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Dislocation dynamics and acoustic emission during plastic deformation of crystals
ELSEVIER
WaveMotion
(1995)343-35.5
Dislocation dynamics and acoustic emission during plastic deformation of crystals B. Polyzos, A. Trochidis Universir?, of Thessaloniki,
School
ofEngineering, Department ofPhysics and Mathematics, Thessaloniki 54006, Greece Received 20 August 1994
Abstract A soliton approach to acoustic emission during plastic deformation of crystals is presented. The approach is based on a microscopic Frenkel-Kontorova model where the rigidity of the substrate is removed in order to establish the interaction mechanism between a dislocation and both longitudinal and transverse acoustic waves. It is shown that this interaction is described by a sine-Gordon-d’Alembert system. Within the framework of this system, two basic mechanisms of acoustic emission are investigated both analytically and numerically. One mechanism is related to nonstationary dislocation motion and the other one to the annihilation of dislocation kink-antikink pairs during Frank-Read source operation. In both cases, computer simulations are obtained which illustrate graphically the analytical considerations and model the acoustic radiation. The obtained results are in agreement with existing experimental data and may provide a better physical insight to the acoustic emission mechanisms during plastic deformation of crystals.
1. Introduction Most of the experimental observations of acoustic emission (AE) during plastic deformation of crystals are interpreted in relation to nonstationary motion of dislocations [l-3]. Since dislocation theory was originally motivated by classical electrodynamics [ 41 it was natural to view the AE associated with the nonstationary motion of dislocations as analogous to the electromagnetic radiation of accelerated (or decelerated) charged particles ( “bremsstrahlung” type of acoustic radiation). Another possible mechanism of AE generation, also in analogy to classical electrodynamics, is the process of dislocation annihilation. The first analysis of this type of AE (“transition” type of acoustic radiation) was made by Natsik and Chishko [ 51. Experimental observations of AE during copper crystal tensile deformation [ 6,7], showing an increase (decrease) of AE intensity induced by a sudden increase (decrease) of strain rate, support the hypothesis that dislocation annihilation is an additional source of AE during plastic deformation. Eventhough the existence of AE measurements from a wide range of metals has been known for many years, there is not yet a commonly acceptable quantitative understanding and theoretical interpretation of its origin and of the specific mechanisms causing it even for relatively simple systems such as pure aluminum. A common feature of all existing AE models is their phenomenological character since they are based on continuum elastic theory and do not deal with the atomic nature of dislocations. Pawelek [ 81 made an attempt to consider AE in terms of lattice-dynamics. He pointed out some quantum-mechanical aspects of AE and proposed a 0165-2125/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDfO165-2125(95)00007-0
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B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
qualitative soliton description of AE during plastic deformation of crystals. The proposed model, however, is rather qualitative and cannot account for the variation of the AE activity as deformation advances and furthermore cannot relate the emitted energy to the strain or external shear stress. It is the aim of this work to outline a new possibility of a more general interpretation of the AE phenomenon within the soliton framework for dislocation dynamics. Since we are interested in the emission of elastic waves by dynamical processes involving solitons (dislocations) it is reasonable to investigate the interaction between elastic waves and solitons (dislocations). For that purpose, a rather simple microscopic lattice model is used. It is a FrenkelKontorova (FK) model distinguishing between two configurations of atoms: one configuration represents the elastic substrate (perfect lattice) while the other, above the slip plane, the atoms making up a dislocation. By removing the rigidity of the substrate energetic exchanges between dislocation and substrate are allowed and the mechanism of interaction between dislocation and both longitudinal and transverse acoustic waves is established in terms of lattice dynamics. In the continuum limit, the basic equations describing the interaction is shown to be a sine-Gordond’Alembert (sGdA) system, i.e. two wave equations for the elastic displacements nonlinearly coupled to a sineGordon (sG) equation which governs the dislocation dynamics. On the basis of the obtained system of coupled nonlinear equations the two basic mechanisms of AE are investigated. The acoustic radiation by an accelerating dislocation cannot be treated analytically and thus is studied numerically. On the contrary, the emission of acoustic waves by annihilating dislocation kink-antikink pairs during Frank-Read source operation is studied both analytically and numerically for small and high velocities prior to the annihilation. Computer simulations are obtained which illustrate graphically the analytical results and model the acoustic radiation. The obtained results are discussed in relation to existing experimental data and may provide physical insight to AE mechanisms during plastic deformation of crystals.
2. Interaction between a dislocation and acoustic waves In this section the interaction between an edge dislocation and both longitudinal and transverse acoustic waves is considered. In our model, the dislocation is represented by replacing the atoms above the slip plane by a chain of N+ 1 identical mass points. The other atoms, making up the elastic substrate, are represented by a chain of N mass points. Two types of motion are considered for the atoms of the substrate: (i) a longitudinal motion of the lattice points and (ii) a transversal motion of the same points. The corresponding elastic displacements are U, and u,, respectively. For the atoms above the slip plane making up the dislocation a displacement y,, is used (Fig. 1). The forces acting on a lattice point n of the substrate result from the interaction with neighbouring points only and are modelled by means of springs with spring constants KL and KT for the longitudinal and transversal directions respectively. For the atoms making up the dislocation only the stretching spring KL is considered. In case of a rigid substrate, a periodic substrate potential of the form A [ 1- cos( 27ry,,l a) ] is used, where A is the constant amplitude and cy is the lattice spacing in the longitudinal direction. According to the continuum theory of elasticity [ 91, the amplitude A is inversely proportional to the lattice spacing d in the transversal direction. By assuming transverse vibrations of the substrate atoms, the lattice spacing d at the point n changes to d - ( u, + , - u,) resulting in a spatial modulation of the constant amplitude A. On the other hand, the longitudinal vibrations of the Y” :-
000000~6000
10
d
l
v?
a
h A- ““0
Fig. 1. The Frenkel-Kontorova
.x
j
I
0
i
0
0
0
model with nonrigid substrate.
34.5
B. Polyzos, A. Trochidis / Wave Motion 21 (I 995) 343-355
substrate modify the lattice spacing CYin the longitudinal direction and the substrate potential becomes non-periodic. Taking into consideration the local compression v, + , - v, at the point n, the actual displacement of the dislocation atoms relative to the potential minimum of the rigid substrate becomes y,, - (v, + , - v,) . It is further assumed, that the interaction potential is of cosine type and should reduce to that of the FK model in case of a rigid substrate (u,, u, = 0). In order to keep our model mathematically tractable, the following simple form of the interaction potential is assumed
A
v=c n
1_
1
_cos 2TYYn -
(“,+I --“A a
un+1-4! d
where the spatial compression of the lattice in both directions has been taken into account. When the displacement v,, U, slowly depend on n, i.e. when 1u,+ , - u, 1 < d and 1u,+ , - v, 1 +c a, the interaction potential given by Eq. ( 1) can be written, after expansion in Taylor series restricted to the first order, in the form V=
CA n [ l-cos
p)]+
+
(u,,,
-0,)
The potential of Eq. (2) leads to the following Hi=
C ~ n
The Hamiltonian H,=
(“,+I
-
u,)
interaction
7
(u,,,
-u,)
sin2(T).
