THE THERMAL SHOCK RESISTANCE OF SOLIDS

Transcription

1 Act mter. Vol. 46, No. 13, pp. 4755±4768, 1998 # 1998 Act Metllurgic Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Gret Britin PII: S (98)00127-X /98 $ TE TERMAL SOCK RESISTANCE OF SOLIDS T. J. LU{ nd N. A. FLECK Cmbridge University Engineering Deprtment, Cmbridge CB2 1PZ, U.K. (Received 1 December 1997; ccepted 28 Mrch 1998) AbstrctÐThe therml shock resistnce of brittle solid is nlysed for n orthotropic plte suddenly exposed to convective medium of di erent temperture. Two types of plte re considered: (i) plte contining distribution of ws such s pores, for which stress-bsed frcture criterion is pproprite, nd (ii) plte contining single dominnt crck ligned with the through-thickness direction, for which criticl stress intensity fctor criterion is pproprite. First, the temperture nd stress histories in the plte re given for the full rnge of Biot number. For the cse of cold shock, the stress eld is tensile ner the surfce of the plte nd gives rise to mode I stress intensity fctor for pre-existing crck t the surfce of the plte. Alterntively, for the cse of hot shock, the stress eld is tensile t the centre of the plte nd gives rise to mode I stress intensity fctor for pre-existing crck t the centre of the plte. Lower bound solutions re obtined for the mximum therml shock tht the plte cn sustin without ctstrophic filure ccording to the two distinct criteri: (i) mximum locl tensile stress equls the tensile strength of the solid, nd (ii) mximum stress intensity fctor for the pre-existing representtive crck equls the frcture toughness of the solid. Merit indices of mteril properties re deduced, nd optiml mterils re selected on the bsis of these criteri, for the cse of high Biot number (high surfce het trnsfer) nd low Biot number (low surfce het trnsfer). The reltive merit of cndidte mterils depends upon the mgnitude of the Biot number, nd upon the choice of filure criterion. The e ect of porosity on therml shock resistnce is lso explored: it is predicted tht the presence of porosity is generlly bene cil if the filure is dominted by pre-existing crck. Finlly, the nlysis is used to develop merit indices for therml ftigue. # 1998 Act Metllurgic Inc. Published by Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION A common mesure of therml shock resistnce is the mximum jump in surfce temperture which brittle mteril cn sustin without crcking. The subject is old nd the literture lrge, yet existing theoreticl models re not ble to rnk the shock resistnce of mterils in the observed mnner. It is generlly ccepted tht the therml shock frcture resistnce of mteril depends on number of mteril properties including the therml expnsion coe cient, therml conductivity k, therml di usivity k, elstic modulus E, frcture toughness K IC, tensile strength s f, nd upon the dditionl prmeters of het trnsfer coe cient h, specimen size, nd durtion of therml shock (which is often overlooked). A mteril with high frcture resistnce under one set of therml shock conditions my become de cient under other conditions. For instnce, when quenched in ir, BeO (beryllium oxide) exhibits much better shock resistnce thn luminium oxide (Al 2 O 3 ), but the order of merit switches when both mterils re wter quenched [1]. Additionlly, experimentl experience suggests tht porosity is detrimentl to the cold-shock resistnce of cermics but is bene cil to hot-shock resistnce [2]. Current {To whom ll correspondence should be ddressed. knowledge of the underlying mechnisms behind these phenomen ppers to be rther limited. A commonly used therml shock prmeter is the merit index of s f =E. This prmeter only cptures the initition of therml shock crcking in brittle mterils under extreme conditions where the Biot number Bi h=k is in nite. An lterntive therml shock prmeter, suggested by sselmn [3], mesures the rtio of the frcture energy for crck initition to the frcture energy for continuous crck propgtion. This prmeter neglects the therml conductivity of the mteril, prmeter considered centrl to therml shock response. Other prmeters for vrious geometries nd therml shock environments re lso proposed [2, 4±10], nd it ppers tht the detils of the therml elds must be coupled with mteril properties nd geometricl prmeters in order to successfully predict the frcture behviour of component subjected to therml shock. The present pper revisits the old problem of plte of nite thickness, with fces suddenly exposed to convective medium of di erent temperture. The min feture tht di erentites this work from most previous studies (cf. for instnce, Emery et l. [11], Nied [8], nd Rizk nd Rdwn [9] who nlyse the therml shock frcture of n edge-crcked elstic plte) is tht new non- 4755

