Adhesion between an elastic body and a randomly rough hard surface

I have developed a theory of adhesion between an elastic solid and a hard randomly rough substrate. The theory takes into account that partial contact may occur between the solids on all length scales. I present numerical results for the case where the substrate surface is self-affine fractal. When the fractal dimension is close to 2, complete contact typically occurs in the macro-asperity contact areas, while when the fractal dimension is larger than 2.5, the area of (apparent) contact decreases continuously when the magnification is increased. An important result is that even when the surface roughness is so high that no adhesion can be detected in a pull-off experiment, the area of real contact (when adhesion is included) may still be several times larger than when the adhesion is neglected. Since it is the area of real contact which determines the sliding friction force, the adhesion interaction may strongly affect the friction force even when no adhesion can be detected in a pull-off experiment. Received 3 April 2002     

粗糙界面板Lamb波的传播特征

弹性波在粗糙界面板中的传播问题对无损检测、地震波传播等问题有实际意义,但是多年来的研究限于相屏近似等方法。本文从严格的理论出发,把粗糙界面起伏看作空间位置变量的随机过程,在粗糙表面起伏很小的条件下,用多次散射相互独立假设得到了平均意义下的Rayleigh-Lamb频散方程。通过对方程进行求解,得到了Lamb波在粗糙界面板中传播时的衰减。结果表明Lamb波的衰减和界面的粗糙起伏方差近似成正比,和粗糙起伏相关长度关系不大。在理论分析的基础上,进行了实验验证。     

A review of boundary elements methodologies for elastic and viscoelastic rough contact mechanics

In this paper, we review the main contributions developed by the authors to solve problems involving contact between bodies with a roughness covering several orders of magnitude. A periodic self-equilibrated formulation is introduced to avoid any border problem. Furthermore, when focusing on the steady-state rolling or sliding contact mechanics between viscoelastic materials, we develop a mathematical formulation, parametrically dependent on the rolling/sliding speed, which employs the same computational techniques introduced for elastic rough contacts. In particular, in both cases, we adopt an ad hoc discretization technique enabling us to reach the full convergence.     

Vibrations of a rectangular membrane with an eccentric inner circular boundary: A comparison of approximate methods

Three different approaches are used in order to obtain, independently, fundamental eigenvalues of the mechanical system under study: the Ritz method, the finite element algorithm and a Fourier expansion-collocation scheme. The agreement can be considered as very reasonable, especially in view of the relatively simple formulation of the Ritz approach in which two polynomial co-ordinate functions are employed. The results may also be of interest to acousticians and microwave specialists since they are applicable to soft-walled acoustical waveguides and TM modes in electromagnetic waveguides.     

Thermal radio emission of a homogeneous half-space with a rough boundary

Khar'kov State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 32, No. 8, pp. 927–932, August, 1989.     

A note on a circular panel sound absorber with an elastic boundary condition

Panel sound absorbers are typically used to absorb low-frequency noise in concert halls, auditoriums, recording studios, and other architectural applications. These systems are composed of flexible panels mounted over an air space that can be either partly or completely filled with a porous material. In this paper, a theoretical model is derived for predicting the sound-absorption coefficient of a cylindrical low-frequency absorber made of a circular plate. The theory takes into account the mass, bending stiffness, damping loss and the elastic boundary condition of the circular plate. The effects of the stiffness of an air-back cavity and of partially adding a porous material into the cavity are also considered. It is observed that the low-frequency resonances of such a system are dependent upon the clamping condition, the width of the air-back cavity, and mechanical properties of the plate. There is good agreement between theoretical and experimental results.     

Resonances of the Rayleigh waves in an elastic semi-infinite strip

Kirchhoff’s theory of plates is used to study forced harmonic vibrations of a semi-infinite strip when the latter is in the generalized stressed state or experiences flexural deformation. The forced vibrations are excited by a load applied to the strip end. Cross-boundary conditions are imposed on the strip’s sides, which allows one to obtain a closed solution. The presence of an infinite real frequency spectrum corresponding to the edge resonances is revealed. The relation of these resonances to the Rayleigh planar and flexural waves is established.     

Vibrations of axially loaded stiffened cylindrical panels with elastic restraints

A method has been developed for analysis of the vibration and buckling of preloaded stiffened cylindrical panels with different boundary conditions along the straight edges, including elastic restraints. In the analysis linear, “smeared” stiffener, Flügge type theory is used. A computer program VIBUPAL has been developed. The method has been verified by comparison with results available for limited cases and very good agreement has been obtained.     

