Yang Xu Mechanical Engineering Department, Auburn University, Auburn, AL 36849 Amir Rostami G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 Robert L. Jackson Mechanical Engineering Department, Auburn University, Auburn, AL 36849 Elastic Contact Between a Geometrically Anisotropic Bisinusoidal Surface and a Rigid Base In the current study, a semi-analytical model for contact between a homogeneous, iso- tropic, linear elastic half-space with a geometrically anisotropic (wavelengths are differ- ent in the two principal directions) bisinusoidal surface on the boundary and a rigid base is developed. Two asymptotic loads to area relations for early and almost complete con- tact are derived. The Hertz elliptic contact theory is applied to approximate the load to area relation in the early contact. The noncontact regions occur in the almost complete contact are treated as mode-I cracks. Since those cracks are in compression, an approxi- mate relation between the load and noncontact area can be obtained by setting the corre- sponding stress intensity factor (SIF) to zero. These two asymptotic solutions are validated by two different numerical models, namely, the fast Fourier transform (FFT) model and the finite element (FE) model. A piecewise equation is fit to the numerical sol- utions to bridge these two asymptotic solutions. [DOI: 10.1115/1.4029537] Keywords: periodic elasticity, sinusoidal normal contact, geometrically anisotropic, fracture mechanics, FFT model, finite element model 1 Introduction The problem of contact between linear elastic periodic sinusoi- dal surfaces has been extensively studied for more than 70 years, beginning with the two-dimensional (2D) planar case done by Westergaard [1]. Westergaard [1] first obtained a closed-form so- lution of sinusoidal waviness contact from first touch to complete contact using complex potentials. Friction and adhesion are not included in his model. The elastic contact body is a half-plane with a 2D sinusoidal waviness on the nominally flat boundary. The amplitude of the 2D sinusoidal waviness must be small com- paring with the wavelength, which guarantees the validity of the application of the linear theory of elasticity. This solution of 2D waviness contact is referred to as the Westergaard solution and will be discussed later in detail. Dundurs et al. [2] solved the same problem based on the Papko- vich–Neuber potentials and their closed-form solution consists of the Fourier series. Manners [3] formulated the plane contact between a periodic arbitrary profile and the rigid flat using com- plex form. The contact stress distributions of some simple peri- odic profile (single and double sinusoidal terms) were obtained. Johnson [4] analytically solved the sinusoidal waviness contact problem considering adhesion by transforming the corresponding problem into a problem of fracture mechanics. Adams [5] solved the same problem by embedding the Maugis adhesion model [6] in the Papkovich–Neuber potentials presented by Dundurs et al. [2]. Carbone and Mangialardi [7] modeled the adhesive sliding friction between a slightly wavy rigid surface and an elastic half- space. The contact interface is modeled as the propagation of two mode-I cracks on the leading and trailing edges, respectively. Block and Keer [8] reviewed and developed closed-form solutions of the frictional sinusoidal waviness contact problem in full-stick, perfect slip, and partial slip conditions for both elastically similar and dissimilar materials. Difficulties of obtaining analytical solutions arise in the three- dimensional (3D) periodic bisinusoidal waviness contact problem since contact area is unknown in advance [9]. 1 For the frictionless, nonadhesive, geometrically isotropic 2 bisinusoidal waviness con- tact, only two asymptotic solutions for early and almost complete contact are available in closed-form and derivation of these asymptotic solutions was provided by Johnson et al. [9]. Johnson [4] also obtained two asymptotic solutions for the adhesive case. Jackson and Streator [10] fit an empirical equation to the numeri- cal and experimental solutions by bridging between Johnson’s asymptotic solutions. Krithivasan and Jackson [11] further devel- oped a similar empirical solution for the elastoplastic contact between a geometrically isotropic sinusoidal waviness and a rigid flat based on the FE solutions. Jackson et al. [12] provided the critical parameter defining the boundary between elastic and elas- toplastic geometrically isotropic sinusoidal contact. Rostami and Jackson [13] developed empirical equations for the average surface separation for both elastic and elastoplastic geometrically isotropic sinusoidal surface contact from the first touch to the complete contact. The viscoplasticity is recently introduced by Rostami et al. [14] in the numerical modeling of the sinusoidal waviness contact. Yastrebov et al. [15] revisited the classical solu- tions of Johnson–Greenwood–Higginson [9], experimental results [9], and the empirical solutions [11]. Yastrebov et al. [15] pointed out an overlooked transition regime between the early and almost Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received August 22, 2014; final manuscript received December 31, 2014; published online February 5, 2015. Assoc. Editor: Sinan Muftu. 1 This is also true for the plane problem. However, the shape of the contact area in the plane problem is known. For the 3D problem, the shape of the contact area depends on the external average contact pressure [15]. 2 In the rest of the article, the material is always linear, isotropic, and homogeneous. “Geometrically isotropic” is reserved for the geometrical property of the rough surface. Readers may find the terminology, “isotropic,” inappropriate here because it is usually applied to describe the rough surface not the smooth one. A rough surface is called isotropic, if every profile of the rough surface, measured in the arbitrary direction, is statistically the same. Axisymmetric is an equivalent terminology used for smooth surfaces. In this study, however, in order to be consistent with the terminology used by Johnson et al. [9], geometrically isotropic is used here to describe the bisinusoidal waviness with the same wavelengths on two principal axes. If the wavelengths are different, the bisinusoidal waviness is called geometrically anisotropic. Journal of Tribology APRIL 2015, Vol. 137 / 021402-1 Copyright V C 2015 by ASME Downloaded From: http://gasturbinespower.asmedigitalcollection.asme.org/ on 03/06/2015 Terms of Use: http://asme.org/terms