: This paper considers discrete mass-spring structure identification in a nonlocal continuum space-fractional model, defined as an optimization task. Dynamic (eigenvalues and eigenvectors) and static (displacement field) solutions to discrete and continuum theories are major constituents of the objective function. It is assumed that the masses in both descriptions are equal (and constant), whereas the spring stiffness distribution in a discrete system becomes a crucial unknown. The considerations include a variety of configurations of the nonlocal parameter and the order of the fractional model, which makes the study comprehensive, and for the first time provides insight into the possible properties (geometric and mechanical) of a discrete structure homogenized by a space-fractional formulation.

In this paper, the Riesz-Caputo fractional derivative of variable order with fixed memory is considered. The studied non-integer differential operator is approximated by means of modified basic rules of numerical integration. The three proposed methods are based on polynomial interpolation: piecewise constant, piecewise linear, and piecewise quadratic interpolation. The errors generated by the described methods and the experimental rate of convergence are reported. Finally, an application of the Riesz-Caputo fractional derivative of space-dependent order in continuum mechanics is depicted.

In this paper, the application of non-local fractional continuum model for plane strain and plane stress elasticity is presented. The kinematics and stress concepts are discussed, and governing equations in terms of displacements for both plane problems are defined. The numerical implementation utilising generalised finite difference method is shown in detail. Three cases are solved to indicate the role of order of fractional continua and length scale: biaxial tension, pure shear and complex state. Classical (local) solution is obtained as a special case.

In this paper the numerical implementation of two-scale modelling of bone microstructure is presented. The study is a part of long-term project on bone remodelling which drives bone microstructure change based directly on trabeculae surface energy. The proposed approach is based on a first-order computational homogenization technique. The coincidence of macro- and micro-model kinematics is done with the use of uniform displacement and traction boundary conditions. The computational homogenization procedure is driven by a self-prepared manager which is coded in Python. The computation on real bone structure (a piece of female Wistar rat bone) is performed as well.

Screw loosening is a critical aspect of an implant design, as it can lead to implant failure. This study proposes a methodology and qualitatively assesses the potential of screw loosening risk for various types of screw heads and implant fixture-abutment connections. It is assumed that the risk of screw loosening is related to the drop in loosening moment under occlusal loads. The methodology and an assumption is verified by confronting the results with laboratory tests. Numerical simulations supplemented with semi-empirical equations were employed to estimate a loosening moment change under occlusal loads. The loosening risk was estimated by comparing the value before and after application of a compressive occlusal load of 150N. The procedure was carried out for 289 implant designs with smooth transition between flat to tapered shape of a screw head and an fixture-abutment connection. All analyses were conducted using Abaqus software. Pearson and Spearman correlation coefficients for normalised change in a screw loosening moment drop has been computed for numerical and laboratory tests. The statistical analysis (Pearson, ρ = 0.8, Spearman, r_s = 0.85) indicates very high correlation and confirms that the general tendencies observed in laboratory tests are reflected in the proposed procedure. The procedure was used for various geometries and the following results are presented: a screw loosening drop, implant stiffness and a tightening moment. The loosening moment drop achieves the extreme values of 6% and 24%. The biggest drop is an effect of a conical implant-abutment connection and a flat screw head while the lowest was recorded for a flat implant-abutment interface regardless of a screw head type. A low drop is also observed for a strongly conical screw head. The proposed methodology exhibited very good correlation when confronted with laboratory tests, supporting a screw preload reduction under occlusal load as a key factor in screw loosening. Analysis across a wide spectrum of implant designs revealed geometry significantly impacts loosening potential under occlusal loads. Two key features were identified as favourable - an abutment-fixture butt joint and a tapered screw. The results also enable prediction of qualitative geometry effects on loosening risk.