A novel di-convolution nonlocal elasticity with applications to beam bending

Abstract

Traditional nonlocal elasticity models often rely on empirical mono-convolution kernels, which may limit their ability to capture complex size-dependent phenomena or result in mathematical inconsistencies. To address these limitations, this study introduces a Di-Convolution Elasticity framework that generalizes constitutive behavior using two independent convolution kernels acting on stress and strain fields. Unlike purely empirical approaches, this framework is rigorously derived via functional equation theory, resulting in versatile additive and multiplicative kernel structures that accommodate both identical and non-identical kernel pairs. An integro-differential governing equation for beam statics is developed and solved analytically through Laplace transformations. Numerical applications to simply supported and cantilever beams demonstrate the model’s ability to predict size-dependent deflections, maintain nonparadoxical behavior, and seamlessly recover classical elasticity in the local limit. Comparative analysis against existing mono-convolution models highlights the superior accuracy and mathematical robustness of the Di-Convolution approach, providing a more comprehensive and physically insightful tool for the analysis of small-scale structures.