Modelling the occurrence of disorientations in dislocation structures

COMPUTATIONAL MATERIALS SCIENCE

ELSEVIER

ComputationalMateriak science 5 (1996) 111-125

Modelling the occurrence of disorientations in dislocation structures D. Francke, W. Pantleon*, P. Klimanek Institute of Physical Metallurgy,

Fniberx

University of Mining and Technology Gustav-Zeuner-Strasse

5. D-09596

F&berg,

Germany

Received 15 July 1995; accepted 31 August 1995

Abstract

The appearance of local disorientationswithin the interior of a crystal volume is described in terms of inhomogeneities in the dislocation stmcture. For this purpose the evolution of the microstructure is considered as a consequence of the interaction of a dislocation cell struchue with dislocation fluxes of opposite sign containing different mobile dislocation densities. With the proposed model the formation of disorientations due to several origins can be predicted.

1. Introduction Plastic deformation of a monocrystalline object volume (i.e. a macroscopically extended single crystal or a grain of a polycrystalline material) is generally accompanied by the formation of a substructure (e.g. dislocation cell walls or subboundaries, respectively) and, consequently, the occurrence of local disorientations [ 11. The reason for this phenomenon occurring even in initially undisturbed crystals is the development of deformation inhomogeneities connected with the accumulation of an excess of dislocations with the same sign of the Burgers vector. In the case of plastic strain gradients due to inhomogeneities of the geometry of loading (e.g. local bending) the accumulation of the dislocation excess is achieved by the generation of “geometrically necessary” dislocations [ 21. Another reason for the process is locally inhomogeneous deformation behaviour of the material, e.g. due to the differences in the yield stresses of a hard inclusion in a softer matrix [ 31. But disorientations arise also for (mesoscopically) homogeneous deformation conditions as realized, for instance, in tensile tests of single crystals [4]. There, the occurrence of a dislocation excess is triggered by statistical fluctuations in the dislocation dynamics leading to spatial differences within the microstructure [ 51. The microst~ctural approach presented here allows the examination of both plastic strain gradients and structure inhomogeneities (e.g. particles) as reasons for the formation of disorientations in a quantitatively satisfactory manner. * Correspondingauthor. 0927-0256/%/$15.00 @ 1996 Elsevier Science B.V. AU rights reserved SSDf 0927-0256(95)00062-3

D. Fran&

112

et al. /Computational Materials Science 5 (1956) I1 l-125

2. Modelling 2. I. Microstructural considerations

The dislocation arrangement considered in the present work is a well developed dislocation cell structure with cell walls (width w) containing high densities Q” of forest dislocations and virtually dislocation-free cell interiors. During deformation an evolution of this microstructure takes place, but the cells are still characterized by the following properties (As a point of reference the tensile deformation of copper single crystals in the (100) direction at 873 K is used and all given material data refer to this conditions.): l For the cell diameter d and the total dislocation density etot the scale relation

d=+

l

l

(1)

of Holt [ 61 remains valid. For copper the proportionality factor K is 16 [ 71. According to the “principle of similitude” [ 81 the volurn~ fraction’ 6 = 2w/d of the cell walls is kept constant. This is experimentally confirmed and 5 is determmed to 0.45 for room temperature [ 9,101. Finally, observations of the slip line length on surfaces of (100) orientated single crystals show that the mean free path of dislocations is correlated to the cell diameter, a second manifestation of the “principle of similitude”, and that mobile dislocations are able to pass on average through three cell walls before immobilization [ 11,121. This allows the assumption of an immobilization probability

for a flux of mobile dislocations passing the cell walls [ 13,141. Though the examined microstructure corresponds to cell formation under polyslip conditions, for simplification of the calculations and in order to gain more insight into the behaviour of the structure, only dislocation fluxes due to glide in a single system are investigated instead of those occurring in the (more realistic, but complicated) geometry of multiple slip. As illustrated in Fig. 1 the plastic deformation is assumed to be maintained by fluxes of mobile edge dislocations which penetrate a chain of dislocation cells. During their passage of the cell walls these dislocations are stored as forest dislocations according to the immobilization probability. Supposing that the deformation temperature is sufficiently high for intensive cross slip and subsequent annihilation processes, the role of screw dislocations is ignored. In this paper, the dislocation content in the cell walls is characterized by two different quantities: the commonly used volume density ew = fetot of the dislocation length with dime’” = L-’ and the integrated dislocation density N = w@ of individual cell walls with dim N = L-‘. 2.2. Balance ofjhes According to the Orowan relation p = bj = bvem ,

(3)

a flux j of mobile dislocations carries the plastic strain rate 9. The moving dislocations are created from dislocation sources in the cell walls and move throughout the cell interiors unchanged In the cell walls some of them become immobilized through their interaction with the forest dislocations and stored in the walls or they annihilate with dislocations of opposite Burgers vector. The flux can be decomposed into a part (j-) ’ The factor two is introduced because the cells are equiaxed in the plane perpendicular to the deformation direction only.

