Homogenized models of mechanical metamaterials
- Creators
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Ulloa, J.1
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Ariza, M.P.2
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Andrade, J. E.1
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Ortiz, M.1
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1.
California Institute of Technology
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2.
University of Seville
Abstract
Direct numerical simulations of mechanical metamaterials are prohibitively expensive due to the separation of scales between the lattice and the macrostructural size. Hence, multiscale continuum analysis plays a pivotal role in the computational modeling of metastructures at macroscopic scales. In the present work, we assess the continuum limit of mechanical metamaterials via homogenized models derived rigorously from variational methods. It is shown through multiple examples that micropolar-type effective energies, derived naturally from analysis, properly capture the kinematics of discrete lattices in two and three dimensions. Moreover, the convergence of the discrete energy to the continuum limit is shown numerically. We provide open-source computational implementations for all examples, including both discrete and homogenized models.
Copyright and License (English)
© 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Funding (English)
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, Spain 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 (English)
J. Ulloa: Software, Investigation, Conceptualization, Writing – original draft. M.P. Ariza: Software, Investigation, Funding acquisition, Writing – review & editing. J.E. Andrade: Supervision, Resources, Funding acquisition, Writing – review & editing. M. Ortiz: Writing – review & editing, Writing – original draft, Methodology, Investigation, Conceptualization.
Supplemental Material (English)
- MMC S1. Auxiliary Python function to rotate the cubic elasticity matrix around an arbitrary axis.
- MMC S2. Auxiliary MATLAB function to compute discrete energies in frame structures.
- MMC S3. Direct numerical simulation of a perforated plate with a 2D honeycomb microstructure; Stabil implementation.
- MMC S4. Homogenized model of a perforated plate with a 2D honeycomb microstructure; FEniCS implementation.
- MMC S5. Direct numerical simulation of a single-edge notched plate with a 2D honeycomb microstructure; Stabil implementation.
- MMC S6. Homogenized model of a single-edge notched plate with a 2D honeycomb microstructure; FEniCS implementation.
- MMC S7. Direct numerical simulation of an L-shaped plate with a 2D honeycomb microstructure; Stabil implementation.
- MMC S8. Homogenized model of an L-shaped plate with a 2D honeycomb microstructure; FEniCS implementation.
- MMC S9. Direct numerical simulation of a cube-torsion test with a 3D octet-truss microstructure; Stabil implementation.
- MMC S10. Homogenized model of a cube-torsion test with a 3D octet-truss microstructure; FEniCS implementation.
- MMC S11. Direct numerical simulation of an L-shaped solid with a 3D octet-truss microstructure; Stabil implementation.
- MMC S12. Homogenized model of an L-shaped solid with a 3D octet-truss microstructure; FEniCS implementation.
- MMC S13. Mesh files.
Data Availability
We include the code and relevant files as supplementary materials to enable the reproduction of the numerical results.
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Additional details
- United States Army Research Office
- Deutsche Forschungsgemeinschaft
- 390685813 - GZ 2047/1
- Deutsche Forschungsgemeinschaft
- 441211072 - SPP 2256
- Deutsche Forschungsgemeinschaft
- 211504053 - SFB 1060
- Ministerio de Ciencia, Innovación y Universidades
- PID2021-124869NB-I00
- DEVCOM Army Research Laboratory
- Department of Defense Multidisciplinary University Research Initiative (MURI) W911NF-19-1-0245
- Accepted
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2024-10-08Accepted
- Available
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2024-11-01Published online
- Available
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2024-11-01Version of record
- Publication Status
- Published