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AMPLITUDE-DEPENDENT INTERNAL FRICTION AND DISLOCATION MOBILITY IN CRYSTALS
K. Ishii
To cite this version:
K. Ishii. AMPLITUDE-DEPENDENT INTERNAL FRICTION AND DISLOCATION MO- BILITY IN CRYSTALS. Journal de Physique Colloques, 1985, 46 (C10), pp.C10-191-C10-194.
�10.1051/jphyscol:19851044�. �jpa-00225428�
JOURNAL DE PHYSIQUE
Colloque CIO, supplément au n"12, Tome 46, décembre 1985 page C10-191
AMPLITUDE-DEPENDENT INTERNAL FRICTION AND DISLOCATION MOBILITY IN CRYSTALS
K. ISHII
Department of Physics,.Nagoya Institute of Technology, Gokiso-ch6, Showa-ku, Nagoya 466, Japan
Abstract - The amplitude-dependent internal friction due to dislocations oscillating in the lattice is described in terms of dislocation mobility.
This kind of intemal friction is expressed in elementary forms in the two limiting cases, i.e. at hish and low frequencies, and with the results ob- tained, the behavior of relaxation peaks due to dislocations are discussed 1 - INTRODUCTION
The amplitude-dependent internal friction will be caused in m y cases by the dislo- cations oscillating in the lattice, where obstacles are dispersed. In previous papers 11-31, this kind of internal friction and the modulus defects associated with it were described in terms of dislocation mobility, and the results ccinpred with data on metals. In this paper, the behavior of relaxation peaks due to dislocations will be mainly discussed.
II - EXPRESSIONS IN TWO LIMITING CASES
The internal friction due to dislocations oscillating in the lattice, denoted by Ap, can be expressed in elementary forms in the following two limiting cases:
At high frequencies, the motion of dislocation will be determined by frictional forces, and the internal friction is expressed in a form
111
-
[*: ( O
) sin0 ü.0 ,.
A
-
a. w (1)where N is the density of mobile dislocations, 00 the stress amplitude, v(a) the dislocation velccity as a function of applied stress a, o = a. sinwt, and ut = 0 . At low frequencies, the frictional stress is small, and the dislocation will bow out under applied stress, to which a small frictional stress is superposed. It is a s s d that the frictional stress of is equal to the effective stress needed for the dislocation to move freely at the same velocity. That is
where oe(V) is the inverse function of v(ae), the velccity as a function of effective stress a,. With this assurnption, the intenial friction is 131
( ' ) Present adress: Nagoya Nutrition College, 2-1 Sasatsuka-cho, Nishiku, Nagoya 451
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19851044
JOURNAL DE PHYSIQUE
Low frequency
Fig. 1
-
The motion of dislocation is : High frequency determined by frictional forces at high. . freguencies, and by tension forces at low
. . frequencies. (c.f. 151)
O L
Viscous forces
---
A special case is that the velccity is determined by viscous forcesV = (b/B)(a-a;), ( 4 )
where B is the drag constant, and ai the internal stress. In this case, the inter- na1 friction is obtained by substituting eq. (4) in eq. (1 ) ,
at high frequencies. This is the same equation that is obtained fran the Granato-
~6cke the- of amplitude-independent internal friction /5/, at high frequencies around the resonant frequency under the condition of overdamping.
The frictional stress for the line-tension limited motion isgiven by
fran eq.(4), where the internal stress is iqnored. Then, the internal friction be- canes
Ap = T N L ' B W / ~ ~ U ~ ~ , ( 7 ) 'frm eqs.(3) and (6). Apart fran a difference in the num.ical factor, this is the
same equation as is given by the ~ranato-~Ücke theory of amplitude-independent inter- na1 friction at low frequencies 151.
