Abstract
In this expository Note, it is shown that the Griffith phase-field theory of fracture accounting for material strength originally introduced by Kumar, Francfort, and Lopez-Pamies (J Mech Phys Solids 112, 523–551, 2018) in the form of PDEs can be recast as a variational theory. In particular, the solution pair \((\textbf{u},v)\) defined by the PDEs for the displacement field \(\textbf{u}\) and the phase field v is shown to correspond to the fields that minimize separately two different functionals, much like the solution pair \((\textbf{u},v)\) defined by the original phase-field theory of fracture without material strength implemented in terms of alternating minimization. The merits of formulating a complete theory of fracture nucleation and propagation via such a variational approach — in terms of the minimization of two different functionals — are discussed.
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Notes
We will come back to the more general case of non-monotonic and dynamic loading conditions in Section 4 below.
A FEniCS code of the resulting phase-field theory is available in GitHub: https://github.com/farhadkama/FEniCSx_Kamarei_Kumar_Lopez-Pamies.
“Large” refers to large relative to the characteristic size of the underlying heterogeneities in the solid under investigation. By the same token, “small” refers to sizes that are of the same order or just moderately larger than the sizes of the heterogeneities. In practice, as a rule of thumb, “large” refers to large relative to the material length scale \(G_c/\mathcal {W}_{\texttt {ts}}\).
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Acknowledgements
Support for this work by the National Science Foundation through the Grants DMS–2206114 and DMS–2308169 is gratefully acknowledged. This work began during the 2024 workshop “Fracture as an emergent phenomenon” at the Mathematisches Forschungsinstitut Oberwolfach, who we thank for hosting us.
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Christopher J. Larsen: Conceptualization, Formal analysis, Writing – review & editing. John E. Dolbow: Writing – review & editing. Oscar Lopez-Pamies: Conceptualization, Formal analysis, Writing – original draft, Writing – review & editing.
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Larsen, C.J., Dolbow, J.E. & Lopez-Pamies, O. A variational formulation of Griffith phase-field fracture with material strength. Int J Fract (2024). https://doi.org/10.1007/s10704-024-00786-3
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DOI: https://doi.org/10.1007/s10704-024-00786-3