2 4756 LU nd FLECK: TERMAL SOCK RESISTANCE dimensionl prmeters cpble of chrcterizing the therml shock resistnce of brittle mteril re obtined in closed form over the full rnge of Biot number. The pper begins by reviewing the trnsient temperture nd stress distributions in homogeneous orthotropic plte over the full rnge of Biot number, Bi h=k. Closed-form expressions re obtined for the mximum stress, s well s its time of occurrence, ttined t the surfce nd t the centre of the plte. Next, the frcture response of the plte is ddressed, by ssuming the plte contins mode I crck extending perpendiculr to the plte surfce. For cold shock, the most dmging crck geometry is tken to be n edge crck, wheres for hot shock centre-crcked plte is considered. It is resonble to ssume tht the presence of these crcks hs no e ect on the one-dimensionl temperture distribution within the plte. The mode I stress intensity fctor for ech crck geometry is clculted from the trnsient therml stress eld. Two distinct filure criteri re considered for therml shock resistnce: (i) A locl tensile stress criterion, corresponding to the initition of tensile frcture in solid contining distribution of ws. For the ske of simplicity, the sttistics of w distribution is neglected nd it is ssumed tht the solid hs deterministic strength s f. Frcture occurs when the mximum tensile stress s mx ttins the strength s f. (ii) A frcture toughness criterion, whereby the lrgest pre-existing w dvnces when the mximum stress intensity fctor K mx ttins the frcture toughness K IC. It is ssumed tht this w is of the sme length scle s the thickness of the structure. In most prcticl circumstnces it will be demonstrted tht this criterion is more restrictive thn (i); this toughness criterion is, however, relevnt to cermic component contining mnufcturing ws or service-induced ws which scle with the size of the structure. For ech filure criterion, the mximum jump in surfce temperture DT mx to withstnd frcture is clculted for single cold shock nd for single hot shock. The vlue of DT mx is sensitive to the mgnitude of the surfce het trnsfer coe cient, vi the Biot number Bi. Approprite non-dimensionl groups re identi ed tht govern the therml shock resistnce of brittle solids over the full rnge of Biot number; these non-dimensionl groups contin mteril, therml nd geometric prmeters. The pper concludes with discussion of the potentil of porosity for incresing the therml shock resistnce of solid, nd on ppliction of therml shock nlysis to therml ftigue. 2. EVOLUTION OF TEMPERATURE AND STRESS A crck-free in nite plte of thickness is considered, with Crtesin coordintes embedded t the centre of the plte, s shown in Fig. 1. Initilly, the plte is t uniform temperture T i, nd t time t = 0 its top nd bottom fces (t z =) re suddenly exposed to convective medium of temperture T 1. Surfce het ow is ssumed to k ˆ 3h T 1 T, t z ˆ 1 where k z is the therml conductivity of the solid in the z-direction, h the coe cient of het trnsfer, nd T(z,t) the temperture of the solid. et ow within the solid induces trnsient temperture distribution T(z,t) nd stress stte s(z,t). The plte is ssumed to comprise uniform, liner thermo-elstic solid with xes of orthotropy ligned with the Crtesin coordintes (x,y,z) given in Fig. 1. The strin stte e is given by e xx ˆ 1 s xx xy s yy xz s zz x T T i 2 E x E y E z e yy ˆ xy E y s xx 1 E y s yy yz E z s zz y T T i 2b e zz ˆ xz s xx yz s yy 1 s zz z T T i : 2c E z E z E z ere, E x,e y,e z re the elstic moduli in the x,y,z directions, respectively, xy, xz, yz the Poisson rtios, nd x, y, z the coe cients of therml expnsion. Symmetry dicttes tht the sher stress nd sher strin components vnish. The con gurtion in Fig. 1 is nlysed under the constrint tht the plte is free to expnd with vnishing xil force nd s xx dz ˆ 0 s yy dz ˆ 0 nd vnishing norml stress in the through-thick- Fig. 1. A nite-thickness plte suddenly exposed to convective medium of di erent temperture.

3 LU nd FLECK: TERMAL SOCK RESISTANCE 4757 ness direction s zz ˆ 0. The geometry nd boundry conditions re such tht the strin e is independent of ll sptil dimensions including z, nd depends only on time t: the plte stretches uniformly but does not bend. It follows from equtions (2) tht the trnsient thermlly-induced stress s xx z,t ssocited with the temperture distribution T z,t is E s xx z,t ˆ E T T i T T i dz 3 where E 1 2 xy, nd x xy y 3b E x E y The two elstic prmeters, E nd, re extensively used in this pper. A prllel reltion exists for the therml stress s yy z,t, nd cn be found directly from eqution (3) upon rotting the plte by 908 bout the z-xis. ere, it is ssumed without loss of generlity tht the stress component inititing frcture is s xx z,t nd ttention is restricted to this stress component only Temperture distribution et ow in the through-thickness direction for the orthotropic plte shown in Fig. 1 is governed 2 2 ˆ k jzjr 4 where k z is the therml di usivity of the solid in the z-direction. Upon introducing the dimensionless sptil vrible z ˆ z= nd dimensionless time t ˆ k z t= 2, eqution (4) simpli es 2 jzjr1: 4b It cn be seen immeditely tht the therml di usivity k z dicttes the time-scle for the trnsient stress stte within the plte, but does not ect the level of therml stresses. Eqution (4) is solved with het trnsfer boundry condition (1) by stndrd seprtion-of-vribles technique [12], giving T z,t T i T i T 1 ˆ 1 2 X1 exp b 2 k z t n nˆ1 sin b n cos b n z= b n sin b n cos b n where b n re the roots of b n tn b n ˆ Bi nd the non-dimensionl het trnsfer coe cient Bi ˆ h=k z is the Biot number for the orthotropic solid. In the limit of perfect therml insultion Bi =0, b n ˆ np nd T z,t T i. At the other limit of perfect het trnsfer, Bi = 1, b n ˆ n 1=2 p nd eqution (5) simpli es to T z,t T i T i T 1 ˆ 1 4 p X 1 nˆ0 1 n 2n 1 exp 2n 1 2 p 2 4 2n 1 p cos 2 z k z t 2 : 7 For nite vlues of Bi, the coe cient b n is determined numericlly nd is bounded by the two limiting vlues given bove np<b n < n 1=2 p, n ˆ 0,1,2,... 7b 2.2. Trnsient stress distribution The therml stress s z,t s xx z,t is obtined directly from equtions (3) nd (5), nd is written in non-dimensionl form s s z,t s E T i T 1 ˆ T z,t T i 1 T z,t T i dz T i T 1 T i T 1 ˆ 2 X1 exp b 2 k z t sin b n n nˆ1 2 b n sin b n cos b n z cos b n sin b n : 8 b n In the limit Bi = 0, the plte is everywhere stress free (i.e. s z,t 0) while, if Bi = 1 s z,t s E T i T 1 ˆ 4 X 1 exp 2n 1 2 p 2 k z t p 4 2 nˆ0 1 n 2n 1 p 2n 1 cos z 2 2 2n 1 2 : 9 p When Bi is smll (Bi < 2), it hs been estblished tht the stress distribution is dominted by the rst two terms of eqution (8). The evolution of dimensionless stress, s z,t s z,t =E T i T 1, is plotted ginst dimensionless time, t k z t= 2 in Fig. 2, t selected loctions through the thickness of the plte, z z= nd for Bi = 1, 10 nd 1. Under cold shock conditions (T i >T 1 ), the surfce lyers experience tensile stress trnsient while compressive zone is developed t the centre of the plte. For ll vlues of Bi, the mximum tensile stress is ttined t the surfce nd the compressive stress is lrgest t the centre of the plte. The overll mgnitude of the stresses increses with incresing Bi. The trnsient tensile stress t the surfce explins the common observtion tht, during cold shock, crck initites t the surfce nd grows unstbly until it enters the centrl compressive region. Alterntively, during hot shock event (T 1 >T i )

