Problems

The list of problems will be updated as problems are submitted and further information about the present problems and additional problems becomes available. Please keep watching these pages. ɳ°ÍÌåÓý¹ÙÍø_2024Å·ÖÞ±­²©²Êapp@ five problems will be selected depending on the expertise and interests of the participants.

1. Problem: Analysis of the potential mechanisms of rockbursts
   
   Industry: Mining industry
   
   Industry representative: Mr Duncan Adams
   
   Moderator: Professor John Napier
   
   Description:
   The violent failure of rock in deep-level mining operations has been an ongoing deterrent to the optimal extraction of valuable mineral resources in many countries. In South Africa this problem has plagued the gold mining industry for many years and has been the subject of extensive research efforts for at least four decades. Although production from South African gold mines has now decreased significantly, considerable economic interest is currently vested in the extraction of large-scale chrome and platinum reserves concentrated in the Bushveld Complex situated west and north of Pretoria in South Africa.
   
   One of the pioneering efforts to explain the mechanics of rockbursts was made in the early 1960?s by N.G.W. Cook (Cook, 1965). He proposed a simple stability analysis in which the unstable region is considered to be coupled to an ?external? loading region. By comparing the effective load-deformation response of the external region (loading ?stiffness?) to the load-deformation characteristics of the ?internal? region one can state whether the ?system? will respond in a stable or an unstable manner. This approach is essentially a simplified form of more detailed analyses of stability examining changes in the structure of the partial differential equations of motion at a point of instability (for example, Rice, 1976, Vermeer, 1990, Gajo et al., 2004). Further elaborations of these concepts have been made by considering numerical models of fault slip and accompanying earthquake cycles that depend on the nature of the fault cohesion breakdown physics and on friction laws that are dependent on the rate of fault slip (Rice, 1993, Lapusta and Rice, 2003).
   
   Other significant approaches to the problem of rapid rock failure and ongoing cycles of failure, associated with earthquakes or intermittent volcanic eruptions, have been numerous attempts to link these phenomena to theories of critical phenomena and the determination of critical exponents that are associated with first and second order phase transitions. This has been linked closely to the concept of ?self-organized criticality? that can be used to explain the universality of power law statistics that apply to a wide variety of phenomena including the frequency-magnitude relationships associated with earthquakes and deep mine seismic activity (Bak et al., 1987, Tang and Bak, 1988, Anghel, 2004). A linked consideration is the predictability of earthquakes that might be related to ?log-periodic? oscillations that are imposed on acoustic emission or seismic activity records prior to failure (Moura, 2005).
   
   For the purposes of the study group, it is suggested that a review should be made of the intrinsic formulation of the equations of motion at a point of failure transition and that it should be assessed whether a Lie/ Symmetry analysis could be applied to these systems when the equation structure is in transition.
   
   References
   Anghel, M. ?On the effective dimension and dynamic complexity of earthquake faults?, Chaos, Solitons and Fractals, Vol. 19, 2004, pp. 399-420.
   
   Bak, P., Tang, C. and Wiesenfeld, K. ?Self-organized criticality: An explanation of 1/f noise?, Physical Review Letters, Vol. 59, 1987, pp. 381-384.
   
   Cook, N.G.W. ?The basic mechanics of rockbursts?, In proceedings: Symposium on Rock Mechanics and Strata Control in Mines, published by The South African Institute of Mining and Metallurgy, Johannesburg, 1965, pp. 56-66.
   
   Moura, A., Lei, X. and Nishisawa, O. ?Prediction scheme for the catastrophic failure of highly loaded brittle materials or rocks?, J. Mech. Phys. Solids, Vol. 53, 2005, pp. 2435-2455.
   
   Gajo, A., Bigoni, D. and Muir Wood, D. ?Multiple shear band development and related instabilities in granular materials?, J. Mech. Phys. Solids, Vol. 52, 2004, pp. 2683-2724.
   
   Lapusta, N. and Rice, J.R. ?Nucleation and early seismic propagation of small and large events in a crustal earthquake model?, Journal of Geophysical Research, Vol. 108, 2003, pp. ESE 8-1 to ESE 8-18.
   
   Rice, J.R. ?The localization of plastic deformation?, In: Theoretical and Applied Mechanics ? 14^th Congress, International Union of Theoretical and Applied Mechanics, W.T. Koiter, ed., North-Holland Publishing Company, 1976, pp. 207-220.
   
   Rice, J.R. ?Spatio-temporal complexity of slip on a fault?, Journal of Geophysical Research, Vol. 98, 1993, pp. 9885-9907.
   
   Tang, C. and Bak, P. ?Critical exponents and scaling relations for self-organized critical phenomena?, Physical Review Letters, Vol. 60, 1988, pp. 2347-2350.
   
   Vermeer, P.A. ?The orientation of shear bands in bi-axial tests?, Geotechnique
   
   
2. Problem: Polar plots of diamond surface energy
   
   Industry: Element Six Technologies
   
   Industry representative: Joh Hansen
   
   Moderator: Open
   
   Description:
   The Wulff-Gibbs Theorem in crystal growth states that the equilibrium shape of a crystal can be derived by constructing normals to the polar plot of the surface energy. Consider two dimensions for simplicity. Derive an efficient algorithm to compute the surface energy for any value of the normal to the surface, on at least two major sectioning planes.
   
