- Email: [email protected]
Link between the individual and continuum approaches of the description of the collective behavior of dislocations
MATESCIENCE ENGINEERING ELSEVIER
Materials Science and Engineering A234-236
&
A
(1997) 249-252
Link between the individual and continuum approaches of the description of the collective behavior of dislocations I. Groma *, P. Balogh Department
of General
Physics,
Eottaos
University
Budapest,
Budapest
Museum
krt.
6-8, POB
323, H-1445
Budapest,
Hungary
Received 29 January 1997; received in revised form 2 April 1997
Abstract The properties of a system of parallel edge dislocations is investigated by numerical and analytical methods. Although the considered 2D assembly is a strong simplification compared to the dislocation systems which develop in real crystals, computer simulations show that it can reproduce several important features of the plastic behavior of single crystals. In the second half of the paper a self consistent field description is derived from the equation of motion of the individual dislocations. The outlined method creates a link between the individual and continuum descriptions of the behavior of the dislocations without any ad hoc assumption. In the model the precise form of the elastic interaction of the dislocations is taken into account. Within this framework the stability of the homogeneous dislocation distribution is investigated. 0 1997 Elsevier Science S.A. Keywords:
Plasticity; Dislocation structures; Stability
1. Introduction As it is well known the dislocations formed during plastic deformation of crystalline materials may arrange themselves in different ordered structures. Because of the long range nature of the dislocation interaction and the high degree of freedom of the system, the problem is extremely difficult to deal with. The existing models can be divided into two groups, the microscopic and the mesoscopic approaches, depending on the length scale applied. In the microscopic models the collective behavior of individual dislocations is investigated in most casesby computer simulations [l-6]. In the mesoscopic (or continuum) models the dislocation system is described by a few continuous variables, like the dislocation density [7,9], or mobile and inmobile dislocation density [8]. The main common disadvantage of the continuum models is that each is based on several ad hoc assumptions which are difficult to derive from the properties of individual dislocations. Therefore, for a clearer understanding of dislocation pattern formation, the link between the two length scale model should be established. * Corresponding author. Tel.: + 36 1 2667927; fax: + 36 1 2667927; e-mail: [email protected] 0921-5093/97/$17.00 0 1997 Elsevier Science S.A. All rights reserved. PII SO921 -5093(97)00150-O
The purpose of the present work is to reduce this gap between the two length scale by investigating the dynamics behavior of a system of straight parallel edge dislocations. The assembly considered is certainly a strong simplification of the real problem. However, in the first part of the paper it will be demonstrated by computer simulation that it is already able to reproduce several features of plastic properties of single crystals. In the second part, a continuum description will be derived from the equations of motion of the dislocations and the stability of the homogeneous dislocation distribution will be investigated.
2. Computer simulations Let us consider N straight edge dislocations positioned at the point J,, i = 1, 2,..., N in the xy plane perpendicular to the dislocation lines. Assuming a drag like motion of dislocations the dynamics of the system is determined by the following equation [5]:
where bi is the Burgers vector of the ith dislocation, tij is the stressfield at the ith dislocation created by thejth
250
I. Groma,
Applied
P. Balogh /Materials
Science
and Engineering
A234-236
(1997)
249-252
Stress
Plastic & Appl. Current
Strain
E Time
Fig. 1. ‘Measured’ stress-stain curve and the final dislocation configuration
(left upper box), (right box).
