ON

THE

OF

FINITE V.

PROPAGATION

G.

OF

VISCO-ELASTIC

WAVES

AMPLITUDE Karnaukhov

UDC 539.374

w The fundamental equations of the nonlinear theory of elasticity and visco-elasticity in the Lagrangian and I~ulerian representations in an arbitrary curvilinear system of coordinates is given in [2, 4, 9]. To within quantities of the second order of smallness inclusive these equations, written in cartesian Eulerian coordinates, have the form

~ii,i=

P (~'+

O~.kvk);

(i.i)

P = P0 (1 - - e); o~ = us + uka,,~; (1.2) tlti = LsSli -t- 2~sLi "-I-(~,E q- ~le 2 -~ln) 6~i d- 2 ~ E q "-I- 2mes~i "+

(1.3)

4pe~ke~/-P"~* e6tlq- 2~*eii,

where 2e~/--= ui, i --l- ui.i;

2Eii = - - ua.iuk.i;

2n = - - I F - 1 2 ;

e = ut,l;

2E = - - ut.ku~ tr

H e r e It and I2 a r e the f i r s t and second i n v a r i a n t s of the d e f o r m a t i o n t e n s o r ; (*) indieates the operation of convolution. The r e m a i n i n g notation is that e o m m o n l y used. The boundary conditions have the f o r m %,hi = PI' w h e r e nj is the n o r m a l to the d e f o r m e d s u r f a e e of the body. Since solution of these equations involves g r e a t m a t h e m a t i c a l difficulties, it b e c o m e s n e c e s s a r y to simplify the initial s y s t e m of equations at the expense of r e s t r i c t i n g the c l a s s of p r o b l e m s . Significant m a t h e m a t i c a l simplifications can be achieved for p r o b l e m s involving the propagation of w a v e s in r o d s , p l a t e s , and shells; in addition, to take into account the e f f e c t s of g e o m e t r i c d i s p e r s i o n use can be made of refined t h e o r i e s of a different s o r t , say, of T i m o s h e n k o type, and to take into account d i s p e r s i o n of a p h y s i c a l nature and of dissipation use can be made of the theory of v i s c o - e l a s t i c i t y . F o r a study of the b a s i c p e c u l i n r i t i e s of the i n t e r a c t i o n of the effects of nonlinearity, d i s p e r s i o n , and d i s s i p a t i o n we e x a m i n e the p r o p a g a t i o n of w a v e s in v i s c o - e l a s t i c r o d s of a r b i t r a r y c r o s s section. We use the following method for d e r i v i n g solutions of the equations; this method can a l s o be used in a n u m b e r of other p r o b l e m s . A s s u m i n g that the e f f e c t s of d i s p e r s i o n , dissipation, and nonlinearity a r e s m a l l quantities of the s a m e o r d e r , we can, in d e r i v i n g the equations, take into account at f i r s t only the nonlinearity and v i s c o - e l a s t i c i t y and then c o n s i d e r the t e r m s c r e a t i n g g e o m e t r i c d i s p e r s i o n as b e i n g known f r o m m o r e refined l i n e a r r o d t h e o r i e s . In a c c o r d a n c e with the n a t u r e of the motion under study we put Oi1 = 1~22 = O12 = O13 = O ~ = 0;

033 =/=0.

(1.4)

Institute of Mechanics, A c a d e m y of Sciences of the Ukrainian SSR. T r a n s l a t e d f r o m P r i k l a d u a y a Mekhanika, Vet. 9, No. 4, pp. 36-44, April, 1973. Original a r t i c l e submitted June 13, 1972. 9 19 75 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15.00.

376

T h e n f r o m the e q u a t i o n s of m o t i o n we have Ul,2 -----"//1.3 ~

U2,| ~

/22.3 ----- U3.] ~

/g3,2 ~

0.