Lamb waves propagation in elastic plane layers with a joint strip

The Lamb waves are used for the ultrasonic characterization of welds because of their relative long-range propagation. In this paper, a simplified model of a weld-strip between two identical semi-infinite elastic layers is investigated. The reflected and transmitted ultrasonic fields are expressed by modal series whose coefficients are obtained by application of orthogonality relation. Comparisons with solutions obtained by finite elements wave propagation simulations are made. The energy balance between the incident and the scattered waves is also used to verify the accuracy of the obtained modal amplitudes.     

Diffraction of sound by an elastic or impedance sphere located near an impedance or elastic boundary of a halfspace

An exact solution is obtained to the problem of sound diffraction by an elastic or impedance sphere located near an impedance or elastic boundary of a halfspace. The problem is solved using the Helmholtz integral equation in which the field of a point source in the halfspace with an elastic boundary is used as the Green function. The diffracted field is represented as a series expansion in spherical harmonics. The expansion coefficients are determined from a set of independent algebraic systems of equations. The matrix coefficients of these systems are determined as integrals of the products of the associated Legendre polynomials on the complex plane with respect to the real and complex angles of the sound incidence on the halfspace boundary. To decrease the number of such integrals, expansions using the Klebsh-Gordon coefficients are applied. As a result, algorithms for calculating the scattered field in the halfspace are obtained.     

Vibrations and buckling of a beam on a variable winkler elastic foundation

Two methods for solving the eigenvalue problems of vibrations and stability of a beam on a variable Winkler elastic foundation are presented and compared. The first is based on using the exact stiffness, consistent mass, and geometric stiffness matrices for a beam on a variable Winkler elastic foundation. The second method is based on adding an element foundation stiffness matrix to the regular beam stiffness matrix, for vibrations and stability analysis. With these matrices, it is possible to find the natural frequencies and mode shapes of vibrations, and buckling load and mode shape, by using a small number of segments. It is concluded that the use of the element foundation stiffness approach yields better convergence at lower computation costs.     

Adhesion between elastic bodies with randomly rough surfaces

I have developed a theory of adhesion between an elastic solid and a hard randomly rough substrate. The theory takes into account the fact that partial contact may occur between the solids on all length scales. I present numerical results for the case where the substrate surface is self-affine fractal. When the fractal dimension is close to 2, complete contact typically occurs in the macroasperity contact areas. For a fractal dimension larger than 2.5, the area of (apparent) contact decreases continuously when the magnification is increased.     

Strong scattering of radio waves by a two-layered medium with a rough boundary

Radiowave scattering by a two-layered medium with a rough boundary is studied in the Kirchhoff approximation. We consider the case of oblique incidence of waves and mean square heights of the upper boundary roughnesses, which are nonsmall compared with the wavelength λ. An analytical expression and the conditions under which P₁ turns out to be large are derived for the term P₁ of the two-frequency correlation function P = (E_m(w₁)E_n(w₂)), which is proportional to the first degree of the coefficient of reflection from the lower boundary. Such effect is shown to emerge only for the resonant frequency relation. In particular, the following resonant relationship is written for the mirror reflection direction: w_2,1(√ε-sin²θ/cosθ-1), where θ is the incidence angle and ε is the layer permittivity. Research Institute of Physics, Rostov-on-Don, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 40, No. 6, pp. 733–743, June, 1997.     

Scattering and diffraction of a plane wave from a randomly rough strip

This paper deals with plane wave scattering and diffraction from a randomly rough strip using a combination of three tools: the perturbation method, the Wiener-Hopf technique and a group-theoretic consideration based on the shift-invariant property of the homogeneous random surface. The D^a-Fourier transformation associated with the shift invariance is defined instead of the conventional complex Fourier transformation. For a slightly rough case, Wiener-Hopf equations for the zero-, first- and second-order perturbed fields are derived. They are reduced to a common Wiener-Hopf equation, an exact solution of which is obtained formally by means of the Wiener-Hopf technique. Using the inverse D^a-Fourier transformation, the scattered wavefield is obtained as a stochastic field. When the strip width is large compared with the wavelength, a uniformly asymptotic representation of the scattered far field is obtained by the saddle point method. For a Gaussian roughness spectrum, several numerical results are calculated and illustrated in figures, based on which the characteristics of scattering and diffraction are discussed.