THE VISCOELASTIC CONTACT BETWEEN HIGH-ORDER POLYNOMIAL SURFACES

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1 THE VISCOELASTIC CONTACT BETWEEN HIGH-ORDER POLYNOMIAL SURFACES Sergiu Spinu 1,, Deli Cerlinc 1, 1 Deprtment of Mechnics nd Technologies, Stefn cel Mre University of Sucev, 13th University Street, 79, Romni, e-mil: sergiu.spinu@fim.usv.ro, deli@fim.usv.ro Integrted Center for Reserch, Development nd Innovtion in Advnced Mterils, Nnotechnologies, nd Distributed Systems for Fbriction nd Control (MANSiD), Stefn cel Mre University, Sucev, Romni. Abstrct: The improvement of the lod-crrying cpcity of mchine elements subjected to contct lod cn mke use of high-order surfces, providing centrl plteu of uniform pressure surrounded by peripherl region of pressure continuously decresing to zero. The solution of the viscoelstic contct problem is difficult to obtin becuse (1) both contct re nd pressure distribution re priori unknown, nd () the contct prmeters keep chnging with time. The ltter difficulties re overcome in this pper by conducting numericl nlysis bsed on both sptil nd temporl model discretiztion. The strong points of the newly proposed lgorithm consist in: (1) the bility to incorporte complex models of viscoelsticity, () the cpbility to tret rbitrry contct geometry, nd (3) the cpcity to simulte rbitrry loding histories. The history of pressure distribution, s well s evolution of contct re nd of the rigid-body pproch in the contct between rigid indenter bounded by high-order polynomil surfces nd polymethyl methcrylte viscoelstic hlf-spce, re obtined. These results prove the dvntges of high-order surfces for the contct in the viscoelstic domin. Keywords: numericl simultion, viscoelstic displcement, high-order polynomil surfces 1. Introduction The design of mechnicl contcts involving mchine elements mde by viscoelstic mtrix nd bounded by high-order surfces cn be improved by numericl nlysis. The ltter cn overcome the limittions existing in most nlyticl solutions: (1) the constitutive lw of the contcting mteril is usully elstic, nd () the contct geometry is qudrtic, thus obeying the Hertz contct theory. Hsslinger [1] proved tht the optiml lodcrrying cpcity of the mechnicl contct cn be ttined with uniform pressure over the whole contct re. The ltter configurtion is incomptible with Liner Elsticity, which requires continuity of stresses in order to fulfill the comptibility conditions. The solution is to dmit peripherl region of pressure continuously decresing to zero, bordering centrl plteu of uniform pressure. The ltter pressure distribution cnnot be chieved in the frme of the Hertz contct, which exhibits semi-ellipsoidl pressure distribution. However, the shpe prmeters of indenters bounded by high-order polynomil surfces cn be chosen to fltten the pressure distribution in centrl zone of contct re. The closed-form equtions for the contct involving high-order surfces were developed by Diconescu []. Diconescu nd Glovne [3] further derived the correltion between pressure distribution nd the eqution of high order xisymmetric surfces nd estblished procedure to fltten the pressure distribution in the centrl zone of contct re. More recently, these uthors [4] further derived the indenter equtions which led to n optimized pressure 15

