On the Problem of the Determination of Force Distributions in Granular Heaps Using Continuum Theory

Hill, James M.; Cox, Grant M.

doi:10.1093/qjmam/55.4.655

Cited by 14 publications

(10 citation statements)

References 7 publications

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“…We have utilized recently determined exact parametric solutions, derived in Hill and Cox [18] and Cox and Hill [20], to determine the stress distributions within two-dimensional wedge and three-dimensional conical shaped stock piles, assuming a highly frictional granular solid. This work extends that of Hill and Cox [14] to the case of φ = π/2 .…”

Section: Results and Conclusionmentioning

confidence: 99%

Stress distributions in highly frictional granular heaps

Thamwattana

Cox

Hill

2004

Zeitschrift f�r Angewandte Mathematik und Physik (ZAMP)

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The practice of storing granular materials in stock piles occurs throughout the world in many industrial situations. As a result, there is much interest in predicting the stress distribution within a stock pile. In 1981, it was suggested from experimental work that the peak force at the base does not occur directly beneath the vertex of the pile, but at some intermediate point resulting in a ring of maximum pressure. With this in mind, any analytical solution pertaining to this problem has the potential to provide useful insight into this phenomenon. Here, we propose to utilize some recently determined exact parametric solutions of the governing equations for the continuum mechanical theory of granular materials for two and three-dimensional stock piles. These solutions are valid provided sin φ = 1 , where φ is the angle of internal friction, and we term such materials as "highly frictional". We note that there exists materials possessing angles of internal friction around 60 to 65 degrees, resulting in values of sin φ equal to around 0.87 to 0.91. Further, the exact solutions presented here are potentially the leading terms in a perturbation solution for granular materials for which 1 − sin φ is close to zero. The model assumes that the stock pile is composed of two regions, namely an inner rigid region and an outer yield region. The exact parametric solution is applied to the outer yield region, and the solution is extended continuously into the inner rigid region. The results presented here extend previous work of the authors to the case of highly frictional granular solids. Mathematics Subject Classification (2000). 73B10, 73B99, 73Q05.

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Section: Results and Conclusionmentioning

confidence: 99%

Stress distributions in highly frictional granular heaps

Thamwattana

Cox

Hill

2004

Zeitschrift f�r Angewandte Mathematik und Physik (ZAMP)

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“…It is evident from the experiments of Reference [4] that localization is the main source of the dip in stress. Therefore, efforts using classical Sokolovoski type rigid plastic continuum formulations to account for this effect would entail the use of unrealistic values of 908 for granular material friction angle [7]. It is noted that some elastic-plastic models have predicted the existence of the local stress minimum at the centre [8].…”

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confidence: 99%

Stress distribution in granular heaps using multi-slip formulation

Hattamleh

Muhunthan

Zbib

2005

Int. J. Numer. Anal. Meth. Geomech.

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SUMMARYMany experiments on conical piles of granular materials have indicated, contrary to simple intuition that the maximum vertical stress does not occur directly beneath the sand-pile vertex but rather at some distance from the apex resulting in a ring of maximum vertical stress. Some recent experiments have shown that the observed stress dip is very much dependent on construction history. A multi-slip model has been proposed to investigate the stress dip phenomenon in granular heaps. The double-slip version of the model was implemented into ABAQUS and used to study the vertical stress distribution along the base of a granular pile. The numerical simulations show that plastic deformation is confined within the localized region around the apex while the rest of the pile is in an elastic state of deformation. Within the plastic region the stress distribution differs significantly depending on the initial active slip orientation. The results show that for homogenous state of granular materials such as those produced by a raining procedure the vertical stress profile along the base reached its peak at the apex (i.e. no dip was observed). On the contrary, granular heaps constructed by the use of a localized source such as a funnel resulted in a significant reduction in the stress distribution within the ring with the minimum attained beneath the peak (i.e. a dip). Therefore, we believe that the initial microstructure and thus the initial slip orientation resulting from sand deposition is the source of the stress dip phenomenon.

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“…But, in general, we observe that Equation (2.14) can only satisfy one of the necessary boundary conditions. We also note from Hill and Cox [8] that the vertical and horizontal force distributions acting along a plane x = constant = h in a two-dimensional wedge-shaped stock pile are respectively…”

Section: Boundary Conditions For the Hopper Problemmentioning

confidence: 78%

“…Measured values of β = sin φ Coal 0.939 0.958 0.973 0.985 Alumina cake 0.941 Waste rock 0.974 Silica 0.979 converging wedge-shaped hopper, an exact parametric solution of these equations is given by Hill and Cox [7] for the special case of the angle of internal friction equal to 90 degrees. This exact parametric solution was later exploited by the same authors in Hill and Cox [8] to determine an exact parametric solution for a two-dimensional wedge-shaped stock pile. This exact solution is the first exact solution of the highly nonlinear coupled ordinary differential equations which involves two arbitrary constants.…”

Section: Granular Materialsmentioning

confidence: 99%

Some Exact Mathematical Solutions for Granular Stock Piles and Granular Flow in Hoppers

Cox

Hill

2003

Mathematics and Mechanics of Solids

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The flow of granular materials in the presence of gravity in hoppers and the storage of granular materials as a stock pile occur in many industrial situations. The governing ordinary differential equations for two-dimensional wedges and three-dimensional cones are highly nonlinear and there are no known general solutions, apart from that we have given for a special angle of internal friction. Here, we give the overall picture relating to those special cases which give rise to analytical solutions for the two problems of granular flow through hoppers and the stress distributions at the base of stock piles. These equations are fundamental to granular mechanics and previously only some special isolated exact solutions have been known. We list here a number of new exact analytical solutions applying for the two special cases of β= ±1, noting that β= sin φ where φ is the angle of internal friction. The case β= −1 corresponds to a non-physical material, but there are materials such as silica and alumina cake which do indeed exhibit large angles of internal friction and the case β= 1 is by no means unrealistic. However, all the solutions presented are meaningful mathematical solutions of the governing equations and constitute the only known general solutions of these important equations. For certain cases, a full independent numerical solution has been obtained and shown to coincide with the appropriate exact analytical solution.

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On the Problem of the Determination of Force Distributions in Granular Heaps Using Continuum Theory

Cited by 14 publications

References 7 publications

Stress distributions in highly frictional granular heaps

Stress distributions in highly frictional granular heaps

Stress distribution in granular heaps using multi-slip formulation

Some Exact Mathematical Solutions for Granular Stock Piles and Granular Flow in Hoppers

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