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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. 

The internal length scale relating to the cell size plays a crucial role in predicting the response of architected structures when subjected to external stimuli. A Volterra derivative-based approach for arriving at the non-local derivative-free continuum laws for architected structures is proposed. A mainstay of the work is the derivative-free directionality term, which recovers its classical counterpart in the infinitesimal limit. Using this approach, we derive the non-local integro-differential governing equations of a shear deformable plate. We also suggest a physical basis for the consideration of energy for nonaffine deformations and accurately estimate it by performing buckling analysis. This discards the requirement of the additional energy to be incorporated in an arbitrary manner for suppressing the unwanted spurious oscillations induced from zero energy modes. The numerical results demonstrate the efficacy of the proposed framework in precisely capturing the mechanical response of web-core shear deformable plate, thereby, manifesting the supremacy of the reduced model in shrinking the cost and computational time. To bolster our claim, various numerical models with different loading conditions have been analysed and compared against the three-dimensional FEM results.