BOUNDARY ELEMENT METHOD FOR SOLVING DYNAMICAL RESPONSE OF VISCOEI]ASTIC THIN I~LATE (I)* Ding Rui (7 ~:)~ Zhu Zhengyou (~i~)": Cheng Changjun (~)":

Transcription

1 Applied Mamatics and Mechanics (English Edition, Vol. 18, No. 3, Mar. 1997) Published by SU, Shanghai, China BOUNDARY ELEMENT METHOD FOR SOLVING DYNAMICAL RESPONSE OF VISCOEI]ASTIC THIN I~LATE (I)* Ding Rui (7 ~:)~ Zhu Zhengyou (~i~)": Cheng Changjun (~)": (Received Abstract Jan ) The one-dimensional problem motion Abstract a rigid flying plate under explosive attack has an analytic solution In this only paper, when a bounda~ polytropic 3" element index method for detonation solving dynamical products response equals to three. In general, a numerical viscoelastic analysis thin plate is is required. giren, bl In Laplace this paper, domain, however, we propose by utilizing two methods "weak" to shock behavior reflection shock in explosive products, and applying small parameter purapproxhnate ftmdamental solution and derelop corresponding bounda O" element terbation method, an analytic, first-order approximate solution is obtained for problem flying method. Then using hnproved Belhnan's n,unerical htversion Laplace plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities transform, flying solution plate obtained agree orightal very problem well with is obtabted. numerical The results namerical by computers. resuhs Thus an analytic show formula that this with method two parameters has higher accuracy high and.faster explosive com'ergence. (i.e. detonation velocity and polytropic index) for estimation velocity flying plate is established. Key words dynamic response, viscoelasticity. BEM I. Introduction The dynamic response viscoelastic structures is one important research directions in materials solid mechanics. under intense Because impulsive loading, complexity shock synsis constitutive diamonds, and relations explosive welding viscoelastic and cladding materials, metals. problem The method solving estimation dynamic flyor response velocity and is very way difficult. raising There it are questions are some available common numerical interest. methods, such as finite difference method and" finite element method (FEM) Under etc., to assumptions solve this" problem. one-dimensional The boundary plane detonation.element and method. rigid (BEM) flying plate, is an efficient normal approach numerical method solving developed problem in motion recent years. flyor As is BEM to solve reduces following problem system in equations region to governing one on flow boundary, field it detonation has obvious products advantages behind comparing flyor (Fig. with I): FEM. Now re are many papers by using BEM to discuss static problems in elasticity and plasticity. But it is more difficult to solve structural dynamic problems ap +u_~_xp + au by means BEM. The main troubles lie in that it is not easy to obtain time-dependent fundamental solution or fundamental solution problem derived by Laplace transformation. For a viscoelastic structure, this problem becomes much more complicated. The basic stage for BEM solving dynamic response viscoelastic structure is as follows: First, using Laplace transformation, original a--t problem may be transferred to a problem in Laplace plane, and basic equation dynamic problem viscoelastic structure in Laplace space is yielded. Next, applying BEM, where solution p, p, S, u basic are pressure, equation density, in Laplace specific domain entropy is obtained. and particle Finally, velocity by means detonation products Laplace respectively, inversion or with numerical trajectory Laplace R inversion, reflected shock solution detonation original wave problem D as can a boundary be obtained. and In trajectory order to conveniently F flyor as apply anor this boundary. method on Both practical are unknown; problems, position is necessary R and to give state a proper parameters on it are governed by flow field I central rarefaction wave behind detonation wave * Project supported by National Natural Science Foundation China i Department Mechanics, Lanzhou University, Lanzhou , P. R. China 2 Shanghai Inst. Appl. Math. and Mech., Shanghai , P. R. China 229

