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Applied Mathematics and Mechanics (English Edition, Vol 22, No 7, Jut 2001)
Published by Shanghai University, Shanghai, China
Article ID: 0253-4827(2001)07-0747-
N O N L I N E A R B E N D I N G T H E O R Y OF D I A G O N A L S Q U A R E
P Y R A M I D R E T I C U L A T E D S H A L L O W S H E L L S ~
XIAO Tan ( N • ), LIU Ren-huai (~lj Jk.'I~ ) (institute of Applied Mechanics, Jinan University, Guangzhou 510632, P R China) (Paper from LIU Ren-huai, Member of Editorial Committee, AMM)
Abstract: Double-deck reticulated shells are a main form of large space structures. One of the shells is the diagonal square pyramid reticulated shallow shell, whose its upper and lower faces bear most of the load but its core is comparatively flexible. According to its geometrical and mechanical characteristics, the diagonal square pyramid reticulated shallow shell is treated as a shallow sandwich shell on the basis of three basic assumptions. Its constitutive relations are analyzed from the point of view of energy and internal force equivalence. Basic equations of the geometrically nonlinear bending theory of the diagonal square pyramid reticulated shallow shell are established by means of the virtual work principle. Key words: diagonal square pyramid reticulated shallow shell; sandwich shell; nonlinearity CLC numbers : 0322 ; T H 7 0 2 Document code : A
Introduction
Double-deck reticulated shells as a main form of large space structures, are widely used in civil engineering and many other fields ~1'21. One of them is the diagonal square pyramid reticulated shallow shell. The shell consists of reverse square pyramids whose vertexes are linked by lower chord rods to form square grids, the angle between an upper chord rod and a lower chord rod is 45 ~. Its geometrical and mechanical characteristics is that it is built on stilts with rods, so its upper and lower faces bear most of load but its core is comparatively flexible. Compared with a homogeneous shell, the diagonal square pyramid reticulated shallow shell has considerably greater overall bending stiffness but smaller overall shearing stiffness, the effect of shearing deformation increases apparently so that it must be taken into account at the time of analysis. Otherwise, because the directions and dimensions of corresponding rods in the upper and lower faces of the diagonal square pyramid reticulated shallow shell are different, there is no structural symmetrical middle face, but coupling of bending deformation and plane deformation. But as we know, nonlinear bending tbr the shell has not yet been studied because of the difficulty
Received date: 2000-05-23; Revised date: 2001-02- Foundation item: the National Natural Science Foundation of China(19972024); the Natural Science Foundation of Guangdong Pr6vince(960197, 994451) Biography: XIAO Tan(1965 - ), Engineer, Doctor
Abstract
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 pur-
terbation 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 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
index) for estimation of the velocity of flying plate is established.
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
governing the flow field of detonation products behind the flyor (Fig. I):
--ff ap +u_~_xp+ au =o,
au au (^) y (^1) =0,
aS a s
a--T =o, p =p(p, s),
(i.
2 9 3
where 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
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para-
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
748 XIA0 Tan and LIU Ren-huai
due to the complexity of structures. This paper is a further work of the previous paper [3 - 11]. By using the principle of equivalence and continuity basic equations of the geometrically nonlinear bending theory of the shell are obtained.
1 Basic Assumptions and Equivalent Model
Let us now consider a diagonal square pyramid reticulated shallow shell. A structure of the
L 3 ,
Fig. l(a)
y' (^) g
Structure of a diagonal pyramid reticulted shallow shell
shell is shown in Fig. 1 ( a ) and Fig. 1 ( b ). We
use two Cartesian coordinate systems x'y'z and xyz
in the shell. Here the x' and y' axes are taken to be paralleled to upper chord rods, the x and y axes are taken to be paralleled to lower chord rods, the z-axis is perpendicular to shell surface, and the angle between the two systems is 45 ~. Lengths of upper and lower chord rods are L3 and L respectively, corresponding cross-section areas are A 3 and A1, relative moments of inertia at the centroid of the cross-section a r e 13 and 11. The length of web rods is L 2 , the cross-section area is A 2 , the distance between upper and lower faces is h. Xl and x2 are curvatures of the reticulated shell along directions x , y respectively, E is Y o u n g ' s modulus. We suppose: 1 ) The distance between upper and lower faces of the diagonal square pyramid reticulated shallow shell is very small as compared with the curvature radius of the shell. All rods are placed densely, uniformly and regularly. 2) All cross-section dimensions of the rods are very small as compared with those of grids and with distance between upper and lower faces of the shell, which each rod is rigidly jointed. 3) The material of rods is absolutely elastic. According to the above assumptions, we equate the diagonal square pyramid reticulated shallow shell with a shallow sandwich shell that has the same geometrical dimensions. That is, we equ'ate upper and lower faces of the reticulated shell with upper and lower faces of the shallow sandwich shell, equate its web rods with core of the shallow sandwich shell. Then we suppose sandwich shell faces are in membrane-stresses, the straight line in the core of the shell keeps straight after deformation, the core can not bear loads in the direction parallel to the shell surface
and it is incompressible in z-direction [3- 11]. We adopt an upper face element ABCD from the
reticulated shell as shown in Fig. 2. In the coordinates x'y'z internal force components of the
upper chord rods of the reticulated shell a r e N'13 , N i 3 , N'123 a n d N'213. The dashed square in Fig.
