ON THE METHOD OF ORTHOGONALITY CONDITIONS FOR SOLVING THE PROBLEM OF LARGE DEFLECTION OF CIRCULAR PLATE* Dai Shi-qiang ( ~ )

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1 Applied Mathematics and Mechanics (English Edition, Vol. 12, No. 7, July 1991) Published by SUT, Shanghai, China ON THE METHOD OF ORTHOGONALITY CONDITIONS FOR SOLVING THE PROBLEM OF LARGE DEFLECTION OF CIRCULAR PLATE* Dai Shi-qiang ( ~ ) (Shanghai Institute of Applied Mathematics and Mechanics: Shanghai University of Technology. Shanghat3 Received July 20, 1990) In this paper, we reexamine the method of successive approximation presented by Prof. Chien Wei-zangfor solving the problem of large deflection of a circular plate, and.find that the method could be regarded as the method of strained parameters in the singular The one-dimensional perturbation theory. problem In terms of the of motion the parameter of a rigid representing flying plate the under ratio of explosive the center attack has an analytic defleetion solution to only the when thickness the of polytropic the plate, index we make of detonation the asymptotic products expansions equals of to the three. In general, a numerical deflection, membrane analysis is stress required. and the In parameter this paper, of load however, as in -Ref. by [i], utilizing and then the give "weak" the shock behavior of orthogonality the reflection conditions shock in (i.e. the the explosive solvability products, conditions) and for applying the resulting the equations, small parameter by purterbation method, which the an stiffness analytic, characteristics first-order approximate of the plate could solution be determined. is obtained It is for poh:ted the problem out that of flying plate driven with by various the solutions high explosives for the small with deflection polytropic problem indices of, other the than circular but plate nearly and equal the to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus orthogonality conditions, we can derive the third order approximate relations between the parameter of load an'd the center deflection and the first-term approximation of membrane stresses at the center and edge of the plate without solving the differential equations. For some special cases (i.e. under uniform load, under compound load, with different boundary conditions), we deduce the specific 1. expression~ Introduction and obtain the results in agreement with the previous ones given by Chien Wei-zang, Yeh kqi-yuan and Hwang Chien in Refs_ [1-4]. Key words cladding of metals. The method circular of estimation plate, large of flyor deflection, velocity method and the of way strained of raising parameters, it are questions of common interest. orthogonality condition, perturbation theory I. Introduction approach of solving the problem of motion of flyor is to solve the following system of equations governing Several the decades flow field ago, of detonation Prof. Chien products Wei-zang behind succeeded the flyor in solving (Fig. I): the large deflection problems for elastic circular plate under uniform load or under a concentrated load at the center in terms of the parameter representing the ratio of center deflection to thickness, which inspired a series of related work that was comprehensively ap +u_~_xp reviewed + au by Prof. Yeh Kai-yuanln. It was generally acknowledged by scholars abroad and au at home au that 1 y Chien's =0, solution is much better than the other ones (for reviews, see Refs. [3,4]). Recently the author has meticulously studied Chien's method (i.0 and found that it should be regarded as as as the method of strained parameters in the singular a--t perturbation theorytsj. Starting from physical consideration, Chien examined and changed the procedure of using directly dimensionless p =p(p, parameter s), of load as perturbation.parameter which brought about unsatisfactory results, chose the center deflection as the small parameter and where expanded p, p, the S, u parameter are pressure, of load density, as the specific asymptotic entropy series and particle of the small velocity parameter of detonation as well, products whose trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para- *Project-- supported by the National Natural Science Foundation of China. meters D and by initial stage of motion of flyor also; the 617 position of F and the state parameters of products