4 4758 LU nd FLECK: TERMAL SOCK RESISTANCE not considered in the nlysis presented below.) It is convenient to chnge the sign in the de nition of s for hot shock: from now on, s is re-de ned for hot shock s s z,t s z,t =E T 1 T i so tht both s nd s re positive t the centre of the plte. The trnsient tensile stress t the surfce of the plte in cold shock, nd t the centre of the plte in hot shock, is plotted in Figs 3() nd (b), respectively, for selected vlues of Bi. For the limiting cse of n idel cold shock (Bi = 1), mximum vlue of s ˆ 1 is chieved t the surfce of the plte t t ˆ 0. For n idel hot shock (Bi = 1), the mximum tensile stress chieved t the plte centre is s mx ˆ 0:3085 t time t * ˆ 0:115 by eqution (9). The mximum tensile stress s mx chieved t the surfce during cold shock is plotted ginst 1/Bi in Fig. 3(c); for comprison purposes, the mximum tensile stress t the centre of the plte during hot shock is included in the gure. It is cler tht the mgnitude of s mx increses with incresing Bi for both hot shock nd cold shock. Further, the mgnitude of s mx is lwys less for hot shock thn for cold shock, t ny give vlue of Bi. The mximum surfce stress in cold shock is dequtely described by the reltion s mx,t * ˆ 1:5 3:25 1 0:5e 16=Bi 10 Bi Fig. 2. Evolution of dimensionless stress s z,t s function of dimensionless time t t selected loctions z/ for: () Bi = 1; (b) Bi = 10; (c) Bi =1. the centre of the plte is under tension nd is prone to crcking; splling of surfce lyer due to lrge compressive stress is lso possibility [2, 5]. This study will focus on filure due to tensile stress t the surfce of the plte in cold shock, nd t the centre of the plte in hot shock. (Notice tht n edge crck my lso grow under hot shock if the crck is su ciently long [13], but this scenrio is wheres, to n excellent pproximtion, the mximum stress developed t the centre in hot shock is given by s mx 0,t * ˆ 0: b 1 2=Bi s shown by the comprison in Fig. 3(c). The semiempiricl eqution (10) ws suggested by Mnson [14], bsed on n erlier result of Buessem [15]; this reltion hs subsequently been widely used, together with the mximum tensilestress criterion, to clculte the resistnce of both brittle nd ductile mterils to crck initition under cold shock conditions [1, 4, 5]. For completeness, the time of occurrence of s mx t the surfce during cold shock nd t the centre during hot shock is plotted ginst 1/Bi in Fig. 3(d). The time t* needed for the surfce lyer to rech the mximum stress in cold shock is given pproximtely by t * ˆ k z t * = 2 ˆ 0:48 1 1:8Bi : 11 Similrly, the time for the centre of the plte to ttin the mximum tensile stress under hot shock is pproximted by t * ˆ k z t * = 2 0:45 ˆ 0: :25Bi : 11b equtions (11) nd (b) describe closely the time dependence of mximum therml stress t the sur-

5 LU nd FLECK: TERMAL SOCK RESISTANCE 4759 Fig. 3. () Surfce stress s surfce in cold shock nd (b) centre stress s centre in hot shock s function of time t for selected vlues of Biot number, Bi; (c) mximum surfce stress nd mximum centre stress, s mx, nd (d) their time of occurrence, t *, s functions of 1/Bi. The dshed lines in (c) nd (d) represent the empiricl reltions (10) nd (11). fce nd t the centre, respectively, s shown by the comprison in Fig. 3(d). So fr mximum tensile stress criterion for frcture initition hs been discussed. In the cse of structure contining defects on the order of the structurl dimension it is more pproprite to determine the temperture jump for which pre-existing crck will grow. 3. CRACKING DUE TO COLD SOCK Consider gin the in nite plte of Fig. 1 subjected to cold shock. If the plte contins number of lrge crcks on the scle of the plte thickness then it is expected tht crcking will commence from the ``worst w''. A rtionl de nition of ``worst w'' is the one which hs the lrgest trnsient mode I stress intensity fctor. The problem is idelized to the highly simpli ed cse of plte contining n isolted mode I edge crck of depth, s shown in Fig. 4(). For this crck the stress intensity fctor K is clculted during cold shock event, for 0 < BiR1. Since the crck plne is norml to the fce of the plte, it does not perturb the trnsient temperture distribution. The stress intensity fctor K ssocited with the therml stress s z,t is derived by strightforwrd numericl integrtion mking use of the pproprite weight function, to give 1 K K 0 ˆ l 3=8 2 p p n p = 1 = F 1 z=,= s z=,t z p d : 1 = 1:5 1 z =Š 2 12 p ere, K 0 pe T i T 1 is reference stress intensity fctor, nd F 1 z=,= is non-dimensionl function de ned in eqution (A1) of Appendix A. The concept of orthotropic rescling [16] hs been used to ccount for mteril nisotropy. With the id of the orthotropic stress± strin reltion (2), the two non-dimensionl elstic prmeters l, n re de ned s

