COMBUSTION, EXPLOSION, AND SHOCK WAVES 187

LAW OF ATTENUATION OF A PLANE SHOCK WAVE IN A DENSE MEDIUM

V. M. Lyubosh i t s

F i z i k a G o r e n i y a i Vzryva ,

UDC 532.593

Vol. 3, No. 2, pp. 299-307 , 1967

A p lane wave of in f in i t e ly s m a l l amp l i t ude i s p r o p - aga ted wi thout d i s t o r t i o n of the p r o f i l e . The p r o b l e m of a t t enua t ion of a shock wave of f in i te a m p l i t u d e a f t e r a su f f i c i en t ly long p ropaga t i on t i m e has been so lved by Landau in the i s e n t r o p i e a p p r o x i m a t i o n [1]. This p a p e r p r e s e n t s in the s a m e a p p r o x i m a t i o n ano ther d e r - iva t ion of the law of a t t enua t ion of a p lane shock wave f r o m the m o m e n t of i t s o r i g i n a t i o n in a s l i g h t l y c o m - p r e s s i b l e m e d i u m (meta l , r o c k ) fo l lowing an exp los ion a t t h e s u r f a c e of the m e d i u m . The e x i s t e n c e o f a " c h e m - i c a l p r e s s u r e sp ike" i s d i s r e g a r d e d , s ince i t s in f lu - ence i s fe l t only at d i s t a n c e s of l e s s than 1 c m f rom the s u r f a c e [2].

The b a s i c p a r a m e t e r s of the e x p l o s i v e and the dense m e d i u m tha t af fec t the r a t e of decay of the shock wave a r e d e t e r m i n e d , a f t e r f i r s t f inding the law of mot ion of the de tona t ion p r o d u c t s - - d e n s e m e d i u m in - t e r f a c e and the law of d iminu t ion of the e x p l o s i v e load for d i r e c t con tac t be tween the e x p l o s i v e and the s l i g h t - ly c o m p r e s s i b l e m e d i u m . The r e s u l t s ob ta ined make i t p o s s i b l e to e s t i m a t e the l i m i t s of a p p l i c a b i l i t y of the l i n e a r ( acous t i c ) a p p r o x i m a t i o n in so lv ing t w o - d i m e n - s i ona l p r o b l e m s of the i n t e r a c t i o n of a de tona t ion wave and a s l i gh t ly c o m p r e s s i b l e h a l f - s p a c e [3].

JUSTIFICATION OF ISENTROPIC APPROXIMATION

Shock waves in dense media are low-intensity

waves. Even when metals are compressed by a pres-

sure of half a million atmospheres their densities in-

crease by only 15-25%, and the Mach number of the

flow created by a shock wave of this amplitude is less

than 0.25. However, compression by even a low-in-

tensity shock wave is, in principle, irreversible,

since as a result of friction (viscosity) and heat con-

duction the energy of the shock wave is dissipated in the m e t a l .

A n o n s t a t i o n a r y o n e - d i m e n s i o n a l shock wave i s d e s c r i b e d by a s y s t e m of t h r e e f i r s t - o r d e r p a r t i a l d i f f e r e n t i a l equa t ions , c o n s e r v a t i o n of m o m e n t u m , c on t i nu i t y and a d i a b a t i c i t y of mot ion, i f in add i t ion the equa t ion of s t a t e p = p(p, 8) i s known,

c) u + tt O u l r) p )

d t 0 x ? c) x ;0; O0 + t t " ~ + r , O u = O t d x ' O x

o s 4_ u OS r ) t O x

So fa r , i t ha s not p roved p o s s i b l e to find an e x a c t so lu t ion of th i s s y s t e m , s i n c e i t m u s t con ta in t h r e e a r b i t r a r y func t ions of the c o o r d i n a t e and t i m e d e t e r - m i n e d f r o m the b o u n d a r y cond i t ions [4].