D. Fran& et al./Computational Materials Science 5 (19%) 111-125

S

113

i

z

-I _A

T

A

A

T i

v-

i-2

i

i+l

i+2

Fig. 1. A chain of cell walls consisting of forest dislocations.

moving to the right with dislocations of positive Burgers vectors and another one (j+) moving to the left with dislocations of negative sign. For a spatially homogeneous system with a constant dislocation density Q” of forest dislocations, the density of mobile dislocations2 Q is reduced due to their absorption

ae e _=---_* ax A

(4)

In the absence of dislocation sources under steady state conditions a position dependent density e(x) = Q(O)e-X/A

(5)

will be achieved leading to a mean free path of dislocations i=Jxp(x)dx/&(x)dx 0

=A.

(6)

0

After passing a region of width W, the mobile dislocation density is reduced to e(x + w) = Q(X) e+‘* = Q(X) ebP E e(x) q ,

(7)

where q E exp( -P) is an abbreviation for convenience only. The proportionality factor P = w/A = l/3 is a constant reflecting the passage of three cell walls on average by any dislocation. It is identical to the immobilization probability mentioned above. If the sources of dislocation loops are homogeneously distributed in a wall i only, the continuous initial density ei leads to a mobile dislocation density with positive Burgers vectors at a point x within this wall given by x

a(x)

=

d

I

eie-(x-Y)/“dy = $ (1 _ ,-dr)

(8)

0

while its mean value is ei(x)dx = $ 0 2 For convenience, only the dislocations moving to the right hand side ate consideed in this subsection.

(9)

D. Fran&eet al./ ComputationalMaterials Science 5 (19%) I1 I-l 25

114

The density leaving this region at the right boundary is then & = pi(w) = ei( 1 - q)/P. The outcoming density $i of mobile dislocations is assumed to be identical to the incoming one in the adjacent wall i + 1; here absorption reduces the density by a factor q, and the mean mobile density is given by Pit1 -4)lp* Th e same holds for all other cell walls far away, so that the remaining mean mobile dislocation density originating from the wall i in a wall j is given by &+j,i

=

*

Oi+l,i9i-’

(10)

The consistency of this approach can be shown by the total density of absorbed dislocations including the amount of dislocations (Qi - Fiji)absorbed in the wall i itself

(ei-b)+~~~i+j,i=(ei-A)+Q(l-4)~~-’ j=l

j=l = &I; -

gji +

gi

=

Qi

which is indeed the created dislocation density. 2.3. Calculation of the fiuxes The flux j,? of dislocations in a cell wall i moving to the right consists of mobile dislocations (&,j) originating from all (infinite in number) cell walls j on the left, but the absorption in several walls must be taken into account. Additionally, there is some contribution of mobile dislocations created in the wall i itself. In the cell wall the glide velocity of all dislocations should be the same. Therefore, the dislocation flux 3 jEy = u(i)

C

Qi,j = u(i)A$ej

j=-a

the definition

~5

=

t1 -qj2 qi-j-l

lJ

AS

a correlation

P2

A; =0,

~

.+d-j-l

I

*

i>j,

(12)

i
Exactly the same arguments can be used to determine the flux to the left hand side j,? with a correlation matrix A- = (A+)T .

(13)

Assuming a glide velocity v(i) independent of the direction of motion, the total plastic strain rate can be calculated from the created mobile dislocation densities from the Orowan equation for each cell wall separately pi = bji = b( j,? + ji'_ ) =bu(i)

(A++A-)ij~j.

Inversion of this expression allows the evaluation of the total flux j as well as the bias Aj 3 Here and in the remainderof the paper Einstein’s summationconventionwill be used.

