A MICROSCOPIC DAMAGE MODEL CONSIDERING THE CHANGE OF VOID SHAPE AND APPLICATION IN THE VOID CLOSING. Zhu Ming (J~ ~) Jin Quan-lin (~Jj~]~k)

Transcription

1 Applied Mamatics and Mechanics (English Edition, Voi. 13, No. 8, Aug. 1992) Published by SUT, Shanghai, China A MICROSCOPIC DAMAGE MODEL CONSIDERING THE CHANGE OF VOID SHAPE AND APPLICATION IN THE VOID CLOSING Zhu Ming (J~ ~) Jin Quan-lin (~Jj~]~k) (Laboratory for Nonlinear Mechanics Continuous Media, Institute Mechanics, Chinese Academy Sciences, Beijing) (Received July 29, 1991; Communicated by Hsueh Dah-weD,4 microscopic damage model ellipsoidal body containing ellipsoidal void for nonlinear matrix materials is developed under a particular coordinate. The change void The shape one-dimensional is considered problem in this model. The motion viscous restrained a rigid equation flying plate obtained under from explosive model attack has an analytic is affected solution by only stress when : ~ i i, void volume polytropic fraction index f, material detonation strain rate products exponent equals m as well to as three. In general, a numerical void shape. analysis Gurson's is equation required. is In modifwd this paper, from however, numerical by solution. utilizing The modified "weak" shock behavior equation reflection is suitable shock for case in nonlinear explosive matrix products, materials and and applying changeable small voids. Lastly, parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying model is used to analyze closingprocess voids. plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula Key words with two nonlinear parameters material, high ellipsoidal explosive void, (i.e. void detonation shape, void velocity closing and polytropic index) for estimation velocity flying plate is established. I. Introduction Since GursonOl built microscopic model 'limited matrix containing a void, microscopic damage mechanics has greatly been developed. Gurson's model was xs;idely used becse Explosive its perfection driven flying-plate and practicality. technique Yamamoto ffmds its important t2j used Gurson's use in equation study to behavior analyze materials conditions under for shear intense localization impulsive loading, ~e ductile shock synsis fracture diamonds, void-containing and explosive materials. welding Gurson's and cladding constitutive metals. model was The used method by Aravast3] estimation in flyor analysis velocity void and growth way that raising leads it to are central questions bursts common interest. during extrusion. On or hand, some modifications to Gurson's model were made. Under assumptions one-dimensional plane detonation and rigid flying plate, normal approach Tvergaardt41 solving considered problem void interaction motion flyor by. means is to solve finite following element system analysis. equations Strain governing hardening was flow indirectly field detonation taken:into products account behind by Yamamotot2j flyor (Fig. through I): introducing matrix average flow stress and Wang Tze-chiang SJ considered directly affection strain hardening matrix materials. In industral production, manufacturing ap +u_~_xp + process many parts such as forge heavy ingots and forming powder metallurgy is to eliminate 1 y =0, all kinds internal cavities in materials. So study on void closing in materials becomes significant. We t~ applied Gurson's model to analyze closing process voids in as rigid-plastic matrix materials. It was found that entire a--t closing voids requires infinite hydrostatic stress or equivalent strain rate. This is becse void shape change was not taken into p =p(p, account s), in model. In microscopic damage mechanics, studies were greatly, concentrated on void nucleation, growth, coalescence and forming a macrowhere p, p, S, u are pressure, density, specific entropy and particle velocity detonation products respectively, crack at last. with Therefore, trajectory change R reflected void shock shape detonation was ignored wave in D Gurson's as a boundary model and and modified trajectory Gurson's F flyor model. as anor However, boundary. void Both shape are unknown; plays an important position part R in and void state closing, parameters on it are governed by flow field I central 755 rarefaction wave behind detonation wave D and by initial stage motion flyor also; position F and state parameters products

