The computer simulation of the internal friction peaks

The computer simulation of the internal friction peaks associated with the oscillation of geometric kinks along the nonscrew dislocation in the atmosphere of nonlinear power dissipating interstitials Tarik Ö. Ogurtani and Alfred K. Seeger Citation: Journal of Applied Physics 65, 4679 (1989); doi: 10.1063/1.343243 View online: http://dx.doi.org/10.1063/1.343243 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/65/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Low-temperature α peak of the internal friction in niobium and its relation to the relaxation of kinks in dislocations Low Temp. Phys. 27, 404 (2001); 10.1063/1.1374728 Cherenkov‐type sharp energy dissipation associated with kinks (solitons) moving harmonically in the atmosphere of paraelastic interstitial atoms J. Appl. Phys. 66, 5274 (1989); 10.1063/1.343716 Nonlinear theory of the dislocation‐enhanced Snoek effect and its connection with the geometric and/or thermal kink oscillations on nonscrew dislocations in body‐centered‐cubic metals J. Appl. Phys. 62, 3704 (1987); 10.1063/1.339252 Internal friction and viscosity associated with mobile interstitials in the presence of a kink harmonically or uniformly moving in anisotropic body‐centered cubic metals J. Appl. Phys. 55, 2857 (1984); 10.1063/1.333317 Internal Friction Peak in Iron due to the Dragging of Dislocations by Interstitial Impurities J. Appl. Phys. 39, 2473 (1968); 10.1063/1.1656583

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The computer simulation of the internal friction peaks associated with the oscillation of geometric kinks along the nonscrew dislocation in the atmosphere of nonlinear power dissipating interstilials Tarik

o.

Ogurtani and Alfred K. Seeger

Middle East Technical University, Ankara, Turkey, and Max-Planck-Institutfur Metaliforschung, Institut fur Physik, Stuttgart, Federal Republic of Germany

(Received 21 October 1988; accepted for pUblication 18 February 1989) The effects of the nonlinear power dissipation on the strength and the position of the cold-work internal friction peaks associated with the oscillation of geometric kinks in the atmosphere of paraelastic interstitials have been investigated by the novel numerical study of the related mathematical macromodel. The present macroscopic model of internal friction is described by a second-order ordinary differential equation with a nonlinear Debye-type dissipative drag term. This system shows not only dissipative resonance behavior, but it also results in unusually sharp energy dissipations, like the Cherenkov radiation, when the kink velocity exceeds the Snoekjump velocity of interstitiaIs at high external driving force amplitudes.

10 INTRODUCTION The present authors have successfully presented 1-4 a series of computer modeling experiments that reveals the fine details of the kink interstitial interactions in terms of the power dissipation formalism which clearly indicates the nonlinear viscosity behavior with well-defined sub-Snoek (below the Snoek velocity) and super-Snoek (above the Snoek velocity) dragging regimes. However, all these theories and simulations are microscopic in character, and they deal with the dynamics of an individual kink moving rigidly, and interacting with the smeared out interstitial clouds. Only in two important occasions,2.5 the linear and nonlinear dislocation-enhanced Snoek effects (DESE), have we given extensive treatments of the collective motion of geometric kinks for the Newtonian viscosity region. We should mention here very clearly that the geometric kink model has an identical mathematical macrostructure 6 •7 in comparison with the dislocation string mode1 8- 12 which was extensively employed in the dislocation damping relaxation phenomena, as long as one stays in the domain oflinear Newtonian viscosity regime. 13 In this paper, we introduce a simple but nonlinear macroscopic mathematical model of dislocation damping due to the presence of dragging interstitial clouds. This macroscopic model carries all the main features of the underlying physical model of geometric kinks which are mutually interacting and oscillating periodically in the nonlinear dissipative atmosphere of interstitials. In this presentation, in order to keep the mathematical picture as simple as possible we will introduce only the simple one-particle model 14 with sub- and super-Snoek viscosity regimes. This simple approach can still account for the strain amplitUde dependence (anomalous) of the relaxation strength of the internal friction peak as well as its saturation behavior with the increase in the concentration of the dragging interstitia! species. In Sec. II, we will present our mathematical macromodel, and show how it can be transformed into a dimensionless form which involves only three system (universal) param4679