2 230 Ding Rui, Zhu Zhengyou and Cheng Changjun fundamental solution which is easily calculated and an efficient method carrying out numerical Laplace inversion. In paper [I], authors discuss BEM for transient response viscoelastic structure, fundamental solution in that paper consists second kind n-order Bessel functions that is too complicated to do calculation. Especially, it is very difficult to use Durbin's numerical inversion. It is possible that difficulty and complexity mentioned above are avoided if we employ approximate fundamental solution in this paper. The numerical Laplace inversion transform plays an important role in dynamic viscoelastic problem. Up to now, though re are some methods about numerical Laptace's inversion, such as Durbin's inversion technique Isl and Miller's numerical inversion Abstract transform 161 etc., se methods do not fit our problem. Here we give an improved Bellman's inversion The one-dimensional formula that is problem more easy to motion be implemented. a rigid flying In Section plate under II, we explosive describe attack basic has an equation analytic in solution Laplace only domain when for polytropic dynamic response index detonation viscoelastic products thin plate, equals and to three. establish In general, corresponding a numerical boundary analysis is integral required. equations. In this paper, In Section however, III, by an utilizing approximate "weak" fundamental shock behavior solution with reflection form shock in double explosive trigonometric products, series and applying and a modified small parameter approximate purterbation method, an analytic, first-order approximate solution is obtained for problem flying fundamental solution are given. In Section IV. we discuss improved Bellman's inversion plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final technique. velocities The numerical flying plate examples obtained and agree conclusions very well are with shown numerical in Section results V. by computers. Thus an II. analytic The Boundary formula with Integral two parameters Equations high explosive (i.e. detonation velocity and polytropic index) for estimation velocity flying plate is established. The governing equation d~tnamical problem viscoelastic thin plate in Laplace domain is given as t3j sd~,~w + Phs~W =q (2.1) materials where//7" under and h intense express impulsive deflection loading, and shock thickness synsis diamonds, plate respectively, and explosive q(x, welding Y, t) is and cladding transverse metals. loading, The l,] 7 method and q are estimation Laplace flyor transform velocity and W and way q, raising respectively, it are questions s is " transformation common interest. variable. And Under assumptions one-dimensional plane detonation and rigid flying plate, normal D----gha/12(l-s"-fiz), E=9OK/(3_~ +~) approach solving problem motion flyor is to solve following "1 system equations (2. governing flow field detonation 2) sfi = (3R - products 20)/2 behind (3K + ~) flyor (Fig. I): J" where G(t) and K(t) are shear and volume relaxation functions, respectively. Assume that W ~ (P, Q. s) is ap fundamental +u_~_xp + au solution (2.1), n from definition, W~ satisfies equation sdv'w ~ + PhszW~--'(~ (P - Q) ( 2.3 ) ~vhere coordinates points P, a--t Q are (.\, y), (~. q). Using Green's formula, we may Obtain boundary integral equations as follow 9 IS,'!,.cv~ ~ (P,O ) 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. OW(Q') Both W(Q')I~'*(P',QQ]dl-'(Q') are unknown; position R and state parameters on it are ~-~:(P', governed by 0') flow o.(q') field I central " rarefaction wave behind detonation wave T ~ ([./~,,,'W~]~., - [/~:,i. WJa,) (2.4) i=1 c,. ow (F') =it,fl(o) ow.~p,,e~ ow.(p',e') r162 o,,(p,) ' d~:'+l,.[~"(o') ~ -~.(e')

3 BEM Viscoelastic Thin Plate (I) 231 a~w.(p, Q').,. +om. (P',Q') ow(q') on(p')on(q') On(F) On(Q') "([ [~ "'d ) + y], 3i/.,,. On(P') ~, - On(P')" ;,, i-1 W(Q') 0I~'.(P',Q') ]df (Q') o.(p') in which M,, and R,, are bending moment and shear force, M.,~ =01~l./dt, M~,t = OM~/or, n is outer normal vector edge, t is tangent vector. [q0]xt (i= 1, 2,... m) express jumping value ~0 at Abstract corner point ~, along edge. Hence we have ( (VP'Et2) The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when C,~= 0,~/2:r polytropic index (VPtEF) detonation products equals to three. (2.6) In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock 0 (VP' ~ ~ and.p, ~/-') behavior reflection shock in explosive products, and applying small parameter purterbation Once we method, get an boundarv analytic, value, first-order value approximate inner solution point is can obtained be obtained for problem by means flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. inner integral formula Isl. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an III. analytic Approximate formula with Fundamental two parameters Solution high explosive (i.e. detonation velocity and polytropic index) for estimation velocity flying plate is established. In order to compute conveniently, in present paper, we derive two approximate fundamental solutions (namely solution i;i7~ in (3.1) and solution W~ in (3.2)) to approximate fundamental solution IV *t~. Without loss generality, assume that thin Explosive plate is driven square flying-plate one, that is, technique region ffmds 12 is its [-at, important st] use [-~r, in :r] study. Substituting behavior materials fundamental under solutions intense impulsive (3.1) and loading, (3.2) into shock synsis equation diamonds, (2.3), and and using explosive orthogona!iiy welding and cladding property metals. trigonometric The method series, estimation we can obtain flyor velocity following and way two kinds raising it are approximate questions fundamental common interest. solutions: Under (i) Approximate assumptions fundamental one-dimensional solution with plane detonation type double and rigid trigolometric flying plate, series normal (DTS) approach solving problem motion flyor is to solve following system equations governing flow field detonation products behind flyor (Fig. I): ^, 1 + 1,~ cosm(~-x) 1 ~' ap +u_~_xp + +~",,.~..: au (3.t) [~D(m~+nb~+~] au 1 (ii) Modified approximate fundamental solution (MAFS) a--t W;(_P, Q)=W~(P, Q)+W:(P, Q) (3.2) where/4/'~ is fundamental solution equation /~V41,V=0, namely, where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products respectively, with W~(P, trajectory Q)=~l/8gD)r~lnr, R reflected shock r=l_tr'-qi--d(x.~)'-+ detonation wave D as (g-r/) a boundary i and trajectory W~ (P, F Q) flyor is as modified anor part boundary. having Both are form unknown; position R and state parameters on it are governed by flow field I central rarefaction wave behind detonation wave 1 ~s~d 0 OO W~(P, O)=" 4n'~l ~Y-~,[(I s)cosm(~-x)l (d~jsinmx m-! au oo + d ~). cosmx) ]/2n ~ [sere' + ~n ] ~ [ ( 1 - s) cosn (r/- x) - ( d ~.')-sinny 1191

4 232 Ding Rui, Zhu Zhengyou and Cheng Changjun c~ q-d~ ~ oosnv) ]12a3[sDn ' +m] + ~, ~, [ (1 - s)cosrn~--x) Oosn (r/- y) m-i ~-1 - ( d ~)sinmx. sinnv + d~)~sinmxcosny + d~,~.cosmx, sinnv " + d ~cos rex. co Shy) ]/~r ~ [sd (m ~ + n~) ~ + m] (3.3) where Abstract The one-dimensional ~=Phs z, L=~, problem do=lilor2]nrdxdy, 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 d~)=l reflection II r2[nr shock - sinmxdxdy, explosive products, and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven d~'}=l by various llz~inrsinnydxdy, high explosives with polytropic d. =L indices l lar~inrcosnydxdy, 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 parameters high explosive (i.e. detonation velocity and polytropic index) for estimation do) =L rqnrsinmxsinnvdxdv, velocity flying plate is d established. ~a)-r ~t(~) 9,,,,=z, -- it rqnreosmx.sinnydxdy ' d(')--l ""-- ~ torqnrc~176 materials under intense impulsive loading, shock synsis diamonds, and explosive welding and here g2-=[-~, zr] [--:,r, a]. cladding metals. The method estimation flyor velocity and way raising it are questions common Inserting interest. previous series W~(W~) into boundary integral equations (2.4) and (2.5), Under and using assumptions constant one-dimensional element to discrete plane detonation obtained and equations, rigid flying n plate, we get normal a set approach equations in solving IV, OW/On, problem ~;1. and motion R.. flyor Given is any to solve two in following four variables system W, equations OW/On, governing M, and /~,, flow. field n we detonation can obtain products behind ors by flyor solving (Fig. I): set discreted boundary integral equations. Thereupon, deflection W{x, y, s) at any point p(x,y) in plate can be calculated with help discreted integral equations for any given s. ap +u_~_xp + au IV. Numerical Inversion au Laplaee au Transform 1 In Section III. we have obtained solution original equation in Laplace domain. In order to get solution as in as time region, we have to use an efficient inversion a--t Laplace transform. Here we use Bellman's inversion technique tt1 Let ~(s) be Laplace transform p =p(p, f(t), s), namely. 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 Introducing on it are variable governed transform by flow x~e field -~, I thus central it yields rarefaction that wave behind detonation wave D and by initial stage motion flyor also; 1 position F and state parameters products ](s)= t X~-lf(-lnX)dX (4.2) 0 Using Gauss-Legendre numerical quadrature formula to approximate above integral, we have

