THERMAL CONTACT STRESSES OF BI-METAL STRIP THERMOSTAT. Chang Fo-van(~U~) (Tsing-hua University, Beijing) (Received Sept. 16, 1981)

Transcription

1 Applied Mathematics and Mechanics (English Edis Vol.4 No.3 Jun. 1983) Published by HUST Press, Wuhan, China THERMAL CONTACT STRESSES OF BI-METAL STRIP THERMOSTAT Chang Fo-van(~U~) (Tsing-hua University, Beijing) (Received Sept. 16, 1981) ABSTRACT The distribution shearing and normal stresses on the contact surface the two strips composing a thermostat is found in closed form. They are local type and concentrated near the ends the strip along a length almost equal to thethickhess the strip. an analytic solution only when the polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior I. Introduction the reflection shock in the explosive products, and applying the small parameter pur- terbation A bi-metal method, an thermostat analytic, first-order is made approximate up two narrow solution strips is obtained welded for the together problem and flying plate different driven by coefficients various high explosives expansion with polytropic (Fig.l). indices When other subject than to but uniform nearly equal heat- to three. ing, thermal stresses are produced and bending occurs. strip.! I_t ~ FIG. 1 materials Pr. under Timoshenko intense impulsive obtained loading, by the shock method synthesis elementary diamonds, strength and explosive materj~is welding and cladding metals. The method estimation flyor velocity and the way raising it are questions the stresses at cross sections distant from the two ends the strip. However, the problem analysing the stresses in the plane contact the two strips approach is not so simple. solving the The problem contact stresses motion including flyor is to shearing solve the following and normal system stresses equations governing are concentrated the flow field at the detonation ends along products a length behind almost the flyor equal (Fig. to I): the thickness the As a result, the maximum contact stress can be the same order that in the strip. This conclusion is practical importance in choosing the elap +u_~_xp + au tic properties the two strips. II. Thermal Stresses the Bi-Metal Strip Thermostat Let a and a, be the coefficients thermal expansion the strips M and B shown in Fig.l, and we sume %>a. E and El are the moduli elticity, and ~i the Poisson's ratios, G and G, the moduli elticity in shear, h where and h, p, the p, S, thicknesses u are pressure, density, the two specific strips. entropy and particle velocity detonation products respectively, As the with temperature the trajectory rises R uniformly reflected shock from 0 detonation f, owing wave to D the a different, boundary and the trajectory F flyor another boundary. Both are unknown; the position R and the state paracoefficients expa1~sion, thermal stresses will be produce~ and bending occurs. meters on it are governed by the flow field I central rarefaction wave behind the detonation wave D During and by expansion initial stage the two motion strips flyor.cannot also; the change position their F lengths and the state freely, parameters and thermal products

2 364. Chang Fo-van.... i....,, stresses develop on the surface contact. Generally, this thermal contact stresses include shearing and normal stresses, shown in Fig.2. t [, _a E " ' C M-- 'J -J: r, /% I a, E, C " /i'.-- ]-- _z FIG. 2 Use sine and cosine series to express respectively the shearing stress r o and normal stress o,, follows: n.tx an analytic solution only ;'o= when E 0. the si.--f-- polytropic l index detonation products equals to three. In general, a numerical analysis "'' is required. ~ / In this paper, however, by utilizing (2.1) the "weak" shock behavior the reflection shock in COS the "-fexplosive products, and applying the small parameter purw-i! terbation method, an analytic, first-order approximate solution is obtained for the problem flying plate driven by various high explosives in which the coefficients o. with polytropic indices other than but nearly equal to three. and b, are unknown. As the normal stresses will constitute a couple, b0=0, index) (A) for estimation Analysis the internal velocity force flying components plate is established. the two strips Using two sections dx apart we take out an element from the strip A (Fig. 3). Denoted by S, 0, ~I are the axial force, shearing force and bending moment acting at the cross sections. In exp. we have: ^ bl.---, a. J n=x From the equilibrium 9 / ~=-;-- 2-,--~1, "" =--t c~176 n~r / FZa. Z M= bl.~/ I b,. h a= \l nrcx. (2.3) 7g i- the Explosive element driven d~ we flying-plate get technique ffmds its important use in I_ I the study ~ behavior materials under intense impulsive loading, shock synthesis diamonds, and explosive welding and cladding metals. The method estimation flyor velocity and the way raising it are questions S M M+dM dq= --bcr. (2.2) ~' g approach solving the problem motion flyor is to solve the following system equations governing the flow field detonation dm=o_ products bh behind the flyor (Fig. I): ap +u_~_xp + From strip B we also cut out an element dx (Fig.4) and its equilibrium gives us where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products ~ -----bro respectively, with the trajectory R reflected shock detonation wave D a boundary and the trajectory F flyor another boundary. Both are unknown; the position R and the state parado, == bao (2.4) meters on it are governed by the flow field I central rarefaction wave behind the detonation wave D and by initial stage motion flyor also; the position F and the state parameters products dm1,-, bhl au ty

