA, FEM MED EXP 0 01 4 02 . . %. The error of 4.02 % is acceptable. However, the experi- mental results also have large variability due to the aniso- tropic and stochastic properties of the material and due to the large variability of the reaction forces, see Fig. 19. 7 Conclusions This paper combines the SBRA (Simulation Based Reli- ability Assessment) method and FEM as a suitable tool for simulating the hard rock (ore) disintegration process. All ba- sic factors have been explained (2D boundary conditions, material nonlinearities, mechanical contacts and friction be- tween the cutting bit and the ore, the methodology for de- activating the finite elements during the ore disintegration process, application of parallel computers). The use of finite element deactivation during the ore disintegration process (as a way of expanding the crack) is a modern and innovative way of solving problems of this type. The error of the SBRA-FEM results (i.e. in comparison with the experiments) is acceptable. Hence, SBRA and FEM can be a useful tool for simulating the ore disintegration process. Because the real material of the ore (i.e. yield limit, frac- ture limit, Young’s modulus, Poisson’s ratio etc.) is very vari- able, stochastic theory and probability theory were applied (i.e. the SBRA method). The SBRA method, which is based on Monte Carlo simu- lations, can include all stochastic inputs and then all results are also of stochastic quantities. However, for better applica- tion of the SBRA method (for simulating this large problem in mechanics) it is necessary to use superfast parallel com- puters. Instead of 500 Monte Carlo simulations (wall time cca 70 hours, as presented in this article), it is necessary to cal- culate >104 simulations (wall time cca 58 days), or more. Our department will be able to make these calculations when faster parallel computers became available. All the results presented here were applied for optimizing and redesigning the bit. In the future, 3D FE models (instead of 2D plane strain formulation), will be applied for greater accuracy. Other methods for simulating the ore disintegration pro- cess are presented in [6] and [7]. Acknowledgment This work has been supported by the Czech project FRVŠ 534/2008 F1b. References [1] Frydrýšek, K.: Výpočtová zpráva styku nože a platinové rudy při těžbě (Calculation Report on Contact between the Bit and Platinum Ore During Mining), 2007, Czech Repub- lic, 2007, p. 17 (in Czech language). [2] Frydrýšek, K., Gondek, H.: Finite Element Model of the Ore Disintegration Process, In: Annals of the Faculty of En- gineering Hunedoara – Journal of Engineering, Tome VI, Fascicule 1, ISSN 1584-2665, University Politechnica Timisoara, Faculty of Engineering – Hunedoara, Roma- nia, 2008, p. 133–138. [3] Frydrýšek, K.: Performance-Based Design Applied for a Beam Subjected to Combined Stress, In: Annals of the Faculty of Engineering Hunedoara – Journal of Engi- neering, Tome VI, Fascicule 2, ISSN 1584-2665, Univer- sity Politechnica Timisoara, Faculty of Engineering – Hunedoara, Romania, 2008, p. 129–134. [4] Marek, P., Brozzetti, J., Guštar, M., Tikalsky, P.: Probabil- istic Assessment of Structures Using Monte Carlo Simulation Background, Exercises and Software, (2nd extended edi- tion), ISBN 80-86246-19-1, ITAM CAS, Prague, Czech Republic, 2003, p. 471. [5] Marek, P., Guštar, M., Anagnos, T.: Simulation-Based Reliability Assessment For Structural Engineers. CRC Press, Boca Raton, USA, ISBN 0-8493-8286-6, 1995, p. 365. [6] Zubrzycki, J., Jonak, J.: Numeryczno-eksperymentalne bada- nia wplyvu kstaltu powierchni natarcia ostrza na obciaženie © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 57 Acta Polytechnica Vol. 48 No. 4/2008 Fig. 19: Experimental measurement, compared with the SBRA-FEM 58 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 4/2008 noža skrawajacego naturalny material kruchy (in Polish lan- guage), Lubelskie Towarzystwo Naukowe, Lublin, Poland, 2003, ISBN 83-87833-42-8, p. 90. [7] Podgórski, J., Jonak, J.: Numeryczne badania procesu skra- vania skał izotropovych (in Polish language), Lubelskie Towarzystwo Naukowe, Lublin, Poland, 2004, ISBN 83-87833-53-3, pp. 80. MSc. Karel Frydrýšek, Ph.D., Ing-Paed IGIP phone: +420 597 324 552 e-mail: karel.frydrysek@vsb.cz Department of Mechanics of Materials VŠB-TU Ostrava Faculty of Mechanical Engineering 17. listopadu 15 708 33 Ostrava, Czech Republic << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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