Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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the tangential stiffness coefficient (6.68) 2 defines the upper limit for the value of Poisson’s ratio ν ( 2 D ) = 1 / 3, which implies ν max = 1 / 3 for the plane stress and ν max = 1 / 4 the plane strain. Similar expressions for determining the meso-parameters of the DEM model are available in the literature for different spatial arrangements of particles. For example, Masuya et al. [127] and Potyondy and Cundall [128] derived k n = Et , which corresponds to a simple cubic lattice; while Wang and Mora [129] obtained k n = E ( 3 D ) √ 2 ( 1 − 2 ν ( 3 D ) ) , k t = 1 − 3 ν ( 3 D ) 1 + ν ( 3 D ) k n (6.69) for a face-centered cubic lattice. Equations (6.66) are usually solved by different finite-difference schemes. The Verlet and Störmer algorithms, summarized in Chapter 6.2.3, are often used for this purpose, and the force-displacement law (6.67) and Newton’s second law of motion (6.1) are used in each computational cycle. In order to obtain the correct motion of each discrete element by integration, the time step must be chosen carefully as already discussed in Chapter 6.2. By analogy between the contact (with stiffness k n ) and the oscillating material point m , it can be shown that the time step must be chosen as a small part (usually the tenth or twentieth) of the half-period of oscillation [10]. Numerical stability can, for example, be improved by using local, contact dissipative damping as ( η n , η t ) in Equations (6.67). Finally, the macroscopic constitutive laws of continuum mechanics connect the stress tensor and the specific deformation tensor, while the meso-scale contact constitutive relations connect the contact force with the relative displacement at the points of contact of the particles. Kruyt and Rothenburg [130] derived micromechanical expressions for stress tensors and relative deformations as a function of microscopic contact parameters. The same authors also developed statistical theories of elastic modules for plane groups of particles [131]. 6.5.4 DEM Modeling of Solid Materials The defining property of solid (cohesive) materials is the ability to transfer tensile force between connected particles. Thus, the Cundall’s original concept [99], [132], developed for blocky rock systems, has been extended to take into account the interface tensile strength [81], [133], [134]. This DEM adaptation for solid materials was achieved usually by adding a bond at the point of contact of two discrete elements. This bond mimics the presence of a cement matrix attached to the contacting particles, which is able to impart cohesion [135]. This approach has been used to model a wide variety of classes of hetero geneous cohesive materials such as sedimentary rocks, concrete, ceramics, grouted soils, solid rocket propellants, explosives, biomaterials. All these materials, in principle, can be represented by the simple model shown in Chapter 6.5.3 (Figure 6.19) with an important proviso that in the case of cohesive (adhesive) contact, the spring in the normal direction provides resistance to both compressive and tensile loads. If, during the deformation process, the bonded contact between two discrete elements is broken (according to some prescribed bond-rupture criterion), the contact becomes purely compressive and frictional (as illustrated in Figure 6.19), if it survives at all (two grains could be separated instead of pressing against each other). Regarding the simple central and angular interaction (pre sented in Figure 6.18c and generalized in Chapter 6.4.3) as a combination of a viscoelastic