Search

Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion along with accompanying production/ annihilation processes of dislocations lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result into the development of various types of dislocation micro-structures, such as dislocation cells, slip and kink bands, persistent slip bands, labyrinth structures, etc., depending on the externally applied loading and the intrinsic lattice constraints. The Walgraef-Aifantis (WA) (Walgraef and Aifanits, J. Appl. Phys., 58, 668, 1985) model is an example of a reaction-diffusion model of coupled nonlinear equations which describe 0 formation of forest (immobile) and gliding (mobile) dislocation densities in the presence of cyclic loading. This paper discuss two versions of the WA model, the first one comprising linear diffusion of the density of mobile dislocations and the second one, with nonlinear diffusion of said variable. Subsequently, the paper focus on a finite difference, second order in time Cranck-Nicholson semi-implicit scheme, with internal iterations at each time step and a spatial splitting using the Stabilizing, Correction (Christov and Pontes, Mathematical and Computer 0, 35 , 87, 2002) for solving the model evolution equations in two dimensions. The discussion on the WA model and on the numerical scheme was already presented on a conference paper by the authors (Pontes et al., AIP Conference Proceedings, Vol. 1301 pp. 511-519, 2010). The first results of four simulations, one with linear diffusion of the mobile dislocations and three with nonlinear diffusion are presented. Several phenomena were observed in the numerical simulations, like the increase of the fundamental wavelength of the structure, the increase of the walls height and the decrease of its thickness.

Spatially or temporally modulated systems are very common in nature, and the effect of external fields on these systems has long been studied. For example, the forcing of a large variety of nonlinear oscillators, from the pendulum to Van der Pol or Duffing oscillators, has led to detailed studies of the different temporal behaviors that were obtained. It has been shown that the resonant couplings between the forcing and the oscillatory modes may lead to several types of complex dynamical patterns, including quasi-periodicity, frequency lockings, devil’s staircases, chaos, and intermittency [8.1] and [8.2].

It is now well documented that, despite their complexity, the dynamics of physicochemical systems driven away from thermal equilibrium may be reduced, close to bifurcation points, to much simpler forms describing the universal properties of the spatio-temporal patterns. One of the most celebrated instabilities in this framework corresponds to the Hopf bifurcation that induces oscillations of the limit cycle type, which are of particular interest in nonlinear chemical systems, nonlinear optics and biology. The description of real oscillatory media is based on a combination of generic aspects that depend only on the characteristics of the bifurcation and of the symmetries of the problem and of nongeneric aspects that arise through experimental set-ups, boundary or geometrical effects, and so forth. Hence, it is interesting to discuss both aspects with the simplest description of oscillatory media close to a supercritical Hopf bifurcation, namely the complex Ginzburg-Landau equation.

As emphasized earlier in this book, stationary spatial structures may be induced by the interplay between reaction and diffusion in nonlinear chemical systems, and this pattern formation mechanism was proposed by A. Turing as early as 1952 [6.1]. Since then, various kinetic models have been dedicated to the justification of this hypothesis and to the existence of pattern-forming instabilities in these systems [6.2]. Theoretical analysis has shown the existence of universal behaviors that do not depend on the details of the kinetics but on more general properties such as, for example, the presence of autocatalytic interactions between molecules of different mobilities. Here also, the full dynamics may be reduced to much simpler forms near the instability, which capture the dominant effects of the asymptotic behavior of the system and allows a simplified analysis of pattern selection and stability [6.3].

The Rayleigh-Benard instability, which is perhaps the most popular one in the field of pattern formation, was discovered at the beginning of the twentieth century and triggered an enormous amount of work. It played an essential role in the development of new theoretical and experimental methods in this field, triggered by the possibility of precise comparisons between theoretical and experimental data. Let me recall here that this instability occurs in a thin layer of fluid (water or oil, for example), of thickness d, placed between flat horizontal plates and heated from below (see Fig. 5).

