HYSTERETIC INTERNAL FRICTION

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HYSTERETIC INTERNAL FRICTION

J. Roberts, D. Barnett

To cite this version:

J. Roberts, D. Barnett. HYSTERETIC INTERNAL FRICTION. Journal de Physique Colloques,

1985, 46 (C10), pp.C10-199-C10-204. �10.1051/jphyscol:19851046�. �jpa-00225430�

HYSTERETIC INTERNAL FRICTION

J.M. ROBERTS AND D.M. BARNETT*

Department of Mechanical Engineering and Materials Science, EJilliam Marsh Rice University, Houston, Texas 77251, U.S,A.

Department of Materials Science and Engineering, Stanford University, Stanford, California,94305, U.S.A.

~ 6 s u m 6

-

q u i depend 2 l a f o i s des dimensions de l a boucle de d i s l o c a t i o n e t de l a r e s i s t a n c e de l ' o b s t a c l e a f i n de d6veloper une t h e 6 r i e q u a n t i t a t i v e pour l a d l s s i p a t i o n d ' e n e r g i e hyst'er&ique. Ceci se p r o d u i t l o r s q u e une d i s l o c a t i o n passe sur un champ i n t e r n e de c o n t r a i n t e s o s c i l l a t o i r e s e t l e de'tachement de l a d i s l o c a t i o n e s t contenu par une f o r c e de r e s t o r a t i o n que l ' o n suppose " e r e une l i g n e de t e n s i o n de d i s l o c a t i o n . Le f a i t que l a c o n t r a i n t e moyenne de f r o t t e m e n t q u i s'oppose au mouvement ge l a d i s l o c a t i o n depend de l a s t r u c t u r e des d i s l o c a t i o n s e t des i m p u r i t e s e s t mis en evidence dans l e cas du cuiure. 0! montre que l e f r o t t e m e n t i n t e r n e , qui depend de l'amplitudt?, e s t hyst%retique n n a t u r e da s l e cas du c u i u r e dans 1 'i n t e r v a l l e de decroissement 3 x 10'$ 1 2 x lo-' e t peut z t r e analyse par l e s t h e o r i e s de Kressel-Brown and Roberts pour f o u r n i r des paramstres r 6 a l i s t e s r e l a t i f s a l a s t r u c t u r e des d i s l o c a t i o n s .

Abstract

-

That i s when d i s l o c a t i o n surmounts an u n d u l a t i n g i n t e r n a l s t r e s s f i e l d and t h e break-away i s r e s t r i c t e d by a r e s t o r i n g f o r c e considered t o be t h a t o f d i s l o c a t i o n 1 in e tension. The s i g n i f i c a n c e o f t h e average f r i c t i o n s t r e s s r e s i s t i n g d i s l o c a t i o n motion being re1 ated t o t h e d i s l o c a t i o n s t r u c t u r e and i m p u r i t y e f f e c t s i n copper i s demonstrated. The a m p l i t de dependen i n t e r n a l f r i c t i o n i n copper i n t h e decrement range o f 3 X 10.' t o 2 X 10-

1

i s shown t o be h y s t e r e t i c i n nature and can be analyzed by t h e Kressel- Brown and Roberts t h e o r i e s y i e l d i n g r e a l i s t i c d i s l o c a t i o n s t r u c t u r a l parameters.

I

-

The well known low temperature mechanical h y s t e r e t i c break-away t h e o r y f o r d i s l o c a t i o n s t e a r i n g away from s o l u t e p i n n i n g p o i n t s by Granato and Liicke /1/ has been extended by the same authors r e c e n t l y t o t a k e i n t o account t h e r m a l l y assisted depinning e f f e c t s over an extended temperature range /2/. There are a s u b s t a n t i a l number o f i n t e r n a l f r i c t i o n observations which suggest t h e decrement i s independent o f temperature (except p o s s i b l y f o r t h e temperature dependence o f t h e e l a s t i c modulus), independent o f frequency except f o r t h e h i g h megacycle range t o t h e low giga c y c l e domain and e x h i b i t s amplitude dependence. Although t h e Granato-LBcke h y s t e r e t i c model apparently agreed w i t h some o f t h i s data, i t i s now r e a l i z e d i t d i d so f o r f o r t u i t i o u s b u t erroneous reasons /2/. The present authors b e l i e v e a t t h i s time t h e Kressel-Brown /3/ t h e o r y d e a l i n g w i t h t h e h y s t e r e t i c

