Modelling architected beam using a nonlocal derivative-free shear deformable beam theory

Abstract

It has been well established that the internal length scale related to the cell size plays a critical role in the response of architected structures. It this paper, a Volterra derivative-based approach for deriving nonlocal continuum laws directly from an energy expression without involving spatial derivatives of the displacement is proposed. A major aspect of the work is the introduction of a nonlocal derivative-free directionality term, which recovers the classical deformation gradient in the infinitesimal limit. The proposed directionality term avoids issues with correspondences under nonsymmetric conditions (such a unequal distribution of points that cause trouble with conventional correspondence-based approaches in peridynamics). Using this approach, we derive a nonlocal version of a shear deformable beam model in the form of integro-differential equations. As an application, buckling analysis of architected beams with different core shapes is performed. In this context, we also provide a physical basis for the consideration of energy for nonaffine (local bending) deformation. This removes the need for additional energy in an ad hoc manner towards suppressing zero-energy modes. The numerical results demonstrate that the proposed framework can accurately estimate the critical buckling load for a beam in comparison to 3-D simulations at a small fraction of the cost and computational time. Efficacy of the framework is demonstrated by analysing the responses of a deformable beam under different loads and boundary conditions. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.