Analysis of the frictional slip between a layer and a half-space

Analysis of the frictional slip between a layer and a half-space R. Abascal, J. Dommguez Escuela Superior de Ingenieros Industrials, Universidad de Sevilla, Av. Reina Mercedes, s/n, 41012-Sevilla, Spain ABSTRACT The numerical analysis of a boundless elastic layer on an elastic halfspace with different material properties under the effects of an uniform surface pressure and a cyclic tangential surface force is presented. Frictional contact conditions are assumed. The study is focussed on the evaluation of the maximum amplitude of the tangential load which produces localized slip between the two regions during the first load cycle but not the subsequent ones. The more simple limit for which no slip exist even for the first cycle is also established INTRODUCTION The study of the slip and separation that may take place between two surfaces in contact when they are under tangential cyclic loading conditions is very important to prevent the failure known as "fretting". This failure mechanism is initiated by localized slip between two surfaces from which some particles are detached. These particles act as an abrasive in subsequent load cycles and deteriorate the material rapidly. The damage mechanism may be combined with corrosion in the case of metals[l]. The slip between the two surfaces can be avoided by introducing a normal pressure between them. If this pressure is enough the prevent slip during the first load cycle then the system behaves linearly and slip will not take place during subsequent cycles of the same amplitude. This welded contact problem can be solved easily. The limit for which the first slip takes place can be obtained from the welded contact model. It corresponds to the tangential load for which the shear traction at a point becomes equal to the normal pressure times the friction coefficient. There are still higher values of the tangential load for which there is partial slip during the first load cycle but not in the subsequent ones. Therefore no fretting will take place for this Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 210 Boundary Elements c 0 ' 0 0 0 0 0 0 , 0 11111. c, ,, ** *: 1,1 1 1 1 , Q ^FrkHoi Sufeca Figure 1. Elastic layer on elastic half-space under tangential load Q and normal load PO load. The slip of the first cycle leaves residual stresses which prevent the two surfaces to slip during the next cycles. The evaluation of the higher value of the load for which the system has this kind of behavior is important since it is the actual limit of the loads which do not produce fretting. The study requires a more complicated model and a numerical solution approach as shown below. The problem analyzed in this work refers to a semi-infinite domain consisting of a boundless horizontal elastic layer with depth "a" on an elastic half-space. The surface of the layer is subject to a uniform constant in time pressure p<, and to a concentrated tangential load of amplitude Q which has a cyclic time dependence. The variation of the load with time is assumed to be sufficiently slow so that inertial effects are negligible. Therefore, a quasistatic analysis is carried out. The finite region with the boundary conditions shown in Figure 1 is used for the study. The problem at hand remains linear for small values of the load Q. The expressions for the tractions along the interface in such case can be found for instance in Ref.[2]. The study of a similar problem with two materials of identical properties and a single loading process which produces first, slip at a point then, one or two slip zones and finally, separation, was done analytically be Schmueser, Comminou and Dundurs [3]. Comminou and Barber [4] extended this analysis to the case of cyclic loading and assuming a friction coefficient JJL = 0.5. These authors identified a value X, of the Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 211 parameter X = Q/(po a) such that for loads X < X, the problem remains linear, and a load range X, < X < X] for which there is slip but only in the first load cycle. Loads corresponding to X > X? produce multiple slip and/or separation which may become general along the interface during the subsequent load cycles. The purpose of this paper is to carry out a study of these ranges of the load for the more general case when the layer and the half-space have different material properties. The study is done numerically by means of the Boundary Element Method (BEM) which is very well suited for this kind of problems. A parametric study is done using as a parameter to define the relative stiffness of the two materials, the square root of the ratio between the shear modula RCs = (Gj/GJ^, the Poisson's ration being the same for both materials. NUMERICAL APPROACH The analysis of the problem in Figure 1 starts by discretizing the boundaries into constant elements and computing the usual BEM system of equations for the boundary nodes of each one of the two subregions where u* and;/ are the displacement and traction vectors respectively, along the boundary of the "z" subregion. The coupling between the two regions is done by means of the compatibility and equilibrium conditions along the interface corresponding to one of the following situations: bonding, separation or sliding with friction. A Coulomb type friction is assumed. Sliding: Pt = pf u? + w»* = 0 Bonding: P?* I < I M P? I .< . ui + «/ = 0 (3) Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 212 Boundary Elements Separation: P?* - 0 P?' = 0 5 > 0 W where the superindex indicates the region to which the variable refers and the subindexes s and n stand for tangential and normal direction to the interface, respectively; /x is the friction coefficient and 5 the distance along the normal direction between two points which could be in contact. The sign of the shear traction during the sliding is determined by the condition of negative work (dissipation of energy) during the process. In the three cases above four equations can be written for each couple of points on the contact surfaces. Those equations plus the two BEM equations for each domain determine the values of the two displacement components and two traction components for each one of the two points of a couple which are or may be, in contact. The solution of the system of equations requires of an iterative process within each load step. This process for a load step N starts by writing the B.E. equations (1) for each boundary node as where x^ and /„ are the boundary unknown vector and the known vector, respectively. The latter is obtained from the product of the known boundary values and the corresponding columns of the H and G matrix. The system matrix A contains