Crack nucleation in heterogeneous bars: h- and p-FEM of a phase field model

Maxime Levy, Francesco Vicentini, Zohar Yosibash*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Failure initiation and subsequent crack trajectory in heterogeneous materials, such as functionally graded materials and bones, are yet insufficiently addressed. The AT1 phase field model (PFM) is investigated herein in a 1D setting, imposing challenges and opportunities when discretized by h- and p-finite element (FE) methods. We derive explicit PFM solutions to a heterogeneous bar in tension considering heterogeneous E(x) and GIc(x), necessary for verification of the FE approximations. GIc(x) corrections accounting for the element size at the damage zone in h-FEMs are suggested to account for the peak stress underestimation. p-FEMs do not require any such corrections. We also derive and validate penalty coefficient for heterogeneous domains to enforce damage positivity and irreversibility via penalization. Numerical examples are provided, demonstrating that p-FEMs exhibit faster convergence rates comparing to classical h-FEMs. The new insights are encouraging towards p-FEM discretization in a 3D setting to allow an accurate prediction of failure initiation in human bones.

Original languageEnglish
JournalComputational Mechanics
StateAccepted/In press - 2024


FundersFunder number
Horizon 2020 Framework Programme
H2020 Marie Skłodowska-Curie Actions861061


    • 1D heterogeneous bar
    • Crack nucleation
    • Phase field model


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