( 14)

(15)

D. Fran& et al./Computatianal Material Science 5 (19%) Ill-125

115

i >

t

_L Y t

Y

_L Fig. 2. Cell wall with spontuneouslyunnihihing dislocation dipoles. _ii

=j,Y+ji+=f,

A ji = jfT - j,? =

= ~

(16) o(i) (A+ -

(A’ - A-)ij

A-)ijej

(A+ +

A-),~’ %,

(17)

from the local distribution of plastic strain rates A. 2.4. Evolution equations A homogeneous cell structure of equiaxed cells with the same dislocation-density in each cell wall is taken as the initial microstructure. Additionally, at the beginning of the deformation the initial net Burgers vector of the dislocations within a cell wall is assumed to be zero. That means, the numbers of dislocations with opposite signs are equal and no disorientation exists between the cells. Now, the evolution of the forest dislocation densities in the walls and the excess of dislocations of positive Burgers vector over that with negative has to be considered in some detail. Neglecting any annihilation the density Nt of forest dislocations with positive Burgers vector (I) in the cell wall i may be increased due to the accumulation of mobile dislocations from the flux to the right

(18) and the density NT of dislocations with negative Burgers vector (T) from the flux to the left dNT dt

1

= pj,? .

(19)

If, additionally, annihilation occurs, mobile dislocations with negative sign (T) can be captured by forest dislocations with positive Burgers vectors or vice versa and can annihilate mutually. This process as shown in Fig. 2 will take place if a moving dislocation passes a forest dislocation within a slip plane distance y less than 1.6 nm [ 151 leading to a cross-section of 2y for the reduction of the dislocation densities. A second contribution to the reduction of the dislocation density Nk is due to the fact that the mean free path of edge dislocations should not be affected by possible annihilation. Those dislocations of a flux to the right hand side which lead to an annihilation event and immediately disintegrate with dislocations of negative Burgers vector can not contribute to the storage of forest dislocations.

116

D. Fran&e et al./Computational Materials Science 5 (19%) Ill-125

Therefore, the total changes in dislocation densities are given by %=(P-2yN:)j,y

-2yNfjr,

dNr = (P - 2yNf)jr dr

- 2yNTjT.

I

Introduction of the total dislocation density Ni = NF + NT and the excess dislocation density ANi = Nf - NT leads to dNi =(P_2yN;)ji+2yANiAji, dt dANi -=PAji. dt

(22) (23)

Returning to the usual dislocation density in the cell walls $” = N/w, introducing the disorientation angle [ 161 cu=bAN

(24)

and taking into account the dependence of the wall thickness on the dislocation density (Wi = Kfi/2@> more suitable form of the equations can be achieved

a

(25) d&i = bPAji . dt This equation system of coupled ordinary differential equations will be studied in detail numerically in Section 4. Beforehand, the evolution of the disorientations will be investigated in an analytical manner. Such a treatment is possible without specifying the annihilation length, because the formation of disorientations is not directly influenced by the annihilation of dislocations. In view of the fact that the annihilation event involves both dislocation densities simultaneously, this may be expected. For homogeneous deformation conditions with no gradients in plastic strain the changes in disorientations disappear and there exists a stationary dislocation density from the equation P

-

W

-2y#

=o.

The stationary density becomes Q:~ = (P/K&y)*

(26) = 3.8 x 1014 m-* for the above given values.

3. The analytical solution 3.1. Mathematical aspects of Toeplitz-matrices

The solution of the differential Eqs. (25) requires the calculation of the flux bias Aj and therefore operation with the involved infinite matrices. A look to the correlation matrix A in more detail shows that every diagonal consists of equal elements AG=ALjo=ai+_,

and they can be constructed from a single (one-dimensional) column vector a,‘.

(27)

D. Fran& et al. / ComputationalMaterials Science 5 (1SW) I1 I -I 25

117

The most important feature of such matrices of the Toeplitz type [ 171 is that the elements of the defining column vector can be interpreted as the Fourier coefficients of a generating function Q(8)

2

=

a,e’“”

(28)

It=-ca

This representation of the Toeplitz matrix as a Fourier series allows certain simplifications of the basic matrix operations. For example the inversion of a Toeplitz matrix can simply be undertaken by

An analogous argument holds for multiplication of Toeplitz matrices [ 171. For solving the equation system considered here, a multiplication as well as an inversion step is essential C E (A+ -A-)(A++A-)-I

=T(e+_-O-)/(O++.+).

(30)

In this way the generating function of the matrix C is defined. The matrix elements themself are then derived from the backward transformation

@+-@J +Ce 97

-in@d@

-97

.

