Search Results (8)

In this paper, we study the creep phenomena for self-similar models of viscoelastic materials and derive a generalization of the Kelvin–Voigt model in the framework of fractal continuum calculus. Creep compliance for the Kelvin–Voigt model is extended to fractal manifolds through local fractal-continuum differential operators. Generalized fractal creep compliance is obtained, taking into account the intrinsic time $\tau$ and the fractal dimension of time-scale $\beta$. The model obtained is validated with experimental data obtained for resin samples with the fractal structure of a Sierpinski carpet and experimental data on rock salt. Comparisons of the model predictions with the experimental data are presented as the curves of slow continuous deformations. Full article

A novel analytical model based on the generalized ubiquitiformal Sierpinski carpet is proposed which can more accurately obtain the normal contact stiffness of the grinding joint surface. Firstly, the profile and the distribution of asperities on the grinding surface are characterized. Then, based on the generalized ubiquitiformal Sierpinski carpet, the contact characterization of the grinding joint surface is realized. Secondly, a contact mechanics analysis of the asperities on the grinding surface is carried out. The analytical expressions for contact stiffness in various deformation stages are derived, culminating in the establishment of a comprehensive analytical model for the grinding joint surface. Subsequently, a comparative analysis is conducted between the outcomes of the presented model, the KE model, and experimental data. The findings reveal that, under identical contact pressure conditions, the results obtained from the presented model exhibit a closer alignment with experimental observations compared to the KE model. With an increase in contact pressure, the relative error of the presented model shows a trend of first increasing and then decreasing, while the KE model has a trend of increasing. For the relative error values of the four surfaces under different contact pressures, the maximum relative error of the presented model is 5.44%, while the KE model is 22.99%. The presented model can lay a solid theoretical foundation for the optimization design of high-precision machine tools and provide a scientific theoretical basis for the performance analysis of machine tool systems. Full article

Due to the fast advancement of technology and industry, miniaturization has become an important research area. Also, all wired systems are shifting into wireless, and thus, there is a need for antennas to transmit and receive data in and out of gadgets. Fractal geometries provide many benefits when used to manufacture microstrip antennas, including features like size filling, multiband, low profile, and compact size. In this study, four fractal antennas, the Sierpinski carpet, Sierpinski gasket, circular patch, and Koch fractal, were designed. Three iterations of the above four antennas were completed. The size of the antennas was 20 mm × 26 mm × 1.6 mm. FR4 epoxy with a full ground was used here for antenna generation. These antennas can be used for 5 GHz band wireless applications. They provide a good return loss at 5.2 GHz. The maximum return loss was achieved using the Koch fractal at its 3rd iteration of −39.85 dB with a gain of 3.6 dB. In order to reduce cross-polarization, a square slot was added in all antennas’ feed lines, and cross-polarization was reduced by up to 60 dB. For simulation purposes, Ansys-HFSS, using FEM for the analysis of complex EM problems, provided accurate results. Also, 3D and 2D radiation patterns were analyzed, and it was found that they were directional in nature with low radiation toward the back side. Full article

In this paper, the phenomenon of creep compliance and the creep Poisson’s ratio of a 3D-printed Sierpinski carpet-based fractal and its bulk material (flexible resin Resione F69) was experimentally investigated, as well as the quantification of the change in the viscoelastic parameters of the material due to the fractal structure. The samples were manufactured via a vat photopolymerization method. The fractal structure of the samples was based on the Sierpinski carpet at the fourth iteration. In order to evaluate the response of both the fractal and the bulk material under the creep phenomenon, 1 h-duration tensile creep tests at three constant temperatures (20, 30 and 40 °C) and three constant stresses (0.1, 0.2 and 0.3 MPa) were conducted. A digital image correlation (DIC) technique was implemented for strain measurement in axial and transverse directions. From the results obtained, the linear viscoelastic behavior regime of the fractal and the bulk material was identified. The linear viscoelastic parameters of both fractal and bulk materials were then estimated by fitting the creep Burgers model to the experimental data to determine the effect of the fractal geometry on the viscoelastic properties of the samples. Overall, it was found that the reduction in stiffness induced by the fractal porosity caused a more viscous behavior of the material and a reduction in its creep Poisson’s ratio, which means an increase in the compliance of the material. Full article

