My research develops mathematically rigorous models for solids with defects by combining continuum mechanics, variational methods, partial differential equations and numerical approximation. The long-term objective is to establish incompatibility-based formulations capable of bridging microscopic defect physics with macroscopic constitutive theories of elasticity, plasticity and fracture.
A key contribution, developed in collaboration with S. Amstutz (University of Avignon), is the development of incompatibility-governed models for elasticity and elasto-plasticity, where incompatibility naturally links dislocation densities to elastic and plastic strain fields.
A small-strain formulation based on incompatibility as the primary kinematic descriptor of plastic deformation.
These developments culminate in A second-order model of small-strain incompatible elasticity, which introduces the current mathematical framework together with preliminary numerical simulations,
and in our recent manuscript Incompatibility-governed deformations: towards a new model of small-strain elastoplasticity, written in collaboration with Thien-Nga Lê (École Polytechnique, France), where the theory is extended to elasto-plasticity.
The idea of considering the incompatibility operator in plasticity goes back to the pioneering works of Ekkehart Kröner, whose work is acknowledged in the fields of theoretical physics and the geometrical
theory of defects in solids.
The origins of this research can be traced back to my PhD work on crystal growth under the supervision of François Dupret at the Université catholique de Louvain, where a global model for crystal growth, with particular emphasis on the Czochralski method, was developed. One fundamental challenge raised by this model was the proper definition of the physical fields, particularly with regard to the principle of objectivity. Since the crystal is continuously grown from the melt, there is no natural undeformed, stress-free reference configuration, unlike in classical continuum mechanics. This led us to formulate the theory directly in terms of the strain tensor and its derivatives, rather than displacement or velocity fields, an idea that later evolved into the distributional approach to dislocations and, more recently, incompatibility-based models of elasticity and elasto-plasticity.
One paper illustrating our research on crystal growth is Dynamic Prediction of Point Defects in Czochralski Silicon Growth: An Attempt to Reconcile Experimental Defect Diffusion Coefficients with the V/G Criterion.
In elasticity, my research investigates how the presence of crystalline defects modifies the classical theory of elasticity. Building on Kröner's theory of strain incompatibility, I have developed intrinsic formulations in which the elastic strain, rather than the displacement field, becomes the primary kinematic variable. This provides a natural framework for describing solids containing dislocations and other lattice defects.
Following the pioneering work of Philippe Ciarlet on intrinsic elasticity, I established several mathematical links between incompatibility, the Frank tensor and the continuum description of dislocations. These ideas led to new analytical results on the incompatibility operator and the Beltrami decomposition, including a compatible–incompatible decomposition of symmetric tensor fields and existence results for intrinsic models of incompatible elasticity. Together, these contributions provide a rigorous mathematical foundation for incompatibility-based elasticity.
More fundamentally, incompatibility is the linearization of the Riemann curvature tensor, while dislocations are associated with torsion. Their interaction therefore naturally belongs to the framework of non-Riemannian differential geometry, a subject explored in several of my works, including The Non-Riemannian Dislocated Crystal: A Tribute to Ekkehart Kröner. The origins of this line of research date back to my PhD work, although several of the resulting publications appeared only later.
Analysis, finite-element approximation, and numerical simulation of second-gradient incompatible elasticity and elasto-plasticity, with discretizations tailored to the incompatibility operator and benchmarking against classical formulations.
Current developments rely on finite-element discretizations derived from differential complexes and compatible with the structure of the incompatibility operator. Numerical simulations investigate size effects, strain localization, dislocation-driven deformation and benchmark problems against established elastoplastic models.
In the context of plasticity, my recent work on BDdev(Ω) (fields of bounded deviatoric deformation), in collaboration with M. Caroccia (University of Florence), explores the mathematical properties of deviatoric strain in modeling plastic deformations. The paper Rigidity and Functional Properties of BDdev(Ω) (Archive for Rational Mechanics and Analysis, 2026) investigates the functional framework for deviatoric strain fields, highlighting their role in capturing the essential features of plastic behavior while excluding volumetric changes. This work provides a rigorous foundation for understanding how deviatoric strain contributes to the development of consistent models in plasticity theory.
In the context of elasticity, the paper On Integral Representation of Local Energy Functionals on BD, published in the SIAM Journal on Mathematical Analysis and co-authored with M. Caroccia and M. Focardi (University of Florence), provides a framework for addressing homogenization problems through integral representation techniques. The notion of rigidity, central in the aforementioned work, also plays a fundamental role in this context.