The Penny-Shaped Interface Crack With Heat Flow

Transcription

1 C.J. Martin-Moran Department of Mechanical Engineering, University of Newcastle-Upon-Tyne, NE1 7RU, England J. R. Barber Department of Mechanical Engineering and Applied Mechanics. M. Comninou Department of Civil Engineering. Mem. ASME University of Michigan, Ann Arbor, Mich The Penny-Shaped Interface Crack With Heat Flow Part 1: Perfect Contact A solution is given for the thermal stresses due to a penny-shaped crack at the interface between dissimilar materials loaded in tension for the case the heat flux is into the material with higher distortivity. Regions of separation and perfect thermal contact are developed at the crack faces. A harmonic potential function representation is used to reduce the problem to a three-part boundary value problem which is formulated as a pair of coupled Abel integral equations using the method of Green and Collins. These equations are further reduced to a single Fredholm equation which is solved numerically. Results are presented illustrating the effect of heat flux and applied tractions on the contact radius and the stress intensity factors for various combinations of material constants. The effect of heat flux is profoundly influenced by the relative signs of Dundurs constant 0 and a constant y describing the mismatch of distortivities. If the more distortive material is also the more rigid, the contact region at the crack face is reduced by heat flow; otherwise it is increased. In the latter case, solutions involving separation are obtained even for applied compressive tractions if the latter is within a certain range. The solution also exhibits nonuniqueness in this range. Introduction If a body conducting heat contains a crack, the crack will act as an obstruction to the heat flux, producing a local perturbation in the temperature field [1, 2]. Thermal stresses will therefore be developed, the stress field being influenced by the inability of the crack to transmit noncompressive tractions [3-7]. If the crack occurs at an interface between dissimilar materials, further complications are introduced - we anticipate the development of contact regions between the crack faces [8] and for one direction of heat flow, regions of "imperfect contact" may be required [9, 10]. In certain circumstances, these contact regions are small, but they have a substantial influence On the stress intensity factor at the crack tip and hence on the mechanics of fracture. The only thermoelastic interface crack problem that has been treated to date is the axisymmetric external crack [11] which may be regarded as two dissimilar semi-infinite bodies bonded over a circular region of their common plane. In this paper we consider the related problem of the penny-shaped crack in which the two bodies are bonded throughout the plane except in a circular region. Contributed by the Applied Mechanics Division for presentation at the 1983 Applied Mechanics, Bioengineering, and Fluids Engineering Conference, Houston, Texas, June 20-22, 1983 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 two months after final publication of the paper itself in the JOURNAL OF APPLIED MECHANICS. Manuscript received by ASME Applied Mechanics Division, May Paper No. 83-APM-10. Copies will be available until February, Statement of the Problem The penny-shaped interface crack in a uniform tension field was treated by Keer et al. [12], who showed that the crack opens except for a small annular region of contact adjacent to the crack tip. If the crack is first loaded in tension and then subjected to a heat flux, it seems reasonable to anticipate that a similar configuration would be maintained with a possible change in the extent of the contact region. The geometry is illustrated in Fig. 1. The interface is taken as the plane z = 0 in cylindrical polar coordinates, the crack extending over the region 0<r<b. We assume that separation occurs in a central circular region 0<r<a and that the faces of the crack are in frictionless