Stress exponent --- The dislocation velocity is ofta expressed formally as
2r = 2 r , ( ~ l o l ) ~ , (8)
where '& , al and m are the constants. The internal friction at high frequencies then becornes
Ap = 4 pb N
W F(m) oom-zl (9)
from eq.(l), where F ( m ) is a Eunction of m. For the velocities expressed in eq.(8), the frictional stress is given by
and thus the internal friction,
m i l
A ? = 2NL2a1 3 00
( Ila6tf:0s0 y d 0 , m (11)
.
fran eq. (3). The two equations, eqs. ( 9) and (1 1 ); show the relaxation type be- havior of the internal friction, in the sense that it increases with the frequency at low frequencies, while it is inversely proportional to the frequency at high frequencies. Such behavior is obtained by Schlipf and Schindlmayr /6/ over the whole range of frequencies fran their theoretical treatment.If we write the dislocation velocity in a p w e r series
7/ =
3
b ( + C l ' s : +ego:+ ..*
1, (12)where B', ê 2 , C J ,
...
are the constants, then the frictional stress at l m frequen- cies is equal to the effective stress obtained £rom eq.(12). At m l 1 amplitudes, the frictional stress becornesIn this case, the specimen used can be treated as a standard anelastic solid, and the intemal friction has a Debye peak
where and
Equation (5) is obtained £rom eq.(14) for U T > > 1 and w i t h B r = B. ALso, eq.(7) is obtained for U T < ( 1. The displacement determined by viscous forces b a o /Brw is equal to that permitted by tension forces aoL2/ 6 pb2 at the peak maximum w .r = 1.
Amplitude increased
-r-+
Fig. 2
-
Schemtic decrement vs frequency. Curve 1 shows the peak at m l 1 ampli- tudes where the internal friction is amplitude-independent, curve 2 in the ampli- tude dependent region, g ~ d m e 3 the peak for which the relaxation t his deterinined by the &ag constant B.As the amplitude is increased, the shape of the peak will be d e f o d . Under the condition that the lcop length L is kept unchanged, the peak maximum will be dis- placed to higher values of frequencies, as schemtically shown in fig.2, because the constant B', and accordingly the relaxation t h , in effect decreases as the amplitude increases.
In the low frequency side of the peak, the internal friction taken at constant fre- quency decreases as the amplitude increases, as a result of the peak shift. This is consistent with the result obtained in eq.(11). At sufficiently low frequencies, the internal friction will be close to that obtained frcm eq.(14), for the velocity is so small that the frictional stress will be proportional to the velocity, accord- ing to eq.(12).
In the high frequency side of the peak, the internal friction measured at constant
C10-194 JOURNAL DE PHYSIQUE
frequency increases as the amplitude increases, king consistent with eq.(9). In the limit of high amplitudes, the peak will approach the Debye pak for which the constant BI is equal to the drag conçtant B. It is because the velocity of disloca- tion is determined by viscous forces at high stress /1,4/. (See fig. 3.)
Fig. 3
-
Schematic velocity vs stress.It approaches = ( b / B
'
) ue at lowstresses, and = ( b / B ) at high stress.
Ccanparison with experimental data --- Both the frequency dependence and the ampli- tude dependence as shown in eqs.(9) and (11) are usually observed in the decrement vs temperature with a relaxation peak. As to the frequency dependence, it is gener- ally the case that when the frequency is increased the pak maximum is displaced to higher values of temperatures. (for example, ref ./7/ ) As a result , the internal friction taken at constant temperature decreases as the frequency increases at temperatures below the peak maximum, while it increases with the frequency above the maximum, as schematically shown in fig.4. This is consistent with the present results, eqs.(9) and (11 ) , since the motion of dislocation will be determined by frictional forces at low temperatures and by tension forces at higher tenpratures.
Frequency increased
l
+
Fig. 4 - Schmtic decrement vs tempera-ture. It is displaced to higher values of temperatures as the frequency increases.
As a result, the internal friction taken at constant temperature changes as shown by the arrows.
\
REFERENCES
/1/ Ishii, K., Proc. 6th ICIFUAS, Tokyo, University of Tokyo Press, Tokyo (1977) p.91.
/2/ Ishii, K., J. Phys. Soc. Japan 2 (1983) 141.
/3/ Ishii, K., J. Phys. Soc. Japan (1983) 149.
/4/ Granato, A. V. Dislocation üynamics, McGraw Hill, New York (1968) p.117.
/5/ Granato, A. V. and ~iicke, K., J. Appl. Phys. 27 (1956) 585.
/6/ Schlipf, J. and Schindhyr, R., mot. 5th ICIFUAS, Aachen, Springer-Verlag, Berlin, Heidelberg (1975) Vo1.2, p.439.
/7/ Schaefer, H. E., Schultz, H. and Stark, H. P., Proc. 3rd Euro. Conf. of IFUAS Manchester, Pergamon Press, Oxford (1980) p.27.
