The flow of granular materials in the presence of gravity through converging wedges and cones arises in many industrial situations. For both wedges and cones, and assuming an ideal cohesionless granular material which satisfies the Coulomb-Mohr yield condition, the number of simple analytical solutions is limited and generally the governing coupled ordinary differential equations need to be solved numerically. Here we show that for plane wedge flow, an exact parametric solution may be determined for the special case of the angle of internal friction φ is assumed to be π/2 . This is the only known exact solution of these important equations involving two arbitrary constants. A general numerical solution obtained by previous authors is shown to coincide with the special exact solution. Mathematics Subject Classification (2000). 73B10, 73B99, 73Q05.

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.

In this paper, we model the mechanics of a collagen pair in the connective tissue extracellular matrix that exists in abundance throughout animals, including the human body. This connective tissue comprises repeated units of two main structures, namely collagens as well as axial, parallel and regular anionic glycosaminoglycan between collagens. The collagen fibril can be modeled by Hooke's law whereas anionic glycosaminoglycan behaves more like a rubber-band rod and as such can be better modeled by the worm-like chain model. While both computer simulations and continuum mechanics models have been investigated the behavior of this connective tissue typically, authors either assume a simple form of the molecular potential energy or entirely ignore the microscopic structure of the connective tissue. Here, we apply basic physical methodologies and simple applied mathematical modeling techniques to describe the collagen pair quantitatively. We find that the growth of fibrils is intimately related to the maximum length of the anionic glycosaminoglycan and the relative displacement of two adjacent fibrils, which in return is closely related to the effectiveness of anionic glycosaminoglycan in transmitting forces between fibrils. These reveal the importance of the anionic glycosaminoglycan in maintaining the structural shape of the connective tissue extracellular matrix and eventually the shape modulus of human tissues. We also find that some macroscopic properties, like the maximum molecular energy and the breaking fraction of the collagen, are also related to the microscopic characteristics of the anionic glycosaminoglycan.

One approach to modeling fully developed shear flow of frictional granular materials is to use a yield condition and a flow rule, in an analogous way to that commonly employed in the fields of metal plasticity and soil mechanics. Typically, the yield condition of choice for granular materials is the Coulomb–Mohr criterion, as this constraint is relatively simple to apply but at the same time is also known to predict stresses that are in good agreement with experimental observations. On the other hand, there is no strong agreement within the engineering and applied mechanics community as to which flow rule is most appropriate, and this subject is still very much open to debate. This paper provides a review of the governing equations used to describe the flow of granular materials subject to the Coulomb–Mohr yield condition, concentrating on the coaxial and double-shearing flow rules in both plane strain and axially symmetric geometries. Emphasis is given to highly frictional materials, which are defined as those granular materials that possess angles of internal friction whose trigonometric sine is close in value to unity. Furthermore, a discussion is provided on the practical problems of determining the stress and velocity distributions in a gravity flow hopper, as well as the stress fields beneath a standing stockpile and within a stable rat-hole.

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