P u t t i n g u 3 = F(x~), we obtain a s y s t e m of n o n l i n e a r i n t e g r a l e q u a t i o n s for d e t e r m i n i n g ul, 1 and ua, 2 f r o m the conditions (1,4) and the e q u a t i o n s of state (1.3). Solving this s y s t e m by the method of s u c c e s s i v e a p p r o x i m a t i o n s , we obtain %3 = EF.3 + (2 vco - c~) F23"~ (2,~do~ do)*F,3;

1

Co = -~- k (I + 2v2)- ~l (1 - - 2v )z + Iv (2 - - v) + ~ v2 + 2my ( l - - 2v) - - 4pv2 :

(z.5) = + ~ (1 + 2v~) - - ~i (1 - - 2v)2 Jr Iv (2 ~ v) -k ~t - - 2rn (1 --2v) - - 4p;

We substitute e q u a t i o n s (1.5) into the equation of motion (1.1). T a k i n g into a c c o u n t the equations (1.2) and (1.4) and i n t r o d u c i n g the Love l i n e a r c o r r e c t i o n to the g e o m e t r i c d i s p e r s i o n [7] and a l s o the d i m e n s i o n less quantities 1

~=-U:

~ = - h - ; ~=-h-

; z-

h

(R is a c h a r a c t e r i s t i c l i n e a r d i m e n s i o n ) , we obtain, to within s e c o n d o r d e r q u a n t i t i e s of s m a l l n e s s i n c l u sive, the following i n t e g r o d i f f e r e n t i a l equation:

~' -3-F- + ~- --~ .~-c + ~'.*--gir- + oz, --

O~ ~

+

at

OzOw + 2 v ~ -

a~:~ ,

(1.6)

where 1 ~

1

1 (2vd0__~o).

TO within q u a n t i t i e s of the s e c o n d o r d e r of s m a l l n e s s we w r i t e the equation (1.6) in the f o r m of the system Ov

o--~- +

v Ov

-~- +

c~(H) OH

/4

O~H

0z + [ ~

~ ,OH

+

~ -&-=~

OH

0

~+~(v/-/)=0.

(1.7)

Here v=--~-+v

; H = 1 - - --0--~-9

For/33 = 0 t h e s e e q u a t i o n s a r e known as the ]3oussinesq equations (see [6]). If we n e g l e c t the e f f e c t s of d i s p e r s i o n and d i s s i p a t i o n , these e q u a t i o n s a r e of the s a m e f o r m as the g a s d y n a m i c e q u a t i o n s , so that all the r e s u l t s valid f o r t h e s e l a t t e r e q u a t i o n s b e c o m e applicable (Riemann i n v a r i a n t s , s i m p l e w a v e s , etc). M o r e o v e r the n o n l i n e a r e f f e c t s lead to a d i s t o r t i o n of the wave p r o f i l e , as a r e s u l t of which the function v(z, 1-) b e c o m e s a m b i g u o u s for sufficiently l a r g e r . H o w e v e r even b e f o r e this time the solution b e c o m e s unsuitable since when the wave p r o f i l e b e c o m e s sufficiently steep the d i s s i p a t i v e p r o c e s s e s b e c o m e i m p o r t a n t , l e a d i n g to a s p r e a d i n g out of the wave p r o f i l e and, as a final r e s u l t , a b a l a n c i n g of the n o n l i n e a r d i s t o r t i o n . D i s p e r s i o n a l s o leads to a s m e a r i n g out of the p r o f i l e , which m a y c o m p e n s a t e the n o n l i n e a r d i s t o r t i o n . T h e r e f o r e in d i s s i p a t i v e and d i s p e r s i v e media the p r o p a g a t i o n of n o n l i n e a r s t a t i o n a r y w a v e s is p o s s i b l e , aIthough t h e i r s t r u c t u r e s in e a c h of these media are not identical [6]. M o r e o v e r in such m e d i a , f o r sufficiently s m a l l (but finite) a m p l i t u d e s , solutions e x i s t which can be c o n s i d e r e d as the analog of s i m ple w a v e s , n a m e l y , the s o - c a l l e d " q u a s i - s i m p l e w a v e s . " In o r d e r to obtain the e q u a t i o n s for " q u a s i s i m p l e w a v e s " we use the method given in [6J, w h e r e i n f o r s i m p l i c i t y we put ~ = ~0 a/0~-; fi = P0 ~/0~r; ~3 = fl30 8/aT;/3 3o = 1/l~ (2ua0--d0); d o = c o a s t . Changing to a fixed s y s t e m of c o o r d i n a t e s x = z--~', with the aid of this method we obtain, f r o m the e q u a t i o n (1.6) and a l s o the s y s t e m (1.7), the K o r t e w e g - - d e V r i e s - - ] 3 u r g e r s (KVB) e q u a t i o n

