A phase field approach for damage propagation in periodic microstructured materials

  • F. FantoniEmail author
  • A. Bacigalupo
  • M. Paggi
  • J. Reinoso
Original Paper


In the present work, the evolution of damage in periodic composite materials is investigated through a novel finite element-based multiscale computational approach. The proposed methodology is developed by means of the original combination of asymptotic homogenization with the phase field approach to nonlocal damage. This last is applied at the macroscale level on the equivalent homogeneous continuum, whose constitutive properties are obtained in closed form via a two-scale asymptotic homogenization scheme. The formulation considers different assumptions on the evolution of damage at the microscale (e.g., damage in the matrix and not in the inclusion/fiber), as well as the role played by the microstructural reinforcement, i.e. its volumetric content and shape. Numerical results show that the proposed formulation leads to an apparent tensile strength and a post-peak branch of unnotched and notched specimens dependent not only on the internal length scale of the phase field approach, as for homogeneous materials, but also on microstructural features. Down-scaling relations provide the full reconstruction of the microscopic fields at any point of the macroscopic model, as a simple post-processing operation.


Periodic microstructured material Asymptotic homogenization scheme Phase field approach 



AB would like to acknowledge the financial support by National Group of Mathematical Physics (GNFM-INdAM). MP would like to acknowledge the financial support of the Italian Ministry of Education, University and Research to the Research Project of National Interest (PRIN 2017: “XFAST-SIMS: Extra fast and accurate simulation of complex structural systems” (CUP: D68D19001260001)). The authors would like to thank the IMT School for Advanced Studies Lucca for its support to the stays of FF and JR in the IMT Campus as visiting researchers in 2019, making possible the realization of this joint work.


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© Springer Nature B.V. 2019

Authors and Affiliations

  • F. Fantoni
    • 1
    Email author
  • A. Bacigalupo
    • 2
  • M. Paggi
    • 2
  • J. Reinoso
    • 3
  1. 1.DICATAM, Universitá degli Studi di BresciaBresciaItaly
  2. 2.IMT School for Advanced Studies LuccaLuccaItaly
  3. 3.Department of Continuum Mechanics and Structural Mechanics, School of EngineeringUniversidad de SevillaSevilleSpain

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