Past

Understanding the fatigue of metals under cyclic loads is crucial for some fields in mechanical engineering, such as the design of wheels of high speed trains and aero-plane engines. Experimentally it has been found that metal fatigue induced by cyclic loads is closely related to a ladder shape pattern of dislocations known as a persistent slip band (PSB). In this talk, a quantitative description for the formation of PSBs is proposed from two angles: 1. the motion of a single dislocation analised by using asymptotic expansions and numerical simulations; 2. the collective behaviour of a large number of dislocations analised by using a method of multiple scales.

It is an inherent premise in Boltzmann's formulation of linear viscoelasticity, that for shear deformations at constant pressure and constant temperature, every material has a unique continuous relaxation spectrum. This spectrum defines the memory kernel of the material. Only a few models for representing the continuous spectrum have been proposed, and these are entirely empirical in nature.

Extensive laboratory time is spent worldwide in collecting dynamic data from which the relaxation spectra of different materials may be inferred. In general the process involves the solution of one or more exponentially ill-posed inverse problems.

In this talk I shall present rigorous models for the continuous relaxation spectrum. These arise naturally from the theory of continuous wavelet transforms. In solving the inverse problem I shall discuss the role of sparsity as one means of regularization, but there is also a secondary regularization parameter which is linked, as always, to resolution. The topic of model-induced super-resolution is discussed, and I shall give numerical results for both synthetic and real experimental data.

The talk is based on joint work with Neil Goulding (Cardiff University).

Tsunami asymptotics: For most of their propagation, tsunamis are linear dispersive waves whose speed is limited by the depth of the ocean and which can be regarded as diffraction-decorated caustics in spacetime. For constant depth, uniform asymptotics gives a very accurate compact description of the tsunami profile generated by an arbitrary initial disturbance. Variations in depth can focus tsunamis onto cusped caustics, and this 'singularity on a singularity' constitutes an unusual diffraction problem, whose solution indicates that focusing can amplify the tsunami energy by an order of magnitude.

Initial stage of the flow with a free surface generated by a vertical

wall moving from a liquid of finite depth in a gravitational field is

studied. The liquid is inviscid and incompressible, and its flow is

irrotational. Initially the liquid is at rest. The wall starts to move

from the liquid with a constant acceleration.

It is shown that, if the acceleration of the plate is small, then the

liquid free surface separates from the wall only along an

exponentially small interval. The interval on the wall, along which

the free surface instantly separates for moderate acceleration of the

wall, is determined by using the condition that the displacements of

liquid particles are finite. During the initial stage the original

problem of hydrodynamics is reduced to a mixed boundary-value problem

with respect to the velocity field with unknown in advance position of

the separation point. The solution of this

problem is derived in terms of complete elliptic integrals. The

initial shape of the separated free surface is calculated and compared

with that predicted by the small-time solution of the dam break

problem. It is shown that the free surface at the separation point is

orthogonal to the moving plate.

Initial acceleration of a dam, which is suddenly released, is calculated.

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The mechanisms for the selection of the propagation speed of waves

connecting unstable to stable states will be discussed in the

spatially non-homogeneous case, the differences from the very

well-studied homogeneous version being emphasised.