(31)

@J+

3.2. Calculation of the matrices For the matrices of Subsection 2.3 the characteristic function of C can be constructed. First of all, the generating function Gp+of the matrix A+ has to be formed

&J+=2

a,+e’“” = 4 + 4(&-l).

(32)

4

n=--oo

Since the left and the right part of the system are equivalent but only the indices of the latter have reversed signs, the function @- is the complex conjugate of @+. The characteristic function of C is found by a+--@@++a-

isin@(l

-q)* = (P(1 +q2) - 1 +q*> +(l-2Pq-q*)cos@

Using integral No. 3.613 3 from Ref. [ 181, the backward transformation leads to C, = sign(n)

-(I

-q)*

1 - 2Pq - q*

&l

Vn # 0,

(34)

co=o. The dependence of the parameter introduced above: (35) with s

=

P(l+q?-1+q* 1 - 2Pq - q*

(36)

118

D. Fran& et al. /Computational Materials Science 5 (19%) I I1 -125

-0.250 ,-0.255 4 k

-0.260 -0.265 -0.270

I: 0.00

0.25

0.50

0.75

1.00

P Fig. 3. Plot of the parameter a vs. capture probability P.

on the capture probability P is only smooth and shown in Fig. 3. As can be seen from the graph, a becomes always negative which leads to an alternating sign of C,. Furthermore, Ial < 1 holds, i.e. as n increases C, vanishes quickly. Since the formation of the disorientation in a wall i depends on the strain rate of all other walls j =p dri

44 u(j)

Ci-j

-

yj

(37)

,

their influence turned out to be determined by the distance i-j only. As one would expect, next neighbours strongly influence the occurrence of disorientation while distant walls have a negligible influence. 3.3. Various deformatiorl profiles In order to calculate the evolution of the disorientations dr, analytically, certain assumptions about the (inhomogenous) relative deformation of each wall have to be made. Thereby, a constant deformation of all cell walls could be superimposed, but would not contribute to the formation of disorientations. The simplest profile in the strain rate distribution one can think of is a “step” (see Fig. 4a) in p. All walls with non-negative indices are deformed by a certain slip A+ while the remaining walls are not strained at all, i.e. the “step” occurs at the wall 0 indicated by a superscript 0. The corresponding function of this vector in Fourier-space is then given by (38) Now, the calculation of the expression (A+ - A-) (A+ + A-)-‘j con -coa0 n 9 = &j&l-

where

cg stands

leads to a vector

nLO, = cpnla-’ ,

n <0,

(39)

for

and a means the same as above. From this the disorientations can be easily calculated, if in all cell walls the same glide velocity is assumed &” = PC,.

(41)

D. Fran& et al./Computational Materialv Science 5 (19%) I1 l-125

.i

A oalo-~

_.-.-.-.-.-._.

_x.=.=,=.=.=.=.=.z. 1.1 IY

119

A

I

;

. g 0.005 .’ *Cl Y O*O~._._._.___.-._._,r -03

- - - - - - I

0

0

I

k+l

03

k+Zl

co

wall index n Fig. 4. Schematic plot of different deformation profiles h: (a) a “step”, (b) a “plateau”, (c) a “‘mmp”.

In the following, it will be shown how several of those steps can be combined to form rather complex deformation profiles. 3.4. Combined deformution profiles As a first example a combination of two steps in opposite directions will be considered here. This is equivalent to the assumption of a finite band of width 1 in which the deformation is localized while no slip occurs in its neighbourhood. As shown in Fig. 4b this “plateau” can be modelled as a subtraction of two steps in the distance I, i.e. cn = c; - ct, = c$ - c;_, , 1 = c; (a-“-’

- J-y,

n
II = c; (a” - a’-“-‘),

OSn
III = $ (a” - u”-‘) (

nil,

(42)

Similarly, a gradient in the strain rate of the width 1 can be understood as a combination of I equally orientated steps. Each of these elementary steps has a strain rate increment of A?/1 and their onset is shifted to the next

D. Fran&

120

et al. / Computational Materials Science 5 (I 994) 1 I1 -125

wall. Another treatable situation is a soft area in the wall chain not only characterized by a region with higher strain rates, but also accompanied by gradients in the neighbourhood. This case is illustrated in Fig. 4c. It can be interpreted as two gradients in j,, each of which consist of elementary steps. Both ramps have then to be subtracted from each other as was already done in the case of the plateau. In this way, one gets five distinguished intervals in the deformation pattern, namely

cn = 1 =

c;- c; g.a-(“+l)(

1 -

II = 3 (&(“+“(

&+l)(J

_

1)

(

1 - &+l + ak) + a”+’ _ 2) )

111= g (a”+‘( 1 - a-‘) - ak+l-(n+qal _ 1)) ) IV = 2 (u”+‘( I - U-’ - a-‘-*) V = j$I(_a”+i(]

- u~+*~-(~+‘)+ 2),

_a-(k+l))(a-‘_l)),

n
(43)

k + 21,

n> k-+21.