(2)
Hamiltonian
sin
(3)
of the elastic substrate is
&n(L:;+ti;)+ n 2
and the Hamiltonian
sine)+
+.,,
of the dislocation
-uJ2+
1 ,K,(un+,
-u”)*,
(4)
is (5)
Thus, the Hamiltonian
of the whole system is
H=H,+HdfHi. Owing to Eqs. (3)-( are obtained
(6) 6)) the following
discrete equations
of motion for the dislocation
and the elastic substrate
(7)
(8) fJ2U m n
at*
-KT(u,+,
+u”_~
-2~“)
= 7
sin
(9)
In the continuum approximation the “point” quantities v,, u,, y,, can be expanded about n as a function of x and the following system of partial differential equations for the continuum variables is deduced
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B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
m-
d2Yn
(10)
at2
a20
rnp
a20
2
-K,a
1 ax
=~ITA-
(11)
a214 t 12) Considering
the change of variables and functions
t=2n-&ii%,
2TY
cp= -t
cy
x= $&i7&, 2mJ “= (Y
the system of Eqs. ( lo)-( p,,-c~,+
and
(13)
12) can be written in dimensionless
sin cp= -er_u,
UlI - uxx = c(sin
2n-u u= -, ff
cos cp- TuX
sin2((p/2),
qo>,,
form as (14) (15)
u,-v~u,=ET(sin2(cp/2)),
(16)
with
and the subscript x denoting differentiation with respect to x. It proves that the dynamics of the system considered is described by a sG equation which governs the dislocation dynamics nonlinearly coupled to two acoustic d’Alembert equations for the elastic displacements. Similar sGdA systems have been obtained in other contexts also like the magnetization in elastic ferromagnets [ lo], domain walls in ferroelectrics [ 11,121, charge-density-waves interacting with sound [ 131 and nonlinear elastic crystals with internal degrees of freedom [ 141. In the absence of the right hand coupling terms, Eqs. (15) and (16) are the classical wave equations while Eq. ( 14) is the pure sG equation for solitary waves. In our case, its elementary solution cp(x, t) =4 tan-‘{exp+
(1 -ut)ldl
-u*)
,
(17)
represents the motion of a dislocation kink (or antikink), depending on the f sign, along the dislocation line with velocity u. Thus, the motion of the dislocation as a whole in a given direction (y direction) may be executed in result of a soliton motion in the perpendicular direction (x-direction). We know [ 15-l 71 that the coupling of the sG equation with one or two wave equations make that pure solitons (i.e. solitary waves recovering their full identity after interaction) are not possible, being accompanied by wavelets that propagate proceeding or tail trailing the solitons. Hence, as well as pure sG equation the sGdA system admits solitary-wave solutions which are kink-like in the dislocation system and hump-like shadows in the elastic systems. 3. AE related to nonstationary
dislocation motion
Most of the existing experimental relation to non-stationary dislocation
observations of AE during plastic deformation of crystals are interpreted in motion. For instance, Fisher and Lally [ 1] suggested that AE is a consequence
B. Polyzos. A. Trochidis/ Wave Motion 21 (1995) 343-355
347
of fast collective motion of dislocations while Sedgwick [ 21 has considered the sudden release of dislocation pileups as possible AE sources. In the same spirit, James and Carpenter [ 31 suggested that AE is related mainly to dislocation breakaway from the pinning points. All the aforementioned explanations assume the “bremsstrahlung” type of acoustic radiation caused by accelerated dislocations to be the main source of AE. On the basis of the coupled Eqs. (14)-( 16) one can study analytically only the motion of a single soliton (dislocation) in steady regime looking for propagative-solution functions of the form cp(z) = cp(x - ut). Such solution can be found by substituting u,, u, from Eqs. ( 15)-( 16) into Eq. ( 14) in which case a double sine-Gordon (DsG) equation for cp(z) is obtained [ 121. This problem will be discussed in Section 4. Thus, in order to illustrate the problem of acoustic radiation by an accelerating dislocation the sGdA system is treated numerically. The numerical scheme adopted is a simple finite difference technique usually called “leap-frog” method. We consider first the medium to be originally at rest, containing a dislocation kink (i.e., at t = 0, v = 0, u = 0, duldtl,=,,=O, &ddt~,~O=O and cp(x, 0) =4 tan-’ exp(x -x0)). Then the dislocation kink is suddenly put in motion travelling from left to right as shown in Fig. 2a. Due to the coupling between dislocation and substrate, the motion of the kink induces nonlinear elastic waves in the substrate. In Fig. 2b (which represents the transverse elastic displacement) one can notice two types of waves: a hump-like “shadow” moving with the dislocation from left to right with the speed of the dislocation and the emitted acoustic radiation moving from right to left with the speed of the transverse acoustic waves u.,.. The same picture results for the longitudinal elastic displacement u (Fig. 2c) where the emitted acoustic radiation moves from right to left with the speed of the longitudinal acoustic waves uL. Thus, the results of the numerical treatment of the sGdA system illustrates a situation that is related to the acoustic emission generated by the nonstationary dislocation motion during plastic deformation of crystals. 4. AE related to dislocation annihilation during Frank-Read
source operation
It is very well known that during plastic deformation of crystals a dislocation segment can act as a Frank-Read source. In every stage of the source operation, associated with a generation of successive dislocation loops, sections of the dislocation line of opposite orientations are annihilated and elastic energy is released. In this sense, a FrankRead source is also a source of the transition type of acoustic emission. A qualitative, schematic soliton description of this annihilation mechanism was described by Pawelek [ 81. For the sake of clarity, this description is briefly repeated here. During the initial stage of source operation two rectilinear dislocation sections A,A,? and BIB, lying in adjacent Peierls valleys and approaching one another are considered (Fig. 3a). Their interaction leads to their annihilation (Fig. 3b) and due to the disappearance of the rectangular loop A { A iB\ B 4 elastic energy is emitted. The important dynamical process described above, can be studied with the coupled Eqs. (14)-( 16). Within a similar framework for a ferroelectric system, this process has been studied both numerically and analytically by Pouget and Maugin [ 11,121. In that paper (where the kinks were representing domain walls), the collisions of both kink-kink and kink-antikink pairs have been developed in the adiabatic approximation, i.e. ignoring radiation effects. Next, emission of sound waves by colliding kinks were investigated by means of the Green’s function technique and an expression for the emitted wave field has been obtained. Our analysis is based on the perturbation theory for solitons [ 181 assuming that the coupling between dislocation and elastic substrate is small, i.e. eL, % +Z 1. Thus, to calculate the energy emitted during the annihilation of a dislocation kink-antikink pair we follow the lines of Ref. [ 191. 4.1. Analytical
results
A. Small initial velocities We consider first the collision of a kink-antikink pair with small initial velocities ( u 2 +Z 1) prior to the collision. During the collision annihilation of the kink-antikink pair into a breather can take place provided the velocity is lower than a certain threshold [ 191. The kink and antikink come of their shadows under the action of strong mutual
B. Polyzos, A. Trochidisl Wave Motion 2I (1995) 343-355
348
(b) U X
Fig. 2. AJZ from a moving dislocation.
