Multi-stage contact model between fractal rough surfaces based on ellipsoidal asperity deformation

In this paper, a normalized band-limited Weierstrass function is presented for modelling 2D fractal rough surfaces. Some conventional statistical parameters, namely the root mean square and the correlation length of rough surfaces, are used to assess between fractal parameters and the roughness of surfaces. An analytic solution of the scattered light field from these fractal surfaces is derived based on Kirchhoff theory. Three statistical parameters, namely the average scattering coefficient, the average intensity of scattered field, and the root mean square of scattered field, are introduced to study the influence of various fractal parameters on the scattered field by theoretical analysis and numerical calculations.

Functionally graded surface has been a domain of research interest due to its use in various technologically advanced applications. The work focuses on static and dynamic analysis of rough fractal surface with exponentially graded material properties. The rough fractal surface is modeled in ANSYS with the coordinate points generated using the two variable modified Weierstrass–Mandelbrot function. Material gradation is applied to the rough surface model in such a way that, for any value of the gradation or inhomogeneity parameter, material properties at the top of the rough surface always remain constant. Force–displacement behavior of the fractal surface in contact with a rigid flat surface is determined through finite element simulation. A parameter representing the nonlinearity of the system is extracted from the force–displacement plot. It is found that higher is the inhomogeneity parameter, higher is the nonlinearity of the system. Furthermore, for a certain change in inhomogeneity parameter, the change in nonlinearity is higher for rougher topography. The dynamic contact system is found to be softening in nature, and the softening nature increases with higher inhomogeneity parameter. The phase plot of the vibrating contact system becomes more asymmetric with respect to the velocity axis for higher value of inhomogeneity parameter.

Two-dimensional contact of asperities to multi-stage contact between curved fractal surfaces based on attitude angle

To numerically calculate bistatic scattering from a fractal rough surface with high permittivity, a comprehensive approach of the forward-backward method (FBM) with spectral acceleration algorithm (SAA) combined with the physics-based two-grid (PBTG) method is developed. Bistatic scattering from and transmitting through one-dimensional (1-D) fractal rough surface with high permittivity, under TE and TM tapered wave incidence, are numerically simulated. Angular reflectivity and transmittivity demonstrate the energy conservation and the functional dependences upon the surface parameters.

The authors wish to explain the similarity between some figures in the above paper (hereafter called the JPD paper) and in their other publication, Lee S, Jang Y H and Kim W 2008 Effects of nanosized contact spots on thermal contact resistance J. Appl. Phys.103 074308 (hereafter called the JAP paper), and to explain the differences between the two papers, which are not explicitly stated in the JPD paper.The main objective of the JAP paper is to calculate the thermal contact resistance of the nanosized contact spots in multiscale contact. During the process of multiscale analysis, the thermal conductivity varies, especially below the phonon mean free path. The JPD paper deals with the electrical contact resistance in the multiscale contact distribution with an assumption of constant electrical resistivity, which is known as a different kind of physics in a larger characteristic length scale.There are similar figures in the JPD paper and the JAP paper: figures 6, 7 and 8 in the JPD paper and figures 3, 4 and 5 in the JAP paper. Two research works were performed on the basis of a specific microcontact distribution. In the JAP paper, the scale of the contact distribution is in the range of the phonon mean free path of Si, which is a very small size of contact distribution. In the JPD paper, the scale of contact distribution is in the continuum scale, which is larger than the phonon mean free path. In addition, due to the characteristics of a fractal surface which repeatedly generates a similar shape of contact distribution in the different length scales, the shape of contact distribution looks similar, but the total sizes of domain in the JPD and JAP figures are different. The projected areas L × L of fractal surface of the JAP paper and JPD paper are 10 μm × 10 μm and 10 mm × 10 mm, respectively. The length scale is already stated in the JAP paper, but not in the JPD paper. Thus, we have to state that the figures were adapted from the JAP paper without clear attribution and that credit should be given to the original source of the images. Most significantly, the two most important features of the two works are based on totally different physics and the calculation methods for contact resistance.The text of section 4.3 of the JPD paper should be modified. The left column of page 5 and the first two lines of the right column should be replaced by the following text.The projected area of a self-affine fractal surface is plotted in figure 6. In order to find the microcontact spot distribution from the fractal surface, numerical simulations are applied to the model of multiscale rough surface. Several contact models are constructed according to the resolution of 1/24 to 1/28. Figure 7 shows a typical finite element model in which the rough elastic surface is in contact with a perfectly flat rigid surface. Due to the fine meshes in the contact interface, strong mesh gradations are required. The contact simulation enforcing the impenetrability constraint on the mating surfaces is performed by a commercial finite element package, Abaqus/HKS ([22] of the JPD paper). The resulting microcontact spot distributions for different resolution of the rough surface are obtained in figure 8. As reported by Borri-Brunetto et al ([23] of the JPD paper), the actual contact area decreases and the number of contact spots increases as the resolution of the discretization escalates.A more detailed description for the model can be obtained in the JAP paper.