the plastic
current
one, Gext(t) is the external stress, and B is the dislocation mobility. In the computer simulations presented in this paper Eq. (1) is solved by numerical integration at different initial configurations and deformation modes. Fig. 1 shows a typical simulation result. In order to be able to compare the results with experimental evidence obtained on unidirectional deformation of single crystals, in the simulation the external deformation rate was kept constant. The external stressnecessary for this was calculated as a function of time (see [5] for further details). The dislocations could move in two slip systems enclosing 45” and 105” with the loading direction. This corresponds to a single slip orientation becausethe Schmid factor is much larger for the first one. The initial configuration consisted of 150 randomly distributed relatively narrow dislocation dipoles. A random dislocation creation was allowed by adding a new dipole to the vicinity of an existing one with a probability proportional to the local stress. In order to investigate whether the observed stress increase corresponds to real hardening, at a certain moment the deformation direction was reversed and later on it was turned back to the original one. (The external rate is plotted as a function of time in the the left bottom box on Fig. 1). The ‘measured’ stress-strain curve and the plastic current as a function of time are plotted in the upper and bottom left boxes in Fig. 1, respectively. Since the calculation was carried out with dimensionlessvariables [5] the units are arbitrary in Fig. 1. The final dislocation
and external
deformation
rate as a function
of time (left bottom
box)
configuration containing 992 dislocations can be seenin the right hand side box. As can be seenin Fig. 1, the initial dipole configuration prevents dislocation motion (at the beginning of the simulation the plastic current is practically zero and the stress increases linearly) until the external stress reaches the level which is large enough to unbound some dipoles, i.e. the system has a finite flow stress. Subsequently there is a short period when some dislocation can move nearly free. As a result of this the slope of the stress-strain curve becomes very small. This part shows a number of similarities with the easy glide (stage I) deformation regime. It has to be mentioned that with other initial configurations we could observe a much longer stage I [5]. At a certain deformation level a much more compact dislocation configuration starts to form. It leads to a faster stress increase, so it can be associated with the onset of the stage II deformation regime. For demonstrating that the final configuration is more ordered than the initial one, Fig. 2 shows the 25 largest dislocation free circles one can draw at the initial (top box) and the final (bottom box) dislocation configuration. Due to the random dislocation distribution the circles have more or less the same radii at the starting configuration. In contrast with this there are much larger circles in the final one. Another important property of the model system can be observed if the deformation direction is reversed in the stage II regime. As it can be seen in Fig. 1, after a
I. Groma,
P. Balogh /Materials
Science
short relaxation period the plastic current becomes zero and the stress decreases linearly with the same slope as it has in the initial ‘elastic’ regime. If the loading direction is reversed back there is no observable dtslocation motion until the stress reaches the same level as it was when the unloading started, i.e. the system undergoes work hardening.
and Engineering
A234-236
(1997)
249-252
251
in the following we assume that the_system consists of only one type of dislocation (with b parallel to the x axis) and the external stress is zero. As a first step let us investigate the case when the number of dislocation is conserved. As a consequence of this fN has to fulfill the relation fN(4 i,, ?2...tN) dil d&. . . dtLV
= fN(t + At, i, + &At, +Z + i$At...L;V + CNAt) 3. Self consistent field description
x d(u’, + 6,A.t) d(L; + i&At)... d(PN + 6,vAt).
As it is well known from statistical mechanics, instead of describing an N particle system with co&jnzky LT ZDC @TZT~ spmi, v?lt” cz-LT&a>LT pwkiy’ he same information from time evaluation of the N particle distribution function f,. For the sake of simplicity
(2)
From Eq. (1) and Eq. (2) one obtains that [lo] (3) where z is the shear stress field of a dislocation. By integrating Eq. (3) over the subspace tZ2,t3,...t,V one finds that
--P++
(i,, L;, t)&(i,
- L;) dt2
(4)
- &) dC2;.
(5)
and
- p _ _ (i,, g2;,t)&(i,
where P+,Pare the one, p+ +,p+ ~ and pP _ are two particle density functions with the corresponding positive and negative dislocation signs. (We obtain two equations because the result depends on the sign of the first dislocation.) Eq. (4) and Eq. (5) contain the second order density functions, so they do not form a closed system of equations. However, if the fluctuation of the dislocation density is small enough one can neglect the correlation functions. It means that the two particle density functions can be given as [lo] Pi,jCil, 32) =
Pi(al)Pj(ti?;),
i, j = + , - .
With this assumption, after adding Eq. (4) and Eq. (5) one arrives at k(F,,
t)&(i,
(6) and subtracting
- g2) dtZ (7)
and
Fig. 2. The 25 largest dislocation free circles at the initial and at the final (bottom box) dislocation configurations.
(top
box)
in which the total dislocation p ~ (v’) and the sign dislocation (i) were introduced. Since
density p(i) = p +(i) + density k(i) = p + (i)-p.
I. Groma,
P. Balogh
/Materials
Science
and Engineering
A234-236
(1997)
249-252
The instability criterion (10) makes it possible to investigate the influence of stress truncation. Fig. 3 shows T(4) if one truncates z at ]i( = R. Since T(g) can be negative the stress truncation always leads to artificial pattern formation. This is in agreement with the numerical results of Gulluoglu et. al. [l].
4. Conclusions
o0
Fig.
5
3. The
k(i,,
T(4)
function
10
15
in case of stress truncation
t)z(i - iI) di, = r,,,(3).
20
at radius
R.
