EXPERIMENTAL EVIDENCE AGAINST ZENER'S THEORY : ON THE POSSIBILITY OF RE-INTERPRETING DAMPING AS A MATERIAL PROPERTY

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EXPERIMENTAL EVIDENCE AGAINST ZENER’S THEORY : ON THE POSSIBILITY OF

RE-INTERPRETING DAMPING AS A MATERIAL PROPERTY

A. Capecchi, M. Capurro

To cite this version:

A. Capecchi, M. Capurro. EXPERIMENTAL EVIDENCE AGAINST ZENER’S THEORY : ON

THE POSSIBILITY OF RE-INTERPRETING DAMPING AS A MATERIAL PROPERTY. Journal

de Physique Colloques, 1983, 44 (C9), pp.C9-447-C9-453. �10.1051/jphyscol:1983965�. �jpa-00223415�

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JOURNAL

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Colloque C9, suppliment au n012, Tome 44, d k e m b r e 1983 page C9-447

EXPERIMENTAL EVIDENCE AGAINST ZENER'S THEORY : ON THE POSSIBILITY OF RE-INTERPRETING DAMPING AS A MATERIAL PROPERTY

A. Capecchi and M. Capurro

I s t i t u t o d i Scienza delZe Costruzioni, Universitd

di

Genova, I t a l y

R6sumB : Ce travail donne les conclusions d'une etude expdrimentale conduite dans le but de caracteriser le frottement interieur dans les metaux en rapport a sa dependan ce avec les paramstres des ondes. Dans ce cadre, des essais 2 tempdrature ambiante, sur barres d'Anticoroda1 excitees en flexion et longitudinalment, viennent d'stre exa- mines vis-2-vis soit des r6sultats obtenus d'autres Auteurs, soit des previsions the2 riq~es. A ce regard, nombreux resultats prdvus par la thdorie thermodlastique se mon- trent en contradiction avec l'experience. Introduisant une correlation entre le frottement interieur et la v6locit6 de phase des ondes, on peut atteindre un meilleur po- sitionnement des points experimentaux. Tout cela B l'avantage de l'interprdtation se- lon laquelle l'attenuation des grandes longueurs d'onde dans les metaux est relide, plut8t q u 1 8 des effets thermo&lastiques,

B

phdnomenes d'inthraction des ondes avec

les dgfauts et les autres het6rogkn6itks microscopiques.

Abstract: An experimental investigation has been carried out with the aim of charac- --

terizing material damping in metals as a function of the wave parameters. Tests on Anticorodal(Si 1%;Mg 0.6%;Mn 0.3%)bars at room temperature, in both flexural and lon- gitudinal small-amplitude vibrations, are discussed with reference as well to results by other Authors as to theoretical predictions. Many aspects of the thermo-elastic

theory seem herewith to be disappointed, in particular the dependence of damping on the specimen size. A new chart of damping coefficient vs. the phase velocity of waves, has been successfully introduced, thus enhancing the view that attenuation of large wavelengths in metals is related, rather than to thermo-elastic effects, to mechanical interaction of waves with defects and microscopical inhomogeneities.

1. Foreword

A quantitative knowledge of material damping in its correlation with the wave parameters would be, even if confined to orders of magnitude, of paramount importan- ce in many branches of engineering. Data on internal friction are scarcely to be found in technical handbooks and, as to specialized literature, a general insight into the matter is, at the present time, missing. For this reason the writers, some years ago, have started an experimental investigation to the purpose of characteri- zing internal friction of some metals commonly used in engineering practice. Preli- minarly, tests have been carried out at room temperature with very small amplitu- des on Anticorodal(A1 alloy with nom.comp.:Si 1%;Mg 0.6Z;Mn 0.3%)extruded bars, corn mercially available in a wide range of size. In addition, data by other Authors'were also at hand for akin materials.

On the side of theoretical interpretations of internal friction,a theory that seems to keep on widely acknowledged reputation, is the theory of thermo-elastic diffusion formulated by ZENER as early as 1937. The aim of the present work is,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983965

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C9-448 J O U R N A L D E PHYSIQUE

a c c o r d i n g l y , a t r i a l of t e s t r e s u l t s , p a r t l y o b t a i n e d by t h e w r i t e r s , p a r t l y taken f r o m t h e l i t e r a t u r e , i n t h e h o p e of a r r i v i n g a t some c e r t a i n c o n c l u s i o n s concerning t h e i r correspondence w i t h t h e p r e d i c t i o n s of Zener's t h e o r y .

