91影视破解版

Welcome to the website of the ERC/UKRI-funded project "Concentration Phenomena in Nonlinear PDEs and Elasto-plasticity Theory" (CONCENTRATE)!

Objectives

Understanding the properties of crystalline materials, such as metals, has been a central part of humanity's march of technological advancement since the end of the Stone Age. However, despite thousands of years of practical experience, the microscopic mechanism by which metals deform was only discovered in the first decades of the 20th century: Volterra and Taylor correctly identified the movement of defects in the crystal lattice, in the form of so-called "dislocations", as the fundamental process that allows crystalline solids to deform. When a dislocation moves, it causes a net displacement of crystal planes and thus a (very small) deformation of the macroscopic specimen; see Fig. 1 for an illustration. You could not bend a spoon if not for the movement of dislocations!

Fig 1- Dislocation movement as the mechanism of plastic deformation.

Yet, despite a century of research, we still do not have a comprehensive, reliable, and rigorous theory that describes the plastic deformation of macroscopic crystalline solids from its micro-mechanical principles. This goal is of such practical, as well as theoretical, importance that it is sometimes referred to as the 鈥淗oly Grail鈥� of Solid Mechanics.

Questions such as these have a mathematical counterpart in the study of so-called 鈥渄iffusely concentrating鈥� sequences of functions; see Fig. 2 for some examples.

Fig 2- Examples of diffuse concentrations

The study of these singular structures poses many challenges and complications. For instance, dislocation defects always occur along curves, which can get "entangled" with each other like spaghetti in a bowl, directly affecting the properties of the material. When one tries to understand the movement of such entangled dislocations, new types of singularities, related to the well-known "devil's staircase" of Cantor, can form and need to be understood thoroughly, for instance the "flat mountain singularity" illustrated in Fig. 3.

Fig 3- The flat mountain singularity

Moreover, a very large number of dislocation defects are usually present, hence a "field description" is desirable to avoid having to consider all these lines separately. But how can one effectively describe a "field of lines" without losing crucial bits of information? Such a model of dislocation fields and the resulting plastic dynamics has so far proved elusive.

While predominately sitting at the intersection of the theoretical fields of Nonlinear PDE Theory, Geometric Measure Theory, and the Calculus of Variations, applications to Solid Mechanics and Theoretical Material Science will also form a crucial part of this research programme.

Publications

[14] Rate-dependent gradient flows of integral currents via varicurrents (Anton铆n 膶e拧铆k, Malte Kampschulte, Filip Rindler),

[13] On the strict approximation of currents and space-time trajectories (Paolo Bonicatto, Filip Rindler, Harry Turnbull),

[12] Scale-invariant bounds on thin sets for measures satisfying a first-order PDE and the anisotropic Michael-Simon inequality (Guido De Philippis, Luca Gennaioli, Alessandro Pigati, Filip Rindler),

[11] Shape optimization of light structures with general Hooke's laws (Dimitrios Andreakis, Flaviana Iurlano, Filip Rindler),

[10] Improvement of the dimension bound for unweighted RCD spaces (Camillo Brena, Luca Gennaioli),

[9] Concentration phenomena and the Vanishing Mass Conjecture (Luca Gennaioli, Filip Rindler),

[8] Lipschitz continuity of harmonic maps between RCD(K,N) spaces and CAT(魏) spaces (Luca Gennaioli),

[7] Sets of finite perimeter on Riemannian manifolds and stochastic completeness (Luca Gennaioli),

[6] A new Duhamel-type principle with applications to geometric (in)equalities (Michele Caselli, Luca Gennaioli),

[5] Fluid-Structure interactions with Navier- and full-slip boundary conditions (Anton铆n 膶e拧铆k, Malte Kampschulte, Sebastian Schwarzacher),

[4] The Space-Time Connectivity Theorem for Normal Currents (Paolo Bonicatto, Filip Rindler, Harry Turnbull), ,

[3] Sharp conditions for the BBM formula and asymptotics of heat content-type energies (Luca Gennaioli, Giorgio Stefani), ,

[2] Homogenization of elasto-plastic evolutions driven by the flow of dislocations (Paolo Bonicatto, Filip Rindler),

[1] Fourier transform of BV functions and applications (Thomas Beretti, Luca Gennaioli),

(also see the publications of the predecessor ERC SINGULARITY project)