The Contact Between Smooth Surfaces 55 36. 37. Mossakovski, V. I., “Compression of Elastic Bodies Under Conditions of Adhesion,” PMM, 1963, Vol. 27, p. 418. Pao, Y. C., Wu, T. S. and Chiu, Y. P., “Bounds on the Maximum Contact Stress of an Indented Layer,” Trans. ASME Series E, Journal of Applied Mechanics, 1971, Vol. 38, p. 608. Sneddon, I. N., “Boussinesq’s Problem for a Rigid Cone,” Proc. Cambridge Philosphical Society, 1948, Vol. 44, p. 492. Vorovich, 1. I., and Ustinov, I. A., “Pressure of a Die on an Elastic Layer of Finite Thickness,” Applied Mathematics and Mechanics, 1959, Vol. 23, p. 637. 38. 39. 3 Traction Distribution and Microslip in Frictional Contacts Between Smooth Elastic Bodies 3.1 INTRODUCTION Frictional joints attained by bolting, riveting, press fitting, etc., are widely used for fastening structural elements. This chapter presents design formulae and methods for predicting the distribution of frictional forces and micro- slip over continuous or discrete contact areas between elastic bodies sub- jected to any combination of applied tangential forces and moments. The potential areas for fretting due to fluctuation of load without gross slip are discussed. The analysis of the contact between elastic bodies has long been of considerable interest in the design of mechanical systems. The evaluation of the stress distribution in the contact region and the localized microslip, which exists before the applied tangential force exceeds the frictional resis- tance, are important Factors in determining the safe operation of many structural systems. Hertz [l] established the theory for elastic bodies in contact under normal loads. In his theory, the contact area, normal stress distribution and rigid body approach in the direction of the common normal can be found under the assumption that the dimensions of the contacting bodies are significantly larger than the contact areas. Various extensions of Hertz theory can be found in the literature [2-151, and the previous chapter gives an overview of procedures for evaluating the area of contact and the pressure distribution between elastic bodies of arbi- trary smooth surface geometry resulting from the application of loading. 56 Traction Distribution and Microslip in Frictional Contacts 57 An important class of contact problems is that of two elastic bodies which are subjected to a combination of normal and tangential forces. The evaluations of the traction distribution and the localized microslip on the contact area due to tangential loads are important factors in deter- mining the safe operation of many structural systems. Several contributions are available in the literature which deal with the analytical aspects of this problem [16-191. The contact areas considered in all these studies are, how- ever, limited to either a circle or an ellipse, and a brief summary of the results of both cases is given in the following section. This chapter also presents algorithmic solutions which can be utilized for the analysis of the general case of frictional contacts. Three types of interface loads are to be expected: tangential forces, twisting moments, and different combinations of them. When the loads are lower than those neces- sary to cause gross slip, the microslip corresponding to these loads may cause fretting and surface cracks. The prediction of the areas of microslip and the energy generated in the process are therefore of considerable interest to the designer of frictional joints. 3.2 TRACTION DISTRIBUTION, COMPLIANCE, AND ENERGY DISSIPATION IN HERTZIAN CONTACTS 3.2.1 Circular Contacts As shown in Fig. 3.1, when two spherical bodies are loaded along the common normal by a force P, they will come into contact over an area with radius a. When the system is then subjected to a tangential force T < fP, Mindlin’s theory [ 161 for circular contacts defines the traction dis- tribution over the contact area and can be summarized as follows: 113 a* = a( I -;) F,, = 0 over the entire surface 58 Chapter 3 lp 1 -T Figure 3.1 The contact of spherical bodies. where F,, l$ = traction stress components at any radius p a = radius of a circular contact area p = (x2 +y2)'I2 = polar coordinate of any point within the contact area T = tangential force P = normal load f = coefficient of friction G = shear modulus of the material U = Poisson's ratio a* = radius defining the boundary between the slip and no-slip regions Figure 3.2 illustrates the traction distribution as defined by Eqs (3.1) and (3.2). It can be seen that a* = a for T = 0 and no microslip occurs; U* = 0 for T = fP and the entire contact area is in a state of microslip and impending gross slip. Traction Distribution and Microslip in Frictional Contacts 59 I t 1 a* fqo - a 6- PQO i I I b a* a T P Figure 3.2 Traction distribution for circular contact (90 =maximum contact pressure = 3P/(2rra2)). The deflection S (rigid body tangential movement) due to the any load T >fP can be calculated from: Consequently, at the condition of impending gross slip, T =fP: (3.4) The traction distribution and compliance for a tangential load fluctuating between fT* (where T* cfP) can be calculated as follows (see Figs 3.3 and 3.4): 60 Chapter 3 P Figure 3.3 Traction distribution for decreasing tangential load T -= T*. h* = inner radius of slip region U* = inner radius of slip region at the peak tangential load T* I /z F, = -jiqO[ 1 - (!)