CONSERVATION INTEGRALS AND DETERMINATION OF HRR SINGULARITY. Wang Ke-ren (:Ej~_Z) Wang Tzu-chiang (71: ~1~[)

Transcription

1 Applied Mathematics and Mechanics (English Edition, ol.8, No.9, Sep 1987) Published by SUT, Shanghai, China CONSERATION INTEGRALS AND DETERMINATION OF HRR SINGULARITY FIELDS Wang Ke-ren (:Ej~_Z) Wang Tzu-chiang (71: ~1~[) (Institute of Mechanics, Academia Sinica,.Beifin) (Received Oct. 13, 1986; Communicated by Yeh Kai-yuan ) The angular distribution functions of HRR singularity fields are analyzed via conservation integrals. Two fi~nctional equations are proved for the angularadiztrib:z,qon--- functions and can be used for their solutions. The detailed forms of the functional equations The one-dimensional and the final governing problem equations of the for motion solutions of a are rigid given flying for the plate cases under of plane explosive strain and attack has an analytic plane solution stress. only Accurate when numerical the polytropic results are index also given of detonation for some typical products parameters equals and to the three. In general, a numerical equivalence analysis of different is governing required. equations In this paper, is proved. however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter purterbation I. Introduction method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities The study of of flying elasto-plastic plate obtained stress-strain agree very fields well at with the crack numerical tip has results attracted by computers. great attentirn Thus an during analytic the formula development with two of parameters fracture of mechanics. high explosive In (i.e. 1968, detonation Hutchinson[~)and velocity polytropic Rice and index) Rosengrenl2] for estimation derived of the well velocity known of HRR flying singularity plate is established. fields and thus paved an efficient way for the development of elasto-plastic fracture mechanics. The J integral introduced by Rice [3] was proved to reflect the intensity of the sigularity 1. fields Introduction and became a very attractive controlling parameter. For cracks of mixed mode, some attempts were made to find similar solutions. Among them, Shih's Explosive work/41 is driven worth flying-plate mentioning. technique He tried ffmds to its find important the angular use in distribution the study of functions behavior of of the materials sigularity under fields intense using the impulsive same methods loading, as shock those synthesis used in [l], of but diamonds, without and success, explosive the reason welding ofwhich and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions is found in this paper to be due to some intrinsic characteristics of the governing equations for the of common interest. singularity Under the fields, assumptions which have of one-dimensional not yet been well plane studied. detonation and rigid flying plate, the normal approach This of paper solving makes the further problem analysis of motion of the of fundamental flyor is to solve equations the following of the singularity system of equations fields. Two governing functional the ei:luations flow field are of detonation proved via products conservation behind integrals, the flyor (Fig. and I): by them several new governing equations are derived. Their detailed forms are given for the cases of plane strain and plane stress. Accurate numerical results are also given for some typical parameters. ap The new governing equations are one +u_~_xp + au order lower than the ones given in [1,2] and have quite different forms, but they are all equivalent au au in general 1 y cases. =0, II. The Case of Plane Strain as as a--t Assume the material behaviour can be described by the plastic deformation theory and its uniaxial stress-strain relation is given p =p(p, by[q s), p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected eo--~oo+ shock ate'o} of detonation wave D as a boundary and (2.1) the on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position 839 of F and the state parameters of products

