Dynamical Approach To The Protevin-Le Chatelier Effect

Abstract

Materials when subjected to deformation exhibit unstable plastic flow beyond the elastic limit. In certain range of temperature and strain rates many solid state solutions, both interstitial as well as substitutional, exhibit the phenomenon of serrated yielding which also goes by the name, the Portevin - Le Chatelier (PLC) effect. The origin of this plastic instability is due to the interaction of dislocations with solute atoms. The objective of the thesis is to provide a dynamical systems approach to the study of this plastic flow instability. The thesis work discusses, within the framework of a model, the connection between microscopic dislocation mechanisms and macroscopic mechanical response of the specimen as stress drops in stress-strain curves. An extension of the model to the associated deformation bands is also considered. The emphasis is on the dynamical aspects of the instability. The methods of nonlinear dynamics like geometrical slow manifold and Poincare map formalism are applied for the first time to study the PLC effect. However, the approach and techniques transcend this particular application as the techniques are equally well applicable for many other physical systems as well, in particular, systems involving multiple time scales. The material covered should be of interest to investigators in the materials science, in particular, those, involved in the dislocation patterning and self organization of dislocations. Many theoretical models for the PLC effect exist in literature. Although the physical phenomenon is inherently dynamic, the conventional theoretical models do not involve any dynamical aspect. A dynamical model for this effect, due to Ananthakrishna, Sahoo and Valsakumar provides an explanation in terms of the dynamic interactions between different dislocation species and evolution of densities of these dislocation species. This model is known to reproduce several of the experimental results. It is within the perspective of this model and its extensions we analyze the PLC effect. The macroscopic manifestation of the PLC effect is the repeated load drops or serration in stress-strain curves (beyond the yield point). Each of the load drop is associated with the formation of a spatial dislocation band and its subsequent propagation. From the perspective of a dynamical system, the changeover from the stress-strain curve with single yield drop to repeated yield drops (the PLC effect) corresponds to a Hopf bifurcation wherein equilibrium state changes over to a periodic steady state. These repeated load drops correspond to auto oscillations of the applied stress (in the absence of any periodic driving force). In particular, as implied by the slow loading and sudden load drops, these oscillations are classified as relaxation oscillations. Relaxation oscillations are a result of disparate time scales of dynamics of the participating modes. Within the context of the model, this refers to very different time scales of evolution of densities of mobile (fast), immobile (slow) dislocations and those with a cloud of solute atoms (not too slow). The focus of attention in the thesis work is on these auto relaxation oscillations. There are several methodologies in nonlinear dynamical systems to study the oscillatory behavior of multidimensional systems with multiple time scales. An effective way is to study the reduced dynamical system in an appropriate space without sacrificing the required dynamical information. To this end, we discuss two techniques which compliment each other. 1.Slow manifold approach: This method utilizes the presence of multiple time scales dynamics. Advantage is that the information on the nature of evolution of the periodic orbit is retained. The limitation is that the transition from one stable state to another as parameter is varied cannot be dealt with. 2.Poincare maps:This approach utilizes the recurrent behavior of the period orbit. This is a convenient methodology to study the nature of stability of periodic orbits. However, in this, the information about the nature of evolution is lost. Both the above techniques provide good description in the presence of high dissipation or larger separation of time scales of the participating modes. For slow manifold analysis, this leads to exact slow manifold structure while in the case of Poincare maps, it leads to simpler, lower dimensional attractors. Specific issues that are dealt with using these approaches and others in this thesis are the following. To start with, we first provide a comprehensive overview of the dynamical behavior as envisaged by the model system in physically relevant two parameter space. The existence of relaxation oscillations bounded by back-to-back Hopf bifurcation is a good representation of the fact that the PLC effect manifests only in a window of strain rates. Within this boundary of Hopf bifurcations relaxation oscillations destabilize to give rise to new states of order, including the chaotic