2 756 Zhu Ming and Jin Quan-lin especially in entire closing voids. In this paper, a microscopic damage model limited ellipsoidal body containing ellipsoidal void for nonlinear matrix materials is developed and change void shape is taken account. Then, void closing nonlinear materials is analyzed numerically. II. Microscopic Damage Model Considering Change Void Shape The microscopic damage model finite ellipsoidal body containing ellipsoidal void is shown in Fig. 1. In order to take account change void shape, model is built in socalled ellipsoid-rotating hyperboloid coordinates (see Fig. 1). The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection shock in explosive products, and applying small parameter purterbation method, an analytic, (a) first-order approximate solution is obtained (b) for problem flying plate driven by various Fig. high I Microscopic explosives with damage polytropic model indices and or ellipsoid-rotating than but nearly equal to three. Final velocities flying plate hyperboloid obtained agree coordinates very well with numerical results by computers. Thus an analytic The coordinates formula with are two defined parameters as follows: high explosive (i.e. detonation velocity and polytropic index) for estimation velocity flying plate is established. x=achlsinocosq~ y=ach~sinosinqg~ z=ash2coso (2.1) ~ [0~ ~)~, Oz 1. [0~ Introduction ~'3, ~p~ [0~, 2ar] Equation ellipsoid: Explosive driven flying-plate technique X 2 y2. ffmds its important Z2 use in study behavior materials under intense impulsive a~ch~ loading, +a~ shock synsis +a~sh22 diamonds, = 1 and explosive welding and cladding The ellipsoid-rotating metals. The method hyperboloid estimation coordinates flyor are velocity orthogonal and way coordinates. raising it From are questions Fig. l, it can common be seen that interest. ellipsoid shape is a function 2 and smaller 2, flatter ellipsoid. The Under assumptions one-dimensional plane detonation and rigid flying plate, normal ellipsoid approach becomes solving a circle problem in x-g plane motion when 3.=0 flyor is. This to solve change following voids is system similar to equations real process governing flow void field closing. detonation products behind flyor (Fig. I): In this problem, following boundary condition is only met when an upper bound approach is adopted. ap +u_~_xp + v, la=/~,jxs (2.2) 1 y =0, where v, is microscopic velocity field, S is outer surface model, ~j is macroscopic strain rate and x~ is as position as a material point in Cartesian coordinates. Considering axisymmetric deformation, a--t ~ ~1~ ~ 22 and ~,~ = 0 (i 4= ]), eq. (2.2) after coordinates transform becomes ash22ch2z..~ 9 z-.-'- 2- ) where p, p, S, u are pressure, v~ =#ch22z_sin2#'f,~'l,sm density, specific entropy and ~-e~'ss particle cos velocity O) detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary and (2 3) trajectory F flyor as anor asinocos6 boundary. Both are unknown; position R and state parameters on it are governed ve by = MchZ2z_ flow field sinz0 I (gl central lchz2z-- rarefaction Ssash~2z) wave behind detonation wave D and by initial stage v~=o motion flyor also; position F and state parameters products

3 A Microscopic Damage Model and Application 757 where v~, vo and v~, are microscopic velocity field in ellipsoid-rotating hyperboloid coordinates. When incompressible condition and eq. (2.3) are met, a kine-possible velocity field can be solved a(2tg n + tg3d v, = 3ch~/ch~2-sin2 O (shs22+3sh2ac~ vo ---" j V~,~=0 + a(sin20- ecos'o)sh2 ch3.~/ch~a-sin20 t. asinocoso ehz2- sin20 (Sncha2-8sssh2)') so that microscopic strain rates ~,, do, d~, "txo. are g,----{ (sin'0-- 2cos'O) [-gn +sha2 (chz2- (1/3) shag) ~ + cz,) -- 8,,sh2~sh2 ( (1/3) sh222 + co s20) }/ch22 (chug-- sin20) The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution - { 8,sh22sh2 only when ( (1/3) sh22, polytropic +cos~0) index + (/~n detonation +0.58~ products sh22 equals to three. In general, a numerical 9 (sin~0- analysis 2cos*0) is + required. si n2ocos~o(81~ In this paper, + 1.5B,sh*2) however, by }/(ch*2 utilizing "weak" shock behavior -sin*0) reflection 2 shock in explosive products, and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven ~0--- by {(8n various +1.58,sh~2) high explosives (ch*2cos*0- with polytropic ch*~,sin*8 indices + or sin'0) than but nearly equal to three. Final velocities + (8n flying +0.5B~ plate obtained 9 shz2. agree (sin*0,- very well 2cos20) with numerical results by computers. Thus an analytic formula +8, ( (1/3) with two sh~2, parameters + co s*0) s h;t,sh2 high explosive }/(c h ~.;t (i.e. - sin*0) detonation ~ velocity and polytropic index) for estimation velocity flying plate is established. ~+,= { g,((1/3) sh~2z +cos~0) sh2~sh2 + (8n +0.5g,sh~2~ (2.5) 9 sh'a (sin~o- 2cos'O) }/cha2 (ch~,;.