NUMERICAL SIMULATION OF DISLOCATION DYNAMICS BY MEANS OF PARAMETRIC APPROACH

Transcription

1 Proceedings of Czech Japanese Seminar in Applied Mathematics August -7,, Czech Technical Universit in Prague pp NUMERICAL SIMULATION OF DISLOCATION DYNAMICS BY MEANS OF PARAMETRIC APPROACH VOJTĚCH MINÁRIK1, JAN KRATOCHVÍL, AND KAROL MIKULA 3 Abstract. The aim of this contribution is to present the current state of our research in the field of numerical simulation of dislocations moving in crstalline materials. The simulation is based on recent theor treating dislocation curves and dipolar loops interacting b means of forces of elastic nature and hindered b the lattice friction. The motion and interaction of one dislocation curve and one or more dipolar loops placed in 3D space is considered. Results of numerical eperiments made b parametrical approach to the problem (using flowing finite volume method) are presented and behaviour of the motion is described in detail. Ke words. FDM, flowing finite volume method, method of lines, parametric curves AMS subject classifications. 3K, U, 7M, 7S1 1. Introduction. Plastic deformation of crstalline solids is a result of the motion of dislocations. The theor of dislocations is described in number of tet books, e.g. [7]. Here we recall onl the basic mobile properties of dislocations and the nature of their mutual interactions. A dislocation is a line defect of the crstal lattice. Along the dislocation line the regular crstallographic arrangement of atoms is disturbed. The dislocation line is represented b a closed curve or a curve ending at the surface of the crstal. At low homologous temperatures the dislocations can move onl along crstallographic planes (the slip planes) with the highest densit of atoms. The motion results in mutual slipping of the neighbouring parts of the crstal along the slip planes. The slip displacement carried b a single dislocation, called Burgers vector, is equal to one of the vectors connecting the neighbouring atoms. The displacement field of atoms from their regular crstallographic positions around a dislocation line can be treated (ecept the close vicinit of the line) as elastic stress and strain fields. On the other hand, a stress field eerts a force on a dislocation. The combination of these two effects causes the elastic interaction among dislocations. There are two tpes of dislocations having much different characteristic length scales and mobile properties: dislocation curves and dipolar loops. Detailed description of these two tpes of dislocations can be found in [7] or [1]. The aim of this contribution is not to describe theor of dislocations; we would like to present results of our simulations and to comment them. Readers interested more in mathematical model and more details about the flowing finite volume method ma find information in [1]. 1 Department of Mathematics, Facult of Nuclear Sciences and Phsical Engineering, Czech Technical Universit in Prague, Trojanova 13, 1 Praha, Czech Republic. Department of Phsics, Facult of Civil Engineering, Czech Technical Universit in Prague, Thákurova 7, 1 9 Praha, Czech Republic. 3 Department of Mathematics and Descriptive Geometr, Slovak Universit of Technolog, Radlinského 11, 13 Bratislava, Slovakia. 1

2 Numerical simulation of dislocation dnamics 19. Dislocation curve and dipolar loop. We consider a plane dislocation segment with fied ends; the segment represented b a plane curve can bow in a slip plane which is identified with the z-plane of the coordinate sstem, i.e. =. If the dislocation segment approaches a loop, the curve can pass b or the curve and the loop start to move together or the curve is stopped b the loop [9]. F b F l F z h (a) Force interaction (b) DL geometr Fig..1. Dislocation dnamics problem geometr: (a) Dislocation curve and dipolar loop interaction; (b) Dipolar loop geometr As the loop is allowed to move along the direction parallel to its Burgers vector b onl (see Figure.1a) just the force component in that direction causes the loop motion. Additionall, the lattice friction acts against the motion. The detailed condition for the dislocation curve and the loop is specified in Section 3.3. The position of the loop is represented b 3 coordinates of its centre. There are two tpes of dipolar loops: a vacanc dipolar loop and an interstitial dipolar loop; each tpe has two stable configurations [1] with normals 1 [1, 1, ] and 1 [1, 1, ]. In vacanc loops a strip of atoms in regular crstallographic positions is missing. On the other hand; in interstitial loops an etra strip of atoms is added. For that reason the vacanc and interstitial loops produce different stress fields. The Burgers vectors of vacanc and interstitial loops have opposite directions. This fact we incorporate into our model b using negative value for the -ais component of the Burgers vector for interstitial loop. According to [1] dipolar loops are small rectangles denoted V 1, V, I 1, and I with longer sides parallel to the z-ais of the coordinate sstem. Average size of a dipolar loop is denoted b l and h. 3. Mathematical model Interaction of dislocation lines. The interaction force per unit length of dislocation line is given b the Peach-Koehler equation, which written for the i-th component reads (3.1) f i = ε ijk σ jm b m s k, where we denote: f i i-th component of the interaction force per unit length of the dislocation line ε ijk Levi-Civita smbol σ jm components of the stress field tensor at the dislocation position b m components of the Burgers vector components of unit tangential vector of the dislocation line s k 3.. Stress field of dipolar loop. Stress field produced b a dipolar loop is given b formula presented b Kroupa et al [1, 13, 1] and is ver comple, especiall