2 is the face element of the equivalent shallow sandwich shell. In the coordinates x'y'z its internal
forces are N'x3, N'y3, N'xy3 and N'yx39 Considering the flat property of the shell, we have N'xy3 =
N'yx3. According to the principle of internal force equivalence, in the coordinate system x'y'z the
relations between internal force components of upper chord rods of the diagonal square pyramid reticulated shallow shell and those of the upper face of the equivalent shallow sandwich shell are as follows :
N,x3. N;,. N'y3. N'23. N~y3 N;23 N213 ( 1 ) L3 ' L3 '. L3 L
Abstract
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 pur-
terbation 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 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
index) for estimation of the velocity of flying plate is established.
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
governing the flow field of detonation products behind the flyor (Fig. I):
--ff ap +u_~_xp+ au =o,
au au (^) y (^1) =0,
aS a s
a--T =o, p =p(p, s),
(i.
where 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
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para-
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
750 XIAO Tan and LIU Ren-huai
x
E..... ~ F z , w 3
q O, 1
Ny
HL ~. ~ G " ?'{~
Q11 Nyt
Fig. 3 Web element of the reticulated shell and core Fig. 4 Element of the equivalent
element of the equivalent sandwich shell shallow sandwich shell
the core
152 = 11, + "Y'~Jx, 732 = y + Z r W 2 = W. (6)
The geometrical equations of faces of the equivalent shallow sandwich shell are
1 2 t
gxi = lZi, x "4- ~ W i , x -- 1r 1 W i
gyi : ?3i,y -t- - ~ W i , y^1 2 -- 1~2W i )/xyi : V i , x + U i , y + W i , x W i , y "
i = 1,3. (7)
The geometrical equations of the core of the equivalent shallow sandwich shell are
}'xz2 -~ U 2 , z + W 2 , x , ~'yz2 = V 2 , z + W Z , y " ( 8 )
Substituting Eqs. (4) - (6) into (7) and ( 8 ) , we get strain components of the equivalent
shallow sandwich shell
the upper face
the lower face
the core
Ex3 = l t , x q- ~ W , x -- I r ,
g y 3 ---- 73 y + ~ W , y^1^2 -- 1 r ~/xy3 = V , x -t- U , y "4- W x W , y.
Cxl = U ~ "4- h r ..1- - ~ W , x^1 2 - K 1 w ,
% = 73~ + h e , , , + ~ w 2 ,^1 - ,~2w,
Y x y l = ?3,x "1- ls y -1" h ( ~b,,~ + Cx,x) + w , w , ,.
)'x.-2 = r + w. , , 7y..2 = Cy + w.y. (11)
2 C o n s t i t u t i v e E q u a t i o n s
In the coordinates x'y'z as shown in Fig. 2, suppose the strain components of the upper face
i r
of the diagonal square pyramid reticulated shallow shell are e'x3, ex3, Y,y3, the normal forces of
Abstract
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 pur-
terbation 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 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
index) for estimation of the velocity of flying plate is established.
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
governing the flow field of detonation products behind the flyor (Fig. I):
--ff ap +u_~_xp+ au =o,
au au (^) y (^1) =0,
aS a s
a--T =o, p =p(p, s),
(i.
