NONLINEAR WAVES IN HOMOGENEOUS AND HETEROGENEOUS ELASTIC SOLIDS

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NONLINEAR WAVES IN HOMOGENEOUS AND HETEROGENEOUS ELASTIC SOLIDS

A. Kluwick, A. Nayfeh

To cite this version:

A. Kluwick, A. Nayfeh. NONLINEAR WAVES IN HOMOGENEOUS AND HETEROGE- NEOUS ELASTIC SOLIDS. Journal de Physique Colloques, 1979, 40 (C8), pp.C8-207-C8-211.

�10.1051/jphyscol:1979835�. �jpa-00219540�

NONLINEAR WAVES IN HOMOGENEOUS AND HETEROGENEOUS ELASTIC SOLIDS A. Kluwick and A.H. Nayfeh

Institut fiir Strdmungslehre Technische Universitat Wien, Karlsplatz IS, A 1040, Vienna, Austria.

department of Engineering Science and Mechanics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061.

Abstract.- The method of multiple scales is used to determine second-order uniform expansions for the displacements describing nonlinear plane waves propagating into an elastic half space. The material properties may be homogeneous or they may vary slowly in the direction of propagation. A discussion is presented for the interaction between dilatational and shear waves as well as the interaction between the nonlinearity and the heterogeneity. The present results and method of solution are compared with those obtained earlier for homogeneous and slightly heterogeneous media by using the analytic method of characteristics.

1. Introduction.- The method of multiple scales /l/ is used to analyse the propagation of finite- amplitude longitudinal and shear waves in a half space whose material properties vary slowly with position. The material constitutive relations are assumed to be elastic but nonlinear. For a comprehensive review of nonlinear propagation in heterogeneous materials, we refer the reader to Nayfeh and Mook /2/.

The problem of finite-amplitude longitudinal and shear waves propagating in a homogeneous iso- tropic half space was studied by Davison /3/. He obtained a second-order uniform expansion by using the analytic method of characteristics /4,1/. Nayfeh /l/ extended Davison's work to the case of anisotropic materials, while Nair and Nemat-Nasser /5/ extended Davison's work by including the effect of a small heterogeneity.

The purpose of the present paper is to analyze the propagation of finite-amplitude longi- tudinal and shear waves in a heterogeneous half spacejwhose material properties vary slowly with position but the heterogeneity need not be small.

The analysis is carried out by using the method of multiple scales rather than the analytic method of characteristics ; it is not clear yet how the latter method can be applied to this problem.

2. Problem Formulation.- We consider the propa- gation of plane longitudinal and shear waves in

heterogeneous, nonlinear elastic solids. We introduce dimensionless quantities by using a reference length xr, a reference wave speed c and a reference density p . If (x,y) and (X.Y) denote the positions of a particle in the undefor- med and deformed states, respectively, and (u,v) denote the displacements of this particle, then

where the deformation is assumed to depend on x only. The equations governing the deformation are statements of the conservation of mass and linear momentum ; that is,

where p0 is the density of the medium in the unde- formed state and a and T are the longitudinal and shear stresses. In what follows, we assume that they are analytic functions of the strains u and vx so that they can be expanded in Taylor series as JOURNAL DE PHYSIQUE Colloque C8, supplément au n° 11, tome 40, novembre 1979, page C8-207

Résumé.- La méthode des échelles multiples est utilisée pour déterminer des développements uniformes au second ordre de déplacements qui décrivent des ondes planes non linéaires se propageant dans un demi-espace élastique. Les propriétés du matériau peuvent être homogènes ou varier lentement selon la direction de propagation. L'interaction entre les ondes de dilatation et de cisaillement ainsi que l'interaction entre propriétés non linéaires et hétérogénéité du milieu sont discutées. Les résultats et les méthodes de résolution proposées sont comparés aux résultats antérieurs obtenus en utilisant la méthode analytique dite des caractéristiques.

Article published online by EDP Sciences and available at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979835

~ 8 - 2 0 8 JOURNAL DE PHYSIQUE

where c and cs a r e t h e l i n e a r l o n g i t u d i n a l and P

shear wave speeds. I n what f o l l o w s , we assume t h a t t h e p r o p e r t i e s of t h e medium v a r y s l o w l y w i t h x so t h a t we can consider p o , c

,

cs, a and bi t o be

P i

functions o f a l o n g s c a l e xl = E X , where E i s a small b u t f i n i t e dimensionless q u a n t i t y .

