COMBINED ANELASTICITY AND NONLINEARITY : THE LONGITUDINAL RESONANCES OF A BAR

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COMBINED ANELASTICITY AND NONLINEARITY : THE LONGITUDINAL

RESONANCES OF A BAR

H. Jeong, D. Beshers

To cite this version:

H. Jeong, D. Beshers. COMBINED ANELASTICITY AND NONLINEARITY : THE LONGITUDI- NAL RESONANCES OF A BAR. Journal de Physique Colloques, 1987, 48 (C8), pp.C8-317-C8-322.

�10.1051/jphyscol:1987846�. �jpa-00227150�

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COMBINED ANELASTICITY AND NONLINEARITY : THE LONGITUDINAL RESONANCE OF A BAR

H.S. JEONG and D.N. BESHERS

Henry Krumb School of Mines, Columbia University, New York, NY 10027, U.S.A.

Résumé - Nous avons ajoute un terme quadratique peu important I

l'équation du modèle du solide anélastique standard et nous avons étudie son effet dans l'approximation du premier ordre. La résonance fondamen- tale ne change pas, en première approximation, mais uneharmonique de deuxième ordre apparaît. En fait, la. deuxième harmonique résulte de deux ondes spatiales à même fréquence. Comme l'amortissement linéaire tend vers zéro, la forme mathématique de la solution change, les deux ondes dégénèrent en une seule qui n'est pas exactement sinusoïdale.

Abstract - The addition of a small quadratic term to the standard anelastic solid model has been investigated to first order approximation. The fundamental resonance does not change, to first order, but a second harmonic wave appears. The second harmonic actually consists of two spatial waves with one frequency. As the linear damping goes to zero, the mathe- matical form of the solution changes, the two waves degenerating into one that is not entirely sinusoidal.

I - INTRODUCTION

The longitudinal resonance of a bar has often been exploited for the measurement of internal friction. The analysis of the experiments has usually been con- ducted in an elementary fashion, and often that suffices. However, when the internal friction is nonlinear, that is to say dependent on the amplitude of oscillation, then there is a need for more careful consideration of the problem, a need universally neglected for lack of a suitable theory. A first step towards such a theory, an approximate treatment for the case of a small nonlinear effect co-existing with a linear damping, is presented in this paper.

II - THE MODEL

Because nonlinear internal friction usually occurs in materials that also exhi- bit linear damping, it is important to treat the linear and nonlinear aspects together. The basic model for linear damping is the standard anelastic solid

(SAS) /1/. The longitudinal resonance of a bar made from a SAS has been treated already /2/. We consider here a bar of homogeneous material obeying a consti^

tutive equation based on the standard anelastic solid, but augmented by one nonlinear term:

* ~ 2

a + ba - ae + ce + de , (1 ) where a is the longitudinal stress, e the corresponding strain, and a, b, c, d are material constants; in the usual notation, a = 1/(JU+6J), c = t a > and b - J c. The coefficient d is assumed subsequently to be small. For a long, thin bar the equations of stress equilibrium reduce to

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987846

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JOURNAL

DE

PHYSIQUE

where p d e n o t e s t h e d e n s i t y o f t h e m a t e r i a l and u = u ( x , t ) t h e d i s p l a c e m e n t of a c r o s s - s e c t i o n o f t h e b a r . S t r a i n and d i s p l a c e m e n t a r e r e l a t e d by

We e l i m i n a t e s t r e s s i n f a v o r o f u by d i f f e r e n t i a t i n g ( 1 ) w i t h r e s p e c t t o x , t h e n u s i n g ( 2 ) , a n d r e p l a c i n g ^Ew i t h u by ( 3 ) , o b t a i n i n g t h e e q u a t i o n o f m o t i o n t o be s o l v e d :

putt ⁺b p u t t t = a u ⁺c u + 2duxuxx.

XX x x t ^{( 4 )}

The b a r is c o n s i d e r e d t o be d r i v e n a t o n e e n d , x = 0 , by a p e r i o d i c s t r e s s o f a m p l i t u d e 20 a n d c i r c u l a r f r e q u e n c y w a n d t o be f r e e a t t h e o t h e r e n d , x = L.