(a) Motion of a dislocation
kink. (b) Associated
radiation of elastic waves.
B. P~lyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
349
(4
(b)
Fig. 3. Schematic soliton description of dislocation section annihilation during Frank-Read source operation (Ref. [ 81)
attraction. After the collision they develop their shadows anew, while the former shadows decay into acoustic waves of the form u (x f u,_t), u (x 31uTt) propagating in the elastic substrate. The continuum dimensionless Hamiltonian of the system is
with
(18)
Hi= $ux sin (p+e,,u,
7(P sin-?.
(20)
To calculate the energy of the emitted waves, first the profile of the shadows in the elastic system is required. Considering the acoustic shadows of the kink given by Eq. ( 17) we look for solutions in the form u(z) = u(x - ut) , u(z) =u(x-uf).In thiscaseEqs. (15) and (16) yield
350
B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
l)u,
(u*-
=e,(sin
cp), ,
(21)
= lT sin2 z
(z+u~)u,
(
(22)
1z
The above equations integrate once with respect to z to give 0Z=-
Asin’p, ET
u, = -
(G - v2)
(23) sin2 ‘p 2’
(24)
which are the profiles of the shadows travelling with the kink. The energy of the emitted waves due to the decay of the shadows is given by (25) the multiplier 2 in front of the integral taking into account the fact that we have two shadows (one for the kink and one for the antikink). Inserting Eqs. ( 18)) (21) and (22) into Eq. (25) one finds m
Eem=2
l+u2 (l-L?)2
I
-cc
8 v;+Ll* % + TE (u$-u2)2‘
(26)
The emitted energy during the annihilation of a pair of dislocation kinks proves to increase with increasing kink velocity. This energy is related to the dislocation annihilation component of AE per event. Our result is in agreement with the relation obtained by Natsik and Chishko [ 201 who used the dislocation model in a continuum medium. According to their relation the mean energy per annihilation event is proportional to the square of the dislocation velocity prior to annihilation. During stage I of deformation only a single active glide system operates and the dislocations move long distances unobstructed. The kink velocity increases tending asymptotically to the limiting “power balance” velocity and consequently, the emitted energy increases giving rise to an ascending AE branch. B. The ultrarelativistic case
Explicit calculation of the acoustic waves is also possible in the case when the velocities f u of the interacting dislocation kinks are ultrarelativistic, i.e. 1 - v2 -=K1. In this case the kink-ant&ink pair may be approximated by the linear superposition of the waveforms of the free kinks given by Eqs. ( 17) [ 211 a2
=
401 +
v2
(27)
9
where the subscript 1 and 2 are realized as kink and antikink. The longitudinal acoustic wave field can be represented in the following form m
U(X,t) s (29-r)--I
I
dk fi(k, t) eiKr .
(28)
--cc
A similar expression holds for the transverse acoustic field. If F,_(x, t) and FT(x, t) stand for the right-hand side of Eqs. ( 15) and ( 16)) the Fourier amplitude C(k, t) can be brought in the form
B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
dq,(k t> dt
= -
ikq,_(k,t) + cL&(k,
351
(29)
t) ,
where qL( k, t) = 86( k, t) /at - ikfi(x, t) and Fc( k, t) is the Fourier transform of FL(x, t) . To find the amplitudes of the emitted acoustic waves, we assume that the free waves were absent prior to the collision, i.e. at t= - ~0 and we define the final amplitudes as follows dt dbw(k, dt
bLsT(k) = I
t)
(30)
’
with bL.T( k, t) ) = q-J k, t) exp( ikt) . In particular, sin( ‘p, + (~2) gives rise to the following
14sech’($$)
F,_(x, t) = i
emission generating terms
sect?(s)+
x [4o sech(E)
tanh(e)
]
sech(-$$)
tanh(-$$)]
,
(31)
where CJ= cr,cr, = + 1 is the relative polarity of colliding kinks (in our case the - sign is of interest). Taking the Fourier transform of the expression (3 1) and inserting it into Eqs. (29) and (30) we arrive at the following expression for the spectral density of the emitted waves eL( k) = ; n-‘e;( 1 - u*) 4k6
sinh*[ rk( 1 -u*)
1
“*I + 1
sinh4[ rk( 1 -d,*)“*]
(32)
Finally, the density (32) gives rise to the following total emitted power Ecrn.~ = ~t~(1+8*/21)(1-u*)“~. For the transverse generating terms
h-(x, t>= i
sound, sin’( ‘pi + (p2) on the right hand of Eq. (16)
[ 4 sech(5)
X[ -,sech($$=) Expression
sech($=!=)
tanh()
tanh(s
gives rise to the following
emission
se&(-$%)]
(34)
(34) leads to the spectral density
CT(k) = t r”et( the corresponding E ~~,T=
(33)
1 - v2) 4k6/sinh2[ rk( 1- u2) “*I ,
(35)
total emitted energy being
$ A+(
1 - u2) I’* .
(36)
In this case, the shadows survive the collision and consequently the acoustic losses are small. It proves that the energy of the emitted waves decreases with increasing velocity. The results are in agreement with that of the problem considered in Ref. [ 191, where also E,,, N ( I- u 2, “2.
352
B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
At the end of stage I and during stage II of deformation where the dislocation velocity, i.e. the velocity of the interacting kinks becomes higher, tending to the limiting velocity of the elastic waves, the energy emitted per annihilation event decreases giving rise to a descending AE branch. This fact is in agreement with experimental observations reported in Ref. [ 71. 4.2. Numerical
study
The annihilation process of a dislocation kink-antikink pair into a breather on account of acoustic losses can be illustrated graphically by numerically treating the obtained sGdA system. Indenting to compare the numerical results with the analytical ones obtained in the previous section, we shall consider the kink-antikink interaction with coupling to transverse elastic waves only, i.e. we consider the system cp,-cp,+sin
~=(er/2)~,sin
cp,
(37)
% - u, =er(sin2(rp/2))x.
(38)
Looking for solution u(z) = u(x - ut), Eq. (38) integrates once with respect to z to give u, =
4
sin* o 2’
On substituting rp,--rp,+sin
(39)
U, into Eq. (37), one obtains rp=
E: 4(u;
-u2)
.
4
en ‘-
8(u+-u2)
sin 2~.