To complete t h e problem formulation, we need t o s p e c i f y t h e i n i t i a l and boundary c o n d i t i o n s . I n t h i s study, we c o n s i d e r t h e s o - c a l l e d s i g n a l 1 in g problem; t h a t i s ,

P(x,O) = Q(x,O) = R(x,O) = S(x,O) = 0 f o r x > 0

~ ( 0 , t ) = ~ @ ( t ) , Q(0,t) = E $ J ( ~ ) f o r t > 0 (6)

S u b s t i t u t i n g equations (9)

-

(13) i n t o equations (3)

-

( 5 ) and (a), u s i n g equation

(7),

and equating t h e c o e f f i c i e n t s o f E on b o t h sides, we o b t a i n

1 aR1

+1_

& + ~ P ~ + ~ P ~ = L ~ ( P ~ , R ~ )

= O --

c asl C, as, as,

as2

P

where

so t h a t

3. Approximate Solution.- We n o t e t h a t t h e i n i t i a l c o n d i t i o n s a r e taken t o be small t o enable us t o determine a s o l u t i o n f o r small b u t f i n i t e amplitude waves. F o l l o w i n g t h e method o f m u l t i p l e s c a l k s /I/, we l e t

where

a r e t h e 1 i near c h a r a c t e r i s t i c s f o r r i g h t - r u n n i ng waves.

Since i n t h e i n i t i a l s t a t e t h e medium was a t r e s t and s i n c e i t i s assumed t o be s e m i - i n f i n i t e , t h e s o l u t i o n o f t h e f i r s t - o r d e r problem c o n s i s t s o f r i g h t - r u n n i n g waves o n l y ; t h a t i s ,

S u b s t i t u t i n g equations (9)

-

(13) i n t o equa- t i o n s (3)

-

(5) and (8), u s i n g equations ( 7 ) and ( l a ) , and equating t h e c o e f f i c i e n t s of E' on both sides, we o b t a i n

A. Kluwick and A.H. Nayfeh c8-209

+ r a

b l

a

¹

a 2

2

a 2

-+ J ( s 2 , x,) (21) G1 =

- -

( f g l l -

asl

Cs as2

P

where

H(s,, x,) = 2c2

af

+ c i f

d

[gn(poc3)

1

axl

^{dx I P}

S

-

^d C~n(poc:)]

J(s2, x,) = 2C2 ag

+

c2 g -

ax ^S dx ¹

The second and t h i r d terms on t h e r i g h t - h a n d s i d e o f equation (19) account f o r t h e e f f e c t o f t h e shear wave on t h e l o n g i t u d i n a l wave, w h i l e t h e

and AR i s t h e domain shown i n F i g . l . I n equations (25) and (262, t h e r e a r e cumulative ( s e c u l a r ) terms

2

which make E P2 and E

R 2

l a r g e r than t h e order o f EPI and E R I f o r a l l x

,

⁰

⁽¹⁾

^{o r t}

,

⁰

^(1).

E E

I f @ ( t ) and J, ( t ) a r e p e r i o d i c o r i f they a r e pulses, then F1 and G1 a r e bounded and hence n o t cumulative. I n t h i s case, t h e o n l y cumulative terms remaining terms account f o r t h e e f f e c t of t h e a r e t h e ones p r o p o r t i o n a l t o s2H(s1, x,). Hence f o r l o n g i t u d i n a l wave on i t s e l f . S i m i l a r l y , t h e second a uniform expansion, ~ ( s ~ , x,) = 0. F o l l o w i n g and t h i r d terms on t h e r i g h t - h a n d s i d e o f e q u a t i o n

Nayfeh /6/, we f i n d t h a t t h e s o l u t i o n o f H(sl, x,) (21) account f o r t h e e f f e c t o f t h e l o n g i t u d i n a l

= 0 i s wave on t h e shear wave, w h i l e t h e remaining terms

account f o r t h e e f f e c t o f t h e shear wave on ( 0 ) ~ (0)

y2

i t s e l f . p =

-LR

=E[%] .

$(5) + ....