contact in the annulus a<r<b. The significant material properties of the two half spaces are coefficients of thermal expansion a, thermal conductivity k, modulus of rigidity /x, and Poisson's ratio v and we use subscripts 1, and 2 to refer to half spaces z>0, z<0, respectively. A uniform heat flux, q z =q 0, and a uniform tension, a zz = <j 0, are imposed distant from the crack. From an asymptotic analysis [10], we know that the transition from perfect thermal contact to separation is only possible if the heat flows into the more distortive material - i.e., (?0(o )>0 the distortivity 8 is defined by [13] a(l + c) For the opposite direction of heat flow, we anticipate that an annulus of imperfect contact will be interposed between (1) (2) Journal of Applied Mechanics MARCH 1983, Vol. 50/29 Copyright 1983 by ASME

2 1 t t t I t I )' I I I I I I 1 i4r,z) = w\(r,z) = w 2 (r,-z), 4>(r,z) 4i(r,z) = -<t> 2 (r,-z), (7) \p(r,z) = i,(r,z) = -fa(r,-z). The remaining conditions at the interface define a mixed boundary value problem for the functions w, <, i/-, in the half space z>0. Boundary Conditions In the separation region (0<r<a), the heat flux and the normal stress both contain a component from the uniform solution that must be cancelled by the perturbation. We therefore require d 2 w dz 2 a 0-0</-<a, z = 0, (8) d 3^ ~dz^ = q 0; Q<r<a, z = 0, (9) I I \ I I I I I I 1 I I I I' Fig. 1 Geometry of the problem separation and perfect contact regions [9, 10]. In the present paper we restrict attention to the simpler case of perfect contact condition (1) is satisfied. The solution with imperfect contact will form the subject of a second paper. Mathematical Formulation Following Keer et al. [12], we construct the solution as the sum of two parts - a state of uniform tension and heat flux and a perturbation stress and temperature field due to the crack that tends to zero at large distances from the origin. We note that even without the crack, a uniform tension and heat flux will generally require nonzero radial and circumferential stresses (o rr,<j m ) because of differential thermal expansion and a difference of Poisson's ratio. As in [11], we represent the perturbation field in each half space in terms of three harmonic functions, 4>, $, ", the displacement and temperature being (1 "), z V0+ V 2fi d<$> ~dz (3-4i>) 2/x d4> 3$ (l-2c) + 25k -5V^ + Vco dz 2/x Z do> (3 - Av) dco 2JX dz 2/t dz T= 1 d 2^ k dz 2 ' The perturbations in stress and heat flux at the plane z = 0 are d 2 co d 2 <f> d } t dz 3 (6) and continuity of these quantities at the interface is satisfied by imposing the symmetry conditions 30/Vol. 50, MARCH 1983 (3) (4) (5) from equations (5) and (6). The crack surfaces must also be free of shear tractions giving d 2 4> drdz d<t> = 0; 0<r<b, z = 0, (10) = C,; Q<r<b, z = 0, (11) lz C t is an arbitrary constant of integration. Notice that this condition extends to r = b because we assume that the contact in a < r < b is frictionless. The normal displacement u z must be continuous (i.e., no gap is opened) in both the bonded region (r>b) and the contact region (a<r<b), while radial displacements are continuous in the bond only. Following the same procedure as in [11], we find that this requires A - ^ d<t> dz 3co dr 3co + (S -(fii+82) A = (1- Mi 1 - ~dr "0 + 3 V = 0; (1 " - =0; r>b, -v 2 ) (1-2.,) (1-2^2) B = 2/*i 2^2 r>a, z = 0, z- = 0 (12) (13) (14) (15) Finally, we assume that neither the bond nor the contact region offer any resistance to heat flow -i.e., there is "perfect thermal contact" [10]. This region of the interface is therefore an isothermal surface that must be the zero of temperature in the perturbation solution, i.e., from equation (4). The Temperature Field d 2^ dz 2 = 0; r>a, z = 0, (16) The solution for i/< and hence T can be obtained independently of 0, u from equations (9) and (16). The problem is of standard form and we merely record the solution which can be written Transactions of the ASME

