A distributional approach to crystalline defects, from isolated dislocations and evolving line networks to diffuse incompatibility and intrinsic continuum models.
This research programme has developed over nearly two decades, from my doctoral work on distributional models of dislocations to recent variational and geometric approaches to defect evolution and incompatibility-based continuum theories. It combines continuum mechanics, geometric measure theory, partial differential equations and the calculus of variations to develop mathematically rigorous descriptions of crystalline defects across multiple length scales.
The programme was strongly influenced by the pioneering work of the theoretical physicists Ekkehart Kröner and Hagen Kleinert, whose geometric and field-theoretic formulations of crystal defects have profoundly shaped the modern continuum theory of dislocations. Throughout this work, dislocations are viewed not merely as perturbations of a smooth deformation, but as intrinsically singular objects carrying essential geometric and topological information.
The common thread is distributional: isolated defects are represented by singular measures, dislocation lines by currents, and diffuse defect structures by incompatible tensor fields. In this progression, incompatibility is not an unrelated continuum concept, but the natural large-scale trace of the underlying crystalline defect geometry.
One of the starting points of this research was the development of a distributional description of two-dimensional Volterra dislocations. Classical elasticity is formulated in terms of smooth displacement fields, but such fields cannot globally represent a crystal containing dislocations: the displacement may be multivalued, while the elastic strain remains meaningful away from the defect set.
The distributional formulation avoids introducing artificial branch cuts and describes defects directly through singular measures and differential operators. The Burgers vector and the dislocation density appear as intrinsic distributional quantities, while compatibility conditions are replaced by equations involving curl and incompatibility.
This approach provided a conceptual bridge between the geometry of isolated defects and continuum models with distributed dislocation densities. It also led naturally to later work on intrinsic elasticity, the incompatibility operator and incompatibility-based models of elasto-plasticity.
While two-dimensional models provide valuable insight into the geometry and mechanics of isolated defects, real crystalline dislocations are inherently three-dimensional objects. They form curved line defects that bend, interact and organize into complex networks, giving rise to mathematical challenges with no analogue in two dimensions. Modeling these phenomena requires a geometric framework capable of describing evolving singular structures while remaining compatible with variational methods.
Geometric measure theory provides the natural mathematical language for this purpose, allowing geometric, topological and variational properties to be treated within a unified framework. In this setting, dislocation lines are represented by Cartesian currents: oriented, measure-valued geometric objects that retain multiplicity, Burgers-vector information and boundary data while remaining stable under weak convergence and singular limits.
This line of research originated from a seminal paper by Stefan Müller and Mariapia Palombaro, which proposed a variational formulation of dislocations based on currents. Motivated by this work, Riccardo Scala and I initiated a systematic investigation of this approach, which culminated in his PhD thesis at SISSA.
Together, we developed a variational framework in which complex dislocation networks can be studied through Cartesian currents. This representation preserves their geometric structure and conservation laws and makes it possible to formulate rigorous quasistatic evolution problems.
The evolution is described by incremental minimization. At each stage, the crystal selects a new configuration by balancing stored elastic energy against the dissipation associated with dislocation motion. Related work investigates the Peach–Koehler force, constraint reactions in dislocation networks, front migration and Cahn–Hilliard-type dynamics for the dislocation strain.
A more recent direction concerns topological singularities in maps of bounded variation. Classical Jacobian determinants are well suited to smooth or Sobolev maps, but they do not directly capture the singular structure of discontinuous fields arising in defect theory.
In collaboration with L. De Luca and R. Scala, I introduced a weak notion of Jacobian for BV maps. The construction combines distributional derivatives, currents and geometric measure theory to identify topological singularities even when the underlying map has jumps or other singular components.
The resulting framework extends the distributional philosophy developed for dislocations: defects are encoded by intrinsic measures rather than by globally smooth fields. It provides a rigorous tool for studying vortices, defects in ordered media and singular structures whose topology survives weak convergence.
Dislocations have a natural interpretation in non-Riemannian geometry. At the continuum level, they are associated with torsion, while strain incompatibility is related to the linearized Riemann curvature tensor. A defective crystal therefore cannot, in general, be described by a globally compatible Euclidean reference configuration.
My work explores this geometric viewpoint through intrinsic formulations of elasticity, the Frank tensor, Kröner's incompatibility formula and compatible–incompatible decompositions of tensor fields. These results clarify how defect densities generate incompatible elastic strains and how the geometry of the crystal is encoded in observable continuum fields.
At the mesoscopic scale, individual dislocation lines are no longer resolved. Their collective effect is represented by distributed defect densities and incompatible strain fields. In this passage, the incompatibility operator becomes the natural bridge between singular dislocation geometry and continuum mechanics.
Compatible strains remain generated by displacement fields, whereas the incompatible component records the part of the strain that cannot be eliminated by any change of displacement. This provides the conceptual basis for treating incompatibility as an intrinsic state variable rather than merely as a compatibility constraint.
A central long-term problem is the derivation of effective continuum theories from large systems of interacting dislocations. When the number of defects increases and their characteristic scale decreases, dislocation measures may develop oscillations, concentrations and complex network structures.
The mathematical challenge is to identify the effective energy and dissipation produced by these clusters and to determine under which scaling regimes they lead to macroscopic plastic strain, hardening or strain-gradient effects. This requires combining compactness, relaxation, Γ-convergence and homogenization with the geometric constraints carried by the dislocation currents.
The theory of weak Jacobians and singular currents offers a natural route toward this objective by providing stable representations of topological charge under weak convergence. More generally, this research seeks to establish rigorous mathematical links between discrete defect mechanics, geometric measure theory and continuum theories of elasticity and plasticity across multiple length scales.
The distributional viewpoint progressively transforms the description of isolated crystalline defects into a continuum theory governed by incompatibility. Rather than representing the endpoint of dislocation theory, incompatibility appears as its natural continuum-scale continuation.
This perspective motivates current work on higher-order incompatibility-based elasticity and elasto-plasticity, where the canonical incompatible part of the strain is promoted to a constitutive internal variable. The resulting models retain the geometric content of defect theory while providing a continuum framework suitable for analysis, computation and comparison with classical plasticity.
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