2 618 Dia Shi-qiang coefficients were determined in the process of solving the perturbation equations. If the expansion were not made, there would be no solutions for higher-order perturbation equations. This kind of techniques of straining parameters is now called the method of strained parameters or the Lindstedt-Poincar~ method or the Rayleigh-Schr6dinger method in quantum mechanics. No one seemed to have applied the method in the field of solid mechanics before Chien's paper [1] was published. When the above method is used to solve the large deflection problems of a circular plate, there appears a series of nonhomogeneous linear ordinary differential equations, and the equilibrium equations of higher-order terms are with homogeneous boundary conditions. Thus it is possible to give the orthogonality conditions or solvability conditions for these equations and then to determine the coefficients of load expansions. In the next section, we shall outline the basic equations and their perturbation equations for the problems under general symmetrical load and general boundary conditions and derive the adjoint equation and orthogonality conditions for the perturbed equilibrium equations. In Sections III to V, we shall carry out specific calculations for three special cases (i.e. with uniform load and rigidly clamped edge, with uniform load and general boundary conditions, with compound load and rigidly clamped edge), and without solving the equations, give out the third-order relations between the load parameter and central deflection and an analytic solution only when the polytropic index of detonation products equals to three. In general, the first-term a numerical approximations analysis is for required. the membrane In this paper, stresses however, at the by center utilizing and the edge, "weak" which shock agree behavior completely of with the reflection the known shock results in the presented explosive in Refs. products, [I -4]. and applying the small parameter purterbation In comparison method, an with analytic, the procedure first-order of approximate directly solving solution the recurrence is obtained equations, for the problem the method of flying of plate rthogonality driven by conditions various high proposed explosives herein with polytropic seems to indices be relatively other than simple but nearly which equal involves to three. only Final elementary velocities integrations. of flying plate If only obtained the typical agree elastic very well behavior with numerical of plates results and shells by computers. is needed Thus to be understood, our method has its advantages. II. Basic Equations and Orthogonality Conditions We assume that the radius of circular 1. plate Introduction is R, the uniform thickness is h, the.lateral load is qf(r) (where q is the parameter of load), the radial and tangential membrane stress resultants is Nr, N,, and they satisfy the well-known K~rmfin equations: of common interest. dr r ~r--d-~r =iv" +r,, F(r)dr approach of solving the problem of motion of flyor is to solve the following system of equations r dr r dr (.2.1) governing the flow field of detonation products behind the flyor (Fig. I): N,.= d--~-(rn,) where E is Young's modulus, D=Eha/['12( 1-vZ)] and v is Poisson ratio. The boundary conditions are I at r=r, as dzw as + v dw _. ki dw w=0'7 -;-.'-tit-r- D dr a--t (z.z) kt F dn,. N, E-rLr-d7 + ( 1 - v)n, ) where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the drd3 L trajectory at r=o, R of reflected and shock N, remain of detonation finite wave D as a boundary and the trajectory F of flyor as another boundary. dr Both are unknown; the position of R and the state parameters (i.0