(9)
s
is the total shear stress created by all the dislocations, Eq. (7) and Eq. (8) are a self-consistent field description of the time evolution of the problem. It is easy to see that the external stress has to be taken into account by substituting rint with rint + z,,~, and dislocation creation can be allowed for by adding a source term g(p, k) to the right hand side of Eq. (7). The linear stability analysis of Eq. (7) and Eq. (8) leads to the instability criterion
$ - N”(4)
The properties of a 2D dislocation assembly were investigated. It was shown by numerical simulations that the system undergoes work hardening and is able to reproduce several properties of the deformation stages I and II observed on single crystals. Starting from the equations of motion of individual dislocations a self-consistent field description was derived. Its stability analysis shows that the elastic dislocation interaction is not enough to introduce pattern formation, but already an infinitesimally small dislocation production term or the introduction of stress truncation can lead to growing perturbations. The obtained instability criterion does not give a definite modulation wavelength, so to be able to compare the result we obtained on the individual dislocation system and from the continuum approach, further numerical investigations are necessary.
Acknowledgements We are grateful to Professors Lendvai and Kovacs for discussions and for their stimulating support of this work. We also acknowledge the financial support of OTKA under the contract numbers T 014987 and T 017609.
References
where q is the wave number of the periodic perturbation and T(d) = iq,
z(t) exp(@)di. (11) s By substituting the shear stress field of an edge dislocation into Eq. (11) one obtains that r(q) is proportional to q:qz/q4, This means that without the source term there is no perturbation which could lead to instability. However, since T(a) vanishes along the x and y axises if dg/dp has any positive value there are growing modulations.
A. Gullouglu, D. Srolovity, R. LeSar, P. Lomdahl, Scripta Met. 23 (1989) 1347. 121N.M. Ghoniem, R.J. Amodeo, Phys Rev. B 41 (1989) 6958. J.A. Blume, A. Needleman, Acta Met. Mat. 41 [31 V.A. Lubarda, (1993) 625. C.S. Hartley, Modeling Simul. Mater. Sci. Eng. [41 A.N. Gullouglu, 1 (1992) 1. G.S. Pawley, Mat. Sci. Eng. A 164 (1993) 306. [51 I. Groma, t61 R. Fournet, J.M. Salazar, Phys. Rev. B 53 (1996) 6283. [71 D.L. Holt, J. Appl. Phys. 41 (1970) 3179. PI D. Walgraef, E.C. Aifantis, J. Appl. Phys. 15 (1985) 688. S. Libovicky, Scripta Metall. 20 (1986) 1625. [91 J. Kratochvil, [lOI I. Groma, Phys. Rev. B. (1997) in press.
Ul
Materials Science and Engineering A234-236
&
A
(1997) 249-252
Link between the individual and continuum approaches of the description of the collective behavior of dislocations I. Groma *, P. Balogh Department
of General
Physics,
Eottaos
University
Budapest,
Budapest
Museum
krt.
6-8, POB
323, H-1445
Budapest,
Hungary
Received 29 January 1997; received in revised form 2 April 1997
Abstract The properties of a system of parallel edge dislocations is investigated by numerical and analytical methods. Although the considered 2D assembly is a strong simplification compared to the dislocation systems which develop in real crystals, computer simulations show that it can reproduce several important features of the plastic behavior of single crystals. In the second half of the paper a self consistent field description is derived from the equation of motion of the individual dislocations. The outlined method creates a link between the individual and continuum descriptions of the behavior of the dislocations without any ad hoc assumption. In the model the precise form of the elastic interaction of the dislocations is taken into account. Within this framework the stability of the homogeneous dislocation distribution is investigated. 0 1997 Elsevier Science S.A. Keywords:
Plasticity; Dislocation structures; Stability
1. Introduction As it is well known the dislocations formed during plastic deformation of crystalline materials may arrange themselves in different ordered structures. Because of the long range nature of the dislocation interaction and the high degree of freedom of the system, the problem is extremely difficult to deal with. The existing models can be divided into two groups, the microscopic and the mesoscopic approaches, depending on the length scale applied. In the microscopic models the collective behavior of individual dislocations is investigated in most casesby computer simulations [l-6]. In the mesoscopic (or continuum) models the dislocation system is described by a few continuous variables, like the dislocation density [7,9], or mobile and inmobile dislocation density [8]. The main common disadvantage of the continuum models is that each is based on several ad hoc assumptions which are difficult to derive from the properties of individual dislocations. Therefore, for a clearer understanding of dislocation pattern formation, the link between the two length scale model should be established. * Corresponding author. Tel.: + 36 1 2667927; fax: + 36 1 2667927; e-mail: [email protected] 0921-5093/97/$17.00 0 1997 Elsevier Science S.A. All rights reserved. PII SO921 -5093(97)00150-O
The purpose of the present work is to reduce this gap between the two length scale by investigating the dynamics behavior of a system of straight parallel edge dislocations. The assembly considered is certainly a strong simplification of the real problem. However, in the first part of the paper it will be demonstrated by computer simulation that it is already able to reproduce several features of plastic properties of single crystals. In the second part, a continuum description will be derived from the equations of motion of the dislocations and the stability of the homogeneous dislocation distribution will be investigated.