2 840 Wang Ke-ren and Wang Tzu-chiang The corresponding three dimentional relations can be obtained as p 3. ~ ~_ 1 t" (2.2) in which the deviatoric stresses are given by and S,j=cr, ~-cs, j( cr,,/3 ) (2.3) 3 Cr~ =T S'JS" (2.4) m is the power hardening.exponent and a is a material constant. In this paper, all stress and strain components without bars are non-dimentionat.ones (equal to the actual ones divided by the yield stress a0 and Yield strain eo=(ro/e, respectively.) As in [1], we introduce the nondimentional stress function ~ as The one-dimensional problem of the q~= motion Kr*+2~ of a ( 0) rigid flying plate under explosive attack (2.5) has an analytic solution only when the polytropic index of detonation products equals to three. In general, a ~(0) numerical is the analysis angular is distribution required. In function this paper, to however, be determined by utilizing and s the is the "weak" singularity shock behavior exponent of given the reflection by 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 plate driven by various high explosives with polytropic indices other than but nearly equal to three. s= (l+m) (2.6) an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic The non-dimentional stress and strain components can thus be expressed by a,=kr'(~"" + (s+2)~) 1. Introduction ae=kr'(s+ 2) (s+ 1)~ r,o=--kr*(l+s)~" Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials e, = ak"r"*~ under intense, impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions eo = akmrr#*8 o of common interest. (.2.7) Y,e Under = ak the "r"~ assumptions ~,e of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve E the following, system of equations g,=a~-,-, [~ 9 "-s(s+ 2)~ ] O 9,,= 3~7-'~,,= - 3~7-'(s+ 1)~" ap +u_~_xp + au The compatibility equation for the strains is as as ~," "--ms(2+ms)~,--(l a--t +ms)~" o= 0 The governing equation can thus be obtained asm y =0, Fig. l Integral path (2.8) p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the on it are governed by the 4-(l flow +ms)[~7-'(l+s)~" field I of central rarefaction ] "=0 wave behind the detonation (2.9) wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

3 Conservation Integrals and Determination of HRR Singularity Fields 841 To obtain new governing equations, we evaluate the J integral along the colitour of ABCDEA in Fig. 1. We have ~ -a~eo~a OuL In equation (2.10), the integral on EA is equal to zero. The integrals on AB and.on CDE are cancelled, with only higher order small quantities left. Therefore, as r~ and G tend to zero, we have On the other hand, we have I.o ( Wn,--P, -~t )ds ~O. I.*( W.,--p,~-~ )ds~ll(o)la(r~/ro Let r, and r 2 tend to zero simultaneously with r2/r t remaining constant, it follows that The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when m the polytropic ~2+'sin$+ index [~,o(e8-- of detonation ~; )-- ~o~o]sin$ products equals to three. In general, a numerical /7($)-- analysis (l+m) 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 + (1 + ms) ('~,eii, + ~,) cose= 0 (2.11) purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. an analytic formula with two ~,=~',/(1 parameters +ms), of high ~0-- explosive (?,0-'~;)/ms (i.e. detonation velocity and polytropic index) Equation for estimation (2.11) can of be the rewritten velocity as of flying plate is established. //lsint~ +//'t 1. cos0 Introduction + ( 1 + ms) ~0//s= 0 (2.12) Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive //1 = loading, m shock ~l synthesis (~e~e+u of diamonds, and explosive welding (2.13) and cladding of metals. The method of estimation 9 (l+m) of flyor velocity and the way of raising it are questions of common interest. /I2= (l+ms)~,#,, (2.14) Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem //s-----~,coso+~,,sino of motion of flyor is to solve the following system of equations (2.15) cr,l----kr~ ) ~, 3.~ ~ t ~, ~ (2.16) o-,=~-~o',--~p) +3,, ap +u_~_xp + au Note that in //~, Tlz and //s,only ~) ~" and ~'" are ~" involved with no derivatives of ~ higher than the second order. From, equation y (2.12) =0, it follows that ~; =ms(//~sin0 as + H~cos0)///s as + (1 + ms) ~,, (2.17) a--t Equation (2.17) is a new governing equation and is different from equation (2.9) in form. It is a nonlinear ordinary differential eqltation p =p(p, of s), ~ of order 3, while equation (2.9) is that of order 4. When //s~:0' 9 equatigll (2.17) i~ obtained through the conservation of the d integral, p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, therefore, it with is equi_x~lent the trajectory to eqtmmon R of reflected (2.9). To be shock more of exact, detonation it should wave be D mentioned as a boundary that during and the the trajectory derivation F of of equation flyor as another (2.13), the boundary. t~e condition Both are of unknown; f0rce on the the crack position is used. of R If and side the EA state in Fig. para- I is meters not free on of it forces, are governed we will by have the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