(i .8)

377

where

l=

2v

Co

l--v

J~----2": P'="2""

W;

w We c o n s i d e r the p r o b l e m c o n c e r n i n g the p r o p a g a t i o n of t w o - d i m e n s i o n a l w a v e s of finite a m p l i t u d e in a v i s c o - e l a s t i c s p a c e . A s s u m i n g that the e f f e c t s of n o n l i n e a r i t y and v i s c o s i t y a r e of the s a m e o r d e r , we h a v e , to within q u a n t i t i e s of the second o r d e r of s m a l l n e s s i n c l u s i v e , the following b a s i c r e l a t i o n s in a L a g r a n g i a n s y s t e m of c o o r d i n a t e s [2, 3, 9]: the e q u a t i o n s of m o t i o n %.. ----PoU~;

(2.1)

the e q u a t i o n s of s t a t e [3] 1 A) (ut.iut.i ~)u,.18,, + (" + --4-

a,,=,(u,.t +Ul.,) + ( K - - 2

+ -~ K-- "-E" + B (ut.mu~.,,8.

+ ul,zu~s + uzsut.i) 1

1

+ 2uta~t,t) + -~.-Aui.tut.t + -~ B (ut.,nu,ny.8~f+ 2ui.r 2

+,,,.t)+

"~

(2.2)

w h e r e A, B, C a r e third o r d e r m o d u l i of e l a s t i c i t y [3]. We c o n s i d e r o n e - d i m e n s i o n a l d i l a t a t i o n w a v e s (u 2 = u 3 = 0, u 1 ~ 0), r e s t r i c t i n g o u r s e l v e s to two m o d e l s of v i s c o - e l a s t i c i t y t h e o r y , n a m e l y , % + % % = 13oeu + 13~e~,+ Voe~;

(2.3) (2.4)

O'll --~ (~1(~11 -~ 0~2~i1 ~--- ~Oell + [~lell -~- [~2ell --{- '~oe121, where eu=u1.,; 8 0 = K + - ~ P ; ~1, a2,

3C;

270=380+2A+6B+

Hi, ~2 a r e c o n s t a n t s c h a r a c t e r i z i n g the v i s c o - e l a s t i c p r o p e r t i e s of the m a t e r i a l [4, 8].

With the a c c u r a c y adopted, we have f r o m the e q u a t i o n s (2.1),

(2.3), and (2.4)

u" = "u"+ Vtu" + ~12u(Iv) -- V3u:u".

(2.5)

F o r the e q u a t i o n (2.3) the q u a n t i t i e s Yl, 72, and 73 a r e given by 71 = al - - a2; u = a~ (a 2 - - al); 7z ----a3; f o r the e q u a t i o n (2.4) they a s s u m e the f o r m : ~t = bl - - b3; Y2 = ba (On

-- bl) + b2 -- b4; "~a = -- bs"

Here

! u = L--; x = -s ; t =

1

\-~o--o/ ~; a~ = ~-\-'P~7/

I

; [

a~=~oL \-~o ] ;a3=2--~--o : b , = - - L - \ - ~ o ] I b,

=

-Co ; b 3 =

po/

" b4 =

'

,(,,) Zv

" 05=

'

2--

,~ 8~

A dot i n d i c a t e s d i f f e r e n t i a t i o n with r e s p e c t to the t i m e ; a p r i m e i n d i c a t e s d i f f e r e n t i a t i o n with r e s p e c t to a coordinate. I n t r o d u c i n g the change of v a r i a b l e s f = 1 / 2 73 u,, z = x--% we have, analogous to that p r e s e n t e d above, the KVB e q u a t i o n

378

f, + i f , - Pf,= = ~t=.

