The workshop will be held in the Warwick Mathematics Institute (Zeeman Building) on the Central Campus of Warwick University. A map of the Central campus can be found here (web-page) and here (PDF document); the Institute is building #35 on both maps.
All talks will be held in room MS.03. The entrance to MS.03 is located on the second floor balcony overlooking the main atrium of the Institute.
Registration will take place from 1000 on the morning of Monday 17 September 2007 at the offices of the Mathematics Research Centre (MRC) on the first floor of the Institute. The Mathematics Common Room is also located on the first floor of the Institute, immediately opposite the MRC.
The dinner on the evening of Monday 17 September 2007 will be held at Harringtons on the Hill in the nearby town of Kenilworth. Directions to the restaurant can be found using, for example, Google Maps.
The nearest main-line railway station is Coventry; less frequent local trains serve Canley, which is slightly closer. National Rail Enquiries offers timetable information and the facility to purchase tickets.
The nearest major airport is Birmingham International Airport (BHX), which is approximately 20 minutes away from Warwick campus by taxi. Coventry Airport (CVT) is closer to campus, but is much smaller and offers a more limited service.
Directions for driving to Warwick campus can be found here. A large multi-storey car park, Car Park 15, is located just north-west of the Institute (immediately north of building #26 on the map).
All participants are encouraged to register online with the MRC. This creates an electronic record of your visit to Warwick and will help the MRC staff in making any necessary arrangements for your visit.
The discrete dislocation (DD) approach, initiated by Van der Giessen and Needleman, has elucidated a range of phenomena in plastically deformed solids. Being a "brute force" computational method, it motivates the question of whether its essential features can be captured by averaging the dislocation motion in some way. Using the homogenisation approach we propose a system of transport equations for the n-point dislocation densities, which possesses a number of such features. We illustrate this in the setting of the shear of a film, where size effects similar to those in the DD model are observed.
I will explain some aspects related of the dynamics of an interface, e.g. a phase boundary, in heterogeneous media. The interface is driven by the desire to reduce its free energy (gradient flow), but the heterogeneity of the environment creates a very complex energy landscape. First I will present results on pinning and de-pinning for a semilinear PDE in a periodic environment. Then I move to a genuinely geometric example, forced mean curvature flow with periodic forcing, and explain our geometric construction of special solutions, so-called pulsating waves, for small forcing. Finally I conclude with some remarks on random environment.
We study the role of defects in the quasistatic evolution of a martensitic phase boundary. Martensitic phase transformations involve a change in shape of the underlying crystal, and thus the propagation of the phase boundary is accompanied by an evolving mechanical stress and strain field. This gives rise to a nonlocal free boundary problem, one where the evolution of the free boundary is coupled to an elliptic partial differential equation. Often, real materials contain defects which can potentially pin the interface and contribute to the hysteresis, and understanding this is the goal of this work. We present a mathematical model and a proof of existence in the sense of sets of finite perimeter. We then present numerical simulations and analysis of both the fully nonlocal problem as well as a linearized version, and draw conclusions on the role of defects on hysteresis.
Plastic deformation of crystalline materials involves disparate length scales, from nanometers to microns, due to the non-local and long-range characteristics of the dislocation interactions. These interactions are responsible for a series of features typical of plastic deformation as intermittence, localization and the Hall Petch effect.
We will present results from a continuum theory of dislocations analogous to pinning/depinning models which have been widely used to model earthquakes, magnetic systems and fluid invasion in porous media. The present theory shows that plastic deformation under slow external loading occurs in a sequence of avalanches involving the collective motion of many interacting dislocations.
We will also show the formation of structures and their influence in macroscopic deformation, for example the dependence of the yield stress on the characteristic size of the sample. We compare our simulations with gradient theories of plasticity and derive the dependency of the length scale parameters introduced in those models on the characteristic sample size and dislocation structure.
The talk covers ongoing joint work with Marisol Koslowski (Purdue), Michael Ortiz (Caltech) and Florian Theil (Warwick) on an application of a time-incremental variational scheme to the study of a dissipative system in contact with a heat bath. Specifically, the system will be a toy model for a light particle moving in an N-dimensional state space, restrained by an elastic potential and experiencing dry friction. This deterministic set-up is then perturbed by an interior-point regularization (an additive entropy-like term) that represents the effects of the heat bath and introduces some randomness into the system. An interesting result is that although the variational scheme generates a process that is, in principle, stochastic, a deterministic gradient flow (in a qualitatively different effective kinetic potential) is obtained in the limit as the time step tends to zero.
Cancelled due to unforseen circumstances
The problem of correlation within an array of parallel dislocations in a crystalline solid is addressed. The first two of a hierarchy of equations for the multi-point distribution functions are derived by treating the random dislocation distributions and the corresponding stress fields in an ensemble average framework. Asymptotic reasoning, applicable when dislocations are separated by small distances, provides equations that are independent of any specific kinetic law relating the velocity of a dislocation to the force acting on it. The only assumption made is that the force acting on any dislocation remains finite. The hierarchy is closed by making a standard closure approximation. For the particular case of a population of parallel screw dislocations of the same sign moving on parallel slip planes the solution for the pair distribution function is found analytically. For dislocations having opposite signs the system of equations suggests that in ensemble average only geometrically necessary dislocations correlate, while balanced positive and negative dislocations would create dipoles or annihilate. Direct numerical simulations support this conclusion. In addition, the relation of the dislocation correlation to strain gradient theories and size effect is shown and discussed.