2 . P r e d i c t i o n s of Zener's theory

ZENER [ 1

1 ,

[ 21, [ 3

1 ,

[ 4 1 , [ 51

,

has i n t e r p r e t e d i n t e r n a l damping i n s o l i d s a s an e f f e c t of thermal d i f f u s i o n . A s i m i l a r concept was p r e v i o u s l y a p p l i e d t o t h e absor- p t i o n of sound waves i n f l u i d s [ 6 ]

.

A f i r s t - a p p r o a c h s o l u t i o n of t h e thermo-elastic problem i n t h e c a s e of f l e x u - r a l waves i n p r i s m a t i c rods was worked o u t (op. c i t . 1211, l e a d i n g t o e s t i m a t e

f o r t h e damping c a p a c i t y ; AE i s a f a c t o r depending on t h e p h y s i c a l p r o p e r t i e s of t h e m a t e r i a l ( A E = 0 , 0 0 4 6 i s suggestedforAluminium i n op. c i t . [ 5 ] p. 9 0 ) ; v i s t h e wave frequency and

being s t h e specimen's t h i c k n e s s and D t h e c o e f f i c i e n t of d i f f u s i o n ( e s t i m a t e d a s 0.88 sqcm/s f o r A 1 i n op. c i t , [ S ] ) , i s t h e i n v e r s e of t h e r e l a x a t i o n time and c o i n - tides w i t h t h e peak frequency; ~ r / 2 i s t h e c o e f f i c i e n t v a l i d f o r r e c t a n g u l a r c r o s s

se

c t i o n s . Eq. ( 1 ) r e p r e s e n t s , i n t h e p l a n e ~ - l v s . v a f a m i l y of bell-shaped c u r v e s , one f o r each t h i c k n e s s , w i t h t h e maxim h o r i z o n t a l l y a l i g n e d t o t h e v a l u e AE/2.

I n a b i - l o g a r i t h m i c p l o t , we have a c h a r t a s t r a c e d on f i g . 2, where t h e two descending branches of each curve reduce v e r y s o o n t o s t r a i g h t l i n e s sloped r e s p e c t i v e l y 1 _and

-

!L

.

4 4

For l o n g i t u d i n a l v i b r a t i o n s , Zener had t o modify t h e t h e o r y , s i n c e t h e macrosco - p i c a 1 s t r e s s g r a d i e n t i n t h e d i r e c t i o n of t h e wave would r e q u i r e exceedingly l a r g e r e l a x a t i o n times with t o o low v a l u e s of damping. The theory of i n t e r g r a i n thermal d i f f u s i o n was t h e n worked o u t , making e l a s t i c a n i s o t r o p y r e s p o n s i b l e f o r i n t e r g r a i n s t r e s s g r a d i e n t s , which should c a u s e l o c a l thermal c u r r e n t s t o flow a c r o s s g r a i n boundaries with r e l a x a t i o n t i k e s of t h e o r d e r of D over t h e squared g r a i n s i z e . Again a bell-shaped c u r v e f o r Q-l i s p r e d i c t e d , with a maximum a t

with d t h e mean diameter of c r ~ s t a l l i t e s , independently of t h e specimen's s i z e ( o p . c i t . [5Ip.92).

I n t h e w r i t e r s ' knowledge, t h e c a s e of t o r s i o n a l waves h a s n o t been examined by Zener and, s i n c e t h e volume s t r a i n i s zero and t h e n n o e l a s t i c e f f e c t s could n o t be produced,one would be compelled t o r e s o r t t o some d i f f e r e n t i n t e r p r e t a t i o n .

The main f e a t u r e of Zener's theory i s t h a t i n t e r n a l damping, f a r from being

re

garded a s a p h y s i c a l p r o p e r t y of t h e m a t e r i a l , depends on t h e type of e x c i t e d waves and on t h e s i z e of specimen. According t o t h i s view, i t would b e impossible t o spe- c i f y howsoever a v a l u e of damping f o r a given m a t e r i a l .

Concerning t h e shape of t h e c u r v e s damping v s . frequency i n b o t h f l e x u r a l and l o n g i t u d i n a l waves, a t y p i c a l p r e d i c t i o n of Z e n e r ' s theory c o n s i s t s i n t h e presence of maxima. The c u r v e s f o r f l e x u r a l waves and t h e l o c a t i o n s of maxima, s h i f t marke- d l y toward t h e domain of h i g h e r f r e q u e n c i e s w i t h d e c r e a s i n g t h e specimen depth. For l o n g i t u d i n a l waves we have, a t a given g r a i n s i z e , a s i n g l e curve w i t h a s i n g l e peak

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given by eq. ( 3 ) .

