On the Barber Boundary Conditions for Thermoelastic Contact

Transcription

1 Maria Comninou Associate Professor, Department of Applied Mechanics and Engineering Science, University of Michigan, Ann Arbor, Mich Mem. ASME J. Dundurs Professor, Department of Civil Engineering) Northwestern University, Evanston, III Fellow ASME On the Barber Boundary Conditions for Thermoelastic Contact A dilemma arising from the conventional boundary conditions for thermoelastic was observed by Barber in treating the indentation of an elastic half space by a rigid sphere. If the sphere is colder than the half space, the interface tractions are necessarily tensile near the periphery of the region. In order to overcome this difficulty, Barber introduced the idea of an imperfect zone. An asymptotic analysis of the transitions between the different zones is carried out in this article. It is found that, if heat flows into the body with the larger distortivity, a direct transition from perfect (no resistance to heat flow) to separation (no heat flow) is possible, the zone of imperfect (vanishing pressure and some resistance to heat flow) is automatically excluded, and the heat flux is square-root singular at the transition. If heat flows in the opposite direction, no direct transition from perfect to separation is possible, there must be an intervening zone of imperfect, and the heat flux is logarithmically singular at the transition from perfect to imperfect. The transition from imperfect to separation is always possible, and it is smooth. These conclusions are direct consequences of the inequalities that must be enforced because of the unilateral nature of thermoelastic. 1 Introduction A number of very interesting observations have been made by Barber about the between bodies that conduct heat [1-7]. Among them is a result showing that, if set up in the conventional manner of thermoelastic problems, the indentation of an elastic solid by a cow rigid sphere is not a well-posed problem because it necessarily leads to tensile tractions near the periphery of the region [3]. Moreover, as also shown by Barber, the situation cannot be rectified by assuming an array of annular and separation zones [5, 7]. It may be recalled in this connection that the customary formulation of thermoelastic problems assumes that the interface offers no resistance to heat flow in the zones, and that no heat transfer takes place between the solids in the separation zones. These conditions are clearly the extremes that could be used in idealizing the thermoelastic problem, but they have the advantage of providing the simplest mathematical model that still portrays some physical reality. As might be suspected, the difficulties with the conventional formulation of thermoelastic arise from the discontinuity in the Contributed by the Applied Mechanics Division and presented at the Winter Annual Meeting, New York, N. Y., December 2-7, 1979, of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Discussion on this paper should be addressed to the Editorial Department, ASME, United Engineering Center, 345 East 47th Street, New York, N. Y , and will be accepted until March 1,1980. Readers who need more time to prepare a discussion should request an extension from the Editorial Department. Manuscript received by ASME Applied Mechanics Division, July, Paper No. 79-WA/APM-33. thermal boundary conditions (abrupt change from no resistance to heat flow in the zone to thermal insulation in the separation zone). An ingenious modification of the boundary conditions has recently been proposed by Barber [7]. It pays a penalty in that an intermediate zone is needed, but avoids the nonlinearities'that