where 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
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para-
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
Bending Theory of Pyramid Shells 751
its rods are
,
N13 : EA3r N'23 : EA3ey 3. (12)
Considering self-equilibrium of the rods, we have N'123L 3 = N'213 L 3. According to the
assumption 2) we have the shear strain
' N'123 L2 Nil3 L2 (13)
9',y3- 12EI 3 + 12EI~-~-"
Substituting Eq. (12) into (1), we obtain the internal force components of the upper face of
the equivalent shallow sandwich shell in the coordinates x'y'z'
N~.3 EA3 , N'y3 EA3 , 6 EI3 ,
- L3 5x3 , = --L3 EY3 ' N~y3 - L~ ?',y3- (14)
Assuming that in Fig. 4, the internal force components of the upper face are N,:3 , Ny 3 ,
Nxy3, Nyx3 in the coordinates xyz, and in Fig. 2 the internal force components are N~3, < 3 ,
,
N'xy3 , Nyx3 in the coordinates x'y'z', their transform relations are
1 , 1 , N'xy3 , )t
N~3 = ~ Nx3 + 2 Ny 3 -
Ny 3 ~ N x 3^1 ,^ + ~^1^ Ny3 ,^ + N,xy3,^ (15)
Nxy 3 2 Nx3 - ~ Ny 3 ;
and the transform relations of the strain components from the coordinates xyz to the coordinates
x y z^!^ r^ rare
e: 3 1 1 1 ',l
= 2 E x 3 + 2 E Y 3 + g~lxy
, 1 1 1 j (16)
%3 5-~3 + ~-zy3 - 5-7xy
~/xy3 = - Ex3 q" ~'y3"
Substituting Eq. (14) into (15) and using ( 1 6 ) , we obtain the physical equations in the
upper face of the equivalent shallow sandwich shell
( E A 3 6 E I 3 ) ( E A 3 6 E I 3
N.,:3 = ~ + L~3-3 G3 + 2L 3 L~ %3,
Ny3 = 2L3 L~ r + 2 ~ 3 + L~T r
EA 3
Nxy3 - 2 L 3 Yxy3 9
In the same manner, we obtain the physical equations of the lower face of the equivalent
shallow sandwich shell
EA j EA 1 6 Ell
Nxl = L1 e x l , NYl - L1 EYl' Nxyl - L~ Yxyl" (18)
Let us consider the web element in Fig. 3. According to the assumption 2) and loading
characteristics of the web structure, we only consider its global shear rigidity but neglect its global
Abstract
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 pur-
terbation 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 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
index) for estimation of the velocity of flying plate is established.
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
governing the flow field of detonation products behind the flyor (Fig. I):
--ff ap +u_~_xp+ au =o,
au au (^) y (^1) =0,
aS a s
a--T =o, p =p(p, s),
(i.
where 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
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para-
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
Bending Theory of Pyramid Shells 753
Nx,x + N~y.y + f x = 0,
Nxy,x + Ny,y + fy = 0 ,
M .... + 2M,,y,,,y + My,yy + N~(w xx + Xl) +
( 2 7 a - e)
2NxyW ~y + Ny(w,yy + x2) - w x f, - W,yfy + f. = O,
M x , ~ + M ~ y , y - Q ~ = 0 ,
M~y,~: + My,y - Qy = 0;
and the boundary conditions
lN, + rnN~y = N~ or u = ~, lNxy x mNy = ~Vy or v = 9, ]
l(Mx,x + M x y , y ) + m(Mxy.x + M y , y ) + w,xNxn + w ) g y n = ~[z o r w = w ' l ( 2 8 )
lMx + mM~y = M~,, or Cx = Cx, lM,:y + mMy = ~]yn o r ~y = ~y,
where l, m are cosines of the external normal to boundary C, r and ~y are boundary values of
r and Cy respectively, N~, Ny, Nxy are global equivalent membrane forces of the equivalent
shallow sandwich shell, Mx, My, M,~y are moments produced by upper and lower membrane
forces of the equivalent shallow sandwich shell. Here
N x = Nxi + N~3, Ny = Nyl + Ny3, Nxy = N:~yl + N~y3,~
M~ = hN~l, My = hNyl, M,y = hN~yl. J
Substituting Eqs. (17) and (18) into ( 2 9 ) , and using Eqs. (9) and ( 1 0 ) , at the same time
substituting Eq. (11) into ( 2 3 ) , we have
Nx = blEx3 + b2r + C l ~.... Ny = b2Ex3 + blEy 3 + Cl@y,y , ~rxy = b3~xy3 + c 2 ( ~ x , y + @ y , x ) , M, = c I Ex3 + hc I ~h.... My = C 1Ey 3 + hc 1 ~ y , y , (30a - h)
M,~y = cz?',r3 + hc2(r + r
Qx = g(r + w , ~ ) , Qy = g ( r + w , y ) ,
where
EAI EA3 6EI3 EA3 6EI3 EA3 6EI1 ]
b l - L , + + b2 - 2 L 3 ' b , = +
EA 1 h 6Ell h
cl - L1 ' c2 - L~
If the stress function @ is introduced as follows:
IVx = q),yy - F x , Ny = @,xz - F y , Nxy = - @,xy,
Eqs. ( 2 7 a , b ) will be satisfied. Here
Fx= ffxdX, F,= flay.
