My research in fracture and damage mechanics develops variational and computational methods for describing crack initiation, propagation and material degradation. The central objective is to formulate models in which the geometry of a crack is not prescribed in advance, but emerges from an energy principle, a sensitivity analysis or an evolving damage field.
This work lies at the interface of the calculus of variations, shape and topology optimization, continuum mechanics and numerical analysis. A recurring theme is the use of geometric perturbations to detect where a defect should nucleate and how it should evolve, thereby connecting mathematical analysis with computational procedures for brittle fracture, ductile damage and coupled multi-physics problems.
An important part of this research originated during my postdoctoral work at École Polytechnique in collaboration with G. Allaire and F. Jouve. We developed a shape-optimization approach in which the crack is treated as an evolving geometric object and its propagation is driven by variations of the mechanical energy.
In contrast with methods based on a prescribed crack path, this formulation allows the geometry of the damaged region to adapt to the evolving stress field. The topological derivative is used to identify energetically favourable locations for crack nucleation, while a level-set method tracks the subsequent evolution of the crack geometry without explicit remeshing. The resulting framework provides a direct connection between fracture mechanics, shape and topology optimization, and the mathematical analysis of evolving interfaces.
The paper Damage and crack evolution by shape optimization methods established this methodology for brittle materials. It also initiated a broader programme in which crack evolution is interpreted as a free-boundary problem governed by variational and geometric criteria.
Shape derivatives describe the motion of an existing boundary, but crack initiation requires a mechanism capable of creating a new defect inside an initially sound body. The topological derivative provides such a mechanism: it measures the leading-order variation of an objective functional produced by inserting an infinitesimal cavity, inclusion or crack at a given point.
In collaboration with A. A. Novotny, I developed a crack-nucleation sensitivity analysis based on this concept. The method produces a spatial indicator of the locations and orientations at which the creation of a crack is energetically most favourable. It therefore offers a mathematically systematic alternative to empirical crack-initiation criteria.
This idea was subsequently extended into computational models in which the same topological information drives both nucleation and propagation. In collaboration with Marcel Xavier, whose PhD I co-supervised, and other colleagues, we applied this approach to brittle fracture in bulk solids and plates, obtaining models that remain geometrically flexible while avoiding the need to prescribe the crack path beforehand.
Besides sharp-interface descriptions based on topological sensitivity, I have also worked on diffuse-interface formulations of brittle fracture based on phase-field models. During the postdoctoral stay of Marco Caroccia at the University of Lisbon, we investigated the variational foundations of damage-driven fracture models through Γ-convergence and asymptotic analysis.
Our paper Damage-driven fracture with low-order potentials: asymptotic behavior and applications establishes the Γ-convergence of a family of low-order phase-field approximations towards the corresponding sharp-interface fracture model. The analysis provides a rigorous mathematical justification for computationally efficient damage formulations while preserving the variational structure of brittle fracture.
This work naturally complements my earlier contributions on topological derivatives. Whereas topological sensitivity provides rigorous criteria for crack nucleation, phase-field models offer a regularized description of crack growth through diffuse damage variables. Together, these approaches provide complementary variational descriptions of fracture initiation and propagation.
This work naturally complements my earlier contributions on topological derivatives. Whereas topological sensitivity provides rigorous criteria for crack nucleation, phase-field models offer a regularized description of crack growth through diffuse damage variables. Together, these approaches provide complementary variational descriptions of fracture initiation and propagation.
A further direction concerns hydraulic fracture, where crack evolution is coupled with the pressure and transport of a fluid inside the fractured medium. The interaction between elasticity, fluid pressure and changing geometry creates a particularly demanding free-boundary problem.
Our topological-derivative formulation incorporates this coupling directly into the fracture indicator. The crack is generated and extended according to the energetic effect of introducing new fractured regions under the combined action of mechanical loading and fluid pressure. This led first to a simplified model of fracking and then to a more complete hydro-mechanical formulation.
These models illustrate how sensitivity analysis can be used not only as an optimization tool, but also as an evolution law for complex physical systems. They provide a bridge between rigorous asymptotic analysis and practical numerical simulations of propagating fractures.
These developments naturally connect topological derivatives, phase-field regularizations and finite-element approximation within a unified variational framework for fracture mechanics.
In collaboration with P. Areias and other engineering colleagues, I developed computational methods for ductile fracture based on explicit crack representations and adaptive remeshing. These formulations address large deformations, configurational forces and coupled multi-physics effects, with particular emphasis on robust finite-element algorithms for engineering-scale simulations.
This line of research complements the energetic framework developed for brittle fracture while relying on a fundamentally different numerical strategy.
Across these different models, the main mathematical issue is the competition between stored elastic energy, dissipation and the energetic cost of creating new surfaces or damaged regions. The calculus of variations provides the natural framework for formulating this competition and for analysing the stability of the resulting evolutions.
Shape and topological sensitivities translate these variational principles into local computational indicators. Their numerical approximation requires the solution of elasticity or coupled multi-physics systems, together with reliable procedures for updating geometries, regularizing singular fields and tracking evolving defects.
This programme therefore combines analytical derivation, asymptotic methods and finite element computation. The objective is not only to predict fracture paths, but also to understand which mathematical structures make such predictions robust and physically meaningful.
My current work places fracture and damage within a broader theory of defective solids, alongside dislocations, incompatibility and plasticity. From this perspective, cracks and distributed damage are different manifestations of the loss of compatibility and regularity in a continuum body.
A long-term objective is to connect geometric crack models, diffuse damage and higher-order continuum theories in a unified variational framework. This includes the analysis of defect interaction, strain localization, multi-physics coupling and structure-preserving numerical methods for materials containing several classes of evolving singularities.