Fractal Surface Research Articles

Temperature Rise in Frictional Sliding Contact of Elastic–Plastic Solids with Fractal Surface

Temperature Rise in Frictional Sliding Contact of Elastic–Plastic Solids with Fractal Surface

New insight into pore characteristics for cake layers formed on nanocomposite membranes: Effect of membrane surface fractality

New insight into pore characteristics for cake layers formed on nanocomposite membranes: Effect of membrane surface fractality

Fractal surface states in three-dimensional topological quasicrystals

We study topological states of matter in quasicrystals, which do not rely on crystalline order. In the absence of a band-structure description and spin-orbit coupling, we show that a three-dimensional quasicrystal can nevertheless form a topological insulator. It relies on a combination of noncrystallographic rotational symmetry of quasicrystals and electronic orbital space symmetry, which is the quasicrystalline counterpart of the topological crystalline insulator. The resulting topological state obeys a nontrivial twisted bulk-boundary correspondence and lacks a good metallic surface. The topological surface states, localized on the top and bottom planes respecting the quasicrystalline symmetry, exhibit a different kind of multifractality with probability density concentrated mostly on high-symmetry patches. They form a near-degenerate manifold of immobile states whose number scales proportionally to the macroscopic sample size. This can present an opportunity for a platform for topological surface physics distinct from the crystalline counterpart. Published by the American Physical Society 2024

On bivariate fractal interpolation for countable data and associated nonlinear fractal operator

Abstract Fractal interpolation has been conventionally treated as a method to construct a univariate continuous function interpolating a given finite data set with the distinguishing property that the graph of the interpolating function is the attractor of a suitable iterated function system. On the one hand, attempts have been made to extend the univariate fractal interpolation from a finite data set to a countably infinite set. On the other hand, fractal interpolation in higher dimensions, particularly the theory of fractal interpolation surfaces (FISs), has received increasing attention for more than a quarter century. This article targets a two-fold extension of the notion of fractal interpolation by providing a general framework to construct FISs for a prescribed set consisting of countably infinite data on a rectangular grid. By using this as a crucial tool, we obtain a parameterized family of bivariate fractal functions simultaneously interpolating and approximating a prescribed bivariate continuous function. Some elementary properties of the associated nonlinear (not necessarily linear) fractal operators are established, thereby allowing the interaction of the notion of fractal interpolation with the theory of nonlinear operators.

Strongly Different Adhesion Reduction for 1D or 2D Random Fractal Roughness, and an Extension of the BAM Model to Anisotropic Surfaces

The influence of roughness on adhesion has been studied since the time of Fuller and Tabor, but recently there has been debate about how roughness exactly seems to kill (but sometimes enhance!) adhesion, particularly with reference to the accepted model of fractal roughness. We show that the Persson–Tosatti criterion does not depend on anisotropy of the surface for a typical power law PSD if written in terms of rms roughness and magnification. Instead, a very simple extension of the Bearing Area Model (BAM) of Ciavarella to anisotropic fractal surface shows a weak but clear dependence on the anisotropy, with higher adhesion in the 1D case, showing better agreement than the Persson–Tosatti criterion to actual numerical results of Afferrante Violano and Dini. However, neither of the two models permit to capture the strong hysteresis found in experiments between loading and unloading, which is very likely to enhance adhesion more as we move from the isotropic to the full 1D case. This suggests the mechanism of load amplification along contact lines and the associated elastic instabilities, is not captured by either the Persson–Tosatti or the BAM model applied to anisotropic surfaces.

Heart Trabeculae‐Inspired Superhydrophilic Electrode for Electric‐Assisted Uranium Extraction from Seawater

AbstractUsing nuclear power to replace electricity generated from fossil fuels is an effective strategy to reduce global carbon dioxide emissions and also spurs the search for new sources of nuclear fuel. Extracting uranium from seawater has a significant reserve advantage, although its ultralow concentration presents substantial challenges. Here, inspired by the fractal structure of cardiac trabeculae on the inner surface of the heart, a uranium enrichment electrode with a superhydrophilic and uranium‐affinitive fractal surface is developed. This innovative design enhances rapid charge/ion transfer, ensures complete surface wetting, and provides numerous adsorption sites. By synergistically integrating the advantages of electric‐assisted processes and bioinspired microstructures predicated on chemical coordination principles, the electrode demonstrates a uranium adsorption capacity of 13.2 mg g−1 following a 7‐d exposure to natural seawater. This research not only demonstrates an effective strategy for the development of advanced uranium enrichment electrodes but also provides more possibilities for innovative approaches in sustainable energy technology.