states. The changes in the nature of these oscillations with control parameters is projected through the bifurcation diagrams and analyzed using techniques like Floquet multipliers, Lyapunovs exponents etc. After the identification of the relevant parameter space for the monoperiodic relaxation oscillations, we focus our attention on the time scales involved in these relaxation oscillations and its connection to the time scales apparent in serrations of the stress-strain curve of the PLC effect. This characteristic feature of the PLC effect, the stick-slip nature of stress-strain curves, is believed to result from the negative strain rate dependence of the flow stress. The latter is assumed to arise from a competition of the relevant time scales involved in the phenomenon. However, in the previous works, the identification and the role of the time scales in the dynamical phenomenon is not clear. The motivation of this part of the work is to identify the time scales involved in the stress drops of the time series and their origin. Since the dynamics involves distinct time scales, in the long time limit, the evolution is controlled only by the slow modes. Hence, the adiabatic elimination or quasi-steady state approximation of the fast modes leads to an invariant manifold, the slow manifold which is useful for the analysis of time scales. The geometry of the slow manifold which is atypical with two connected pieces is shown to be at the root of the relaxation oscillations. The analysis of the slow manifold structure helps to understand the time scales of the dynamics operating in different regions of the slow manifold. The analysis also helps us to provide a proper dynamical interpretation for the negative branch of the strain rate sensitivity of the flow stress. The slow-fast dynamical nature manifests itself through multiperiodic oscillations also, in the form of mixed mode oscillations (MMOs), which are oscillations with both large amplitude excursions as well as small amplitude loops. In MMOs, the small amplitude oscillatory loops are confined to one part of the slow manifold (around the fixed point) and the large amplitude excursions arise as jumps from one piece of the slow manifold to the other. More generally, MMOs are a characteristic feature of a family of dynamical systems which also exhibit alternate periodic-chaotic sequences in bifurcation portraits. Usually, the origin of these features is explained in terms of either the approach to a homoclinic bifurcation duo to a saddle fixed point (Shilnikov scenario) or a saddle orbit (Gavrilov-Shilnikov scenario). However, the dynamical model exhibits features from both the above scenarios. The emphasis of this study is on explaining the origin of the incomplete approach to a global bifurcation in the dynamical model. Apart from attempting to understand the complex bifurcation sequences, an additional motivation for this study is the apparent lack of systematic investigation into the incomplete approach to global bifurcation exhibited by a variety of physical systems. The method of the analysis is general and applicable to the family of MMO systems. In the model, using the structure of the bifurcation sequences, and the equilibrium fixed point, a local analysis shows that the approach to homoclinicity is asymptotic at best, and is a result of the ‘softening' of eigenvalues of the saddle equilibrium point. This softening, in turn, is a consequence of back-to-back Hopf bifurcation which reflects the constraint of the physical phenomenon, namely, the occurrence of the multiple stress drops only in an interval of the strain rates. The characteristic features, namely, MMOs, alternate periodic-chaotic sequences, and incomplete approach to homoclinicity are related to each other and arise as a consequence of the atypical slow manifold structure. The slow manifold structure analysis assumes that the evolution of the system is constrained within the neighborhood of the slow manifold which also implies that the dynamical system involves high dissipation. Hence, the dimension of the effective dynamics in the long time limit is reduced. The analysis reveals information regarding the structure of the periodic orbit for a given set of parameter values but does not provide any information regarding the nature of stability of the periodic orbits. However, any insight into the mechanism of the instability of the periodic orbits in the model may lead to a better understanding of the underlying physical phenomenon. Poincare maps and equivalent discrete dynamical systems provide a convenient means to obtain such an insight on the nature of the periodic solutions of the dynamical system. This methodology compliments the invariant slow manifold analysis, since in Poincare maps, the nature of the stability information is preserved at the expense of the structure of the periodic orbit. However, these two methodologies are not exclusive to each other, since the slow manifold structure as well as Poincare maps may be constructed using a