(40)
By setting I/I= 2~+ rr, Eq. (40) can be written in the form -sin;
&c1,-$~+2(1--C)
+
[
c
with C = &/4( ut - u ‘) and introducing
ccI,-+4x+
&
(
-sin-
; +2q
sin $
2(1-C)
1
=0
(41)
a new parameter 77such that C = 4r)l( 1+ 477)) Eq. (4 1) yields sin II, . 1
(42)
Eq. (42) is formally the double sine-Gordon equation in a form which is very convenient for numerical studies [ 22,231. On the basis of Eq. (42), the interaction of a dislocation kink-antikink pair has been studied numerically for various initial velocities prior to the collision. The numerical scheme used to solve Eq. (42) was the same simple finite difference technique adopted in the case of a moving dislocation already discussed in Section 3. The starting function was a widely separated kink and antikink moving towards each other with velocity u. This supplied +(x, t = 0) and $(n, t = At), i.e. the first two rows needed for the finite difference equation. The precise definition of the starting function was [ 231 @(x, t) =4 tan-’
sinh(x+ut-x0) cosh( R)
-4
tan-’
sinh(x-
ut+x,)
cash(R)
+ 1 ’
where the parameter R is defined by r) = sinh’( R/4) and the separation distance x0 was chosen to be 7. Fig. 4 gives the dislocation kink-ant&ink collision case with high initial velocities prior to the collision that was studied analytically in Section 4.1. The first panel, Fig. 4a, shows the two kinks meeting and after the collision, each going on, but with reversed amplitude. The corresponding radiation of elastic waves is given in Fig. 4b. The principal peaks follow the evolution of the solitons of Fig. 4a. One can also see a “collision” of these humps which behave
B. Polyzos. A. Trochidis / Wave Motion 21 (1995) 343-355
09 ”
t
Fig. 4. Dislocation
kink-antikink
collision with high initial velocities.
(a) Collision of the kinks. (b) Associated radiation of elastic waves.
like solitons because, after the collision, they pursue their way unaltered. In addition, we note two small humps at the left and right of the graph. These are the radiations at speed - uT and + ur respectively. Fig. 5 illustrates the case of a dislocation kink-antikinkcollision with small initial velocities prior to the collision. Fig. 5a shows that the interaction leads to the formation of an oscillatory “breather-like” bound state localized in x which decays slowly. The corresponding radiation in u is illustrated in Fig. 5b. It can be seen that the decay of the breather is followed by acoustic radiation in the elastic system. This numerical study illustrates efficiently the main features of the analytical results obtained in the previous section.
354
B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
Fig. 5. Dislocation kink-ant&ink collision with small initial velocities. (a) Formation of a localized “breather-like” state. (b) Associated radiation of elastic waves.
5. Conclusions The AE during plastic deformation of crystals has been considered on the basis of a lattice-dynamics approach. Using a FK model, where the rigidity of the substrate were removed, the interaction of a dislocation with both longitudinal and transverse acoustic waves has been investigated. It was shown that this interaction can be described by two d’Alembert wave equations nonlinearly coupled to a sG equation. Within the framework of the sGdA system obtained, the two basic mechanisms of acoustic emission during plastic deformation of crystals have been considered. The mechanism of AE related to nonstationary dislocation motion has been investigated numerically and it was shown that the sudden motion of dislocations is a source of acoustic radiation. The source of AE related to the annihilation of dislocation kink-ant&ink pairs during Frank-Read source operation has been studied both analytically and numerically and the annihilation AE component per event wase calculated. It proves that dislocation annihilation can also be a mechanism of considerable AE during plastic deformation. Computer simulations obtained illustrate graphically the acoustic radiation and confirm the analytical considerations. Obviously, the model of the annihilation process considered here, i.e. the collision of a kink-antikink pair is “ideal” since in real situations many dislocations are present and thus a multisoliton interaction takes place. The investigation of this multisoliton interaction, however, will require more elaborated mathematical techniques [ 12,241. Despite the simplifications already mentioned in the text, the results are in agreement with experimental observations and it is believed that the proposed model contains the necessary physics to provide a better understanding of the AE mechanisms during plastic deformation of crystals. The model developed here allows the investigation of the AE considering specific loading conditions (cycling loading for example) by adding appropriate forcing terms in the sG equation. Furthermore, it allows the calculation of the power spectra of the emitted energy which can be substantially contribute to the identification of the specific emission mechanisms. These problems will be tackled in a future publication.
B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
35s
Acknowledgements This work was supported by the Greek Ministry Collaborative Grant CRG93 1330.
of Research and Technology
Grant 91ED146 and the NATO
References [ 1 1R.M. Fischer and J.S. Lally, “Microplasticity detected by an acoustic technique”, Can. J. Phys.45, 1147-llSZ(l967). 12 I R.T. Sedgwick, “AE from single crystals of LiF and KCI”, J. Appl. Phys. 42, 1728-1732 ( 1968). [ 3 1D.R. James and H. Carpenter, “Relationship between AE and dislocation kinetics in crystalline solids”, .I. Appl. Phys. 42, 46854689 (1971). 14 1J.D. Eshelby, “The interaction of kinks and elastic waves”, Proc. R. Sot. A 226, 222-246 ( 1962). IS ) V.D. Natsik and K.A. Chishko, “Sound emission during annihilation of dislocations”, Fiz. tverd. tela 14, 3126-3129 ( 1972). [ 6 1V.S. Boiko, R.I. Garber, V.K. Kivshik and L.F. Krivenko, “Experimental investigation of transition sound radiation emitted by dislocations emerging to surface”, ZHETF 72, 708-713 (1976). 17 I V.S. Boiko, R.I. Garber, L.F. Krivenko and S.S. Krivulya, “Dislocation description of dynamical behaviour of slip bands”. Fiz. tverd. tela 15, 3426-3428 (1973). 18 I A. Pawelek, “Possibility of a soliton description of AE during plastic deformation of crystals”, J. Appl. Phys. 63. 5320-5325 ( 1988). [ 9 I S. Boffi, G. Caglioti, G. Rizzi and F. Rossito, “Glide systems and Peierls stresses in fee and bee metals from phonon energies”, J. Appl. Phvs. 44, 603610 ( 1973). [ 10 I G.A. Maugin and A. Miled, “Solitary waves in elastic ferromagnets”, Phys. Rev. B 33, 4830-4842 ( 1986). 1I I I J. Pouget and G.A. Maugin, “Solitons and electroacoustic interactions in ferroelectric crystals, 1. Single solitons and domain walls”, Phys. Rev. B 30. 5306-5325 (1984). I I? I J. Pouget and G.A. Maugin, “Solitons and electroacoustic interactions in ferroelectric crystals. 2. Interaction of solitons and radiations”, Phys. Rev. B 3/,4633-4649 (1985). 