Shape preserving aspects of a novel class of bi-cubic partially blended rational zipper fractal interpolation surfaces

Shape preserving aspects of a novel class of bi-cubic partially blended rational zipper fractal interpolation surfaces

Environmental analysis and comparison of the conventional and fractal glass textured surface photovoltaic panels

There are two major forms of solar energy that are typically utilized: photovoltaic and concentrated applications. The application of fractal glass texture to photovoltaic solar panels is a cutting-edge technique in the field of solar panels that generate electricity from exposure to light. When it comes to studying the environmental implications of a product during its development and commercialization, the life cycle assessment (LCA) approach is an excellent technique that can be utilized. The purpose of this study is to offer a thorough understanding of the product’s effects on the environment by taking into consideration a wide range of criteria, including environmental, economic, and other evaluations. Through the utilization of the life cycle assessment (LCA) methodology and the SimaPro software, this paper presents a comparative analysis of conventional solar panels and fractal glass texture panels. During the course of this research, 18 midpoint indicators and three endpoint indices were investigated. In addition, a sensitivity analysis has been carried out on the fractal property of the panel in order to evaluate the impact that it has on environmental impacts and damages. For the purpose of this study, three distinct levels of fractal coating were applied to the panel surface: one percent, three percent, and five percent. As a result of the data, it was determined that the “Photovoltaic cell single-Si wafer” and the “Transport, freight, sea, transoceanic tanker” had the most significant impact on the midpoint and endpoint indices for both panels, respectively. Furthermore, there is a direct association between the rise in fractal coating on panels and the reduction in environmental repercussions, approximately. This correlation exists because of several factors.

A Comparison of Evaluation Methodologies of the Fractal Dimension of Premixed Turbulent Flames in 2D and 3D Using Direct Numerical Simulation Data

AbstractA Direct Numerical Simulation (DNS) database of statistically planar flames ranging from the wrinkled flamelets to the thin reaction zones regime and DNS data for a Bunsen premixed flame representing the wrinkled flamelets regime have been utilised to evaluate the fractal dimensions of flame surfaces using the filtering dimension method, the box-counting algorithm and the correlation dimension approach. The fractal dimension evaluated based on the fully resolved three-dimensional data has been found to be reasonably approximated by adding unity to the equivalent fractal dimension evaluated based on two-dimensional projections irrespective of the methodology of extracting fractal dimension. This indicates that the flame surface can be approximated as a self-similar fractal surface for the range of Karlovitz and Damköhler numbers considered here. While all methods, provide results identical to each other for benchmark problems, it has been found that the fractal dimension evaluation based on box-counting method provides almost identical results as that obtained using the filtering dimension method for both three and two dimensions, while the fractal dimensions based on the correlation dimension tend to be slightly smaller. The findings of the current analysis have the potential to be used to reliably estimate the actual fractal dimension in 3D based on experimentally obtained 2D binarised reaction progress variable field. The inner cut-off scales estimated based on all three methodologies yield comparable results in terms of order of magnitude with the box-counting method predicting a smaller value of inner cut-off scale in comparison to other methods. The execution times for fractal dimension extraction based on filtering dimension and box-counting methodologies are found to be comparable but the correlation dimension method is found to be considerably faster than the two alternative approaches and provides results consistent with theoretical bounds in all cases.

Stability of Fixed Points of Partial Contractivities and Fractal Surfaces

In this paper, a large class of contractions is studied that contains Banach and Matkowski maps as particular cases. Sufficient conditions for the existence of fixed points are proposed in the framework of b-metric spaces. The convergence and stability of the Picard iterations are analyzed, giving error estimates for the fixed-point approximation. Afterwards, the iteration proposed by Kirk in 1971 is considered, studying its convergence, stability, and error estimates in the context of a quasi-normed space. The properties proved can be applied to other types of contractions, since the self-maps defined contain many others as particular cases. For instance, if the underlying set is a metric space, the contractions of type Kannan, Chatterjea, Zamfirescu, Ćirić, and Reich are included in the class of contractivities studied in this paper. These findings are applied to the construction of fractal surfaces on Banach algebras, and the definition of two-variable frames composed of fractal mappings with values in abstract Hilbert spaces.

