Towards Optimal Designing of Complex Non-local Bar-like Structures
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The pronounced trend toward miniaturization has led to a substantial increase in interest in nano- andmicro-scale structures in recent years. Owing to the characteristic physical dimensions of nanodevices, local effects often must be explicitly taken into account. This, in turn, poses significant challenges for the development of reliable mathematical models. Many phenomenological nonlocal models have been proposed to account for scale effects, including integral-type theories, general nonlocal theories, strain-gradient theories, and micropolar theories, among others. In the present study, however, we employ the Fractional Continuum Mechanics (FCM) model [1]. The primary objective of the present study is to quantify the influence of the icrostructure of individual structural elements on the global response of the entire structure. Since the microstructure governs the nonlocal behavior of the material, the parameters of the fractional model—namely, the fractional order of the material α and the characteristic length scale ℓf—are systematically varied to emulate microstructural modifications. The application of fractional calculus in mechanics is inherently complex and computationally demanding [2]. Consequently, an additional key objective of this work is to develop a computationally efficient procedure for the analysis of complete structures. To this end, a multilevel framework is proposed, employing fractional bar reduction to a surrogate model [3]. The procedure was evaluated using full-scale models of truss-like structures that incorporate local effects. Several distinct models were analyzed, with particular emphasis on variations in the displacement field arising from different configurations of nonlocal effects. The results clearly demonstrate the influence of the microstructure on the global structural response, including the occurrence of asymmetric deformation in geometrically symmetric and symmetrically loaded/supported models. REFERENCES [1] Sumelka, W., ”Thermoelasticity in the Framework of the Fractional Continuum Mechanics, Journal of Thermal Stresses, Vol., 37/61, pp. 678–706, 2014. [2] Szajek K., Sumelka W., Structural and Multidisciplinary Optimization , Structural and MultidisciplinaryOptimization, Vol., 59/1, pp. 185-200, [3] Szajek K., Sumelka W., Surrogate model for statics of fractional thin bar element and its equivalence with mass-spring metamaterial, Scientific Reports, Vol., 15/1, pp.39114, 2025.