3 Thermal Contact Stress o[,bi-metal Strip Thermostat 365 From exps.(2 4) and (2.1) we get: (/" 21uU L ~ ~ 11"I"7 [7 17[- S ~ bl ~a.t,-- COS~i- n~x... COS n.."r ' /"~ ) l.r.-1 glx I. r I h,la. bl ~1 1 b. h, a.\,, n#x \ (2.5) M,= 7--~t-~-.~+-~-~-)t ~o~7---~o~.. ) ~ (B) The connection between the coefficients o. and b. FIG. 4 /'0 / M,+dM, Ot +dot S, +d~ i That the two strips after bending should have the same curvature entails M M, E1 =E,--~, an Substituting analytic solution M and only M, when into eqs. the polytropic (2.3) and index (2.5) in detonation the above products expression, equals we to ob- three. In general, tain : a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior the reflection 1 V-,{lb.+ha,.~( shock in the nrcx "l 1 %-,( I b. COS-'7"---- explosive COS products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem flying 2 n/x cos-~--cosnn / an From analytic the identity formula with the two corresponding parameters high coefficients explosive (i.e. detonation the two series, velocity we and get polytropic 1 [ I b.+h a.~ 1 { I b.+h, a.~ E I \-~'~" -{"~ I=E'?[, \----~ ~; -2-.-/ ' hi h # E~,,-.~-E~ b. =---~- " o,, n Eft, materials under intense impulsive loading, shock synthesis diamonds, and explosive welding and cladding or metals. The method estimation flyor velocity and the way raising it are questions b.=k a.. (2.6) approach in which solving the problem motion flyor is to solve the following system equations governing the flow field detonation h, h products behind the flyor (Fig. I): # E,I,-- E'-E-/- (2.7) k=~ J ) i E,I, E1 ap +u_~_xp + au h, h From (2.7) we can know that in the ce E-~-, =~E~ there exists only shearau au 1 ing stress in the contact surface the two y strips. =0, Since f GO" 0 2r rlrx ~-" ~ 2, GBn COS dx 1.-, 1 from exps. (2.1) and (2.6), we have n~x COS'~'~ where p, p, S, u are pressure, c,0= Eb. co~ -7-- =k ~a.. "~ i-1 density, specific i-! entropy and particle velocity detonation products respectively, with the trajectory R reflected shock detonation wave D a boundary and the Then trajectory F flyor another boundary. Both are unknown; the position R and the state parameters on it are governed by the c%=k/, flow field dr,dg I central rarefaction wave behind (2.8) the detonation wave D and by initial stage motion flyor also; the position F and the state parameters products

4 366 Chang Fo-van Eq. (2.8) means that the normal stress G0 is proportional to the slope the ~0 curve at the corresponding point. (C) Analysis stresses the two strips A and Having obtained the axial forces S and bending moment M, we can express the stresses in strip A using eq. (2.3) follows: through a,. The normal stress G, can be obtained by G_ S My 1 a.( mrx ) "... h=.~-1 7 CO$--i... COS ha" +bl t./ ~ll b., ha.\/ nsrx "~ ~-" T ) '. -~. -if-l- ~---i R cos --T----cos n.,r ) ~. l ~k Jr rl ~ tt." x I By eq. (2.7) we can eliminate b, and obtain: bt, k t + ~T'Tt, g 2/j2~.,nLC~176 an Substituting analytic solution G. into only the when equilibrium the polytropic equation index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior the reflection shock in the explosive products, and applying the small parameter purterbation method, an f". analytic, + first-order 2.. approximate solution is obtained for the problem flying Final we get velocities the shearing flying stress plate obtained rxw : agree very well with numerical results by computers. Thus index) for estimation r=,=l(_g+_k) the velocity I Y b flying kl xzh' plate is,i established. v.,. nux D-(T + -~--)(T-v~ )t...~ ~ a. nn =- (2. J o) Substituting r=, into the equilibrium equation (2.9) Explosive driven flying-plate A technique ffmds its important use in the study behavior oo OG, dy+ [ /'or='j ^ materials under intense impulsive loading, shock synthesis diamonds, and explosive welding and Io, 8.V J v -~-xau=u cladding metals. The method estimation flyor velocity and the way raising it are questions we common get the interest. normal stress G, : approach Ga, j solving the ar problem 3.V yt motion b h flyor. hi \/h is to s solve h~y the y" following system nl. ~x equations ],+r[(y,, governing the flow field detonation products behind the flyor (Fig. I): (2.1z) For the part B, from eqs. (2.5) and (2.6) we get u~ : ap +u_~_xp + au Substituting a" S, M,y J 1,_ y au bll/h au kl \i 1 v-~a./ n~rx..... T "" COS~--COS into the equilibrium equatiqrk,/2 ~.,e act;d, I p =p(p, Or', s), du=o -a-i- Y- J ou y "t f,, where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products respectively, we have the with shearing the trajectory stress R r', reflected : shock detonation wave D a boundary and the trajectory F flyor another boundary. Both are unknown; the position R and the state parameters on r;, it Jl I ~ y\. b [kl h,w k~,2\1 =1\2--'-~)-'- are governed _~77,%,7-T/~,"T'---" by the flow field I )] central Y-~-a, rarefaction.._, sin,~rx wave behind the detonation wave 7- (2.13) D and by initial stage motion flyor also; the position F and the state parameters products