Obvious ly , behind the shock f ront the e n t ropy of the m e t a l i s v a r i a b l e . It i s known, however , tha t the en - t r o p y j u m p in a l o w - i n t e n s i t y shock wave i s a s m a l l quant i ty of the t h i r d o r d e r as c o m p a r e d with the j u m p s in the o the r p a r a m e t e r s

r ~ S =

This m e a n s that a l o w - a m p l i t u d e shock wave m a y be c o n s i d e r e d i s e n t r o p i c in the second a p p r o x i m a t i o n [1] ( a s suming the f i r s t to be a wave of in f in i t e ly s m a l l ampl i tude ) .

In a c c o r d a n c e with t h e r m o d y n a m i c s , the spec i f i c en tha lpy i n c r e m e n t is

A w = T A S + ap (1) ?

To d e t e r m i n e the e n t r o p y j ump it is n e c e s s a r y to know the law of c o m p r e s s i b i l i t y of the m e t a l at high p r e s s u r e s . Since the ampl i tude of the shock wave is 1 - 2 o r d e r s g r e a t e r than the y ie ld point of the meta l , s t r e n g t h e f fec t s a r e a s s u m e d neg l ig ib ly s m a l l and c r y s t a l l i n e bod i e s a r e r e g a r d e d a s q u a s i - f l u i d with the s c a l a r p r e s s u r e f ie ld c h a r a c t e r i s t i c of a f lu id . In r e - a l i ty , however , the s ta te of s h o c k - c o m p r e s s e d bod i e s i s d e s c r i b e d by the s t r e s s t e n s o r of s t r e s s e s d i f f e r ing by 2 /3 of the dynamic y ie ld s t r e s s , i . e . , 2 - 2 6 thousand arm. The c o r r e s p o n d i n g e r r o r at p r e s s u r e s of s e v e r a l hundred thousand a t m o s p h e r e s i s not g r e a t [2].

The r e l a t i o n be tween p r e s s u r e and the d e n s i t y of c e r t a i n s o l i d s {rocks, m e t a l s ) can be a c c u r a t e l y e x - p r e s s e d by the Ta i t f o r m u l a g iven by Stanyukovich in [4]. In th i s f o r m u l a A(S) i s a v a r i a b l e depending on the en- t ropy , and the exponent n m a y be a s s u m e d cons t an t ove r a b r o a d r ange of v a r i a t i o n of p r e s s u r e (for m e t a l s n = 4). At p r e s s u r e s l e s s than a m i l l i o n a t m o s p h e r e s i t i s p o s s i b l e to t a k e A = c o n s t . F o r s t e e l A = 5 . 1 0 5 a tm, for d u r a l u m i n A = 2.03 . 10 ~ a tm.

02v n + ! 1 - -

S n~ go A~

R e t u r n i n g to (1), we obta in

A w = - Ap [1 ~- n + l ( - ~ ) ']

It i s e a s y to check tha t in the w o r s t c a s e the c o r r e c - t ion for n o n i s e n t r o p i c i t y of shock c o m p r e s s i o n of a m e t a l i s 1.5% (RDX, d e n s i t y 1.6 g /cm~) .

P a s s i n g to the de tona t ion p r o d u c t s (DP), we note tha t up to the m o m e n t of r e f l e c t i o n f r o m the m e t a l

188 F I Z I K A OORENIYA I VZRYVA

the i r motion is descr ibed by a centered ra re fac t ion wave. At the moment of ref lect ion the surface of the meta l abruptly acquires the veloci ty Un. If u n is less than the velocity of the DP behind the detonation front, equal to D/4, where D is the detonation velocity, then a ref lec ted shock passes through the DP. Its condition of format ion is

Pexp D ~ ~ D , 4

- - v -

where E 0 is the speed of sound in the undisturbed med- ium, P0 is the density of the undisturbed medium, and

R - p~xvD ~<1. Pc co

The number R de te rmines the rigidity of the dense medium in re la t ion to the given type of explosive. For meta l s and quartz R < 1; for an absolutely rigid body H - - 0 .

