Plastic deformation in metals—the permanent bending, stretching, or denting that happens when you hit a piece of copper wire with a hammer—remains mathematically impossible to model from first principles because the equations break down at the microscopic scale where individual dislocation lines move. The problem is that as you zoom in on a deforming crystal, the dislocation lines form an increasingly dense, tangled network whose structure is constrained by a hidden mathematical rule (divergence-freeness), and no existing theory can describe what happens to that network as the length scale shrinks to zero. This project develops new mathematical tools to solve that problem, aiming to produce the first rigorous model that connects the motion of individual dislocations to the macroscopic behaviour of plastically deformed materials—a result that has been called the Holy Grail of plasticity theory. If successful, the work would give engineers a physically accurate way to predict when and how metal components fail under stress, improving the design of everything from aircraft turbine blades to nuclear reactor pressure vessels. The research is fundamentally curiosity-driven mathematics, but solving this long-standing conjecture would also open the door to realistic simulations of metal forming, fatigue, and fracture that current empirical models cannot provide.
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Numerous important open problems in Analysis, from such diverse areas as the compensated compactness theory of PDEs, the shape optimization of elastic materials, or the transport of geometric structures like vortex filaments in fluids and dislocation lines in crystalline materials, have at their core deep questions about "diffusely concentrating" sequences of maps, measures, or currents. Prototypical sequences of this kind display an increasing number of thin and repetitive structures as the typical length scale goes to zero. The challenge is to understand the asymptotic configurations that this "network" of structures can exhibit, which are usually highly restricted by the presence of a (linear) PDE constraint like divergence-freeness. Despite much progress in the related study of singularities in measures over the last decade, diffuse concentrations have remained shrouded in mystery. Building on the recent groundbreaking advances by the PI at the intersection of PDE Theory, Geometric Measure Theory, and the Calculus of Variations, the CONCENTRATE proposal aims at transformative progress in this highly active and rapidly evolving research area. As an application and guiding light to the theoretical investigation, the project will furthermore tackle the micro-to-macro homogenization of large-strain elasto-plasticity driven by the motion of dislocations, thus furnishing a rigorous and realistic model of plastic deformations. Often referred to as the "Holy Grail" of plasticity theory, such a homogenization result has so far proved elusive, despite much collective effort, since it requires a fine understanding of the diffuse concentrations encountered when passing from discrete dislocation lines to fields of dislocations. The PI's research leadership in these areas makes him uniquely placed to tackle the ambitious goals of this proposal through the development of novel mathematical tools and the solution of long-standing conjectures of both pure and applied character.
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