‘I F, = +,,[ 1 - (5)2]”2+2j4!3 [ 1 - 02] U* 5 p 5 b* F, = -JqO [ 1 - (92]1f2 - +2jq0 . (:)[ - 1 - (;)*]”’-fq0@[l - h* 5 p 5 a 2 10 - (5) ] P < a* Traction Distribution and Microslip in Frictional Contacts 61 Figure 3.4 Hysteresis loop. where 3P yo = maximum contact pressure = - 2na2 The deflection can be calculated from: Sd = deflection for decreasing tangential load - 3(2 - u)fP [ 2 ( I T* - T)2'3 - ( I 7'7')2'3] - 16Ga 2fP for T decreasing from T* to -T*; Sj = -6&T) -_ - 3(2 -8)fP[ 2 ( 1 T* + q" ( T*)2'3 - - l-fp 16Ga 2fP for T increasing from -T* to T*. 62 Chapter 3 The frictional energy generated per cycle due to the load fluctuation can therefore be calculated from the area of the hysteresis loop as: M’ = work done as a T’ = I (Sd - &)dT - T’ result of the microslip per cycle (3.5) ( T*)5’3 ST* [ 1- ( I T*)2’3]] - I-fp 6fP fP 3.2.2 Elliptical Contacts A similar theory was developed by Cattaneo [17] for the general case of Hertzian contacts where the pressure between the two elastic bodies occurs over an elliptical area. Cattaneo’s results for the traction distribution in this case can be summarized as follows: on slip region F,, = 0 for the entire surface where a, b = major and minor axes of an elliptical contact area a*, b* = inner major and minor axes of the ellipse defining the boundary between the slip and no-slip regions 3.3 ALGORITHMIC SOLUTION FOR TRACTION DISTRIBUTION OVER CONTACT AREA WITH ARBITRARY GEOMETRY SUBJECTED TO TANGENTIAL LOADING BELOW GROSS SLIP This section presents a computer-based algorithm for the analysis of the traction distribution and microslip in the contact areas between elastic Traction Distribution and Microslip in Frictional Contacts 63 bodies subjected to normal and tangential loads. The algorithm utilizes a modified linear programming technique similar to that discussed in the previous chapter. It is applicable to arbitrary geometries, disconnected con- tact areas, and different elastic properties for the contacting bodies. The analysis assumes that the contact areas are smooth and the pressure distri- bution on them for the considered bodies due to the normal load is known beforehand or can be calculated using the procedures discussed in the previous chapter. 3.3.1 Problem Formulation The following nomenclature will be used: x, y = rectangular coordinates of position U, v = rectangular coordinates of displacement in the x- and y-directions respectively E = Young’s modulus v = Poisson’s ratio G = modulus of rigidity P = applied normal force T = applied tangential force f = coefficient of friction N = number of discrete elements in the contact grid Fk = discretized traction force in the direction of the tangential force uk = discretized displacement force in the direction of the tangential F,., F,: = rectangular components of traction on a contact area at any point k force at any point k force at point k point k ylk = displacement slack variables in the direction of the tangential yuc = force slack variables in the direction of the tangential force at The contact area is first discretized into a finite number of rectangular grid elements. Discrete forces can be assumed to represent the distributed shear traction over the finite areas of the mesh. Since the two bodies in contact 64 Chapter 3 obey the laws of linear elasticity, the condition for compatibility of defor- mation can therefore be stated as follows: uk = p uk < p in the no-slip region in the slip region where the difference between the rigid body movement /? and the elastic deformation uk at any element in the slip region is the amount of slip. The constraints on the traction values can also be stated as: Fk <f’Pk Fk =fPk in the no-slip region in the slip region (3.7) where Fk = the discretized traction force in the x direction at any point k Pk = the discretized normal force at any point k f’ = the coefficient of friction The condition for equilibrium can therefore be expressed as: Introducing a set of nonnegative slack variables Ylk and Y2k, Eqs (3.6) and (3.7) can be rewritten as follows: where )‘I, = 0 ?‘lk ’ 0 where Y2k ’ 0 U,, = 0 Uk + Y,k = #? in the no-slip region in the slip region Fk 4- Y2k = fpk in the no-slip region in the slip region (3.9) (3.10) Since a point k must be either in the no-slip region or in the slip region, therefore: [...]