J. Appl. Phys. 65 (12), 15 June 1989

eters. In Sec. III, the results of our computer studies in connection with the internal friction coefficient Q - I versus the three system parameters wiH be presented. Similarly, the internal friction coefficient Q -1 will also be given in the other parametric space representation which is more easily explorable by the present experimental techniques. The closely associated phenomena, the dislocation-induced Snoek effect, will be reexamined numerically in Sec. IV. The results of our previously presented microscopic theory of DESE 1-.5 will be utilized in Sec. IV, where the newly developed macroscopic model of dislocation damping is employed fully in the numerical calculations ofthe mean square kink oscillation amplitude denoted by (A i >. These last quantities are extremely important input for the microscopic theory of DESE. 5 Our recent computer simulation work on the dislocation-induced Snack effect (DISE) shows that there is no clear distinction between the DISP associated with the given dragging point defects and the dislocation damping peak (DDP) generated by the same interstitials as long as one deals with the strong kink-interstitial coupling comparing with the kink-kink mutual interaction in the sub-Snoek viscosity regime. On the other hand, in the weak kink-interstitial dragging comparing again with the kink-kink coupling, the situa~ tion becomes completely different in character. Even in the sub-Snaek viscosity domain, there are clear and fundamental discriminations between DISP and DDP relaxations. For example, one observed that the DDP due to the geometric kink collective oscillations in the atmosphere of dragging point defects is shifted to a temperature range even lower than the ordinary Snoek peak temperature of the relevant interstitials without a substantial reduction in the peak strength. Still, the most interesting nonlinear case occurs in the super-Snoek viscosity regime, where the DDP now becomes very small in height and shifts back to higher temperatures, even higher than the Snoek peak temperature. The DISP, on the other hand, develops fully in strength, and now dominates the overall damping behavior. The peak temperature of DISP coincides exactly with the peak temperature of the ordinary Snoek relaxation of the interstitial species in the super-Snoek viscosity regime.

0021-8979/89/124679-09$02.40

© 1989 American Institute of Physics

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II. THE MACROMATHEMATICAL MODEL FOR DISLOCATION DAMPING

The present macromode1 which is mathematical in character relies heavily on our treatment of collective geometric kink chain oscillations in the atmosphere of paraelastic interstitials in body-centered-cubic (bce) metals and alloys. First, we reexamine this physical model in some detail, and give the sound justification for the present oversimplication.

Ao The kink chain dynamics for the non-Newtonian power dissipation regime According to our extensive analytical as well as computer modeling experiments Z•5 in regards to the drag force acting on uniformly moving kinks in the cloud ofSnoek - and/or Cottrell-type heavy interstitials in anisotropic bec metals, one can use the Debye type of function for the power dissipation term in the equation of motion of kinks which are geometric in character. The following equation of motion of an individual kink in the geometric kink chain can be wri.tten: M.a2 i+B

ttY

I
o;o.·.·.·.~.~

unusual type of radiation if the velocity of the particle exceeds the phase velocity of waves of that frequency in the medium concerned. In the present case, according to our microscopic theory, the kink moving in an atmosphere of interstitials absorbs energy from the external driving force field as soon as its velocity exceeds the mean jump velocity of interstitials in the crystaL IV. THE DISLOCATION-INDUCED SNOEK RELAXATION {DISR} It should be emphasized that there is a fundamental distinction between the dislocation-enhanced Snaek effect and the dislocation-induced Snoek relaxation phenomena. The first effect is static and involves the power dissipation behavior of interstitials in the presence of inhomogeneous strain field of kinked dislocations. DISR, on the other hand, is the power dissipation of the same interstitial which is now excited by the temporal (time-varying) component of the inhomogeneous field of kinks moving or oscillating along the dislocation line. Therefore, the first is a spatial relaxation, and the second effect is a temporal relaxation phenomena. In this paper, we are going to deal with the temporal relaxation of interstitial due to the interaction between interstitials and collectively oscillating kinks along the dislocation-geometric kink system. According to the Appendix, the DISR has the following dynamic Q - I factor which is obtained by combining the microscopic theory ofDISP and the macromodel developed in Sec. II: (19)