2 distribution in the ellipticl contct, nd provided numericl vlidtion of the theoreticl frmework. Spinu nd Grdinru [5] investigted the lod-crrying cpcity of the contct between high-order surfces under conditions of gross-slip nd prtil-slip. However, ll these reserch efforts were limited to the elstic domin. In this pper, more in-depth study of this remrkble type of surfces is conducted, by nlyzing the pressure developing in the contct between bodies mde by liner viscoelstic mtrix, bounded by fourth nd sixth-order surfces. To this end, technique originlly developed for the elstic contct of rough bodies [6] is combined with method for viscoelstic displcement computtion [7].. The contct of high-order surfces Generl formule governing the elliptic contct of high-order surfces were derived by Diconescu []. The ltter uthor obtined the polynomil punch profiles tht induce the following pressure distributions: n i i i p( ) p 1 c, i 1 c 1, ci ci 1, i (1) where p is the centrl pressure, n the degree of the polynomil expressing the corresponding punch profile, nd y the system coordintes in the common plne of contct, nd x y b, with nd b the contct re hlf-xes. In this frmework, the fourth order polynomil surfce tht induces n xisymmetric pressure distribution, possessing flt centrl pressure plteu surrounded by monotonous decrese to zero, cn be expressed [3] in rdil coordinte system s: 4 3 r r hi( r) p , () x with p the centrl pressure induced by norml lod W, with p 5 W (4 ). The optimized pressure distribution induced by punch of eqution () is [3]: p ( ) p (3) In the sme mnner, n optimized pressure distribution of the form: p 1 3, (4) 4 ( ) p with p 7 W (6 ), results in the indenttion by xi-symmetric punch bounded by sixth-order polynomil surfce: 15 hi() r p r r r Contct model (5) The contct model employed in this pper is similr with the one presented in [8], consisting in three type of equtions: (1) the eqution of the surfce of deformtion between the two bodies, () the boundry conditions, nd (3) the sttic equilibrium. The difficulty in solving the contct model stems from the fct tht neither the contct re, nor the pressure distribution re known in dvnce, nd moreover, keep chnging during the contct process, together with the complince of the viscoelstic mteril. The numericl tretment of the forementioned contct involves therefore sptil discretiztion of the contct surfce, s well s n dditionl temporl discretiztion. The sptil discretiztion employs rectngulr uniform mesh lying in the common plne of contct, on which ll problem prmeters re ssumed piecewise-constnt, bsed on the discrete vlues computed in the control points. The temporl discretiztion ssumes tht the 16

3 [, t] loding window is divided into smll time steps, nd the problem prmeters re ssumed piecewise-constnt in the time dimension s well. The min dvntge of this discretiztion process is the substitution of integrtion of rbitrry functions over rbitrry sptil or temporl domins with summtion, which cn be performed for prescribed input. The nottion of problem prmeters cn then mke use of the discrete indexes covering the discrete surfce ptches, i.e. is the uniform pressure predicted for the ptch of the surfce sptil grid, fter k time steps. Within this frmework, solution for the viscoelstic displcement induced by prescribed, but otherwise rbitrry loding history, t prescribed time in the observtion window, is redily vilble [7]: vs 3 Nt N1 N n1 l 1 m1 1 p( i, j, k) ( i, j) u ( i, j, k) K ( i l, j m, k n) where p( l, m, n) p( l, m, n 1), i 1 K N, j 1 K N, k 1 K N, N 1 nd N vs t (6) re the number of sptil grids nd N t the number of temporl steps. Kvs( i l, j m, k n) denotes the viscoelstic influence coefficient [7], expressing the displcement induced fter k time steps in the sptil cell, by uniform pressure of mgnitude 1 ( 1 ) P, tht cted in the cell th ( l, m) in the time step of the observtion ( i, j) n window, with n k. Eqution (6) proves tht viscoelstic displcement computtion requires the entire history of pressure distribution in the viscoelstic contct. To overcome the memory effect specific to viscoelstic mterils, the contct model is solved successively t every time step, thus ssuring the simultion of the loding history. The contct solver is bsed on the Conjugte Grdient-type scheme originlly dvnced by Polonsky nd Keer [6] for the elstic contct of rough surfces. In