2. Computer simulations Let us consider N straight edge dislocations positioned at the point J,, i = 1, 2,..., N in the xy plane perpendicular to the dislocation lines. Assuming a drag like motion of dislocations the dynamics of the system is determined by the following equation [5]:
where bi is the Burgers vector of the ith dislocation, tij is the stressfield at the ith dislocation created by thejth
250
I. Groma,
Applied
P. Balogh /Materials
Science
and Engineering
A234-236
(1997)
249-252
Stress
Plastic & Appl. Current
Strain
E Time
Fig. 1. ‘Measured’ stress-stain curve and the final dislocation configuration
(left upper box), (right box).
the plastic
current
one, Gext(t) is the external stress, and B is the dislocation mobility. In the computer simulations presented in this paper Eq. (1) is solved by numerical integration at different initial configurations and deformation modes. Fig. 1 shows a typical simulation result. In order to be able to compare the results with experimental evidence obtained on unidirectional deformation of single crystals, in the simulation the external deformation rate was kept constant. The external stressnecessary for this was calculated as a function of time (see [5] for further details). The dislocations could move in two slip systems enclosing 45” and 105” with the loading direction. This corresponds to a single slip orientation becausethe Schmid factor is much larger for the first one. The initial configuration consisted of 150 randomly distributed relatively narrow dislocation dipoles. A random dislocation creation was allowed by adding a new dipole to the vicinity of an existing one with a probability proportional to the local stress. In order to investigate whether the observed stress increase corresponds to real hardening, at a certain moment the deformation direction was reversed and later on it was turned back to the original one. (The external rate is plotted as a function of time in the the left bottom box on Fig. 1). The ‘measured’ stress-strain curve and the plastic current as a function of time are plotted in the upper and bottom left boxes in Fig. 1, respectively. Since the calculation was carried out with dimensionlessvariables [5] the units are arbitrary in Fig. 1. The final dislocation
and external
deformation
rate as a function
of time (left bottom
box)
configuration containing 992 dislocations can be seenin the right hand side box. As can be seenin Fig. 1, the initial dipole configuration prevents dislocation motion (at the beginning of the simulation the plastic current is practically zero and the stress increases linearly) until the external stress reaches the level which is large enough to unbound some dipoles, i.e. the system has a finite flow stress. Subsequently there is a short period when some dislocation can move nearly free. As a result of this the slope of the stress-strain curve becomes very small. This part shows a number of similarities with the easy glide (stage I) deformation regime. It has to be mentioned that with other initial configurations we could observe a much longer stage I [5]. At a certain deformation level a much more compact dislocation configuration starts to form. It leads to a faster stress increase, so it can be associated with the onset of the stage II deformation regime. For demonstrating that the final configuration is more ordered than the initial one, Fig. 2 shows the 25 largest dislocation free circles one can draw at the initial (top box) and the final (bottom box) dislocation configuration. Due to the random dislocation distribution the circles have more or less the same radii at the starting configuration. In contrast with this there are much larger circles in the final one. Another important property of the model system can be observed if the deformation direction is reversed in the stage II regime. As it can be seen in Fig. 1, after a
I. Groma,
P. Balogh /Materials
Science
short relaxation period the plastic current becomes zero and the stress decreases linearly with the same slope as it has in the initial ‘elastic’ regime. If the loading direction is reversed back there is no observable dtslocation motion until the stress reaches the same level as it was when the unloading started, i.e. the system undergoes work hardening.
and Engineering
A234-236
(1997)
249-252
251
in the following we assume that the_system consists of only one type of dislocation (with b parallel to the x axis) and the external stress is zero. As a first step let us investigate the case when the number of dislocation is conserved. As a consequence of this fN has to fulfill the relation fN(4 i,, ?2...tN) dil d&. . . dtLV
= fN(t + At, i, + &At, +Z + i$At...L;V + CNAt) 3. Self consistent field description
x d(u’, + 6,A.t) d(L; + i&At)... d(PN + 6,vAt).