(2.6)

We obtain f r o m this, with the aid of the t r a n s f o r m a t i o n s f ~ --f, z ~ --x, ~ --* ~-, 0 ~ --~, an equation of the f o r m (1.8). It follows f r o m physical considerations that a 2 > th, b 3 > b I, b 4 > b 2. T h e r e f o r e for the equation of state (2.3) we always have/3 > 0 (positive dispersion) while for the equation (2.4) the parameter/~ can be both positive as well as negative (negative dispersion). We note that in a fixed s y s t e m of coordinates the following equation c o r r e s p o n d s to the equation (1.8):

After substituting the v a r i a b l e s f(z, "r) = --w(z, ~), ~ = ~---z, we have, with the adopted a c c u r a c y , the KV]3 equation

w, + ww; -- ~w;~ = ~tw;c. With the aid of this equation we can solve boundary, value problems for a semiinfinite rod and a halfspace. w We note the fundamental features in the behavior of the solutions of the equations obtained for > O. If we neglect the nonlinearity, dispersion, and dissipation, we obtain from equation (1.8) t,---- 0.

(3.1)

This equation has a solution of the form f(x), which r e p r e s e n t s a p r o g r e s s i n g wave that undergoes no distortion. Upon taking intoaecount d i s p e r s i o n effects and neglecting the nonlinearity and dissipation, we have a linearized K o r t e w e g - - d e V r i e s (KV) equation

f~ -}- ~1,~xx

=

O.

(3.2)

Its general solution is of the f o r m +co

t(x,~)=~-~-(3p~) -~-=,

Ai - - - - L - | f ( x , O ) d x ' , (3p,,:) s J

(3.3)

where Ai(z) is the Airy function; f(x, O) is an initial perturbation. A s suming that f(x, 0) decays with sufficient rapidity as x --:L ~ , and using the asymptotic behavior for a r b i t r a r y Airy functions, we can show [6] that for large T the p e r t u r b a t i o n has the form of a q u a s i stationary wave with the variable wave number k(x, "r) = Jx/3fl'rJl/2, with the frequency w = --ilk 3, and with amplitude p r o p o r t i o n a l to Iw" (k)-r I-1/2 i" (k), where f (k) is a F o u r i e r component of the initial p e r t u r b a tion. Moreover the local wave n u m b e r k(x, T) is displaced in space with the group velocity u = w'(k). T h e r e is a domain of rapid oscillations when x < 0. When dissipation and d i s p e r s i o n are taken into account the linearized equation yields a damped q u a s i - s t a t i o n a r y wave. Thus in the linear approximation without dissipation being taken into account we always have solutions in the f o r m of s o - c a l l e d stationary waves. The nonlinear effects lead to a distortion of the wave profile. When d i s p e r s i o n is p r e s e n t but dissipation is absent, stationary waves, in the nonlinear c a s e , can be periodic or s o l i t a r y waves. We c o n s i d e r stationary solutions of equation (1.6) for /33 = 0; r = ~(z--v'r). In finding these solutions we follow the method given in [6], r e g a r d i n g v--1 as a small first o r d e r quantity. Introducing the new independent variable f = (f12--2--2v/2)~' and limiting o u r s e l v e s to t e r m s of the second o r d e r of s m a l l n e s s , we obtain

[lf~-l-,f~.--(v--1),~ =O (~=z--v%

[~=~).