113I A.S. Rozhavsky, Y.S. Kivshar and A.V. Nedzvetsky, “Dependence of the elastic characteristics of charge-density-wave conductors on the DC electric field”, Phys. Rev. B 40, 41684170 (1989). I14 I J. Pouget and G.A. Maugin, “Nonlinear dynamics of oriented elastic solids I, II”, J. ofElasticity 22, 135-1155, 157-183 ( 1989). I IS 1 G.A. Maugin and J. Pouget, Continuum modelsfor discrete systems, Ed. A.J. Spencer, Balkema, Amsterdam ( 1985). [ 16 I G.A. Maugin, Trends in the Application of Pure Mathematics to Mechanics, Eds. K. Kirchgassner and E. Kroener, Springer, Berlin ( 1986). I 17 I G.A. Maugin, “Applications of an energy-momentum tensor in nonlinear electrodynamics: pseudomomentum and Eshelby stress in solitonic elastic systems”, J. Mech. Phys. Sol. 40, 1543-1558 (1992). I 18 I Y.A. Kivshar and B.A. Malomed, “Dynamics of solitons in nearby integrable systems”, Rev. Mod. Phys. 61,673-915 (1988). 1191 Y.A. Kivshar and B.A. Malomed, “Dynamics of domain walls in elastic ferromagnets and ferroelectrics”, Phys. Rev. B 42. 8561-8570 (1990). I20 I V.D. Natsik and K.A. Chisko, “Dynamics and radiation of dislocation Frank-Read source”, Fiz. tverd. tela 117. 342-345 ( 1975). (211 Y.A. Kivshar and B.A. Malomed, “Sine Gordon solitons in external fieldS”, Physica D 24, 125-136 (1987). interactions in the double sine-Gordon equation”, Phvsica 19 D. 165-20.5 I22 I D.K. Campbell, M. Peyrard and P. Sodano, “Kink-antikink (1986). interaction in a modified sine-Gordon model”, Physica 9 D, 33-S I ( 1983). 1231 M. Peyrard and D.K. Campbell, “Kink-antikink I24 I G.A. Maugin, “Analytical and numerical problems for nonlinear wave propagation problems in “nearly” integrable systems, in: Muthemuticul and Numerical Aspects of Wave Propagation, Eds. R.E. Kleinman et al., SIAM, Philadelphia, USA ( 1993).
WaveMotion
(1995)343-35.5
Dislocation dynamics and acoustic emission during plastic deformation of crystals B. Polyzos, A. Trochidis Universir?, of Thessaloniki,
School
ofEngineering, Department ofPhysics and Mathematics, Thessaloniki 54006, Greece Received 20 August 1994
Abstract A soliton approach to acoustic emission during plastic deformation of crystals is presented. The approach is based on a microscopic Frenkel-Kontorova model where the rigidity of the substrate is removed in order to establish the interaction mechanism between a dislocation and both longitudinal and transverse acoustic waves. It is shown that this interaction is described by a sine-Gordon-d’Alembert system. Within the framework of this system, two basic mechanisms of acoustic emission are investigated both analytically and numerically. One mechanism is related to nonstationary dislocation motion and the other one to the annihilation of dislocation kink-antikink pairs during Frank-Read source operation. In both cases, computer simulations are obtained which illustrate graphically the analytical considerations and model the acoustic radiation. The obtained results are in agreement with existing experimental data and may provide a better physical insight to the acoustic emission mechanisms during plastic deformation of crystals.
1. Introduction Most of the experimental observations of acoustic emission (AE) during plastic deformation of crystals are interpreted in relation to nonstationary motion of dislocations [l-3]. Since dislocation theory was originally motivated by classical electrodynamics [ 41 it was natural to view the AE associated with the nonstationary motion of dislocations as analogous to the electromagnetic radiation of accelerated (or decelerated) charged particles ( “bremsstrahlung” type of acoustic radiation). Another possible mechanism of AE generation, also in analogy to classical electrodynamics, is the process of dislocation annihilation. The first analysis of this type of AE (“transition” type of acoustic radiation) was made by Natsik and Chishko [ 51. Experimental observations of AE during copper crystal tensile deformation [ 6,7], showing an increase (decrease) of AE intensity induced by a sudden increase (decrease) of strain rate, support the hypothesis that dislocation annihilation is an additional source of AE during plastic deformation. Eventhough the existence of AE measurements from a wide range of metals has been known for many years, there is not yet a commonly acceptable quantitative understanding and theoretical interpretation of its origin and of the specific mechanisms causing it even for relatively simple systems such as pure aluminum. A common feature of all existing AE models is their phenomenological character since they are based on continuum elastic theory and do not deal with the atomic nature of dislocations. Pawelek [ 81 made an attempt to consider AE in terms of lattice-dynamics. He pointed out some quantum-mechanical aspects of AE and proposed a 0165-2125/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDfO165-2125(95)00007-0
344
B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
qualitative soliton description of AE during plastic deformation of crystals. The proposed model, however, is rather qualitative and cannot account for the variation of the AE activity as deformation advances and furthermore cannot relate the emitted energy to the strain or external shear stress. It is the aim of this work to outline a new possibility of a more general interpretation of the AE phenomenon within the soliton framework for dislocation dynamics. Since we are interested in the emission of elastic waves by dynamical processes involving solitons (dislocations) it is reasonable to investigate the interaction between elastic waves and solitons (dislocations). For that purpose, a rather simple microscopic lattice model is used. It is a FrenkelKontorova (FK) model distinguishing between two configurations of atoms: one configuration represents the elastic substrate (perfect lattice) while the other, above the slip plane, the atoms making up a dislocation. By removing the rigidity of the substrate energetic exchanges between dislocation and substrate are allowed and the mechanism of interaction between dislocation and both longitudinal and transverse acoustic waves is established in terms of lattice dynamics. In the continuum limit, the basic equations describing the interaction is shown to be a sine-Gordond’Alembert (sGdA) system, i.e. two wave equations for the elastic displacements nonlinearly coupled to a sineGordon (sG) equation which governs the dislocation dynamics. On the basis of the obtained system of coupled nonlinear equations the two basic mechanisms of AE are investigated. The acoustic radiation by an accelerating dislocation cannot be treated analytically and thus is studied numerically. On the contrary, the emission of acoustic waves by annihilating dislocation kink-antikink pairs during Frank-Read source operation is studied both analytically and numerically for small and high velocities prior to the annihilation. Computer simulations are obtained which illustrate graphically the analytical results and model the acoustic radiation. The obtained results are discussed in relation to existing experimental data and may provide physical insight to AE mechanisms during plastic deformation of crystals.