As it is well known from statistical mechanics, instead of describing an N particle system with co&jnzky LT ZDC @TZT~ spmi, v?lt” cz-LT&a>LT pwkiy’ he same information from time evaluation of the N particle distribution function f,. For the sake of simplicity
(2)
From Eq. (1) and Eq. (2) one obtains that [lo] (3) where z is the shear stress field of a dislocation. By integrating Eq. (3) over the subspace tZ2,t3,...t,V one finds that
--P++
(i,, L;, t)&(i,
- L;) dt2
(4)
- &) dC2;.
(5)
and
- p _ _ (i,, g2;,t)&(i,
where P+,Pare the one, p+ +,p+ ~ and pP _ are two particle density functions with the corresponding positive and negative dislocation signs. (We obtain two equations because the result depends on the sign of the first dislocation.) Eq. (4) and Eq. (5) contain the second order density functions, so they do not form a closed system of equations. However, if the fluctuation of the dislocation density is small enough one can neglect the correlation functions. It means that the two particle density functions can be given as [lo] Pi,jCil, 32) =
Pi(al)Pj(ti?;),
i, j = + , - .
With this assumption, after adding Eq. (4) and Eq. (5) one arrives at k(F,,
t)&(i,
(6) and subtracting
- g2) dtZ (7)
and
Fig. 2. The 25 largest dislocation free circles at the initial and at the final (bottom box) dislocation configurations.
(top
box)
in which the total dislocation p ~ (v’) and the sign dislocation (i) were introduced. Since
density p(i) = p +(i) + density k(i) = p + (i)-p.
I. Groma,
P. Balogh
/Materials
Science
and Engineering
A234-236
(1997)
249-252
The instability criterion (10) makes it possible to investigate the influence of stress truncation. Fig. 3 shows T(4) if one truncates z at ]i( = R. Since T(g) can be negative the stress truncation always leads to artificial pattern formation. This is in agreement with the numerical results of Gulluoglu et. al. [l].
4. Conclusions
o0
Fig.
5
3. The
k(i,,
T(4)
function
10
15
in case of stress truncation
t)z(i - iI) di, = r,,,(3).
20
at radius
R.
(9)
s
is the total shear stress created by all the dislocations, Eq. (7) and Eq. (8) are a self-consistent field description of the time evolution of the problem. It is easy to see that the external stress has to be taken into account by substituting rint with rint + z,,~, and dislocation creation can be allowed for by adding a source term g(p, k) to the right hand side of Eq. (7). The linear stability analysis of Eq. (7) and Eq. (8) leads to the instability criterion
$ - N”(4)
The properties of a 2D dislocation assembly were investigated. It was shown by numerical simulations that the system undergoes work hardening and is able to reproduce several properties of the deformation stages I and II observed on single crystals. Starting from the equations of motion of individual dislocations a self-consistent field description was derived. Its stability analysis shows that the elastic dislocation interaction is not enough to introduce pattern formation, but already an infinitesimally small dislocation production term or the introduction of stress truncation can lead to growing perturbations. The obtained instability criterion does not give a definite modulation wavelength, so to be able to compare the result we obtained on the individual dislocation system and from the continuum approach, further numerical investigations are necessary.
Acknowledgements We are grateful to Professors Lendvai and Kovacs for discussions and for their stimulating support of this work. We also acknowledge the financial support of OTKA under the contract numbers T 014987 and T 017609.
References
where q is the wave number of the periodic perturbation and T(d) = iq,
z(t) exp(@)di. (11) s By substituting the shear stress field of an edge dislocation into Eq. (11) one obtains that r(q) is proportional to q:qz/q4, This means that without the source term there is no perturbation which could lead to instability. However, since T(a) vanishes along the x and y axises if dg/dp has any positive value there are growing modulations.
A. Gullouglu, D. Srolovity, R. LeSar, P. Lomdahl, Scripta Met. 23 (1989) 1347. 121N.M. Ghoniem, R.J. Amodeo, Phys Rev. B 41 (1989) 6958. J.A. Blume, A. Needleman, Acta Met. Mat. 41 [31 V.A. Lubarda, (1993) 625. C.S. Hartley, Modeling Simul. Mater. Sci. Eng. [41 A.N. Gullouglu, 1 (1992) 1. G.S. Pawley, Mat. Sci. Eng. A 164 (1993) 306. [51 I. Groma, t61 R. Fournet, J.M. Salazar, Phys. Rev. B 53 (1996) 6283. [71 D.L. Holt, J. Appl. Phys. 41 (1970) 3179. PI D. Walgraef, E.C. Aifantis, J. Appl. Phys. 15 (1985) 688. S. Libovicky, Scripta Metall. 20 (1986) 1625. [91 J. Kratochvil, [lOI I. Groma, Phys. Rev. B. (1997) in press.
Ul