This Workshop forms a part of the programme Mathematical Developments in Solid Mechanics and Materials Science. It will explore the modelling of interactions between defects at several different scales. Topics to be treated will include the modelling of single and groups of dislocations (e.g. persistent slip bands); phenomenological theory of damage and underlying physical mechanisms; derivation of non-local macroscopic laws from micromechanics and homogenization theory; the influence of microstructure on localization and instability, and similar topics. The objective is to advance understanding and to develop mathematical technique to make allowance at any given scale for detail at an adjacent scale. For instance, although there is advantage in treating dislocations within the theory of elasticity, certain effects, such as the Peierls stress, can only be understood in terms of the underlying lattice: the elastic theory thus requires augmentation by parameters "passed up'' from the lower scale description. Conversely, events on the lower scale may be influenced by long-range interactions best described at the higher level (macroscopic stress influencing dislocation configurations providing an example). Similar problems of interactions between scales appear at other levels. Often, phenomenological models are postulated ab initio, or perhaps may be derived from physical assumptions coupled with thermodynamic restrictions. Some computational schemes introduce "non-local'' terms, primarily to stabilize the numerics, though with an underlying assumption that such terms really do represent the physics. The introduction of terms inherited from a lower scale usually introduces a characteristic length and permits the development of patterns. The Workshop will include presentations from specialists in physical modelling, numerical computation and in mathematics, to facilitate at least a common understanding, and hopefully the development of mathematical theory for problems associated with passing from one scale to another.
The formal timetable of the workshop is kept deliberately light, so that participants can spend a high proportion of their time actively considering, discussing and perhaps solving outstanding problems. Additional talks and/or discussion sessions will be arranged during the Workshop according to participants' wishes.
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