UNLOADING OF AN ELASTIC-PLASTIC LOADED SPHERICAL CONTACT

2004 AIMETA International Tribology Conference, September 14-17, 2004, Rome, Italy UNLOADING OF AN ELASTIC-PLASTIC LOADED SPHERICAL CONTACT Yuri KLIGERMAN( ), Yuri Kadin( ), Izhak ETSION( ) Faculty of Mechanical Engineering, Technion Israel Institute of Technology, 32000, Haifa, Israel Fax +972-4-829-5711 Phone +972-4-829-2075, E-mail: mermdyk@tx.technion.ac.il Phone +972-4-829-2090, E-mail: skadin@t2.technion.ac.il Phone +972-4-829-2096, E-mail: etsion@tx.technion.ac.il ABSTRACT The process of unloading an elastic-plastic loaded sphere in contact with a rigid flat is studied by the Finite Element Method. The sphere material is assumed isotropic with elastic-linear hardening. The numerical simulations cover a wide range of loading interference deformations for various values of Young's modulus and Poisson ratios of the sphere material. The contact loads, and deformations in the sphere during both loading and unloading, are calculated for the range of interferences. Empirical dimensionless expressions are presented for the unloading load-deformation relation, and the residual axial displacement. NOMENCLATURE a P P c P P r ω ω c ω ω res ω : contact radius : contact load : contact load at the inception of plastic deformation : imum contact load before unloading : dimensionless contact load, P/ P c : radial coordinate : contact interference : critical interference at the inception of plastic deformation : imum contact interference before unloading : residual contact displacement after complete unloading : dimensionless interference, ω/ ω c INTRODUCTION Repeated loading and unloading of rough surfaces is an important problem, especially in the technology of micro/nano-systems, such as MEMS micro switches [1, 2], for example, or head-disk interaction [3] in magnetic storage system. Hence, the interest in loading/unloading

of a sphere in contact with a flat, which can simulate a single asperity of a rough surface, is obvious. Johnson [4] offered one of the first simple analytical models of unloading an elastic plastic spherical indentation contact by assuming the unloading process to be perfectly elastic based on Tabor s [5] observations of spherical indentation for hardness measurement. Mesarovic and Johnson [6] examined the process of decohesion of two adhering elastic plastic spheres following mutual indentation beyond their elastic limit. It was assumed that during unloading the deformation is predominantly elastic. Vu-Quoc et al. [7] presented a simple model for the normal force-displacement relation for contacting spherical particles, accounting for the effects of plastic deformation. Nonlinear Finite Element simulations were carried out by Lin and Hui [8], Yan and Li [9], and Ye and Komvopoulos [10] to study the effect of loading and unloading spheres. Li et al. [11] presented a theoretical model for frictionless contact of a rigid sphere with an elastic perfectly plastic half-space or an elastic perfectly plastic sphere with a rigid wall. This model can be considered a modification of Johnson s model since it is based on a finite element analysis and considers the variation in the curvature of the contact surface during the contact interaction. The main goal of the present paper is to develop a model for the unloading of an elastic-plastic loaded sphere in frictionless contact with a hard smooth plane. Since a spherical contact may simulate a single asperity of a rough surface, the present analysis can help understand the behaviour of rough surfaces during repeated loading and unloading. UNLOADED SPHERICAL CONTACT MODEL A schematic representation of the contact problem is shown in Fig 1. A compliant hemispherical body of an original undeformed radius R comes into contact with a rigid flat surface. The problem is axisymmetric (2D) about the z axis. Therefore, it is sufficient to consider only one half of the axisymmetric hemisphere section. The boundary conditions consist of constrain in the vertical direction at the hemisphere base and in the radial direction at its z axis. The spherical surface is free elsewhere except from the axial constriction enforced by the contacting rigid flat. The contact is assumed frictionless (perfect slip condition). The dashed and solid horizontal lines in Fig. 1 show the rigid flat positions before and after loading, respectively. The interference, ω, and contact area with a radius, a, (see Fig.1) correspond to a normal load, P, applied to the contact. We assume that the contact interference, ω, is much smaller then the undeformed sphere radius, R, but larger than the critical interference c at the inception of plastic deformation [12]. The contact radius, a, is also very small compared to R. While the elastic contact problem of a sphere and a rigid flat has an analytical solution (the classical Hertz solution, see e.g., [4]), the elastic-plastic contact problem requires a numerical solution. In the present work this problem was solved using the approach of Kogut and Etsion [13] for a frictionless contact between an elastic-perfectly plastic sphere and a rigid smooth flat. The present numerical model is based on a Finite Element Analysis (FEA), using the commercial package ANSYS 8.0. The material of the sphere was modeled as an elastic linear hardening material. A 2% linear hardening was selected since it still yields results that are very close to an elasticperfectly plastic case, thus enabling comparison with existing loading results [13], yet providing a much better convergence of the numerical solution. The von Mises yielding