2. Interaction between a dislocation and acoustic waves In this section the interaction between an edge dislocation and both longitudinal and transverse acoustic waves is considered. In our model, the dislocation is represented by replacing the atoms above the slip plane by a chain of N+ 1 identical mass points. The other atoms, making up the elastic substrate, are represented by a chain of N mass points. Two types of motion are considered for the atoms of the substrate: (i) a longitudinal motion of the lattice points and (ii) a transversal motion of the same points. The corresponding elastic displacements are U, and u,, respectively. For the atoms above the slip plane making up the dislocation a displacement y,, is used (Fig. 1). The forces acting on a lattice point n of the substrate result from the interaction with neighbouring points only and are modelled by means of springs with spring constants KL and KT for the longitudinal and transversal directions respectively. For the atoms making up the dislocation only the stretching spring KL is considered. In case of a rigid substrate, a periodic substrate potential of the form A [ 1- cos( 27ry,,l a) ] is used, where A is the constant amplitude and cy is the lattice spacing in the longitudinal direction. According to the continuum theory of elasticity [ 91, the amplitude A is inversely proportional to the lattice spacing d in the transversal direction. By assuming transverse vibrations of the substrate atoms, the lattice spacing d at the point n changes to d - ( u, + , - u,) resulting in a spatial modulation of the constant amplitude A. On the other hand, the longitudinal vibrations of the Y” :-
000000~6000
10
d
l
v?
a
h A- ““0
Fig. 1. The Frenkel-Kontorova
.x
j
I
0
i
0
0
0
model with nonrigid substrate.
34.5
B. Polyzos, A. Trochidis / Wave Motion 21 (I 995) 343-355
substrate modify the lattice spacing CYin the longitudinal direction and the substrate potential becomes non-periodic. Taking into consideration the local compression v, + , - v, at the point n, the actual displacement of the dislocation atoms relative to the potential minimum of the rigid substrate becomes y,, - (v, + , - v,) . It is further assumed, that the interaction potential is of cosine type and should reduce to that of the FK model in case of a rigid substrate (u,, u, = 0). In order to keep our model mathematically tractable, the following simple form of the interaction potential is assumed
A
v=c n
1_
1
_cos 2TYYn -
(“,+I --“A a
un+1-4! d
where the spatial compression of the lattice in both directions has been taken into account. When the displacement v,, U, slowly depend on n, i.e. when 1u,+ , - u, 1 < d and 1u,+ , - v, 1 +c a, the interaction potential given by Eq. ( 1) can be written, after expansion in Taylor series restricted to the first order, in the form V=
CA n [ l-cos
p)]+
+
(u,,,
-0,)
The potential of Eq. (2) leads to the following Hi=
C ~ n
The Hamiltonian H,=
(“,+I
-
u,)
interaction
7
(u,,,
-u,)
sin2(T).
(2)
Hamiltonian
sin
(3)
of the elastic substrate is
&n(L:;+ti;)+ n 2
and the Hamiltonian
sine)+
+.,,
of the dislocation
-uJ2+
1 ,K,(un+,
-u”)*,
(4)
is (5)
Thus, the Hamiltonian
of the whole system is
H=H,+HdfHi. Owing to Eqs. (3)-( are obtained
(6) 6)) the following
discrete equations
of motion for the dislocation
and the elastic substrate
(7)
(8) fJ2U m n
at*
-KT(u,+,
+u”_~
-2~“)
= 7
sin
(9)
In the continuum approximation the “point” quantities v,, u,, y,, can be expanded about n as a function of x and the following system of partial differential equations for the continuum variables is deduced
346
B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
m-
d2Yn
(10)
at2
a20
rnp
a20
2
-K,a
1 ax
=~ITA-
(11)
a214 t 12) Considering
the change of variables and functions
t=2n-&ii%,
2TY
cp= -t
cy
x= $&i7&, 2mJ “= (Y
the system of Eqs. ( lo)-( p,,-c~,+
and
(13)
12) can be written in dimensionless
sin cp= -er_u,
UlI - uxx = c(sin
2n-u u= -, ff
cos cp- TuX
sin2((p/2),
qo>,,
form as (14) (15)
u,-v~u,=ET(sin2(cp/2)),
(16)
with
and the subscript x denoting differentiation with respect to x. It proves that the dynamics of the system considered is described by a sG equation which governs the dislocation dynamics nonlinearly coupled to two acoustic d’Alembert equations for the elastic displacements. Similar sGdA systems have been obtained in other contexts also like the magnetization in elastic ferromagnets [ lo], domain walls in ferroelectrics [ 11,121, charge-density-waves interacting with sound [ 131 and nonlinear elastic crystals with internal degrees of freedom [ 141. In the absence of the right hand coupling terms, Eqs. (15) and (16) are the classical wave equations while Eq. ( 14) is the pure sG equation for solitary waves. In our case, its elementary solution cp(x, t) =4 tan-‘{exp+
(1 -ut)ldl
-u*)
,
(17)
represents the motion of a dislocation kink (or antikink), depending on the f sign, along the dislocation line with velocity u. Thus, the motion of the dislocation as a whole in a given direction (y direction) may be executed in result of a soliton motion in the perpendicular direction (x-direction). We know [ 15-l 71 that the coupling of the sG equation with one or two wave equations make that pure solitons (i.e. solitary waves recovering their full identity after interaction) are not possible, being accompanied by wavelets that propagate proceeding or tail trailing the solitons. Hence, as well as pure sG equation the sGdA system admits solitary-wave solutions which are kink-like in the dislocation system and hump-like shadows in the elastic systems. 3. AE related to nonstationary
dislocation motion
Most of the existing experimental relation to non-stationary dislocation
observations of AE during plastic deformation of crystals are interpreted in motion. For instance, Fisher and Lally [ 1] suggested that AE is a consequence
B. Polyzos. A. Trochidis/ Wave Motion 21 (1995) 343-355
347
of fast collective motion of dislocations while Sedgwick [ 21 has considered the sudden release of dislocation pileups as possible AE sources. In the same spirit, James and Carpenter [ 31 suggested that AE is related mainly to dislocation breakaway from the pinning points. All the aforementioned explanations assume the “bremsstrahlung” type of acoustic radiation caused by accelerated dislocations to be the main source of AE. On the basis of the coupled Eqs. (14)-( 16) one can study analytically only the motion of a single soliton (dislocation) in steady regime looking for propagative-solution functions of the form cp(z) = cp(x - ut). Such solution can be found by substituting u,, u, from Eqs. ( 15)-( 16) into Eq. ( 14) in which case a double sine-Gordon (DsG) equation for cp(z) is obtained [ 121. This problem will be discussed in Section 4. Thus, in order to illustrate the problem of acoustic radiation by an accelerating dislocation the sGdA system is treated numerically. The numerical scheme adopted is a simple finite difference technique usually called “leap-frog” method. We consider first the medium to be originally at rest, containing a dislocation kink (i.e., at t = 0, v = 0, u = 0, duldtl,=,,=O, &ddt~,~O=O and cp(x, 0) =4 tan-’ exp(x -x0)). Then the dislocation kink is suddenly put in motion travelling from left to right as shown in Fig. 2a. Due to the coupling between dislocation and substrate, the motion of the kink induces nonlinear elastic waves in the substrate. In Fig. 2b (which represents the transverse elastic displacement) one can notice two types of waves: a hump-like “shadow” moving with the dislocation from left to right with the speed of the dislocation and the emitted acoustic radiation moving from right to left with the speed of the transverse acoustic waves u.,.. The same picture results for the longitudinal elastic displacement u (Fig. 2c) where the emitted acoustic radiation moves from right to left with the speed of the longitudinal acoustic waves uL. Thus, the results of the numerical treatment of the sGdA system illustrates a situation that is related to the acoustic emission generated by the nonstationary dislocation motion during plastic deformation of crystals. 4. AE related to dislocation annihilation during Frank-Read