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19851046

C10-200 JOURNAL DE PHYSIQUE

d i s s i p a t i o n o f energy by motion o f a d i s l o c a t i o n through a f l u c t u a t i n g i n t e r n a l s t r e s s f i e l d against a l i n e t e n s i o n o r l o n g range r e s t o r i n g f o r c e stands as t h e b a s i s f o r developing a s e m i - q u a n t i t a t i v e t o q u a n t i t a t i v e h y s t e r e t i c i n t e r n a l f r i c t i o n theory. I n e f f e c t , Roberts /4,5/ has considered t h e l o w frequency h y s t e r e t i c d i s l o c a t i o n damping asociated w i t h t h e 1 in e t e n s i o n 1 im i t e d energy d i s s i p a t i v e l o s s due t o the breakdown o f a t t r a c t i v e d i s l o c a t i o n j u n c t i o n s . The l a t t e r

is

-Y

t o e x p l a i n the s t r e s s amplitude dependent decrement ( i n t h e range 1 t o 6 X 10 ) data f o r n i c k e l . The Kressel-Brown t h e o r y presents an i n s i g h t i n t o t h e average f r i c t i o n s t r e s s a c t i n g upon d i s l o c a t i o n when t h e deformation i s i n t h e m i c r o s t r a i n domain.

I t t h e purpose o f t h e present paper t o ( a ) p o i n t - o u t t h e s i g n i f i c a n c e o f t h e Kressel-Brown 131 t h e o r y i n l a y i n g t h e groundwork t o understand d i s l o c a t i o n h y s t e r e s i s , ( b ) t o e x p l a i n t h e physics and more general a p p l i c a b i l i t y o f t h e Roberts theory /4,5/ t o understand h y s t e r e t i c d i s l o c a t i o n damping and apply t h i s t o some data we have taken on copper, and ( c ) t o show how both ( a ) and (b) l e n d i n s i g h t i n t o understanding t h e average f r i c t i o n s t r e s s a c t i n g upon d i s l o c a t i o n d u r i n g m i c r o s t r a i n deformation.

The Kressel-Brown /3/ theory p o i n t s out t h a t a l a r g e h y s t e r e t i c l o s s associated w i t h a closed s t r e s s c y c l e damping l o o p i s pos'sible when a d i s l o c a t i o n segment moves over a s i n s u s o i d a l l y v a r y i n g u n d u l a t i n g s t r e s s f i e l d o f amplitude ri

,

a g a i n s t a r e s t o r i n g f o r c e which s a t i s f y t h e r e l a t i o n [I]

1.41 <

5

1

( 1 )

where R i s the r e s t o r i n g f o r c e per u n i t l e n g t h per u n i t displacement a c t i n g upon d i s l o c a t i o n o f Burgers v e c t o r b, ^T. i s t h e shear s t r e s s amplitude o f a f l u c t u a t i n g i n t e r n a l s t r e s s f i e l d o f w a d l e n g t h A > b, which i s o f s h o r t e r range than t h a t o f R. Kressel and Brown go on t o show t h a t t h e energy l o s s A E per u n i t l e n g t h o f d i s l o c a t i o n t e a r i n g over a s t r e s s u n d u l a t i o n as per r e l a t i o n (1) i s given by

AE = bX6 r

i ^{( 2 )}

where 0 i s constant o f t h e order o f 0.21 f o r R A / r j b = n. When RA/rib = 2n, 6 = 0 and when RA/rib = 1.41 0 = 0.45 ( a maximum value). The present authors apply t h e Kressel-Brown/3/ r e l a t i o n ( 1 ) t o one elementary d i s l o c a t i o n model f o r t h e sake o f c l a r i f i c a t i o n and evidence o f a p p l i c a b i l i t y t o r e a l d i s l o c a t i o n c o n f i g u r a t i o n s . Consider a random network o f d i s l o c a t i o n dip01 es o f h e i g h t h and 2 average d i s t a n c e apart. T h i s d i s l o c a t i o n c o n f i g u r a t i o n has been discussed by F r i e d e l / 7 / and i s somewhat analogous t o t h e T a y l o r l a t t i c e discussed by L a i r d /8/ w i t h respect t o d i s l o c a t i o n concepts i n f a t i g u e . For t h i s model, t h e s t r e s s amplitude ri i s approximately given by

-1 .9 -1

&

^(F)

r e l a t i o n ( 1 ) i s v a l i d f o r 3($2 <

(e)

[I] I n c h e c k i n g t h e K r e s s e l - B r o w n / 3 / t h e o r y , t h e p r e s e n t a u t h o r s f i n d t h e l o w e r l i m i t i n r e l a t i o n ( 1 ) t o b e 1.33 n o t 1.41.