criterion was used to detect local transition from elastic to plastic deformation, and the Prandtl-Reuss constitutive law governed the stress-strain state in the plastic zones. The solution consists of two stages (see Fig. 2). In the first one the sphere is loaded by the rigid flat to a dimensionless interference =. During this stage the interference is gradually increased up to a desired imum value and the contact load and contact radius reach teir imum value P and a, respectively. The dimensionless contact load P Pc, as a function of the dimensionless interference = c during the elasticplastic loading stage were given by Kogut and Etsion [13] in the form: P P c 1.03( = 1.40 ( ) ) 1.425 1.263 for for 6 c 1 6 110 where P c is the contact load at the inception of plastic deformation. The second stage consists of the unloading process, where the interference is gradually reduced. When the unloading process is completed, the contact load, contact radius, and real contact area fall to zero. However, the original undeformed spherical geometry is not fully recovered. Residual stresses and strains remain locked in, and result in a deformed shape of the unloaded sphere. This deformed shape is limited to a relatively narrow zone of the final loaded contact prior to unloading and its immediate vicinity. The deformed profile of the unloaded sphere may be characterized by a residual interference res and a residual nonuniform curvature with a radius R res at its summit as shown schematically in Fig. 2. The residual interference res and residual deformed profile with its summit residual curvature R res depend on the final loading level in terms of or P from which the unloading process started. (1) DISCUSSION OF THE NUMERICAL RESULTS The model in Fig. 1 had a finite element mesh that consisted of 1920 six-node triangular elements comprising a total of 3987 nodes. The sphere was divided into four different mesh density zones where zones I, II, and III were within 0.01R, 0.05R, and 0.1R distance, respectively, from the sphere tip. Zone IV, outside the 0.1R distance had gradual coarser mesh at increasing distance from the sphere tip. Different materials and hemispherical radii were analyzed (see Table 1) to study the effect of these variables on the elastic-plastic contact loading-unloading behavior. These parameters were selected in such a way that the critical interference, ω c, which seems to be the most characteristic parameter of elastic plastic contact problems, varied over a wide range of values from about 0.88x10-6 R to 60.5x10-6 R. The ratio of E/σ y also covered a wide range from about 297 to 2464. The hemisphere was loaded to various contact interference values with an upper limit of = 170. Loading beyond this value is outside the scope of the present study. It requires the treatment of very large deformations that may be relevant in plastic forming. Fig. 3 presents the dimensionless elastic plastic load-displacement results for the loadingunloading process in terms of P vs. ω. The figure demonstrates results for two very different cases (case 1 and case 2 in Table 1) one of a high E/σ y ratio combined with a low ω c /R ratio

indicated by the rhombs and the other with a low E/σ y ratio combined with a high ω c /R ratio indicated by the pluses. As can be seen from Fig. 3 scaling the variables with their corresponding critical values at plastic yield inception renders the dimensionless solution a universal nature that is independent of the physical and geometrical properties of the sphere. The dashed line in Fig. 3 presents the elastic-plastic loading results obtained by Kogut and Etsion [13] for an elastic perfectly plastic material (Eq. (1) above). As can be seen these results correlate well with the present model. A best fit of the current numerical results for the elastic-plastic loading over the entire range of ω values up to 170 is presented by the solid line in Fig. 3, and by the following empirical relation: 1.28 ( 1) + 1 P = 1.24 (2) The numerical results of the unloading process initiated from three representative values of 50, 100, and 150, respectively, are also shown in Fig. 3. It is clear from these results that the present dimensionless solution is universal not only for the loading but also for the unloading stage. It is also clear from Fig. 3 that the load-displacement behavior during unloading is not linear and is strongly affected by the level of initial loading. The residual interference res for any given initial value of can be found from the unloading process at its completion when the contact load vanishes. A best fit that was carried out on many unloading cases from various initial values resulted in a useful empirical relation for res / in the form: ( ) ( ) 1 1 1 (3) 0.28 0. c c res 1 = 69 Similarly, from many unloading numerical results like the typical ones shown in Fig. 3 the following empirical relation was found for the load-displacement behavior: where P P = = P P c res res 0.0331 p = 1.5 ( ) n p (4) n (5) Hence, by using Eqs. (3) together with the proper relations in Eq. (1) or (2) for =, in Eq. (4), it becomes obvious that is indeed the major parameter affecting the contact load behavior during the unloading process. Fig. 4 presents the numerical results from which the relation of Eq. (3), which is also presented in the Figure by the solid line, was derived. As can be seen the residual interference res decreases with a decreasing level of the initial elastic-plastic loading and it completely vanishes for pure elastic loading when 1. On the other hand, as the level of elasticplastic loading increases, the residual interference approaches asymptotically an upper limit.

