Applied Mathematics and Mechanics (English Edition, Vol. 13, No. 2, Feb. 1992)

Published by SUT, Shanghai, China

A NUMERICAL CALCULATION OF DYNAMIC BUCKLING OF A THIN SHALLOW SPHERICAL SHELL UNDER IMPACT Mu Jian-chun ( ~ 1 ~ )

Wu Wen-zhou ( ~ 3 ~ )

Yang Gui-tong (~J~Jl~)

( Taiyuan University o f Technology, Taiyuan)

(Received Sep. 8, 1990) Abstract Assuming the deformation o f the shell has an axial symmetrical form, we transform Marguerre's equationsI~1into difference equations, and use these equations to discuss the

Abstract buckling o f an elastic thin shallow spherical shell subjected to impact loads. The result shows when impact load acts on the shells, a jump o f the shell takes place dependent on the

The one-dimensional problem of the motion of a rigid flying plate under explosive attack has values 2 and the critical buckling load increases with the enlargement o f the loading area. 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 words shock shell, impactproducts, and applying the small parameter purbehavior of theKey reflection inbuckling, the explosive terbation method, an analytic, first-order approximate solution is obtained for the problem of flying I. Idriven n t r o dby u c tvarious i o n high explosives with polytropic indices other than but nearly equal to three. plate Final The velocities of flyingsnap-through plate obtainedofagree veryelastic well with results byshell computers. Thus axisymmetric clamped thin numerical shallow spherical under uniform an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic distributed impulsive loads was first reported by Humphreys and Bodnert~J. Using Rayleigh-Rith index) for estimation of the velocity of flying plate is established. methods, they found critical impulse. Afterwards, Budiansky and Rotht3~ solved the same problem based on Galerkin's methods. However, the results from both Rayleigh-Ritz's and Galerkin's 1. with Introduction methods have considerable error compared the exact resultt4]. In 1969, N. C. Huang studied the axisymmetric buckling of clamped elastic shaliowsphericalshell under uniformly distributed step Explosive driven flying-plate technique ffmds its important use in the study of behavior of loads andunder impact loads.impulsive His results makeshock it clear that the shell haswelding a jumpand to a materials intense loading, synthesis of deformation diamonds, andofexplosive definite value of the load when the shell was loaded by uniformly distributed step loads, whil~e the cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions of shell increases abruptly only in a very small area, but no jump occurs when the shell ofdeformation common interest. Under the assumptions of one-dimensional plane detonation rigid flying plate, the normalthe is under uniformly distributed impact loads. Afterwards, someand other researchers calculated approach of solving the problem of motion of flyor is to solve the following system of equations dynamic buckling problem of shell [SL16J.All these researches were limited on either the shell under governing the flow field of detonation products behind the flyor (Fig. I): uniformly distributed loads or concentrated load at the apex of the shell. In 1969, W. B. Stephens and Fulton RoberteLTJwent on studying the problem of static and dynamic buckling of shallow --ff =o,various axisymmetric step loads. They spherical shell with different boundary conditions under ap +u_~_xp + au found that the buckling load increases basically with the increase of the size of loading area. The au au y1 loads =0,has been little studied, up to now. The buckling problem of shell under instantaneous impact (i.0 problem of variation of buckling load, when the size of loading area is different, has not been aS as reported. In this paper, some investigation a--T of those =o, problems will be reported. II.

Fundamental Equations

p

=p(p, s),

of unloaded shell is shown Fig. 1. and particle velocity of detonation products whereThe p, p,shape S, u are pressure, density, specificinentropy respectively, 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 by the flow field I of central 125 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 293

126

Mu Jian-chun, Wu Wen-zhou and Yang Gui-tong

si:

t

k.~_

dO

T

t l ' O ~'

R: the radius of curvature of the shell a : the radius of the base circle H: the central height of the shell