would come from introducing a pressure-dependent thermal resistance in the zone. (The simple one-dimensional model used by Barber [7] to motivate and explain his boundary conditions will show that a constant thermal resistance does not eliminate the difficulties.) Thus Barber proposes that the zone possibly consists of two parts: a zone of perfect in which the conventional assumption of no thermal resistance holds, and a zone of imperfect in which the pressure vanishes and the interface offers some resistance to heat flow. In the present article, we explore using an asymptotic analysis the transitions from perfect to separation, from perfect to imperfect, and from imperfect to separation. An interesting feature of the analysis is that the functions with powers of the local distance, which are traditionally associated with the Williams technique [8], are not enough to treat the thermoelastic problem. These functions must be supplemented with a set containing logarithmic terms, such as considered by Dempsey and Sinclair [9] and Sinclair [10]. The analysis shows quite convincingly that the Barber boundary conditions work: If the flow of heat is in a direction such that a transition from perfect to separation is possible, a zone of imperfect is automatically excluded. If, in contrast, the direction of heat flow precludes a direct transition from perfect to separation, the transition from perfect to imperfect becomes Journal of Applied Mechanics DECEMBER 1979, OL. 46 / 849 Copyright 1979 by ASME

2 possible indicating that a zone of the latter type will be present. However, in view of the fact that existence and uniqueness theorems for unilateral thermoelastic are not available, the ensuing analysis merely indicates that it is possible to find local solutions consistent with the Barber boundary conditions. 2 Analysis The asymptotic analysis (r 0) of the transitions between the different zones is based on eight sets of elastic fields and four sets of thermal fields. Pour sets of the elastic fields corresponding to complex potentials of the type z 1-x and denoted symbolically as \A], \B\, Cj, and \D\ are given in [11]. The additional four sets \E\, \F\, \G\, and \H\ of elastic fields derived from complex potentials of the form z 1_x log 2 are cataloged in the Appendix of this article. Thus the potential 0(z) [12, Section 37] for body 1 would be written as 4>i(z) = (Ai + (Si)z 1 - x + (E l + if 1 )z l ~ x log z (1) The thermal fields \K\, L), \M\, and \N\ correspond to the complex functions z -x and z _x log z, and they are tabulated in the Appendix. For instance, the temperature in body 1 is written as Kir " x cos A0 + Li/-~ x sin A0 + Mir~ x (log r cos \9 + 6 sin A0) + A~ X flog r sin A0-0 cos A0) (2) \\\\\\'< <««,«v 2 H-2-0.2, k z perfect perfect imperfect Fig. 1 separation imperfect separation Arrangement of the transitions The catalogs allow one to write at sight the simultaneous equations that follow from the boundary conditions. It may be noted that the logarithmic potentials yield dimensionally inconsistent terms containing logr, or the logarithm of a length. This should cause no concern, however, as some characteristic length in an actual problem can always be used to render the argument of the logarithm dimensionless. The placement of the coordinate axes in relation to the two solids and the labeling of the materials are shown in Fig. 1. The shear moduli are denoted by ft, Poisson's ratios by v, the coefficients of thermal expansion by a, and thermal conductivities by k. A key quantity in the analysis is the distortivity of a material [13]. It is defined as for plane strain, and as <x(l + v) 5 = for plane stress. The distortivity of a material is intimately related to the curvature change caused by heat flux [14]. 