3 13 V. Minárik, J. Kratochvíl and K. Mikula in the closest neighbourhood of the dipolar loop. Under the assumptions that restrict the motion of dislocation curve and dipolar loops in some directions, and the fact the dipolar loop width h is ver small comparing to the other sizes, we can simplif the original Kroupa s formula b using Talor epansion and integration, and obtain an algebraic formula for the stress field of a dipolar loop: σ (,, z) = µhb { [ l z π(1 ν) ϱ [ l z (3.) + ϱ 3 + l + z ϱ 3 + where (3.3) + [ l z ϱ + l + z ϱ + + l + z ϱ + ] [ ±ν + ] [ ± ( + ) ( ± + + ] ( ( + ) 3 ) ] }, ] [ 3 ( ± ) + ϱ = + + (l z), ϱ + = + + (l + z), µ is shear modulus, ν is the Poisson ratio, and [,, z] is the position relative to the centre of dipolar loop. With the upper sign in (3.) we get the stress formula for the dipolar loops V 1 and I 1, while with the lower sign we get the formula for V and I dipolar loops. Readers more interested in the origin of (3.) can find detailed information in [1]. The compleit of the stress field is demonstrated in Figure Governing equations. The dnamics of the sstem of a dislocation curve and a dipolar loop is governed b a sstem of two equations describing their motion. The motion law for the dislocation curve Γ t can be parameterized b a smooth time dependent vector function X : I S R, i.e., at an time t it is given as the Image( X(., t)) = { X(u, t), u S} where S is a fied parameterization interval and I is a time interval. For a smooth curve X u > and for unit arc-length parameterization s, ds = X u du. Then X s and X s represent unit tangent and normal vectors, respectivel. Using the Frenet formulae, the evolution equation can be written in the form of intrinsic diffusion equation [, 1,, 11] )] (3.) B X t = ε X ss + F X s for the position vector X. Drag coefficient B and line tension ε are material constants and F represents eternal driving forces. If we denote b X (t, s) and X z (t, s) the components of the dislocation curve position vector in the z-plane, then X s = [Xs z, Xs ]. The equation (3.) is solved numericall to model complicated dislocation curve dnamics. The motion law of the dipolar loop can be easil obtained from (3.1) where all stress contributions of the whole dislocation curve should be included: (3.) F c = σ b curve n ds. Γ t The term F c is the -ais component of the total force acting on a dipolar loop, n = Xs z is the -ais component of the dislocation curve s normal vector. Incorporating lattice friction F depending on material, the equation governing the gliding of the dipolar loop is (3.) d(t) dt = 1 BP F,total(Γ t, (t)),

