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FOURIER ANALYSIS OF HYSTERETIC DAMPING MECHANISMS
D. Beshers, V. Coronel
To cite this version:
D. Beshers, V. Coronel. FOURIER ANALYSIS OF HYSTERETIC DAMPING MECHANISMS.
Journal de Physique Colloques, 1985, 46 (C10), pp.C10-171-C10-173. �10.1051/jphyscol:19851039�.
�jpa-00225423�
JOURNAL DE PHYSIQUE
Colloque CIO, supplément au n012, Tome 46, décembre 1985 page C10-171
FOURIER ANALYSIS OF HYSTERETIC DAMPING MECHANISMS
D.N. BESHERS AND V.F. CORONEL
Henry Krumb School of Mines, Columbia University, New York.
NY 10027, U.S.A.
RésumC: Une analyse de F o u r i e r du modele d'arrachement s u i v a n t Granato- Lücke p r é d i t des amplitudes des harmoniques acoustiques q u i o n t des ampli- tudes comparable à c e l l e s observées dans l e f e r . Para1 lelement, 1 'analyse de F o u r i e r de 1
'
h y s t é r é s i s de Rayleigh sou1 i g n e l e s d i f f e r e n c e s e s s e n t i e l l e s e n t r e 1 es deux types de h y s t é r é s i S.Abstract: A F o u r i e r a n a l y s i s o f Granato-Lücke breakaway p r e d i c t s amplitudes o f a c o u s t i c harmonics t h a t a r e comparable i n order o f magnitude t o those observed i n i r o n . A s i m i l a r a n a l y s i s o f Rayleigh-law h y s t e r e s i s b r i n g s out t h e e s s e n t i a l d i f f e r e n c e s between these two examples o f h y s t e r e s i s .
1
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INTRODUCTIONRecent experimental r e s u l t s have shown t h a t a t s u f f i c i e n t l y h i g h s t r a i n amplitudes s u b s t a n t i a l generation o f acoustic harmonics may occur i n t h e kHz range o f f r e
-
quencies /1-3/. ThB harmonic spectrum depends on the s t r a i n amplitude E, i n a complex fashion, b u t t h e p a t t e r n o f v a r i a t i o n i s d i f f e r e n t from one m a t e r i a l t o another /3/.
The harmonics appear t o a r i s e from t h r e e p r i n c i p a l sources: l a t t i c e anharmonicity /4,5/, d i s l o c a t i o n motion /1,4/, and processes o f magnetization
/ 3 / .
L a t t i c e anhar- m o n i c i t y c o n t r i b u t e s c h i e f l y t o t h e second harmonic (wzth amplitude A2, and s i m i l a r l y f o r t h e others), and indeed dominates A2 u n t i l co: 10- o r h i q h e r /1,3,4/. The t h i r d , f o u r t h , and f i f t h harmonics a r e o f t e n r e a d i l y observable f o r co<lO-,
a r i s i n g c h i e f l y from t h e motion o f i m p e r f e c t i o n s . 0 b s e r v a t i o n s o f these spactra o f f e r us i n f o r m a t i o n t h a t complements the i n t e r n a 1 f r i c t i o n and modulus d e f e c t . Since t h e mechanisms o f generation a r e i n h e r e n t l y nonlinear, t h e i n f o r m a t i o n contained i n the harmonic amplitudes cannot be e x t r a c t e d by any l i n e a r i n v e r s i o n scheme. Rather, i t seems necessary t o proceed by p o s t u l a t i n g models, guided by experience, andeval u a t i n g t h e i r harmonic generati on numerically. An a n a l y t i c approach was taken by Hikata and Elbaum (HE) /4/, studying the v i b r a t i n g s t r i n g model i n f i r s t o r d e r p e r t u r b a t i o n theory. The nonl i n e a r elements i n t h e i r model are t h e dependence o f d i s l o c a t i o n s t r a i n ( t h e area swept by t h e d i s l o c a t i o n ) on t h e a p p l i e d s t r e s s and t h e dependence o f 1 in e t e n s i o n on d i s l o c a t i o n character, which changes as t h e d i s - l o c a t i o n s bow out. HE.found t h a t A n en, and a l s o showed phase d i f f e r e n c e s . Although HE worked o n l y t o f i r s t oraer, both t h e n o n l i n e a r e f f e c t s they considered p e r s i s t a t l