the beginning of the observtion window, no loding history is ssumed, so the initil contct stte is clculted s purely elstic process, i.e. the initil displcement is generted by the current pressure only. In the subsequent time increments, the pressure history, which is obtined in step-by-step pproch, is superimposed in the eqution of the surfce of deformtion. The instntneous contct re nd pressure distribution re determined with tril-nderror pproch. A guess contct region is ssumed, nd the resulting pressure distribution is then computed bsed on this ssumption. If ll constrints in the contct model re verified, contct problem solution is chieved. This solution is unique bsed on the theorem of uniqueness of solution for the elstosttic problem. Otherwise, the process is restrted with different initil guess. 4. Results nd discussions The generl formule for the contct of high-order surfces, optimized for centrl plteu of uniform pressure, re first verified numericlly. The pressure profiles in rdil plne, depicted in figure 1, induced by xisymmetric punches of eqs () nd (5), mtch well their nlyticl counterprts (3) nd (4), respectively. Dimensionless pressure is defined s rtio to, the centrl pressure obtined when the sme lod is trnsmitted through the sme contct re in Hertz-type indenttion. Dimensionless rdil coordinte is defined s rtio to the contct re, which is fixed but otherwise cn be rbitrrily chosen. The norml lod, s well s the contct complince, re lso fixed. The pressure profiles in figure 1 prove clerly tht, when the sme norml lod is trnsmitted through the sme contct re, the use of optimized high-order polynomil punches led to more uniform pressure distribution. The ltter is expected to improve the lod-crrying cpcity of the mechnicl contct. p H 17

4 1.8 t= (Hertz) t=.5* t= p / p H.6.4 t=* Figure 1: The elstic contct of high-order surfces The constitutive lw of the viscoelstic mteril employed in this pper is tht of the polymethyl methcrylte (PMMA), thermoplstic polymer whose relxtion modulus under unixil compression in window of observtion of 1 s ws obtined experimentlly by Kumr nd Nrsimhn [9]. The creep complince of PMMA, needed in the computtion of the influence coefficients, results by inverse Lplce trnsform, s described in [1]: K vs 4 5 ( t) exp(.1 t) exp( t), [1/ MP]. (7) r / H Figure : Mxwell hlf-spce sphericl indenttion: continuous lines - nlyticl dt; dshed lines numericl predictions Figure 3 shows the pressure profiles ttined in the indenttion of PMMA viscoelstic hlf-spce indented by rigid fourth-order polynomil punch of eqution (), wheres the results for the sixth-order surfce given by eq. (5), re shown in figure 4. The rdil coordinte is normlized by the contct rdius chieved t the beginning of the loding process (i.e., t distribution by the initil mximum (centrl) pressure. t ), nd pressure The viscoelstic prt of the computer progrm ws benchmrked ginst the implicit solutions derived in the clssicl literture [11-15] of the viscoelstic contct. The step loding sphericl indenttion of viscoelstic hlf-spce described by the Mxwell rheologicl model with the relxtion time is nlyzed in figure. The Hertz contct prmeters (contct rdius H, centrl pressure p H ), re used s normlizers. The pressure profiles chieved t vrious time moments from the loding history gree well with the clssic solution. Figure 3: PMMA fourth-order polynomil punch indenttion 18