4 Numerical simulation of dislocation dnamics 131 where (t) is the -ais position of the dipolar loop, P = (l + h) is the perimeter of the dipolar loop, and F c F if F c > F (3.7) F,total (Γ t, (t)) = if F c < F F c + F if F c < F.. Numerical scheme. For discretization of the problem described earlier, we use the flowing finite volume method [11] in space and the method of lines [, 3, ] in time. Discrete solution is represented b a moving polgon given, at an time t, b plane points X i, i =,..., M. The values [ X and X M are prescribed in case of fied ends of the curve. The segments Xi 1, X ] i are called flowing finite volumes. [ We construct also dual volumes V i = Xi 1, ] [ X i Xi, X ] i+ 1, i = 1,.., M 1, where X i 1 = X i 1+ X i (see Fig..1). di di+1 X i 1 X i 1 X i X i+ 1 X i+1 Fig..1. Piecewise linear approimation of the dislocation curve. Integrating evolution equation (3.) in dual volume V i followed b some other straightforward steps described in [1] we get a sstem of ordinar differential equations: B d X ( i Xi+1 X = ε i X i ) X i 1 X + F i+1 X i 1 (.1) i, dt d i + d i+1 d i+1 d i d i + d i+1 i = 1,..., M 1, where d i denote distances between neighbouring nodes of the dislocation curve s discretization. In discretization of the governing equation (3.) for the dipolar loop we sum contributions of ever curve segment to obtain (.) M 1 F c = i= ( ) σ X i+ (t),, X z ( 1 i+ z b 1 curve X z i+1 Xi z ), where [(t),, z] is the centre of the dipolar loop at time t. Net, we use formula (3.7) applied to discrete F c defined above and get (.3) d dt = 1 BP F,total( X,..., X M, ). The complete discrete problem consists of (.1) and (.3) with accompaning initial and boundar conditions. The initial conditions simpl describe positions and shapes of the dislocation curve and dipolar loops at the beginning of the computation. As we consider a plane dislocation with fied ends to model a particular phsical process, the boundar condition prescribes positions of the endpoints of the dislocation curve. Thus, no tangential velocit is evaluated for the endpoints.

5 13 V. Minárik, J. Kratochvíl and K. Mikula. Eperimental results..1. Simulation 1 - standalone dislocation curve. In this simulation we present results we got for a standalone dislocation curve eerted to a periodic eternal stress. Starting with a straight dislocation line fied at ends, periodic eternal stress is applied to the sstem. The dislocation curve allowed to glide onl in the z-plane is either attracted or repelled in the z-ais direction. As positions of the curve end points are fied, initial straight line becomes curved as the time flows. We observe curvature of the curve (elastic stress of the dislocation) to partl eliminate the eternal stress, so the curve is not moving with the same speed everwhere. The bigger the curvature is, the slower is the response to the eternal stress. Of course, the motion also depends on dimensions of the dislocation curve and the eternal stress. For a longer curve it takes obviousl more time to achieve a higher curvature state (such as a semicircle) than for a shorter curve. Moreover, if the eternal stress is not high enough (will be precised later), the dislocation curve never transforms to a semicircle. When the eternal stress is switched off, the curvature pushes the dislocation curve back to its initial position, i.e. straight line. This is in agreement with the well-known minimal potential energ principle. To precise the phrase eternal stress is high enough, we should take line tension of the dislocation curve into account. Depending on the ratio of the line tension of the dislocation curve and the eternal stress, as well as the length of the time period of the eternal stress, we can get one of the following behaviours:.1.1. Behaviour 1 - equilibrium state prior reaching a semicircle shape. If the eternal stress is small comparing to the line tension of the dislocation curve, the motion stops after a certain time, and the dislocation curve will not reach a semicircle shape. It stops in a position where the eternal stress and the dislocation curve s line tension are in an equilibrium state. The rest of the time period, before the eternal stress is decreased or reversed, the dislocation remains in this equilibrium position, no matter how long the time period is (see Figure.1 (a) (d))..1.. Behaviour - time period of the eternal stress is too short. If the time period of the eternal stress is to short, the dislocation curve will not reach the equilibrium state prior the eternal stress is reversed. After reversing, dislocation curve starts to move backwards, i.e. in the opposite direction it was moving before reversing of the eternal stress Behaviour 3 - eternal stress is too large. When the eternal stress is too large and the time period long enough, the dislocation curve will create a socalled Frank-Read source. It blows up through the shape of a semicircle, grows and spreads in a much larger area than was occupied b the semicircle (see Figure.1 (e) (f))... Simulation - dislocation curve and one dipolar loop. The second simulation was computed for a single dipolar loop positioned at [,, z ], while the dislocation curve was again gliding in the z-plane ( = ). Note the dipolar loop was positioned in a different laer of the atomic lattice than the dislocation curve, i.e.. Two different settings for the eternal stress were used. In the first one a constant eternal stress was applied for the whole time of computation, while in the second a periodic eternal stress was applied. We confirmed several our epectations during the

