Modeling of static deformations and dynamics of localized rotation in a chain of finite size particles

In beam-like structures and granular media the rotations are usually quite small. Linear generalized continuum models were developed and successfully applied for the modelling of such deformations. But, they do not describe full continuous rotations. The development of generalized continuum models describing not only small oscillations, but full continuous rotations as well is quite challenging and an open problem. This manuscript extends the results by Vasiliev A.A, Dmitriev S.V. Discrete and Multifield Models of a Cosserat Chain: One-Period and Two-Period Solutions. Russian Physics Journal. 59, 961 (2016). We derive and analyse three types of discrete models: i) nonlinear springs model, ii) linearized model with respect to displacement of particles and full sine-dependence of the angle of rotations; and iii) linearized one with respect to both, displacements and rotations. The latter is the discrete analogue of the classical Cosserat model. As a test example, we consider a chain of finite size particles with fixed boundaries in viscous medium with an applied momentum load to the central particle. By employing continuous Cosserat model, we find an approximated stationary solution for the full and reduced discrete Cosserat models. Numerically calculated static deformations of the chain under small momentum loads by using all three models are very similar. Under strong load localized rotation is observed. It is not described by the linear model (iii). Our results indicate the possibility of modelling of localized rotations in the nonlinear model (i) and the model containing nonlinear sine-dependence of the angle of rotation (ii).