6 4760 LU nd FLECK: TERMAL SOCK RESISTANCE isotropic in plnes norml to the y-xis nd l = z =1. The normlized p stress intensity fctor K K=K 0 ˆ K= pe T i T 1 is plotted ginst dimensionless time t k z t= 2 in Fig. 4(b), for selected normlized crck length =. For illustrtive purposes, results re presented only for the cse Bi = 10; results over the full rnge of Biot number (0 < BiR1) re qulittively similr to those shown. At ny given crck length, K displys pek vlue fter nite time nd decreses to zero s t 4 0 nd s t 4 1. It is further noted tht the mgnitude of K depends upon crck length, nd chieves pek vlue for crck of length =11=3. Thus, for given Biot number Bi, K chieves globl mximum vlue K mx t time t* nd t crck length *. Non-dimensionl vlues of K mx nd the corresponding non-dimensionl vlues t * tk z = 2 nd * * = re plotted in Fig. 4(c) s function of 1/Bi. Simple curve ts to the plots of K mx nd t *, * ginst 1/Bi re given by K mx K mx K 0 ˆ 0: : Bi t * kt* 2 ˆ 0:08 0:4 1 1:4Bi 14b Fig. 4. () Geometry nd conventions for single edge crck under cold shock, (b) dimensionless stress intensity fctor K s function of dimensionless time t for Bi = 10, (c) K mx nd the corresponding non-dimensionl vlues t * nd * plotted s functions of 1/Bi. The dshed lines in (c) nd (d) represent the empiricl reltions (14). l ˆ E z E x, n ˆ p E x E z z ˆ 2G xz r 1 z where 2 G xz 1 2 xz 1 E z xy yz E z E y xz 13 nd E x E, E z 1=E z 2 zy =E y 1. The sher modulus in the x±z plne is denoted by G xz. Positive de niteness of the strin energy density requires l>0 nd 1<z < 1. For the ske of brevity only the cse of trnsversely isotropic solid will be considered for which the mteril is * * ˆ 1 14c 3 nd re in stisfctory greement with the precise vlues. It is cler from Fig. 4(c), nd from equtions (14) tht K ttins its mximum vlue pproximtely t = ˆ 1=3 nd kt * = 2 ˆ 0:1, for Bi>5. It is importnt to note tht the limiting vlue K mx ˆ 0:222K 0 is the lrgest stress intensity fctor ttined for ny crck length, under the most severe therml shock boundry condition (Bi = 1). This limit lso pplies to n edge crck under hot shock [13] nd centre crck under hot shock (see the results in the next section). For the purposes of mteril selection for cold shock, it is ssumed the plte contins ``worst w'': this w is tken s n edge crck of length * ˆ =3 which mximizes K during the cold shock. Assume tht filure occurs when K mx given by eqution (14) equls K IC for the solid. The subsequent pth of propgtion of the edge crck remins to be discussed. After propgting strighthed towrds the centre of the plte, the crck incresingly feels the presence of compressive stresses in the centrl portion of the plte nd K drops. As soon s the crck enters the centrl portion of the plte under compression, the T-stress t the crck tip chnges from negtive to positive [17] which, ccording to Cotterell nd Rice [18], cuses the stright-hed dvnce of the crck to become con gurtionlly unstble. More speci clly, in the

7 presence of positive T-stress, the crck my de ect prllel to the surfce, resulting in crck brnching or splling. De ection is encourged by compressive residul stresses in cermic lmintes (nd by other crck de ectors such s pores nd wek interfces); this hs been the subject of severl recent studies [16, 17, 19, 20]. Development of these concepts is left to lter study. LU nd FLECK: TERMAL SOCK RESISTANCE CRACKING DUE TO OT SOCK Now consider the plte of Fig. 1 subjected to hot shock: initilly the plte is t uniform temperture T i. At time t = 0 the top nd bottom surfce of the plte re exposed to n environment t temperture T 1 (>T i ), nd the surfce het trnsfer condition is gin given by eqution (1). As discussed bove, the hot shock induces trnsient tensile stresses t the centre of the plte, with pek vlue speci ed by eqution (10b). A plte under hot shock is most likely to develop mode I crcks in the centre of the plte where mximum tensile stress is ttined. In order to develop therml shock criterion, the mgnitude of hot shock required to propgte pre-existing centre crck of length 2, symmetricl with respect to z =0 is determined [Fig. 5()]. The mode I stress intensity fctor ssocited with the therml stress trnsient (9) is given by integrtion of s z,t over the crck fce, with respect to the pproprite weight function r K K 0 ˆ l 3=8 2 p p tn n p 2 3 = F 2 z=,= s z=,t z s cos p cos p 2 d : z 5 2 The dimensionless function F 2 z=,= is given by eqution (A2) of Appendix A. Note tht under hot shock, p the prmeter K 0 is de ned by K 0 ˆ pe T 1 T i since T 1 >T i. For brevity, gin ttention is restricted to trnsversely isotropic pltes (l = z = 1). The non-dimensionl stress intensity fctor K/K 0 is evluted by numericl integrtion of eqution (15), nd is plotted ginst time in Fig. 5(b), for selected vlues of crck length nd for the prticulr choice Bi = 10. The qulittive shpe of the response is similr for other Biot numbers: K increses from zero to mximum vlue, nd then decys bck to zero. It cn be noted from Fig. 5(b) tht K ttins n overll mximum vlue K mx t prticulr time t* nd t prticulr crck length 2*. The vlues of K mx,t *, * depend upon the Biot number, s shown in Fig. 5(c). Fig. 5. () Geometry nd conventions for centre crck under hot shock, (b) dimensionless stress intensity fctor K s function of dimensionless time t for Bi = 10, (c) K mx, nd the corresponding non-dimensionl vlues t * nd * plotted s functions of 1/Bi. The dshed lines in (c) nd (d) represent the empiricl reltions (16). As expected, K mx chieves pek vlue K mx K mx =K 0 ˆ 0:177 for the cse of perfect het trnsfer Bi = 1. With decresing Bi, K mx decreses in monotonic mnner. A curve t to the plot of K mx K mx =K 0 vs 1/Bi is included in Fig. 5(c), nd is dequtely pproximted by K mx ˆ 0: :12 1 : 16 Bi For the full rnge of Biot number (0 < BiR1) K is lrgest for crck length of pproximtely = ˆ 0:5. Accurte curve ts for t *, * re