The ref lec ted shock may be considered isentropic, s ince even on ref lect ion f rom an absolutely rigid wall the entropy of the DP inc rea se s by only 8% [4]. This wave is descr ibed by the general solution of the equa- tions of gasdynamies for one-dimensional isentropic motions

x = (u + c) t; ] x (u - c) t + F, (u - c}./ (2)

In a dense medium we get a t r ave l ingwave descr ibed by the par t icu lar (Riemannian) solution

x = ( ~ + c) t + ~ , (~ ) ; ]

~=___.L_~ (7 i0). n - - 1

(3)

Starting f r o m s ys t em (2), (3) and the obvious equali- ty of the p r e s s u r e and velocity on ei ther side of the in- t e r f ace u = ~, p = p, we find the law of motion of the interface, the fo rm of the function F2(u) , and then the law of attenuation of the shock wave.

DETERMINATION OF MOTION OF DP-METAL IN- TERFACE

We will consider the s t r ic t ly one-dimensional p rob- lem (without allowance for la te ra l expansion of the ex- plosive). In accordance with the equation of the ]sen: t rope for the DP,

64 (_~__f 64 _ / x l t - u \' P = - - - ~ P , = - ~ - P " t - - - 5 - - - ) "

We rewr i t e the Taft equation for a meta l with al lowance for the dependence of speed of sound on density,

d p i . rt ~ 1 p \n-1 - 2 / p \n-~.

, , , . t o) '

In

Lk ~o/ (4)

F r o m the constancy of the Riemann invar iant for a

shock wave t ravel ing to the r ight in a meta l it follows that

- .) _ p = l + ~ - _ _ _ . L ! ~-1 1 2 ~ "

Equating the p r e s s u r e s on both sides of the interface, we obtain a differential equation for the law of motion of the interface,

( 1 +n-12 ----c0x)n-i--l= '--5-n --27 n Po-~Co ~ ~xpD~[x/t--~D )8;

w h e n n - - 4 ; x - - - - - t [ x + 3 V T ~ x R

+ 2 z 0 ]

This is the fami l i a r differential equation of La- grange. An exact solution is possible only in the p a r a - me t r i c fo rm (4). However, this solution is c lumsy and cannot be used for subsequent calculations. We will s implify Lagrange ' s equation by l inear]zing the depen- dence of the speed of sound on the p r e s s u r e in the m e t - al. Retaining the f i r s t t e r m of the expansion of the Newton binomial, f rom (4) we have

/ 7 = A l l + 3_ .~ n c-~----2_ _ l / [ n - - 1 Co J

2 (c -Co) = - - (5) ----r.. Pc Co Pc Co U. n - - 1

The same approximation is used in the theory of Wshort waves, " which, as distinct f rom acoustics, takes into account the p r e s s u r e dependence of the speed of sound, but a s sumes that this dependence is l inear [5]. F rom (4) i t is c lear that the l inear i ty is re tained only when p<A.

Solving the simplif ied equation of motion of the in- t e r f ace

we obtain the t ime dependence of the veloci ty (and hence the p re s su re ) at the in ter face in implici t form,

In t 1/31n--2--u - - ~ / ' 2 ~ ( V ( ~ - ~ - - I ) *g Ull

where ~ = u . . At the initial moment t = r t ; D D

tt= un. Denoting ~/'2 ~= R -- a, we obtain s imple p a r a - me t r i c express ions for the t r a j ec to ry of the interface,

, -

as t --- r162 the in ter face moves through the finite d is - tance

x-- t=__a ~e~ - 1 . (6)

COMBUSTION, EXPLOSION, AND SHOCK WAVES 189

Table 1

Ref lec t ion of Detonat ion Wave f rom Slightly C o m p r e s s i b l e Medium |