source operation
It is very well known that during plastic deformation of crystals a dislocation segment can act as a Frank-Read source. In every stage of the source operation, associated with a generation of successive dislocation loops, sections of the dislocation line of opposite orientations are annihilated and elastic energy is released. In this sense, a FrankRead source is also a source of the transition type of acoustic emission. A qualitative, schematic soliton description of this annihilation mechanism was described by Pawelek [ 81. For the sake of clarity, this description is briefly repeated here. During the initial stage of source operation two rectilinear dislocation sections A,A,? and BIB, lying in adjacent Peierls valleys and approaching one another are considered (Fig. 3a). Their interaction leads to their annihilation (Fig. 3b) and due to the disappearance of the rectangular loop A { A iB\ B 4 elastic energy is emitted. The important dynamical process described above, can be studied with the coupled Eqs. (14)-( 16). Within a similar framework for a ferroelectric system, this process has been studied both numerically and analytically by Pouget and Maugin [ 11,121. In that paper (where the kinks were representing domain walls), the collisions of both kink-kink and kink-antikink pairs have been developed in the adiabatic approximation, i.e. ignoring radiation effects. Next, emission of sound waves by colliding kinks were investigated by means of the Green’s function technique and an expression for the emitted wave field has been obtained. Our analysis is based on the perturbation theory for solitons [ 181 assuming that the coupling between dislocation and elastic substrate is small, i.e. eL, % +Z 1. Thus, to calculate the energy emitted during the annihilation of a dislocation kink-antikink pair we follow the lines of Ref. [ 191. 4.1. Analytical
results
A. Small initial velocities We consider first the collision of a kink-antikink pair with small initial velocities ( u 2 +Z 1) prior to the collision. During the collision annihilation of the kink-antikink pair into a breather can take place provided the velocity is lower than a certain threshold [ 191. The kink and antikink come of their shadows under the action of strong mutual
B. Polyzos, A. Trochidisl Wave Motion 2I (1995) 343-355
348
(b) U X
Fig. 2. AJZ from a moving dislocation.
(a) Motion of a dislocation
kink. (b) Associated
radiation of elastic waves.
B. P~lyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
349
(4
(b)
Fig. 3. Schematic soliton description of dislocation section annihilation during Frank-Read source operation (Ref. [ 81)
attraction. After the collision they develop their shadows anew, while the former shadows decay into acoustic waves of the form u (x f u,_t), u (x 31uTt) propagating in the elastic substrate. The continuum dimensionless Hamiltonian of the system is
with
(18)
Hi= $ux sin (p+e,,u,
7(P sin-?.
(20)
To calculate the energy of the emitted waves, first the profile of the shadows in the elastic system is required. Considering the acoustic shadows of the kink given by Eq. ( 17) we look for solutions in the form u(z) = u(x - ut) , u(z) =u(x-uf).In thiscaseEqs. (15) and (16) yield
350
B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
l)u,
(u*-
=e,(sin
cp), ,
(21)
= lT sin2 z
(z+u~)u,
(
(22)
1z
The above equations integrate once with respect to z to give 0Z=-
Asin’p, ET
u, = -
(G - v2)
(23) sin2 ‘p 2’
(24)
which are the profiles of the shadows travelling with the kink. The energy of the emitted waves due to the decay of the shadows is given by (25) the multiplier 2 in front of the integral taking into account the fact that we have two shadows (one for the kink and one for the antikink). Inserting Eqs. ( 18)) (21) and (22) into Eq. (25) one finds m
Eem=2
l+u2 (l-L?)2
I
-cc
8 v;+Ll* % + TE (u$-u2)2‘
(26)
The emitted energy during the annihilation of a pair of dislocation kinks proves to increase with increasing kink velocity. This energy is related to the dislocation annihilation component of AE per event. Our result is in agreement with the relation obtained by Natsik and Chishko [ 201 who used the dislocation model in a continuum medium. According to their relation the mean energy per annihilation event is proportional to the square of the dislocation velocity prior to annihilation. During stage I of deformation only a single active glide system operates and the dislocations move long distances unobstructed. The kink velocity increases tending asymptotically to the limiting “power balance” velocity and consequently, the emitted energy increases giving rise to an ascending AE branch. B. The ultrarelativistic case
Explicit calculation of the acoustic waves is also possible in the case when the velocities f u of the interacting dislocation kinks are ultrarelativistic, i.e. 1 - v2 -=K1. In this case the kink-ant&ink pair may be approximated by the linear superposition of the waveforms of the free kinks given by Eqs. ( 17) [ 211 a2
=
401 +
v2
(27)
9
where the subscript 1 and 2 are realized as kink and antikink. The longitudinal acoustic wave field can be represented in the following form m
U(X,t) s (29-r)--I
I
dk fi(k, t) eiKr .
(28)
--cc
A similar expression holds for the transverse acoustic field. If F,_(x, t) and FT(x, t) stand for the right-hand side of Eqs. ( 15) and ( 16)) the Fourier amplitude C(k, t) can be brought in the form
B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
dq,(k t> dt
= -
ikq,_(k,t) + cL&(k,
351
(29)
t) ,
where qL( k, t) = 86( k, t) /at - ikfi(x, t) and Fc( k, t) is the Fourier transform of FL(x, t) . To find the amplitudes of the emitted acoustic waves, we assume that the free waves were absent prior to the collision, i.e. at t= - ~0 and we define the final amplitudes as follows dt dbw(k, dt
bLsT(k) = I
t)
(30)
’
with bL.T( k, t) ) = q-J k, t) exp( ikt) . In particular, sin( ‘p, + (~2) gives rise to the following
14sech’($$)
F,_(x, t) = i
emission generating terms
sect?(s)+
x [4o sech(E)
tanh(e)
]
sech(-$$)
tanh(-$$)]
,
(31)
where CJ= cr,cr, = + 1 is the relative polarity of colliding kinks (in our case the - sign is of interest). Taking the Fourier transform of the expression (3 1) and inserting it into Eqs. (29) and (30) we arrive at the following expression for the spectral density of the emitted waves eL( k) = ; n-‘e;( 1 - u*) 4k6
sinh*[ rk( 1 -u*)
1
“*I + 1
sinh4[ rk( 1 -d,*)“*]
(32)
Finally, the density (32) gives rise to the following total emitted power Ecrn.~ = ~t~(1+8*/21)(1-u*)“~. For the transverse generating terms
h-(x, t>= i
sound, sin’( ‘pi + (p2) on the right hand of Eq. (16)
[ 4 sech(5)
X[ -,sech($$=) Expression
sech($=!=)
tanh()
tanh(s
gives rise to the following
emission
se&(-$%)]
(34)
(34) leads to the spectral density
CT(k) = t r”et( the corresponding E ~~,T=
(33)
1 - v2) 4k6/sinh2[ rk( 1- u2) “*I ,
(35)
total emitted energy being
$ A+(
1 - u2) I’* .