3 dh 2q 0 dz 2 Im[-~Jr 2 +(z + ia) 2 + zlog(vr 2 + (z + id) 2 +z + id)]. (17) We also record expressions for certain derivatives at the surface, ~dr~ ~ 3-irr d 2 j, ~3?~ ; r>a, z = 0 (18) -2q^a 2 -r 2 0<r<a, z = 0 (19) which are required in subsequent calculations. We note that (18) is obtained from (17) by integration, arbitrary constants being assigned to ensure continuity of derivatives at r = 0, a. Development of Two Integral Equations We anticipate that the stress field will contain discontinuities at both r = a (due to the discontinuous heat flux) and at / = b (due to the crack tip). It is convenient to isolate these discontinuities in separate functions and this can be achieved by making the change of variable (5, -S 2 ) x=co + /3</> \p, all r,z, A P = B/A (20) (21) is Dundurs constant [14]. Using equation (20) to eliminate co in the conditions (8), (12), and (13) we obtain, respectively, d 2 X d 2 <t> : dz 2 dz 2 o- d<t> (1-/3 2 ) ~dr dr dx dz (6i-fi 2 ) &J!_ A dz 2 0<r<a, z = 0, = 0; r>a, z = 0, ( hzm,r i/ * ^ r>b, z = 0, / fii -&2 \ V 5,+6, / (22) (23) (24) (25) Following Green [15] and Collins [16], we now satisfy the conditions (11) and (12) identically by the representations dz Jo Vr 2 + (z + /'0 2 d(j> 2Ci n, -T = Im [log(v r 2 + (z + ia) 2 + z + id)] OZ 7T (26) "Mi g 2(t)dt (27) )b Vr 2 + (z + (7) 2 g,, g 2 are real functions of / to be determined. Differentiating with respect to z and interpreting the square roots in accordance with references [15, 16], we find d 2 x 1 d i- «(r,b) mi tg ( t )dt z = 0, (28) dx 2 r dr I 0 Vr 2 -/ 2 d 2 <j> 2C l 1 d f- ~dz T w\lb 2 -r 2 r dr ib tg 2 (t)dt ; 0<r<6, z = 0 s/1 2^? 1 d f tg 2 U)dt r dr Jr Jl 2^? r>b; z = 0 (29) Hence, on substituting (19), (28), and (29) into (22) d_r tg x (t)dt _ fid_ C >tg 2 (t)dt _ 2&C x r dr Jo Vr dr Jb V/ 2 -r 2 w-^lb 2 -r 2 2^o(5I - h)r^la 2 -r 2 + a 0 r + KA 0<r<a. (30) We integrate this equation from zero to r obtaining f tgi(t)dt f»r ( ") 2 -^(^?-b) + ^, 2<5r 0 (5,-5 2 ) r 3, 2 2\3/2i n^ - + [a -(«-r 2 ) 3/2 ]; 0<r<o 37T/1 (31) which can be treated as an Abel equation for g, in terms of g 2 and known functions. A similar equation for g 2 in terms g x is obtained by applying the same procedure to equation (24). Expressions for dx/dr, d<j>/dr are obtained from (26) and (27) by integration as a</> ~dr~ dx 1 f min( ''' a, t gl (t)dt dr r Jo Vr 2 -/ 2 2C x b i r + g 2 U)dt ir r Jb 1 f " tg 2 (t)dt_ r Jr ^ll 2^?' (32) r>b, z = 0 (33) constants of integration have been assigned to ensure continuity of derivatives of the harmonic functions (and hence of displacements) at r = 0, a, b. The resulting Abel equation is s; tgl U)dt V7 2 -r 2 ~ JY lc x b + ' 2<?0 ( )(ff-l/ T )g 3 g 2 (t)dt + 3TT.4(1-3 2 ) ir (1-/3 2 ) Solution of the Integral Equations,-,. ; r>b. (34) J o Vr Equations (31) and (34) can be inverted by standard methods (see e.g. (15)), but a bounded solution for g 2 is obtained only if the right-hand side of (34) tends to zero at larger, i.e., if ""- 2<7 0 (5,-«2 )( 8-l/7)fl 3 g 2 (t)dt + 3TTV4(1 -/3 2 ) 2C x b = 0 (35) If this condition is satisfied, we obtain after some manipulations from (34) and s: 2/3 rtgaodt g 2 (x) = ; x>b (36) TT(1 -/3 2 ) JO (t 2 -x 2 ) Journal of Applied Mechanics MARCH 1983, Vol. 50/31

4 2fix [ >g 2 (t)dt 2a 0 x 9O(5I -8 2) TTM [2ax+(a 2 -x 2 )log(^^)] 2j8C,, / b + x \ n x<a (37) SQ-- a 0 A /3 2 «r 0 («i-«2)6 ^2(1-J3 2 ) -(7 6 2^ -[( f)( f)] (a-s) from (31). We can now obtain a single Fredholm equation for g, by substituting (36) into (37) and reversing the order of the double integral. Notice that the ranges of the integrals in (36) and (37) do not overlap and hence no singularities will occur in the range of the Fredholm equation except perhaps at the end points. This is a consequence of change of variable (equation (20)). It will also be observed from equation (36) that g, is an odd function of x, and some simplification can be achieved both at this stage and in the subsequent numerical solution by using this fact to extend the range of the integral equation to -a<x<+a. We then obtain the equation 0 f +0, IY b+x \/ b ~ s "\ 7T 2 (l-/3 2 ) 2a 0 x q 0 (.8j -5 2 ) llax + (a 2 -;c 2 )log ( -^^-) ir w 2 A Auxilary Conditions 2/3C, -r-^logl- ); -a<x<+a 7T 2 \ b-x S gi(j) ds (38) To complete the solution, we need two auxiliary conditions to find the unknown constant C l and the inside radius of the contact zone a. One condition (35) has already been found as a requirement for the regular inversion of the Abel equation (34). We write (35) as a condition on g { by substituting for g 2 from (36) obtaining? +a f b + t \ 8] _ a gl (Olog^-j^-Jrf/ = 4 C,6(l -P 4 go ( )(/3-l/ 7 )g 3 ZA The second condition follows from the inequalities at the transition from contact to separation (r = a) and can be imposed by demanding that the contact traction tend to zero at r a +. (The condition that the gap close smoothly at r a ~, used by Keer et al. [12] is equivalent.) From equations (5) and (20), we note that a discontinuity of contact tractions at r = a, z = 0 could only arise through the term d 2 x/dz 2, since equations (17) and (29) guarantee continuity of d 2 i/-/dz 2,d 2 4>/dz 2, respectively. Further, it can be shown that (28) will define an expression that is discontinuous at r = a unless and hence gi(a) = 0. g 2 TT 2 (l-/3 2 ) S>[( i)( f)] gi(s) (a-s) (40) 2a n a ds = + 2 q 0(6 i -6 2 )a 2 2(3C PC, / b + a \ K 2 A V M T ), (41) 7r 2 V b-a / from (38) with x = a. This is the required additional condition. ^(^MtS) g(s)fifc Kb 0(0- l/ 7 )a 2 /Z> 2 log( ±il) 3ir(l-(3 2 ) AgiQc) g(x) = <?o(5, -5 2 )fe 2 " (42) (43) This equation and (39) are then used to eliminate a 0, C,, g(x)+^k^yy^8{s)ds respectively, in (38) which becomes /3 2 K(x,s) = (PCS) = h(x), -cts:x<+a (44) >[(4^)(w)] (a-s) 1 r / b + x \ x ( b + a M / b + s \ TbHl^)-^<T^)MTs-) (45) (39) and h(x) = (a2 -x 2 ) ir 2 b 2 log(^l) V a-x / 2/%3-l/ 7 ) a 3 3TT 2 (1-/3 2 ) "ft 3 MIS) X F)] (46) In practice, SQ would be given while the contact radius a would be unknown. However, it is numerically more convenient to solve (43) for various values of a and the compute appropriate values of SQ from (41). Note that SQ can be regarded as a dimensionless ratio of mechanical and thermal loading. Interface Tractions and Stress Intensity Factors The interface tractions can be obtained from equations (5) and (20) by substituting for the appropriate derivatives and using (36) and (43) to express the results in terms of g(x). The calculations are routine but tedious and only the final expressions are given here which are Solution Procedure We first use (39) to eliminate C, in (41) obtaining «/<7o = l- 2C,0 o WF^P 32/Vol. 50, MARCH 1983 Transactions of the ASME

5 j«jtitanium -5 alloy (II) V> t" nickel silver T Pb (1-2) S 5- m/w - Al ' r i T a> In gva -e o u Ni *Nb Ta Cr \w Mo W SiC zinc wbronze a H y s Zn $_ grey iron 870/30 " aluminum Be/Cu alloys Al cu AQ 0 10 (l-2i/)//i Sn 15 (MPa)~ magnesium alloys Mg Fig. 2 Distortivity (6) and (1-2i>)/,i for a range of common structural materials. The slope of a straight line joining any two points will have the same sign as the product 0y for the corresponding interface. y SQ (3= 25/ I / 1 1 2(1-I3 2 )SQ I- -I D 2I arc tan inr s a/b I -2-3 i T i C*^-^5^-- 4 ^^^ 5 = <? 8 r 'E ^02^ X = 5 Fig. 3 Variation of the contact radius a/b with SQ = a oai{q 0(6f & 2)b! for various material combinations 8 7=1/ 9/ 1 0 r +a r 2/3 2-2C.6 06 J-«LTTV* 2 -/- 2 CT^/CTQ = "o wn/r2 -b 2 TT(1 - /3 2 )Sg J -a I r(r-s)-jr 2 - b 2 1 "I g(s)ds Vr 2 -s 2 ; a<r<b, z = 0 (47) J (r-s) ( r2_,,2)3/2 /9-I rlog[wr 2 -s 2 1:1 -W/ ] g(s)rfs ; / >, z = 0. (49) b f g(s)(xs' r>b~ (48) The normal traction is bounded at r = b +, but singular at tf«/ffo = 1 + 2(l-(3 2 )SeJ-«(r-jW^-s (r-j)v. 2 ' ' / = ft~ with dimensionless stress intensity factor Journal of Applied Mechanics MARCH 1983, Vol. 50/33

