#### Year

2001

#### Degree Name

Doctor of Philosophy

#### Department

Faculty of Informatics

#### Recommended Citation

Cox, G. M., Stress distributions beneath heaps and rat-hole problems for granular materials, Doctor of Philosophy thesis, Faculty of Informatics, University of Wollongong, 2001. http://ro.uow.edu.au/theses/1851

#### Abstract

Granular materials are widely used throughout the world in many industries, and problems such as stable obstructions or uncontrolled flooding that occur impact significantly on the economies of such industries. Using the proper continuum mechanical theory of granular materials, two such problems are examined in this thesis. Firstly, the phenomenon of stable vertical cylindrical cavities known as rat-holes in stockpiles and hoppers that impede the flow of the granular material through the reclaim hole is examined. Secondly, the stress distribution at the base of a two and three-dimensional sand pile is considered in search of the peak vertical pressure, which may not be located directly beneath the vertex of the sand-pile.

The rat-hole problem is well known but is not properly understood, and existing theory is unsatisfactory, in that it is believed not to properly incorporate actual material properties. Here the classical rat-hole theory of Jenike and his co-workers is re-examined, with a view to examining the validity of the so-called "stable rat-hole equation", which is widely used in practice. Jenike's original theory assumes a symmetrical stress distribution which is independent of height. However in practice, rat-holes tend to exhibit some tapering with height, and here the stress profiles corresponding to a symmetrical but slightly tapered circular cavity are determined. Existing theory for rat-holes applies only to the Coulomb-Mohr yield function. Here for an existing rat-hole, and assuming a shear-index granular material, the limiting stress profiles are determined which extends existing theory to a wider constitutive law.

The determination of the horizontal and vertical force distributions at the base of a sand-pile is by now a famous problem in granular theory. In 1981 it was suggested from experimental work that the peak vertical force at the base does not occur directly beneath the vertex of the pile, but at some intermediate point so that there is a ring of maximum vertical pressure. Practising engineers have some reservation this result and numerous discrete theoretical and computational models of granular sand-piles have been proposed to explain this phenomenon, with varying degrees of success. Here for two and three-dimensional sand-piles the horizontal and vertical force distributions are estimated using the Jenike solutions for converging hoppers. For a two-dimensional sand-pile, a formal exact parametric solution is presented for the special case of the angle of internal friction equal to ninety degrees. Next, a sand-pile that is not entirely at yield is proposed, which has an inner dead region and an outer yield region. From this model a solution is determined which is not unique but does predict the dip in the vertical force as suggested from experimental work.

Finally, the exact parametric solution for the two-dimensional sand-pile problem for an angle of internal friction equal to ninety degrees is exploited to solve the of granular materials in the presence of gravity through a converging wedge shaped hopper, which are used in many industrial situations. This is the only known exact solution of these important equations which involves two arbitrary constants.