One dimension band-limited fractal surfaces with different fractal dimension and different rms slope are created by Monte Carlo simulation. The curves of the single scattering shadowing function of the band-limited fractal surface with varying of the incident angle are obtained by utilizing a ray-tracing algorithm, the influence of different fractal dimension and different rms slope on the fractal surface shadowing function is analyzed. The curve of the shadowing function with varying of the incident angle for the fractal surface and Gaussian rough surface is compared with same rms slope. The results show that the curves of the shadowing function for the fractal surface with different fractal dimension and three Gaussian rough surface are basically same.

Due to inherent nonlinear stiffness and damping characteristics, dry friction interfaces have a significant impact on the dynamics of jointed structures. When two surfaces are brought into purely normal contact, contact pressure distribution is of high concern. This work focused on the statistics of the contact pressure distribution between a rough surface and a smooth rigid plane, which provides new insight into the interface contact behaviour. First, the fractal rough surface is generated using the Weierstrass–Mandelbrot function with measured roughness parameters. With meshed rough surfaces, an elastic–plastic finite element contact analysis is performed to determine the contact pressure distribution. Then, the results of contact pressure are statistically analysed. The effects of roughness and contact load on contact area, contact stiffness and mean contact pressure are thoroughly investigated. The probability distribution of contact pressure is determined by fitting a continuous function using a twofold Weibull mixture model. The proposed probability distribution function is found to be capable of describing contact pressure. The contact pressure distribution is affected by the surface fractal characteristics and evolves with the contact load.

This work investigates interfacial electro-mechanical properties, including electrical contact resistance, interfacial capacitance and characteristic frequency of contacts formed with various surface structures. Fractal rough surfaces were generated and characterised by fractal dimension and root-mean-square (RMS) roughness. The rough surface with a thin oxide layer was compressed by the rigid flat to form a capacitor. Electrical impedances of this contact capacitor were simulated using the finite element method across a wide range of frequencies. A power-law relationship was found between the electrical contact resistance and applied compression load. An analytical model is proposed to capture the interfacial capacitance behaviour with increasing contact loads, revealing a transition of predominated modes for the capacitance. Higher fractal dimension yields smaller overall capacitance in the gap dominant and transition zones. The dependence of the characteristic frequency on compression was found to follow a power-law function at the low load range. It is found that the exponent and magnitude of obtained power-law relations show strong correlations to the fractal dimension and RMS roughness, respectively. Results of this work provide insights into developing a potential impedance measurement protocol to determine the thickness of the oxide layer on conductive fractal rough surfaces.

Fractal geometry function provides a practical way to simulate natural rough surface for electromagnetic scattering calculation of microwave remote sensing. In our study, we generate three-dimensional random rough surface based on band-limited Weierstrass-Mandelbrot model and carry out a series of quantitative analyses of the relation between fractal parameters and rough surface parameters. In order to eliminate the artifical reflection from the truncated boundary of rough surface model, a tapered incident wave model is built to illuminate the surface profile model with the use of sampling strategy. The tapered incident wave is introduced into the classical Two-Scale Model to calculate backscattering coefficient. The proposed model is applied to simulate the scattering of sea clutter based on the fractal sea surface model. Its simulation results are compared with real sea surface scatterometer measurement experiments in Beihai, Guangxi province and the overall deviation is less than 10%. The results validate the efficiency of the proposed model in obtaining the scattering simulation data on sea surface for parameter inversion of active microwave remote sensing.