   The basis of the computation is the density by area of the carbon-carbon bonds which are intersected and therefore broken by unit area of the chosen plane. The energy of the carbon-carbon bond is assumed to be the same for all orientations of the carbon-carbon bond in diamond.
   
   A good general reference is:
   
   S. Terentiev. Molecular-dynamics simulation of the effect of temperature of the growth environment on diamond habit. Diamond and Related Materials, 8 (1999), 1444-1450.
   
   A reference to the Wulff-Gibbs theorem is:
   http://www.dur.ac.uk/sharon.cooper/lectures/
   
   
3. Problem: Optimization of deliveries from distribution points

   Industry: CaterPlus

   Industry representative: Mr Martin Engelbrecht

   Moderator: Professor Montaz Ali

   Description:
   CaterPlus is a broad line distributor of a comprehensive range of food service products, which serves numerous customers in southern Africa via strategically located, independent business units.

   CaterPlus has embarked on a systematic process to determine the geographic extent of its total customer base in southern Africa. Thus far, the process has involved dividing South Africa into clearly demarcated geographic regions. This was done firstly by province, then by municipal district, main-places and lastly sub-place, as defined by Statistics South Africa in the 2001 census. As far as possible a postal code has been allocated to the sub-place. CaterPlus is now planning to divide its entire customer base into these defined regions to assist with the general optimization of its business. This will be done by utilising the postal code provided with the customer delivery address. By doing this, CaterPlus is hoping to define its current customer base to determine the geographic extent of its involvement within the industry.
   
   The initial study embarked upon was the customer base of the Catering Supplies Division, which is South Africa?s leading food service distributor of dry groceries and consumables operating eighteen decentralised business units, run by independent, entrepreneurial management teams operating under various trade names. It is the intention of CaterPlus to expand the study to include the customer base of its Frozen Foods division which operates an additional fourteen strategically located business units.
   
   By embarking on this process CaterPlus has encountered several challenges, as well as opportunities that it would like to investigate including, but not limited to:

   • the optimization of its deliveries from the points of distribution in all the major centres in South Africa, including Johannesburg, Cape Town, Durban, Bloemfontein, Port Elizabeth, East London, George, Pietermaritzburg, Empangeni, Polokwane and Nelspruit
   • the determination of the extent of the clientele shared between its business units
   • the determination of the extent of the market that does not currently form part of its national customer base which will possibly be done by utilizing the VAT numbers of its clientele that it currently has on record at its various business units.

   CaterPlus would like correct guidance in this process.

   A broad overview of the Companies in CaterPlus can be found on the web site
   http://www.bidvest.co.za/index.asp within the category Food Service Products.

4. Problem: HIV modelling in a labour force

   Industry: Mining industry

   Industry representatives: Medical practitioners in the industry

   Moderators: Dr Londiwe Masinga and Professor David Sherwell

   Description:
   The impact of HIV/AIDS in the labour force is a major area of concern for industry. Predicting future health and ability to return and continue working in a physically demanding environment, for employees starting Antiretroviral Therapy (ART), is an important element for human resource management.

   The Study Group is asked to

   • examine company medical data for robust indicators for the successful placement of workers on ART in categories of physical labour severity
   • examine optimal placement of a worker, given his/her fitness and the needs of the company
   • consider predictor modelling of this problem
     
     
     
5. Problem: Modelling temperature, moisture content and maturity in concrete dams

   Industry: Cement and Concrete Institute

   Industry representative: Professor Yunus Ballim

   Moderator: Open

   Description:
   At the first Mathematics in Industry Study Group a mathematical model was developed to analyse the temperature, moisture content and maturity in concrete dams. Maturity describes the degree of hydration which occurs when water and cement are combined. By considering heat conservation, hydration chemistry and water conservation, three coupled nonlinear partial differential equations for the temperature, moisture content and maturity were derived. Since the moisture diffusivity of concrete depends on the moisture content, the moisture content satisfies a nonlinear diffusion equation. The Study Group is asked to investigate analytical and numerical solutions of this system of equations subject to suitable boundary and initial conditions for models of hydration of concrete dams.

6. Problem: Modelling airblasts in a long tunnel with surface roughness

   Industry: Mining

   Industry representative: Professor Richard Stacey

   Moderator: Eunice Mureithi

   Description:
   Airblasts due to the collapse of rock are extremely hazardous occurrences and can cause death and damage to mining equipment and infrastructure.

   The Second Mathematics in Industry Study Group in 2005 developed models for the increase of air pressure due to the collapse of rock in an underground excavation. It found that the rise in pressure is not very large.

   The next step is now to model the flow of air in a tunnel with surface roughness connected to the excavation. The tunnel is sufficiently long for turbulent dissipation to be significant. A model due to Fanno has been developed for turbulent compressible air flow in a tunnel. In the Fanno model the mass and energy equations are the same as in the classical inviscid model but the momentum equation has an extra term due to wall drag.

   The Study Group is asked to:

   • investigate if the parameters for typical mining problems lead to the Fanno regime
   • investigate a Lie group analysis of the full Fanno model and of asymptotic reductions of the model that have been performed.

   Reference:
   H Ockendon, JR Ockendon and SAEG Falle. The Fanno model for turbulent compressible flow. J Fluid Mechanics, 445 (2001), 187-206