3 Transition From Perfect Contact to Separation In the region of perfect thermal we insist that the temperature and normal component of heat flux be continuous. The first requirement implies that there is no resistance to heat flux. Moreover, we assume that the is frictionless. Placing the zone of perfect on the negative x-axis, the boundary conditions are TAr, IT) = T 2 (r, -ir) k qe m (r, ir) = q e^(r, u 0 m (r, -ir) ir) = u 9 < 2 '(r, -ir) (Jre w (r, TT) = <r r0^(r, -ir) = 0 <r S g m (r, TT) = <W 2 >(r, -ir) (3) (4) (5) (6) (7) (8, 9) (10) where qe denotes heat flux in the (^-direction. In the region of separation where the bodies are out of, we assume that no heat is transmitted. Placing the separation zone on the positive A:-axis, the boundary conditions are qmr,0) = q,>(r, 0) = 0 <r re (i'(r,0) = <r re <«(r, 0) = 0 (11, 12) (13,14) <r»<f (1) (r, 0) = W 2 >(r, 0) = 0 In addition, attention must be paid to the inequalities N{r) = a ee m {r, ir) = aoo (2) (r, -TT) < 0 in the zone 6 = ±7r, and g(r) = u 0 w (r, 0) - u 0 W{r, 0) > 0 (15,16) (17) (18). in the separation zone 6 = 0. As there are four boundary conditions on the thermal quantities, the temperature distribution is not coupled to the elastic fields in this case, and it can be determined at the outset. Excluding A = 0, since this root leads to a nonintegrable heat flux, (11) and (12) yield L\ =, L 2 = N l = N 2 = 0, and (5), (6) reduce to Mi cos Air = M 2 cos A-ir (19) \ Ki cos Xir + Mi w sin Air - K 2 cos Air + M 2 TT sin Air (20) : Miki sin Air = M 2 k 2 sin \ir (21) Kiki sin Air M\h\ cos Air = K 2 k 2 sin Air + M 2 k 2 cos Air :: (22)1 These equations are contradictory unless either sin AT = 0 or cos Air = 0. The case of sin Air = 0 corresponds to no heat flow through the zone, and is of no interest in the present context. Hence we take immediately t cos Air = 0 (23) J which leads to considerable simplifications in the simultaneous! equations arising from" the mechanical boundary conditions (7) (10).- and (13)-(16). It may be noted that the temperature coefficients enterj these equations through (7). Although the temperature changes beingjj harmonic functions induce no stresses, they distort the surfaces of thej. bodies (see the Appendix). jj Solving the simultaneous equations for the coefficients we get j Ai = A 2 = d = C 2 = 0 B 2 = Si Di = -D 2 = ABi- 2r 1-A Ki (24)1 (25.(26)1 850 / OL. 46, DECEMBER 1979 Transactions of the ASME

3 Ei = E 2.= G 1 = G 2 = Q Fi F 2-2X Hi = Hi = Ki K 2 =-^Ki (27) (28) (29) (30) [15]), because an initial gap that is reasonably smooth analytically cannot interact with terms with singular second derivatives. In contrast, a direct transition from perfect to separation is possible if, for one reason or another, no square-root singularity occurs in the heat flux. This follows by taking the root X = 3/2 of (23), writing the expressions forg(r), N(r), and q(r) corresponding to this root, and adding to g(r) and N(r) the terms with B\ from the rootx = -1/2. where and Li = L 2 = Mr = M 2 = Ni = N 2 = 0 ki(s x ) Q Kl + 1 K2+ 1 Ml M2 (31) (32) (33) with K = 3 Av for plane strain. The coefficients B\ and Ki remain free. Next we compute the gap g(r) = u 9 d)(r, 0) - u«< 2 >(r, 0) Wr 1 Ki log r + fli in the separation zone, and the normal tractions N(r) = o-08 (1) (r, TT) = <r 0(,< 2 >(/-, -w) = 2r x sin XT Ki log r + K 1 + B 1 (l-\) (34) (35) in the zone. The heat flux in the positive y-direction through the zone is q{r) = ~qo a Hr, w) -qs y "(r, 7r) = k{kk\r l x sin XTT (36) It is seen from (36) that it is necessary to have X < 0 