where one has the following identity in the linear Newtonian viscosity region, Q D'~P = (t? /8) Q D- I, if the kink-interstitial interaction dominates the kink-kink coupling, that is, K> 10. Physically, this situation corresponds to the mostly observed behavior of the dislocation relaxation phenomenon, where the peak temperature is well above the Snoek relaxation peak temperature (1's = 1), and the strength of the dislocation relaxation peak is fully developed (saturation).

1075

GKMPEAK

In Fig. 6, the dislocation damping peak COOP) calculated from Eq. (A7) and the dislocation-induced Snoek peak (DISP) which is evaluated from Eq. (19) are shown for different values of the K coupling ratio which is directly proportional with the uniform concentration of interstitials for a fixed value of the driving force Fs = 0.01. We should mention that the above presented system is completely in the sub-Snoek dissipation regime since the largest value of Fe which occurs when K = 0.1 is equal to 0.2, according to the expression Fe = 2FJK. This figure clearly shows that the dislocation damping shifts to higher temperatures without any change in the strength when the interstitial concentration increases (or if the coupling ratio K increases from 0.1 to 1000) as long as the power dissipation is sub-Snoek in character. On the other hand, the dislocation-induced Snoek peak can never have a peak temperature below the peak temperature of the ordinary Snoek relaxation. With decreasing concentration, DISP first shifts to lower temperature without any shape change, but later its strength drops gradually and pins down at the temperature where 'f's = 1. We should emphasize that the observed internal friction which we call the "cold work peak (CWP)" is the linear summation ofthe DDP and the OISP. These latter two peaks are almost identical in shape and behavior when the whole system is in the saturation level and the dissipation is of a Newtonian linear viscosity type which corresponds to the initial stage of the sub-Snoek damping regime with high kink-interstitial interaction compared to the kink-kink coupling. V. DISCUSSION

According to our extensive computer studies which will be discussed in this section, the crucial difference which distinguishes the DOP from the nISP lies in the fact that in the DDP case any variation in the line tension parameter (stiffness) Kk--due to the increase or decrease in the density of geometric kinks along the dislocation line segment-does not affect the shape of the peak but shifts it along the temperature ('f'saxis) coordinate. This homomorphic behavior is valid not only for the sub-Snaek regime, but also for the super-Snoek domain of power dissipation, since Fn is constant of K k •

THE SATURATION BEHAVIOUR

DESPEAK \,50

FIG. 6. Internal friction coefficients associated with the dislocation damping peak as well as the dislocation-induced Snoek peak are given as a function of the normalized Snoek relaxation time for various values of the coupling ratio K which is directly proportional with the concentration of interstitials. During this computer experiment, the external force amplitude is kept constant (F, = 0.01) and the macrosystem stayed well in the sub-Snoek regime (F8 ,,0.2).

LOG 10 (NORMAl! ZED SNOEK FREQUENCY)

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J. Appl. Phys., Vol. 65, No. 12, 15 June 1989

T.

O. Ogurtani and A. K.

Seeger

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GKMPEAK 1.15

DESPEAK

FIG. 7. Internal friction coefficients related to the DISP as well as the DDP associated with the geometric kink chain oscillations are given as a function of the normalized Snoek relaxation time (the reciprocal temperature scale) for a fixed value of Fe;, and for various values of the stiffness or the kink density adjustment parameter, The kink density is decreased by factors of2, 4, 6, 8, and 10. The strain amplitude and the interstitial concentration are kept oonstant. Fa = iO, the initial values of K = 0.1 and ! = 0.03.

1.50

....