4 842 Wang Ke-ren and Wang Tzu-chiang Thus equatior, (2. t7) becomes H(O) = cons tant.=h0 (2.18) ~; =ms(h,sino+hzeoso--ho)/ff[3+ (i +ms)y, o (2.19) Accurate numerical results show that when H3~:0, equations (2.19) and (2.9) are indeed equivalent, that is, they lead to the same results with the same initial conditions, or with the same initial values for ~, ~', ~'" and ~"". In other words, H0 is determined by the initial value of 1-I(0). In the above disoussion, the conservation of J integral is used. Similarly, the use of the conservation of J2 integral will lead to another functional equation. We have I-~1.[~=~ (W.nz--a,jnjm,z)ds=O (2.20) Referr!ng The one-dimensional again to Fig. 1, problem we evaluate of the the motion J2 integral of a rigid on the flying contour plate under of ABCDEA. explosive Assume attack has the an contour analytic to solution be very only close when to the the crack polytropic tip so that index only of detonation the main singularity products equals terms to have three. to In be general, considered. a numerical On CDE, analysis we have 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 Ou, approximate Our f solution Ou, is obtained +_~ for the } problem of flying plate driven by various cr~ high explosives with +l~r-ff-'-=p'~ polytropic "---4-=-'sinoy k indices OS other O+ than but COS0 nearly equal to three. an analytic formula with two parameters OU of high explosive 1 OU (i.e. detonation velocity t, and polytropic + P,{~r sin O+r---eosOfl~o}=sinO[p,u, +p,u,] COS 0. + r ' LP, 1. u; Introduction + Pou~ +u,po--uop,] Explosive driven flying-plate = ak"+ technique 'r~ +~'{ (1 ffmds + ms) its (~,~, important + u use sino the study of behavior of materials under intense impulsive + [~,(~; loading, "~e) shock +~,o synthesis (~ + of ~,).]eos0} diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest. Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following 0 system of equations governing the flow field of detonation ( products )'= behind ( ), the flyor ( )'= (Fig. 0--~0--( I): ) The at2 integral on CDE can thus be obtained as ap +u_~_xp + au Io~x au { W.n2--cr,,nsu~,a}ds=aK au 1 y =0, ~+1I~( 0) (2.21) as as a--t i~(6)= I~{ m -1+,. (l+mf or, szn0--[(l+ms)(~,~,+~,o~0)sin 0 p, p, S, u are pressure, density, + (~,(~; specific -~e) entropy + "~,e(~$ and +~,))eos0] particle velocity }do of detonation products (2.22) respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the Note that equation (2.21) does not dependent on r, therefore, the main part of the J2 integral on AB trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parashould meters on be equal it are to governed that on CDE by the in flow magnitude field I but of central with opposite rarefaction signs. wave In other behind words, the detonation the main wave parts D of and j2 integral by initial on stage CDE of and motion AB of are flyor concelled also; the and position the remaining of F and the parts state will parameters tend to of zero products when