In order to make the illustration more transparent, a soft area in a rather hard matrix has been assumed, whereas the practically more important opposite case where a hard “particle” is embedded in a soft matrix results from the preliminary variant by simply swapping the signs of all matrix elements. Assuming a constant dislocation velocity in all walls, the excess dislocation density in each wall ANi as well as the disorientation angle ai can now be derived from Eq. (23). As can be seen from Fig. 5 the highest disorientations occur at the walls next to a boundary between gradients and areas with constant strain rate. This is due to the kinks in the deformation profile at this eight points. In the areas of the gradients one observes a reasonable high and almost constant excess dislocation density, i.e. disorientations accumulate in such bands unrestrictedly. For example for 1 CC n <<1 in the region II one = -A-y/l which is in good agreement with This leads to a disorientation (Y,,= -Ajt/l gets c, = -Aj/Pl. Ashby’s formula saying that the density of geometrically necessary dislocations [ 191 AQ= -kgrady, respectively the disorientation angle LY=-wgrady=

-7

AY

(45)

is proportional to a gradient in plastic strain. In Ref. [ 141 it is discussed that these areas finally accumulate such high disorientations that dynamic recrystallization may occur there. If there is no gradient of such kind as e.g. in areas with homogeneous deformation or no deformation at all, the change in the excess dislocation density quickly drops to negligible values. 4. Numerical calculations 4. I. Extensions of the model

The equation for the determination of the disorientations is not as simple as it seems to be, because the bias of the flux incorporates the ratio of the glide velocities of moving dislocations at different places. In order to

D. Fran&

et al./ Computational Materials Science 5 (19%) 111-125

121

Tm 0.010 \

I

. ?-0.005

-20

-15

-10

-5

0

5

10

15

L

20

,.J 3.6*10U

‘E

3.5*10"

'3.4910U a t '@ 3.3voU

:::::i:L -20

0

0.40

1

0.30 6

I

-15

-10

-5

0

5

10

15

20

-15

-10

-5

0

5

10

15

20

0.20 0.10 0.00 -0.10 -0.20 -0.30 -0.40 -20

wall index n Fig. 5. Simulated disorientations for the case of a ramp in 9”. In order to convert the matrix elements cn into evolved dislocation densities and disorientation angles a time integmtion over 20 s was undertaken.

take into account this fact, further assumptions have to be made. The glide velocity is assumed to depend on stress [ 201 as 0 =

u0

-s

exp (

,-~(r--rJ B

(

(

(46)

>)

with an applied shear stress r and the athermal component of the stress To given by the Taylor relation The values of the activation energy and the prefactor ua are unimportant for the present rP = a,pb@. approach since only the fraction

-49

u(k)

= exp

-gu,pb(

fl-

&!I

(47) >

is needed in the calculations. Finally, the activation volume has to be determined. For the purpose of this paper, which is to give an idea for the formation of disorientations, we have chosen AV to be a constant given by the distance 1/N, = 2y/P of dislocations in the wall under stationary conditions

AVz+&$ =37%' 00

(48)

D. Francke et al. / Computational Materials

122

Science5 (19%) 1 I I-125

A second extension of the model results from a closer inspection of the evolution density. Eq. (25) shows that care has to be taken, if one of the densities N1 or NT In this case, fewer dislocations are assumed to be captured from the traversing involved in the annihilation process. To circumvent this in case of N’ > P/2y equations dNL -=(P-2yNT)j’-Pj’, dt dNT ~ = -2yNT j’ dt

equation for the dislocation exceeds P/2y = 1 x 108m-‘. dislocation fluxes than are we change to the following

(49) ,

which allows an annihilation ol’ the totally captured dislocations without any storage. However, a constant fraction of the flux is assumed to pass the cell wall undepleted. These changes leave the equation for AN unaffected, the only differences are in the evolution of N dN = PAj - 4yNTj’. dt