⁽²⁹⁾

A p a r t i c u l a r s o l u t i o n o f equations (19) and (20) c

P POC;,

i s

where

PzP = c s s2H(s13 xl)

-

^{a3 9}

cp(cp

-

cs) 2(c;

-

c:) ^{E = s +} 2

^{m ( E )} i' ^L

cp ^[PLJ P o C ^{(0)cp(O) 3}

^%

^dx,+

^{. ..}

(25) 0 P

+

F1 (SI .s2 ,XI

s ¹ (30)

s2H(s1, x i )

-

cp

J ^"

^(cpf)dsl Thus, t o f i r s t o r d e r t h e shear wave does n o t a f f e c t

Rzp =

-

_(cP

-

^{c s )} t h e 1 ongi t u d i n a l wave. However, equations ( 2 5 ) and

(26) show t h a t t h e shear wave does a f f e c t t h e l o n -

+

^{g2 +GI} (s1 $ 5 2 ,XI) (26) g i t u d i n a l wave a t second order.

2 2

2(cp

-

^{c S )} The a n a l y s i s o f equations (21) and (22) i s

analogous t o equations (19) and (20). Again t h e

where w a v e - i n t e r a c t i o n terms do n o t g i v e r i s e t o cumula-

c8-210 JOURNAL DE PHYSIQUE

FIG.l : Domain o f i n t e g r a t i o n f o r l o n g i t u d i n a l wave.

t i v e terms e f f e c t s and t h e e l i m i n a t i o n o f cumulative terms leads t o

J

(sl, xl) = 0. Hence

where

x 1

The domain As i s shown i n F i g . 2. Again, if @ ( t ) and q~ ( t ) a r e p e r i o d i c o r pulses, t h e i n t e g r a l s i n equations (33) and (34) a r e bounded and hence n o t cumulative. Therefore, t h e expansion (31) and (32) i s u n i f o r m and t h e shear wave i s n o t a f f e c t e d by t h e l o n g i t u d i n a l wave t o f i r s t order.

FIG.

2 : Domain o f i n t e g r a t i o n f o r shear wave.

Moreover, 4. Shock Waves.- The r e s u l t s o f t h e preceding

2 s e c t i o n show t h a t t o f i r s t o r d e r t h e r e i s no i n t e r - b l f

Q Z P = + FZ (33) a c t i o n between T o n g i t u d i n a l and shear waves. Hence,

2 ( c i

-

^c:) i t i s s u f f i c i e n t t o analyze one o f them, say l o n g i -

s t u d i n a l waves.

b l c D Equation (29) and (30) show t h a t t h e speed o f

S2 P = -cs

Ik

( ~ ~ g 1 d . 2

-

2(c;

-

c:) f 2 + G2 a l o n g i t u d i n a l wave depends on i t s l o c a l amplitude.

0 Thus, a t some d i s t a n c e x, t h e s o l u t i o n (29) and

(34) (30) becomes m u l t i - v a l u e d due t o t h e steepening o f

where t h e wave-form. Since n e i t h e r t h e s t r e s s n o r t h e

s t r a i n can be muti-valued i n space o r time, a shock wave forms. The shock f o r m a t i o n d i s t a n c e i s t h e

F2= s m a l l e s t d i s t a n c e a t which t h e f i e l d v a r i a b l e s

A e x h i b i t i n f i n i t e slopes i n e i t h e r space o r time.

D i f f e r e n t i a t i o n o f equation (29) w i t h r e s p e c t t o x (35)

shows t h a t aP/ax and aR/ax f i r i t become i n f i n i t e , and hence a shock develops, a t a d i s t a n c e x g i v e n by

A. Kluwick and A.H. Nayfeh

where 5 = Es i s t h e c h a r a c t e r i s t i c f o r which €al+' (5,) > 0 and

I@' ( E ) I

a t t a i n s i t s maximum. S i m i l a r - l y , a shear shock develops a t an x given by

X

where 0 =qs i s the c h a r a c t e r i s t i c f o r which €bl

$ ' (n,) > 0 and

I Y 1 ( q ) (

a t t a i n s i t s maximum.

Shock waves can a l s o be l o c a t e d by determining t h e deformations u and v from P and Q. Then t h e shoks can be l o c a t e d by f o l l o w i n g Nayfeh and Kluwick /7/

and imposing the c o n d i t i o n t h a t t h e displacements are continuous across t h e shocks.