6 Numerical simulation of dislocation dnamics 133 (a) T =. (c) T =. (e) Frank-Read source, T = (b) T = (d) T = (f) Frank-Read source, T = Fig..1. (a) (d) Reaching an equilibrium state: at T. = an equilibrium between the eternal stress and the line tension of the dislocation curve is reached and remains for the rest of the eternal stress period; (e) (f) Creation of Frank-Read source: In the case of large eternal stress the dislocation curve blows up and a new Frank-Read source is growing further in its neighbourhood simulations. The dnamics of the sstem is highl dependent on the mutual relative position of the dislocation curve and the dipolar loop, on the shape of dislocation curve, and, of course, on the stress applied. The dipolar loop produces its own stress field that interacts with the dislocation curve. Vice versa, the dislocation curve, as a line dislocation, produces a stress field which interacts with the dipolar loop. Both stress fields are of short-range tpe, and normall the dislocation curve and the dipolar loop do not interact as the are far enough from each other. However, as the dislocation curve moves according to the applied eternal stress, it can happen that the dislocation curve moves closer to the dipolar loop and the interaction between them gains more influence over the consequent motion of the sstem. Tpicall, the dislocation curve bows around the position of the dipolar loop, and is forcing to move the dipolar loop out of its place as in Figure.. (d) and.. On the other hand, the dipolar loop is restricted to move onl in the -ais direction, and there is also some amount of energ needed to eceed in order to start the motion at all. Force corresponding to this minimal energ is called friction force...1. Positional dependence. The most significant for the behaviour of the sstem is the relative position of the dipolar loop and the dislocation curve. One can sa there is a distance threshold behind which there is almost no interaction. This threshold depends on the direction in space because of the shape of the dipolar loop. For the -ais, if the dipolar loop is positioned in a laer too far from the laer (plane) of the dislocation curve, the interaction is negligible. However, for the other two aes the situation is different. Depending on the eternal stress and its size, as well as on the line tension of the dislocation curve, there are following possibilities:

7 13 V. Minárik, J. Kratochvíl and K. Mikula (a) _DL = 3 nm (b) _DL = nm (c) _DL = 1 nm (d) _DL = 1 nm Fig... Dependence on the distance of the dipolar loop laer. Several time levels of the simulation are shown for the dipolar loop positioned initiall at [, DL, ]. For the nearest case where DL = 1 nm the dipolar loop sweeps to the right The dislocation curve stops prior to reach the interaction region around the dipolar loop (like in Section.1.1). In this case the motion of the dislocation curve is not influenced b the presence of the dipolar loop in the sstem. The dislocation curve is not moving towards the dipolar loop. In this case again the motion of the dislocation curve is not influenced b the presence of the dipolar loop - under the additional assumption both dislocations were not interacting at the beginning of the simulation. The dislocation curve is moving (in its glide plane =, of course) towards the dipolar loop, and will interact with it at some time of evolution.... Dependence on the shape of the dislocation curve. The dipolar loop can glide out of the dislocation curve s interaction region onl if there is a clear wa. If, for eample, the dipolar loop is surrounded b two hills (see Figure. (a)) of the dislocation curve, and these two hills are further growing due to the stress applied, the dipolar loop cannot move through the energetic walls of the hills and resides between them. Consequentl, the dislocation curve s motion is locall hindered and the curve distorts around the dipolar loop (Figure.). Such a situation is not rare. Even for an initiall straight dislocation curve it can happen that the forced motion of the dipolar loop is not fast enough to leave the interaction region. It can be either due to the large friction force (most energ is consumed for overcoming friction, see Figure.3), or due to the strong eternal stress (simpl the dislocation curve is moving faster than the dipolar loop can react, see Figure. (b))...3. Dependence on the applied eternal stress. Consider the situation from one of the previous sections: the dislocation curve is moving towards the dipolar loop. The stronger the eternal stress is, the closer to can the dislocation curve

8 Numerical simulation of dislocation dnamics 13 (a) T =. (b) T =.3 (c) T = (d) T =. (e) T =. (f) T = Fig..3. Dependence on the friction force of the dipolar loop. Several time levels of the simulation are shown for the dipolar loop positioned initiall at [, 3, ]. The motion of the dipolar loop is hindered b a large friction Fig... The dnamics of the sstem depends on all parameters. The rightmost image uses times bigger dipolar loop than the middle one. The leftmost image uses. times bigger dipolar loop and times bigger shear modulus than the middle one. Dipolar loop fied at [,,.1] is not shown in the images approach the dipolar loop. The closer it can approach, the stronger the interaction is. Finall, the stronger is the interaction, the faster the dipolar loop is forced to sweep awa (Figure.). The achieved results seem to be close to the real phsical processes in materials. However, during the computations we observed that the dislocation curve bows so fast that the dipolar loops are unable to respond to it. This fact is not in agreement with real phsical eperiments []. We tried to remove this drawback from our model b adding an additional term of elastic relaation to the governing equation of the dislocation curve. It is a question of further investigations and simulations, how will this term work in the model.