(51)

The case of NT > P/2y can be handled in a similar manner. 4.2. Numerical

results

For the purpose of time-integration the Runge-Kutta algorithm of fourth order has been used. Also, the problem (A+ -A-) ( Ai +A-) -’ was not calculated numerically; instead the results of the analytical modelling have been used. Some results of such calculations will now be presented, thereby taking Fig. 6 as an example. Here, the above introduced plateau in 9, with a minimum to maximum strain rate ratio of 0.7 was assumed while its maximum value was chosen to be IO-* s- ’ . The disorientation profile is simulated for a continuous deformation of 20 s. In contrast to the assumption of a constant glide velocity leading to the dashed-dotted curves the full lines of Fig. 6 are calculated according to the ratio of the glide velocities (47). As can be seen, this leads to a spreading out of the disorientations while its maximum values decrease. This can be understood as follows: In the areas of a gradient in i/i disorientation formation will occur as already explained. This in turn raises the dislocation densities in the walls according to (25). In comparison with the surrounding areas, there are locally more obstacles for dislocation glide leading to lower effective stresses and lower glide velocities. This will slow down the process of further disorientation formation in those areas and finally result in a stationary state. On the other hand, mobile dislocations attempt to leave regions with relatively high dislocation content. Since they are predominantly of one sign this will give rise to secondary disorientation formation in the walls where they are finally accumulated. Therefore, one gets an asymmetric spreading out of the disorientation profile. As a final result, areas consisting of several cells are linked together, which show large disorientations between each other. Those high and spreaded disorientations could not be calculated if one assumed fluctuations in the initial local dislocation content only. The simulation of a homogeneous deformation in this case leads to a neutralization of locally developing disorientations and tends to a final arrangement where the mean disorientation vanishes. Another case showing the effects of a locally inhomogeneous microstructure is based on an approach of Ref. [ 51 for the dislocation evolution at higher temperatures. There, it is shown that under hot-working conditions two different steady states will be achieved dependent on the initial conditions. Such a behaviour can be introduced in the present, more simplified model by using a shorter annihilation length in some cell walls. Without any coupling to adjacent walls they will develop a higher stationary dislocation density. In order to study the influences of such an assumption on the disorientations in the chain of cell walls, a numerical study is performed, where only the annihilation length in the wall 0 is decreased by a factor of five.

D. Fran&

et al. / Computational Materials Science 5 (19%) 111-l 25

123

9.10"

‘:

E \

6*10" 7*10"

'a 6*10U 5*10M 4*10@ 3*10" -20

-15

-10

-5

0

5

0 \

II

0.0 -0.5

I

i

I

1

wall index n Fig. 6. Simulation showing the influence of the glide velocities of mobile dislocations. The dashed-dotted curves are calculated under the assumption of a constant dislocation velocity, whereas for the full lines the velocity ratio (47) is taken into account.

The calculation with a homogeneous strain rate + = 10s2 s-* for 20 s (but considering local glide velocities) leads to a broad region of increased dislocation density around the inhomogeneity as well as two disorientation bands of opposite signs with disorientation angles about 3-4”. Such a disorientation band is suggested to be a preferred nucleation site for the initiation of dynamic recrystallization [ 141.

5. Conclusions The proposed model of a chain of dislocation cell walls based on the examination of the dislocation fluxes under single slip conditions allows the prediction of the occurrence of disorientations due to inhomogeneities in the plastic deformation or the microstructure. Finally, some precautions have to be stated before applying the model: l In the analytical examination an infinite set of cell walls is considered, and the inversion and calculation makes use of the properties of infinite matrices. In the numerical treatment, of course, only a finite set of walls is investigated, and there are strong effects from the boundaries, because effectively a vanishing strain rate in the outer space is involved. These boundary effects have no serious influences on the numerical calculation because of the rapid spatial damping of disorientations. Therefore, we restricted the presentation of our calculations to an inner zone neglecting five cell walls near the boundary on either side.