3 . F l e x u r a l waves: t e s t r e s u l t s

S t r u c t u r a l t y p e s s o f a r i n v e s t i g a t e d by t h e w r i t e r s were s t r a i g h t b a r s , e i t h e r clamped a t one end o r suspended a t t h e n o d a l p o i n t s of t h e wave-shapes f o r t h e free-.

f r e e beam. Most of Authors have employed e i t h e r of t h e s e t y p e s . GRANICK & STERN [7]

have e x c i t e d i n vacuum beams clamped t o t h e moving head of a s h a k e r , a t midspan.

This d i s p o s i t i o n , e q u i v a l e n t t o a c o u p l e of tuned c a n t i l e v e r s , conveys b u t s h e a r f o r c e s t o t h e s u p p o r t . To reach t h e lowest f r e q u e n c i e s , we equipped c a n t i l e v e r s w i t h end masses, a s o p e r a t e d a l s o by o t h e r Authors. Low-frequency v i b r a t i o n s were always e x c i t e d i n t h e h o r i z o n t a l p l a n e t o avoid t h e e f f e c t s of g r a v i t y , a s o u t l i n e d i n

[a].

The damping c o e f f i c i e n t Q-I was g e n e r a l l y measured from t h e decay of free v i b r a t i o n s . Frequencies i n v e s t i g a t e d by t h e w r i t e r s range from 0 , 3 Hz t o f a r beyond 1000 Hz and t h e corresponding phase v e l o c i t i e s of waves from a few meter-per-sec. t o c a .

1000 m/s. R e s u l t s from t e s t s on c a n t i l e v e r s of r e c t a n g u l a r c r o s s - s e c t i o n a r e i l l u - s t r a t e d on f i g . 1 t o g e t h e r w i t h r e s u l t s f r o m s u s p e n d e d beamsin t h e same frequency r a n ge. Damping c o e f f i c i e n t Q -1 i s p l o t t e d v s . frequency. A l l t e s t s were c a r r i e d o u t i n

vacuum (2.10 Torr) and f r e e v i b r a t i o n s i n t h e 1 s t mode were e x c i t e d through a i r i n s u f f l a t i o n . The displacementwas measured by means of a c o n t a c t l e s s e<

dy-currents t r a n s d u c e r . There i s e v i - dence f o r c o n s t a n t Q-l between 0 , 5 a n d 2 Hz, a s measured on beams b o t h with and without end masses. On both s i d e s beyond t h i s range t h e r e i s an i n c r e a s e of damping: a t very low f r e q u e n c i e s t h e i n c r e a s e i s s u r e l y owed t o 2nd o r - d e r e f f e c t s r e v e a l e d by t h e appearance of t o r s i o n a l motions.

The i n c r e a s e of (2-I a t h i g h e r f r ? q u e n c i e s , must be a t t r i b u t e d , a t a l l evidence, t o r a d i a t i o n i n t o t h e suppor t i n g frame through t h e clamp. This i s Fig. 1. F l e x u r a l waves. R e s u l t s f r o m t h e w r i demonstrated by t h e r e s u l t s from su- t e r s ' (blank symbols) and from o t h e r A u t h o r s l spended beams, t h a t d i s p o s e themselves ( f u l l symbols) t e s t s . on t h e lowest v a l u e s measured on c a n t i

l e v e r s , without showing any f u r t h e r i~

c r e a s e . R a d i a t i o n damping of c a n t i l e v e r through t h e s u p p o r t h a s been c a l c u l a t e d by t h e w r i t e r s and may be expressed by t h e formula:

where c p i s t h e d i l a t i o n a l wave v e l o c i t y , c t h e b a r v e l o c i t y , 6 i s a modal c o e f f i - c i e n t of t h e o r d e r of u n i t y , b and h a r e r e s p e c t i v e l y t h e c r o s s - s e c t i o n a l width and d e p t h , w t h e a n g u l a r frequency, a a c o n s t a n t depending on t h e n a t u r e of t h e

ra

d i a t i n g medium. For r a d i a t i o n i n an unbounded h a l f - s p a c e , cx = 2,5. This v a l u e d e c r e a s e s , however, i f r e f l e c t i o n s of t h e i n c i d e n t waves t a k e p l a c e , a s i n r e a l i t y cto i n bounded media. I n o u r t e s t i n g c o n d i t i o n s we have e s t i m a t e d , from an a n a l y s i s of t h e p r e s e n t and o t h e r unpublished r e s u l t s (on d i f f e r e n t m a t e r i a l s ) , a = 1.14. I t i s seen t h a t t h e r a d i a t i o n damping i s a f u n c t i o n of t h e specimen s i z e ( n o t e , however, t h a t a may b e lower than I ) , and i n c r e a s e s with frequency. Accordingly, t h e w r i t e r s sug-