The b o u n d a r y O c o n d i t i o n s a r e t h e n , c h o o s i n g t h e p h a s e a p p r o p r i a t e l y ,

o ( ~ , t ) = ooeiWt + 0 e - i w t a t x = O

0 ( 5 )

We s e e k a s t e a d y - s t a t e s o l u t i o n o f t h i s p r o b l e m , e x p e c t i n g t o f i n d r e s o n a n t modes, a n d t o e x a m i n e t h e b e h a v i o r i n t h e v i c i n i t y o f r e s o n a n c e . By s t e a d y - s t a t e we mean t h a t e v e r y t e r m i n t h e s o l u t i o n m u s t r e p e a t i n t i m e w i t h t h e f u n - d a m e n t a l f r e q u e n c y ; t h a t is we assume t h a t h a r m o n i c s o f t h e f u n d a m e n t a l w i l l o c c u r , b u t n o t s u b h a r m o n i c s . T h i s a s s u m p t i o n i s c o n s i s t e n t w i t h e x p e r i m e n t a l r e s u l t s i n t h i s l a b o r a t o r y .

The n o n l i n e a r term n e c e s s i t a t e s c a r e i n t h e u s e o f complex e x p o n e n t i a l func- t i o n s , w h i c h a r e o t h e r w i s e t h e s i m p l e s t r e p r e s e n t a t i o n w i t h w h i c h t o work. T h e r e a l p a r t o f a n y complex number Z i s g i v e n by

b u t R , a s a n o p e r a t o r , d o e s n o t commute w i t h m u l t i p l i c a t i o n . We s h a l l t h e r e f o r e i n e f f e c t r e q u i r e t h a t R o p e r a t e f i r s t by w r i t i n g t h e t r i a l s o l u t i o n i n t h e f o r m

m

u ( x , t ) = z [ u n ( x ) e i n w t +

u

* ( x ) e-inwt]. ( 8 ) n=O

I n t h i s way, a l l o u r e x p r e s s i o n s a r e r e a l , a l t h o u g h s o m e t i m e s we w i l l n o t b o t h e r t o w r i t e o u t t h e c o n j u g a t e p a r t .

I11

-

THE SOLUTION

S u b s t i t u t i o n o f ( 8 ) i n ( 4 ) g i v e s a v e r y l e n g t h y e q u a t i o n . B e c a u s e e x p ( i n w t ) a n d e x p ( - i n w t ) a r e l i n e a r l y i n d e p e n d e n t f u n c t i o n s , we r e q u i r e t h a t t h e c o e f f i c i e n t o f e a c h b e z e r o , which h a s t h e e f f e c t o f c o n v e r t i n g o u r p a r t i a l d i f f e r e n t i a l e q u a t i o n i n t o a n i n f i n i t e s e t o f c o u p l e d o r d i n a r y d i f f e r e n t i a l e q u a t i o n s . Each o f t h e s e o r d l n a r y e q u a t i o n s may be viewed a s c o n s i s t i n g o f two p a r t s . One p a r t c o n t a i n s f o u r t e r m s w h i c h c o r r e s p o n d t o t h e f o u r t e r m s o f t h e s t a n d a r d l i n e a r s o l i d a t t h e f r e q u e n c y nw a n d t h e o t h e r p a r t i s an i n f i n i t e s e r i e s which h a s t h e c o e f f i c i e n t d o f t h e n o n l i n e a r t e r m i n ( 1 ) . T h e s e o r d i n a r y e q u a t i o n s a r e t h e r e - f o r e inhomogeneous. Now t h e s o l u t i o n t o a n inhomogeneous e q u a t i o n i s t h e gem- e r a 1 s o l u t i o n o f t h e c o r r e s p o n d i n g homogeneous e q u a t i o n p l u s a n y p a r t i c u l a r s o l u t i o n o f t h e inhomogeneous e q u a t i o n . We c a n o b t a i n t h e s o l u t i o n s t o t h e homogeneous e q u a t i o n s i m m e d i a t e l y , and t h e n a p p r o x i m a t e t h e i n f i n i t e s e r i e s i n t e r m s o f t h e s e known s o l u t i o n s , a f t e r which p a r t i c u l a r s o l u t i o n s c a n b e f o u n d . T h i s s t r a t e g y i s f a c i l i t a t e d by t h e f a c t t h a t t h e terms o f t h e i n f i n i t e series, w h i l e n o n l i n e a r , become p r o d u c t s o f t r i g o n o m e t r i c f u n c t i o n s o f x a n d t h e r e f o r e

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The homogeneous e q u a t i o n s

The reduced e q u a t i o n f o r any n has t h e form

where U' i n d i c a t e s dU/dx, and a n o t h e r e q u a t i o n which i s t h e complex c o n j u g a t e of ( 9 ) . Equation ( 9 ) h a s t h e well-known form