more columns than rows as both tractions and displacements are unknown along the interface. The contact equations for the nodes on the interface are written as CN *N = ° (6) The above system of equations (5) plus (6) allows for the solution of the problem for the load step N provided that the contact conditions for all the points on the interface (represented by €„) remain constant during the load step. Since these conditions will, in general, change during the load step, each load step must include several iterations with different contact Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 213 conditions CN*. For each load step one works with increments of the variables with respect to the previous step f» -/«- 0 = A . (7) The increments of the variables are subdivided into M possible subincrements and written as A *„ = £ A ^* (8) where each sub-increment corresponds to different contact conditions Gj. The iteration process for load step N begins by solving the system (7) assuming the same contact conditions of the end of the previous step and applying the complete load increment A/^. A scale factor /?„' is computed from the solution of this system such that it represents the part of the load which can be applied without change in the contact conditions. This load can be written as and A 4 (9) where Ar* is the solution of (7) with A/*. Next the contact conditions are modified and the load (1 - #/) A/%, applied. The process continues up to the point when /?/ > 1. Then the computation for load step N is finished. A similar iterative process was applied by the authors for dynamic problems in Refs. [6] and [7]. LAYER ON A HALF-SPACE The following values of the parameters have been used for the model in Figure 1: a = 1 m; L = 10 m ; H = 10 m; Poisson's ratio u, = ^ = Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 214 Boundary Elements 1/3 ; po=l Nw/nf ; friction coefficient p = 0.5 and a shear modulus of the half-space 62 = 1 Nw/nf. The parametric study is done by changing the shear modulus of the layer material and the load amplitude Q. The load is assumed to be applied in 100 steps following the variation shown in Figure 2. • It 32 41 64 LaW Sup Figure 2. Tangential load variation Constant Boundary Elements are used for the study. The discretization consists of 50 equal elements on each side of the contact interface and on the layer free surface, 3 equal elements on each lateral boundary of the layer, 5 on each lateral boundary of the half-space and 10 equal elements for the bottom of the model. In order to test this model which extends to a finite region around the load to represent an infinite domain and includes a constant Boundary Element approximation, a problem with known analytical solution is solved. Such solution was obtained by Comninou and Barber [5] for the case of two identical material properties for the layer and the half-space. Figure 3 shows a comparison between the analytical and the present numerical results for the maximum load of the second load cycle (point C of Figure 2) with a load amplitude X = 2.353. Slip between the layer and the half-space takes place for this value of X during the first cycle but not in the subsequent ones. The normal and shear tractions along the interface versus the horizontal distance x to the point load are shown in Figure 3. The normal tractions have a smooth variation and are almost the same as in the linear case. On the other hand, the shear tractions show a clear difference with those of the linear case because of the residual tractions existing in the zones where slip Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 215 took place in the previous load cycle. These residual tractions avoid the sliding in the second and subsequent load cycles. The results presented in Figure 3 show a very good agreement between the analytical and the present numerical solution. - Normal Theor. * * Lo.d r • Normal B.E.M. « {?Sl£ —Shear Theor. ^ « Rd - l • Shear B.EJ*. * . ,.oj -4-3-2' -1 Figure 3. Results for two identical materials 20 18 16 14 I*" O 10 Zone 1 0 Oj 1 lj 2 2J 3 3J 4 4j Ra« Figure 4. Fretting zones Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 216 0.4 0 3 \ 02 4 « •0.4 Boundary Elements -5 -4 3 -2 -1012345 «/* 0.4 035 O " & 025 g 02 I 0.15 3 0.1 * 0.05 0 4X05 LoadC #-03 XX - RCf-0.25 -RCs-0.50 RCm-l -RCs-2 - RC«-4 -5-4-3-2-1012345 0.4 035 0.15 0.1 0.05 0 •0.05 LoftdA P-O5 X • X - RCs-025 RCs-OJO -RCs-2 -RCm-4 5 -4 3 -2 -1012345 %/* OJ5 03 025 02 0.15 0.1 0.05 0 -0.05 5 -4 -3 -2 -1012345 %/a Figure 5. Tractioo5 distribution along the interface Once the B.E. model and' the iterative approach have been validated the parametric study for a range of the stiffness ratio RCs = (GJG^ going from 0 to 4.5 is carried out. For each stiffness ratio two limiting values of the load parameters X = Q/PO a are determined. The first one, X, is the limit of the loads which do not produce non-linear effects on the contact interface. This limit define the linear zone (zone 1) in Figure 4. The second limit X? defines the load zone (zone 2) for which there is slip along the interface Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 217 during the first load cycle but the residual tractions avoid slip in the subsequent load cycles. Values of X > X% (zone 3) correspond to loads which produce slip in all the load cycles and therefore the fretting process can not be controlled. The values of X, and X% obtained analytically by Comninou and Barber [5] for the case of RCs = 1 are shown as dots in the figure. Figure 5 shows some of the traction distributions which appear along the interlace during the loading process. Only in one case are the normal tractions represented (Figure 5a) since those tractions show very little difference with the linear ones for zone 2. The shear tractions represented in Figure 5b correspond to the second cycle (Point C) and X = X,. Therefore, no slip has taken place and the curves in this figure represent the shear traction distribution for a linear problem. Figures 5c and 5d show the shear traction distribution for X = X2 during the first and second load cycles (Points A and C, respectively). In the former case, there have been slip only in the negative part of the x-axis and residual tractions have appeared. It can be seen in Figure 5d that the residual tractions corresponding to point A remain and some more appear along the positive part of the x-axis which have been produced during the negative part of the first load cycle (point B). 2.0 1.5 1.0 7« 3O'O & -OJ -1.0 -1J -zo - RCs-0.25 RCs-0.50 -RCs-1 -RCs-2 -•RCs -4 LoadC ft -0.5 A - \ -5-4-3-2-1012345 x/a Figure 6. Relative tangential displacements along the interface Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X