common factor, namely, extremal values of the fast variable of the dynamical system. The methodologies adopted for the analysis assumes large dissipation arising out of the multiple time scale behavior such that the next maximal amplitude (NMA) maps can be modeled by one dimensional discrete dynamical systems. The dynamical portrait of the model shows differing nature of dynamics and consequently Poincare maps with different geometrical shapes in the {m,c) plane. Within the framework of one dimensional maps, these shapes can be schematically reconstructed using minimal information regarding the principal periodic orbit embedded in higher dimension and its nature of stability. This suggests that one dimensional maps might be sufficient to represent the higher dimensional dynamical system. For most of the parameter space, the NMA maps of the dynamical model possess characteristic features of a locally smooth maximum and asymptotically long tail. These features have been observed in many other physical systems, both experimental and model systems. Hence, this analysis is focused on a broader issue of Poincare maps in a family of dynamical systems with multiple time scale dynamics and mixed mode oscillations. Here, the dynamical model has been used as a representative dynamical system for this family. The scope of the study is to understand the dynamical features of the MMO systems within the framework of one dimensional systems. Specifically, by using some general constraints on the one dimensional map, we first analyze the basic mechanism that is responsible for the reversal of periodic sequences of RLk type which corresponds to the dominant periodic states of the MMO systems. This in turn allows us to understand the period adding sequences as well. The analysis also helps to demonstrate that the width of the periodic states contained within the chaotic regions bounded by two successive periodic states of the form RLk is smaller than that for RLk .To this end, we first construct a model map which mimics the dominant bifurcation sequences of MMO systems. This map is utilized to verify the analytical results for the parameter width of the periodic windows. This analysis also throws light on the origin of the ordered structure of the isolas of RLk periodic orbits, in MMO systems, which was shown to be the result of a back-to-back Hopf bifurcation. The results indicate the ubiquity in the qualitative dynamical features of physical systems from widely differing origin, exhibiting alternate periodic-chaotic sequences. Although the model for the PLC effect is successful in describing the features of the phenomenon, a shortcoming of the dynamical model has been the absence of the spatial aspect. A dominant process in the PLC effect is the movement of dislocations (mainly through cross glide) which is essentially nonlocal. This feature has been incorporated into the dynamical model through a 'diffusive' term for the mobile dislocations. Preliminary results indicate that various types of band propagation, as seen in experiments, are recovered. It is known that the solute atmosphere aggregation occurs primarily during the waiting time of the mobile dislocations after its arrest. As another extension, the present model has been revised to incorporate these aging effects also. An outline of the thesis is as follows. Focus of this thesis work is on the dynamical aspects of the PLC effect. The phenomenology and few techniques in nonlinear dynamics are introduced in Chapters 1 and 2. Chapter 3 provides a comprehensive tour of dynamical behavior of the model in physically relevant two-parameter space. The rest of the work is presented in three parts (six chapters). In the first part of the thesis, the structure of the relaxation oscillations in the phase space is analyzed using the topology of the slow manifold. A connection between the slow manifold structure and the negative strain rate sensitivity of the flow stress is attempted using this analysis (Chapter 4). As a natural extension, the approach is utilized for the analysis of multiperiodic relaxation oscillations also. The emphasis is on the connection between the dynamical behavior of the model and incomplete approach to a global bifurcation (Chapter 5). In the second part of the thesis, the stability properties of periodic orbits are analyzed in detail using the Poincare map formalism, complimenting the study on the structure of periodic orbits using slow manifold. The structure and gross features of the Poincare map are reproduced utilizing only minimum information regarding the principal periodic orbit in the multidimensional space (Chapter 6). Within the framework of one dimensional systems, we analyze the mechanisms responsible for the structure of bifurcation portraits of MMO systems (Chapter 7). Third and the last part, of work focuses on modeling the spatial aspect of the PLC effect and refinement of the dynamical model (Chapters). The last chapter, Chapter9, is devoted for discussion of the results and scope for future work.