(36)
In this case, the shadows survive the collision and consequently the acoustic losses are small. It proves that the energy of the emitted waves decreases with increasing velocity. The results are in agreement with that of the problem considered in Ref. [ 191, where also E,,, N ( I- u 2, “2.
352
B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
At the end of stage I and during stage II of deformation where the dislocation velocity, i.e. the velocity of the interacting kinks becomes higher, tending to the limiting velocity of the elastic waves, the energy emitted per annihilation event decreases giving rise to a descending AE branch. This fact is in agreement with experimental observations reported in Ref. [ 71. 4.2. Numerical
study
The annihilation process of a dislocation kink-antikink pair into a breather on account of acoustic losses can be illustrated graphically by numerically treating the obtained sGdA system. Indenting to compare the numerical results with the analytical ones obtained in the previous section, we shall consider the kink-antikink interaction with coupling to transverse elastic waves only, i.e. we consider the system cp,-cp,+sin
~=(er/2)~,sin
cp,
(37)
% - u, =er(sin2(rp/2))x.
(38)
Looking for solution u(z) = u(x - ut), Eq. (38) integrates once with respect to z to give u, =
4
sin* o 2’
On substituting rp,--rp,+sin
(39)
U, into Eq. (37), one obtains rp=
E: 4(u;
-u2)
.
4
en ‘-
8(u+-u2)
sin 2~.
(40)
By setting I/I= 2~+ rr, Eq. (40) can be written in the form -sin;
&c1,-$~+2(1--C)
+
[
c
with C = &/4( ut - u ‘) and introducing
ccI,-+4x+
&
(
-sin-
; +2q
sin $
2(1-C)
1
=0
(41)
a new parameter 77such that C = 4r)l( 1+ 477)) Eq. (4 1) yields sin II, . 1
(42)
Eq. (42) is formally the double sine-Gordon equation in a form which is very convenient for numerical studies [ 22,231. On the basis of Eq. (42), the interaction of a dislocation kink-antikink pair has been studied numerically for various initial velocities prior to the collision. The numerical scheme used to solve Eq. (42) was the same simple finite difference technique adopted in the case of a moving dislocation already discussed in Section 3. The starting function was a widely separated kink and antikink moving towards each other with velocity u. This supplied +(x, t = 0) and $(n, t = At), i.e. the first two rows needed for the finite difference equation. The precise definition of the starting function was [ 231 @(x, t) =4 tan-’
sinh(x+ut-x0) cosh( R)
-4
tan-’
sinh(x-
ut+x,)
cash(R)
+ 1 ’
where the parameter R is defined by r) = sinh’( R/4) and the separation distance x0 was chosen to be 7. Fig. 4 gives the dislocation kink-ant&ink collision case with high initial velocities prior to the collision that was studied analytically in Section 4.1. The first panel, Fig. 4a, shows the two kinks meeting and after the collision, each going on, but with reversed amplitude. The corresponding radiation of elastic waves is given in Fig. 4b. The principal peaks follow the evolution of the solitons of Fig. 4a. One can also see a “collision” of these humps which behave
B. Polyzos. A. Trochidis / Wave Motion 21 (1995) 343-355
09 ”
t
Fig. 4. Dislocation
kink-antikink
collision with high initial velocities.
(a) Collision of the kinks. (b) Associated radiation of elastic waves.
like solitons because, after the collision, they pursue their way unaltered. In addition, we note two small humps at the left and right of the graph. These are the radiations at speed - uT and + ur respectively. Fig. 5 illustrates the case of a dislocation kink-antikinkcollision with small initial velocities prior to the collision. Fig. 5a shows that the interaction leads to the formation of an oscillatory “breather-like” bound state localized in x which decays slowly. The corresponding radiation in u is illustrated in Fig. 5b. It can be seen that the decay of the breather is followed by acoustic radiation in the elastic system. This numerical study illustrates efficiently the main features of the analytical results obtained in the previous section.
354
B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
Fig. 5. Dislocation kink-ant&ink collision with small initial velocities. (a) Formation of a localized “breather-like” state. (b) Associated radiation of elastic waves.
5. Conclusions The AE during plastic deformation of crystals has been considered on the basis of a lattice-dynamics approach. Using a FK model, where the rigidity of the substrate were removed, the interaction of a dislocation with both longitudinal and transverse acoustic waves has been investigated. It was shown that this interaction can be described by two d’Alembert wave equations nonlinearly coupled to a sG equation. Within the framework of the sGdA system obtained, the two basic mechanisms of acoustic emission during plastic deformation of crystals have been considered. The mechanism of AE related to nonstationary dislocation motion has been investigated numerically and it was shown that the sudden motion of dislocations is a source of acoustic radiation. The source of AE related to the annihilation of dislocation kink-ant&ink pairs during Frank-Read source operation has been studied both analytically and numerically and the annihilation AE component per event wase calculated. It proves that dislocation annihilation can also be a mechanism of considerable AE during plastic deformation. Computer simulations obtained illustrate graphically the acoustic radiation and confirm the analytical considerations. Obviously, the model of the annihilation process considered here, i.e. the collision of a kink-antikink pair is “ideal” since in real situations many dislocations are present and thus a multisoliton interaction takes place. The investigation of this multisoliton interaction, however, will require more elaborated mathematical techniques [ 12,241. Despite the simplifications already mentioned in the text, the results are in agreement with experimental observations and it is believed that the proposed model contains the necessary physics to provide a better understanding of the AE mechanisms during plastic deformation of crystals. The model developed here allows the investigation of the AE considering specific loading conditions (cycling loading for example) by adding appropriate forcing terms in the sG equation. Furthermore, it allows the calculation of the power spectra of the emitted energy which can be substantially contribute to the identification of the specific emission mechanisms. These problems will be tackled in a future publication.
B. Polyzos, A. Trochidis / Wave Motion 21 (1995) 343-355
35s
Acknowledgements This work was supported by the Greek Ministry Collaborative Grant CRG93 1330.
of Research and Technology
Grant 91ED146 and the NATO
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