3 where kl = 0 and k~=oo and hi =co and hi =0 and kl~ 0 and Large Deflection of Circular Plate k==0 tbr the simply supported edge; kz=o for the simply hinged edge; k2=e~ for the rigidly Clamped edge; kz =c,o for the clamped but freely-slipped edge; kz~o for the other elastically supported edge. Introduce the following dimensionless quantities: x=rz/r z, W=,,, S=3( 1 -vz)rzn,/et?, 3( 1-v z) w/h T =3( 1 -vz)rzn,/em,1 O 0=-2-- (1-v z),,,/3(1-vz)qr4/eh ' 619 (z.3) Thus the Kfirmfin equation and boundary conditions are turned into (2.4a) an analytic solution! d z only when the 1 dw polytropic : index of detonation products equals to three. (2.4b) In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of 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 of flying Final and velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high dw" explosive,dzw (i.e. detonation ds velocity and polytropic index) for estimation l of atx=l, the velocity W=0, of flying -~=-Z--~-,z-, plate is established. S=-# d-----~ at x = 0, --dw ~,/x --d'~' 1. Introduction S finite, (2.4e) (2.s) where of common interest. Therefore Under the the problem assumptions is reduced of one-dimensional to solving Eqs. plane (2.4a, detonation b, c) under and the rigid boundary flying plate, conditions the normal (2.5). approach Let of W(0) solving = ~ the and problem in terms of of motion the small of flyor parameter is to solve W make the following the asymptotic system of expansions equations governing the flow field of detonation Q =q~w,, products + behind q3w ~. +.'. the flyor (Fig. I): t W =wl( x )W,~ + ~o3( x )W?, + "'" (2.7) S=S~(x)W~ S,(x)W; +... T=Tz( au x)w~ au + 1 T,( x)w',, + "" Substituting them into Eqs. (2.4) an.d (2.5) and equating the coefficients of W" (n = 1, 2, 3 (i.0,...) on both sides of the resulting expressions, as we as obtain the recurrence equations a--t where p, p, S, u are pressure, wl(1)=0, density, specific wi(1)+aw"(1)=0, entropy and particle wl(0)=l velocity of detonation products respectively, with the trajectory d z R of reflected. 1 shock,3 of =H~(x) detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed by Sz( the 1 flow ) + us field ~ ( 1 I ) of = central 0, S,( rarefaction 0 ) finite, wave behind the detonation wave { -EU (xs') =-T.w' (2.9)

4 620 Dia Shi-qiang dz (x dw3"~=s2w'+q3g(x)=ha(x) "-d-ff~x z d x ] ' wa(1)=o, w',(1)+kw~(1)=o, wa(0)=0 (2.1o) and T2=Sz + 2x dsz dx (2.1I) As a preliminary procedure, we shall give the solvability conditions (i.e., orthogonality conditions) for the problem of the type (2.10). Consider the problem x-~-x )=H(x) (xei-0,1]) q)(0)=0, q)(1)=0, q)'(1)+2q)"(1)=0 (2.12a) (2.1zb) We prove the following theorem: Theorem I The solvability condition (or orthogonality condition) for the problem (2.12a,b) is an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is <e, required. H>=J'0 In this paper, however, by utilizing the "weak"(2.is) shock behavior of the reflection shock in the explosive products, and applying the small parameter purterbation where method, an analytic, first-order approximate solution is obtained for the problem of flying ~=xln-~-2x (2.14) Final velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic Proof formula Integrating with two by parts parameters and using of high the explosive boundary (i.e. conditions detonation (2.12b), velocity we and have polytropic <r,lr =[ ( I -,~)~( 1 ) + 2r J~o" (1) - ~(0)r --<~,L~0> (2.15) where 1. Introduction Explosive driven flying-plate technique ffmds its important use in the study of behavior (2.18) of cladding If ~ satisfies of metals. the adjoint The method differential of estimation equation of of flyor Eq. velocity (2.12a) and the way of raising it are questions of common interest. Under the assumptions of one-dimensional Lip plane -=0 detonation and rigid flying plate, the normal (2.17) approach and the boundary of solving conditions the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I):,P(0) =0, (1--2)~p(1) +2~p'(1) =0 (2.18) then the right side of Eq. (2.15) is turned into zero. It can be easily verified that ~b given in Eq. (2.14) satisfies Eqs. (2.17) and (2.18). Then it follows that (qj,l~pf=<~,h) =0 (2.19) (i.0 The proof of the theorem has been completed. as as a--t Corollary 1 As 2=0, or as the boundary conditions (2.12b) become ~0(0) =~o(1) =r =0 (2.20) where the orthogonality p, p, S, u are condition pressure, (or density, the solvability specific entropy condition) and particle for the problem velocity of (2.12a,b) detonation is reduced products to trajectory F of flyor as another <xlnx, boundary. H>=Ii Both are H(x)xln unknown; the =0 position of R and the state (2.21) parameters

