EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 32 (2026) ESAIM: COCV, 32 (2026) 12 References
Open Access
Issue
ESAIM: COCV
Volume 32, 2026
Article Number 12
Number of page(s) 38
DOI https://doi.org/10.1051/cocv/2025101
Published online 11 February 2026
  1. A.A. Griffith, The phenomena of rupture and flow in solids. Phil. Trans. Roy. Soc. London 18 (1920) 163–198. [Google Scholar]
  2. M. Negri, A comparative analysis on variational models for quasi-static brittle crack propagation. Adv. Calc. Var. 3 (2010) 149–212. [Google Scholar]
  3. M. Negri and C. Ortner, Quasi-static propagation of brittle fracture by Griffith's criterion. Math. Models Methods Appl. Sci. 18 (2008) 1895–1925. [Google Scholar]
  4. G.A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998) 1319–1342. [Google Scholar]
  5. A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures. Calc. Var. Pari. Differ. Equ. 22 (2005) 129–172. [Google Scholar]
  6. A. Mielke and T. Roubicek, Rate-Independent Systems: Theory and Application., vol. 193 of Applied Mathematical Sciences. Springer, New York (2015). [Google Scholar]
  7. K. Friebertshausen M. Thomas, S. Tornquist, K. Weinberg and C. Wieners, Dynamic phase-field fracture in viscoelastic materials using a first-order formulation. Proc. Appl. Math. Mech. 22 (2023) e202200249. [Google Scholar]
  8. C.J. Larsen, C. Ortner and E. Suli, Existence of solutions to a regularized model of dynamic fracture. Math. Models Methods Appl. Sci. 20 (2010) 1021–1048. [Google Scholar]
  9. T. Roubicek, Coupled time discretization of dynamic damage models at small strains. IMA J. Numer. Anal. 40 (2020) 1772–1791. [Google Scholar]
  10. G. Lazzaroni and R. Toader, A model for crack propagation based on viscous approximation. Math. Models Methods Appl. Sci. 21 (2011) 2019–2047. [CrossRef] [MathSciNet] [Google Scholar]
  11. B. Grossman-Ponemon, E. Maggiorelli, A.J. Lew and M. Negri, How do cracks in brittle materials evolve, and how do we compute them? IACM Bull. 54 (2025) 2–7. [Google Scholar]
  12. B. Bourdin, G.A. Francfort and J.-J. Marigo, Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000) 797–826. [Google Scholar]
  13. P. Farrell and C. Maurini, Linear and nonlinear solvers for variational phase-field models of brittle fracture. Int. J. Numer. Methods Eng. 109 (2017) 648–667. [Google Scholar]
  14. M. Ambati, T. Gerasimov and L. De Lorenzis, A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comp. Mech. 55 (2015) 383–405. [Google Scholar]
  15. B. Bourdin, G.A. Francfort and J.-J. Marigo, The variational approach to fracture. J. Elasticity 91 (2008) 5–148. [Google Scholar]
  16. F. Vicentini, C. Zolesi, P. Carrara, C. Maurini and L. De Lorenzis, On the energy decomposition in variational phase-field models for brittle fracture under multi-axial stress states. Int. J. Fract. 247 (2024) 291–317. [Google Scholar]
  17. T. Wick, Multiphysics phase-field fracture: modeling, adaptive discretizations, and solvers, vol. 28 of Radon Series on Computational and Applied Mathematics. De Gruyter, Berlin (2020). [Google Scholar]
  18. S. Almi and M. Negri, Analysis of staggered evolutions for nonlinear energies in phase field fracture. Arch. Ration. Mech. Anal. 236 (2020) 189–252. [Google Scholar]
  19. D. Graeser, C. ad Kienle and O. Sander, Truncated nonsmooth newton multigrid for phase-field brittle-fracture problems, with analysis. Comput. Mech. 72 (2023) 1059–1089. [Google Scholar]
  20. H. Amor, J.-J. Marigo and C. Maurini, Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J. Mech. Phys. Solids 57 (2009) 1209–1229. [Google Scholar]
  21. C. Miehe, F. Welschinger and M. Hofacker, Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int. J. Numer. Methods Eng. 83 (2010) 1273–1311. [Google Scholar]
  22. A. Chambolle, S. Conti and G.A. Francfort, Approximation of a brittle fracture energy with a constraint of non-interpenetration. Arch. Ration. Mech. Anal. 228 (2018) 867–889. [Google Scholar]
  23. L. Ambrosio and V.M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via r-convergence. Comm. Pure Appl. Math. 43 (1990) 999–1036. [Google Scholar]
  24. A. Braides, Approximation of Free-Discontinuity Problems. Springer-Verlag, Berlin (1998). [Google Scholar]
  25. G. Dal Maso, An introduction to T-Convergence. Birkhauser, Boston (1993). [Google Scholar]
  26. E. Maggiorelli, Griffith criterion for steady and unsteady-state crack propagation, in Mathematical Modeling in Cultural Heritage, Springer INdAM Series. Springer (2025) 99–116. [Google Scholar]
  27. J.-F. Babadjian, V. Millot and R. Rodiac, On the convergence of critical points of the Ambrosio-Tortorelli functional. Ann. Inst. H. Poincare Anal. Non Linéaire 41 (2023) 1367–1417. [Google Scholar]
  28. M. Negri, From phase-field to sharp cracks: convergence of quasi-static evolutions in a special setting. Appl. Math. Lett. 26 (2013) 219–224. [Google Scholar]
  29. P. Sicsic and J.-J. Marigo, From gradient damage laws to Griffith's theory of crack propagation. J. Elasticity 113 (2013) 55–74. [Google Scholar]
  30. F. Rorentrop, S. Boddin, D. Knees and J. Mosler, Approximation of balanced viscosity solutions of a rate-independent damage model by combining alternate minimization with a local minimization algorithm (2022) Preprint arXiv:2211.12940. [Google Scholar]
  31. A.A. Leon Baldelli and C. Maurini, Numerical bifurcation and stability analysis of variational gradient-damage models for phase-field fracture. J. Mech. Phys. Solids 152 (2021) Paper No. 104424, 22. [Google Scholar]
  32. L. Minotti and G. Savare, Viscous corrections of the time incremental minimization scheme and visco-energetic solutions to rate-independent evolution problems. Arch. Ration. Mech. Anal. 227 (2018) 477–543. [Google Scholar]
  33. D. Knees and M. Negri, Convergence of alternate minimization schemes for phase field fracture and damage. Math. Models Methods Appl. Sci. 27 (2017) 1743–1794. [Google Scholar]
  34. R. Herzog, C. Meyer and G. Wachsmuth, Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions. J. Math. Anal. Appl. 382 (2011) 802–813. [Google Scholar]
  35. D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23 (2013) 565–616. [Google Scholar]
  36. M. Kimura and M. Negri, Weak solutions for unidirectional gradient flows: existence, uniqueness, and convergence of time discretization schemes. NoDEA Nonlinear Differ. Equ. Appl. 28 (2021) Paper No. 59, 46. [Google Scholar]
  37. C. Kuratowski. Topologie. I and II. Editions Jacques Gabay, Sceaux (1992). [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.