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The paper reports on the results of theoretical and experimental investigations on the spontaneous formation of shear bands in sand bodies. The phenomenon is considered as a bifurcation problem. Consequently, material response and configuration-dependent loading determine the bifurcation mode. Both Coulomb's and Roscoe's solutions of inclination of the shear band can be correct theoretically and experimentally. The first one holds for non-rotating stress axes, the second one for co-rotating stress and strain increment axes during failure. Values in between can occur if the rotation of principal stress axes is not equal to one of these limits. If Coulomb's inclination of shear band occurs, there is a thin deforming material layer separating rigid bodies. Inside the shear band non-coaxiality of strain increment and stress holds from the beginning. If Roscoe's inclination of shear band occurs, it is separating two deforming bodies. Inside the shear band strain increment and stress are coaxial at peak.

Evolutions which are equal in parallel planes are called plane-parallel, this is not always the same as plane strain. Soil-structure interactions (SSIs) are left aside in this chapter, solids occur only as rigid base. Initial and boundary conditions are more complex than in Chap. 11 and more specific than in Chap. 10. Attractors in the large help to justify approaches by calculations and model tests, or at least to delimit their range of validity. Other than with RSEs such asymptotes are non-uniform fields of state variables, but they are again driven (exogeneneous) and/or thermally activated (endogeneous). According to the involved RSEs they belong to monotonous deformations (SOM-states and state limits) or cyclic ones and ratcheting (asymptotic state cycles). As state fields are at best compounds of such states our attractors in the large cannot be treated with mathematical rigour. The generic term 'strange attractor' will sometimes be used for the spontaneous loss of symmetry by localization or diffuse bifurcation.Psammoid heaps on a solid base serve for introduction (Sect. 12.1). Even without water they cannot fully be captured by models with grain skeletons as these can arise and decay. With peloids in heaps skeleton viscosity and pore water diffusion play a dominant role (Sect. 12.2). We will see why conventional limit equilibria can at best yield design estimates. As always evolutions are dominated by skeleton stress obliquities and relative void ratios or consolidation ratios, these state quantities change with space and time.The ground yields usually more when placing fills if it consists partly of peloid (Sect. 12.3). We will meet the same dominant factors also with excavations (Sect. 12.4). Simplified design models will again be discussed. Some field observations support more sophisticated numerical models although the assumed plane-parallelity is rarely given.We will see in Sect. 12.5 how the same symmetry can be employed for seismically activated evolutions. Hypoplastic simulations are confirmed by shake box tests and field observations, but also delimited by critical phenomena.It is discussed what could be done with seismically activated viscosity and entropic pressures. Slow tectonics are considered in Sect. 12.6, therein initial Psammoid heaps upon a solid baseThe granular heaps considered in this section may consist of simple grain skeletons in the sense of Sect. 2.2, and may be surrounded by and contain more or less water. Their base may be rigid and fixed if not stated otherwise, and rough or smooth with respect to the grain size (Sect. 10.3). Heaps get shape, density and stress by placement, and can spread or collapse so that shape and state are changed. Such evolutions depend also on the water in the pores and around the heap. Grain skeletons can arise by sedimentation or placement and can get lost by decay or removal (Sect. 12.4), and they can take up or lose pore water. Only evolutions between rise and fall of skeletons can be captured with methods of soil mechanics. For simplicity we assu...

SUMMARYThe paper deals with numerical investigations of the behaviour of granular bodies during shearing. Shearing of a narrow layer of sand between two very rough boundaries under constant vertical pressure is numerically modelled with a "nite element method using a hypoplastic constitutive relation within a polar (Cosserat) continuum. The constitutive relation was obtained through an extension of a non-polar one by polar quantities, viz. rotations, curvatures, couple stresses using the mean grain diameter as a characteristic length. This relation can reproduce the essential features of granular bodies during shear localization. The material constants can be easily determined from element test results and can be estimated from granulometric properties. The attention is laid on the in#uence of the initial void ratio, pressure level, mean grain diameter and grain roughness on the thickness of shear zones. The results of shearing are also compared to solutions without the polar extensions. The FE-calculations demonstrate that polar e!ects manifested by the appearance of grain rotations and couple stresses are signi"cant in the shear zone, and its thickness is sensitive to the initial void ratio, mean grain diameter and layer height. The e!ect of the pressure level is rather low within the considered range.