Un [ Pn I o Un

I - - - E m/sec Type of interaction R +'l a m/sec Pnr l r ~

SteeI-TNT, t .3 g/era 3

SteeI-RDX, 1.6 g/cm 3

Duralumin-TNT, 1.3 g/cm 3

RDX, 1,6 g/cm 3

Quartz-tryst. TNT, 1.3 g/cm 3

0.I98

0.333

0.520

0.88

0.58 i

).088

0.13

0.17~

0.23

o.19 i

),145

0.22

0.33

0.45

0.35

530 0,78

1070

1050

1880

1140

r Or

210 1.27

0.66 420

0.56 155

O.44 ] 282

0.55 153

0.91

1.41 0.89

1.62 0.86

1.87 i 0.82

1.65 0.85

207 505

405 885

154 830

285 14C0 +

Remark. In Tables 1-3 the detonation velocity of TNT at 1.3 g/cm 3 was taken as 6000 m/sec, that of RDX (1.6 g/cm 3) as 8200 m/sec; co for quartz is 5000 m/sec, for steel 5050, and for duralumin 5 5 0 0 m/sec.

For an in f in i te ly r ig id wall [4]

- - = ( ~ = 0).

The coe f f i c i en t ce t akes into account the ef fec t of c o m p r e s s i b i l i t y of the m e d i u m on the r a t e of fal l in p r e s s u r e , Owing to the f ini te c o m p r e s s i b i l i t y of the m a t e r i a l , the in i t ia l p r e s s u r e is l e s s than (64/27)pg, but then the p r e s s u r e at the i n t e r f a c e fa l l s m o r e s lowly.

On r e f l e c t i o n f r o m a r ig id wall

Pn

with a l lowance for c o m p r e s s i b i l i t y

0 _=- t l , ~ . e - - ": -- ": [exp ('/;, + : -- ae -2'3) -- 1].

The in i t ia l va lues of the d i m e n s i o n l e s s ve loc i t y and

p r e s s u r e Un/D and Pn/Pnr , l ike the exponent ~, a r e uniquely d e t e r m i n e d by the number R, i . e . , by the

r a t i o of the acous t i c i m p e d a n c e s of the DP at the Chap- m a n - J o u g u e t point and the med ium.

The Lagrange equation, wr i t t en for the in i t ia l m o -

ment of t i m e t = T, g ives a cubic equat ion for d e t e r - min ing ~ = t l ( R )

( 1 - - ~ q ) = ~ / 7 ,~ 3 - - 3 / 2 ~ 16 R ; ~ = I /~2 R ~:~ 1 - - -q"

F r o m ~? we find the d i m e n s i o n l e s s va lue of the in i t ia l

p r e s s u r e

Pn po ~o u 27 +i 1 - ---- - - = - (1 - - ~):~

16 16 R (1 + '-'/3a )3 �9 Pnr 27 '%xpD='

The spec i f i c i m p u l s e en r e f l e c t i o n of a de tonat ion

wave f r o m a c o m p r e s s i b l e m e d i u m is

] I

](t)= f.at; - ~ 2 7 x ( t ) - - l I ( t ) F,o c,, i' ud t - - - -

lr " . ~ ; F , ~ I '- 8 IR

Accord ing to (6),

I __ 27 Ir 8 R

3 e ~ ) 3 e ~ -- 3 -- 2a .

The impu l se depends only s l igh t ly on c o m p r e s s i b i l i t y .

Table 2

At tenuat ion of Shock Wave in Steel P r o - duced by Exploding a Contact Charge of

TNT, Dens i ty 1.3 g / c m ~.