5 Thermal Contact Stress Bi-Metal Strip Thermostat 367 Substituting r~, into the equilibrium equation we obtain G~ : y k rrf3h, 1/ _y'~+ b thl h,%[ h, _h I + ~ a, ncos T U - (2. i 4) (D) Solution the problem by the principle minimum strain energy method The strain energies the two strips are denoted respectively by Ui and I h/2 ] # 1, I J~ ~ I (2. is) an analytic solution only when the polytropic index detonation products equals to three. In general, a numerical -h,/z- analysis is required. In this paper, however, by utilizing the "weak" shock behavior Let /(x) be the the reflection horizontal shock displacement in the explosive point products, C for and strip applying A at the a small distance parameter x purterbation from the method, origin an due analytic, to the first-order action approximate stresses solution r0 and is a,. obtained Owing for to the the problem rise flying temperature (t--re), the total horizontal displacement C will be an analytic formula with f two (x) +a(t--t,)x parameters high explosive (i.e. detonation velocity (2.16) and polytropic Similarly, for strip B with the coefficient expansion a, the horizontal dis- placement point C is -/,(x) +a,(t-to)x (2.17) in which --/l(x) is the horizontal displacement C under the action r0 and materials under intense impulsive loading, shock synthesis diamonds, and explosive welding and cladding ~G - As ~0 metals. is negative The method for strip estimation B, a negative flyor velocity sign and is the needed way before raising it /,(x) are questions Since common point interest. C belonging to the upper and lower strips h no relative displace- ment, we have: approach solving the problem motion flyor is to solve the following system equations governing the flow field / detonation (x) + f, ( X) products -- (a, --a) behind (t--to) X=O the flyor (Fig. I): The above function when expanded into sine series should have the coefficients all equal to zero- That is I ' o {[(x) +fl(x) " au (at--a) (t--to)x} au 1 sin-- i-ax=o ".=x. (2.18) From the Ctigliano's theorem, we have (2.19) a U : =f /,(~),i..~-~:, where p, p, S, u are pressure, b density, 8a. J. specific entropy and particle velocity detonation products respectively, with the trajectory R reflected shock detonation wave D a boundary and the Besides, we have: trajectory F flyor another boundary. Both are unknown; the position R and the state parameters on it are governed by the flow field I central rarefaction wave behind the detonation wave D and by initial stage motion = xsln --T---ax=-- n~x. flyor also; the lz m* cosnx position F and the state parameters (2.20) 9 products I ap +u_~_xp + au