(3.4)

An analogous equation follows f r o m the equation (2.5). The relation (3.4) coincides with the equation for stationary waves resulting from the Boussinesq equations. It was studied in detail in [6]. The main features of the solution a r e the following. After integration, the equation (3.4) is reduced to the form 3pf~ -~ (b, -- fl (b~--/) (t~3 --t),

(3.5)

379

where the bi(i p i, 2, 3) are constants, being certain combinations of the velocity v and the two constants of integration; in addition,v = 1/3(b i + b 2 + b 3) + 1. Since/3 > 6, it follows from the condition of boundedness of the solution that all the b i must be real (bi > b 2 - b3) , moreover b i > u -> b 2. Ifb 2 = b 3, then the solution has the form of a solitary wave ("soliton") (see Fig. la)

t (~) -

fo

with amplitude f0 = bl--b2, with breadth of order ~

and with speed v = 1 + 1/3 f0 + b3.

W h e n b 3 < b2, the solution of equation (3.4) represents a periodic wave and is described by the expression

(3.6)

f (~) = ~ dn ~ (z a, s) + bs,

where dn(zl, s) are the Jacobi elliptic functions with modulus s, the wave amplitude is a = bl--b2/2; s 2 = b i-bx/b t-b3; z i = x/'(a/6 ~ ~(~/s). The wave length of the periodic solution (3.6) is equal to l

(2K(s) is the period; K is the complete elliptic integraI of the first kind). The m e a n vaule of the amplitude F = )t-i ~ f(})d} and the velocity are equal to 0

2a E(s) Fbs;

-- s~ K (s)

v-----I +

2a (2 - - #) + b s ,

3s~

where I~ is the complete elliptic integral of the second kind. W h e n s --*O the expression (3.6) becomes the solution of the linearized equations.

If s << 1, then

,(~) = 7 + a[cos k~ + s~--~cos 2k~ + O ( # ) l, w h e r e k = 2~/X is the w a v e n u m b e r ;

T h e d i s p e r s i o n e q u a t i o n h a s the f o r m

v= l +[-~k'

l -2--~/r

+o(,e~.

A s s --"i, K(s) --,o, dn z t = i/oh zl, and the periodic wave asymptotically approaches a sequence of solitons with amplitudes f0 = 2a (relative to the level f = f), the distance between which is given by the formula (see Fig. lb) )t = 412~/f01ln(l--s2)[. The soliton propagation speed is always greater than one. For periodic waves of small amplitude v < 1 (for i~ = 0). O n the phase plane (f, f') the integral curves of equation (3.5) have the form shown in Fig. 2, where A is a saddle point and B is a center. The curve 1 corresponds to a soliton; curves 2 correspond to periodic waves. Moreover only those branches are realized which are situated to the right of the point A. W h e n no dispersion is present (p = 0) the equations (I.8) and (2.6)go over into the Burgers equation, which is transformed through the substitution [11] f = --2/J0/Dx)ln .q(x, T). Therefore its general solution, subject to the initial condition C v = #r is easily obtained, since

qD(x, ,) =

1 V 4n~---}

x - - xx ~

exp

1 2~

to (x2) dx2 dXr 0

The p r i n c i p a l m o m e n t M = ~

f(x, T) dx = ~ f0(})d} < ~ does not v a r y with the t i m e , i . e . , --oo

" i n t e g r a l of m o t i o n . " 38O

it is an

a/\

~

7

Fig. 1

2

/l'

Fig. 2

As T - - ~ , the profile f(x, ~) tends towards some u n i v e r s a l a s y m p t o t i c value, d e t e r m i n e d only by the constant M (see [5], [6]). In p a r t i c u l a r , as/z --* 0 and M > 0, it has the f o r m of an expanding triangle with a shock wave on the f o r w a r d portion of the profile:

liml(x,,r ~-,o

0 (O V'-UM~).