~

us 1.00

ifi0:: 0.15

OISPeoks

U

I.I.! 00.50 0.25

ODO.3

-2

fi"

-l~~~~~~O~;;;;~::!~----~

LO G10 (NORMALIZED SNOEK RELAXATION TIME)

The Q r;r:p factor of the macrosystem in the second scheme of the parametric space has the following format, according to the Appendix:

Qi)Jp

= (2KhsF;) {y'2/(l +y,2» ,

(20)

which may be reduced to the expression given below, which is identical for the initial stage of the sub-Saoek OinearNewtonian viscosity regime) region2 :

QDdp-1"B/[(l +flA-2)2+1'~]

=l'J[ (1

+ fli 2)2 + 1(21';] .

(21)

The maximum value ofEq. (21), which is called the relaxation strength of the macrosystem, represented by the ordinary differential equation Eq. (6), may be obtained from the expression of d DE = 1/ [2 (1 + n A- 2) 1 where the peak position value of 1'" = (1 + fli 2)IK. For the relaxation mode of the system (G A » 1), the relaxation strength becomes exactly equal to 112; it has also been verified by our computer studies that this value c-Orresponds to the upper saturation limit of the peak, not only for the sub-Snoek but also for the super-Snoek regime. In order to investigate the effect of the kink density along the dislocation segment which is assumed to be firmly pinned at both ends, on the DDP as well as DISP, we change K k • which is proportional to the kink density. for a given factor that may be called the stiffness parameter. The results of the stiffness parameter adjustment on the DDP and DISP are given in Fig. 7. This figure shows very dearly and unambiguously that the decrease in the stiffness has a profound effect on the strength of DISP as an enhancement without changing the peak position which occurs exactly at the ordinary Snoek peak temperature. On the other hand, the variation in stiffness has no effect on the shape of the DDP other than the shifting along the the "i"" axis. This stiffness alternation experiment is done under constant Fa, which corresponds to the experimental conditions of constant interstitial concentration and fixed strain amplitude. In order to see whether our macromodel shows any effect of the strain amplitUde on the strength and position of the DISP, we have performed a series of computer experi4685


ments where the K parameter is kept fixed but the external force amplitude represented by Fs varied according to our wish, under the constant normalized natural frequency (the relaxation mode). We should mention that the above computer simulation work coresponds to normal laboratory experiments done under constant low frequency by measuring the internal friction as a function of temperature for different values of the strain amplitUde, assuming that the dislocation structure and interstitial concentration stay invariant. Our computer studies indicated that DISP is not affected by the variation of the strain amplitude in terms of the peak position and the peak strength regardless the nature of the power dissipation. On the other hand, the DDP is affected from the strain amplitUde variations if the system in the super-Snoek regime. As one can expect a priori that the anharmonic contribution to the kink-kink interaction which is represented by the elastic storage term - Kkk U 3 is an extremely important source for the experimentally (ordinary) observed strain amplitude sensitivity of the DISP.19 Our lump model l4 already includes an anharmonic term as a first-order approximation to the nonlinearity in the elastic energy associated with the coloumbic kink-kink long-range interaction. However, in our future macromode1 studies we are going to add a new term in Eq. (6) in order to take care of the kink-kink nonlinear interaction properly. 7 The nonlinear power dissipation with sub- and superSnaek viscosity regimes can explain almost all (saving the strain amplitUde dependence of DISP) of the important main features of the internal-friction-associated geometric kink oscillation (GKO) in terms of the line shape and position. First of aU, this macrotheory illustrates how and why the strength of the internal friction peak associated with the dislocation damping is increased from a zero value up to a saturation level with the increase in the concentration of interstitials. One could even use the theory of nonlinear diffusion of interstitials in the strain field of kinks in the B k term in order to take care of the contribution ofthe Fermi-Diractype distribution of interstitials20 in the strong field of the embedding dislocation line! T.

O.