( Un=530 m/sec. ~ = 9.55, ~-r =1.27, T =

u m/see ~ #sec

1.I 1.2 1.5

482 440 353

0.95 2.1 6.3

2.3 4 . 8

14.5

t~o, x.Cm cm

1.1 1.25 2.4 2.7 7,3 8.0

* l~2.75 cm

0.45 0 . 9 8 2.9

The r e s u l t s of the ca lcu la t ions a r e p r e s e n t e d in Table 1. The las t t h r ee colulnns conta in va lues of the ve loc i t y and p r e s s u r e ca lcu la ted f r o m the exac t v e l o c - i t y - p r e s s u r e r e l a t i on (4). It should be noted that for

RDX exploded at the su r f ace of du ra lumin the s i m p l i - f ica t ion of the p -u d i a g r a m is known to be i n c o r r e c t ,

s ince Pn > A = 20 300 a tm. N e v e r t h e l e s s , even in that c a s e the in i t i a l ve loc i ty d i f fe r s f r o m that ca l cu la t ed f r o m the exac t r e l a t i o n by only one th i rd . In all c a s e s the in i t ia l va lues of the p r e s s u r e a lmos t co inc ide with those ca lcu la ted f r o m the exac t equat ion of mot ion of the i n t e r f a c e , This j u s t i f i e s s imp l i f i ca t ion (5) above.

ATTENUATION OF SHOCK WAVE IN METAL

Tile D P - m e t a l i n t e r f a c e is a pis ton d i sp lac ing the m e t a l . As shown above, the ve loc i ty of this p is ton fa l l s f r o m the in i t ia l va lue ~n in a c c o r d a n c e with an app rox - i m a t e l y exponent ia l law with t ime cons tan t 0 equal to

190 FIZIKA GORENIYA I VZRYVA

~ _ 0,485 a 0.395 {3.54e -- 2.54),where l is the effect ive length

of the charge with al lowance for la tera l expansion. The p r e s s u r e at the in ter face va r i e s accord ing to an analo- gous law (in the " shor t wave" approximat ion) . The fall in p r e s s u r e at the in te r face affects the ampli tude of the shock wave, s ince the propagat ion veloci ty of the p e r - turba t ions behind the f ront (velocity of c h a r a c t e r i s t i c s ) is g r e a t e r than the ve loc i ty of the shock wave.

TaMe 3

Attenuation of Shock Wave in Steel P r o - duced by Exploding a Contact Charge of

RDX, Dens i ty 1.6 g /cm 3.

t 0 ' U n = 1070 m/see, "*=4 72, ~-r =1 .41 )

Un u, [ t, $1see u m/see T 0=2.25, tco em x. cm x/l*

/dsee

1.1 975 0 .52 1.15 I 0.58 } 0.73 0.22 1.2 890 1.15 2.6 [ 1.3 I 1.6 0.48 1.5 715 3.5 7.8 3.9 4.8 1.45

* 1~3.30 em

The veloci ty of propagat ion of the c h a r a c t e r i s t i c c a r r y i n g a given value of ~ (and a co r r e spond ing value

of the p r e s s u r e p0cou) is c+u = Co + ~ u in a c c o r d - ance with the a s sumed condit ion of i sen t rop ic i ty and hence cons tancy of the minus of the Riemann invar iant in a shock wave t r ave l ing f r o m left to r ight .

In the f i r s t approximat ion the shock wave ve loc i ty is equal to hal f the sum of the ve loc i t ies of propagat ion of the pe r tu rba t ions behind and ahead of the front, i . e . ,

n + l -- e 0 + t t f r + .c = C'0 "4- - - ' ~ Ufr " C h a r a c t e r i s t i c s c a r r y i n g a

2 value of the ve loc i ty l e s s than at the f ront over take the f ront and weaken it. In the i sen t rop ic approx imat ion the at tenuation of the shock wave due to the en t ropy jump at the f ront is taken into account in p r e c i s e l y this way.

