September - December 1999

**Organisers**: K Bhattacharaya (*Pasadena*), P Suquet (*Marseille*), JR Willis (*Cambridge*)

Workshop on

**Defect Mechanics and Non-locality
**

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.