On the fractal dimension of a fractal surface with one single unbounded variation point

On the fractal dimension of a fractal surface with one single unbounded variation point

In-situ measurements of contact evolution for fractal rough surfaces under normal compression

The contact interface plays a key role in the overall functionality and stability of structures. Understanding the evolution of the contact interface over time and its dependency on materials and load is crucial for functional integrity and operational safety assessment. In this study, we employ in-situ three-dimensional X-ray computed tomography (3DXRCT) to examine the creep behavior of 3D-printed surfaces exhibiting various roughness under constant normal compression. We observe that the overall contact area enlargement during the contact creep decreases with roughness amplitude and fractal dimension. The variation of interfacial separation distance is found to increase with roughness amplitude and decrease with fractal dimension. Correlation analysis reveals that the microcontact size played a more important role than the asperity shape in determining the microcontact enlargement. By examining the calculated interfacial strains extracted from XRCT measurements, significant deformations are found to occur at the non-contacting zones, indicating strong asperity interactions. This study offers high-resolution experimental measurements and unravels the asperity micromechanics for contact creep on rough surfaces, providing insights into understanding and optimizing the performance of rough interfaces.

Modeling and identification of clamping contact stiffness

The dynamics of thin-walled parts are highly affected by the clamping conditions. Clamping stiffness is a function of c lamping force and surface roughness profiles of the clamp and part. Since the surface profiles cannot be altered, estimating clamping stiffness as a function of the clamping force is essential to simulate vibrations of the machined thin-walled parts. This paper presents the modeling of clamping stiffness as a function of the applied clamping force and material properties. Surface profile parameters are estimated from the identified contact stiffnesses evaluated using the Finite Element (FE) model. The contact stiffnesses are either predicted directly from the proposed mechanics model of the part or estimated from the Fractal surface parameters. It is shown that an average clamping stiffness can be predicted from the Fractal surface parameters, or directly and more accurately from the static model of the clamped part.

An approximation theoretic revamping of fractal interpolation surfaces on triangular domains

An approximation theoretic revamping of fractal interpolation surfaces on triangular domains

Development of superhydrophobic fractal surfaces using Silica nanoparticles based on polylactic acid through molding technique

This research introduces a novel approach to creating superhydrophobic surfaces on polymer substrates. The objective is to develop a general method for producing various types of superhydrophobic polymer surfaces using modified nanoparticles, as well as making Unplasticized PolyVinyl Chloride(UPVC) and polylactic acid hydrophobic. To achieve this goal, we investigate the circulation of a rotating cover, pressure, and pressure time. We embed modified silica nanoparticles onto UPVC boards, which confirms surface hydrophobicity with a contact angle of 155° and a sliding angle of 6–7° for water droplets. Under specific conditions, including a suspension concentration of 10 g/L, coating time of 5 seconds, temperature of 155°C, rotation speed of 400 rpm, and pressure of 2 MPa with a pressure time of 4 minutes, we achieve superhydrophobicity on PVC boards, resulting in contact angles of 150–160°. Additionally, we find that the sliding angle of water droplets is less than 10°.

A fractal model of rough surfaces based on ellipsoidal asperities

PurposeThis study aims to use fractal theory to investigate the contact mechanics of two bidirectional anisotropic surfaces, taking into account the friction coefficient of the contact interface. This study introduces a model that centers on normal contact load and contact stiffness, providing an extensive framework to elucidate the interactions between these surfaces.Design/methodology/approachThe research adopts the Weierstrass–Mandelbrot (W-M) function for simulating bidirectional surface profiles. Through the application of elastic-plastic contact theory, it evaluates the contact area and load of a singular asperity across elasticity, elastoplasticity and plasticity phases. The contact load and stiffness of the rough surface are determined using a refined asperity density distribution function, and these findings are juxtaposed with extant models to validate their precision and rationality.FindingsThe study delineates the influence of fractal dimension (D), surface roughness (G), ellipse eccentricity (e) and friction coefficient (µ) on the contact area, load and stiffness. It reveals that the contact area enlarges with the fractal dimension (D) yet diminishes with increased eccentricity (e), roughness (G) and friction coefficient (µ). These elements considerably affect the contact load and stiffness, underscoring their significance in comprehending surface interactions.Originality/valueThis study applies fractal theory to analyze the contact mechanics of bidirectional anisotropic surfaces, considering the geometry and mechanics of ellipsoidal asperities on rough surfaces to develop a contact mechanics model. This model clarifies the deformation of an asperity in normal contact, presenting a more rational alternative to current models.