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C9-450 JOURNAL

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g e s t t h a t measurements of damping from c a n t i l e v e r t e s t s should b e a c c e p t e d only wi$h

- A

specimen s i z e s and f r e q u e n c i e s such a s t o guarantee a r e a s o n a b l e l i m i t a t i o n of A Q

-

¹ r a d

'

say A grad

<

5. lo-'.

While t e s t i n g h i g h e r f r e q u e n c i e s , t h e w r i t e r s have given up t h e c a n t i l e v e r tp pe and adopted t h e f r e e - f r e e beam type v i b r a t i n g i n t h e v e r t i c a l p l a n e . S e v e r a l devL c e s f o r suspension of specimens were examined: t h e b e s t c h o i c e seemed t o c o n s i s t i n t h i n (0

<

0,4 mm) s t e e l wires upon which t h e specimens should l i e h o r i z o n t a l l y . The e x c i t a t i o n was provided i n t h i s c a s e by an e l e c t r o m a g n e t i c e x c i t e r , a c t i n g on a s m a l l s h e e t of magnetic materiaL glued on t h e lower s i d e of t h e specimen, c l o s e l y t o one end where t h e c u r v a t u r e i s p r a c t i c a l l y n e g l i g i b l e . T e s t i n g i n vacuum turned o u t t o be unnecessary, except i n some few c a s e s a t t h e low f r e q u e n c i e s . The s i z e of s p e c i - mens ranged 2

-

100 mm i n depth, 500

-

6000 mm i n l e n g h t , w i t h a c o n s t a n t width of 20 mm. Specimens were c u t from extruded b a r s of commercial p r o d u c t i o n , without any f u r t h e r thermal and mechanical c o n d i t i o n i n g , and without any c o n t r o l of t h e metal- l u r g i c a l p r o p e r t i e s . The h e t e r o g e n e i t y of t h e specimen m a t e r i a l was probably r e s p o l s i b l e , a t l a r g e e x t e n t , f o r t h e s c a t t e r i n g of r e s u l t s . The r e s u l t s a r e p l o t t e d vs.

frequency i n t h e diagram of f i g . 1 , where a comparison with t h o s e by o t h e r Authors i s p o s s i b l e , a n d s u c c e s s i v e l y a l s o i n t h e diagram of f i g . 2 , where they a r e confron- ted with Zener's p r e d i c t i o n s .

Damping should be con- s i d e r e d a s a c o n s t a n t u n d e r about 8 H z , i n f u l l a g r e e - ment w i t h t h e lowest r e s u l

t s from c a n t i l e v e r s . Suc- c e s s i v e l y , a d e s c e n t w i t h c o n s t a n t s l o p e ( i n bi-lo- g a r i t h m i c p l o t ) of ca.-T/4 s e t s o u t . A comparisonwith r e s u l t s from o t h e r e x p e r i - menters l e a d s t o t h e £0112 wing c o m e n t s :

i . GRANICK & STERN measure a damping d e c r e a s i n g w i t h i n c r e a s i n g frequency a t a c o n s t a n t s l o p e of - n / 4 , b ~ t t h e i r v a l u e s a r e h i g h e r by some 20 times than t h e o n e s o b t a i n e d by t h e w r i t e r s . Our i n t e r p r e t a t i o n i s t h a t F i g . 2. R e s u l t s from f l e x u r a l t e s t s (modes I and 111) i n such r e s u l t s a r e , a t a l a r comparison with p r e d i c t i o n s of t h e r m o e l a s t i c t h e o r y . ge e x t e n t , a f f e c t e d by

energy r a d i a t i o n through t h e s u p p o r t i n g d e v i c e t h a t , i n t h e p r e s e n t c a s e , i s a l s o t h e e x c i t i n g system. The

ra

t e of d i s p e r s e d energy, however, does n o t appear t o depend on frequency a s p r e d i c t e d by eq. (41, p r o b a b i l y because t h e mechanical c o u p l i n g of t h e specimen i s n o t w i t h a r a d i a t i n g medium, b u t w i t h an o s c i l l a t o r endowed with a mass and s t i f f n e s s of i t s own' ( a = 0 i n t h i s l i m i t i n g c a s e ) .