6 368 Chan~ Fo-van Substituting (2.19) and (2..20) into (2.18), we get: OU, +OU. bl" aa. ~ + (at-a) (t--to n= eos.==o (2.21) or OU bp Oa---~+ (a,--a) (t--to) " mr cos.~-- 0 (2. e2) in which U=Ui+U i is the total strain energy the thermostat. Substituting eq. (2.15) into eq. (2.21), we have an equation for solving the coefficients (l n 9 Jo J hi2 (15 OO. Oa. 1 #q12 t'~'-~t-'-~ -' G o~. <z~ ( -- +a; an analytic -I-(a,--a) solution (t--to) only h/i-cos when,=----0 the polytropic index detonation products (2.23) equals to three. In lib general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior Substituting the the reflection stress shock components in the explosive ~,, ~i, products, rx~, ~;, and ~;, applying r', the into small eq. parameter (2.23) we purterbation method, an analytic, first-order approximate solution is obtained for the problem flying get ~,... ~ -.,, "' i'.,+-~--.~+-m-, r ~1 " ~.~.,+~ ".h, "' (,~, "' Final velocities flying plate obtained agree very well with, )]} numerical results by computers. Thus +h" =' d l r i, kl ill, M 1 hi r 1. kl # 9 kl 1: h ~ l" I r2 kl,, kl \-I i x lt-2. 6kl] 1, kl \l, 1 h,f2 +#-..t~l~+~, t' +~)J-;" 1. xl-~+-~-t Introduction *~ )J*~v, #L-3- +,,, ~o k, ' )]+--Dr-",,,r 2 0~,, "' "~'.'. ~,~co~.. materials under intense impulsive loading, shock synthesis diamonds, and explosive welding and cladding metals. The method estimation flyor velocity and the way raising it are questions 9 {E-[' +-~# (' *-z-, 1]+ ~ ~[' * ~(~-' 1]}+... Under the sumptions one-dimensional plane detonation and rigid flying (2.24) plate, the normal Eq. (2.24) can be used for solving the coefficients a~ and written in the folapproach solving the problem motion flyor is to solve the following system equations governing lowing simple the flow form: field detonation products behind the flyor (Fig. I): in which ~[n4ac2qnz+pz]=--ccosn= (2.25) ap +u_~_xp + au 2,flFi..l._u ~_/lt (i..l - &l xl i hrl. 3 kl l kl ~,~, -~-,)J+~,-~L T ~h, k~rt, -~ )]} P'=h" =~r~[l, kt tl,13 kl \~,. ~ hl [1'~ kt /J3 ~l ] ~ 1 2, kl 6kl\l ~ p =p(p, lr2-- s), 6kl(. kl ~t' l h, F2 Td-d-[Y* ~-7, ( '+ --~;J-~'-~L~ +-D-~-t ' + --~ zj*t6v, xl~ t- kl t kl "h, r. 2 ::6kl; kl where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products (2.26) respectively, '2"=,,_'.-_'fAr_, with the _, ~L(,,_,_.o trajectory R k,,~_, reflected.,,; shock r, - detonation t,~ ~.. kt wave,,~l D a boundary and the trajectory F flyor another boundary. Both are unknown; the position R and the state parameters on it are governed by the flow field I central rarefaction wave behind the detonation wave D and by initial stage motion flyor also; the position F and the state parameters products

7 Thermal Contact Stress,Bi-Metal Strip Thermostat ~ =.-I n t,/~_, t. =n x kl V1, I htr 3kl/ kl \'1'1 C=h' ='I 1 [ I+ ~k7/l 1+13 hi '~]+ 1. h,' [ 1. kl i.. kl 11u ~'~-61.-~L-~ -~-kt =h /J E, h'~ly~-&-~, ~,~-=~,---3 ]jr In solving eq. (2.24), let us set ~a. cos.==s Jt~l J From eq. (2.25) we obtain (2.2z) --Cncos tiff a. = n4 + 2rln2 + p 2 (2.28) Substituting (2.28) into (2.27) we have: The one-dimensional -CE i =S (2.29),., n~+ problem 2r~n2+# ~ the motion a rigid flying plate under explosive attack h an analytic solution only when the polytropic index detonation products equals to three. In general, AS the sum a numerical the infinite analysis is series required. is In known, this paper, we can however, express by S utilizing in terms the "weak" C 9 shock behavior Substituting the the reflection value shock S into the the explosive third products, part eq. and applying (2.26), the we can small get parameter the purterbation method, an analytic, first-order approximate solution is obtained for the problem flying value C 9 The sum the series is given in the appendix follows: J _.= Y sinh2f/3+fl sin 2=? 1 (2.30] index) for.- estimation ng3r2~nz+p z the -~.#Z--r]2 velocity flying sinh plate 2/!/f+sinZy= is established. 2P l in which materials under intense impulsive loading, shock synthesis diamonds, and explosive welding and cladding When ~ is metals. comparatively The method large, estimation is ten flyor velocity the ce and, the the way sum can raising be it put are questions in a common simpler interest. form: approach solving the problem motion flyor is to solve the following system equations 1 ~ 1 (2.3D governing the flow field n' + 2~n2 detonation + p 2 = =lp~o products 2# 2 behind the flyor (Fig. I): m--t Substituting exp. (2.28) into (2.1) and using the sum another series in the appendix, we get the contact shearing ap +u_~_xp stress + au rg. au au n~x 1 9 n COS n= SIll --~-- sin-f-- = -- C ~ ~+2~n*+P* = r... C= ". o.'.",~ sinh---v--x =/9 cos-"s cosh :rb sin =P 4pY! s~n tlz pft stn-y=;t ~ 9 _ cosh _~--x sin ~Y~x sinh =~ cos =F} (2.32~ where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products respectively, with the trajectory R reflected shock detonation wave D a boundary and the When ~fl is comparatively large, cosh.~fl=sinh=/~; and in thevicinity the ends trajectory F flyor another boundary. Both are unknown; the position R and the state parameters strip on it 9 sinh-7--x=cosh-7 are governed =8 by =8 the ~x. flow Hence field I the central exp. rarefaction (2.32) can wave be simplified behind the detonation follows: wave D and by initial stage motion flyor also; the position F and the state parameters products

