Tridynamic model of the beam with transverse shear deformation

Abstract

Tridynamics is a nonlocal approach to describe lower-scale multi-body interactions at the macroscopic level. Although such a description is sometimes crucial to model the response of a body (for instance carbon nanotube), the computational cost might be very high in the analysis of large structures. Therefore the introduction of basic elements such as beam and plate in this framework could significantly reduce the computational cost. Through this study, we present a set of meaningful and physical quantities that explain the deformation of the beam in this framework. However, the introduced quantities have the potential in describing the variation of field variable in other mechanical problems such as thermal problems. Starting from the kinematic of the beam, a tridynamics beam model (with no restriction on the height of the beam) is developed by using the Lagrangian of the system. The nonlocal parameters are calibrated by carrying out the material correspondences with the classical equations. To see the credibility of the model, a dispersion analysis from low to high wave numbers is performed, and the results are compared with the local and nonlocal models available in the literature.