ii. BENNEWITZ & ROTGER's r e s u l t s 191 f o l l o w t h e curve of our t e s t s on c a n t i l e v e r s f a i r l y w e l l , although t h e s m a l l e r width of specimens determines, a s p r e d i c t e d by eq. (41, a smoother r i s i n g . Successively a maximum i s reached and a descending b r a n ch i s t r a c e d with f a r h i g h e r v a l u e s than measured by Granick & S t e r n (about 5 t i m e s ) . iii. I n s p i t e of t h e l a r g e s i z e of t h e specimens (15 mu t h i c k ) , daka worked o u t by t h e w r i t e r s from LEPORATI's [ l o ] o r i g i n a l t e s t s on c a n t i l e v e r s i n a i r , d i s p o s e them

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s e l v e s on a c u r v e f o l l o w i n g t h e one by Bennewitz & Riitger, but w i t h t w i c e a s high v a l u e s

.

Although t e s t i n g on beams suspended a t t h e nodal p o i n t s has proved more r e l i a - b l e than t e s t i n g on c a n t i l e v e r s , t h i s does n o t prevent t h a t measurements may b e a f f e c t e d by o t h e r s o u r c e s of e x t e r n a l d i s s i p a t i o n , because some i n f l u e n c e of t h e suspen- s i o n i s always t o be expected. Previous experiments [ I l l , c a r r i e d o u t by t h e w r i t e r s on approximately t h e same specimens a s p r e s e n t l y , b u t suspended a t t e x t i l e t h r e a d s , e x h i b i t r e s u l t s of t h e same behavior b u t s h i f t e d toward h i g h e r v a l u e s of damping. I n a d d i t i o n , e f f e c t s of t h e mechanical c o u p l i n g of specimen t o suspensions show u p w i t h d e c r e a s i n g wave-lenghts. The most important of such e f f e c t s seem t o be: e x c i t a t i o n of t h e suspensions due t o e r r o r i n p o s i t i o n i n g w i t h r e s p e c t t o nodes, swinging o f t h e w i r e s due t o motion of t h e suspension p o i n t s i n t h e l o n g i t u d i n a l plane; s t r a i n i n g of t h e suspension induced by l a t e r a l deformation o t h e beam c r o s s s e c t i o n . The t h r e e a s sumed e f f e c t s of coupling produce an increment of damping t h a t t h e w r i t e r s have c a l - c u l a t e d t h e o r e t i c a l l y a s

where 'Q, i s t h e proper damping c o e f f i c i e n t of t h e suspensions, t h e t h r e e terms i n s i de t h e b r a c k e t s r e p r e s e n t i n g each a s i n g l e e f f e c t of d i s s i p a t i o n , i n t h e o r d e r they a r e mentioned above; E i s t h e d i s t a n c e (along t h e a x i s ) of suspension f r o m n o d a l p o i n t , 6 i s t h e v e r t i c a l misalignment from a x i s , 9 t h e a n g l e of w i r e s t o t h e h o r i z o n ;

k,

i s wave number ( f o r t h e n-th mode)

$A

^and^$^: a r e t h e d e r i v a t i v e s of t h e wave shape e v a l u a t e d a t t h e nodal p o i n t s , wo = ( ~ K / M ) $ i s t h e n a t u r a l a n g u l a r frequency of t h e s i n g l e suspension, being K t h e e x t e n s i o n a l s t i f f n e s s and M t h e mass of t h e specimen;

g i s t h e g r a v i t y a c c e l e r a t i o n and t h e Poisson r a t i o . Note t h a t t h e q u a n t i t y w i t h i n t h e b r a c k e t s i s independent of frequency, whereas Qilmay b e supposed t o behave pro- p o r t i o n a l l y t o t h e specimen damping. I t i s very d i f f l i c u l t t o s e v e r experimentally one e f f e c t from t h e o t h e r , o r even t o t h i n k of e l i m i n a t i n g one of them without incremen- t i n g t h e o t h e r s . A favored c a s e should occur w i t h very l i g h t specimens layed on pul- l e d w i r e s ; then s i n 8 ~ 0 , and uo i s very l a r g e , what might c o n s i d e r a b l y reduce t h e i~

p o r t a n c e of t h e l a s t two e f f e c t s r e p r e s e n t e d i n eq. ( 5 ) .

