V-model

Introduction

The article is based on the oncoming paper V.A. Kuzkin and I.E. Asonov "Vector-based model of elastic bonds for DEM simulation of solids".

Discrete Element Method (DEM) is widely used for computer simulation of granular materials both solid and free-flowing. In powders and other free-flowing media interactions between particles usually include contact forces, dry and viscous friction, cohesion, electrostatic forces etc. For simulation of solids particles are additionally connected by so-called bonds~\cite{BPM, Wang}. In general case bond transmits both forces and moments acting between particles. They are responsible for stability, elasticity, strength and other intrinsic properties that distinguish solids from free-flowing materials. The bonds may have different physical meaning. On the one hand they can specify the law of interaction between different parts of one material represented by the particles. On the other hand bonds can be considered as a model of some additional glue-like or cement-like material, connecting particles.

In practice models allowing to represent forces and moments, acting between particles, as a function of characteristics of particle motion (linear and angular velocities, rotational matrixes, quaternions, etc.) are required. According to the review presented in paper~\cite{Wang} only several models presented in literature describe all possible kinds of deformation of the bond. Bonded-particle model (BPM) was proposed in paper~\cite{BPM} for simulation of rocks. The BPM model is widely used in literature for simulation of deformation and fracture of solids in both two and three dimensions. Several drawbacks of BPM model, in particular, in the case of coexistence of bending and torsion of the bond, are discussed in paper~\cite{Wang}. It is noted that the main reason for the drawbacks is incremental algorithm used in the framework of BPM model. Another approach based on decomposition of relative rotation of particles is proposed in paper~\cite{Wang}. Forces and moments are represented as functions of angles describing relative turn of the particles. It was shown that method~\cite{Wang} is more accurate than incremental procedure of BPM model. However in the framework of model~\cite{Wang} potential energy of the bond and its relation to forces and torques are not considered. Though the expression for potential energy is not required for DEM simulations it is still very important. It is required for control of energy conservation and construction of nonlinear elastic force laws. The approach proposed in paper~\cite{Wang} does not guarantee that the forces and moments caused by the bond are conservative. Note that any model for an elastic bond should be perfectly conservative.

Also let us note recent

There are also some other physical drawbacks of the existing bond models that will be highlighted in the paper. Let us summarize them:

* energy conservation is not guaranteed
* bond connects centers of the particles (very far from the reality where particles are glued by their surfaces)  
* calibration model parameters is not always clear. For example, in the framework of BPM model 
* 

V-model

The problem of energy conservation can be solved, in particular, if forces and moments are derived from the potential energy. This approach is used in classical molecular dynamics for both material point and rigid body particles. The approach for construction of potential energy of interactions between rigid bodies was proposed in paper~\cite{Price}. Initially it was applied to simulation of liquids, in particular, liquid crystals~\cite{Allen}. In papers~\cite{Zhilin_FL, IvKrMoFi_MTT_2003, IvKrMo_PMM_2007} similar ideas were applied to solids. In particular, analytical description of elastic properties of graphene was carried out in paper~\cite{IvKrMo_PMM_2007}. However the problem of construction of interatomic potential describing nonlinear deformation and fracture of solids was not considered. The potential allowing to model nonlinear interactions between rigid bodies in two dimensional case was proposed in papers~\cite{BeIvKrMo_MTT_2007, Ivanova_Byzov} and generalized for three dimensional case in paper~\cite{Tovstik}. However the potential proposed in paper~\cite{Tovstik} is not applicable in the case of large relative rotations of particles occurring, for example, in the case of fracture. The potential for simulation of deformation and fracture of graphene was proposed in paper~\cite{Kuzkin_DAN}.

History and acknowledgements

The idea underlining V-model was first formulated on the paper by Vitaly Kuzkin during communication with Michael Wolff in Technical University of Hamburg (March, 2011). The first formulation was very simple and coarse, but it work! The results of some test simulations were presented by Vitaly Kuzkin on APM 2011 conference (July, 2011). At the present moment V-model is much more flexible and physically meaningful than its first version. Now it is developed jointly by Vitaly Kuzkin and Igor Asonov