124

D. Fran&

er al. /Computational Materials Science 5 (I 9%) 11 I-125

0.0 -1.0

-2.0 -3.0 -20

-15

-10

-5

0

5

10

15

20

wail index n Fig.7. Simulationof a cellwall chain with a decreasedannihilation length in the central wall.

l

l

l

The inversion of Eq. (3) is not completely true for regions with zero strain rate. The convention for the velocities leads to an obviously wrong finite velocity and therefore formally dislocation dipoles have to move with this speed, since no plastic strain rate is allowed. Related to this, there should be no mobile dislocation density in regions with zero strain rate. An underlying assumption of the model is that the transport of mobile dislocations is much faster than the considered time-scale and the evolution of mobile dislocations can be eliminated adiabatically. This implies that there is an instantaneous reaction of the mobile dislocations and they are in steady state at every moment. The consequence is that the model is not valid within regions with very low strain rates because there the evolution of the mobile dislocations is very retarded. The dislocations in such regions will move with the given velocity, but the time to reach a second or a third wall is too high so that the idea of steady state is not justified. Possibly, this problem can be overcome by superimposing a constant deformation rate to all cell walls. A third restriction is due to the applied stress which is assumed to be the same in each cell wall. Furthermore, the formula for the velocity ratio is valid only if in all cases the effective stress, the difference r - rcL, is positive allowing the exponential approximation; for low effective stresses the exponential law has to be replaced by a hyperbolic sine law. Finally, the model is not sufficient to describe the generation of new individual1 cell walls or the destruction of existing walls. However, the rearrangement caused by the changes in the cell sizes is incorporated, which requires no additional walls due to the infinite number of cells under consideration.

D. Fran& et al./ComputationalMaterialsScience 5 (19%) III-125

125

Acknowledgements

This work was performed within the Graduiertenkolleg “Werkstoffphysikalische Modellierung” at the Freiberg University of Mining and Technology. The authors gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft DFG. Note added in proof

In the considerations of this paper only changes in the dislocation content Ni of a wall i, which result from additional storage or annihilation of edge dislocations, are taken into account. A variation of Ni due to changes of the wall thicknes can be achieved easily, if an additional term Q’”(dw/dt) is incorporated in the evolution Eqs. (20-23), which will rewrite then

(52) (53) This alteration will lead to some (minor) quantitive changes in the results, whereas the qualitative behaviour is not changed at all. However, the provement of Ashby’s relation for the geometrically necessary dislocations becomes more complicated due to the fact that a constant spatial gradient grad? leads to a strain dependent strain rate gradient A? = wi grad p in each of the cell walls. The factor wi will allow the immediate integration of (53) as before and the application of the formulas stays justified [ 141. References [ I] J. Gil Sevillano, P van Houtte and E. Aemoudt, Prog. Mater. Sci. 25 (1980) 69. [2] M.F. Ashby. Philos. Mag. 14 (1966) 1157. [3] N.A. Fleck, GM. Muller, M.F. Ashby and J.W. Hutchinson, Acta Metail. Mater. 42 (1994) 475. 141 A. Orlova and M. BoEek, Z. Metallkde. 84 (1993) 806. [ 51 W. Pantleon and I? Klimanek, in: Proc. 15th Rise Int. Symposium Numerical Predictions of Deformation Processes and the Behaviour of Real Materials, eds. S.I. Andersen et al., Rise National Laboratory, Roskilde, Denmark (1994) p. 439. [6] D.L. Holt, J. Appl. Phys. 41 (1970) 3197. [71 M.R. Staker and D.L. Hoh, Acta Metall. 20 (1972) 569. [ 81 D. Kuhlmann-Wilsdorf, Metall. Trans. A 16A (1985) 2091. 191 D. Knoesen and S. K&zinger. Acta Metall. 30 (1982) 1219. [ 101 E. Goettler, Philos. Mag. 28 (1973) 1057. [ 111I? Ambrosi and C. Schwink, Scripta Metall. 12 (1978) 303. [ 121 Y. Kawasaki and T. Takeuchi. Scripta Metall. 14 (1980) 183. [ 131 A.S. Argon and F! Haasen, Acta Metall. Mater. 41 (1993) 3289. [ 141 W. Pantleon, Ph.D. thesis, Freibetg University of Mining and Technology (1995). submitted. [ 151 U. Essmann and H. Mughrabi, Philos. Mag. A 40 (1979) 731. [ 161 W.T. Read and W. Shockley, Phys. Rev. 78 (1950) 275. [ 171 H. Widom, Stud. Math. 3 (1965) 179. [ 181 IS. Gradshteyn and I.M. Ryzhii, Table of Integrals, Series, and Roducts (Academic Press, New York, 1980). [ 191 M.F. Ashby, Philos. Mag. 21 (1970) 399. [20] A. Seeger, Philos. Mag. 45 (1954) 771.