M , M8 are bending moments (they are positive when the inner surface of the shell is pulled), N, Neare Abstract Fig. 1 The shape of unloaded shell membrane forces (the pull is positive), and Q, is shear force The one-dimensional problem of the motion of a rigid flying plate Fig. under 2 explosive attack has

an analytic solution only when the polytropic index of detonation products equals to three. In According to the coordinate in Fig. !, the equations o f any meridian line of the shell are: general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive and applying the small parameter (2.1) purz f H [ 1 - ( rproducts, laY] terbation method, an analytic, first-order approximate solution is obtained for the problem of flying R=a~/2H (2.2) plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final The velocities of flying plate obtained agree very well with numerical results by computers. Thus equations are based on the following assumptions: an analytic formula with two parameters of high explosive velocityo fand a) Both the loads acting on the shell (here, it is impact)(i.e. anddetonation the deformation thepolytropic shell under index) for estimation of the velocity of flying plate is established. external load have an axial symmetrical form. b) This discussion is limited within the range of elastic deformation. 1. of Introduction c) The shear deformation and moment inertia o f the shell during the deformation process are not considered. 1) T hExplosive e e q u i l i bdriven r i u m flying-plate equations:technique ffmds its important use in the study of behavior of materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and (rM,)' - M - r O velocity , = 0 and the way of raising it are questions (2.3) cladding of metals. The method of estimation ofiflyor of common interest. (rN,)' -N,=O (2.4) Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal [rN, (w -z)' + rQ,3' = mrhe (2.5 ) approach of solving the problem of motion of flyor is to solve the following system of equations In equation (2.5),ofthe item w--z reflectsbehind the geometric nonlinear governing the flow field detonation products the flyor (Fig. I): o f the shell,( ) / = # ( )/Or, ( ' " ) =az( )/Ot z,m is the mass per unit volume of the shell, and the other items are showed in Fig. l and Fig. 2. --ff =o, ap +u_~_xp+ au 2) T h e p h y s i c a l equations au

au y1 =0, e,=( N,-~,N,) /Eh

aS el ~ a( No s --vN,)/Eh

a--TM , = D (=o, K,+vK,) p =p(p, s), Me=D(KI+vK,)

(3.6) (2.7) (s.s) (2.9)

(i.0

(2.2), density, equations (2.6) entropy to (2.9) and are linear where Based p, p, S,onu assume are pressure, specific particle velocity of detonation products where , E iswith Young's modulus Rand v is Poisson's ratio. respectively, the trajectory 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 paraD = I of E hS /12( lrarefaction --v z) meters on it are governed by the flow field central wave behind the detonation wave 3) The geometrical equations D and by initial stage of motion of flyor also; the position of F and the state parameters of products 293

Dynamic Buckling of a Thin Shallow Spherical Shell

e , f U ' - z ' w' + (w' )*/2 e,=U/r ge~

- - W It

K#= --W' / r

127

(2.10) (2.11) (2.12) (2.13)

4) The govermental equation Let F be the function of stress and it satisfies:

N, =F'/r

(2.14)

NsfF"

(2.1s)

Combining the equilibrium equations, the physical equations, and the geometrical equations in polar coordinate, we obtain the following equations which must be satisfied by the deformation of the shell DV4w = v * F / RAbstract + F ' w" /r + F" w' I r - rnhf~ (2.16)

v 4 F = Eofh (the- Vmotion Z w / R of - - wa 'rigid w" /rflying ) plate under explosive attack (2.17) The one-dimensional problem has an analytic solution only when the polytropic index of detonation products equals to three. In Introducing the following dimensionless quantities general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock ]t=213(l-vS)3~"(H/h) ~'2, X = rand A / a applying , W=wF/2H behavior of the reflection shock in the explosive products, the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying = ( 2explosives H / a Z ) ( E with / m ) 8polytropic 9 ~=24F/4EHSh, ~=tb' plate driven by various rhigh indices other than but nearly equal to three. Final The velocities of flying plate obtained agree very well with numerical results by computers. Thus equations in dimensionless quantities are: an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic Governmental equations: index) for estimation of the velocity of flying plate is established.

v'Wffi( x~), lX +(r

), l X - f f

(2.i8) (2.19)

( X # ' )'1. - @Introduction / X + X W ' = - - ( W ' )'/2 Boundary conditions: when X = ~driven flying-plate technique ffmds its important use in the study of behavior of Explosive materials under intense impulsive loading, shock synthesis of~-)ffio diamonds, and explosive welding and w'(,t, ~)=w,(a. (2.zo) cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions ~'(~, ~)-~(~, ~)I~=o (2.21) of common interest. when X = 0 Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion flyor is to solve the following system of equations W'(0, ofr)--WW(0, r)=0 (2.22) governing the flow field of detonation products behind the flyor (Fig. I):

~(0, r)=0

Initial conditions:

--ff ap +u_~_xp+ au

where ( ' ) = 8 ( III.