9 13 V. Minárik, J. Kratochvíl and K. Mikula T =. T =. T = T = 1. T = 1. T = Fig... Dipolar loop swept b the curve; on the other hand, the curve is distorted b the stress field of the loop. In this test there were used: µ = GPa, ν =.33, B = 1 Pa s, b =.77 nm, l = 3 nm, h = nm, F = MPa m, applied stress σ a = 1.1 MPa. Initial position of dipolar loop was [,, 3]. The subsequent stages are shown for increasing time T Acknowledgements. The first author was partl supported b the NSF grant NSF-13 Award 113 and b the project Applied Mathematics in Technolog and Phsics MSM 771 of the Ministr of Education of the Czech Republic, the second author was partl supported b the project KONTAKT No. ME and Grant Agenc of the Czech Republic No. 1/3/, and the third author was partl supported b the grant VEGA 1/313/3. The conference participation was possible thanks to the Internal Grant of the Czech Technical Universit in Prague No REFERENCES [1] S.B. Angenent and M.E. Gurtin., Multiphase Thermomechanics with an Interfacial Structure. Evolution of an Isothermal Interface, Arch. Rat. Mech. Anal., 1 (199), pp [] M. Beneš., Diffuse-Interface Treatment of the Anisotropic Mean-Curvature Flow, Applications of Mathematics, Vol., No. (3), pp. 37-3, ISSN -79 [3] M. Beneš., Mathematical analsis of phase-field equations with numericall efficient coupling terms, Interfaces and Free Boundaries 3, (1), pp. 1 1, ISSN [] M. Beneš., Mathematical and Computational Aspects of Solidification of Crstallic Materials, Acta Mathematica Universitatis Comenianae, Volume 7 (1), No. 1, pp. 13 1, ISSN -9 [] G. Dziuk., Convergence of a Semi Discrete Scheme for the Curve Shortening Flow, Mathematical Models and Methods in Applied Sciences, (199), pp. 9 [] M. Gage and R.S. Hamilton., The Heat Equation Shrinking Conve Plane Curves, J. Diff. Geom., 3 (19), pp. 9 9 [7] J.P. Hirth and J. Lothe., Theor of Dislocations, John Wille (19)

10 Numerical simulation of dislocation dnamics 137 σ (,, ) σ (,, ) σ (,,.7*l) σ (,,.7*l) σ (,, l) σ (,, l) σ (,, 1.*l) σ (,, 1.*l) Fig... Compleit of the dipolar loop stress field. Graphs showing stress field at planes z =, z =.7l, z = l, and z = 1.l are presented.

11 13 V. Minárik, J. Kratochvíl and K. Mikula [] J. Huang, N.M. Ghoniem and J. Kratochvíl., On the sweeping mechanism of dipolar dislocation loops under fatigue conditions, Modelling and Simulation in Materials Science and Engineering, 1 (), pp [9] J. Kratochvíl, F. Kroupa and L.P. Kubin., The Sweeping of Dipolar Loops in Cclic Deformation: Kinetic Diagrams, Proceedings of the th Risoe Int. Smposium on Material Science: Deformation-Induced Microstructures: Analsis and Relation to Properties. Risoe National Laborator, Roskilde, Denmark (1999), pp [1] F. Kroupa., Long-Range Elastic Field of Semi-Infinite Dislocation Dipole and of Dislocation Jog, phs.stat.sol., Vol. 9 (19), pp. 7 3 [11] K. Mikula and D. Ševčovič., Evolution of Plane Curves Driven b a Nonlinear Function of Curvature and Anisotrop, SIAM J. Appl. Math., Vol. 1, No., pp [1] V. Minárik, J. Kratochvíl, K. Mikula and M. Beneš., Numerical Simulation of Dislocation Dnamics, Numerical Mathematics and Advanced Applications, ENUMATH (3), pp. 31 [13] Š. Verecký, J. Kratochvíl and F. Kroupa., The Stress Field of Rectangular Prismatic Dislocation Loops, Phs. Stat. Sol. (a), Vol. 191 (), pp. 1