... contact A comparison between Mindlin's theory, which is discussed in Section 3.2 (solid line) and the numerical results (symbol s) obtained by the modified linear program is shown in Fig 3.6 and good agreement can be seen The rigid body movement (0.6 619 6 x 10 -4 in. ) was also found to compare favorably with Mindlin's prediction (0.67 13 9 x 10 -4 in. ) with a deviation of 1. 41 O/O EXAMPLE 2: Circular Hertzian... normal and tangential loading are used here: Case 1 P I = lO,OOOpsi, P2 = 10 ,OOOpsi and the tangential load is 800 lbf applied at 45 " inclination Case 2 Pi= 20,OOOpsi, P2 = 10 ,OOOpsi and the tangential load is 800 lbf applied at 45 " inclination Case 3 P I = 20,OOOpsi, P2 = 10 ,OOOpsi and the tangential load is 12 00 lbf applied at 45 " inclination The coefficient of friction on both regions 1 and 2 is... loo0 11 00 12 00 Tr-actionContour Diutribution 700 Wit+ 900 11 00 13 00 15 00 17 00 19 00 210 0 f2= 0 .12 p2= 20000 pui 2300 Figure 3 .1 7 Traction distribution for contact on two discrete square areas under an 8001bf tangential load applied at a 45 " inclination P I = 20,00Opsi, P2 = 10 ,000psi (Case 2) from the center of rotation in the no-slip region and less than that in the slip region Because the bodies in. .. 2701bf, a tangential load of 361bf, and a coefficient of friction of 0.2 are used in this case As shown in Fig 3.7, the traction distribution (symbol s) shows good agreement with Mindlin's theory (solid line) The rigid body movement of the rubber half space (0.3 946 9 x 10 - 3in. ) was found to be 30 times that of the steel sphere (0 .13 156 x 10 - 4in. ) and both agree well with Mindlin's prediction with a 2.29% deviation... slip region with increasing tangential load and the rigid body movement is shown in Figs 3 . 14 and 3 .15 , respectively EXAMPLE 5: Discrete Contact Area on Semi-Infinite Bodies Two disconnected square areas of the same size (0. 6in x 0. 6in) on semi-infinite steel bodies are in contact with uniform pressures assumed on each contact region The centroids of the two squares are placed 1. Oin apart i Traction... Contact area alb = 2.0 a / b = 0.5 a / b = 8.0 a / b = 0 .12 5 Number of grids Resulting figure 11 2 I12 11 6 I16 Fig 3.8 Fig 3.9 Fig 3 .10 Fig 3 .11 x 10 3 0.0 0.2 0 .4 0.6 0.8 [i-] Normalized Distance 1. o Figure 3.8 Traction distribution on the elliptical contact as compared with Cattaneo’s theory (a/b = 2) x 10 3 0.0 0.2 0 .4 [,/m] 0.6 Normalized Distance 0.8 1. o Figure 3.9 Traction distribution on the elliptical... Distribution + 43 0 psi 525 800 675 750 825 m o 975 10 80 11 25 12 00 Figure 3 .12 Contour plot of traction distribution on a uniformly pressed square contact area with T = 8001bf Chapter 3 i Trac on i Contour ; Distribution + ; s30p.i 687 744 8 01 : : i s s s : i : i i 91s 972 10 29 14 3 8 8 Figure 3 .13 Contour plot of traction distribution on a uniformly pressed square contact area with T = 10 001bf Three conditions... Contacts EXAMPLE 4: Square Contact Area on Semi-Infinite Bodies with Uniform Pressure Distribution A hypothetical square contact area between two steel bodies with uniform contact pressure of 10 ,000psi and a coefficient of friction, f = 0 .12 , is discretized with 10 0 square grid elements The equal traction contours are shown in Figs (3 .12 ) and (3 .13 ) for a tangential force, T = 8001bf and 10 001bf, respectively... Distribution and Microslip in Frictional Contacts 3.3.2 General Model for Elastic Deformation Since both bodies are assumed to obey the laws of elasticity, the elastic deformation uk at a point k is a linear superposition of the influences of all the forces 4 acting on a contact area Accordingly: (3 .12 ) j= 1 where a y = the deformation in the x-direction at point k due to a unit force at point j The discrete... Illustrative Examples EXAMPLE 1: Circular Hertzian Contact with Similar Materials The first application of the developed algorithm is finding the traction distribution over the contact area of a steel sphere of 1 in radius on a steel half space The normal load is taken as 216 01bf, the tangential load is 14 41 bf and the coefficient of friction is 0 .1 A grid with 80 elements is used in this case to approximate . The rigid body movement (0.6 619 6 x 10 -4 in. ) was also found to compare favorably with Mindlin's prediction (0.67 13 9 x 10 -4 in. ) with a deviation of 1. 41 O/O. EXAMPLE 2: Circular. = 2.0 a/b = 0.5 a/b = 8.0 a/b = 0 .12 5 11 2 Fig. 3.8 I12 Fig. 3.9 11 6 Fig. 3 .10 I16 Fig. 3 .11 x 10 3 0.0 0.2 0 .4 0.6 0.8 1 .o [i-] Normalized Distance Figure 3.8. (solid line). The rigid body movement of the rubber half space (0.3 946 9 x 10 - 3in. ) was found to be 30 times that of the steel sphere (0 .13 156 x 10 - 4in. ) and both agree well with Mindlin's