Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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the same reason - to keep the formalism as simple as possible, the basic equations are presented only for circular geometry in this introduction.) However, this computational convenience comes at a price. First of all, the circular/spherical shape of discrete elements significantly underestimates the rolling resistance among them. Second, they cannot reproduce more complex configurational rearrangements of particles (such as the particle interlock), which typically occur when discrete elements are of more complex shape. These two computational aspects result in an inherent underestimation of macro-strength. An additional artifact introduced by the circular/spherical particle geometry is the numerical porosity, which should not be confused with actual porosity. Moreover, the discrete elements are often modeled as rigid, but a certain overlap between them (as indicated in Figure 6.19a) is allowed to model the occurrence of relative displacement and localized contact deformation (soft or smooth contact ). The contact dynamics methods based on “non-smooth” formulations, which exclude the possibility of the particle overlap, are not addressed in this introduction; the interested reader is referred to the review paper of Donze and co-authors [10] and the references cited therein. The rigidity assumption is reasonable when the movements along the interfacial surface represent the largest part of the deformation in the assembly of discrete elements, which is typical for loose materials (such as dry sand Figure 6.1c) and industrial transport processes and flow phenomena. Whatever model is adopted or developed for a certain problem, it will naturally be based on a greater or lesser simplification of the actual physical processes on the meso-scale which is not only inevitable but also desirable, given that many details of the meso-scale contact do not have to be significant for the macro-scale response of the system as a whole. Research challenges include not only realistic quantitative simulations of large particulate systems, with the ability to predict responses and their experimental validation, but also the transition from the meso-scale contact properties to the macroscopic properties of materials. This meso-macro transition should make it possible to understand the collective behavior of the large conglomerate of discrete elements as a function of their contact properties [110]. A typical example of a non-cohesive system are dry granular materials (e.g., dry sands) mentioned above. They are characterized by the dominance of non-cohesive interparticle actions of short range: elastic or inelastic contact forces and contact friction between touching grains. The simplest rheological model of such contact interactions is presented in Figure 6.19c with one spring in the normal and tangential directions at the point of contact. According to the more complex approach of Xiang et al. [121], any contact between particles can be rheologically represented by a Kelvin spring-damper element in the normal direction and a spring-damper-slider element in the tangent direction (Figure 6.19d). The contact of the particles in the normal direction is ideally elastic in the case of mutual pressure while the tensile strength of non-cohesive materials is by definition equal to zero. Thus, the generalized contact behavior of particles of a simple central-angular type of interaction (Figures 6.16 and 6.18d represent two isomorphic interaction models) takes into account: normal interactions, shear interactions, and slip. In the general case, the forces acting on the particle j include: the gravitational force ( m j g ) , and normal ( f n ji ) and shear ( f t ji ) components of the elastic contact force between elements i and j (Figure 6.19b). Accordingly, taking into account the basic law of dynamics (6.1) and 2D geometry,