6 K, = lim (^ rs/t^ptp^ i 2C,fl o 0 irb P 2 Hl-P 2 )SQ +a " g(s)ds -. (b-s) from equation (47). Notice that a<r<b is a region of contact and hence for a physically meaningful result (50) must define a compressive traction, i.e., a 0 K x <0. (51) From the definition of SQ (equation (42)), it can be seen that the inequalities (1) and (51) are compatible if and only if SQK { <0, (52) since A, b are necessarily positive. The stress intensity factor in shear at r = b + can be obtained from equation (49) and is related to K x by * 2=lim ^ ^ ^,/g r~b + (T0 as required by the asymptotic stress field at the crack tip. Results and Discussion (53) The solution of the Fredholm equation (44) depends on the material properties of the two half spaces only through the dimensionless ratios (3 and y defined by equations (21) and (25). This is also the case for SQ, the dimensionless tractions, and the stress intensity factors. In discussing the numerical results, it is convenient to take as a point of reference the solution due to Keer et al. [12] for the penny-shaped interface crack without heat flow, which is essentially the present solution in the limiting case q 0 0 and hence SQ oo. We can recover Keer's solution if we choose a/b to be an eigenvalue (C 0 ) of the homogeneous Fredholm equation corresponding to (44). At such a value, the nonhomogeneous equation will give an unbounded solution for g(x) and hence for SQ. In practice, this means that SQ increases without limit as a/b approaches C 0 from either side. The eigenvalue C depends on the kernel of the integral equation and hence on 1/31 only, and the values obtained agree with those given by Keer et al. [12] within limits of numerical accuracy. For all physically possible values of l3(-0.5</3< +0.5), C 0 is very close to 1, the minimum being C 0 = for,Q=±0.5. For given values of 13 and 7, equation (44) can be solved numerically for any value of a/b in the range 0<a/b< 1, but we find that the inequality (52) is always violated for values of a/b on one or the other side of the eigenvalue C 0. If (3 and 7 have the same sign, physically possible solutions are obtained if and only if 0<a/b<C 0, as if /3 and 7 have opposite signs they are obtained only for C 0 < a/b < 1. Thus, if we apply a uniform tension a 0 and gradually increase the heat flux q 0 from zero, the contact region will get larger if 187> 0 and smaller if (37 <0. In view of this striking dependence on the sign of fiy, it is pertinent to ask whether there is sufficient correlation between the properties of real materials to make one sign more common than the other. The product Py will be positive if the material with the higher distortivity has also the higher value of (l-2c)/^. We therefore plot 5 against (l~2c)//i for a range of materials in Fig. 2. If we join the points corresponding to the two materials of the interface by a straight line, (3y will have the same sign as the slope of this line. For example, an interface between copper and zinc gives 187>0, while one between copper and mild steel gives /37<0. It is clear from Fig. 2 that many practical examples of each kind occur. Examples With /3Y>0 We next consider in more detail the behavior when /37>0. It is convenient to choose the material of higher distortivity to occupy the region z > 0 (half space 1) in which case /3 and 7 are n Fig. 4 Stress intensity factors for purely mechanical loading (q 0 = 0) as a function of 0 34/Vol. 50, MARCH 1983 l/sq Fig. 5 Effect of heat flux on K-, for /3 = 0.25 and y = 0.1 Transactions of the ASME

7 1 K q / /3=-25,y = -l I 0 I SQ Fig. 6 Effect of applied traction a 0 on the thermal stress intensity factork lq for/3 = 0.25 andy = l/sq Fig. 7 Effect of heat flux on K-\ for /3 = 0.5 and y- Py < 0 and the contact area is reduced by heat flux -0.25, for which both positive. The inequality (1) then requires that q 0 >0 if imperfect contact is not to occur. Figure 3 shows the relation between SQ and a/b for various combinations of j3 and y. The ratio SQ falls continuously with a/b from the eigenvalue C, reaching zero at alb for (3 = 0.25, 7 = 0.1. This point corresponds to the crack with thermal loading only (i.e., a 0 =0). For alb >0.688, negative values of SQ are obtained corresponding to a 0 <0 in other words, the contact region