Ogurtani and A. K. Seeger

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ACKNOWLEDGMENTS

The authors wish to thank Professor H. Schultz and Professor E. Tekin for valuable discussions. Thanks are also due to Dr. K. Differt and Dr. M. Ozenbas, and Mr. A. AItunoglu for the preliminary computer manipulations, and to Mr. L. Starke and Mr. H. Roessmann of the Computer Center of Max-Planck-Institut, Stuttgart, for their constant assistance. APPENDIX: THE CONNECTION BETWEEN THE INTERNAL FRICTION COEfFICIENT AND THE DYNAMICAL Q-'1 FACTOR OF A DIffERENTIAL EQUATION ASSOCIATED WITH THE MACROMODEL

In this Appendix we will give a universal connection between the internal friction coefficient (IF) ofa sample due to dislocation-kink damping phenomena and the Q - I factor of the DE which describes this physical model according to Eq. (6). The Q -\ factor of the differential equation (6) can be calculated from the power dissipation which is given by the second term:

=B

I D

A. k,

(dU Idt)2 1+ (A; dUldt) 2

= (B k /A;)[y,2/(l

+/2)}.

(Ai)

Similarly, the energy dissipated per cycle which is now a dynamical quantity depends upon time and can be calculated by the foHowing expression:

(A2)

The maximum elastic energy stored in the macrosystem during the one cycle is closely related to the quantity denoted by WE which is given by the following operational definition: (A3)

Even though for the present case the energy dissipation per cycle is a time-dependent quantity due to the initial transient regime, still we will utilize the following generalized definition: (A4)

which can be also written in the following form according to Eq. (13): QiJ J = (S!TBFt) (y'2/(1

+y'2».

(AS)

The temporal value of the expression given by Eq. (AS) may be called the dynamical Q D 1 factor of the DE, which has been also investigated rather extensively by the present authors. It has been found that in the sub-Snoek regime, Q D I reaches to the steady-state value after few cycles such as one or two when the normalized relaxation time is less than unity, On the other hand, for the large values ofthe normalized relaxation time T B and the small values of the FB force, one may need a large number of cycles in order to reach the steady-state value. In the second scheme of the parametric representation 4686


of the DE, the dynamic Q D- 1 can be given by the fonowing expression which may be obtained directly from Eq. (A5): QiJl

= (2KITsF;) (y'2/(l +y'2».

(A6)

A. The internal friction coefficient of the sample in the kink model In this subsection we will obtain the connection between the usual definition of the IF coefficient of a sample Q S 1 and the Q v' ! factor of the associated macromathematical modeL One can easily show that the shear strain due to the movements of kinks may be given by the following expression: (A7)

where A is the dislocation density embedding the geometric kinks, Then the energy dissipated by the displacements of the kinks under the action ofthe applied shear stress has the form (A8)

which may be combined by the maximum stored energy per unit volume of the sample, W~nax = d;,o/2G, to give the following expression for the IF coefficient: Q S·-1_ -

LlW d[ 2W~ax

(ITI"'. dU 2G ) AbNka k J(] dt sm((ut)O",.o-, r?-iJ,.,o (] dt

= ( --,-

(A9)

where G is the usual shear modulus and O"r,O is the amplitude of the harmonic shear stress with frequency w. The above equation can be reduced into the following format using the fact that

Fk hence

= bOkO"r,o

sinew!) ,

2\(-G ) ANk JoCZ Qil = ( r! J c?;,o ()

1Tlm

dU, CAW) dtFk dt which results in the desired connection rigorously from the DE(6) by performing a close orbit integration in the phase space:

Q 8- 1 = (2AGb 2L

2/"rS~I)Q iJ

1,

(All )

The expression in the parentheses is the relaxation strength ofthe internal friction associated with the dislocation damping for the Ritz first-order approximation. The exact analytical expression for the Newtonian viscosity region results in the following rigorous expression 2 ,12:

Q 8- 1 = (16AGb 2L 2hT.4S~I)QDEJ ,

(Al2)

where

QDEJ = TB/[ (1 + n.4 2)2 + 1'1 J ' which is equal to Q D 1 in the linear Newtonian regime, according to our computer plots. Therefore, the Ritz approximation used in our numerical simulation results in only minor deviation from the exact analytical method by a factor of r?-/8, which is well inside the limits of reliability of our results. T.