5 Conservation Integrals and Determination of HRR Singularity Fields 843 r~ and r 2 tend to zero. Now we consider the j2 integral on BC. II~ is easy to show that the main part is Ijo ( Wn2--cr,~n~u,,Dds= -- ln ( r2/r D.1-I*(0) 2-1"(0)= m (l+m)'~]+=eoso-{(l+ms)(#,e~,+uo~dsino (2.23) It follows that + [~,o(~;--~e)+~o(~+~,)]eoso} (2.24) s J { Wnz }as = ~) ano D~,t -- tr~n~ui,z = [ H*(H) --H*(O) ] ln(r~/rl) +o(r) (2.25) Let r~ The and one-dimensional r 2 tend to zero problem with r2[r of t remaining the motion constant, of a rigid we flying obtain plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. H*(O)=H*(~) In this paper, however, by utilizing the "weak" shock (2.26) behavior of the reflection shock in the explosive products, and applying the small parameter pur- Functional equations (2.26) and (2. l l) are very similar. Equation (2.26) can be rewritten as terbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high H,eosO--H2sinO+ explosives with polytropic (l+ms)~0h~=h*(~r) indices other than but nearly equal to three. (2.~7. an analytic H~ formula and Hz with are two given parameters again by of equations high explosive (2.14) (i.e. and detonation (2.15) and velocity and polytropic./7]=--go sin 0+'r,o cos 0 (2.28a) From equation (2.27), the following governing equation can be obtained 1. Introduction ~; =ras(hl coso--h= sino--h*(~r))/h~+ (1+ms) ~,0 ( Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials It is another under new intense gox/eming impulsive equation loading, and shock is also synthesis a nonlinear of diamonds, ordinary differential and explosive equation welding of and cladding 0forder of 3. metals. When The /-/] method # 0, it of is equivalent estimation to of equation flyor velocity (2.17) and on the the way condition of raising that it are the questions crack tip is of enclosed common in interest. the plastic zone. Under Equations the assumptions (2.11) and (2.26) of one-dimensional can be rewritten plane as detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation Fl sin0+f2 products behind cos0=0 the flyor (Fig. I): (9.. 29) Ft cos 0--F~ sino=h*(~r)=f, ~ ap +u_~_xp + au From equation (2.29) it follows that /-"= (l+m) " as as Fz= (1 + ra) (~,t~, + ~o~e) a--t y =0, (2.30) p, p, S, u are pressure, density, FI----F, specific cos0, entropy Fz=--Fosin~ and particle velocity of detonation products (2.31) respectively, From equation with (2.31) the trajectory we can obtain R of reflected the following shock of two detonation governing wave equations: D as a boundary and the on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion ~; ~ (1 of -b flyor m,~) ~,o also; + the (ms) position [/"o sin04- of F and ~,,~,]/~s the state parameters of products (2.32) l

6 844 Wang Ke-ren and Wang Tzu-chiang ~';--'--~'s+(ms)[ ' (1-t-m)m ~r~~ +,,#e~.e_f,eos/~]/~, * (2.33) Equations (2.32) and (2.33) are equivalent to equation (2.9) on condition that ~o~0 and ~,#=k= 0, respectively. III. The Case of Plane Stress The analysis in section 2 can be similarly applied to the case of plane stress. Equations (2.1) to (2.6) remain to be valid. The third part of equation (2.7) should be replaced by ibffia~'-'(~,--~,/2) } (S.1) The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when the polytropic I1~ II index of detonation 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 ~o= (t~,o-- approximate ~; )/rm solution is obtained for the problem of flying plate The driven governing by various equation high in explosives [1] is with polytropic indices other than but nearly equal to three. an analytic formula with two ~g parameters of high explosive (i.e. detonation velocity and polytropic + (1+~).rm~",-~.{ (s+ 2) (2s+ 1)~--~ "" )- 1. Introduction It is different Explosive from driven the corresponding flying-plate technique equation ffmds (2.9) for its the important plane strain. use in But the equations study of (2. behavior l l), (2.26) of materials and (2.31) under are valid intense for impulsive both the loading, cases of shock plane strain synthesis and of plane diamonds, stress. and explosive welding and cladding Similarly, of metals. the governing The method equations of estimation (2.17), of (2.28), flyor velocity (2.32) and and (2.33) the way are of also raising valid it are for questions both the of cases. common interest. Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach I. Numerical of solving Results the problem of motion of flyor is to solve the following system of equations The numerical integrations are performed by fourth order Runge-Kutta method with automatically adjusted steps to control the accuracy. First, the integration is made for equation (2.9) and the validity of functional integral ap +u_~_xp equations + au (2. l l) and (2.26)is checked. In tables 1 and 2 are listed numerical results for two typical au parameters. au 1 It can be seen that the functional integral y =0, equations (2.1 l) and (2.26) are indeed satisfied. During the integration the accuracy is controlled within 10.8, and the equations (2.11) as and (2.26) as are satisfied also to the accuracy of 104. a--t In tables 1 and 2 are also listed the results by the governing equation (2.17), which also fully agree with those by equation (2.9): The p difference =p(p, s), can only be found at the point of 0---~r., /'/'s= 0. In tables I and 2, the first line gives the results by equation (2.9) and the next line the results p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products by equation (2.17), Where the tick t,, resprc~,ats the complete coincidence of the eight significant figures. respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products