-sin'o) +cos'o(gn+l. 58,sh'2)/(cha2- sin'0) Explosive 9~0= { 2g,sin0cos0sh2~(cos~0 driven flying-plate technique + (1/3) ffmds its sh2~l important z) +2 (gx, use +0.58osh~).) in study behavior materials under 9 sin0cos0 intense impulsive (sinzo- 2eos~0) loading, sh2 shock }/ch2 synsis (c h',t - diamonds, sinao) ~ and explosive welding and cladding metals. The method estimation flyor velocity and way raising it are questions common interest. +{ 6 (gn +0.58~ 28,sh2~sinOcosO } Under /ch2 assumptions (chz2- sin'o) one-dimensional -- 2(/~n sh~2) plane detonation sino and rigid flying plate, normal approach 9 solving cososh2ch2/(ch~2- problem motion sin~o) ~ +0.5R flyor is osinocoso to solve following system equations governing flow field detonation products behind flyor (Fig. I):. shilch2/(cn~2 - sin~8) where ~~ is g,= 2Rn +Sss and /~~ (R~- 83D/3 ap +u_~_xp + 9 The microscopic equivalent strain rate 1 y =0, ] = 2[ (~,-- ~o) z + (do-- ~,)a + (~, _ daoa ~ 0]/9 ----AS I, +Bg,,8, as +c8: as (2.6) a--t where A, B and C are all functions ;l and O. The matrix material is an incompressible, isotropic nonlinear material and microscopic where stress and p, p, strain S, u are are pressure, related to density, specific entropy and particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary and trajectory F flyor as anor boundary. ~o/ao---- Both are (&,/~o)" unknown; position R and state (2.7a) parameters or on it are governed by flow field I central rarefaction wave behind detonation wave D and by initial stage motion flyor also; position F and state parameters products (2.7b)

4 758 Zhu Ming and Jin Quan-lin where microscopic equivalent stress a~ (1.5s, js~j) 89, s,j is microscopic stress deviator, ~.= ((2/3) d~s0,j) vl is microscopic equivalent strain rate, ~cj is microscopic strain rate deviator, tr0, $0 and strain rate exponent m are material parameters. The microscopic dissipation w, is defined: The macroscopic dissipation W is given by f~i, g -=+t to,=.}, a,~d~,j =--h---~-ve. (2.8) 1 K W~U-Iv. wodv= V(m+l)Iv. ':+'dv (2.9) where Vis entire volume model and V,~ is matrix volume. The macroscopic stress can be shown that Xi~=OW/ORij (2.10) From eq. (2.6), macroscopic dissipation is The one-dimensional problem motion a rigid flying plate under ra+ l explosive attack has an analytic solution only when W-- (m+l) K polytropic V 1 ~vm(ab ~+BB,Bo index detonation products 2 equals +CR:) dv to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection shock in explosive products, and applying small parameter purterbation method, an analytic, KR:+, ~.jv.(aco,+bco m+, ='(re+l) first-order " approximate solution +C) is obtained 2 dv for problem flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with = two KR'~+I parameters.w* high explosive (i.e. detonation velocity and polytropic (2.11) index) for estimation velocity (re+l) flying plate is established. m+ I W.m~-.Ig=(AcoJ-bBm -FC)--'i-'dV Explosive driven flying-plate 3 ('~sinodo[ ~, (Aco2.bB~q_C) ch22ash2~ technique ffmds its important use in study behavior Jo J~1 materials under intense impulsive loading, shock synsis diamonds, and explosive welding and cladding metals. The method 9 ch2. estimation (sha2 +cos'8)d2 flyor velocity and way raising it are questions (2.12) common interest. Using Under eq. (2.10), assumptions we can get one-dimensional ~g,,/l~. plane detonation and rigid flying plate, normal approach solving problem motion flyor is to solve following system equations OW KR7 +1 OW* KR'~ aw* governing flow field detonation products behind flyor (Fig. I): 2.:~-- m+l "0--g~,, =/n+l OW KR~ OW. = X. ~ ap = +u_~_xp m+l +.O---:b-~+KB,W* 1 m+l (2.13) y ----"m--+tl (m+l)w*- =0, Oco J --- KR~W*-o~X,,, as as a--t The average equivalent stress O~ is defined by o.. i Iv o..dv =-v-jjv K f.. -.+I a "V where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary and trajectory F flyor as anor boundary. m+lw Both are K~*I unknown; W* position R and state (2 para- 14) meters on it are governed by flow field I central rarefaction wave behind 1 detonation wave where D and by initial void volume stage fraction motion f---- flyor (V also; - V.) / position V. Considering F and ~ ~ ---- state (O,/K) parameters m # 9 can products be written

