A phase-field formulation incorporating mode-dependent fracture toughness for fracture modeling in rock-like materials

Research output: Contribution to journalResearch articleContributedpeer-review


For most rock-like materials, including concrete, their failure mechanisms under compression have proven to be rather complex and of great research interest. In this work, a phase-field formulation for brittle fracture is proposed to account for the mixed-mode failure observed in numerous experimental investigations. The tensile failure and shear failure are driven by separate energetic terms based on a directional decomposition. It is assumed that the critical energy release rate is a continuous description that depends on the loading mode. Finally, the different interpolations between the intact state and the crack state for each loading mode, i.e. the degradation functions, is considered to complete the proposed model. Following a consistent derivation, the mixed-mode phase-field model is implemented into an in-house finite element framework. Several experiment-based examples serve to illustrate the capability of the model. Finally, the summary closes this work with some further insights.


Original languageEnglish
Article number109345
JournalEngineering Fracture Mechanics
Early online date26 May 2023
Publication statusPublished - 1 Sept 2023

External IDs

unpaywall 10.1016/j.engfracmech.2023.109345
Scopus 85161341274
ORCID /0000-0002-6115-6493/work/142250914



  • Mixed-mode failure, Phase-field fracture, Rock-like material

Library keywords