8 370 Chang Fo-van C# sinh'-~-x sin =~'(I--'~) r~165 sinh a'fl (2.s3) From eq. (2.32) the shearing "force obtained from r is obtained. r~ Cz r ~ r r ~r]' P=b / r0dx=- -., ~-a-.,., / ~sinh -7--xcos-T-x cost* ~r,8 sin ~Y Jo 4flY(sinn pn-l-sln ~rejoo~ --cosh-~--xsin/y x sinh trfl cos ny} dx Cbl f,81 +sin~:r ~[sinh :~fl cosh zfl--sinh =fl cos aq' " sinh' (2.34) When fl= is a large value, the above result can be simplified follows: an analytic solution only when the polytropic index detonation products equals to three. In general, a numerical analysis p_ Cb( is required. In this paper, however, by utilizing the "weak" shock --4tip (2.35) behavior the reflection shock in the explosive products, and applying the small parameter purterbation From eqs. method, (2.8) and analytic, (2.32), first-order we have approximate the normal solution stress is G0 obtained the for contact the problem surface flying plate driven by various high explosives with polytropic indices the two strips. Generally, ~ is large and sinha',8=cosh=,8. other than but nearly equal to three. Therefore, the normal stress ~0 is: f ~ dr0 kc= G~ =.~ =p =~ 1.. Introduction z)' ) -- (,8 cos.~+~ sin =3') sinh "--I---~xsm-7--xf (2.36),ear the end the strip Ga can be formulated folmaterials under intense impulsive loading, shock synthesis diamonds, and explosive welding and cladding lows : metals. The method estimation flyor velocity and the way raising it are questions kc= sinh ~x { fl sin ny (i --~)--?cos=?(! --r Under (2.3T) G~ the 4~Y sumptions sinh =,8 one-dimensional plane detonation and rigid flying plate, the normal approach solving the problem motion flyor is to solve the following system equations Now let us calculate the couple M formed by o~. From eq. ~z.8) we get governing the flow field detonation products behind the flyor (Fig. I): f t -. I f~ dtqdx M = --J~ bg. x dx= -- ap } +u_~_xp + c, au = b 9 (2,38) ;, 4pp The negative sign before the integral sign comes from the fact that when G0 is tensile stress, the bending moment produced is negative. Ill. Two Illustrative Examples (I) First illustrative example: The thermostat h its two strips equal thickness, h=h,,g=g,, where p, E=E,, p, S, u are pressure, density, specific entropy and particle velocity detonation products #=#,. The coefficients expansion are respectively and a,.the respectively, with the trajectory R reflected shock detonation wave D a boundary and the temperature rises from t o to t. Calculate the thermal contact stresses the trajectory F flyor another boundary. Both are unknown; the position R and the state parameters two strips. on it are This governed example by the w flow used field by I Pr. central Timoshenko rarefaction in wave his behind paper. the detonation wave D and by initial stage motion flyor also; the position F and the state parameters products

9 Thermal Contact Stress Bl-Metal Strip Thermostat 371 From eq. (2.7), k~0. Hence in the contact surface there exists only shearing stress z0 9 Eq. (2.24) gives:.. i' 28.!" 420 s, cosm~ 3~" I--I n ] Written in simplified form, it becomes ~---[.'+2.~+P'] : cosnnc in which / 28 i <<,,-o> <,-,.> +-'. SuppOse The the one-dimensional strip h its problem ~-=10, the then motion a rigid flying plate under explosive attack h an analytic solution only when the polytropic index detonation products equals to three. In / 10'. general, a numerical 28 p=~-~-x420 analysis is required. =207.65t In r/=~-x100x~r~= this paper, however, by utilizing the "weak" shock behavior the reflection shock in the explosive products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem flying /9:41( Jr ) = an analytic formula with : /89 two parameters high explosive (i.e. detonation velocity and polytropic From eq. (2.31), we get: l = I ~' n4+2 X n: ' : X X * = i--1 Substituting Explosive C driven the flying-plate third part technique eq. (2.26) ffmds its into important (2.29) use and in using the study (2.27), behavior we materials under intense impulsive loading, shock synthesis diamonds, and explosive welding and cladding get metals. The method estimation (a _a)~o~ (t_to) flyor + velocity ~,S ~ and the way raising it are questions I ar z 2 I =S Under the sumptions one-dimensional I0' 70 plane 3 E detonation and rigid flying plate, the normal approach solving the S = problem E motion (a,--a)(!--t0) flyor is to solve the following system equations governing the flow field detonation products behind the flyor (Fig. I): From eq. (2.29) we have --C x = E (a,--a) (t--t,) C= E (al --a) ap (t--to) +u_~_xp + au From eq. (2.32) and owing to the sinh =Bfficosh=/9, we have the shearing stress r0 : r, ~ = x3.1416E(o,_a)(t_to) ~ s i n ~ x ~ It0 4X X [~~-9~- cos ~sins.tssszr ~r where p, p, S, u are cosh pressure, ~-~-----x density, specific entropy and particle velocity detonation products sis ~x cos ~} respectively, with sinhl3.219~r the trajectory R reflected shock detonation wave D a boundary and the trajectory F flyor another boundary. Both are unknown; the position R and the state para- The following Table gives the thermal shearing stresses i meters on it are governed by the flow field I central rarefaction wave behind the detonation wave D and by initial stage motion flyor also; the position F and the state parameters products