as, otherwise, the heat flux is not integrable. This precludes a singularity in N(r) and, moreover, the pressure vanishes at the end of the zone. Equations (36) and (23) show that the heat flux may be square-root singular at the transition from perfect thermal to separation. In fact this corresponds to the general case, and we consequently concentrate on the root X = 1/2 of (23). For this value of X, *(r) = %Qr"«-JXxlogr + Bi (37) N(r) = -3r 1>' 1 ' 2 - fld log r + - Ki + Bi (38) q(r) = -yji^ir- 1 ' 2 (39) The direction of heat flow is determined by the sign of Ky.Ki < 0 corresponds to heat flowing into the upper solid, and K\ > 0 to heat transfer in the opposite direction. The two materials can always be labeled so that 5i > 5 2, which makes f > 0 (S r = o 2 is exceptional in that there will be no restriction on the direction of heat flow). Noting that the dominant terms for small r in (37) and (38) are those with the logarithm, and that log r < 0 near the transition point, it follows that Ki must be negative in order to satisfy either of the inequalities (17) and (18). The conclusion is therefore that a direct transition from perfect thermal to separation is possible only if heat flows into the material with the larger distortivity. This observation is precisely the same as what follows from the specific example worked by Barber [3]. It is interesting to note, however, that the difficulty with heat flow in the wrong direction is not only that the interface tractions become tensile, but that one also has overlapping of material. The conclusion about the direction of heat flow remains valid also if there is an initial gap between the bodies (advancing 4 Transition From Perfect to Imperfect Contact If the zone of perfect is placed as before on the negative x-axis, the pertinent boundary conditions in this zone are (5)-(10), supplemented by the inequality (17). The Barber boundary conditions for the zone of imperfect, placed on the positive x-axis are q,m(r, 0) = <7 (2 >(r, 0) (40) u ^(r, 0) = * <«(/, 0) (41) cr r e a Hr, 0) = <r r9 < 2 >(/-, 0) = 0 (42, 43) <Joo {1) {r, 0) = <W 2) (r, 0) = 0 (44, 45) In addition, the following inequality must be obeyed: [T^r, 0) - T 2 (r, 0)]q (r, 0)< 0 (46) These conditions are nothing but obvious generalizations of the set proposed by Barber in his paper discussing problems that involve a cold rigid punch [7], The physical interpretation of (44)-(46) is that the pressure is negligibly small, the resistance to heat flow is considerable, and that consequently the surfaces of the contracting bodies are at different temperatures. The inequality ensures that the direction of heat flow is consistent with the difference in surface temperatures. It is interesting to note in the present case that (5), (6) and (40) only provide three conditions on the thermal quantities, and hence the heat flow problem is indeterminate without the mechanical conditions. The simultaneous equations resulting from the boundary conditions have a nontrivial solution for the coefficients only for sin XTT = 0 (47) Of interest are only the roots X < 0, because X > 0 gives nonintegrable heat flux at the interface. The solution of the simultaneous equations then yields the following results for the coefficients: d = -(2 - \)A h C 2 = -(2 - X)A 2 (48, 49) IK Di = XBi + Ji, D 2 =\B 2 Ji (50, 51) 1 A 1 A Ei E 2 Gi G 2 0 (52) Fi = -F 2 = Ji (53) Hi = -H 2 2Xf N x (54) K 2 = *fi- L 2 M x = -(»9 _ki U ' k 2 M 2 = 0 _ki N 2 -- Ni ' k 2 (55) (56) (57) (58) where Ai, A 2, Bi, Ki, L\, and Ji are free coefficients and B 2 is determined by «2 + l n KX + 1, e t (5i ) ( T 1.. \, cn, B 2 = Bi + : ; \Li-z rji (59) M2 Ml Ni Journal of Applied Mechanics DECEMBER 1979, OL. 46 / 851