As p r e d i c t e d by eq. ( 5 ) , d i s s i p a t i o n i n t h e suspension i n c r e a s e s with d e c r e a s i n g wave l e n g h t . The f a c t y a s a s c e r t a i n e d experimerk t a l l y s i n c e , by t e s t i n g on s h o r t beams (9.<1200 mrn i n our p r e s e n t t e s t c o n d i t i o n s ) o r e x c i t i n g h i g h e r modes even on long specimens, t h e r e s u l t s f a i l t o e n t e r t h e band of f i g . 2 , b u t tend t o i~

c r e a s e s t e e p l y i n v e r s e l y t o t h e wave l e n g t h . T h i s behaviour i s i l l u s t r a t e d i n f i g . 3 , where damping i s p l o t t e d a g a i n s t wave l e n g t h f o r two d i f f e r e n t wave v e l o c i t i e s . From t h e s e c t i o n s t o f o l l o w t h e reasons f o r i n t r o d u c i n g t h i s new p a r a meter w i l l c l e a r l y appear. Therefore we have de- c i d e d t o drop a l l r e s u l t s corresponding t o wave l e n g t h s lower than 1.5 m. I n s p i t e of h i g h e r mo- d e s being unfavoured, some measurements on 3rd modes could be executed on long specimens,with

c r 230 m/sec s a t i s f a c t o r y r e s u l t s .

WLVEIENG!H

-t---t ₁ ₂ +--+-- -

-

3 4 m S

Fig. 3. E f f e c t of wave l e n g t h on dam ping f o r two d i f f e r e n t wave v e l o c i t i e s .

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C9-452 JOURNAL

DE

PHYSIQUE

4 . T o r s i o n a l wave: t e s t r e s u l t s

The w r i t e r s have n o t s o f a r t e s t e d on t o r s i o n a l waves. R e l i a b l e r e s u l t s i n t h i s f i e l d a r e however a t hand from an a c c u r a t e a n a l y s i s by FROMMER & MURRAY ₁₁₂₁

.

The specimens, 100 rmn i n diameter and 900 mm i n l e n g t h , were suspended a t small s i z e w i r e s through a d e v i c e a l l o w i n g a x i a l r o t a t i o n s . Frequencies i n t h e range 800- 12000 H z were t e s t e d , e x c i t i n g harmonics from t h e 1 s t up t o t h e 10th. The r e s u l t was a damping c o n s t a n t w i t h frequency, amounting t o 3 , 7 * 1 0 - ~ t o w i t h i n 3%, b u t t h i s was m i s i n t e r p r e t e d a s t h e only v a l u e of m a t e r i a l damping.

5. L o n g i t u d i n a l waves: t e s t r e s u l t s

T e s t s were c a r r i e d o u t by t h e w r i t e r s on specimens 20x20 m i n s i z e w i t h l e n g t h s ranging 1000-6665 mm suspended a t a V-shaped t h i n w i r e w i t h t h e a x i s on t h e v e r t i c a l l i n e , i n a i r , and i n vacuum up t o 2500 mm. Measurements i n a i r covered a wider f r e - quency range, from c a . 380 t o c a . 11000 Hz, w i t h e x c i t a t i o n of s u c c e s s i v e harmonics.

Some of t h e d a t a a r e given i n t a b . I . The behavior of damping v s . frequency, s i m i l a r l y t o t h e c a s e of c a n t i l e v e r s i n f l e x u r a l t e s t s , shows a tendency t o c o n s t a n c e i n t h e f i e l d of i n t e r m e d i a t e f r e q u e n c i e s , w i t h a s l i g h t increment toward low f r e q u e n c i -

e s and a marked r i s i n g i n t h e f i e l d TAB. I

-

R e s u l t s from l o n g i t u d i n a l t e s t s i n

of very h i g h f r e q u e n c i e s . Both such a i r (Specimen s i z e : 20x20~6665 m ) .