The s t a t i o n a r y solution, bounded at infinity, which d e s c r i b e s the profile moving at constant speed v without d e f o r m a t i o n , a s s u m e s the f o r m (see [5], [6])

f(~)=foq'fl

/

lq-exp

IF

fl ~

; V=foq--~-,

'

~=x--~

(f0 and fl a r e constants). This sohltion r e p r e s e n t s a shock wave with a jump of magnitude fl and width of the transition region equal t o h = 2 M / f 1. A s ~ 0 , 6--0. When ~ = 0 the equations (1.8) and (2.6) t r a n s f o r m into the KV equation. A detailed s u r v e y of p a p e r s devoted to the study of this equation is given in [5] and [6]. We pause to c o n s i d e r the basic f e a t u r e s in the b e h a v i o r of the solution of the KV equation. The stationary solutions a r e solitons and periodic waves. The g e n e r a l picture of the evolution of an initial p e r t u r b a t i o n [1, 5, 6, 10-13] c o n s i s t s of the following. The initial p e r t u r b a t i o n f0q~(~) (4 x/l; f0 and l a r e a c h a r a c t e r i s t i c amplitude and length of the initial perturbation) in the g e n e r a l c a s e splits up into the solitons =

t, Ix, ~) = a seeh~

~

(x - - ~);

v=

and a wave packet (Fig. l c ) . The n u m b e r o f s o l i t o n s , t h e i r amplitudes, the uniquely defined d i m e n s i o n s and speed of the solitons, and also the wave packet s t r u c t u r e , depend on the f o r m of the initial p e r t u r b a tion and the magnitude of the s i m i l a r i t y p a r a m e t e r ~ =/(f0/fl)l/2 [5, 6]. The solitons move forward with speeds p r o p o r t i o n a l to t h e i r amplitudes. In the c o u r s e of time the wave packet splits up; long wave o s c i l lations a p p e a r in its f o r w a r d portion and s h o r t wave oscillations at its r e a r . The amplitudes of the s o i l tons, f o r m e d f r o m the initial p e r t u r b a t i o n and decaying as x - ~: 0o, a r e d e t e r m i n e d by the c h a r a c t e r i s t i c B

f

'-i

,41

,4 t=o a t=K;-

0

a5

l,O

Fig. 3

15

x

cr

c

Fig. 4

381

values of a Sturm--Liou~rille boundary value problem [10]. The manner in which the solitons are formed from the initial perturbation is described in detail in [1, 5, 6, I0, 12, 13]. As an example of the process involving the dispersemeut of a wave into solitons w e cite the evolution of a sinusoidal perturbation of large amplitude, shown in Fig. 3, wherein only one period is shown [13]. At the start the sinusoidal perturbation evolves as a simple wave: its steepness increases and it has a tendency to tip. Before tipping can play a part dispersion c o m m e n c e s and the first soliton begins to separate out. Gradually the perturbation splits into solitons, which themselves form a "simple wave," and at s o m e instant of time their amplitudes lie on a single line (curve C). After this the large amplitude solitons overtake the slower ones and an intersection of soliton trajectories occurs. For comparison, in Fig. 4a w e show the solution of the equation fv + ffx = 0 (a simple wave) as the curve A B D E F , and the solution of the Burgers equation as the curve A B C . The solution of the K V equation for ~ --"0 and q(~) > 0 is shown in Fig. 3b. It consists of a series of solitons whose n u m b e r increases as ~I/2 andwhose width decreases as 51/2. W e consider a progressing stationary wave of the K V B equation: f = f(x--c~r). A detailed study of this equation is given in [5, 6, 12]. Instead of a periodic wave there arises a nonsymmetric train of waves (Fig. 4c). For small viscosity the first waves of oscillating structure are close to solitons, and, if the viscosity is large in comparison with the dispersion, an ordinary shock wave with a monotone structure arises wherein the critical value of the parameter/J, for which a transition occurs from an oscillating to a monotone structure, is given by the expression ~ = 4V~-~, where 2 c = f(_~o)_f(~o).

LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

382

CITED

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