4 The physical quantities of interest are the normal tractions in the zone of perfect (Jos(r, ±ir) = iir^nir the temperatures at the interface and the heat fluxes x cos X T T(r, ±TT) = (Ki - 7rii);-- x cos X T 2 (r, 0) = T^r, 0) = K ir ~ x Ki -4 + r r q 0 (r, ±7r) = ki[nick log r - 1) + XLjr"*" 1 cos XTT qo(r, 0) = -fci[^i(x log r - 1). + XLi]/-"*" 1 The dominant terms correspond to X = 1, which gives a 0 o(r, ±ir) = -Air^N^ Ti(r, 0) - T 2 (r, 0) = ir ^ ixr q»(r,0) = Aiiilogr (60) (61) (62) (63) (64) (65) (66) (67) (68) The materials can again be labeled so that di> & 2 and f > 0. Since log r < 0 in the vicinity of the transition, the inequality (46) is satisfied for both directions of heat flow. From (68) it follows that Ji < 0 corresponds to heat flowing into the upper solid and ii > 0 to flow in the opposite direction. However, the inequality (17) is satisfied only for Ni > 0. The conclusion is, therefore, that the transition from perfect to imperfect is possible only if heat flows into the material with the smaller distortivity. It is also interesting to note that the flux has a logarithmic singularity at the transition from perfect to imperfect. 5 Transition From Imperfect Contact to Separation The boundary conditions for the two zones have already been given in the previous sections. Note, however, that the zone of imperfect is now placed on the negative x -axis (Fig. 1). The Barber inequality is then [Ti(r, TT) - T 2 (r, -w)]q (r, w) > 0 (69) A nontrivial solution for the coefficients is possible only if sin Xir = 0 (70) with X > 0 excluded as before. The inequality (69) is violated if logarithmic terms are retained in the temperature. Thus the nonvanishing coefficients are C 1 = -(2-\)A 1, C 2 = -(2-A)A 2 (72,73) Di = XBi, D 2 ~\B 2 (74,75) The coefficients Ai, A% B\, K\, and K<L remain free. Using these coefficients, it is found that the gap and the heat flux vanish asymptotically, and that none of the field quantities have discontinuous derivatives at the transition from imperfect to separation. Acknowledgment The work by one of the authors (MC) was partially supported by the National Science Foundation under the Grant ENG References 1 Barber, J. ft., "Thermoelastic Instabilities in the Sliding of Conforming Solids," Proceedings of the Royal Society, London, ol. 312, Series A, 1969, pp Barber, J: R., "The Effect of Thermal Distortion on Constriction Resistance," International Journal of Heat and Mass Transfer, ol. 14, 1971, pp Barber, J. R., "Indentation of the Semi-infinite Solid by a Hot Sphere," International Journal of Mechanical Sciences, ol. 15, 1973, pp , 4 Barber, J. R., "Comments on Frictionally Excited Thermal Instabilities," Wear, ol. 26, 1973, pp Barber, J. R., "The Effect of Heat Flow on the Contact Area Between a Continuous Rigid Punch and a Frictionless Elastic Half-Space," Zeischrift fur angewandte Mathematik und Physik, ol. 27,1976, pp Barber, J. R., "Some Thermoelastic Contact Problems Involving Frictional Heating," Quarterly Journal of Mechanics and Applied Mathematics, ol. 29, 1976, pp Barber, J. R., "Contact Problems Involving a Cooled Punch," Journal of Elasticity, ol. 8,1978, pp Williams, M. L., "Stress Singularities From arious Boundary Conditions in Angular Corners of Plates in Extension," ASME JOURNAL OF APPLIED MECHANICS, ol. 19,1952, pp Dempsey, J. P., and Sinclair, G. B., "On the Stress Singularities in the Plane Elasticity of the Composite Wedge," Journal of Elasticity, in press. 10 Sinclair, G. B., "On the Singular Eigenfunctions for Plane Harmonic Problems in Composite Regions," ASME JOURNAL OF APPLIED MECHANICS, in press. 