5 A Microscopic Damage Model and Application 759 m c~~ (W*/(1 -- f))m+ t (2.15) Then, "= ffi m+l To= ~./~.----[(1--/)'W*] ~ -- cotm ~ (2.16) Eq. (2.16) is controlled by material parameters, void size an d shape. The long axis ellipsoidal void is ax-- ach2~ and short one is b~---- ash2~. Define void shape factor a=b:/a~----th21, void volume fraction /~ch22~sh/l,/(ch222shaz). At and /l2can be solved if a and fare given. When co is eliminated between eq. (2.16), restrained equation in T,, and T, can be got cb(t,,~ To~ a~.f~ m) (2.17) The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic The numerical solution Solution only when eq. (2.17) polytropic are shown index Fig. detonation 2. These restrained products equals curves to are three. called In general, yield equation a numerical for analysis matrix materials is required. in obedience In this paper, to however, total strain by ory utilizing and called "weak" vinous shock behavior restrained equation reflection for shock viscous in matrix explosive materials. products, Besides and applying void volume small fraction parameter f and pur terbation method, an analytic, first-order approximate solution is obtained for problem flying material strain rate exponent m, eq. (2.17) also relates to void shape factor a. Fig. 3 shows plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final relation velocities between flying intersecting plate obtained point T,* agree very restrained well with curves numerical in results T,, by coordinate computers. axis Thus and an analytic void shape. formula It can with be found two parameters that smaller high explosive void shape (i.e. detonation factor, velocity greater and influence polytropic on index) restrained for estimation equation. velocity flying plate is established. f=o. Ol, re=l/3 m--l/3 m- 1/5 " Explosive driven flying-plate technique ffmds its important use in study ram0 behavior materials under intense impulsive loading, shock synsis diamonds, and explosive welding and cladding metals. The method estimation flyor velocity and way raising it are questions common interest : Under assumptions one-dimensional plane detonation and rigid flying plate, bt/at normal approach solving 1. a=0.995; problem 2. a=0.447; motion flyor is to solve 0 following 0'.5 system i - equations 3. a=o.1; 4. a=o.05 governing flow field detonation products behind flyor (Fig. I): Fig. 3 Void shape influence on Fig. 2 Restrained curves T= and T. restrained equation ap +u_~_xp + For spherical void with rigid-plastic von Mises matrix material, m = 0 and a , it is 1 easy to be got that eq. (2.17) becomes Gurson's y equation. =0, That is m=o~ a-----l; r a, +2fch(3Tm/2)- (1-t7/2) J'l" as as a--t (2. 18) a-----0~ e----t,:-- 1 In general case, according to results numerical calculation, eq. (2.17) can be given by where p, p, S, u are pressure, r density, ~, +2fqffxch(3qzT,~/2) specific entropy and -- particle (1 +/2)=0 velocity detonation products (2.19) respectively, with trajectory R reflected shock detonation wave D as a boundary and qt~---az-um~ q2~a-t.sa'~, rl~l-kcsch(15a) trajectory F flyor as anor boundary. Both are unknown; position R and state parameters where csch on it (15a) are governed is hyperbolic by flow cosecant field I function. central rarefaction Eq. (2.19) wave is suitable behind for detonation case wave m< D ~. 5 and and by initial small stage f. motion flyor also; position F and state parameters products