The mapping relationships between the conductivity properties are not only of great importance for understanding the transport phenomenon in porous material, but also benefit the prediction of transport parameters. Therefore, a fractal pore-scale model with capillary bundle is applied to study the fluid flow and heat conduction as well as gas diffusion through saturated porous material, and calculate the conductivity properties including effective permeability, thermal conductivity and diffusion coefficient. The results clearly show that the correlations between the conductivity properties of saturated porous material are prominent and depend on the way the pore structure changes. By comparing with available experimental results and 2D numerical simulation on Sierpinski carpet models, the proposed mapping relationships among transport properties are validated. The present mapping method provides a new window for understanding the transport processes through porous material, and sheds light on oil and gas resources, energy storage, carbon dioxide sequestration and storage as well as fuel cell etc. Full article

The bending of self-similar beams applying the Euler–Bernoulli principle is studied in this paper. A generalization of the standard Euler–Bernoulli beam equation in the ${\mathcal{F}}_{d_{\mathcal{H}}}^3$ continuum using local fractional differential operators is obtained. The mapping of a bending problem for a self-similar beam into the corresponding problem for a fractal continuum is defined. Displacements, rotations, bending moments and shear forces as functions of fractal parameters of the beam are estimated, allowing the mechanical response for self-similar beams to be established. An example of the structural behavior of a cantilever beam with a load at the free end is considered to study the influence of fractality on the mechanical properties of beams. Full article

Porous structures exhibiting randomly sized and distributed pores are required in biomedical applications (producing implants), materials science (developing cermet-based materials with desired properties), engineering applications (objects having controlled mass and energy transfer properties), and smart agriculture (devices for soilless cultivation). In most cases, a scaffold-based method is used to design porous structures. This approach fails to produce randomly sized and distributed pores, which is a pressing need as far as the aforementioned application areas are concerned. Thus, more effective porous structure design methods are required. This article presents how to utilize fractal geometry to model porous structures and then print them using 3D printing technology. A mathematical procedure was developed to create stochastic point clouds using the affine maps of a predefined Iterative Function Systems (IFS)-based fractal. In addition, a method is developed to modify a given IFS fractal-generated point cloud. The modification process controls the self-similarity levels of the fractal and ultimately results in a model of porous structure exhibiting randomly sized and distributed pores. The model can be transformed into a 3D Computer-Aided Design (CAD) model using voxel-based modeling or other means for digitization and 3D printing. The efficacy of the proposed method is demonstrated by transforming the Sierpinski Carpet (an IFS-based fractal) into 3D-printed porous structures with randomly sized and distributed pores. Other IFS-based fractals than the Sierpinski Carpet can be used to model and fabricate porous structures effectively. This issue remains open for further research. Full article

Navier–Stokes equations describe the laminar flow of incompressible fluids. In most cases, one prefers to solve either these equations numerically, or the physical conditions of solving the problem are considered more straightforward than the real situation. In this paper, the Navier–Stokes equations are solved analytically and numerically for specific physical conditions. Using $F^{\alpha }$-calculus, the fractal form of Navier–Stokes equations, which describes the laminar flow of incompressible fluids, has been solved analytically for two groups of general solutions. In the analytical section, for just “the single-phase fluid” analytical answers are obtained in a two-dimensional situation. However, in the numerical part, we simulate two fluids’ flow (liquid–liquid) in a three-dimensional case through several fractal structures and the sides of several fractal structures. Static mixers can be used to mix two fluids. These static mixers can be fractal in shape. The Sierpinski triangle, the Sierpinski carpet, and the circular fractal pattern have the static mixer’s role in our simulations. We apply these structures just in zero, first and second iterations. Using the COMSOL software, these equations for “fractal mixing” were solved numerically. For this purpose, fractal structures act as a barrier, and one can handle different types of their corresponding simulations. In COMSOL software, after the execution, we verify the defining model. We may present speed, pressure, and concentration distributions before and after passing fluids through or out of the fractal structure. The parameter for analyzing the quality of fractal mixing is the Coefficient of Variation (CoV). Full article