The equation of propagat ion of the c h a r a c t e r i s t i c s behind the f ront is Eq. (3). (Here and in what follows the ba r has been omit ted for the sake of s impl ic i ty . ) F2(u ) i s de t e rmined by the law of mot ion of the piston, which is given in p a r a m e t r i c fo rm, x = x, (u,), t = = tn (u.). Since

{~ t < O and u = u . % ( t ) e - ' / ' ' , s 0 ( t ) = t > O

Xn : (Un -- it) O, t = OIn (ttn/u),

the equat ion of the c h a r a c t e r i s t i c s takes the f o r m

n + l ) t + x=(Co + --T-u

+ �9 \ ~ ] U

The shock f ront is not a c h a r a c t e r i s t i c ; it p ropaga te s at a ve loc i ty

n + I t t a~ -- D s w = " + ' : + ~ = C o + - - y - , (8) d t

whereas the veloci ty of propagat ion of the c h a r a c t e r - n + l i s t ic fo rmed at the instant t = 0 1 n u , . i s c 0 + ~ u -

/2

The p r e s s u r e at the shock front may be found f rom (7), if one knows the posit ion of the f ront at a given moment , which can be found f rom Eq. (8) re la t ing the shock wave veloci ty with the flow veloci ty (pressure) at the front. Differentiat ing (7) with r e spec t to t ime along the front, we find the equation for the law of at- tenuat ion of p r e s s u r e (flow velocity)

a___Z_t + 2 t_L = du u

[ 4 co 2 , - - 1 1 + 2 ! i n _~.~]" -~-O n + l u2 n +----~ - u u

The solution of this l inear inhomogeneous d i f feren- t ial equation gives the flow veloci ty at the shock f ront as an impl ic i t function of t ime

t =( unt| r 4 eo 3--n ] 4 Co --o- ~ - ~ t . + l . . 2~T-1) .+1-- ,, +

3 -- ~ Un + - - + In 2 ( n + 1) u

We denote c_~o :x: Utt

t = f . . / ' 4~ ( 1 _ a - . ] 4~ o 2 T T C ~ j

Un + 3 - - ~ n + l n ~ . 2 ( . + I) u

(9)

If ~t > 5, and n ~ 4, then to within 3%,

t = - i t , \ u / n + l \ u /

t, = t ( ~ = 2)=~O 8 n + l

F r o m the las t equation it is c l ea r that the wave shape is d i s tor ted the m o r e rapidly, the sma l l e r 0 and the g r e a t e r u n for the same charge length, i . e . , the g r e a t - e r the compre s s ib i l i t y of the medium and the higher the detonation ve loc i ty . For example,

t2 TNT-steel / t2 RDX-steel ,~ 2.

We will find the coord ina te of the f ront at the m o - ment when the p r e s s u r e at the f ront is equal to Pn%U,

( 1 - 3 - .

n 4- 1 2 ( n + l) J (10)

The r e su l t s of ca lcula t ions based on (9) and (10) a re p resen ted in Tables 2 and 3o

As expected, for a contact cha rge the law of s imi - l a r i ty is sa t isf ied: the peak p r e s s u r e of the shock wave is an impl ic i t function of x / t . We note that as t --~ re la t ion (9) goes over into Landau~s asymptot ic law

~-~ 1 / / 7 .

COMBUSTION, EXPLOSION, AND SHOCK WAVES 191

SUMMARY

I. The problem of the one-dimensional expansion of DP in a slightly compressible medium admits of a sim- plified, but sufficiently accurate analytical solution.

2. The time scale of shock wave attenuation is giv- en by the time constant of the DP-medium interface.

3. A plane shock wave in a slightly compressible medium may be treated in the linear approximation, i.e., may be considered nondamping only at a dis- tance of the order of several tenths of a charge length from the surface of the medium. This should be taken into account in analyzing Lamb's linear solution of the problem [3].

REFERENCES

i. L. D. Landau and E. M. Lifshitz, Mechanics of Continuous Media [in Russian], Moscow, Gostekhlzdat, 1954.

2. L. V. Al~tshuler, UFN, 85, 2, 1963. 3. V. S. Nikiforovskii, PMTF, 2, 1964o 4. K. P. Stanyukovich, Nonsteady Motions of a

Continuum [in Russian], Oostekhizdat, 1955. 5. A. A. Grib, O. S. Ryzhov, and So A. Khristian-

ovich, PMTF, I, 1960.

24May1966 Moscow