6 760 Zhu Ming and Jin Quan-lin HI. The Closing Process Voids In closing process internal voids in materials, size and shape voids are changed. When voids are closed entirely, void volume fraction f= 0 and void shape factor a From definition void volume fraction, we have where f, is initial void volume fraction. When ] = 0, macroscopic strain condition entire dosing voids can be written ln(1-/,)-----e, The long axis outer boundary ellipsoid is as /----(1--/)R. (3.1) lne (1-- ]'0) / (1--.f) "] = J':'~',,dt = g,, (3. 2) (3.3), and short one is bs 9 We have The one-dimensional problem motion a rigid flying plate under explosive attack has o, I,.a,,o.:,~= a,; v, I,.a,,,..,s=as an analytic solution only when polytropic index detonation products equals to three. (3.4) In general, a numerical analysis is vxl~.a,,0.o=b,, required. In this o~lx.a,,0.o=b~ paper, however, by utilizing ) "weak" shock behavior reflection shock in explosive products, and applying small parameter purterbation From method, eq. (2.4), an analytic, n first-order approximate solution is obtained for problem flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with 1 two 1-~ parameters I th222 high explosive )_! (i.e. I_R. detonation velocity and polytropic g ~ index) for estimation velocity flying plate is established. ~2/ as= R./3 "~]~./ 2 (3.5),,2 1 /~o(1-- sh~21) ] -~-~ = 3f \ l+ cla~l~ /-2 [.3ch~2, 2 ch~ -J Explosive driven bs/bs=r./3-1~o flying-plate technique ffmds its important use in study behavior materials under intense impulsive loading, shock synsis diamonds, and explosive welding and cladding The above metals. equations The method represent estimation evolution. flyor velocity and way raising it are questions change common void interest. shape in closing process voids. The Under assumptions one-dimensional plane detonation and rigid flying plate, normal approach macroscopic solving stress in problem void closing motion can bc got flyor from is eqs. to solve 0.6 following system equations governing (Zl6) and (3.5). flow field When detonation bl=0, products voids behind are entirely flyor (Fig. I): a= closed. The stress and strain states this moment are significant in engineering application. 0.4 From eqs. (2.1 6) and (3.5), parameters ap +u_~_xp + void closing can be reckoned up if strain rates /~11~ 1 y /~83 =0, 0.z (or /~, ~ /~e ), initial Void volume fraction f0 and //~~E~/R, initial size void at0, b 10 are given. as The as results f fio. O l,m= l/3 a--t J(,~) numerical calculation are shown in Fig * u! I Fig. 4 shows relation p void =p(p, shape s), factor a and void volume fractionf It can be seen that void Fig. 4 Relation a and f where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products shape is flatter.and flatter in dosing proces o voids. respectively, with trajectory R reflected shock detonation wave D as a boundary and trajectory Fig. 5 F and flyor Fig. as 6 show anor boundary. relation Both 7 e are, unknown; Tm and f position in accordan~ R and with state different parameters on it are governed by flow field I central rarefaction wave behind detonation wave D and by initial stage motion flyor also; position F and state parameters products

7 A Microscopic Damage Model and Application I. a,0ffi=l.ll8, b,,==0.5 9, g./b,=-j..o9,j,=o.oz m== g../b.= )',==0.01,m--1/3 a a=o.447 a=0.287 ('a).f (r Fig. 5 Relation T.andf 0~,1 ~ 0.8 ' ' - (b) -1. ~. a,o'=l.ll8,b~o~=0.5 f --,~i 6'---7" ~,./~t. =-1.o9 8 Jo~O.Ol,mwl/3 0~0.0~1 The one-dimensional problem motion a rigid flying plate a==0 under 669 explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, 4 f ~ a by = utilizing "weak" shock behavior reflection shock in explosive products, and applying small parameter purterbation method, an analytic, first-order /(~,) approximate solution is obtained for problem flying plate driven by 0.4 various high O. 8 explosives with polytropic indices or 0.4 than but 0 nearly B equal to three. Final velocities flying (-) plate obtained agree very well with numerical results (b) by computers. Thus an analytic formula with two parameters Fig. 6 Relation high explosive T = and (i.e. f detonation velocity and polytropic index) for estimation velocity flying plate is established. m==l/s,fo=o.ox ~ J,~O.5~,a= C Explosive driven flying-plate technique ffmds its important use in study behavior materials under intense a, impulsive loading, shock synsis 1( diamonds, and explosive welding and cladding 1( metals. The method estimation flyor velocity and way raising it are questions common interest. a m-- 0~,,~,.._~=...~ Under assumptions E.~ x one-dimensional 1o-*) plane detonation and rigid flying plate, Eo( X 10"') normal approach solving,i problem /I io -- motion flyor is to solve ~ i following 8 system lo equations governing flow field (a) detonation products behind flyor (Fig. I): (b) Fig. 7 Relation T=and EoOf entire closing void m and a. When ] decreases, hydrostatic ap +u_~_xp + stress [ T,n I increases and equivalent stress Te 1 decreases. bj oioi1.118,blem0. 