"n ^O" + K 2 ~ n 0 = 0 ( 1 0 )

where t h e complex p r o p a g a t i o n c o n s t a n t K = kn + i a n i s d e t e r m i n e d by

The g e n e r a l s o l u t i o n of ( 1 0 ) is

unO

= C sinK x ⁺DncosK x ^{( 1}2 )

where t h e c o n s t a n t s C and D a r e t o be determined e v e n t u a l l y by t h e boundary c o n d i t i o n s . For t h e Boundary c o n d i t i o n s given by ( 5 ) and ( 6 ) . t h e s o l u t i o n is

and

u ( x , t ) = u l o ( x ) e i w t + C.C. ( 1 4 )

where C . C . s t a n d s f o r complex c o n j u g a t e . T h i s is t h e s o l u t i o n g i v e n p r e v i o u s l y / 2 / f o r t h e r e s o n a n c e o f a bar made of a s t a n d a r d l i n e a r s o l i d .

The Inhomogeneous e q u a t i o n s

R e t u r n i n g t o t h e inhomogeneous e q u a t i o n , we s t a r t t h e p r o c e s s of approximating t o t h e n o n l i n e a r terms by assuming t h a t

and t h a t a l l o t h e r U n a r e s m a l l of o r d e r d , a t l e a s t , and proceed t o s o r t t h e t e r m s by t h e powers of d , i n t h e end n e g l e c t i n g a l l t e r m s c o n t a i n i n g d t o a power h i g h e r t h a n 1 . I n t h e p r o c e s s of a p p r o x i m a t i o n we a r e a l s o helped by assuming t h a t 6J/J i s s m a l l s o t h a t t h e s t r a i n a m p l i t u d e a t r e s o n a n c e i s o r d e r s of magnitude l e s s t h a n u n i t y . The o n l y e q u a t i o n s t h a t s u r v i v e t h i s p r o c e s s a r e t h o s e f o r n = 0 , 1 , 2 . For n = 1 , t h e n o n l i n e a r term v a n i s h e s i n t h i s approxima.- t i o n . T h e r e f o r e t h e p r e s e n t t r e a t m e n t w i l l n o t , i n f a c t , e x t e n d t o t h e c a s e o f n o n l i n e a r damping. R a t h e r , t h e approximation i s one i n which t h e fundamental is u n a f f e c t e d t o f i r s t o r d e r , b u t s m a l l harmonic g e n e r a t i o n o c c u r s . Our approxirnw t i o n s have t h e n reduced t h e i n f i n i t e s e t of e q u a t i o n s e n v i s i o n e d i n ( 8 ) t o t h r e e o n l y . We f o c u s o u r a t t e n t i o n on t h e e q u a t i o n s f o r n = 2, p u t t i n g a s i d e t h e e q u a t i o n f o r n = 0 which w i l l not a f f e c t t h e r e s u l t s . Thereby, we d i s c u s s h e r e a system w i t h o n l y two s t a t e s , a c l o s e d system n o t a n open o n e . We have t h e n t o s o l v e t h e e q u a t i o n s f o r n = 2

and t h e complex c o n j u g a t e of ( 1 6 ) . With U, 0 g i v e n by (131, and d e n o t i n g t h e

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JOURNAL DE PHYSIQUE

c o e f f i c i e n t of cosK1!L-x) i n (13) by A, t h e p a r t i c u l a r s o l u t i o n of ( 1 6 ) i s by t h e method of v a r i a t i o n of p a r a m e t e r s

where 2 2

D = [(2K1)

-

^{K 2}

1

⁽a ⁺ ic2w).

The boundary c o n d i t i o n s , ( 5 ) and ( 6 ) , imply t h a t t h e s t r e s s c o r r e s p o n d i n g t o U2 s h a l l be z e r o a t both ends of t h e b a r , f o r c i n g t h e o t h e r t e r m s i n U t o be p r o p o r t i o n a l t o t h e c o e f f i c i e n t of t h e p a r t i c u l a r s o l u t i o n . L e t P s e n o t e t h e c o e f f i c i e n t of sin2K1 (L-x) i n ( 1 7 ) . Then t h e g e n e r a l s o l u t i o n o f ( 1 6 ) i s

U2 = C sinK (L-x) + C2cosK2(L-x) ⁺Psin2K1(L-x)

1 2 (19)

where

Cl = (,-K2/2K1)P, and

C2 = P ( K ~ / ~ K , ) [ C O ~ ~ K ~ L

-

cosK2Ll/sinK2L.

