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. This is exemplified in A second-order model of small-strain incompatible elasticity (Mathematics and Mechanics of Solids, 2024), which introduces higher-order terms to describe incompatible deformations, as well as in Incompatibility-governed elasto-plasticity for continua with dislocations (Proc. R. Soc. Lond., Ser. A, 2017) and Analysis of the incompatibility operator and application in intrinsic elasticity with dislocations (SIAM J. Math. Anal., 2016). These works provide a rigorous framework for understanding how incompatibility and plasticity interact to influence the mechanical response of solids with defects.
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 (Journal of Elasticity, 2011) offers a perspective on how incompatibility can be incorporated into elasticity models, bridging microscopic defect structures and macroscopic material behavior. This builds on (despite their later publication date) my PhD work with François Dupret (UCL Belgium), including A distributional approach to the geometry of 2D dislocations at the continuum scale (Ann. Univ. Ferrara, 2012) and A distributional approach to 2D Volterra dislocations at the continuum scale (Europ. Jnl. Applied Math., 2012), which explored distributional methods for modeling dislocations. Additionally, The non-Riemannian dislocated crystal: a tribute to Ekkehart Kröner (J. Geom. Mech., 2010) reflects on the geometric foundations of dislocation theory.
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.
For mesoscopic defect modeling in 3D deformable bodies, I have explored variational methods to model the evolution of dislocations in single crystals, as seen in Variational evolution of dislocations in single crystals (J. Nonlinear Sci., 2019) and A variational approach to single crystals with dislocations (SIAM J. Math. Anal., 2019). These works, in collaboration with Riccardo Scala (University of Siena), include: Constraint reaction and the Peach-Koehler force for dislocation networks (Math. Mech. Complex. Syst., 4 (2), 105-138, 2016), Currents and dislocations at the continuum scale (Meth. Appl. Analysis, 23 (1), 1-34, 2016), and Analytic and geometric properties of dislocation singularities (Proc. R. Soc. Edinburgh: Section A Math., 150(4), 1609-1651, 2020). Additionally, A new approach to topological singularities via a weak notion of Jacobian for functions of bounded variation (Indiana Univ. Math. J., 73(2), 723-779, 2024) represents a first step toward the homogenization of dislocation clusters into elastoplasticity. In these works, we made use of Cartesian currents as a fundamental tool in deriving these variational models. Together, they emphasize the role of variational principles in deriving consistent models for defect evolution, where the interplay between energy minimization and dissipation is critical. The transition to elasto-plasticity via homogenization remains an open challenge and a step yet to be fully achieved.
Finally, my work on fracture mechanics and topological optimization began with Damage and crack evolution by shape optimization methods (Journal of Computational Physics, 2011), developed during my postdoctoral research at École Polytechnique in collaboration with G. Allaire and F. Jouve, which was foundational in the use of shape and topological sensitivity for the evolution of damage and cracks in brittle materials. This approach was later revisited in Topological Derivative-Based Fracture Modelling in Brittle Materials (Engineering Fracture Mechanics, 2017), in collaboration with A. Novotny (LNCC, Brazil), focusing solely on the topological derivative. The methodology was further extended to hydraulic fracture in A simplified model of fracking based on the topological derivative concept (International Journal for Numerical Methods in Engineering, 2018) and Hydro-mechanical fracture modelling governed by topological derivatives (Computer Methods in Applied Mechanics and Engineering, 2019), both also in collaboration with A. Novotny. Together, these works demonstrate how topological derivatives and shape optimization can be used to model the nucleation and propagation of cracks, offering a powerful tool for predicting material failure.
In the realm of thermodynamics, my research integrates thermodynamic forces into defect modeling, as in Thermodynamical forces in single crystals with dislocations (Z. angew. Math. Phys., 2014). This contributes to developing constitutive models that are both mechanically and thermodynamically consistent. The paper Is Objectivity really an objective concept? (Ann. Appl. Math., 2022) also examines fundamental aspects of objectivity in continuum mechanics.