a r g e r amplitudes. Numerical modeling using o n l y t h e macroscopic average d i s l o c a t i o n v e l o c i t y /1,6/ showed r i c h harmonic spectra b u t i n s u f f i c i e n t features t o f i t data i n d e t a i l . Next came a study o f two d i s l o c a t i o n models, t h e edge d i p o l e and a g l i d e l o o p s t a b i l i z e d by a center o f d i l a t a t i o n . / 7 / These models i n c l u d e d a L e i b f r i e d viscous damping term. When t h e o s c i l l a t i n g s t r e s s i s l a r g e enough t o per- m i t t h e d i s l o c a t i o n t o escape, temporarily, from i t s p o t e n t i a l w e l l , t h e r e a r e sharp drops i n t h e harmonic generation, as t h e l i n e a r damping term dominates; t h i s escape behavior o f f e r s a p o s s i b l e explanation o f some drops observed f o r A and Ag i n brass.
A t s t r e s s amplitudes below t h e escaoe l e v e l , these models f o l l o w e d anri cn, as f o r HE.
The most prominent aspect o f nonl i n e a r d i s l o c a t i o n damping i s Granato-Lücke (GL breakaway /8/. Here t h e d i s l o c a t i o n escapes a b r u p t l y from a p o t e n t i a l w e l l defined by p o i n t d e f e c t s a t t h e r e s t p o s i t i o n o f the d i s l o c a t i o n , and then moves i n a
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19851039
CIO-172 JOURNAL
DE
PHYSIQUEpotential well defined by t h e l i n e tension. Does breakaway c o n s t i t u t e an escape of the s o r t discussed in reference
7?As a step toward an answer, we have carried out a numerical analysis of harmonic generation i n the GL breakaway model , using the approximation of constant l i n e tension, in which dislocation segments a r e arcs of c i r c l e s , but we have omitted viscous damping. When there a r e only major, unbreakable pinning points, t h e dislocation bows out smoothly with increasing s t r e s s , until the Orowan i n s t a b i l i t y i s reached. Our analysis here i s r e s t r i c t e d t a s t r e s s e s below t h i s c r i t i c a l s t r e s s . The area A swept out by a dislocation with length L between t h e pinning points subject t o a resolved shear s t r e s s
Ti s
A =ST2 s i n - ' (Lbr/ZT) - &-- cos [ s i n - ' ( ~ b r / 2 ~ ) ] (1) where b i s t h e burgers vector, and T i s the l i n e tension. With
T =-cosin
wt,A was expressed a s a Fourier s e r i e s and t h e coefficients found numerically. Absent
v i s c o s i t y , the out-of -phase components a r e negligible. Only t h e odd harmonics appear, by symmetry. A3 varies as
~ 3r i g h t up t o t h e Orowan c r i t i c a l s t r e s s -cc,
0while A5 varies as
T ?u p t o 0 . 8 ~ ~ and then climbs a 1 i t t l e f a s t e r . A t
= T ~A3
,equals 0.21A and A, i s 0.085A1. The r u l e A seeem t o be well-obeyed, and t h e dislocation i s simpjy moving in a potential !$el1 defined by i t s l i n e tension. The GL breakaway damping i s a form of hysteresis. Using computer generated models of pinning, and A(T) given above, but neglecting v i s c o s i t y , we obtained t h e Fourier c o e f f i c i e n t s of typical breakaway s i t u a t i o n s . When several pinning points are present, t h e harmonics associated with A
( T )a r e s i g n i f i c a n t only f o r ~ 0 > 0 . 8 ~ . other- wise, breakaway predominates. There i s a rich spectrum (odd orders only) because of t h e discontinuity, and the A fa11 only slowly with increasing n , p a r t i c u l a r l y f o r
T O
j u s t above breakaway, a w&ll-known e f f e c t in Fourier analysis. The most s t r i king r e s u l t i s t h a t A i s as large as the out-of-phase component of the fundamental. Since the l a t t e r i s a Aeasure of the damping, we can Say t h a t A Q-'Al.