Equation ( 1 9 ) shows t h a t o u r s o l u t i o n i s more c o m p l i c a t e d t h a n we might have e x p e c t e d . While we have o n l y one p e r i o d i c i t y i n t i m e , t h e s p a t i a l v a r i a t i o n i n v o l v e s two c l o s e l y spaced p e r i o d i c i t i e s , K and 2K1. The v a r i a t i o n of t h e e l a s t i c f i e l d of t h e second harmonic a l o n g t g e b a r i s t h u s an i n t e r f e r e n c e phenomenon. Use o f e q u a t i o n s ( 2 ) and ( 3 ) shows t h a t t h e s t r e s s and s t r a i n of t h i s f i e l d a r e n o t q u i t e p r o p o r t i o n a l t o e a c h o t h e r , because t h e p r o p a g a t i o n c o n s t a n t s of t h e two waves d i f f e r s s l i g h t l y . The d i f f e r e n c e between K and 2K1 a r i s e s from t h e f r e q u e n c y dependence of t h e a n e l a s t i c s t r a i n f o r t h e S ~ S . F u r t h e r ( 1 7 ) and (19) s t a t e t h a t t h e a m p l i t u d e of U2 i s p r o p o r t i o n a l t o t h e s q u a r e of t h e a m p l i t u d e of U 1 , and is t h e r e f o r e more s h a r p l y peaked a t resonance t h a n U 1 .

The d e g e n e r a t e Case

The d e g e n e r a t e c a s e when t h e denominator of P v a n i s h e s (K2

5

2K ₁) r e q u i r e s s p e c i a l c o n s i d e r a t i o n . We r e s t r i c t o u r a t t e n t i o n t o t h e neighborhood of t h e fundamental r e s o n a n c e , k l L = n, s o w is not s i g n i f i c a n t l y v a r i a b l e . Using (111, we still f i n d t h r e e c a s e s f o r which K = 2K1: ( i ) when b = c = o , o r ^T = o;

( i i ) when b = c = m , o r T = -; and ( f i i ) c = b a , o r 6 J = o , f o r any TO. Case ( i i i ) c o r r e s p o n d s t o t h e Pemoval of t h e a n e l a s t i c d e f e c t s from t h e m a t g r i a l , a s by p u r i f i c a t i o n o r a n n e a l i n g . The damping is z e r o f o r a l l t h r e e c a s e s , b u t t h e modulus d e f e c t v a n i s h e s o n l y f o r t h e l a t t e r two.

When t h e damping goes t o z e r o , t h e s o l u t i o n does n o t d i v e r g e a s s u g g e s t e d by t h e appearance of t h e denominator of (171, b u t i n t h e l i m i t a s K .-2K g o e s t o z e r o t h e f u n c t i o n a l form changes. A new s o l u t i o n must b e s o u g h t $or t h i s c a s e ; i t is

2 3 2 3

where B1 = dA k / a = dA ( k / 2 ) / a , and A i s t h e a m p l i t u d e of t h e

fundamental. O w e 1 o b t a i n e d o ( 2 0 f by u s i n g the0method of v a r i a t i o n of p a r a m e t e r s , but t h e same r e s u l t is found by p u t t i n g 2K1 = K2 + n , where n i s s m a l l ,

o b t a i n i n g t h e f i r s t o r d e r approximation i n 0 , and t a k i n g t h e l i m i t a s

n

+ 0. TO keep A f i n i t e i n t h i s l i m i t , t h e d r i v i n g s t r e s s must go t o z e r o . The q u a l i t a - t i v e cRange i n t h e s o l u t i o n a s t h e damping g o e s t o z e r o , r e p r e s e n t e d by t h e f a c t o r of x i n t h e second term of ( 2 0 ) , i s noteworthy. The t r u e n a t u r e of t h e s o l u t i o n i s obscured when t h e undamped wave f u n c t i o n s a r e used i n t h e n o n l i n e a r c a l c u l a t i o n .

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body w i t h no damping and no n o n l i n e a r i t y . T h e r e a r e no e n e r g y l o s s e s i n s u c h a model and s o a t s t e a d y ~ s t a t e t h e r e can be no e n e r g y i n p u t e i t h e r . Such a s t a t e might be c r e a t e d by a d r i v i n g f o r c e o p e r a t i n g f o r a c e r t a i n l e n g t h o f t i m e t o b u i l d up some p r e s c r i b e d a m p l i t u d e , a f t e r which t h e d r i v i n g f o r c e would be removed, and t h e s t a t e j u s t c r e a t e d would l a s t f o r e v e r a f t e r . Although s e e w i n g l y c o n v e n i e n t f o r c a l c u l a t i o n , and t h e u s u a l s t a r t i n g p o i n t , t h i s s i t u a t i o n i s q u i t e u n p h y s i c a l .

