There has been a recent surge of interest in the study of the mechanisms underlying fracture and slip dynamics. The transition from static to kinetic friction, stress induced creep phenomena, dynamics of faults in earthquakes, and the rheology of ordered phases of block copolymers and surfactants constitute a few of the presently active research areas. Qualitative similarities in the behaviors observed in these seemingly distinct fields has motivated studies to try and extract some common set of underlying universal principles. Indeed, similarities already exist between the models utilized to quantify earthquake fault dynamics and the stick slip motion of solids. Similarly, models of creep and stress response of solids fall under the broad generic class of ``rate and state'' models, which also encompasses models for the response of confined fluids under shear.

The present work represented an effort in a similar direction, with a fundamental objective to identify some common principles and thereby propose a phenomenological model to describe the rheology and stress response of layered materials. Block copolymers, for instance, display unique characteristics in their rheological response to shear. While the main focus of our work was motivated by considerations pertaining to the lamellar phase of block copolymers, nevertheless we believe that the model we proposed and analyze may possess features generic to describing similar phenomena in other layered materials like laminates, composites etc. At a practical level, this broad class of materials are widely used in a variety of industrial applications. For instance, multilayer polymer mirrors have been fabricated and advanced for applications requiring exceptional birefringent properties. The lamellar phase of surfactant liquid crystalline systems has been used in a broad range of applications in cosmetic and food processing industries. It is to be noted that most of the applications utilizing these materials typically employ flow devices to process and manipulate them. A fundamental understanding of the nonlinear response and rheology of these layered materials would therefore greatly enhance the design and understanding of the processes required to manipulate them.

The basic premise underlying our model for layered materials is the existence of two states within each layer, labeled respectively S1 and S2. In the absence of an external stress or strain, the system is assumed to be at equilibrium with respect to the populations of the states S1 and S2. Upon imposing an external stress or strain, the energy of one of the states, say S1, is assumed to increase in response to the applied strain. In contrast, the energy of S2 is assumed to be unaffected by the applied stress. Therefore, the application of a stress (or a strain) induces a bias in the transition between S1 and S2 in the forward direction. The dynamics of this transition (leading to a change in the relative populations of the states S1 and S2) also impacts upon the rheological properties of the layer. This work was devoted to elucidating the rheological characteristics and implications arising from the above described dynamical transformations.

The above discussion has been abstract, without any reference whatsoever to the actual nature of the variables S1 and S2. Such a generic mode of discussion was effected to enable one to possibly identify such state variables in diverse applications. However, to focus upon a concrete example, in this article we will be concerned primarily with the rheology and response of the lamellar phase of multiblock copolymers --- specifically, diblock, triblock and pentablock copolymers. As will be clarified later, these three cases encompass the different rheological possibilities for a simplified rheological model of the lamellar phase. The rheology and response of the lamellar phases of multiblock copolymers possess a number of applications in the practical context involving the processing of these materials. The effects of shear flow on the response and orientation of these systems have been experimentally studied, revealing quite contrasting features depending upon the specific architecture of the polymers.

Other possible applications of the generic ideas of our model include the description of similar phenomena occurring in confined fluids. A confined fluid typically consists of a few molecular layers each of which involves both frozen, solid like molecules (S1), and fluid, liquid like molecules (S2). One might envision applying the ideas embodied in this research to describe the stress response and the melting of these fluids to shear.

Within the above model, we focused on the specific case of multiblock copolymers, wherein the bridge and loop conformations of the chain constitute the internal states. The numerical results of our model for different polymeric architectures and different rheological constraints indicate a rich variety of phenomena including strain localization and shear banding. We derived an explicit constitutive equation that explains the origin of the inhomogeneous rheological response of the model. The predictions of our numerical results display a very good qualitative agreement with the trends observed in experiments.

Using similar ideas, we also studied the stick-slip lubricated motion of solids. Here again, the model was able to reproduce qualitatively most of the features observed in experiments.