8 4762 LU nd FLECK: TERMAL SOCK RESISTANCE included in Fig. 5(c), nd re given by t * ˆ kt* 2 ˆ 0:08 0:4 1 1:4Bi * * ˆ 0:5 16b 16c In order to select mteril of optiml therml shock resistnce, the plte contining centre crck of length 2* is considered such tht K is mximized during the hot shock. Assume tht filure occurs when K mx given by eqution (16) equls K IC for the solid. It is cler from comprison of equtions (14) nd (16), nd from comprison of equtions (10) nd (b) tht the hot shock resistnce for centre-crcked plte is greter thn the cold shock resistnce for n edge-crcked plte, regrdless of whether filure is strength-controlled or toughness-controlled. In the ltter cse, this is true for ll vlues of Bi, s long s the crck length does not exceed criticl length criticl where the stress intensity fctor under cold shock equls tht under hot shock. For n edge-crcked plte hving > criticl, hot shock then becomes more severe thn cold shock [13]. Also, s K increses continuously with increses in for n edge crck under hot shock, crck propgtion is inherently unstble; under cold shock, crck extension is stble once the crck length reches * therefter dk=d<0 [cf. Fig. 4(b)]. For nite plte with two symmetricl edge crcks subjected to severe therml shock Bi = 1, it hs been found tht criticl ˆ 0:6 [13]. 5. TERMAL SOCK MATERIAL PARAMETERS FOR ENGINEERING CERAMICS Therml shock resistnce is mjor issue in the selection of engineering cermics for therml pplictions, such s furnces nd engine prts. A centrl problem in designing ginst therml shock is the identi ction of pproprite mteril selection criteri in order to select the most shock resistnt mteril for given ppliction. Mteril performnce indices re summrized for both strength-controlled filure nd toughness-controlled filure Merit indices for strength-controlled filure A stress-bsed frcture criterion for cold shock is tht s mx,t * ttins the frcture strength of the solid s f ; similrly, for hot shock s mx 0,t * ttins the vlue s f. The mximum temperture jump sustinble by the solid DT in the extreme cse of perfect het trnsfer (Bi = 1) follows from equtions (10) nd (b) s s f DT ˆ A 1 17 E where A 1 11 for cold shock, nd A for hot shock. ere, nd in the following, the distinction between E nd E, nd between nd hs been dropped, s brod mteril selection criteri re concerned with, nd terms of minor signi cnce re neglected. The temperture jump sustinble increses with decresing Biot number, so tht in the limit of smll Biot number (Bi < 1), DT follows from equtions (10) nd (b) s s f 1 DT ˆ A 2 E Bi A s f k 2 18 E h where A for cold shock, nd A for hot shock. It is cler from eqution (17) tht for perfect het trnsfer the highest temperture jumps re chieved for mterils with lrge vlue of s f =E. In the cse of poor surfce het trnsfer (Bi < 1), the best mterils hve the lrgest vlue of the mteril property group ks f =E. It is instructive to mp engineering cermics on plot with xes ks f =E nd s f =E, s shown in Fig. 6(). The Cmbridge Mterils Selector softwre [21] is prticulrly useful for this purpose. Mterils of high therml shock resistnce under conditions of idel het trnsfer lie to the right of the digrm, nd mterils of high therml shock resistnce under conditions of poor het trnsfer lie to the top of the digrm. Glss cermics nd grphites lie t the extreme top, right portion of the digrm nd hve the highest shock resistnce mong cermics over the full rnge of het trnsfer coe cient. It is lso cler from the gure tht the reltive order of merit cn switch between competing mterils, depending on the mgnitude of Bi. For exmple, beryllium oxide, BeO, hs higher vlue of ks f =E thn lumin, Al 2 O 3, nd is preferble for pplictions of low het trnsfer (smll Bi). When surfce het trnsfer is high (lrge Bi), s f =E becomes the relevnt mteril prmeter, nd lumin hs superior shock resistnce to BeO Merit indices for toughness-controlled filure A similr strtegy cn be employed to rnk mterils on the bsis of filure from dominnt crck by therml shock. The toughness-bsed frcture criterion for hot nd cold shock is tken to be tht K mx *,t * ttins the frcture toughness of the solid K IC. The mximum temperture jump sustinble by the solid DT in the extreme cse of perfect het trnsfer (Bi = 1) follows from equtions (14) nd (16) s K IC DT ˆ A 3 p E p 19 where A for cold shock, nd A for hot shock. The temperture jump sustinble increses with decresing Biot number, so tht in the limit of smll Biot number (Bi < 1), DT follows from equtions (14) nd (16) s K IC DT ˆ A 4 E 1 p p Bi A K IC 4 E k p p h 20

