Vector-based model of elastic bonds for simulation of granular solids

A model (further referred to as the V model) for the simulation of granular solids, such as rocks, ceramics, concrete, nanocomposites, and agglomerates, composed of bonded particles (rigid bodies), is proposed. It is assumed that the bonds, usually representing some additional gluelike material connecting particles, cause both forces and torques acting on the particles. Vectors rigidly connected with the particles are used to describe the deformation of a single bond. The expression for potential energy of the bond and corresponding expressions for forces and torques are derived. Formulas connecting parameters of the model with longitudinal, shear, bending, and torsional stiffnesses of the bond are obtained. It is shown that the model makes it possible to describe any values of the bond stiffnesses exactly; that is, the model is applicable for the bonds with arbitrary length/thickness ratio. Two different calibration procedures depending on bond length/thickness ratio are proposed. It is shown that parameters of the model can be chosen so that under small deformations the bond is equivalent to either a Bernoulli-Euler beam or a Timoshenko beam or short cylinder connecting particles. Simple analytical expressions, relating parameters of the V model with geometrical and mechanical characteristics of the bond, are derived. Two simple examples of computer simulation of thin granular structures using the V model are given.

Figure 1

Scanning electron microscope images of composite consisting of PbS nanoparticles bonded by a copolymer. From Ref. [10].

Figure 2

Two bonded particles in the undeformed state (left) and deformed state (right). Here and below $a$ is an equilibrium distance.

Figure 3

Different kinds of deformation of the bond and corresponding change in vectors, connected with the particles. Dashed lines show the initial states of the particles.

Figure 4

Particles connected by a short cylinder.

Figure 5

Dynamical buckling of the discrete beam. Numbers in the figure are corresponding moments of time. Particles radii equal to 0.5$a$ are used for visualization.

Figure 6

The initial (left) and final (right) distributions of the particles on the half-sphere. Bottom view. Particles of radii $0.125a$ are used for the visualization.

Figure 7

Buckling of the discrete half-spherical shell under point force load. Particle radii equal 0.5$a$ are used for visualization.