5 L1Sz)= lii. Large Deflection of Circular Plate 621' The Case of Uniform Load and Rigidly Clamped Edge This is the case examined in [1 ]. For this case, F= G = 1,,~ = 0 and/a =2/(1 --v).the solution for the problem (2.8) (i.e., the problem of small deflection) is qt can be also derived from the equation w1=(x-1) ~, ql=4 (3.1) 1 (xlnx, Lwl)=--wl(O)=--l=(xlnx, qi)=--~-q~ (3 9 2" ] Thus the problem (2.9) becomes Through integration by parts we obtain d = LISz =--d-~xz (xs2) = - 2(x- 1 )z = H~ { S2(1)+t*S~(1)---0, S~(0)finite an analytic solution only when the = polytropic x dx----t(xs2)dx--s~ index of detonation ( 1 ) products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter purterbation method, an analytic, first-order =<x, approximate H2)=--2I l solution x(x- is 1 obtained ydx = - for 3_ the problem of flying o 6 Final and hence velocities of flying plate obtained agree very well with numerical results by computers. Thus index) for estimation of the velocity S'(1) of flying =----~-, plate 1 is S,(1) established. =~t* (a.3) (a.o (3.s) Similarly, from 1. Introduction Explosive driven flying-plate technique 1 ffmds its important ' 1 St(O) use in the study of behavior of x-- 1--# ' 1--~ I H2) t* of common interest. = x-- l--t* ' 6(l--t*) approach it follows of that solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I): From Eqs. (2.!1), (3.5) and (3.7) we ap get +u_~_xp + au (.3.6) s, (0) =+(3+ u) Ca.7) r,(1) =@t*-2), r~(0)=~-(a (i.0 (a.o as as In this way, we have given the first-term a--t approximation for the membrane stresses at the edge and center of the circular plate. Now we turn to examine the problem (2.10), i.e. where p, p, S, u are pressure, density, specific d z / dtu~'~ entropy,., and, particle velocity of detonation products respectively, with the trajectory { L~,~ R =-2W'=, of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state paraw,(0) =w,( l ) == meters on it are governed by the flow field I of central ~(1 ) =0 rarefaction wave behind the detonation wave kx-e~--)=.~,,,,, +q~=ns(x) (3 9)

6 622 Dia Shi-qiang By means of the orthogonality condition (2. 21), we obtain where Introducing the auxiliary functions (xlnx,.h3) = II + Iz =0 (3.1o) 11= xs~w~lnxdx, I~ =q~ o xlnxdx =----~-qs (3.11) f~ I' 1 z 3 f(x) = jl w:(.,c)inxdx----(x:--2x)lnx----~x + 2x--y ~(x) = f(x)dx, g(1)=--l--- ~ (3.12.) and integrating by parts, we get z, =- c s' ( 1)+ s,(1)~g( 1)+ I'o H,(=)g(,)d= an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, =-Cs;(1) + s~(1)3g(a)-t 1 f(o~ however, 1 [1 by (x--l) utilizing 61nxdx the "weak"... -Tjo (3.13) shock behavior of 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 of flying Furthermore, using Eqs. (3.5) and (3.12), we have Final velocities of flying plate obtained I~=-- agree very 5 (l--p)-~ well with 41 numerical results by computers. Thus (3.14) an analytic formula with two parameters of high 108 explosive (i.e. 360 detonation velocity and polytropic Inserting Eqs. (3,1 I) and (3.14) into Eq (3.10), we find out :1 Introduction v (3.15) qs----~-~-~'~, -#)-~ 270(I--v) materials and hence under the intense third-order impulsive approximate loading, relation shock synthesis between of the diamonds, load parameter and explosive and welding the central and cladding deflection of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest. Under the assumptions of v O one-dimensional = 4W,, + plane detonation.w: + O(W and ~. rigid ) flying plate, the (3.16) normal 270( 1 Y) approach of solving the problem of motion of flyor is to solve the following system of equations governing which is in the agreement flow field with of detonation the classical products result behind given the by flyor Chien (Fig. Wci-zangOl. I): IV. The Case of Uniform Load and General Boundary Conditions In this section, we consider the case of uniform load and general boundary conditions. For this case, we still have F= G = 1, but the values of 3, and y /~ =0, might be some different combinations. The solution for the corresponding problem of small deflection (2.8) is (i.0 as as a--t wl- 1123"[(x--1)z--22(x--1)], ql-----l~ 4 (4.1) In a similar way, we obtain where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters

7 Large Deflection of Circular Plate jt 1 =-- (1+22) z o x(x-l-2)zdx= 6(1+24)" (1+4A+6).') (4.2) and # &(D = ~(1+22)~ ( ~) 1--/2 1 = 6(1+ 22) z E( t)+p(l =)3 (4.3) For the corresponding problem (2.10), according to the theorem in Sec. II., we have the orthogonality condition where f <xlnx-2x, Ha)=JI+ lz=o (4.4) The one-dimensional problem l of the motion of a rigid flying plate under explosive attack has an analytic solution only "[1 = when xszw'~ the polytropic (lnx-2)dx index of detonation products equals to three. In o general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock lz~qa It in the explosive products, (lnx--2)dx and applying (1 +2 X ~/ qs the small parameter pur- (4.5) terbation method, an analytic, first-order 0 approximate solution is obtained ql for the problem of flying Final In order velocities to calculate of flying J,, plate we take obtained the procedure agree very similar well with to that numerical in Sec. results III, that by computers. is, introduce Thus the an auxiliary analytic functions formula with two parameters of high explosive (i.e. detonation velocity and polytropic = (inx-2)w~dx=l't-'4-~j ' (x-l-2)(lnx-2)dx 1. Introduction --1--~2~ [x*inx- 2( l + 2)xlnx-l ( l + 22)x* of and common interest. approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of S: detonation products behind the 1 flyor (Fig. I): +2(1+2)'x-( a+32+2a~ )1 (4.6) a(x) = f(x)dx, o(1)= 18(1+2,~) (5+15,l+ 181 ') (4.7) Hence integration by parts leads to s,=-ts:( ~)-~ s,(~o o( ~ ) + j'fl~=(x)o(x~d. =:-[S~(1)+S,(1)Jg(1)-t as a--t 22' (1+2)' as 3(1+22)aOL1)rk 6(1+22)zj(0) _ 1 I: (x-l-a)5(inx-a;~* (4 8) 3(1 4-22):' where Using p, Eqs. p, S, (4.2), u are (4.6) pressure, and (4.7), density, we obtain specific the entropy final expression and particle for J,, velocity and from of detonation Eqs. (4.3) products and (4.4), trajectory F of qa= flyor ~ as -~22 another,/'t -- boundary. 270(11 + 2/1-) Both ~ E73+ are unknown; 388/l the position 840~.*+ of R 360), and 4 the state parameters D and by initial stage of motion -4- of I0/*(5+ flyor also; 352+ the position I08),z+ of 1622s+ F and the 1084'3 state parameters of products (4.9) (i.0

8 624 Dia Shi-qiang which is exactly the result given by Chien Wei-zang and Yeh Kai-yuanlSL V. The Case of Compound Load and Rigidly Clamped Edge In this section, we consider the large deflection of a circular plate with rigidly clamped edge under the compound load, i.e.under the combined action of uniform pressure and concentrated load at the center. Hwang Chient*l thoroughly treated the problem. He introduced some small parameters different from W and considered the case of arbitrary central deflection, which could be zero. t]ut we have to confine Ourselves to the case of nonzero central deflection. According to Ref. [4], in this case, we have '2 G(x)=/3+ax, 2=0, #'-= 1--v (5.1) where /3 and a are two definite parameters for given load. The solution for the problem (2.8) is { w,=+qefl(x- 1)"-.la(x- i)+.iaxlnxj (5 The one-dimensional problem of 4 the motion of a rigid flying plate under explosive attack has an analytic solution only when q' =i-g+ the polytropic ~ index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior Through of the the reflection procedure shock as in in Sec. the explosive III., we obtain products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various I [~,z 1 S~(1). high explosives ==---2-J0 with xwl polytropic dx=- 288q2~(3~a"+20afl+3flz)" indices other than but nearly equal to three. Final velocities of flying plate obtained agree very well with numerical results by computers. Thus (5.a) index) for estimation of S,.