a<r<b can be further extended by applying compressive tractions at infinity up to the limiting point A SQ= The crack loaded by a uniform compressive traction presents some interesting features. If there is no heat flow (g 0 =0), it will close completely, since the crack is able to transmit compressive tractions. If we now superpose a flow of heat and assume as in the foregoing analysis that the contact region does not impose a resistance to heat flow between the faces of the crack, there is no reason to anticipate any perturbation in either temperature or stress field. Thus, a/b = 0 is always a solution of the problem if o a <0. We show this in Fig. 3 by continuing the SQ curve down the axis a/b = 0 for negative values of SQ. However, it follows that, in the range -0.09<SQ<0, there are three solutions to the problem, for (3 = 0.25, 7 = 0.1 corresponding to the points B, C, and D in the Fig. 3. All these solutions satisfy the boundary conditions and inequalities of the problem. There are precedents for the existence of multiple solutions in thermoleastic contact problems, as here, the heat flows into the more distortive material [17-20]. In simple onedimensional cases, for which a stability analysis can be completed by perturbation methods [17, 20], it is found that, if several steady-state solutions exist, at least two of them are stable and hence the configuration achieved in practice depends on the history of heating and loading. In the present problem, we should anticipate that the point D, with the crack fully closed, could be reached by starting with a 0 <0, <7o =0 and slowly increasing the heat flux. On the other hand, point B should be obtainable by first loading the crack in tension and then slowly changing a 0 to the desired compressive value. If the compressive stress were increased beyond this point, presumably the solution would proceed to A in Fig. 3 and then change discontinuously to the state with a/b = 0. A similar jump from 0 to must occur if a 0 is increased from compressive to tensile values at alb = 0. It is not clear what if any sequence of heating and loading could cause solution C to be established and it is tempting to label it unstable. However, we should remark that the only unstable solutions discovered in [17, 20] were those involving states resembling imperfect contact. In Fig. 3 we also show results for other combinations of (3 and 7 from which it is clear that if /3 is reduced or 7 is increased, the extent of the contact region for a given SQ is reduced. All the curves are asymptotic to the straight line asa/z? 0. Stress Intensity Factor SQ= a/b (54) In discussing the stress intensity factor at the crack tip, it is helpful to distinguish cases the mechanical loading is dominant (SQ large) and those thermal loading is dominant (SQ small). In the former case, the stress intensity factor is best presented in the dimensionless form of equation (50). Values for purely mechanical loading (q a =0) are shown in Fig. 4. For (3<0.3, the eigenvalue C is very close to unity and the numerical accuracy of the limiting solution is questionable. However, the stress intensity factor is a smooth function of l/sq (see Fig. 5) which tends to a limit as a/b approaches C 0. Values for (3<0.3 were therefore obtained by extrapolation from the more reliable solutions with a/b<c 0. This extrapolation procedure was tested on the established points for (3 > 0.3 and was very precise. The effect of heat flux superposed on mechanical loading is illustrated in Fig. 5 K { is plotted against l/sq for (3 = 0.25 and 7 = 0.1. Notice that the heat flux has a very substantial effect in increasing the stress intensity factor above the mechanical value. When thermal loading is dominant, it is more meaningful to relate the stress intensity factor to the heat flux defining a "thermal" stress intensity factor. K., z^l-r 2 /b 2 = lim = SQK, (55) -*-fo 0 (8,-6 2 )6M] We plot K x q against SQ in Fig. 6 for (8 = 0.25, and 7 = 0.1. In the limit SQ = 0 (no applied tractions) AT 