Periodic Report Summary 1 - STP (Scale Transitions in Plasticity)
During the first year of the project, major progress regarding the understanding of pattern formation and microstructure in continuum crystal plasticity was made. In particular, we considered an energetic formulation of strain-gradient elasto-plasticity subject to a class of single-slip side conditions, and derived rigorously that the non-convexity induced by enforcing single-slip can be resolved mathematically: For a large class of plastic deformations, a given single-slip condition (specification of Burgers’ vectors and slip planes) can be relaxed by introducing a microstructure through a two-stage process of mollification and lamination. The resulting relaxed side condition prescribes only certain slip planes but allows for the slip direction in these planes to be arbitrary. This shows that a pure strain-gradient penalty term is not sufficient to regularise the single-slip problem in the sense of making the associated elasto-plastic energy lower semicontinuous. The great advantage of using such a relaxed model comes into play in numerical simulations, as the relaxed model does not require the fine scales of the microstructure to be resolved. Introducing such an ultra-fine microstructure would be prohibitive in terms of computational cost when the model should be used on a larger scale. This work thus has the potential to make numerical simulations of strain-gradient plasticity with single-slip numerically feasible.
For the aforementioned relaxed model, an existence result is current work in progress. The important partial result, namely that the relaxed 'single plane' side condition is in fact stable under the consideration of an energy minimizing sequence, has already been derived. The idea of the proof is to show that under the condition that slip-plane-mixing occurs in such a minimizing sequence, the strain-gradient term in the energy must necessarily become large. This work indeed shows that the relaxation is the correct approach, in the sense that no microstructure below the physically modeled strain-gradient length-scale needs to be resolved. The next, natural steps in this project would be to study the limit of small strain-gradient length-scales as well as an approximation by a numerical phase-field type approach penalizing deviation by the relaxed single slip-plane condition.
During the first 18 months of the project, the fellow has become fully integrated as a staff member at Durham University with a permanent lectureship and a successfully completed probationary period. As a testament to the success of the project, early during the second year of the funding period, the fellow received an offer for a full professorship at the Albert-Ludwigs-University in Freiburg. This offer was realized and therefore the funding of the project unfortunately had to end. The fellow will, of course, continue the promising research on the basis of the outcome of the successful project in order to generate impact both within and outside of academia.
For the aforementioned relaxed model, an existence result is current work in progress. The important partial result, namely that the relaxed 'single plane' side condition is in fact stable under the consideration of an energy minimizing sequence, has already been derived. The idea of the proof is to show that under the condition that slip-plane-mixing occurs in such a minimizing sequence, the strain-gradient term in the energy must necessarily become large. This work indeed shows that the relaxation is the correct approach, in the sense that no microstructure below the physically modeled strain-gradient length-scale needs to be resolved. The next, natural steps in this project would be to study the limit of small strain-gradient length-scales as well as an approximation by a numerical phase-field type approach penalizing deviation by the relaxed single slip-plane condition.
During the first 18 months of the project, the fellow has become fully integrated as a staff member at Durham University with a permanent lectureship and a successfully completed probationary period. As a testament to the success of the project, early during the second year of the funding period, the fellow received an offer for a full professorship at the Albert-Ludwigs-University in Freiburg. This offer was realized and therefore the funding of the project unfortunately had to end. The fellow will, of course, continue the promising research on the basis of the outcome of the successful project in order to generate impact both within and outside of academia.