Substituting Eq. (33) into Eqs. ( 3 0 a , b , c ) , their solutions are
Abstract
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 pur-
terbation 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 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
index) for estimation of the velocity of flying plate is established.
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
governing the flow field of detonation products behind the flyor (Fig. I):
--ff ap +u_~_xp+ au =o,
au au (^) y (^1) =0,
aS a s
a--T =o, p =p(p, s),
(i.
where 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
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para-
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
754 XIAO Tan and LIU Ren-huai
where
~x3 = ( - b2q~,x~ + blq~.yy - blCl~bx,~ + b 2 C l ~ y , r - b l F x + b 2 F y ) / A ' ] ~y3 = ( b l q ~ , ~ - b 2 ~ , y y + bzel~b,,~ - b l e l ~ y , y + b2F~ - b l F y ) / A , ?'~y3 = - (q),~y + c2@~,r + c 2 ~ @ , ~ ) / b 3 ,
/, = b l - Using Eqs. (30) and ( 3 4 ) , Eqs. ( 2 7 c , d , e ) can be written as g ( @ x , x + W , x x ) + g ( @ y , y + W , y y ) + ( q ) , y y -- F x ) ( W , x x ..F / ~ 1 ) - 2p,~yW.~, + (q~,~, - F , ) ( W , y y + x 2 ) - w , ~ A - w , , f y + f , = O,
_- dtq) ..... + deq),xr:~ + d3~b.... + d4~b~,rr + ds@y,~z - d 6 F , , x + d l F r , , = g(~b~ + w. ~ ) ,
- dlq~ y)7 + dzq~ ~y + d3~by,~y + d4(~y,~ x + dc~b~,~z - d 6 F y. y + d l F x , ~ = g(~by + w y ) ,_ (36)
/)1 c
where
b2cl bl cl c~
dl - A ' d2 - A b 3 ' d3 = hcl A '
C2 b 2 C2 C2 b 1 c d4 = hc2 - bZ' d5 = ~ + hc2 - ~ 3 ' d6 - A
Finally, from Eq. (9) the compatibility equation is given by Ex3.yy + Ey3,xx -- Yxy3,xy ~- ( W , x y ) 2 -- W , x x W , ~ 7 -- I r -- 1 ~ 2 W , x x. Substituting Eq. (34) into Eq. ( 3 8 ) , the compatibility equation can be represented as d l ( @..... + ~by,ryy) - d2(~bx,xyy + @y,,,xy) + dT(q~ ..... + ~9,yryy) + da~.:~sz = ( W , x y ) 2 -- W , ' x x W , y y -- Ir W , y y -- Ir W , x x -t- d 7 ( F,;,yy + Fy,~x) - d9( F.... + F y , y y ) , where
(35)
(38)
t)~ 1 262 b 2 d 7 - A ' d 8 - /)3 A ' d 9 - A ' (40)
Now Eqs. (36) and (39) which contain four unknown variables q~, w, ~bx, ~y constitute the nonlinear bending equations of the diagonal square pyramid reticulated shallow shells. R e f e r e n c e s : [ 1 ] DONG Shi-lin, XIA Heng-xi. Analysis of orthogonal and ortho-laid space truss as equivalent (sand- wich) plateI J]. Journal of Civil Structure, 1982,3(2) :14 - 25. (in Chinese) [ 2 ] DONG Shi-lin, FAN Xiao-hong. Analysis of diagonal square pyramid space grids by sandwich plate analogy method[J]. Engineering Mechanics, 1986,3(2) :112 - 126. (in Chinese) [ 3 ] LIU Ren-huai, LI Dong, NIZ Gw) h~a, et al. Non-linear buckling of squarely~latticed shallow spherical shells[ J]. International Journal of Non-Linear Mechanics, 1991,26(5) : 547 - 565. [ 4 ] LIU Ren-huai, NIE Guo-hua. Nonlinear bending theory of reticulated shallow shells[J]. Journal
Abstract
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 pur-
terbation 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 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
index) for estimation of the velocity of flying plate is established.
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
governing the flow field of detonation products behind the flyor (Fig. I):
--ff ap +u_~_xp+ au =o,
au au (^) y (^1) =0,
aS a s
a--T =o, p =p(p, s),
(i.
where 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
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para-
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