5. Comparison w i t h t h e Method o f C h a r a c t e r i s t i c s .

-

The propagation o f f i n i t e - a m p l i t u d e waves i n homo- geneous e l a s t i c s o l i d s was s t u d i e d by Davison /3/

by using t h e a n a l y t i c method o f c h a r a c t e r i s t i c s /4,1/. Davison's r e s u l t s were extended t o s o l i d s w i t h a more general c o n s t i t u t i v e equation by Nayfeh /1/ and t o heterogeneous s o l i d s by N a i r and Nemat-Nasser /5/.

Using t h e a n a l y t i c method of c h a r a c t e r i s t i c s , one expands both t h e dependent and independent v a r i a b l e s i n terms o f t h e exact outgoing characte- r i s t i c s . I n t h e case o f homogeneous s o l i d , one o b t a i n s /1/

P = €4 (5)

+ ...,

R =

-

E C 4(5)

+ ...

P (39)

Q = E$ (rl)

+ ...,

^{S =}

^-

^{ECS $(rl)}

⁺ ...

where

We note t h a t t h e l a s t term\ on t h e r i g h t - h a n d sides of equations (40) and (41) b s c r i b e a n o r l i -

near i n t e r a c t i o n between t h e l o n g i t u d i n a l and shear waves. Other i n t e r a c t i o n terms a r e present i n t h e

second-order q u a n t i t i e s /3/. I f the i n t e g r a l s of 4 and J, a r e bounded, then t h e i n t e r a c t i o n terms a r e small compared w i t h t h e remaining terms i n equations (40) and (41) which a r e unbounded w i t h e i t h e r x o r t. Hence, one can expand equations (39) i n T a y l o r s e r i e s by assuming t h e terms p r o p o r t i o n a l t o a2 and b2 t o be small. The r e s u l t i s t h e same as t h e one obtained from our s o l u t i o n i n t h e case of of a homogeneous medium. Tf t h e i n t e g r a l s o f @ and Il, a r e unbounded, one can i n c l u d e these terms i n t h e c o n d i t i o n f o r t h e e l i m i n a t i o n o f s e c u l a r terms and hence o b t a i n a s o l u t i o n i n agreement w i t h equa- t i o n s (39)

-

(41). A l t e r n a t i v e l y , one can i n s p e c t t h e expressions f o r Pz, Rz, Q2 and SZ, use t h e method of r e n o r m a l i z a t i o n /1/ i n c o n j u n c t i o n w i t h t h e already obtained s o l u t i o n o f Sect. 3, and o b t a i n an expansion t h a t reduces t o (39)

-

(41) f o r a homogeneous s o l i d .

I n t h e case o f t h e weakly heterogeneous medium s t u d i e d by N a i r and Nemat-Nasser /5/, we have

S u b s t i t u t i n g these expressions i n t o equations (29)

-

(32) shows t h a t t h e r e s u l t i n g expansion i s equiva- l e n t t o t h a t o f N a i r and Nemat-Nasser when t h e i n t e r a c t i o n terms a r e n e g l i g i b l e . I f they a r e not, our r e s u l t s can be m o d i f i e d as discussed above t o o b t a i n an expansion t h a t i s t h e same as t h a t o f N a i r and Nemat-Nasser. However, whereas t h e method of m u l t i p l e scales can t r e a t systems w i t h s l o w l y v a r y i n g p r o p e r t i e s which need n o t be small, t h e a n a l y t i c method o f c h a r a c t e r i s t i c s has n o t been a p p l i e d y e t t o cases o f 1 arge h e t e r o g e n e i t i e s

.

References

/1/ Nayfeh, A.H.

,

P e r t u r b a t i o n Methods (Wi 1 ey I n t e r s c i e n c e , New York, 1913).

/2/ Nayfeh, A.H. and Mook, D.T., Nonlinear O s c i l - l a t i o n s (Wiley I n t e r s c i e n c e , New York, 1979).

/3/ Davison, L., I n t . J. S o l i d s S t r u c t . - 4 (1968) 301.

/4/ L i n , C.C., J. Math. Phys.

33

(1954) 117.

/5/ N a i r ,

S.

and Nemat-Nasser, S., I n t . J. Eng.

S c i .

9

(1971) 1087.

/6/ Nayfeh, A.H., J. Sound Vib.

-

42 (1975) 357.

/7/ Nayfeh, A.H. and Kluwick, A., 3 . Sound Vib.

48 (1976) 293.

-