1? =0.17. Superposition of compressive traction reduces the stress intensity factor, but tensile traction exacerbates it. Example With /3-y<0 When 187 <0, the contact region decreases with increasing heat flux q 0 and is always small, a/b being in the range C 0 <a/b<l. This presents problems of numerical accuracy, particularly when (3 is small and C 0 is very close to unity. We therefore give results for the case (3 = 0.5 and 7= -0.25, in Fig. 7 K s is plotted against l/sq. The results obtained are closely approximated by the relation Ki = /SQ (56) Journal of Applied Mechanics MARCH 1983, Vol. 50/35

8 It follows from the definition (55) that # = SQ (57) It is interesting to note that with 187 <0 an increase of the tensile traction a 0 causes an increase in the extent of the contact region. Results for small values of SQ correspond to alb very close to unity-e.g., SQ = 0.5. is obtained with a/6 = l-10~ 8. All calculations were performed with double precision, but results for lower values of SQ must be treated with caution. Acknowledgments The authors gratefully acknowledge support from the U.S. Army under contract DAAG29-82-K References 1 Karush, W., and Young, G., "Temperature Rise in a Heat Producing Solid Behind a Surface Defect," J. Appl. Phys. Vol. 23, 1952, pp Barber, J. R., "The Disturbance of a Uniform Steady-State Heat Flux by a Partially Conducting Plane Crack," Int. J. Heat Mass Transfer, Vol. 19, 1976, pp Florence, A. L., andgoodier, J. N., "The Linear Thermoelastic Problem of Uniform Heat Flow Disturbed by a Penny-Shaped Insulated Crack," Int. J. Eng. Sci., Vol. 1, 1963, pp Shail, R., "Some Steady-State Thermoelastic Stress Distributions in the Vicinity of an External Crack in an Infinite Solid," Int. J. Eng. Sci., Vol. 6, 1968, pp Rubenfeld, L., "Non-Axisymmetric Thermoelastic Stress Distribution in a Solid Containing an External Crack," Int. J. Eng. Sci., Vol. 8, 1970, pp Barber, J. R., "Steady-State Thermal Stresses in an Elastic Solid Containing an Insulated Penny-Shaped Crack," /. Strain Analysis, Vol. 10, 1975, pp. J Barber, J. R., "Steady-State Thermal Stresses Caused by an Imperfectly Conducting Penny-Shaped Crack in an Elastic Solid," J. Thermal Stresses, Vol.3, 1979, pp Comninou, M., "The Interface Crack," ASME JOURNAL OF APPLIED MECHANICS, Vol. 44,1977, pp Barber, J. R., "Contact Problems Involving a Cooled Punch,".1. Elasticity, Vol. 8, 1978, pp Comninou, M., and Dundurs, J., "On the Barber Boundary Conditions for Thermoelastic Contact," ASME JOURNAL OF APPLIED MECHANICS, Vol. 46, 1979, pp Barber, J. R., and Comninou, M., "The External Axisymmetric Interface Crack With Heat Flow," Q. J. Mech. Appl. Math., Vol. 35, 1982, pp Keer, L. M., Chen, S. H., and Comninou, M., "The Interface Penny- Shaped Crack Reconsidered," Int. J. Eng. Sci., Vol. 16, 1978, pp Dundurs, J., and Panek, C, "Heat Conduction Between Bodies With Wavy Surfaces,"//;/. J. Heat Mass Transfer, Vol 19, 1976, pp Dundurs, J., "Discussion on Edge-Bonded Dissimilar Orthogonal Elastic Wedges Under Normal and Shear Loading," ASME JOURNAL OF APPLIED MECHANICS, Vol. 36,1969, pp Green, A. E., and Zerna, W., Theoretical Elasticity, Clarendon Press, Oxford, 1954, pp Collins, W. D., "On the Solution of Some Axisymmetric Boundary Value Problems by Means of Integral Equations. VIII: Potential Problems for a Circular Annulus," Proc. Edin. Math. Soc, Vol. 13, 1963, pp Barber, J. R., Dundurs, J., and Comninou, M., "Stability Considerations in Thermoelastic Contact," ASME JOURNAL OF APPLIED MECHANICS, Vol. 47, 1980, pp Comninou, M., and Dundurs, J., "On Lack of Uniqueness in Heat Conduction Through a Solid to Solid Interface," ASME Journal of Heat Transfer, Vol. 102, 1980, pp Comninou, M., and Dundurs, J., "On the Possibility of History Dependence and Instabilities in Thermoelastic Contact," J. Thermal Stresses, Vol. 3, 1980, pp Barber, J. R., "Stability of Thermoelastic Contact for the Aldo Model," ASME, JOURNAL OF APPLIED MECHANICS, Vol. 48, 1981, pp /Vol. 50, MARCH 1983 Transactions of the ASME