( the 1 ) velocity = - # S; of ( flying 1 ) = 2--~-- plate q is i ( established. 36a 2 + 2oafl+ 3fl ~) Similarly, $2(0), T 2 (0), and T 2(1) could be derived, and their expressions ate omitted here. 1. Introduction To determine q3, we apply the orthogonality condition (2.21), i.e materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding (5.4) and of where common it is easily interest. found that approach of solving the problem of K2= motion.la+t~ of flyor is to _ solve q z_.~ the following system of equations (5.5) governing the flow field of detonation products behind 4 qa= the flyor ql (Fig. I): In a similar way, we have + 2afxln"x--2xlnx--2(x-- 1)J as a--t 1 ' 1 as (5.6) (i ~-al 2xqnZx - 6x"lux + 7x 2-8x-! } (5.7) where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, and with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed Kl=--CS~(1)+S2(1)3,q(1) by the flow field I of central rarefaction -)L[ 2 Jo ~ w'b( " wave x behind )d x the detonation (5.8) wave

9 From the above known results, we obtain Large Deflection of Circular Plate fjl~t s abz t_~_~_6p, 73 ^a\ ) qs=qlkl= (4a+/~)' ~,nk-'m--a 4" 2--~'aZ/~ Verification and Comparison show that our results exactly agree with those given by Hwang Chient4]. VI. Conclusion As is well-known, the procedure of solving the differential equations for higher-order perturbation terms is usually complicated, which could be partially avoided if the method of orthogorality conditions presented above, is used. For the large deflection problems of circular plate treated in this paper, if only the elastic properties at the center and edge are needed to be understood, the application of orthogonality conditions and some elementary integration will make it possible to avoid the procedure.of solving at least two differential equations. This is true for the third-order approximation considered above and aiso true for the higher-order approximation. an analytic solution only when the polytropic index of detonation products equals to three. In general, The a idea numerical of the analysis method proposed is required. above In this expected paper, however, to be used by to utilizing solve the the more "weak" complicated shock behavior problems of of the deflection reflection and shock stability in the of explosive plates and products, shells. and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate Acknowledgement driven by various high The explosives author would with polytropic like to express indices his other thanks than to Prof. but nearly Hwang equal Chien to three. for his Final helpful velocities suggestion of flying and discussion. plate obtained agree very well with numerical results by computers. Thus index) References for estimation of the velocity of flying plate is established. [ 1 ] Chien Wei-zang, Large deflection of a circular clamped plate under uniform pressure, Chinese Journal of Physics, 7 (1947), I 13. Introduction [ 2 ] Chien Wei-zang and Yeh Kai-yuan, On the large deflection of circular plate, Chinese Journal materials of Physics, under intense 10 (1954), impulsive 209- loading, 238. (in shock Chinese) synthesis of diamonds, and explosive welding and cladding [ 3 ] Yeh of Kai-yuan, metals. The Large method Deflection of estimation of Flexible of flyor Structural velocity Elements and the way (Rods, of raising Membranes, it are questions Plates and of common Shells), interest. Lanzhou University (1984). (in Chinese) [ 4 ] Under Hwang the Chien, assumptions Large deflection of one-dimensional of circular plane plate detonation under compound and rigid load, flying Applied plate, Mathematics normal approach and of Mechanics solving the 4, 8 problem (1983), of 791 motion of flyor is to solve the following system of equations governing [5] Dai the Shi-qiang, flow field The of detonation PLK method, products Singular behind Perturbation the flyor (Fig. Theory I): and Its Applications in Mechanics (Edited by Chien Wei-zang), Science Press, Beijing (1981), (in Chinese) as a--t as where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters (i.0