In a chemical system maintained in non-equilibrium conditions (e.g., by appropriate feeding or when some species remain in excess), various types of instabilities may occur. For example, in a continuously stirred tank reactor, or CSTR (see Fig. 14), one may obtain bistability, temporal oscillations, and chaos. In this type of reactor a permanent feeding is maintained via the inflow and outflow of some reactive species. The concentration and the flow of these species may thus be controlled and used as control parameters. These systems are considered as uniform as the result of the mixing of the chemicals, which is realized by appropriate mechanical means, although some properties of the system may sometimes depend on the stirring. Of course, only temporal patterns may observed in this type of reactor, where chaotic behaviors were first reported in 1977 and induced a large amount of more and more quantitative work, leading, as early as 1979, to the construction of the strange attractor of the Belousov-Zhabotinsky reaction, and the determination of its dimension and its Lyapounov exponents.

Besides generic properties that only depend on the symmetries of the system, the type of bifurcation encountered, the structure of the dynamics, and so forth, non-equilibrium patterns may however also present specific properties induced by particular dependences of the kinetic coefficients on the control parameters or driving fields, or geometrical effects induced by particular boundary conditions or experimental set-ups. In this chapter I will review some of these aspects that are more specifically related to hydrodynamical or chemical systems.

In Chapter 1 we saw how the behavior of complex systems can be described by simple amplitude equations near primary spatial and temporal instabilities. These equations are found either by the adiabatic elimination of stable modes or by projecting the original dynamical system on its center manifold and removing non-secular terms in the resulting equations by change of variables, such that it would take the simplest form possible. I already discussed various qualitative features of these simplified equations in relation to pattern selection and stability. Until now, almost ideal or perfect structures with uniform or nearly uniform amplitudes have been considered. However, it is well known from experimental observations and numerical analysis that imperfect structures, characterized by various types of defects, may easily appear beyond pattern-forming instabilities. This is due, on the one hand, to the existence of sets of unstable modes and, on the other hand, to degeneracies in the spectrum of unstable wavevectors. As a result, although the individual spatio-temporal patterns usually break these symmetries (space or time, translation, rotation, parity, etc.), all the patterns that are equivalent with respect to these symmetries may grow in different parts of the system, leading to the formation of various kinds of defects connecting these different states. Although some aspects of such defects were analyzed qualitatively in the preceding chapters in relation with paticular spatio-temporal patterns, I would now like to discuss the basic properties of defects in a more general context.

As discussed in previous chapters, the spontaneous nucleation of spatio-temporal patterns in systems driven far from thermal equilibrium by external constraints is the subject of intensive experimental and theoretical research. These studies, which were at first considered related to fundamental problems, appear now to be also of technological importance, as discussed in previous sections. Despite the complexity of the dynamics which gives rise to this phenomenon, great progress has been achieved in the understanding of pattern formation and stability near instability points where the reduction of the dynamics leads to amplitude equations for the patterns. According to the restricted number of bifurcation types and amplitude equation structures, the way similarities occur between the behaviors of systems of different nature, is reminiscent of the universality classes of phase transitions. It is nevertheless essential to understand how instabilities arising at the microscale are able to organize the system at the macroscale and to determine the stability and selection properties of the patterns for each particular case.

The development of a dislocation inhomogeneity in the form of regular spatial patterns during cyclic deformation is examined from a dynamical instability point of view. A gradient-dependent dislocation dynamics framework is devised and the competition between mobility, interaction, and generation processes is considered. Within such a framework, it is shown that uniform dislocation distributions may become unstable versus spatial modulations leading to the formation of different types of patterns including cellular, ladderlike, and labyrinth structures. In particular, the spatially periodic wall structure of persistent slip bands (PSBs), along with the theoretical determination of the corresponding wavelength, is discussed on the basis of nonlinear dynamical equations of the reaction-diffusion type. The effect of temperature on the wavelength selection process is predicted in accordance with experimental observations. Moreover, it is shown that the possibility of secondary or multiple slip effects and the associated labyrinth structures can conveniently be described by the dynamical model. Finally, a preliminary discussion on the applicability of gradient-dependent disloca­tion dynamics and self-organization techniques, in describing the early stages of the deformation with emphasis on the inhomogeneous development of slip during monotonic loading, is given.