11 Comninou, M., "The Interface Crack," ASME JOURNAL OF APPLIED MECHANICS, ol. 44,1977, pp Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhoff, Groningen, Dundurs, J., and Panek, C, "Heat Conduction Between Bodies With Wavy Surfaces," International Journal of Heat and Mass Transfer, ol. 19, 1976, pp Dundurs, J., "Distortion of a Body Caused by Free Thermal Expansion," Mechanics Research Communications, ol. 1,1974, pp Dundurs, J., "Properties of Elastic Bodies in Contact," in The Mechanics of the Contact Between Deformable Bodies, eds., de Pater, A. D., and Kalker, J. J., Delft University Press, Delft, APPENDIX Elastic Fields With Logarithmic Terms. The elastic fields required in the asymptotic analysis are associated with the complex potentials of the type z 1_x and z 1-x log z. The first of them were cataloged in a previous publication [11]. The second four sets using the same notation as employed by Muskhelishvili [12, Section 37] are \E\: 0(z) = 2 1 -Mog2, }fr(z) = 0 (76,77) 2nu r = r l - x \[{K X) log r - 1] 2/iuo = r 1 - x [-(/c X) log r - 1] sin X0 a rr = r" x [(l - X)(2 + X) logr X] + (K X)0 sin X0) (78) + U X)0 ) (79) + (1 - X)(2 + X)0 sin X0) (80) nri, = r-* [X(l - X) log r X] sin X0 - X(l - X)0 ) (81) (Too = r- x \[(l - X)(2 - X) log r + 3-2X] + (1 - X)(2 - X)0 sin X0 (82) \F\: 0(2) = iz 1 - x log2, 4>(z) = 0 (83,84) 2>xu r = r^mk X) log r - 1] sin X0 - (* X)0 (85) 2nu = r^kk X) log r + 1] + (K X)0 sin X0j o-rr = r- x \ [(1 - X)(2 + X) log r X] sin X0 (86) -U-X)(2 + X)0cosX0! (87) rll = r-* [_x(i - X) log r + 1-2X] - X(l - X)0 sin X0 am = r- x \[{l - X)(2 - X) log r + 3-2X] sin X0 (88) - (1 - X)(2 - X)0 ) (89) G : </>(*)= 0, \p{z) = z 1 ~ x log z (90,91) iu r = /-i-^-log r cos (2 - X)0 + 0 sin (2 - X)0] (92) 2/iu e = r l -*[log r sin (2 - X)0 + d cos (2 - X)0] (93) 852 / OL. 46, DECEMBER 1979 Transactions of the ASME

5 0rr = r~ x \[-(l - X) log r - 1] cos (2 - X)0 ^«= r- x [(l - X) log r + 1] sin (2 - X)0 <r»«= f- x [(l - X) log r + 1] cos (2 - X)0 + (1 - X)0 sin (2 - X)0 (94) + (1 - X)0 cos (2 - X)0 (95) - (1 - X)0 sin (2 - X)0 (96) \H): d>(z) = 0, ^(z) = iz 1 -* log z (97) 2/iu r = r 1 " x [log r sin (2 - X)0 + 0 cos (2 - X)0] (98) 2nu 0 = r 1 " x [log r cos (2 - X)0-0 sin (2 - X)0] (99) o> r = /-- x j[(l - X) log r + 1] sin (2 - X)0 + (1 - X)0 cos (2 - X)0 (100) (Tro = '-" x [(l - A) log r + 1] cos (2 - X)0 - (1 - X)0 sin (2 - X)0 (101) am = r- x \[-(l - X) log r - 1] sin (2 - X)0 - (1 - X)0 cos (2 - X)0 (102) Temperature Fields With Corresponding Displacements. The temperature distributions needed in the asymptotic analysis correspond to the functions z~ x and z~ x log z in complex variables. The components of heat flux follow from the Fourier law of heat conduction The thermal strains 9i = -kd t T eij= titbij, i,j = 1,! (103) derived from these temperature distributions are compatible and can be integrated for displacements u,\ Herein for plane strain, and 7] = a(l + v) r] = a (104) (105) for plane stress. The four sets of fields derived on this basis are \K\: ID: \: \M\: m T = r' x q r = k\r _1 ~ x qo - k\r~ 1_x sin X0 u r = r 1 x 1 X un = r 1 ~ x sin T = r~ x sin X0 q r = k\r~ 1 ~ x qo = k\r~ 1 ~ x sin X0 u r = r 1_x sin X0 uo = r l ~ x T = r~ x (log r + 0 sin X0) q r = kr- l - x {(\ log r - 1) cos'x0 + X0 sin X0] qo = kr- l - x [(\ log r - 1) sin X0 - X0 ] Ur = log r Ug = ) log r -I 1 sin X0 + 0 i-xi T = r-mlog r sin X0-0 ) : sin X0 1 X, q r = kr- l ~ x [(\ log r - 1) sin X0 - X0 ] qo = fer- 1 " x [(-X log r + 1) - X0 sin X0] uo ; logr 1, sin X sin X0 (106) (107) (108) (109) (110) (111) (112) (113) (114) (115) (116) (117) (118) (119) (120) (121) (122) (123) (124) (125) Journal of Applied Mechanics DECEMBER 1979, OL. 46 / 853