Aconparison with Our data f o r magnetically saturated iron shows t h a t Af ~$-'Q-'A f o r 2 x l O - ' < ~ ~
< 3 ~ 1 0 - ~ . The discrepancy may be largely accounted f o r by phase e i f e c t s , since t h e phase changes rapidly j u s t above breakaway, and changes sign a t somewhat higher values of
T O ,while t h e experimental values of A 3 are averaged over segments in d i f f e r i n g conditions and so emitting with d i f f e r ~ n g phases. Ue have n o t performed such an average however. HE found t h a t edges and screws emit t h i r d harmonic with opposite phase, and t h i s finding should p e r s i s t t o high amplitudes such as those studied here, so f u r t h e r cancellation i s t o be expected. Finally, the inclusion of v i s c o s i t y will c e r t a i n l y reduce t h e h y s t e r e t i c harmonics. In view of the many f a c t o r s neglected, we find i t encouraging t h a t our model gives r e s u l t s not incon- s i s t e n t with the data, q u i t e capable of explaining t h e magnitude of t h e observed harmonics.
I I I - MAGNETOMECHANICAL DAMPING
The well-known hysteresis of magnetic 6-H curves has an associated hysteresis in the damping. In the Rayleigh region t h e magnetic permeability increases l i n e a r l y with f i e l d , and t h e magnetomechanical damping increases l i n e a r l y with s t r a i n amplitude.
Kneller /8/ has given an analytical form f o r t h e B-H loop f o r t h i s case, with a Fourier analysis f o r cyclic v a r i a t i o n of
H.ive may take his formalism over comp- l e t e l y t o t h e magnetomechanical case, as Boser has done f o r dislocations /9/. The corresponding s t r a i n - s t r e s s law i s
Where again
T Oi s the amplitude of , t h e cyclic s t r e s s , and the minus sign holds when t h e s t r e s s i s r i s i n g , and t h e plus sign when the s t r e s s i s f a l l i n g . When
T =~ ~ s i n w t and
E = C(An cos
r u t t6, s i n
M W , ~ ) ,m
Kneller's resu t s are: An= (-1)P48rb / n h n (n2-4). where B
= ( a +B T O ) T O ; p
=( n - l ) / 2 ; and al 1 other c o e f f i c i e n t s are zero. f o r odd n ,
fhat i s , t h e s t r a i n harmonics (odd only) a r e always out of phase with t h e driving
s t r e s s . The r e l a t i v e l y rapid drop of A with n , the s t r e s s dependence A a
T,',and
the simple phase relationship distinguiDh the Rayleigh-law hysteresis sh$rply from
t h e GL h y s t e r e s i s . However, i t i s noteworthy t h a t f o r both types t h e F o u r i e r
c o e f f i c i e n t s a r e independent of frequency, as t h e y a r e i n reference 1, and should be f o r a h y s t e r e t i c mechanism. A comparison o f K n e l l e r l s spectrum w i t h observations i s n o t y e t p o s s i b l e because d i s l o c a t i o n c o n t r i b u t i o n s occur over t h e sarne range, and a successful s u b t r a c t i o n demands a knowledge o f b o t h amplitude and phase t h a t i s n o t y e t a v a i l a b l e . The standard p i c t u r e o f t h e Rayleigh r e g i o n i s t h a t Bloch w a l l s move through a randomly v a r y i n g f o r c e f i e l d , ( o r t h e e q u i v a l e n t p o t e n t i a l energy) making sudden h y s t e r e t i c jumps whenever t h e f o r c e from t h e a p p l i e d s t r e s s exceeds t h e l o c a l r e s i s t i n g f o r c e / I O / . F i g u r e 1 o f r e f e r e n c e 3 shows an example i n which a s i n g l e s t r a i g h t l i n e f i t s t h e magnetomechanical damping o v e r t h r e e orders o f magnitude i n
E O . We conclude t h a t t h e random p o t e n t i a l i s e q u a l l y random over a l 1 these scales, e q u a l l y rough on a l l . The motion o f t h e Bloch w a l l s must indeed c o n s i s t o f many small jumps t h a t average t o t h e simple form ( 2 ) . I n o t h e r words, many u n c o r r e l a t e d p o t e n t i a l w a l l s a r e t r a v e r s e d by t h e Bloch w a l l i n goi ng around a E - T loop.