A generalized forward–backward method (GFBM) with the spectral accelerated algorithm (SAA) to solve the magnetic‐field integral equation (MFIE) is developed. It is applied to the numerical simulation of bistatic scattering from a one‐dimensional (1‐D) fractal dynamic sea surface with a ship presence under TE tapered wave incidence at a low grazing angle (LGA). Numerical simulations of bistatic scattering at an LGA show the functional dependence upon the fractal dimension of the conducting rough surface, ship location, and other parameters. © 2001 John Wiley & Sons, Inc. Microwave Opt Technol Lett 31: 146–151, 2001.

Sampling effect on contact and transport properties between fractal surfaces

In this paper, equivalent surface impedance boundary condition (ESIBC), which takes fractal parameters (D, G) into SIBC, is implemented in the 4-component 2-D compact finite difference frequency domain (2-D CFDFD) method to analyze the propagation characteristics of lossy circular waveguide with fractal rough surface based on Weierstrass-Mandelbrot (W-M) function. Fractal parameters’ effects on attenuation constant are presented in the 3 mm lossy circular waveguide, and the attenuation constants of the first three modes vary monotonically with scaling constant (G) and decrease as the fractal dimension (D) increasing.

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.

The role of surface roughness in contact mechanics is relevant to processes ranging from adhesion to friction, wear and lubrication. It also promises to have a deep impact on applied science, including coatings technology and design of microelectromechanical systems. Despite the considerable results achieved by indentation experiments, particularly in the measurement of bulk hardness on nanometre scales, the contact behaviour of realistic surfaces, showing random multiscale roughness, remains largely unknown. Here we report experimental results concerning the mechanical response of self-affine thin films indented by a micrometric flat probe. The specimens, made of cluster-assembled carbon or of sexithienyl, an organic molecular material, were chosen as prototype systems for the broad class of self-affine fractal interfaces, today including surfaces grown under non-equilibrium conditions, fractures, manufactured metal surfaces and solidified liquid fronts. We observe that a regime exists in which roughness drives the contact mechanics: in this range surface stiffness varies by a few orders of magnitude on small but significant changes of fractal parameters. As a consequence, we demonstrate that soft solid interfaces can be appreciably strengthened by reducing both fractal dimension and surface roughness. This indicates a general route for tailoring the mechanical properties of solid bodies.

We have studied the influence of substrate surface roughness on the growth of Ge on self-affine fractal Si surfaces. Self-affine fractal surfaces can be prepared by irradiating surfaces of solid materials with an energetic ion beam, which modifies the surface morphology predominantly via an erosion process. By tuning the ion beam energy, surface parameters such as surface roughness (σ0) and correlation length (ξ), can be controlled. Values of σ0 and σ0/ξ affect the effective free energy of the surface. When materials are grown on surfaces with different values of surface roughness parameters, the morphology of the deposited layer is expected to be influenced. In comparison with an earlier study where 2 MeV Si ion irradiation was used, we have produced surfaces with larger values of σ0 and σ0/ξ by irradiating the surfaces with 45 keV Si ions. We have made a comparison for Ge growth on Si surfaces irradiated with 2 MeV and 45 keV Si ions and have observed remarkably different growth behaviour. Upon Ge deposition, in the former case, the surface roughness decreases at smaller length scales and remains practically unchanged at larger length scales. This indicates that Ge has grown preferentially within the valleys of the rough surface. In the latter case, where σ0 is large and ξ is small on the substrate surface, the roughness increases at all length scales by almost a factor of 2, upon Ge deposition, indicating the lack of preferential growth within the surface valleys. Following Ge deposition, in the former case the roughness exponent increases from 0.82 to 0.99, while in the latter case it remains practically unchanged (∼0.83). Roughness parameters of the substrate surface are found to have a significant influence on the morphology of the overlayers.