9 LU nd FLECK: TERMAL SOCK RESISTANCE 4763 Figures 6() nd (b). Cption overlef.

10 4764 LU nd FLECK: TERMAL SOCK RESISTANCE Fig. 6. () The merit indices for strength-controlled filure ks f =E t low Bi vlues vs s f =E t high Bi vlues, (b) merit indices for toughness-controlled filure kk IC =E vs K IC =E, nd (c) K IC =E vs s f =E, with the guide lines f dded to help in selecting mterils ccording to both strength- nd toughnessbsed frcture criteri. where A for cold shock, nd A for hot shock. From equtions (19) nd (20) it cn be deduced tht, for toughness-controlled therml shock, the best cndidte mterils hve high vlue of K IC / E for idel het trnsfer (Bi = 1), nd high vlue of kk IC =E for poor het trnsfer (Bi < 1). A lrge number of engineering cermics re displyed on mp with kk IC =E nd K IC =E s xes, see Fig. 6(b). The reltive loction on the mp is qulittively similr to tht given in Fig. 6() for strength-controlled therml shock, nd similr conclusions cn be drwn from the mp. For exmple, glss cermics nd grphites hve the best therml shock resistnce mongst the cermics. The reltive order of merit depends somewht on the Biot number: lumin hs superior shock resistnce to beryllium oxide for idel het trnsfer, but is inferior for poor het trnsfer. There re two seprte detrimentl e ects of incresing specimen size on therml shock resistnce, for both cold shock nd hot shock. For toughness-controlled filure, dimensionl considertions dictte tht the therml shock resistnce decreses with incresing plte thickness, t ll Biot numbers; this is cler from exmintion of equtions (19) nd (20). Further, when surfce het trnsfer is imperfect (Bi < 1) the mximum temperture jump for both strength- nd toughnesscontrolled filure decreses with incresing Biot number, see equtions (18) nd (20); thus, DT decreses with incresing. Plte thickness hs no e ect on therml shock resistnce only for the cse of strength-controlled filure with perfect het trnsfer, see eqution (17). The issue of deciding whether mteril selection procedure should be bsed on strength criterion or toughness criterion is delicte one. It is instructive to plot dt for engineering cermics on mp for idel het trnsfer, with xes K IC =E nd s f =E, s shown in Fig. 6(c). Cermics with high shock resistnce from the strength viewpoint lie in the regime of lrge s f =E vlue; nd, cermics with high shock resistnce from the toughness viewpoint lie in the regime of lrge K IC =E vlue. It is cler from the mp tht mteril dt cluster long the leding digonl: mterils such s glss cermics with high s f =E vlue lso hve lrge K IC =E vlue. Equivlently, the rnking of mterils by the strength criterion is the sme s tht given by the toughness criterion.

11 LU nd FLECK: TERMAL SOCK RESISTANCE 4765 Dimensionl nlysis, nd considertion of equtions (17)±(20) revel tht the dmissible temperture jump for both cold shock nd hot shock is less for toughness-controlled frcture thn for strength-controlled frcture, t su ciently lrge plte thickness. A trnsition plte thickness vlue t exists for which DT is equl for toughness-controlled filure nd for strength-controlled filure. Consider rst the cse of idel het trnsfer, Bi = 1. Then, upon equting the DT vlues for the strength criterion (17) nd for the toughness criterion (19), it is found tht t 1 2 KIC s f 21 for both cold nd hot shock. In the other limit of poor het trnsfer (Bi < 1), equting DT ccording to equtions (18) nd (20) gives 2 KIC t 1 Bi 22 s f for both cold nd hot shock. Lines of constnt t ccording to de nition (21) hve been dded to Fig. 6(c), nd my be interpreted s follows. Mterils which lie long line of constnt t hve the sme therml shock resistnce ccording to the strength criterion nd to the toughness criterion, for plte of thickness t. For pltes of this thickness, the strength-bsed criterion is conservtive for mterils lying bove the line, nd the toughnessbsed criterion is conservtive for mterils lying below the line. Mterils with lrge vlue of t cn be considered to hve high dmge tolernce, compred with mterils of low t vlue: thus, the t vlue cn be thought of s useful mesure of dmge tolernce. 6. CASE STUDY: TE POTENTIAL USE OF CERAMIC FOAMS FOR TERMAL SOCK APPLICATIONS At rst sight, it is uncler whether cermic fom hs superior or inferior therml shock resistnce to tht of fully dense cermic. The presence of porosity in fom reduces its therml conductivity, frcture toughness, filure stress, elstic modulus, nd mny other physicl properties. The coe cient of therml expnsion nd therml di usivity re generlly not ected by porosity provided the pores contin gses nd not liquids. In this section, the therml shock resistnce of brittle fom is estimted compred to tht of the solid mteril. Consider, for illustrtion, the in uence of porosity on the therml shock frcture resistnce of n insultion plte mde of cermic fom. On writing r* for the density of the fom, nd r s for the density of the cell wll mteril, the reltive density cn be expressed s r*/r s nd, to rst order, dicttes the properties of the fom. For simplicity, it is ssumed tht the fom consists of open cells nd the surfces of the fom plte re insulted to prevent the penetrtion of convective medium into the fom cells. The elstic modulus of the fom E* derives minly from the bending of struts mking the cell edges, nd is given pproximtely by [22] E * ˆ E s r * =r s 2 : 23 ere nd below, the superscript ``*'' is used to denote fom properties nd the subscript ``s'' for properties of the solid mteril of which the fom is mde. According to Gibbon nd Ashby [22], the strength of the fom in compression s * f is relted to the cell wll strength s fs by s * r *! 3=2 f ˆ 0:2s fs : 24 The frcture toughness of the fom scles with the reltive density, cell size l nd frcture strength of the cell wll mteril ccording to K * IC ˆ 0:65 r* p =r s 3=2 s fs pl : 25 It is convenient to relte the tensile strength of the cell wll mteril s fs to the frcture toughness K s IC nd the intrinsic w size of the cell wll mteril s fs ˆ K s p IC 26 p nd thereby write the frcture toughness of the fom s K * IC ˆ 0:65 r* p =r s 3=2 K s IC l= : 27 This expression is physiclly meningful only for the cse l>>: it is ssumed tht ws within the cell wll mteril re on much smller length scle thn the cell size. The therml expnsion coe cient for the fom is tken to be equl tht of the cell wll mteril, nd the therml conductivity of the fom is tken s r s k * ˆ 2=3 k s r * =r s : 28 ere, the reltively smll contributions from gseous conduction nd therml rdition hve been neglected [22]. To proceed, the estimted therml shock resistnce of the fom DT* is compred to tht of the cell wll mteril DT s from equtions (17) nd (18) for crush strength-controlled filure, nd from equtions (19) nd (20) for toughness-controlled filure. Then, for both cold shock nd hot shock, it is found tht DT *! ˆ 0:2 r* 1=2 29 DT s r s for idel het trnsfer, crush strength-bsed filure