5 BEM Viscoelastic Thin Plate (I) ] = Y2, w,x: -'/( - lnx, s) t-i where W, are weight functions..e are roots Legendre polynomialg order n. Let s~=./+l; j=l, 2,..., n. From (4.2). we obtain n f(-lnx,)= y-~c,(i, j)](sj) (4,4) t-1 where coefficients. Cn (i, j) may be found Abstract in [7]. Once X, is fixed, from {4.4). we may obtain value /it)..when t,=-lnx~ 9 In order The to calculate one-dimensional value problem t at any time. motion we employ a rigid flying following plate under techniques. explosive As attack ~ If (c0] has an analytic solution only when polytropic index detonation products equals to three. In = (;/c)](s/c), where c is arbitary constant, n (4.4) becomes general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection shock in explosive products, n and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven by various high explosives with polytropic./-1 indices or than but nearly equal to three. Final where velocities ~,~=C(j+I), flying plate C~,(i, obtained j) may agree be very calculated well with by numerical means results by method computers. similar Thus to an analytic formula with two parameters high explosive (i.e. detonation velocity and polytropic computing C,, (i, j) in (4.4). Thus. by regulating value c. we can obtain value "f at index) for estimation velocity flying plate is established. given time. We call (4.51 improved Bellman's numerical inversion Laplace transformation. V. Numerical Examples and Conclusion In order to verify accuracy two approximate.fundamental solutions and materials under intense impulsive loading, shock synsis diamonds, and explosive welding and improved Bellman's inversion, we deal with problem dynamic response an elastic cladding metals. The method estimation flyor velocity and way raising it are questions thin common plate. interest. Table Under 1 The value assumptions deflection one-dimensional in Laplace plane domain detonation (~'~=value and x rigid IE-04) flying plate, normal approach solving problem motion flyor is to solve following system equations governing s flow field detonation 1 products behind 2 flyor (Fig. 3 I): 4 (DTS) IV ~ , , (MAFS) W TV ap +u_~_xp au s (DTS) W ~ , (M AFS) W a--t TV " i , H48~26 Remark The values in last volume are oretical ones. where p, p, S, u are pressure, density, specific entropy and particle velocity detonation products respectively, Table 2 9 The with value trajectory deflection R at reflected different shock time fw=valuex detonation 1E-04) wave D as a boundary and,,,,,,..,, trajectory F flyor as anor boundary. Both are unknown; position R and state parameters on t it are governed by flow field I central rarefaction wave behind detonation 1 wave D and by x initial stage motion flyor also; position F and state parameters c= " products (DTS) W I (MAFS) W TV "1 t=--c~nx, here x=o,s

6 234 Ding Rui, Zhu Zhengyou and Cheng Changjun Example 1 Assume that transverse load.q=qo cos cot, where q gx 104 N/m ~, a~=i.5. Letting size plate be lm tin 5m, P----g.9 10ekg/m. 8, E=2 10Zlkg/m3, /~ We take n=m=25 in expression (3.1) and n=m = 15 in expression (3.2). The boundary is divided into 12 elements. In Table 1. we list value ~dr- flection at center point plate in Laplace transform domain. And deflection at center in original problem is given in Table 2. value From Table I, we find that results obtained from modified appro.ximate fundamental solution are better, Table 1 and 'Table 2 also show that using eight-point Bellman's numerical inversion, we can obtain Abstract expected results. Example 2 Assume that materials thin plate is standard linear solid, namely The 131 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, first-order approximate ----~], solution 3K(t)=l_epo 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 in computation, we take an analytic formula with two parameters high explosive (i.e. detonation velocity and polytropic index) for estimation Pl=5x 10-~s, velocity q0=3x flying 103MPa, plate is qj---gx established. I02MPa.s, tz,=0.3, The thin plate is a square plate with size [-~r, ~r],,, [-zr, ~r],~ and h----o.olm, P=2.4 loskg/m s, q(x,y,t)=poh(t), Pp.