Strona 2_redak - Instytut Agrofizyki im. Bohdana Dobrzańskiego ...

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The micropolar model yields results that are convergent with those obtained
with the distinct element method. Comparison of the fields of displacement and
rotation of the material, determined with the two methods for the case of loading of
a random system of spheres with normal and tangential stresses and with couple
stresses, performed by Chang and Liao [32], confirms the good agreement of the
solutions of both the methods (fig. 3.15). However, the methods differ in their areas
of practical application. Models basing on the formalism of the mechanics of
continuum provide a convenient and practical method of solving problems
concerning a large number of granules. The model of micropolar medium combined
with the finite element method constitutes then an effective tool for the description
of even highly complex processes occurring in practice [160]. Limitations of microstructural
models analyzing the motion of each individual granule of the medium
result from the computational capacity. Nevertheless, the models provide deeper
knowledge on the mechanisms of stress transmission and on the occurrence of
deformations on the level of interactions between individual granules [72].
3.6. Localization of shear deformation
In the course of numerous operations performed on granular materials, nondilatational
strain of the material is localized within a small area of the material.
The reasons for this lie both in external conditions of the operations performed, and
in the mechanical properties of the granular material. In the final stage of the shear
process, when the stress is close to the critical stress state, strain usually loses its
initial uniformity and a clearly defined shear band forms, separating the areas of
rigid movement of the material (fig. 3.16). Deformation is mainly localized within
the shear band formed. This phenomenon is commonly observed in silos with rough
walls, during so-called mass flow, when between the silo wall and the flowing
material there forms an intermediate (boundary) layer of granular material. It is in
that layer that shearing of the material occurs, as well as dilation causing silo
overload. It is assumed that the thickness of the boundary layer of a granular material,
in which shear takes place, is constant and does not depend on the dimensions of the
silo [120, 177]. This would imply that with increasing dimensions of the silo the
effect of the boundary layer on dynamic overload of the silo decreases. Analysis of
the scale errors resulting from generalization of results of stress distribution in
model scale studies onto real size objects indicates a significant contribution of the
processes taking place in the shear band to the level of the errors [120, 124, 125].
The theory of the Cosserat brothers, including in the equations of medium motion
the displacements and rotations of granules, permits the analysis of stress and strain
distribution along the shear band thickness [119]. Inclusion of granule rotations in the
theory introduces into the equations of medium motion the dimension of a single