12 4766 LU nd FLECK: TERMAL SOCK RESISTANCE DT *! ˆ 0:133 r* 1=2 30 DT s r s for Bi < 1, crush strength-bsed filure DT *! ˆ 0:65 r* 1=2 r l DT s r s 31 for idel het trnsfer, toughness-bsed filure, nd DT *! ˆ 0:43 r* 1=2 r l DT s r s 32 for Bi < 1, toughness-bsed filure. For the ske of rgument tke l =10 in equtions (31) nd (32). (Note tht the precise vlue of l/ hs only moderte e ect on the expression for DT*/DT s due to the squre root dependence on l/.) Typiclly, the reltive density of cermic foms is in the rnge 0.03±0.3, nd equtions (30) nd (32) suggest tht foms re inferior to their fully dense prent mterils, t low Biot numbers, due minly to their poor conductivities. owever, in the cse of high surfce het trnsfer, eqution (29) revels tht foms hve higher shock resistnce thn the prent solid for r*/ r s <0.04, bsed on the crush strength criterion. In similr fshion, for high surfce het trnsfer, eqution (31) revels tht foms hve higher shock resistnce thn the prent solid for ll r*/r s less thn unity, bsed on the toughness criterion. In conclusion, t lrge Bi numbers, open-cell foms hve promise for improved therml shock resistnce, provided the reltive density is su ciently low. Indeed, from Figs 6() nd (b), it cn be seen tht grphite nd zirconi (ZrO 2 ) foms, mong others, lie to the right of their respective prent solid mterils. If, during therml shocking, the convective medium cn in ltrte into the interior structure of open-celled brittle foms, the sitution is more complicted due to the coupling of globl therml stress nd the therml stress induced t the strut level. A preliminry study of therml shock dmge under such conditions cn be found in Orenstein nd Green [23]. 7. APPLICATION TO TERMAL FATIGUE The results of the current study cn lso be used to estimte the therml ftigue resistnce of cermics, metls nd polymers. Repeted therml shock of plte cn led to the initition nd growth of ftigue crcks. A conservtive pproch is to design for in nite ftigue life, nd to consider two filure criteri: (i) A stress-bsed criterion for the initition of ftigue crcks. It is ssumed tht crcks do not initite when s mx for ech therml shock is less thn the endurnce limit of the mteril s e ; here, s e is de ned s the stress mplitude t ftigue life of 10 7 cycles in fully reversed ftigue test. (ii) A stress-intensity bsed criterion for the propgtion of n existing crck. It is ssumed tht thermlly-shocked plte hs in nite crck growth life provided the stress intensity K mx for ech therml shock is less thn the ftigue threshold DK th of the mteril. ere, the ftigue threshold is de ned s the vlue of the cyclic stress intensity, DK, corresponding to crck growth rte of m/cycle, in test for which the minimum lod of ech cycle equls zero. The merit indices for stress-controlled ftigue crck initition re ks e =E for Bi < 1 nd s e =E for idel het trnsfer Bi = 1. Alterntively, when the plte contins crcks on the length scle of its thickness, the pertinent merit indices become kdk th =E nd DK th =E for Bi < 1 nd Bi = 1, respectively. It is instructive to compre the therml ftigue properties for rnge of cermics, metls, nd polymers with their therml shock resistnce, for the cse of idel het trnsfer, see Figs 7() nd (b). The dt re tken primrily from the Cmbridge Mterils Selector [21] nd from Fleck et l. [24]. Figure 7() tkes s xes the therml shock merit indices s f =E nd K IC =E, nd Fig. 7(b) dopts the equivlent prmeters s e =E nd DK th =E. The prticulr choice of mterils is such s to de ne the outer boundries of mteril behviour for the generic clsses of solid: cermics, metls nd polymers. Consider rst Fig. 7(). It is noted tht the high frcture toughness exhibited by metls ords them high therml shock resistnce for the cse of pre-crcked plte. When strength is the dominnt filure criterion, certin cermics (such glss cermics, grphites nd silic glss) out perform the metls. It is notble tht the eld of cermics covers wide rnge, which prtly explins the need for creful mterils selection for therml shock pplictions. Polymers hve reltively low Young's modulus, nd thereby resonble therml shock resistnce: they lie between the dt for metls nd cermics. Now consider the therml ftigue response, s shown in Fig. 7(b). The eld for metls moves downwrds by bout n order of mgnitude s the ftigue limit for metls is bout n order of mgnitude less thn their frcture toughness. The drop in strength property from s f to s e is less: bout fctor of two. Thus, metls hve signi cntly worse therml ftigue performnce compred to their behviour under single therml shock. In contrst, for polymers nd cermics, there is only