This upper limit, which is, is obtained at very large initial plastic deformations when the elastic recovery becomes negligibly small compared to the initial. The behavior of the residual interference discussed above is very well depicted in Eq. (3) for the extreme cases of = c and. In the first case Eq. (4) becomes: = 3 ( )2 P (6) describing a purely elastic nonlinear force-displacement unloading process. In the second case the force-displacement behavior during unloading (see Fig. 3) degenerates to: = res = (19) which describes a purely plastic unloading process. Note that the discrete numerical results shown in Fig. 4 were obtained for all the different cases of the sphere material and geometrical properties combinations shown in Table 1. It is very clear from the Figure that the dimensionless residual displacement, ω res /ω, is independent of these physical and geometrical properties. This, once again, demonstrates the powerful universality of the present model. CONCLUSIONS Empirical dimensionless expressions for the dimensionless contact load, and residual displacement during unloading of an elastic-plastic loaded spherical contact were derived for a wide range of loading levels in terms of imum contact interference. These expressions were based on fitting numerical results obtained from FEA for various material and geomrtrical properties of the loaded sphere. The dimensionless model was found to be universal in nature and independent of specific material or radius of the sphere. The main conclusion of the present analysis is that unloading a spherical contact after elasticplastic loading is a non-linear and non-elastic phenomenon. REFERENCES [1] Majumder, S., McGruer, N.E., Adams, G.G., Zavracky, A.P., Zavracky, P.M., Morrison R.H., and Krim, J., Study of Contacts in an Electrostatically Actuated Microswitch, Sensors and Actuators A, 93, 2001, pp. 19-26. [2] Majumder, S., McGruer, N.E., and Adams, G.G., Contact Resistance and Adhesion in a MEMS Microswitch, Proceedings of 2003 STLE/ASME Joint International Tribology Conference, 2003-TRIB-270, 2003, pp. 1-6. [3] Peng, W., and Bhushan B., Transient Analysis of Sliding Contact of Layered Elastic/Plastic Solids with Rough Surfaces, Microsystem Technologies, Vol. 9, No. 5, 2003, pp. 340-345. [4] Johnson, K.L., Contact Mechanics, Cambridge University Press, Cambridge, MA, 1985. [5] Tabor, D., A Simple Theory of Static and Dynamic Hardness, Proceedings, Royal Society, A192, 1948, pp. 247 274.

[6] Mesarovic, S. D., Johnson, K.L., Adhesive Contact of Elastic-Plastic Spheres, Journal of the Mechanics and Physics of Solids, 48(10), 2000, pp 2009-2033. [7] Vu-Quoc, L., Zhang, X., and Lesberg, L., A Normal Force-Displacement Model for Contacting Spheres Accounting for Plastic Deformation: Force-Driven Formulation, ASME Journal of Applied Mechanics, Vol. 67, No. 2, 2000, pp. 363-371. [8] Lin, Y. Y., and Hui, C. Y., Mechanics of Contact and Adhesion between Viscoelastic Spheres: An Analysis of Hysteresis during Loading and Unloading, Journal of Polymer Science. Part B - Polymer Physics, Vol. 40, No. 9, 2002, pp. 772-793. [9] Yan, S. L., and Li, L. Y., Finite Element Analysis of Cyclic Indentation of an Elastic- Perfectly Plastic Half-Space by a Rigid Sphere, Journal of Mechanical Engineering Science, Vol. 217, No. 5, 2003, pp. 505-514. [10] Ye, N., and Komvopoulos, K., Effect of Residual Stress in Surface Layer on Contact Deformation of Elastic-Plastic Layered Media, ASME Journal of Tribology, Vol. 125, No. 4, 2001, pp. 692-699. [11] Li, L. Y., Wu, C.-Y., and Thornton, C., A theoretical model for the contact of elastoplastic bodies, Proc. Instn. Mech. Engrs. 216 (Part C), 2002, pp. 421 431. [12] Chang, W. R., Etsion, I., and Bogy, D. B., An Elastic-Plastic Model for the Contact of Rough Surfaces, ASME Journal of Tribology, 109, 1987, pp. 257-263. [13] Kogut, L., and Etsion, I., Elastic-Plastic Contact Analysis of a Sphere and a Rigid Flat, ASME Journal of Applied Mechanics, Vol. 69, 2002, pp. 657 662. Case Symbol E, GPa σ y, MPa ν R, mm ω c, µm ω c / R E/σ y 1 200 210 0.32 10 0.059 5.9x10-6 952.4 2 + 80 220 0.3 2 0.081 40.5x10-6 363.6 3 69 28 0.3 10 0.0088 0.88x10-6 2464.3 4 80 220 0.3 1 0.0403 40.3x10-6 363.6 5 106 357 0.31 10 0.6053 60.53x10-6 296.9 6 100 105 0.32 10 0.059 5.9x10-6 952.4 Table 1. Materials and geometrical properties combinations

P a ω r z Figure 1. Schematic of the contact problem. ω res Sphere Profile ω R res R Fully Unloaded Residual Profile Fully Loaded Profile a Figure 2. Schematic description of the different elastic-plastic stages.

Dimensionless Contact Load, P/Pc 1000 800 600 400 200 0 Ref. [13] 0 50 100 150 200 Dimensionless Interference, ω /ω c Figure 3. Dimensionless contact load vs. dimensionless interference. Dimensionless Residual Displacement, ω res / ω 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 25 50 75 100 125 150 Dimensionless Interference, ω / ω c Figure 4. Dimensionless residual displacement vs. dimensionless imum interference.