Fracture and size effect in mechanical metamaterials
Abstract
We resort to variational methods to evaluate the asymptotic behavior of fine metamaterials as a function of cell size. To zeroth order, the metamaterial behaves as a micropolar continuum with both displacement and rotation degrees of freedom, but exhibits linear-elastic fracture mechanics scaling and therefore no size effect. To higher order, the overall energetics of the metastructure can be characterized explicitly in terms of the solution of the zeroth-order continuum problem by the method of Γ-expansion. We present explicit expressions of the second-order correction for octet frames. As an application, we evaluate the compliance of double-cantilever octet specimens to second order and use the result to elucidate the dependence of the apparent toughness of the specimen on cell size. The analysis predicts the discreteness of the metamaterial lattice to effectively shield the crack-tip, a mechanism that we term lattice shielding. The theory specifically predicts anti-shielding, i. e., coarser is weaker, in agreement with recent experimental observations.
Copyright and License
© 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Funding
M. Ortiz gratefully acknowledges the support of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via project 211504053 - SFB 1060; project 441211072 - SPP 2256; and project 390685813 - GZ 2047/1 - HCM. M. P. Ariza gratefully acknowledges financial support from Ministerio de Ciencia e Innovación under grant number PID2021-124869NB-I00. J. Ulloa and J. E. Andrade acknowledge the support from the US ARO MURI program with Grant No. W911NF-19-1-0245.
Contributions
J. Ulloa: Validation, Software, Investigation. M.P. Ariza: Visualization, Software, Investigation, Funding acquisition. J.E. Andrade: Supervision, Resources, Funding acquisition. M. Ortiz: Supervision, Investigation, Conceptualization.
Data Availability
No data was used for the research described in the article.
Conflict of Interest
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Michael Ortiz reports financial support was provided by Deutsche Forschungsgemeinschaft. M.P. Ariza reports a relationship with Spain Ministry of Science and Innovation that includes: funding grants. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Additional details
- Deutsche Forschungsgemeinschaft
- 211504053 - SFB 1060
- Deutsche Forschungsgemeinschaft
- 441211072 - SPP 2256
- Deutsche Forschungsgemeinschaft
- 390685813 - GZ 2047/1
- Ministerio de Ciencia, Innovación y Universidades
- PID2021-124869NB-I00
- United States Army Research Office
- W911NF-19-1-0245
- Available
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2024-09-10Available Online
- Accepted
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2024-09-07Accepted
- Publication Status
- Published