The r e a l i t y i s t h a t l o s s e s a r e a l w a y s p r e s e n t , b u t t h e mechanism of t h e l o s s may t a k e more t h a n o n e form. The SAS is a l i n e a r model w i t h t h e e n e r g y l o s s occur- r i n g everywhere i n t h e v i b r a t i n g body. The m e c h a n i c a l e n e r g y is c o n v e r t e d t o h e a t , e a c h volume e l e m e n t r e c e i v i n g i t s due p r o p o r t i o n , but t h e d e t a i l e d mechar nism of d i s s i p a t i o n is n o t s p e c i f i e d . S t e a d y c s t a t e o c c u r s when t h e i n p u t of e n e r g y from t h e d r i v i n g s t r e s s e q u a l s t h e sum o f t h e d i s t r i b u t e d l o s s e s . I f , i n s t e a d o f a n e l a s t i c i t y , n o n l i n e a r i t y i s i n t r o d u c e d , i n t h e g e n e r a l c a s e t h e e f f e c t is t o p r o v i d e f o r t h e c o n t i n u a l c o n v e r s i o n of e n e r g y from t h e f u n d a m e n t a l mode t o modes of h i g h e r f r e q u e n c y . We e x p e c t t h a t t h e s e o t h e r modes w i l l a l s o b e s u b j e c t t o t h e n o n l i n e a r i t y and s o t h e i r e n e r g y w i l l be c o n v e r t e d t o s t i l l o t h e r modes, b o t h h i g h e r and lower i n f r e q u e n c y , and s o on u n t i l a t s t e a d y - s t a t e t h e r e w i l l be a n i n f i n i t e ( i n a continuum) number of modes e x c i t e d , w i t h n e t c o n v e r s i o n from t h e f u n d a m e n t a l t o t h e o t h e r modes g i v i n g a n e n e r g y o u t f l o w t h a t w i l l be j u s t e q u a l t o t h e i n p u t by a d r i v i n g f o r c e a t t h e f u n d a m e n t a l f r e q u e n c y . Note t h a t , a l t h o u g h t h e c r e a t i o n o f harmonics draws e n e r g y from t h e f u n d a m e n t a l mode, t h a t e n e r g y is s t i l l i n o r g a n i z e d form i n t h e harmonics, n o t y e t l o s t t o h e a t . T h i s is q u i t e i n c o n t r a s t t o t h e a n e l a s t i c l o s s e s , which go s t r a i g h t t o t h e t h e r m a l b a t h .

The model p r e s e n t e d h e r e f a l l s s h o r t of d e s c r i b i n g t h e g e n e r a l c a s e . With o n l y two modes t r e a t e d , and t h e s e c o n d o n e s m a l l , t r a n s f e r s of e n e r g y t o h i g h e r modes have been n e g l e c t e d a s h i g h e r o r d e r p r o c e s s e s . The s o l u t i o n ( 2 0 ) t o t h e

undamped c a s e r e p r e s e n t s a n a p p a r e n t l o s s - l e s s s t e a d y - s t a t e . We b e l i e v e t h a t t h i s is an a r t i f a c t of t h e a p p r o x i m a t i o n , and t h a t t h e g e n e r a l c a s e w i l l eventu?

a l l y p r o v e t o be a s s u g g e s t e d i n t h e p r e v i o u s p a r a g r a p h .

When we combine t h e two mechanisms, a s i n t h i s problem, a11 modes c r e a t e d by n o n l i n e a r p r o c e s s e s have a n e l a s t i c i t y i n t h e i r own r i g h t a s shown by e q u a t i o n s ( 1 0 ) t o ( 1 2 ) . The e f f e c t is t o c r e a t e s m a l l s h i f t s i n t h e r e s o n a n t f r e q u e n c i e s of t h e modes, s h i f t s a s s o c i a t e d w i t h t h e modulus d e f e c t , s o t h a t t h e e x a c t r e s c n a n t f r e q u e n c i e s a r e n o t r e l a t e d by i n t e g e r s / 2 / , w h i l e t h e harmonic f r e q u e n c i e s a r e . Thus r e s o n a n c e o f t h e f u n d a m e n t a l mode c o r r e s p o n d s t o kl = r / L , w i t h a f r e q u e n c y w l , w h i l e r e s o n a n c e of t h e s e c o n d harmonic c o r r e s p o n d s t o k(w2) =