5.5($/2 <

$

It i s suggested here, t h a t t h e h y s t e r e t i c energy l o s s &Iirr, per d i s l o c a t i o n event surmounting a s t r e s s f i e l d o f amplitude ^T. under t h e i n f l u e n c e o f a constant r e s t o r i n g f o r c e d u r i n g a u n i d i r e c t i o n a l S t r e s s c y c l e i s c l e a r e r t o understand p h y s i c a l l y when w r i t t e n as:

I n t h i s r e l a t i o n , 44 i s t h e c a t a s t r o p h i c area swept out by t h e d i s l o c a t i o n segment a f t e r surmounting t h e s t r e s s h i l l o f amplitude r . . The constant, ~l

,

a measure o f both t h e maximum unbalanced s t r e s s wdich r e s u l t s a f t e r t h e d i s l o c a t i o n segment surmounts T and p r i o r t o becoming h a l t e d by t h e combined r e s t o r i n g f o r c e - f l u c t u a t i n g i n t e r n a l s t r e s s f i e l d , and t h e c a t a s t r o p h i c area swept i o u t by t h e d i s l o c a t i o n segment i n r e t u r n i n g towards i t s i n i t i a l c o n f i g u r a t i o n d u r i n g unloading. Thus, n measures t h e amount t h e unloading s t r e s s - d i s l o c a t i o n s t r a i n curve i s d i f f e r e n t from t h e l o a d i n g curve. When t h e maximum unbalanced s t r e s s i s zero, n = 0 and t h e c a t a s t r o p h i c recovered area = M. When t h e maximum unbalanced s t r e s s ^=T. and t h e c a t a s t r o p h i c recovered s t r a i n 10, then rl = 0.5 which correspondZ t o a Granato-LUcke t y p e o f break-away mechanism f o r unzipping o f t h e d i s l o c a t i o n double loop from a s i n g l e p i n n i n g p o i n t . This range o f ~l from 0 t o 0.5 corresponds t o the Kressel-Brown range o f

0 from 0 t o 0.45.

To complete a theory f o r d i s l o c a t i o n h y s t e r e s i s , one must consider a d i s t r i b u t i o n i n d i s l o c a t i o n loop l e n g t h s (Roberts /5/ employs a t r u n c a t e d Gaussian d i s t r i b u t i o n ) t e a r i n g over u n d u l a t i n g s t r e s s f i e l d s as we1 1 as a d i s t r i b u t i o n f u n c t i o n i n u n d u l a t i n g s t r e s s f i e l d amplitudes (Roberts /5/ considered a square w e l l d i s t r i b u t i o n i n one case). I n a d d i t i o n , Roberts /5/ p o i n t s out f o r l a r g e damping t h e decrement ( A ) must be defined as W i r r / W t o t a l wherein, M i r r i s t h e area enclosed i n the damping loop and Wtota i s t h e t o t a l area under t h e l o a d i n g p o r t i o n o f t h e s t r e s s - s t r a i n curve not merely t h e stored e l a s t i c s t r a i l energy.

I f t h i s i n not done, f o r l a r g e damping t h e decrement values could exceed u n i t y . The present authors wish t o p o i n t out t h a t t h e d e t a i l e d theory o f Roberts /4,5/

w i t h a p p l i c a t i o n t o n i c k e l /6/ i s s u f f i c i e n t l y general t h a t i t could apply t o a l l forms o f i n t e r n a l s t r e s s f i e l d s , ie., a t r e p u l s i v e as well as a t t r a c t i v e j u n c t i o n s f o r example. The only severe r e s t r i c t i o n o f t h e Roberts theory i s t h a t t h e bowing d i s l o c a t i o n surmounts o r t e a r s over only one i n t e r n a l s t r e s s f i e l d undulation, and does not break-away from i t s p i n n i n g points. That i s no unzipping o f t h e d i s l o c a t i o n network occurs. This i s reasonable since Roberts only considered d i s l o c a t i o n t e a r i n g over t h e weakest i n t e r n a l s t r e s s undulations w i t h t h e longest l e n g t h loops. Therefore, once one c a t a s t r o p h i c event has occurred, t h e p r o b a b i l i t y i t i s pinned by a s t r o n g j u n c t i o n o f s h o r t e r l e n g t h i s high. The d e t a i l e d p r e d i c t i o n s f o r t h i s h y s t e r e t i c amplitude dependent damping are r a t h e r involved, b u t i n t e r e s t e d readers can secure those d e t a i 1 s from r e f s . /4,5,6/.