7 Conservation Integrals and Determination of HRR Singularity Fields 845 Table.t m= 3.0 8" ~e r~ (r~ ~*(0)-10 a JY(O). 10 ~ 0 2O v' 1.2~ ,15087?06 C ~.332~87~ T O ? ?1 8O , ~ ~ ? The one-dimensional problem of the motion of 0.6?2?T230 a rigid flying plate T under explosive attack has an analytic solution only when the polytropic index 0.6T of detonation products equals to three. In general, a numerical C ~ analysis is required ~93 In this paper, however, by ~7 utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter purterbation 1~0 method, 6.~ an analytic, first-order 6.109~4867 approximate solution is obtained : for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final 180 velocities ~X10 of flying plate -~ obtained -~. I~34i5 agree 10 -~ very well with numerical results by computers. O, Thus an analytic formula with two I0 -~ parameters --O. 9T~59~ of x 10 high -~ explosive (i.e. detonation velocity and polytropic,table 2 1. Introduction m~ " Explosive driven flying-plate technique ffmds its important O" r ' ~ use I1"(8) in the "103 study of behavior _T'Z(0) 9 101o of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and O 2.? cladding of metals. The method of estimation 0 of flyor velocity and the way of raising it are questions of common interest r Under the assumptions of one-dimensional plane T0 detonation and rigid flying plate, O. the normal approach of solving the problem of motion of flyor is to solve the following system of equations 4O C.3201T ~I~ ~T6gT~ ~t ,,/ v' ap +u_~_xp + au 8O 0.990?~43 ~.3~26E13~ O. 3,il 868 y =0, ~ O ? as as 120 ~4~58 ~.3ooe~ri a--t o.511sro18 O ~.23~081~2 0.~ ,51t" p, p, S, u are pressure, density, specific entropy C.0~ l and particle velocity of detonation products ~ respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor o as another boundary. Both are unknown; the position of R and the state parameters 180 on it are --O.~I209~XZO-~ governed by the "~[ flow field I of central T rarefaction wave behind the detonation O O wave D and by initial -o stage of motion -~.~07~xlo-, of flyor also; the / position 0.5S6dlmST of F and the state parameters of products

8 846 Wang Ke-ren and Wang Tzu-chiang _ References [ I ] Hutchinson, J.W., J. Mech. Phys. Solids, 16 (1968), 13-31, [ 2 ] R):',e, J.R. and G.R. Rosengren, J. Mech. Phys. Solids, 16 (1968), [ 3 ] Rice, J.R., J.'Appl. Mech.. 35 (1968), "[ 4] Shih, C.F., ASTM ST[' 560 (1974), [ 5 ] Knowles~ J.K. and E. Sternberg, Archives for Rat. Mech. Analysis, 44 (1971/72), The one-dimensional problem of the motion of a rigid flying plate under explosive attack has an analytic solution only when 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 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 plate driven by various high explosives with polytropic indices other than but nearly equal to three. an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic 1. Introduction Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of common interest. Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations ap +u_~_xp + au y =0, as a--t as p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the on it are governed by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of flyor also; the position of F and the state parameters of products