Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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equivalent radius. The force on particle j exerted by particle i is obtained by applying Newton’s third law, f n ji = − f n i j . Equation (6.64) is derived under the following assumptions: (i) the spherical elements are ideally smooth, (ii) the materials are elastic and isotropic, (iii) the shear component of the elastic contact force has no effect on the normal force, and (iv) the overlap is small relative to the size of spherical elements. This law is essential for the simulation of certain phenomena in granular materials, such as the propagation of elastic waves. However, it should be remembered that the Hertzian contact model is adequate only for the elastic contact (i.e., when the forces do not exceed the yield strength anywhere in the contact zone). For more complex cases, contact models based on viscoelasticity and elastoplasticity have been developed ([110] and references cited therein).

Figure 6.19: Schematic representation of (dry) contact of two circular elements according to the classical theory. (a) Definition of meso-parameters; (b) contact forces; (c) basic interaction model - bonding contact without damping; and (d) a more complex contact model involving damping and friction. At small deformations, compact geomaterials (e.g., sand in Figure 6.1c) are charac terized by a linear elastic response. The force with which two spherical elements of such material act on each other can be decomposed into the elastic normal force, f n ela , and the incremental shear force, f t , which are (in the classical interpretation) related to the relative normal and the incremental tangential displacements, respectively, through the coefficients of normal and secant shear stiffness ( k n and k t ) ( f n i j ) ela = − k n u n i j , f t i j = { f t i j } updated − k t ∆ u t i j . The elastic response is completely defined by this pair of constants ( k n , k t ). The incremental shear force should reset to zero, { f t i j } updated = 0 , whenever the elements can slide relative to each other, which happens when the Mohr-Coulomb type limit is reached | f t i j | = f t coh + µ f f n i j . (6.65) The limit value (6.65) is defined by the local values of the contact friction coefficient, µ f , and the cohesion, f t coh (which is by definition equal to zero for non-cohesive materials).