13 LU nd FLECK: TERMAL SOCK RESISTANCE 4767 Contours of t re included for the ftigue cse in Fig. 7(b), nd for the sttic cse in Fig. 7() mking use of eqution (21). It is noted tht the vlues of t re consistently smller for ftigue loding thn for sttic loding: the mterils re less dmge-tolernt under ftigue loding thn under sttic loding. AcknowledgementsÐThe uthors re grteful to W. J. Clegg nd M. F. Ashby for insightful discussions. REFERENCES Fig. 7. At high vlues of Bi, the mterils selection chrt of () K IC =E vs s f =E for single therml shock, (b) DK th =E vs s e =E for repeted therml shocks. The guide lines t help in selecting mterils ccording to both strength- nd toughness-bsed frcture criteri, for stble nd for cyclic therml loding. smll drop in vlues of s e =E nd DK th =E compred with the sttic properties s f =E nd K IC =E, respectively. It is cler from Fig. 7(b) tht the eld for metls lies within tht for cermics: there is little dvntge in using metls for therml ftigue pplictions in preference to cermics, unless other fctors dominte (such s cost nd mnufcturbility). For completeness, the trnsition plte thickness t is de ned for the ftigue cse in similr mnner to tht given in eqution (21) 2 DKth t 1 : 33 s e 1. Mnson, S. S. nd Smith, R. W., Trns. ASME, 1956, 78, Kingery, W. D., Property Mesurements t igh Tempertures. Wiley, New York, sselmn, D. P.., J. Am. Cerm. Soc., 1969, 52, Bron,. G., Therml shock nd therml ftigue, in Therml Stresses, ed. P. P. Benhm et l. Pitmn, London, Mnson, S. S., Therml Stress nd Low-Cycle Ftigue. McGrw-ill, New York, sselmn, D. P.., Cerm. Bull., 1970, 49, Evns, A. G., Proc. Br. Cerm. Soc., 1975, 25, Nied,. F., J. Therm. Stresses, 1983, 6, Rizk, A. E.-F. A. nd Rdwn, S. F., J. Therm. Stresses, 1993, 16, Jin, Z.-. nd Mi, Y.-W., J. Am. Cerm. Soc., 1995, 78, Emery, A. F., Wlker, G. E. Jr nd Willims, J. A., Trns. ASME J. Bsic Engng, 1969, 91, Crslw,. S. nd Jeger, J. C., Conduction of et in Solids. Oxford University Press, Oxford, Nied,. F., Engng Frct. Mech., 1987, 26, Mnson, S. S., Behviour of mterils under conditions of therml stress, Nt. Advis. Commun. Aeromut. Rep., 1954, No. 1, Buessem, W., The ring test nd its ppliction to therml shock problem. Metllurgy Group, O ce of Air Reserch, Wright-Ptterson Air Force Bse, Dyton, Ohio, utchinson, J. W. nd Suo, Z., Adv. ppl. Mech., 1992, 29, Lu, T. J., J. Am. Cerm. Soc., 1996, 79, Cotterell, B. nd Rice, J. R., Int. J. Frct., 1980, 16, Prksh, O., Srkr, P. nd Nicholson, P. S., J. Am. Cerm. Soc., 1995, 78, Tu, W.-C., Lnge, F. F. nd Evns, A. G., J. Am. Cerm. Soc., 1996, 79, Cmbridge Mterils Selector, Grnt Design Ltd, Cmbridge, U.K. 22. Gibbon, L. J. nd Ashby, M. F., Cellulr Solids: Structures nd Properties, 2nd edn. Cmbridge University Press, Cmbridge, Orenstein, R. M. nd Green, D. J., J. Am. Cerm. Soc., 1992, 75, Fleck, N. A., Kng, K. J. nd Ashby, M. F., Act mter., 1994, 42, Td,., Pris, P. C. nd Irwin, G. R., Stress Anlysis of Crcks ndbook. Del Reserch, St. Louis, Missouri, APPENDIX OVERLEAF

14 4768 LU nd FLECK: TERMAL SOCK RESISTANCE APPENDIX A De nition of Green's function F 1 z=,= in eqution (12) The dimensionless function F 1 z=,= ppering in eqution (12) is given by [25] where F 1 z, f 1 ˆ f 1 f 2 z f 3 z 2 f 4 z ˆ 0:46 3:06 0: : A1 f 3 f 2 2 ˆ 3: ˆ 6:17 28:22 34:54 14:39 1 1:5 5: : f 4 2 ˆ 6:63 25:16 31:04 14: :5 5: : : De nition of Green's function F 2 z=,= in eqution (15) The dimensionless function F 2 z=,= ppering in eqution (15) is given by [25] s p F 2 z=,= ˆ1 p 1 1 cos p 2 z 1 : A2 p 2 4