Experimental Evidence against Zener’s Theory : on the Possibility of Re-interpreting Damping as a Material Property

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 longitudinal 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 importance 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 characterizing internal friction of some metals commonly used in engineering practice. Preliminarly, tests have been carried out at room temperature with very small amplitudes 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 C9-448 JOURNAL DE PHYSIQUE accordingly, a t r i a l of t e s t r e s u l t s , p a r t l y obtained 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 thehope of a r r i v i n g a t some c e r t a i n conclusions concerning t h e i r correspondence with the predict ions of Zener's theory. 2 . Predict ions of Zener's theory ZENER [ 1 1 , [ 21, [ 3 1 , [ 4 1 , [ 51 , has in te rpre ted 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 fus ion . A s imi la r concept was previously app l ied t o the absorpt ion of sound waves i n f l u i d s [ 6 ] . A first-approach solut ion of the thermo-elastic problem i n the case of f lexur a l waves i n pr ismatic rods was worked out (op. c i t . 1211, leading t o est imate f o r t h e damping capaci ty; AE i s a f a c t o r depending on the physical p roper t i es of the mater ial ( A E = 0 ,0046 i s suggestedforAluminium i n op. c i t . [ 5 ] p. 9 0 ) ; v i s the wave frequency and being s the specimen's thickness and D the c o e f f i c i e n t of d i f fus ion (estimated as 0.88 sqcm/s f o r A 1 i n op. c i t , [ S ] ) , i s the inverse of the re laxa t ion time and coin tides with the peak frequency; ~ r / 2 i s the c o e f f i c i e n t v a l i d f o r rectangular c ross se ct ions . Eq. (1 ) represents , i n the p l a n e ~ l v s . v a family of bell-shaped curves, one f o r each thickness , with t h e maxim hor izon ta l ly al igned t o t h e value AE/2. In a bi-logarithmic p l o t , we have a char t a s t raced 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 respect ively 1 and ! L . 4 4 For longi tudinal v ibra t ions , Zener had t o modify the theory, s ince the macrosco pica1 s t r e s s gradient i n the d i r e c t i o n of the wave would requ i re exceedingly l a r g e relaxat ion times with too low values of damping. The theory of in te rgra in thermal d i f fus ion was then worked out , making e l a s t i c anisotropy responsible f o r in te rgra in s t r e s s gradients , which should cause loca l thermal cur ren ts t o flow across g ra in boundaries with re laxa t ion t i k e s of the order of D over the squared gra in s ize . Again a bell-shaped curve f o r Q-l i s predicted, with a maximum a t with d the mean diameter of c r ~ s t a l l i t e s , independently of the specimen's s i z e (op .c i t . [5Ip.92). In the w r i t e r s ' knowledge, the case of to rs iona l waves has no t been examined by Zener and, s ince the volume s t r a i n i s zero and thennoelast ic e f f e c t s could no 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 physical property of the mate r ia l , depends on the type of exci ted waves and on the s i z e of specimen. According t o t h i s view, it would be impossible t o spec i f y howsoever a value of damping f o r a given mate r ia l . Concerning t h e shape of the curves damping vs. frequency i n both f l e x u r a l and longi tudinal waves, a typ ica l predict ion of Zener's theory cons i s t s i n the presence of maxima. The curves f o r f l e x u r a l waves and the locat ions of maxima, s h i f t markedly toward the domain of higher frequencies with decreasing the specimen depth. For longi tudinal waves we have, a t a given gra in s i z e , a s i n g l e curve with a s ing le peak given by eq. ( 3 ) . 3 . Flexural waves: t e s t r e s u l t s S t ruc tura l types so f a r inves t iga ted by the w r i t e r s were s t r a i g h t bars , e i t h e r clamped a t one end o r suspended a t the nodal po in t s of t h e wave-shapes f o r the free-. f r e e beam. Most of Authors have employed e i t h e r of these types. GRANICK & STERN [7] have exci ted i n vacuum beams clamped t o t h e moving head of a shaker, a t midspan. This d i spos i t ion , equivalent t o a couple of tuned c a n t i l e v e r s , conveys but shear f o r ces t o the support. To reach the lowest f requencies , we equipped c a n t i l e v e r s with end masses, a s operated a l s o by o ther Authors. Low-frequency v ibra t ions were always exci ted i n the hor izon ta l plane t o avoid t h e e f f e c t s of g rav i ty , a s out l ined i n [a] . The damping c o e f f i c i e n t Q-I was general ly measured from the decay of free vibra t ions . Frequencies invest igated by the w r i t e r s range from 0 ,3 Hz t o f a r beyond 1000 Hz and the corresponding phase v e l o c i t i e s of waves from a few meter-per-sec. t o ca. 1000 m/s. Resul ts from t e s t s on c a n t i l e v e r s of rectangular cross-section a r e i l l u s t r a t e d on f i g . 