10 372 Chang Fo-van I 0.851(8.5h) I0 91(9h) 0.92/ i" E(a, --a)(t--tg) ! 1.92kD"cm z [ [ 0.95 ;' (9.8h~ ! i. (lob) The third line indicates the thermal stresses when E=2 X loekg/cm z, (a,--a) =4X10 -e, (t--to) =200~.From the above results it can be seen that the shear- ing stresses are concentrated at the end the strip and the length is about J I0 the thickness.. In Fig.5 the distribution the stresses is also shown. Pr. Timoshenko in his paper on t;~ (al'--~)(t--to) "AnalySis Bi-Metal Thermostat"used 02 c, The Distribution Curve the ordinary method Strength Materials to calculate the maximum 0.2( an analytic solution only when the polytropic index detonation products equals to three. In general, stress at a numerical the cross analysis sections is required. far from In this paper, however, by utilizing the "weak" shock behavior the ends the the reflection strip. shock His in result the explosive is 0.I[ products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem flying I plate driven by u=ax=~e various high (a,--a) explosives (t--to) with polytropic indices other than but nearly equal to three. O.1C an analytic According formula to with our two present parameters analy high explosive (i.e. detonation velocity and polytropic index) sis, we for can estimation first get the the velocity contact flying plate 0.0[ is established. stress and then proceed to figure out the above maximum stress.thus we can O ~ 9/ check whether our method solution FIG.5 The Distribution the Explosive driven flying-plate technique ffmds its important is correct or not. From eq.(2.35), Shearing use in Stresses the study the behavior Bi materials under intense impulsive loading, shock synthesis Metal diamonds, Thermostat and explosive welding and the shearing force in the contact surcladding metals. The method estimation flyor velocity and the way raising it are questions face common is interest. Under the sumptions p Cbl one-dimensional plane detonation and rigid flying plate, the normal approach solving the problem motion flyor is to solve the following system equations In the cross sections distant from the ends the bending moment will be governing the flow field detonation products behind the flyor (Fig. I): k hcbl M= "'--_P = 2 ap +u_~_xp + au The maximum normal stress in the cross sections distant from the ends is equal to P 6M Cl 6Cl Cl Cl,~ m. = ~ ~ r = -4-fi),T + %f ~ = T~T 0 + 3) = ~ p h Substituting C=137.25E (a]--a)(l--to), ~=13.219, p=207.65, ~=I0 into the above equa- tion, we get: where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products I37.25E(a,--a) (t--t0) x I0 = (a,--a) (t--to)e respectively, with the trajectory R X reflected shock detonation wave D a boundary and the trajectory F flyor another boundary. Both are unknown; the position R and the state parameters This on it value are governed coincides by the with flow that field I Pr. central Timoshenko's. rarefaction wave It behind necessarily the detonation indicates wave D and by initial stage motion flyor also; the position F and the state parameters products 1= loh