5 The greater a, greater [ T=[ and Te y ; The greater =0, m-,,l/3,f,=,, ~ m, greater [ T= I and smaller as T 0. as Fig. 7 shows relation hydrostatic a--t stress T,~ and 0.4 equivalent strain E0 at time p entire =p(p, closing s), voids. In same E0, greater m and a, greater [ T= [; In same 0.2 JB, where T,, p, p, greater S, u are m pressure, and a, density, greatere,. specific entropy In paper and [6], particle we velocity detonation products proved respectively, that with hydrostatic trajectory stress R and reflected equivalent shock strain detonation are wave D as a boundary and -Z\-E,;( x lo-=) trajectory main parameters F flyor affecting as anor boundary. closing process Both are unknown; internal position R and state parameters on it are governed by flow field I central rarefaction wave behind detonation 2 wave voids in materials. It can be known from Fig. 7 that greater m D and by initial stage motion flyor also; position F and Fig. state 8 parameters.~./~.~ influence products on and a, more difficult void closing. void closing effect

8 762 Zhu Ming and Jin Quan-lin Fig. 8 shows relation void short axis b~ and strain in compressed directione33. In same E33, greater [ 8=/8e [, better void closing effect. IV. Summary The viscous restrained equation nonlinear materials containing voids relates to void shape besides void volume fractionfand material strain rate exponent m. Especially for small void shape factor, effect void shape cannot be ignored. In process void closing nonlinear materials, greater material strain rate exponent m, more difficult void closing and flatter void shape, easier void closing. The greater I ~',,/8, I, better void closing effect. References [ I ] Gurson, A. L., Continuum ory ductile rupture by void nucleation and growth, Part 1, Yield criteria and flow rules for porous ductile media, J. Eng. Mater. Tech., 98 (1977), 2. [2] Yamamoto, H., Conditions for shear localization in ductile fracture void-containing The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic materials, solution lnt. J. only Fracture, when 347 (1978), polytropic 14. index detonation products equals to three. In.[3] general, Aravas, a numerical N., The analysis is required. void growth In this that paper, leads to however, central burst by utilizing during extrusion, "weak" J. shock Mech. behavior Phys. Solids. reflection 34 (1986), shock 55. in explosive products, and applying small parameter purterbation [4] Tvergaard, method, V., an Influence analytic, first-order void on approximate shear band instabilities solution is obtained under plane for strain problem conditions, flying lnt. plate J. driven Fracture, by various 17 (1981), high explosives 389. with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus [5] Wang Tze-chiang and Qin Jia-liang, Constitutive potential for void-containing nonlinear an analytic formula with two parameters high explosive (i.e. detonation velocity and polytropic index) materials for estimation and void growth, velocity Acta flying Mechanica plate is Solida established. Sinica, 2 (1989), 127. (in Chinese) [6t Zhu Ming and Jin Quan-lin, A constitutive relation for materials containing voids and application in closing voids, Advances in Constitutive Laws for Engineering Materials, Ed. by Fan Jing-hong and Sumio Murakami, Pergamon Press, Oxford (1989), 542. Explosive driven flying-plate technique ffmds its important use in study behavior materials under intense impulsive loading, shock synsis diamonds, and explosive welding and cladding metals. The method estimation flyor velocity and way raising it are questions common interest. Under assumptions one-dimensional plane detonation and rigid flying plate, normal approach solving problem motion flyor is to solve following system equations governing flow field detonation products behind flyor (Fig. I): ap +u_~_xp + 1 y =0, as a--t as where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary and trajectory F flyor as anor boundary. Both are unknown; position R and state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D and by initial stage motion flyor also; position F and state parameters products