1 together with resul tsfromsuspended beamsin the same frequency r a n -1 ge. Damping c o e f f i c i e n t Q i s p lo t ted vs . frequency. A l l t e s t s were c a r r i e d out i n vacuum (2.10 Torr) and f r e e v i b r a t i o n s i n the 1 s t mode were exci ted 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 transducer. There i s evidence f o r constant Q-l between 0,5and 2 Hz, a s measured on beams both with and without end masses. On both s i d e s beyond t h i s range there i s an increase of damping: a t very low frequencies the increase i s surely owed t o 2nd order e f f e c t s revealed by t h e appearance of to rs iona l motions. The increase of (2-I a t higher f r? quencies, must be a t t r i b u t e d , a t a l l evidence, t o rad ia t ion i n t o the suppor t i n g frame through the clamp. This is Fig. 1. Flexural waves. Resul ts f r o m t h e w r i demonstrated by t h e r e s u l t s from sut e r s ' (blank symbols) and from otherAuthorsl spended beams, t h a t dispose themselves ( f u l l symbols) t e s t s . on the lowest values measured on c a n t i l evers , without showing any f u r t h e r i~ crease. Radiation damping of c a n t i l e v e r through t h e support has been ca lcu la ted by the w r i t e r s and may be expressed by the formula: where cp i s the d i l a t i o n a l wave v e l o c i t y , c t h e bar ve loc i ty , 6 i s a modal coef f i c i e n t of the order of un i ty , b and h a r e respect ively the cross-sect ional width and depth, w t h e angular frequency, a a constant depending on t h e na ture of t h e ra d i a t i n g medium. For rad ia t ion i n an unbounded half-space, cx = 2,5. This value decre a s e s , however, i f r e f l e c t i o n s of the incident waves take place, a s i n r e a l i t y cto i n bounded media. In our t e s t i n g condit ions we have estimated, from an ana lys i s of the present and o ther unpublished r e s u l t s (on d i f f e r e n t mate r ia l s ) , a = 1.14. It i s seen t h a t the rad ia t ion damping i s a funct ion of the specimen s i z e (no te , however, t h a t a may be lower than I ) , and increases with frequency. Accordingly, the w r i t e r s sugC9-450 JOURNAL DE PHYSIQUE 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 be accepted only wi$h A specimen s i z e s and frequencies such a s t o guarantee a reasonable l i m i t a t i o n of A Q 1 rad ' say A grad < 5. lo-'. While t e s t i n g higher frequencies , the wr i te r s have given up t h e c a n t i l e v e r tp pe and adopted the f ree f ree beam type v i b r a t i n g i n the v e r t i c a l plane. Several devL ces f o r suspension of specimens were examined: t h e b e s t choice seemed t o cons 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 hor izon ta l ly . The e x c i t a t i o n was provided i n t h i s case by an electromagnetic e x c i t e r , a c t i n g on asmal l sheet of magnetic materiaL glued on t h e lower s ide of the specimen, c lose ly t o one end where the curvature i s p r a c t i c a l l y neg l ig ib le . Test ing i n vacuum turned out t o be unnecessary, except i n some few cases a t t h e low frequencies. The s i z e of specimens ranged 2 100 mm i n depth, 500 6000 mm i n lenght , with a constant width of 20 mm. Specimens were cu t from extruded bars of commercial production, without any f u r t h e r thermal and mechanical conditioning, and without any cont ro l of the metall u r g i c a l p roper t i es . The heterogeneity of the specimen mater ial was probably r e s p o l s i b l e , a t l a r g e ex ten t , f o r the 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 lo t ted vs. frequency i n t h e diagram of f i g . 1 , where a comparison with those by o ther Authors i s possible,and successively a l s o i n the diagram of f i g . 2 , where they a r e confronted with Zener's p red ic t ions . Damping should be cons idered a s a constantunder about 8 Hz , i n f u l l agreement with t h e lowest r e s u l ts from can t i l evers . Success ive ly , a descent with constant slope ( i n bi-logarithmic p l o t ) of ca.-T/4 s e t s out. A comparisonwith r e s u l t s from o ther experimenters leads t o the £0112 wing coments : i. GRANICK & STERN measure a damping decreasing with increasing frequency a t a constant s lope of n / 4 , b ~ t t h e i r values a r e higher by some 20 times than theones obtained 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 Fig. 2. Results 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 red ic t ions of thermoelastic theory. ge ex ten t , a f fec ted by energy rad ia t ion through the supporting device t h a t , i n the present case, i s a l s o the exc i t ing system. The ra t e of dispersed energy, however, does no t appear t o depend on frequency a s predicted by eq. (41, probabily because the mechanical coupling of t h e specimen i s not with a rad ia t ing medium, but with 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 imi t ing case) . ii. BENNEWITZ & ROTGER's r e s u l t s 191 follow t h e curve of our t e s t s on can t i l evers f a i r l y well , although the smaller width of specimens determines, a s predicted by eq. (41, a smoother r i s i n g . Successively a maximum i s reached and a descending bran ch i s t raced with f a r higher values than measured by Granick & Stern (about 5 times). iii. In s p i t e of the l a r g e s i z e of t h e specimens (15 