Original Article Path-dependent analysis of elastic sphere contact subjected to tangential loading with varying directions Shiping Huang 1,2 and Anil Misra 2 Abstract The Cattaneo–Mindlin solutions of contact between elastic spheres and their recent extensions do not consider the sequential application of arbitrarily directed shear forces in the contact tangential plane. For this loading condition, the contact tractions simultaneously undergo loading and unloading. This article presents a path-dependent analysis wherein we use superposition at each loading step to obtain the contact tangential traction, and, subsequently the tangential displacement and compliance. The methodology is illustrated by example calculation of contact shear force–displacement relationship, which shows the formation of hysteretic loop and the noncoaxiality of the shear forces and displacements. Keywords Contact mechanics, stick–slip, elastic spheres, tangential traction–displacement, hysteresis, shear compliance Date received: 13 October 2011; accepted: 7 February 2012 Introduction Contact between solid bodies is ubiquitous in nature and contact problems abound across all areas of engin- eering. For the contact between nonconforming solids, the contact mechanics of elastic spheres has served as a benchmark and as a fundamental basis for rough con- tact mechanics 1–9 and also for granular mechanics. 10,11 The pioneering work of sphere contact problem under normal loading was published by Hertz 12,13 in 1881 and 1882. From then on, the sphere contact problem has attracted many researchers. Cattaneo and Mindlin independently derived the force–displacement relation- ship when considering combined normal and shear loading 14–17 (referred as C–M solutions hereafter). By dividing the loading into infinite loading steps, Mindlin and Deresiewicz 16 made use of the sphere contact solu- tion under constant normal force to derive the force– displacement relationships under varying oblique forces. The solutions given by Mindlin and Deresiewicz are widely cited, although seldom used in their complete form. In most applications, these solu- tions have been modified or simplified to reduce the computational requirements. 18,19 It is noteworthy that the C–M solutions for elastic sphere contacts are incomplete and require additional considerations in a number of aspects. We mention here three fundamental aspects that pertain to the relatively simple loading conditions which comprise the applica- tion of a shear action while an initially applied normal force is held constant. First, the C–M solutions neglect the transverse tangential tractions perpendicular to the direction of applied shear force that arise due to the Poisson’s ratio effect. Second, these solutions do not consider tangential forces that do not pass through the center of the contact area and produce a twisting moment. Third, these solutions only consider contact loading confined within an N–T plane as shown in Figure 1. Subsequent application of out-of-plane tan- gential loading, such as that denoted by T 2 , was not considered. The first two aspects have been investigated in some detail; 20–23 however, the third aspect has 1 School of Civil Engineering and Transportation, South China University of Technology, People’s Republic of China 2 Department of Civil, Environmental and Architectural Engineering, The University of Kansas, USA Corresponding author: Anil Misra, Department of Civil, Environmental and Architectural Engineering, The University of Kansas, Learned Hall, 1530 West 15th Street, Lawrence, KS 66045-7609, USA. Email: amisra@ku.edu Proc IMechE Part J: J Engineering Tribology 226(8) 678–686 ! IMechE 2012 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1350650112440414 pij.sagepub.com