)/a,,

(2.zs)

=o,

w(x, au 0)=0 au 1 ~ ( x , y o)ffi4a =0,

(2.z4) (2.25) (i.0

a=(aZJlz/16I-I*)(mlE)"z(OWlSt) aS as

The Calculating Method

a--T =o, p =p(p, s),

Based on the axisymmetric assumption, the difference method can be used along any meridian where p, shell. S, u are pressure, density, and under particlea concentrated velocity of detonation products line ofp,the In order to grasp thespecific feature entropy of the shell load, a finer grid is respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the used surrounding the centre area of the shell. trajectory F of flyor as another boundary. Both are unknown; the position of R and the state paraHere we take meters 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 293

128

Mu Jian-chun, Wu Wen-zhou and Yang Gui-tong

Al=0.05, Ai=0,25 Then

X,=i(i-I )A,

i~4

i----1, 2, 3, ".., N + I /~=A/Az+3(1--AffA,)

N is the number of the grid.

Suppose the step length of time is c~, then v = 3 j ( j = 0 , 1, 2, ...) The specific calculating procedure is as follows: Suppose W(X, r ) is known. then f ( X , r ) = - X [ X W ' +(W')z/2] is also known So using XZcM + X c y - q ~ = f ( X , r) we can obtain the value of ~ . The two Abstract equations above can be rewritten in central-difference ,formula (3.1) / , = - - Xproblem , [ X o ( Wof, §the motion)IZA (W,+, --m,_, The one-dimensional of a +rigid flying plate )VSA'] under explosive attack has ( 3 . ]In )' an analytic solutionX fonly ( ~ , _when , - Z r 1the 6 2 polytropic+ index X,( r 1of6 2detonation products equals to three. general,( r ai.e. numerical analysis is required. In this paper, however, by utilizing the "weak" shock j is constant in above two equations) behavior of the reflection shock inconditions, the explosive products, and applying the small parameter purAccording to the boundary we obtain: terbation method, an analytic, first-order approximate solution is obtained for the problem of flying (~.2) t:,=0, indices /x+~=O plate driven by various high explosivesr with polytropic other than but nearly equal to three. Final Let velocities of flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic ~, =a,r247, +#, index) for estimation of the velocity of flying plate is established. Then aj = - Xt (2Xi + A )/D,

P,=EzAV,-#,_,X,(2X,1. Introduction A).VD, D , = X , a , _ ~ ( 2 X , - A) - 2 ( A2+ 2X~ ) Hence Explosive driven flying-plate technique ffmds its important use in the study of behavior of a, = i l l = 0 materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and From ( 2estimation , r ) - - v ~ (of ) . ,flyor r ) /velocity 2 = 0 and we the have: cladding of metals. The method~ 'of way of raising it are questions of common interest. ( r +,-r ) lZA-~,r +,l~l=o Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving thewe problem Using equation (3.2) obtain:of motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I): So,bali=2,

.cb~+~='[E(2~'Al,'l)+axJ~x+, + ~ } l~l--E(2vAl,~)--a~-lax+~} =o, 3, .--, N + l ) a n d --ff ap +u_~_xp+ au

~=(~,+L--~i_~)/2A ( i = 2 , 3, .-., N + l ) . c a n b e f o u n d . au au y1 the =0, When i= 1, because ~b0 is not known, we can use following difference formula

r =( r Let

g= (Xr )'/X

(i.0

!A aS= ~ , I aAs

a--T =o, p =p(p, s),

Then g = ~ ' +'-~, g , = ~ , ' + X~ ' limx_,0-----X,--=2W~4:(0" r) where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively,lim, with the Xa trajectory. =R- -of reflected detonation W~4~(0, shock r ) , of lim g , = l i m wave ~ + D as a boundary ~t =2~',and the . . ~ 0flyor as another boundary. Both are unknown; T--*0 X ' * 0position of R and the state paratrajectory F Xof the meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave Equation (2.18) can be simplified as follows: D and by initial stage of motion of flyor also; the position of F and the state parameters of products

(%)

293

Dynamic Buckling of a Thin Shallow Spherical Shell

9

W('.:-k2Ww/X-Wv/X2-bWt/X3=g+qbtW'/X+r

129

;-

8w~'>(o, r)/s=g,+~.4,'(o, r)W.(o, r ) - ~ - f o , r)

X~">0

X=o

(3.3)

(3.4)

From boundary conditions, we known that:

W o = W . W_,=W~, W ~ + , = O ,

W~,.=W~

The second- time derivative of W is approximately expressed by the following difference equation:

W~=(a~W,s§

s-~ +wW, j_z + 3~W, j_~)

(3.5)

(

Parameters

a~,

fls , I,'..,, ,

3~ depend on the step length of time, and

as = 2/8 z, f l j = - - 5 / 3 ~,

v~=4/3 ~, 3 s = - ! l < Y '

So equation (3.5) is simplified as: I)~-~- = ( 2 W ~ j - - S W ~ J-I + 4W~ J-z - W ~ s - s ) / 3 ' (3.6) Abstract Finally, we take the difference equations: WhenThe i= one-dimensional 1 problem of the motion of a rigid flying plate under explosive attack has / ].6of detonation 9,\ an analytic solution only when the polytropic 4index products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection explosive the small parameter pur= ( -shock 5 W ~ inj _ the ,+4W ~ s _ , - Wproducts, , j _ 3 ) / aand " - g tapplying s (3.7) terbation method, an analytic, first-order approximate solution is obtained for the problem of flying When i = 2 plate driven by various high explosives with polytropic indices other than but nearly equal to three. _ 2of flying + X___~ + 1 agree very 1 well with numerical results by computers. ] Thus Final[ +velocities plate (robtained an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established.

[ _ ~ ( ~ + ~ : ) - ~ j w,, +~ + ~ , j

1.

9 (~,+-x:)+~ 1

1

w. + ( - ~ ) w .

Introduction

(,"'-~)]",,+[

1

]W4j

1

Explosive driven flying-plate technique ffmds its important in the study of behavior of A ~ use X,A' materials= under intense impulsive loading, shock synthesis of diamonds, and explosive welding and [ - S W , ;-1 + 4 W , j . , . - W , . , ' - 8 ] / ' Y - g , (a.g) cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions Wcommon h e n / = 3,interest. 4. . . . , N - 2 of Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal ~,- r ~1the j 1problem 2 1 1 1 _ -of-1-solving W ,-, . +of [/~ 4 approach motion of flyor is to solve the following system of equations governing the flow field of detonation products behind the flyor (Fig. I): 6 2 1 2 4 2

-~)]w,_,,+ [--~ --,--~,~.(~,+~)-~.]~,,+[~, ~,~, + X_~_~.V(,~, + .,.~._)

--ff ap +u_~_xp+ 1

,

au

au

aS

as

au

=o,

"t

+ ~-~,~ (~,,-w)],,,,,,,+ y =0, =2(--SW', .,._, + 4W, .,,._~.-W, j_s)/,aZ-.q,, 9 When i = N - I

1

[-

1 ~,

(i.0 (s.o)

a--T =o, p =p(p, s),

[ & g-' - Xx_,AS + X a _ I A ~

where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation 1 products 9. 1 1 + 1 (~_,_ X__~.~_i )] respectively, 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 by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion of parameters of products 9 . . , ,flyor _ , ~ . . also; the +position x - - ; T T _ ,of / - mF_ Iand the" -state "

+[~+ ~._,~,-+~(--,+~._,)

~.~._,~

293

130

Mu Jian-chun, Wu Wen-zhou and Yang Gui-tong =2(--5WLt

~_,-I-4Wx_t j_z--W•_l j_s)/c~Z--g~-t ~

(3.10)

When i = N I +

J

qW

,,'. =2( -SWx

.,_t + 4 W ~ j _ z - W ~ r j_s)/c~z-.qx~

(3.11)

The matrix form: A W - - R , A is an N x N rank matrix with the width of the band equal to .5; R is an N x I rank matrix.

W = { W , W2 "'" W, ... W~r}~

War+l=0

From the above equations it is known Abstract that we must obtain the values of W whenr = ( j - - 1)~, r = ( j - - 2 ) ~ , r = ( j - - 3 ) c ~ in order to take the value of Wwhen r = j 3 . The one-dimensional problem of the motion of a rigid flying plate under explosive attack has Now we solve the problem using the initial conditions an analytic solution only when the polytropic index of detonation products equals to three. In a) r = 0 analysis ( J = 0 ) ,isWrequired. l,=0 general, When a numerical In this paper, however, by utilizing the "weak" shock b) When r = t ~ ( j =shock l) behavior of the reflection in the explosive products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying Tt~r,o=(W~t-2W~o+W~c_l))/~z=O Hence W i t = W , ( _ t ) plate driven by various high explosives with polytropic indices other than but nearly equal to three. , , = plate ( W,, obtained - W~