2 r / L , w i t h a f r e q u e n c y w 2 , s u c h t h a t w2 + 2wl. T h e r e f o r e t h e s e c o n d harmonic i s i m p e r f e c t l y e x c i t e d . The e x c i t a t i o n i n t h i s c a s e comes n o t from t h e d r i v i n g s t r e s s b u t from t h e t e r m on t h e r i g h t hand s i d e o f ( 1 6 ) t h a t r e n d e r s t h e e q u a r t i o n inhomogeneous.

The e x c i t a t i o n o f harmonics i s t h e r e f o r e d i s t r i b u t e d t h r o u g h o u t t h e b a r , a t t h e f r e q u e n c y w l , g e n e r a t i n g a wave o f f r e q u e n c y 2wl and p r o p a g a t i o n c o n s t a n t 2K

1 ' To s a t i s f y t h e boundary c o n d i t i o n s t h e l a t t e r g e n e r a t e s a n o t h e r wave a t t h e same f r e q u e n c y b u t w i t h t h e p r o p a g a t i o n c o n s t a n t K 2 = k2 + i a 2 c o r r e s p o n d i n g t o 2wl, a s g i v e n by ( 1 1 ) .

A s we have d e f i n e d i t , k i s n e i t h e r e q u a l t o 2kl n o r i s i t e q u a l when w = dl t o 2+/L, d i f f e r i n g from b o t 8 by t e r m s of t h e o r d e r of 6J/J = ( c / a b )

-

^1. ^However,

t h e w i d t h o f t h e r e s o n a n c e i s a l s o of o r d e r 6 J / J U , s o t h e mismatch i n f r e q u e n c y is n o t s i g n i f i c a n t i n t h e p r e s e n t a p p r o x i m a t i o n . T h a t i s , i f ( 1 9 ) i s expanded f o r s m a l l

n

[ a s s u g g e s t e d below ( 2 0 ) ] , t h e r e s u l t d i f f e r s from ( 2 0 ) o n l y by

(7)

JOURNAL

DE

PHYSIQUE

s m a l l t e r m s . S i n c e we a r e assuming d s m a l l , t h e s e t e r m s belong t o t h e h i g h e r o r d e r s t h a t l i e o u t s i d e t h i s approximation.

For t h e approximation used h e r e , t h e a m p l i t u d e o f t h e second harmonic i s s m a l l and t h e energy l o s s e s from i t , by e i t h e r i n t e r n a l f r i c t i o n o r mode c o n v e r s i o n , a r e second o r d e r . T h e r e f o r e , a t s t e a d y - s t a t e , t h e a p p a r e n t i n t e r n a l f r i c t i o n i s unchanged t o f i r s t o r d e r . O f c o u r s e , b e f o r e s t e a d y - s t a t e was a c h i e v e d , energy had t o be added t o c r e a t e t h e harmonic wave; once c r e a t e d , t h e n t h e l o s s e s from t h e harmonic a r e second o r d e r .

F u r t h e r work i s n e c e s s a r y t o t r e a t t h e e x p e r i m e n t a l l y o b s e r v e d c a s e s of amplitude-dependent damping. The assumption of s m a l l d w i l l have t o b e dropped, a l t h o u g h t h e assumption of s m a l l s t r a i n a m p l i t u d e w i l l remain e m i n e n t l y v a l i d . I n c o n c l u s i o n , we have shown t h a t t h e i n c l u s i o n of a n e l a s t i c i t y l i f t s a degener- acy between 2K1 and K2, r e v e a l i n g t h a t t h e harmonic wave h a s an u n e x p e c t e d l y c o m p l i c a t e d s p a t l a l form. We e x p e c t t h a t t h e u s e of t h e SAS wave f u n c t i o n s w i l l be e s s e n t i a l i n t h e f u r t h e r development of t h e n o n l i n e a r t h e o r y .

R e f e r e n c e s

/ 1 / A. S. Nowick and B. S. B e r r y , A n e l a s t i c R e l a x a t i o n i n C r y s t a l l i n e S o l i d s , Academic Press,New York, 1972.

/ 2 / D. N. Beshers i n Techniques of M e t a l s R e s e a r c h , Vol. 7 , p a r t 2, Ed. R. Bunp s h a h , J o h n Wiley and Sons, New York, 1976, pp531-707.