11 Thermal Contact Stress Bi-Metal Strip Thermostat 373 that our analysis the thermal contact stresses is correct. (2) Second illustrative example: As an example involving not only shearing stress v0 but also normal stress r0, let us postulate that h-~--h,, /=10h, E=2X 10ekg/cm 2, p+=0.3, G'=8Xl05kg/cm 2. E,=l.05Xl0"kg/cm z,,u,=0.3, Gt----4xl0~kg/cm z, at--a=4xjo"'+ t--to=200~c. From eq.(2.7), '(~;), k= zh El l-- -- =h 1--~ zh 2l 1 (l +~) 21 I +l= 61 E, 2 kl kl 1 From eq.(2.26), {, + (,+~_) ++[+++ The one-dimensional,, _8688_ problem ++ the motion a + rigid (}-')]} flying plate under =+6+4+ explosive attack h ~+,-- l {~+_+ 1 J an analytic solution only when the polytropic (+~+,3)+,. index [~++, detonation (~ ~)]} products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock #='63.85 behavior the reflection shock in the explosive products, and applying the small parameter pur- 1.3 /2 ' \ terbation method, an analytic, + t,+-++~,--t first-order (~- approximate +{) +~ solution is +-+- obtained N0.3(-+-++) for the problem flying plate driven by various 2?7= high explosives ( 1 ~ with polytropic 35 indices l other than but nearly equal to three. Final velocities flying plate obtained agree very +38+~ well with numerical (%~-~)]} results by computers. :-+"+~ Thus an analytic formula r/= with two parameters high explosive (i.e. detonation velocity and polytropic ~= ~ (,63.85+,24.45) =~,44.,5= t~.o07 materials From eq. under (2.30), intense we get: impulsive loading, shock synthesis diamonds, and explosive welding and cladding metals. The method estimation flyor velocity and the way raising it are questions ,6 1.-! n++2x nz ' 4 x,63.85x L x =--'0" approach From the third solving part the problem eq. (2.26), motion we have flyor is to solve the following system equations governing the flow field detonation products behind the flyor (Fig. I): C-- ~(o o>(,,.,o o3,83 + o 4o~28(,+~ 08"18~['+~( ~1]} l 35 i 1 (!'f%-e696 t0,,0 {~+~+~[~+~ (~-~1]} ap +u_~_xp + au ----{E(a,--a) (t--to) I 145S} I0+ X Substituting C into eq. (2.29) and using the sum the series, we have Therefore, J76 104{E(a,--a) (t--to) O l145S}--'--S S= E (al--a) (t--to) where p, p, S, u C=E(,,l--a) are pressure, (t--to) density, 10" x specific 4,.8176[ entropy and I. J particle 145 x ] velocity detonation products respectively, with = the E trajectory (a, -- R a) (t-- reflected to) shock detonation wave D a boundary and the trajectory F flyor another boundary. Both are unknown; the position R and the state para- Then from eq.(2.32) we have the shearing stress: meters on it are governed by the flow field I central rarefaction wave behind the detonation wave D and by initial stage motion flyor also; the position F and the state parameters products

12 374 s 9 Fo-Van lslnh ~ x X (t--to) / n ~e X'I2.007 X E (<z~--a) si~hl2.007~ cos " i ---- msin4.4385n ] cosh~x " ~XCOS4.4385z r -- sinhl2.'007= sis Near the end the strip r, becomes sinh ~07. ~ 12. x r.=l.o523e (a,--a)(t--to) sinh ~ sin :r(1-----~ ) And near the end the strip, from eq.(2.37) the normal stress Go is The one-dimensional problem the motion a rigid flying sinh ~07 ~ 12 plate under x explosive attack h an analytic solution 9 ~ only when s the polytropic index detonation products equals to three. In (T O 6XlO 4X12.007X (a,--a) (t--to) sinhl~.o07= general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior the reflection shock in the explosive products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem flying Final velocities In the folloiwng flying plate Table, obtained the distribution agree very well at with several numerical mints results by To computers. and (T0is Thus an given. analytic formula with two parameters high explosive (i.e. detonation velocity and polytropic index) for estimation 0.St the velocity flying plate is established. X (Sh) ,96 I I~lOh! o.ooo2 je(a,-.)(,-,,) I ~ O. 32krJcm' T I materials t oe under ECa --a)(t-to) intense I U.UU~4 impulsive loading, O.OS4 shock synthesis diamonds, and "0.104 explosive welding and cladding metals. The I method estimation flyor velocity and the way raising it are questions O,88k~cm* ! Under The above the sumptions results show one-dimensional that in the contact plane detonation surface and rigid the two flying strips plate, the the normal approach solving the problem motion flyor is to solve the following system equations stresses are concentrated near the ends, covering a length the same order governing the flow field detonation products behind the flyor (Fig. I): the thickness the strip. The third and fifth lines the Table are respec- O. 103 O tively the values re and Go, when the numerical value is substituted into the two above-mentioned ap +u_~_xp + expressions au concerned. curves for r an~ Go are 'shown R(u,--u).(t--t,) In Fig.6 the The normal stress in the cross sections the strip is composed two parts. One is due to the axial force p producing uniform stress. From eq. (2.35), for strip A, we have P Cl (T'=b--F=T~%- where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products respectively, The other part with is the due trajectory to the R bending reflected moment shock M, detonation which is also wave D formed a boundary by two and the trajectory parts. One F is flyor from another shearing boundary. stress Both r0 are, unknown; namely, the position R and the state parameters on it are governed by the flow field I central rarefaction wave behind the detonation wave D and by initial stage motion flyor also; the position F and the state parameters products