O. Ogurtani and A. K. Seeger

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B. The dislocatlon~induced Snoek relaxation of the sample in the geometric kink chain model The energy dissipated per cycle associated with a single kink that is harmonically and rigidly oscillating along the dislocation line can be written in the following approximate (but very accurate) form according to our microscopic theory developed in Refs. 3 and 4:

(.6.Wkink )

=

-1TCoa6C~a~(Ai)/3kBTD("I"s)'

(AU)

where (A ~) is the mean square value of the kink oscillation amplitude, and for the present notation it is given by

(A

n = (U 2) = (y2) I (Aj(ll)2 ,

(A14)

which simply reads (A

n = a6 (y2)/T;.

The function denoted by D is very complicated in general, and it contains two contributions: the Cottrell cloud (isotropic part of the elastic dipole tensor) and the Snoek cloud (deviatoric or shear part of the elastic dipole tensor). For the present paper, in order to simplify the discussion, we are going to use only the Snoek-type interstitials which yields the following very accurate expression 3- 5 :

CAlS) The internal friction coefficient due to above-mentioned dissipation can be written as Q Drip (sample)

= ANk (A Wk(nk >/1Ta;.o / G ,

(A16)

which can reduce into the following form after some deliberation: QDI~P (sample)

= A( D( 1"s)/.BrJ (8/~) (A17)

where .6. is the relaxation strength defined previously. Hence, the dislocation-induced relaxation peak associated with the macrosystem becomes, according to definition,

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Q Di~p

= (8/ffl) [2K(V)h s F;O +,.,;))

= Q DI~P (sample)/II

X

(A18)

in the second scheme of the parametric representation. In the linear Newtonian regime, one has the following identity, (y'2) = (y2), that yields Q!)i~p

=

(iV~)QD1;P ]j(l

+ '1";) ,

(A19)

which results in QDI~I'

=

(8/~)QDJp

when the coupling ratio K is large compared to unity (the saturation stage of the dislocation damping). We should mention here that for more sophisticated computer simulation studies one can utilize Eq, (A17) directly from the previously calculated form of DC ). This means an enormous amount of computation time! 'T. Ogurtani and A. Seeger, J. App!. Phys. 55, 2857 (1984). 2T. Ogurtani and A. Seeger, Phys. Rev. B 31,5044 (1985). 3T. Ogurtani and A. Seeger, J. App!. Phys. 57, 5127 (1985). ·T. Ogllrtani and A. Seeger, J. App!. Phys. 58,4102 (1985). ST. Ogurtani and A. Seeger, J. A ppl. Phys. 62, 3704 ( 1987). "A. Seeger and P. Schiller, Acta Metal!. 10, 348 (1962). 7T. Suzuki and C. Elbaum, J. Appl. I'hys. 35, 1539 (1964). 8J. S. Koehler, in Imperfections in Nearly Perfect Crystals, edited by W. Schockley, J. H. Hollomon, R. Maurer, and F. Seitz (Wiley, New York, 1952), p. 197, 9 A. Granato and K. Liicke, J. App!. Phys. 27, 583 (1956). lOG. Schoeck, Acta Metall.ll, 617 (1963). "H. M. Simpson and A. Sosin, Phys. Rev. B 5,1382 (1972). 1ZT. Ogurtani, Phys. Rev. B 21, 4373 (1980). HA. C. Eringen, Mechanics ofContinua (Krieger, New York, 1980), p. 88. "T. Ogurtani and A. Seeger (unpublished). "T. Ogurtani and A. Seeger, J. App!. Phys. 57,193 (1985). 16M. Ross, Introduction to Ordinary Differential Equations (Wiley, New York,1975),p.179. IIT.Ogurtani (unpublished). ,gL. D. Landau and E. M. Lifshitz, Electrodynamics o/Colltinuous Media (Pergamon, New York, 1963), p. 357. 19J. Saur, W. Benoit, and H. Schultz (unpublished). 2"T. Ogurtani and A. Seeger, J. App!. Phys. 62, 852 (1987).

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