mu t h i c k ) , daka worked out by the 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 can t i l evers i n a i r , dispose them selves on a curve following the one by Bennewitz & Riitger, but with twice a s high v a lues . Although t e s t i n g on beams suspended a t t h e nodal points 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 not prevent t h a t measurements may be a f f e c t e d by o ther sources of external d i ss ipa t ion , because some inf luence of the suspension i s always t o be expected. Previous experiments [ I l l , c a r r i e d out by the w r i t e r s on approximately the same specimens a s present ly, but suspended a t t e x t i l e threads, e x h i b i t r e s u l t s of the same behavior but s h i f t e d toward higher values of damping. In addi t ion, e f f e c t s of t h e mechanical coupling of specimen t o suspensions show upwith decreasing wave-lenghts. The most important of such e f f e c t s seem t o be: exc i ta t ion of the suspensions due t o e r r o r i n posi t ioning with respect to nodes, swinging o f t h e wires due t o motion of the suspension po in t s i n the longi tudinal plane; s t r a i n i n g of the suspension induced by l a t e r a l deformation o the beam cross sect ion. The three a s sumed e f f e c t s of coupling produce an increment of damping t h a t the w r i t e r s have calculated t h e o r e t i c a l l y a s where Q , ' i s the proper damping c o e f f i c i e n t of t h e suspensions, t h e th ree terms i n s i de t h e brackets represent ing each a s i n g l e e f f e c t of d i ss ipa t ion , i n t h e order they a r e mentioned above; E i s t h e d i s tance (along the a x i s ) of suspension fromnodal po i n t , 6 i s the v e r t i c a l misalignment from a x i s , 9 t h e angle of wires t o the horizon; k, i s wave number ( f o r the n-th mode) $A and $: a r e the der iva t ives of the wave shape evaluated a t t h e nodal po in t s , wo = ( ~ K / M ) $ i s the n a t u r a l angular frequency of the s i n g l e suspension, being K the extensional s t i f f n e s s and M the mass of the specimen; g i s the g rav i ty acce le ra t ion and t h e Poisson r a t i o . Note t h a t the quant i ty within the brackets i s independent of frequency, whereas Qilmay be supposed t o behave propor t iona l ly t o t h e specimen damping. It i s very d i f f l i cu l t t o sever experimentally one e f f e c t from the o ther , or even t o th ink of el iminat ing one of them without increment i n g the o thers . A favored case should occur with very l i g h t specimens layed on pulled wires; then s i n 8 ~ 0 , and uo i s very la rge , what might considerably reduce the i~ portance of the l a s t two e f f e c t s represented i n eq. (5 ) . As predicted by eq. (5 ) ,d i ss ipa t ion i n the suspension increases with decreasing wave lenght . The f a c t yas ascertained experimerk t a l l y s ince, by t e s t i n g on shor t beams (9.<1200 mrn i n our present t e s t condit ions) o r exc i t ing higher modes even on long specimens, the 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 , but tend t o i~ crease s teeply inversely t o t h e wave length. This 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 lo t ted aga ins t wave length 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 the sect ions t o follow the reasons f o r introducing t h i s new para meter w i l l c l e a r l y appear. Therefore we have decided t o drop a l l r e s u l t s corresponding t o wave lengths lower than 1.5 m. In s p i t e of higher modes 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. Effec t of wave length 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 . C9-452 JOURNAL DE PHYSIQUE 4 . Torsional wave: t e s t r e s u l t s The w r i t e r s have not so f a r t es ted on to rs iona l waves. Rel iable 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 accurate ana lys i s by FROMMER & MURRAY 1121 . The specimens, 100 rmn i n diameter and 900 mm i n length, were suspended a t small s i z e wires through a device allowing a x i a l r o t a t i o n s . Frequencies i n t h e range 80012000 H z were t e s t e d , exc i t ing harmonics from t h e 1 s t up t o the 10th. The r e s u l t was a damping constant with frequency, amounting t o 3 , 7 * 1 0 ~ t o within 3%, but t h i s was misinterpreted a s t h e only value of mater ial damping. 5. Longitudinal waves: t e s t r e s u l t s Tests were c a r r i e d out by the w r i t e r s on specimens 20x20 m i n s i z e with lengths ranging 1000-6665 mm suspended a t a V-shaped t h i n wire with the a x i s on the 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 ca. 380 t o ca. 11000 Hz, with exc i ta t ion of successive harmonics. Some of the data a r e given i n tab. I. The behavior of damping vs. frequency, s i m i l a r l y t o the case 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 constance i n the f i e l d of intermediate frequencies , with a s l i g h t increment toward low frequencie s and a marked r i s i n g i n the f i e l d TAB. I Results from longitudinal t e s t s i n of very high frequencies . Both such a i r (Specimen s ize : 20x20~6665 m). increments a r e in te rpre ted by t h e w r i