= 0.5MPa, H(t) is Heaviside function, And Explosive also edge driven flying-plate is technique free. We ffmds divide its each important boundary use in into 12 study elements, behavior n materials deflection under at center intense point impulsive at different loading, time shock is shown synsis in Table diamonds, 3. and explosive welding and cladding metals. The method estimation flyor velocity and way raising it are questions common Table interest. 3 The value deflection plate center at different time Under (W=value assumptions IE-04) one-dimensional plane detonation and rigid flying plate, normal approach solving problem motion flyor is to solve following system equations governing t flow field detonation products behind o flyor (Fig. I): ( M A F S ) W t ,1 ap +u_~_xp + au (MAFS)W , In this paper, BEM for a--t dynamic response viscoelastic structure is given. In Laplace domain, we provide two methods to approximate fundamental ~ solution. The solution original problem is obtained by use improved Bellman's inversion. where Comparing p, p, S, our u are method pressure, with density, existing specific method, entropy one and particle can see velocity that method detonation can products avoid respectively, difficulties caused with by trajectory calculation R reflected and inversion shock transform detonation due wave to D using as a boundary approximate and trajectory fundamental F flyor solution as anor p'resented boundary. in this Both paper. are unknown; One can also position see that R and realization state para- meters numerical on it are inversion governed by transform flow field is I more central convenient rarefaction and wave behind efficient detonation by using wave improved Bellman's inversion technique. It is easy to see that present method can also deal with much more complicated dynamic problems, such as dynamic response for a viscoelastic plate on a viscoelastic foundation, etc.. In next paper, we will give oretical analysis and error estimation boundary element method presented in this paper.

7 BEM Viscoelastic Thin Plate (I) 235 References [1] B. Sun, et al., Boundary element analysis transient response for two-dimensional viscoelastic structures, Shanghai Mechanics, 11, 1 (1990), (in Chinese) [ 2 ] B. Sun, et al. Boundary element analysis transient response for multiphase viscoelastic structures, Computational Structure Mechanics and Applications, 7 (1990), 19~21 (in Chinese) [31 T. Yang, Dynamic responses axisymmetrical problems for a viscoelastic plate on a viscoelastic foundation, Acta Mechanica Sinica, 22,2 (1990), [41 P. Gu and J. Zhu, Study approximate Abstract fundamental solution orthogonal polynomials in dynamical BEM, Computational Structure Mechanics and Applications, 7, The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic 4 (1990), solution only (in when Chinese) polytropic index detonation products equals to three. In general, [5] F. a Durbin, numerical Numbericai analysis is inversion required. In this Laplace paper, transform: however, by An utilizing efficient improvement "weak" shock to" behavior Dubher and reflection Abate's shock method, in The explosive Computer products, Journal, and 17 (1974), applying small parameter purterbation t6] M.K. method, Miller an and analytic, W. T. first-order Guy, Numerical approximate inversion solution is obtained Laplace for transform problem by use flying plate driven Jacobi by polynomials, various high SIAM explosives J. Numer. with polytropic Anal., 3, 4 indices (1966), or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus [7] R. Bellman, R. E/Kalaba and J. Lockett, Numerical brversion Laplace Tratisform, an analytic formula with two parameters high explosive (i.e. detonation velocity and polytropic index) Amer. for estimation Elsevier Publ. velocity Co. (1966). flying plate is established. [81 C. A. Brebbia, Botmdary Element Techniques: Theory and Applications in Engineerhtg, Springer-Verlag (1984). 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 + au 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