increments a r e i n t e r p r e t e d by t h e w r i H a r m o n i c

number

1 3 5 7 9 1 1 16 17 18 19 2 8

t e r s a s t h e e f f e c t s of e x t e r n a l i n f l u ences. I n t h e c a s e of lower f r e q u e n c i e s , always t o be e x c i t e d on t h e f i r s t few harmonics, an i n f l u e n c e of anoma- l o u s wave r e f l e c t i o n s a t t h e end f a - c e s of t h e specimen h a s been suspec- t e d . T h i s e f f e c t should be i n v e r s e l y p r o p o r t i o n a l t o t h e number of e x c i t e d harmonic, s o i t i s expected t o d i s a p - p e a r w i t h i n c r e a s i n g t h i s number be- yond say 2 o r 3 . The i n f l u e n c e of su- spensions shows up with d e c r e a s i n g wa ve l e n g t h s and i s r u l e d by t h e same e f f e c t s a s i n t h e c a s e of f l e x u r a l waves, a p a r t from swinging of t h e b a r , t h a t i s avoided thanks t o i t s v e r t i c a l p o s i t i o n . The e f f e c t of t e s t i n g i n vacuum ap- p e a r s , i n t h e s e measurements, t o b e s i g n i f i c a n t . The lowest v a l u e s o f a r o b t a i n e d amounts t o Q-I = 4 , 9 ~ 1 0 - ~ , b u t t h e r e a r e reasons t o b e l i e v e t h a t such v a l u e i s s t i l l a f f e c t e d by e x t e r n a l i n f l u e n c e s and t h a t i t could f u r t h e r l y be lowered by improving t e s t i n g t h e technique.

6. Comparison of t e s t r e s u l t s w i t h t h e t h e o r y and c o n c l u s i o n s

A comparison of t e s t r e s u l t s w i t h t h e p r e d i c t i o n s of Zener's theory l e a d s t o t h e f o l l o w i n g c o n c l u s i o n s :

i. t h e r e i s no evidence f o r low v a l u e s of damping a s p r e d i c t e d f o r v e r y low frequen- c i e s ( f i g . 2)

i . i . n o p e a k i s t o be found i n t h e whole range of frequency both f o r f l e x u r a l and

102

g i t u d i n a l waves ( f i g . 2 and t a b . I)

i.i.i. t h e r e i s no evidence f o r a dependence of damping on the-specimen t h i c k n e s s i n bending. A l l our r e s u l t s o b t a i n e d from specimens of exceedingly d i f f e r e n t depths a r e grouped i n a r a t h e r narrow band, f o l l o w i n g a s t r a i g h t l i n e sloped ca.

-

^IT

.

S c a t t e - r i n g t h e r e i n amounts t o some few u n i t s ,

i6

c o n t r a s t w i t h t h e t h e o r y , p r e j i c t i n g va- l u e s almost i n t h e e n t i r e l o g a r i t h m i c f i e l d ( f i g . 2 ) .

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Wave l e n g t h Frequency ( H z )

D a m p i n g 0-I

'

1 6 . 2 ~ 1 0 - ~ 12.8 10.8 10.5 9.2 12.4 1 5 . 8 16.6 1 7 . 0 2 0 . 8 5 6 . 0

-

1 3 . 3 4.4 2.7 1.9 1.5 1.2 0.78 0.74 0.70 0.67 0 . 4 6

377 1131 1882 2632 3393 4 1 3 6 6391 6 7 6 8 7143 7518 10897

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I n t h e b e l i e f t h a t a s i n g l e key s h o u l d b e found f o r t h e i n t e r p r e t a t i o n of a l l d a t a i n view of a s i n g l e p h y s i c a l e x p l a n a t i o n , t h e w r i t e s h a d a l r e a d y advanced t h e i d e a o f c o r r e l a t i n g m a t e r i a l damping t o t h e phase v e l o c i t y of waves 1111, [I31

.

The n e w l y o b t a i n e d d a t a , a s b r o u g h t i n t o a diagram of damping v s . phase v e l o c i t y ( f i g . 4 ) , show

a b e h a v i o r q u i t e s i m i l a r t o t h a t a l -

-?'-

^---? -

- ----

I

r e a d y o b t a i n e d v s . f r e q u e n c y i n f l e - x u r a l t e s t s . The a l i g n m e n t of d a t a i s t h e r e w i t h improved, a n d , i n a d d i - i i o n , i t i s p o s s i b l e t o accomodate i n t o t h e s l o p e d band a l s o t h e r e s u l - t s c o n c e r n i n g t o r s i o n a l and l o n g i t u - d i n a l waves ( c s = 3050 m/s and co=

