My research in continuum mechanics centers on the mathematical modeling of elasticity, elasto-plasticity, constitutive behavior, and thermodynamics, with a particular emphasis on defects in solids (e.g., dislocations, fractures, and damage).
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 the incompatibility tensor captures the geometric and physical effects of defects in crystalline materials.
The last version of the model, and some preliminary numerical
simulations can be found in our papers A second-order model of small-strain incompatible elasticity and our latest (and most complete)
Incompatibility-governed deformations: towards a new model of small-strain elastoplasticity (in collaboration with Thien-Nga Lê from école Polytechnique, France).
The idea of considering the incompatibility operator in plasticity goes back to the pioneer works of Ekkehart Kröner, whose work is acknowledged in the fields of theoretical physics and the geometrical
theory of defects in solids.
This line of research also has its origins in my PhD thesis, carried out 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 problem raised by this model was to well define the physical fields, with particular attention to the concept of Objectivity, since the crystal
beeing grown from the melt, there is no natural, undeformed, stress-free reference configuration, as in conventional elasto-plasticity of deformable bodies. One solution was to construct a theory based
on the strain and its derivatives instead of the dispolacement or velocity fields (this led to the development of the to the so-called "distributional approach to dislocations").
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 work revisits classical theories, such as Kröner's formula for strain incompatibility, to address the challenges posed by defects in single crystals.
The paper Strain incompatibility in single crystals: Kröner's formula revisited offers a perspective on how incompatibility can be incorporated
into elasticity models, bridging microscopic defect structures and macroscopic material behavior.
Following the pioneer work of Philippe Ciarlet on intrinsic elasticity, some contributions were proposed, making the link with dislocations through and the Frank tensor whose curl is precisely
the incompatibility operator, that is related
to the dislocation density precisely by Kröner's formula. Intrinsic elasticity means that the main kinematical variable is the elasticv strain instead of the displacement field, being the latter
obtained as a by-product.
This approach was also at the basis of the "distributional approach to dislocations" (see the paper A distributional approach to 2D Volterra dislocations at the continuum scale),
and justified the need to understand better the Beltrami decomposition (see the papers A compatible-incompatible decomposition of symmetric tensors in Lp with application to elasticity and
Existence and asymptotic results for an intrinsic model of incompatible small-strain elasticity).
My first works on the incompatibility were written for my PhD thesis in collaboration with F. Dupret (completed in 2006, but published only in 2012).
The incompatibility is the linearization of Riemann curvature tensor, and the dislocations are related to the torsion, therefore these concepts are related to non-Riemannian differential geometry,
a topic I have discussed in several papers as well (see The non-Riemannian dislocated crystal: a tribute to Ekkehart Kröner).
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.