I 1 1

-

C y l i n d r i c a l s i n g l e c r y s t a l s , 12.7 mn i n diameter and approximately 19 cm long were grown from t h e melt under vacuum from 99.999% pure ASARCO copper using a modified Bridgman technique. One c r y s t a l CM-AL was doped w i t h Al. Chemical a n a l y s i s revealed the undoped c r y s t a l s t o be 99.80 wt.% Cu and t h e doped c r y s t a l t o be 99.75 wt.% Cu w i t h 0.005 wt.% Al. Therefore, t h e A1 doped c r y s t a l had e s s e n t i a l l y 0.01 atomic % A1 introduced t o it. C r y s t a l CM-AL had i t s t e n s i l e a x i s

C10-202 JOURNAL DE PHYSIQUE

o r i e n t e d 26', 26' and 29.5' w i t h respect t o t h e [OOl], [ O l l ] and [Ill] d i r e c t i o n s r e s p e c t i v e l y . The undoped c r y s t a l s CM-1T and CM-2T were o r i e n t e d 26', 23', 30.5' and 28', 20°, and 31.5' w i t h respect t o t h e [OOl], [ O l l ] , and [lll] d i r e c t i o n s r e s p e c t i v e l y .

Closed u n i d i r e c t i o n a l s t r e s s c y c l e ' damping loops were measured on some o f t h e e

3

c r y s t a l s i n c l u d i n g t h e shear p l a s t i c p r e s t r a i n (yn)range o f t o 1.1 X 10'

,

t h e shear s t r e s s amplitude ( T ) range o f 0.4 t o 1.f MPa, t h e temperature range 135 t o 300°K and over t h e freque8cy range 1 t o 6 t i m e s 10" Hz. The d e t a i l s o f t h e m i c r o s t r a i n technique f o r measuring t h e damping loops and t h e i r subsequent a n a l y s i s are given by P i n a t t i and Roberts 161. Roberts 19,101 has made a d e t a i l e d d i s c u s s i o n o f how t w i c e t h e mean f r i c t i o n a l s t r e s s )f'i( as d e r i v e d from m i c r o s t r a i n s t u d i e s [ie., t h e slope o f t h e l o o p area, Wirr, versus t h e maximum l o o p width (W i s c o r r e l a t e d t o p r e c i s e d i s l o c a t i o n models. This w i l l be done i n t h i s i h @ % \ i a t i o n .

( T ~ ) i s d e f i n e d as t h e maximum s t r e s s amplitude t o which damping loops can be observed t o be closed. This s t r e s s l e v e l corresponds t o a m i c r o y i e l d s t r e s s and i n t h e Roberts theory 151 corresponds t o t h e lowest a p p l i e d s t r e s s t o a c t i v a t e Frank-Read sources f o r t h e loop-length and j u n c t i o n - b r e a k i n g s t r e n g t h d i s t r i b u t i o n s employed.

I V

-

Fig. 1 shows t h e decrement versus loop s t r e s s amplitude f o r copper c r y s t a l CM-IT a t d i f f e r e n t p r e s t r a i n s ( y ) a t 300°K. It i s observed t h e decrement i s s t r e s s amplitude dependent and th@ amplitude dependence becomes l e s s s t r o n g as t h e p r e s t r a i n increases.

Low temperature i n v e s t i g a t i o n s i n d i c a t e d t h a t both T~ and A were temperature i n s e n s i t i v e from 135OK t o 300°K (See Fig. 2) f o r CM-2T. The e r r o r i n measuring

T~ i s about 5% and t h u s l y i s w i t h i n t h e l i m i t o f d e t e c t i n g a change i n T through a temperature v a r i a t i o n o f t h e e l a s t i c constants. P r i o r t o the low teAperature t e s t s , t h e existence o f any p o s s i b l e t i m e e f f e c t upon T and A was i n v e s t i g a t e d over a s i x hour period. No such e f f e c t was found. The data o f Fig.

2 s t r o n g l y suggests t h e decrement i n t h i s temperature range i s h y s t e r e t i c i n nature. Fig; 3 i s a p l o t o f Wir vs. W a t v a r i o u s p r e s t r a i n s f o r c r y s t a l CM-lT a t 300 K. The curves are n o t l i n e a k ( % ) t h e r e may be e r r o r s i n WL (max) and i n Wirr. absolute values as much as 10%. This p l o t shows t h a t t h e r e s t r i c t e d a n e l a s t ~ s t r a i n s i n v e s t i g a t e d i n t h e present study are between 5 X

lo-'