5000 m / s ) . The most s t r i k i n g a s p e c t of t h i s new p h i l o s o p h y , t h e r e f o r e , c o n s i s t s i n r e g a r d i n g damping pheno mena a c c o r d i n g t o a s i n g l e u n i t a r y view. T h i s o c c u r s , i n t h e w r i t e r s '

p r e t a t i o n s t o b e assumed f o r each k i n d of waves. Should t h i s i d e a b e F i g . 4 . P l o t t i n g a l l t e s t r e s u l t s v s . wave extended t o a l l m e t a l l i c m a t e r i a l s ,

v e l o c i t y . s o d e f i n i n g damping a s a f u n c t i o n of

wave v e l o c i t y a l o n e , t h e c o n c e p t of i n t e r n a l f r i c t i o n would b e c a r r i e d b a c k t o a m a t e r i a l p r o p e r t y , i n s t e a d of a s i m p l e

m e

c h a n i c a l e f f e c t dependent a s w e l l on s t r u c t u r a l p a r a m e t e r s a s on t h e k i n d of e x c i t e d waves. On o t h e r hand, t h e dependence of damping on t h e wave v e l o c i t y , seems t o i n d i c a t e a mechanical i n t e r a c t i o n of sound-waves w i t h t h e c o n s t i t u t i v e s t r u c t u r e of t h e ma- t e r i a l , and p a r t i c u l a r l y w i t h d i s l o c a t i o n s and o t h e r i n h o m o g e n e i t i e s . I n a p r e v i o u s work (op. c i t . r13]) t h e w r i t e r s have s u g g e s t e d t h a t d i f f u s i o n of waves t h r o u g h mobi- l e d i s l o c a t i o n s c o u l d b e r e s p o n s i b l e f o r energy a b s o r p i o n a t a r a t e i n v e r s e l y propor- t i o n a l t o t h e p h a s e v e l o c i t y . D i f f u s i o n a g a i n s t o b s t a c l e s of d i f f e r e n t k i n d , such a s g r a i n b o u n d a r i e s and i n t e r s t i t i a l s c o u l d b e e q u a l l y w e l l s u s p e c t e d . Note h o w e v e r t h a t , i n c o r r e s p o n d e n c e w i t h a v e l o c i t y of c a . 10 m/s damping a b r u p t e d l y becomes a c o n s t a n t , s o t h a t t h e p r e v i o u s l y e n v i s a g e d e f f e c t , should d i s a p p e a r and some o t h e r mechanism s e t o u t .

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[I] ZENER C., Phys. Rev.,

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( 1 9 3 7 ) 230.

[ 2 ] ZENER C., P h y s . Rev.,

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( 1 9 3 7 ) 90.

1 3 1 ZENER C., O T I S W. E R. NUCKOLLS, P h y s . Rev.,

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5 3 , ( 1 9 3 8 ) 1 0 0 . 1 4 1 RANDALL R.H., ROSE F.C. & C . ZENER, Phys. Rev.,

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1 5 1 ZENER C., E l a s t i c i t y and A n e l a s t i c i t y o f M e t a l s , U n i v . C h i c a g o P r e s s , C h i c a g o ( I l l ) ( 1 9 4 8 ) .

[ 6 ] KIRCHHOFF G., Pogg. Ann., 1 3 4 , ( 1 8 6 8 )

177.

Q u o t e d b y RAYLEIGH J.W., The T h e o r y o f Sound, L o n d o n ( 1 8 9 4 - 9 6 ) 348.

[ 7 ] GRANICK N. & J.E. STERN, N a s a TN-02893 ( 1 9 6 5 ) .

[ 8 ] CAPECCHI A., CAPURRO M., CONTI G. & A. TAFANELLI, I X Conv. Naz. AIAS, T r i e s t e ( 1 9 8 1 ) 545.

1 9 1 BENNEWITZ K . E H. RBTGER, Phys. Z e i t s c h r . ,

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( 1 9 3 6 ) 578.

1 1 0 1 LEPORATI E., C o s t r u z i o n i M e t a l l i c h e ,

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( 1 9 7 0 ) .

[ll] CAPECCHI A., CAPURRO M. & A. TAFANELLI, X Conv. Naz. AIAS, C o s e n z a ( 1 9 8 2 ) 4 2 5 . 1 1 2 1 FROMMER L . & A. MURRAY, J. o f t h e I r e s t o f M e t a l s ,

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1 1 3 1 CAPECCHI A., CAPURRO M. E A. TAFANELLI, V I C o n g r . Naz. AIMETA, Genova,

1,

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