Composite scattering computation from multiple dielectric targets of one-dimensional fractal sea surface using MOM

Bistatic scattering coefficient have important application of ocean active microwave remote sensing. In this paper, a general expression of the composite scattering coefficient for multiple dielectric targets of one-dimensional fractal sea surface is derived by MOM method. Secondly, the bistatic scattering coefficient is computed, there are two class missile targets and one class ship target of one-dimensional fractal sea surface, and the influence of the water dielectric constant, fractal dimension, wind speed, the target dielectric constant and spatial position is discussed, and the corresponding conclusions are given. The proposed algorithm is able to solve some composite scattering problems, such as ‘dielectric or conductive object + rough surface’, ‘dielectric or conductive floating object +rough’ and it has certain generalization and reference, and it lays a foundation for solving the composite scattering coefficient of multiple medium targets in two-dimensional fractal sea surface.

Normal Contact Model of Fractal Surfaces Considering Friction and Asperity Interactions

Friction as well as asperity interactions have significant influence on the contact area and force of rough surfaces. In this work, a single-asperity, elastoplastic contact model, considering friction, is established by means of continuum mechanics and power-exponential functions. Then, the deformation of the asperity interactions is modeled by the displacement of the average line for asperity height. Using fractal theory, the actual deformation and contact stiffness of an asperity are derived, which take the friction and asperity interactions into account. Furthermore, the actual contact area and normal stiffness models for fractal surfaces are developed, accounting for friction and asperity interactions. The accuracy of the developed models is confirmed by using published experimental results. The developed models are closer to experimental results than the reported fractal contact models. Finally, the effects of the friction coefficient and surface morphology parameters on contact characteristics are discussed.

A design of compact wideband substrate integrated waveguide fractal koch surface slot antenna for Ka‐band applications

SummaryThis article analyses a substrate integrated waveguide (SIW) fractal koch surface (FKS) slot antenna performance for Ka‐band applications. This FKS slot shape is obtained by modifying the conventional rectangular slot with the first iteration of the FKS method. The resultant FKS slot behaves as a radiator in the provided space. A single‐ and two‐element FKS slotted SIW antenna characteristics are analysed on a 20 mil thick RT/Duroid 5880 substrate and compared with the conventional rectangular slot. The proposed antenna with dimensions mm3 achieves an impedance bandwidth from 27.4 to 32.1 GHz with a fractional bandwidth of 15.79% and achieves a gain of 8.3 dBi in broadside direction by improving space filling factor in SIW structure. The prototype of the SIW‐fed FKS slotted antenna array measurement results is satisfactory with the simulation results.

Investigations on the modal vibration caused by bolted joint interface contact in the rotor-AMBs systems: Modelling and experimentation

Rotor-active magnetic bearings (rotor-AMBs) systems nowadays have been widely used in fluid machinery and the impeller and the shaft are commonly connected by bolted joint. However, when a certain pre-tightening force is applied to the bolted joint and the interface contact between the shaft and the impeller is formed, causing flexible modal vibrations when rotor is levitated. The mechanism of this modal vibration needs to be revealed to ensure stable operation of the machine. In traditional modelling, the effect of the interface contact is equated to an additional stiffness matrix by massless spring units, whose certain contact stiffness is uniformly distributed over the interface. Particularly, the calculation of contact stiffness and the effect of AMBs are neglected, resulting in the inaccurate response compared to experiments. Based on traditional modelling, we consider that there is partial separation in the contact interface due to the AMBs supporting and a novel additional stiffness matrix related to contact status is proposed by calculating the real-time contact area. The contact stiffness is calculated by microscopic contact model based on fractal theory and surface topography measurement. Finally, numerical simulation shows that the interface contact influences the system robustness and the unstable mode is excited. The increase of pre-tightening torque and contact radius both will increase the steady vibration amplitude, with the former being the major contributor. Moreover, the influence and order of the vibration are quantitatively validated by the experiments.