Surrogate model for statics of fractional thin bar element and its equivalence with mass-spring metamaterial

The paper presents a two-level metametrial homogenization procedure, in which the mechanical behavior of a bar with non-uniform stiffness profile is considered. The connection between the microstructure (spring-mass model) and the surrogate model (simple spring model) is done by an intermediate space-fractional bar element (FBE) modeled utilizing fractional calculus (FC). The impact of the FC parameters, namely a length scale and a material order, on the global stiffness of the bar was analyzed based on 6000 numerical analyses, including various stiffness profiles - distributions of Young's modulus and the cross-sectional area along the element axis. The study reveals that changes in FC parameters significantly influence the stiffness of the resulting bar (up to 20%), and the impact of the change on the stiffness is very difficult to predict, highlighting the need for a detailed analysis. The presented approach with the model that mimics the microstructure efficiently opens a way to analyze large lattice or chain structures expressing scale effect and then to look for their optimal design.