-

V- ANALYSIS AND DISCUSSION

A t t h e r a t h e r l a r g e s t r e s s amplitudes employed i n t h i s study, t h e d i s l o c a t i o n can athermal l y surmount i m p u r i t y p i n n i n g p o i n t s a d t h e Peie 1 s-Nabarro stress. The former mechanism can c o n t r i b u t e about 2 X

lo-'

lo-'

T i n t h e l i g h t o f t h e Roberts t h e o r y /4,5/ f o r t h e c u r r e n t data, as was done by p i h a t t i and Roberts 161 f o r n i c k e l , we f i n d t h e f o l l o w i n g c h a r a c t e r i s t i c d i s l o c a t i o n s t r u c t u r a l parameters f o r c r y s t a l CM-IT. The primary d i s l o c a t i o n l o o p l e n g t h on each s i d e of t h u n d u l a t i n g i n t e r n a l s t r e s s o b s t a c l e i s 2L and i s found t o e 4.61 and 3.10 10'' cm f o r c r y s t a l CM-IT a f t e r p r e s t r a i n s ( Y ) o f 4.15 X 10'' and 5.97 X 10-' r e s p e c t i v e l y . Furthermore; t h e data suggests Pthe average u n d u l a t i n g i n t e r n a l s t r e s s amplitude Ti i s = 0.3 &/2L f o r these p r e s t r a i n s i n c r y s t a l CM-IT. The primary d i s l o c a t i o n d e n s i t y c o n t r i b u t i n g t o t h e hysteresis,

x X.4.15 x l d ' )! = 3.60 x 16' 0 Y, = 2.77 x 16' 0.10 .

A

T..O.S Y ?a CRYSTIL CY-21 To- 0.14 "

A . ,r VS. m r,-0.93 "

A & . I . I I "

-

l.7 5 - l , 2 9 " Q IN Pal

-6 . l I . * l

no 200 250 ⁵⁰⁰

T ('I0

DETERYYATION OF T I E

+

.

5

.

m 1;- 3 . 0 110.' o y, .4.7* 1 10-'

. .

-

&

-

0 2 3

loa ^{WL (Max)}

0 . 7 8 4

-

0 , 5 6 8

.

0.196

t

Fig. 1 Decrement ( A ) versus

(MPa) a t d i f f e r e n t

0

prestrains ( y ) f o r c r y s t a l s CH-1T a t 30O0K. P

Fig. 2

T~ versus T("K) f o r crystal CM-2T a f t e r Y~ = 5 X A versus

T("K) f o r crystal CM-2T a t various Stress amp1 i tudes ( r 0 ) a f t e r Y p = 5

x

Fig. 3 W i r r versus WL (max) f o r crystal CM-1T a t various prestains a t 300°K.

F i g . 4

;F versus y f o r c r y s t a l s CM-IT and P CM-AL a t 300°K.

C10-204 JOURNAL DE PHYSIQUE

dp

lola

l o m 3

The present authors suggest t h a t t h e mean f r i c t i o n a l s t r e s s

(c)

- - -

+

T ~

solute

where ;pN i s t h e athermsl P e i e r l s-Nabarro stress, an athermal i m p u r i t y f r i c t i o n s t r e s s and rStr i s t h e d j s l o c a t i o n s t r u c t u r a l athermal f r i c t i o n s t r e s s . From t h e work o f Young /I?/, ^{T ~ N} appears t o be (0.02 MPa and hence makes an i n s i g n i f i c a n t c o n t r i b u t i o n t o q as shown i n Fig. 4.

Precise knowledge o f i s n o t known b u t according t o L i /13/ o r F r i e d e l 1141 i t c o u l d vary between 1 t o 0.003 MPa r e s p e c t i v e l y , when t h e m i s f i t parameter f o r A1 i n Cu i s taken as 0.1 and t h e atomic c o n c e n t r a t i o n o f A1 i n c r y s t a l CM-AL i s

lom4.

From equation (3), i t i s r e a d i l y shown t h a t t h e average s t r u c t u r a l s t r e s s i s given by nii o r 0.3 n b / 2 L . N o t i n g n = 0.4 'from t h e work o f Roberts f o r CM-IT a t a p r e s t r a i n o f compares f a v o r a b l y t o t h e TF w i t h p r e s t r a i n (Fig. 4 ) i n c r e a s i n g p r e s t r a i n . REFERENCES

/I/ Granato, A.V. and ~ u c k e , K., J. App. Phys. 27, (1956) 583, 789.

/2/ Granato, A.V. and Lucke, K., J. App. Phys. T, (1981) 7136.

/3/ Kressel

,

,

/4/ Roberts, J.M., phys. s t a t . sol. ( a ) 19, (1973) 731.

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