I V
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SUMMARYF o u r i e r a n a l y s i s o f a simple model o f GL breakaway gives r e s u l t s t h a t a r e compatible, w i t h i n an o r d e r o f magnitude, w i t h experimental data. The e s s e n t i a l f e a t u r e s o f the GL model on t h i s viewpoint a r e the presence o f two c o r r e l a t e d p o t e n t i a l w e l l s , one formed by t h e i n t e r a c t i o n w i t h t h e weak p i n n i n g p o i n t s and t h e o t h e r by j o i n t a c t i o n o f t h e major ( o r nodal) p i n n i n g p o i n t s , t h e l i n e t e n s i o n and t h e a p p l i e d s t r e s s . I n c o n t r a s t , Bloch w a l l motion t r a v e r s e s many u n c o r r e l a t e d p o t e n t i a l w e l l s . These differences a r e r e f l e c t e d n o t o n l y i n t h e damping, b u t a l s o i n t h e harmonic spectrum.
The answer t o Our r h e t o r i c a l q u e s t i o n seems t o be, no, breakaway i s n o t an escape i n t h e sense o f Beshers and Oppenheim, b u t d i s t i n c t l y t h e opposite, a t l e a s t i n t h e present model: on escape t h e harmonics drop a b r u p t l y , on breakaway they r i s e abruptly.
The a d d i t i o n of viscous damping t o t h e present model may modify t h i s s i t u a t i o n , b u t w i l l probably n o t reverse t h e concl usion.
References:
/1/ Jon, M.C., Mason, W.P., and Beshers, D.N.,
J:
Appl. Phys. 49 (1978) 5871./2/ Beshers, D.
N.
and Coronel, V. F., " M i c r o s t r u c t u r a l C h a r a c t e x z a t i o n o f M a t e r i a l s by Non-Microscopical Techniques", Ed. Andersen, N. Hesse1 e t al., (Riso N a t i o n a l Laboratory, Roskilde, Denrnark, 205 (1984) ) ./3/ Coronel, V.F. and Beshers, D.N., t h i s conference.
/4/ Hi kata, A., and Elbaum, C., Phys. Rev.
144
(1966) 469./ 5 / Buck, O., and Thompson, D.O., Mater. Sci. Eng. 1 (1966) 117.
/ 6 / Beshers, D. N., Jon, M.C., Beshers, G.M., and Seth, V., " I n t e r n a 1 F r i c t i o n and U l t r a s o n i c A t t e n u a t i o n i n S o l i d s " , Ed. H a s i g u t i , R.R., and Mikoshiba, N., ( U n i v e r s i t y of Tokyo Press, Tokyo) 1977.
/7/ Beshers, D.N., and Oppenheim, Alan, 3 . Appl. Phys. 52 (1981) 6509.
/8/ Knel l e r , E., "Ferromagnetismus" ( S p r i nger, Heidel b e y ) 1962.
/9/ Boser, O., J. Appl
.
Phys. 54 (1983) 2338./ I O / Kronmüller, H., Angew. ph%.