5 Figure 4: PMMA sixth-order polynomil punch indenttion The rdil pressure profiles in figures 4 nd 5 suggest tht the centrl plteu of uniform pressure extends with time, while the contct re increses, s depicted in figure 5. Dimensionless contct re is defined s rtio to the initil contct re A. Figure 5: Evolution of contct re The evolution of the rigid-body pproch is presented in figure 6. The norml pproch is normlized by the initil (i.e., t t ) vlue, which is different for ech n. The sme trend s for the contct re cn be observed. Figure 6: Evolution of dimensionless rigid-body pproch 5. Conclusions The simultion of the contct of liner viscoelstic bodies, bounded by high-order polynomil surfces, is chieved in this pper by combining robust solver for the frictionless norml contct with numericl method for the clcultion of the displcement induced in viscoelstic hlf-spce by prescribed but otherwise rbitrry pressure history. The method requires both sptil nd temporl discretiztion. The viscoelstic contct process simultion is ccomplished by computing series of subsequent contct sttes. Code vlidtion is chieved by comprison with the contct of complex geometries in the purely elstic domin, or with tht of simple geometries of viscoelstic bodies described by bsic rheologicl models. The initil plteu of uniform pressure specific to optimized high-order polynomil surfces is conserved nd extended during the loding progrm. Contct re lso increses with time, prticulrly when the degree of the polynomil is smller. The numericl simultions suggest tht the high-order surfces re lso ttrctive for the contct in the viscoelstic domin, conserving close to uniform pressure in the center of the contct re. 19

6 6. Acknowledgement This work ws prtilly supported from the project Integrted Center for Reserch, Development nd Innovtion in Advnced Mterils, Nnotechnologies, nd Distributed Systems for Fbriction nd Control, Contrct No. 671/9.4.15, Sectorl Opertionl Progrm for Increse of the Economic Competitiveness co-funded from the Europen Regionl Development Fund. 7. References [1] Hsslinger, J., 199, Contct shpe optimistion: the mthemticl theory, Proceedings of the Contct Mechnics Interntionl Symposium, A. Curier (Ed.), pp , Presses Polytechniques et Universitires Romndes, Lussne, Switzerlnd. [] Diconescu, E., 6, Elliptic elstic contct between high order symmetricl surfces, ASME J. Tribol., 18(4), [3] Diconescu, E., nd Glovne, M., 5, Flt Centrl Pressure in Circulr Elstic Contcts Between High Order Surfces, Proceedings of WTC5, World Tribology Congress III, September 1-16, 5, Wshington, D.C., USA, Article ID WTC [4] Glovne, M., Spinu, S., nd Diconescu, E., 1, Improved pressure distribution in elliptic elstic contcts between high-order surfces, Advnces in Tribology, Vol. 1, Article ID 83859, 11 pp. [5] Spinu, S., nd Grdinru, D., 13, Numericl Anlysis of Lod Crrying Cpcity in the Contct between High-Order Surfces, Applied Mechnics nd Mterils, Vol. 371, pp [6] Polonsky, I. A., nd Keer L. M., 1999, A Numericl Method for Solving Rough Contct Problems Bsed on the Multi-Level Multi- Summtion nd Conjugte Grdient Techniques, Wer, 31(), pp [7] Spinu, S., nd Grdinru, D., 15, Semi- Anlyticl Computtion of Displcement in Liner Viscoelstic Mterils, IOP Conf. Ser.: Mter. Sci. Eng. 95, 1111 (7 pges). [8] Spinu, S., 15, Numericl Simultion of Viscoelstic Contcts. Prt 1. Algorithm Overview, J. Blk. Tribol. Assoc., 1(), pp [9] Kumr, M. V. R., nd Nrsimhn, R., 4, Anlysis of Sphericl Indenttion of Liner Viscoelstic Mterils, Curr. Sci., 87, pp [1] Chen, W. W., Wng, Q. J., Hun, Z., nd Luo, X., 8, Semi-Anlyticl Viscoelstic Contct Modeling of Polymer-Bsed Mterils, ASME J. Tribol., 133(4), 4144 (1 pges). [11] Lee, E. H., nd Rdok J. R. M, 196, The Contct Problem for Viscoelstic Bodies, ASME J. Appl. Mech., 7(3), pp [1] Hunter, S. C., 196, The Hertz Problem for Rigid Sphericl Indenter nd Viscoelstic Hlf-Spce, J. Mech. Phys. Solids, 8(4), pp [13] Yng, W. H., 1966, The Contct Problem for Viscoelstic Bodies, ASME J. Appl. Mech., 33(), pp [14] Ting, T. C. T., 1966, The Contct Stresses Between Rigid Indenter nd Viscoelstic Hlf-Spce, ASME J. Appl. Mech., 33(4), pp [15] Ting, T. C. T., 1968, Contct Problems in the Liner Theory of Viscoelsticity, ASME J. Appl. Mech., 35(), pp