13 .... ' Thermal Contact Stress Bi-Metal Strip Thermostat 375 T,. oo Mi =#P h Clb s (o* ~ )(r --li) and the other part is from the couple due to on. From eq. (2.38) we have OAt ~~~. kl Clb ~ ' = -V-" 4--~ The maximum bending stress is: lh+ k/ ~C/b s ~;' =it.'--'~-]4-'~ b-~- For the cross sections distant from the ends, by adding oi and ~z we get the ma- -OA6 7 The one-dimensional problem the motion "0,20 a rigid flying plate under explosive attack h ximum normal stress for the strip: an analytic solution only when the polytropic index "0,:l detonation products equals to three. In general, a araix numerical CI analysis ( is 6kl required. ), In this paper, however, by utilizing the "weak" shock FIG. 6 Shearing Stress ro vs., behavior the reflection shock in the explosive products, and applying the small parameter pur- Positive Stress ao Curve terbation method, an analytic, first-order approximate solution is obtained for the problem flying Since for this problem -~-=~, M 1 we have h=h,, If r0b, Ef2.lXlO'kg/cm', G:gXl0Skg/cm *, #=0.3, Elfl.OS Final velocities flying plate obtained agree very well Gi=4XlOSkg/cm with numerical 1, #:0.3 results by computers. Thus I0 O'mix ~ X 5E (a, --a) (t-- t0) x 5,1 X X OA = E (a, --a) (t--is) Appendix materials The under Sum intense the impulsive Two Series loading, shock synthesis diamonds, and explosive welding and cladding metals. The method estimation flyor velocity and the way raising it are questions (1) We start from the following infinite product: #'" e " ==z(l+z') I+ I+~ -- (I) approach solving -: 7 the problem ( motion :-7)(:') flyor is to solve the following system equations governing the flow field detonation products behind the flyor (Fig. I): Taking logarithm and differentiating both sides, we have For ~<p ~ L_l = ~--~--coth l (2),'+ z' 2z. =z -'~-~ ap +u_~_xp + au, from eq. (2) we can calculate au au the 1 sum the following series: '. ' ] ~,.,+.,~.+p. 2iWp~-~,~ <,'+(.-iwp,-.,)-~ I1-1 rl-i where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products ~cot h,~/~-/-"~'pz -- n' + l I respectively, with the trajectory R reflected shock detonation wave D a boundary and the trajectory F flyor another boundary. Both are unknown; the position R and the state parameters on it are governed by the flow field I central rarefaction wave behind the detonation wave D and by initial stage motion flyor also; the position F and the state parameters products

14 376 Chan~ Fo-van Since J~+~/~-~=,,/~--(~%~-+, ~+Cp-.) the above expression is simplified by equating the real parts, - ""+2~rl+P~ 1 = Fsinh2~z/9+ flsln2xy ] =8P#P " sinhi~ga'-fs-ih~ya -- (3) in which #=,,/+ (?+.), v=~/+cp-.) (2) By contour integration, we have tlcosll~sifl~r~: sinh =z x l... t~+z 2 =(--1) ~' sinh~z (4) For ~<p, from eq.(4) we can write the series follows: The one-dimensional 9 nifx TI."~X tlcos?2ffs;i1 problem --[ the I motion F ticosi/iksii1 a rigid T flying plate r/cosrl under HS 11"1- explosive [ attack h an analytic solution only when the polytropic index detonation products equals to three. In - ~u165 r~-~ ri-j general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior the reflection shock in the explosive products, and applying the small parameter pur-! sinh terbation method, an analytic, first-order approximate ~x solution (/~_,~,) is sinh_~_cb+,v) obtained for the i problem flying Expanding the above expression and equating the real parts, we get Z rl-i tlcosn~' $1H----- l.,+2n,r/+p z 2~/p z-- q~ (slnh2=/?+ slnor) ' ). cosh x sintx sinh:r~ cos=y'--sinh~-.x cos-t~x cosiest# sin=v (5) materials under intense impulsive loading, shock synthesis diamonds, and explosive welding and cladding metals. The method estimation flyor velocity and the way raising it are questions References I. Timoshenko, Under the sumptions S., Analysis one-dimensional bi-me~l plane the2~ostat, detonation Journal and rigid flying the Optical plate, the So- normal ciety America, ii, (1925). approach solving the problem 2. Boley and Weiner, Theory motion thermal stresses, flyor is to solve the following system equations John Wiley and Sons, Inc.(1960). governing 3. Phillips, the flow E.G., field Functions detonation products a Complex behind Variable, the flyor (Fig. Interscience I): Publisher,(1947). 4. Chang Fo-van, Cylindrical bending long rectangular plates, Journal Franklin Institute, April, (1950). ap +u_~_xp + au where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products respectively, with the trajectory R reflected shock detonation wave D a boundary and the trajectory F flyor another boundary. Both are unknown; the position R and the state parameters on it are governed by the flow field I central rarefaction wave behind the detonation wave D and by initial stage motion flyor also; the position F and the state parameters products