This seminar focuses on the variational approach to fracture, often referred to as the phase field method applied to fracture mechanics. It begins with a critical examination of the original formulation for brittle materials—which possesses proven $\Gamma$-convergence properties [1]—along with variants developed over the past decade for quasi-brittle and cohesive materials [2,3], used in statics but also in dynamics. These extensions have introduced different expressions for the degradation and crack density functions, along with different assumptions on the way the material stiffness is degraded.
The presentation then provides an overview of recent methodological advances aimed at addressing open challenges in computational materials science and structural mechanics. In the first area, significant progress has been made in modeling material heterogeneity [4,5], which is essential at those scales of observation. In the second, novel contributions have adapted the method to enable its application to shell structures [6] and to metamaterials based on 3D-printed lattice architectures [7].

Selected references
    [1] G.A. Francfort, J.-J. Marigo (1998) Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46:1319–1342.
    [2] F. Freddi, G. Royer-Carfagni (2010) Regularized variational theories of fracture: a unified approach, J. Mech. Phys. Solids, 58:1154–1174.
    [3] J.-Y. Wu (2017) A unified phase-field theory for the mechanics of damage and quasi-brittle failure, J. Mech. Phys. Solids, 103:72–99.
    [4] M. Paggi, J. Reinoso (2017) Revisiting the problem of a crack impinging on an interface: a modeling framework for the interaction between the phase field approach for brittle fracture and the interface cohesive zone model. Comp. Meth. Appl. Mech. Eng., 321:145–172.
    [5] P. Lenarda, J. Reinoso, M. Paggi (2022) Multi-phase field approach to tensile fracture and compressive crushing in grained heterogeneous materials. Theor. Appl. Fract. Mech., 122:103632.
    [6] J. Reinoso, M. Paggi, C. Linder (2017) Phase field modeling of brittle fracture for enhanced assumed strain shells at large deformations: formulation and finite element implementation, Comput. Mech., 59:981–1001.
    [7] M. Paggi (2026) Mechanics of complex network materials: A formulation based on phase field damage evolution on graphs. Comp. Meth. Appl. Mech. Eng., 450:118637.