Papers
Topics
Authors
Recent
Assistant
AI Research Assistant
Well-researched responses based on relevant abstracts and paper content.
Custom Instructions Pro
Preferences or requirements that you'd like Emergent Mind to consider when generating responses.
Gemini 2.5 Flash
Gemini 2.5 Flash 87 tok/s
Gemini 2.5 Pro 51 tok/s Pro
GPT-5 Medium 17 tok/s Pro
GPT-5 High 23 tok/s Pro
GPT-4o 102 tok/s Pro
Kimi K2 166 tok/s Pro
GPT OSS 120B 436 tok/s Pro
Claude Sonnet 4 37 tok/s Pro
2000 character limit reached

Homogenization and continuum limit of mechanical metamaterials (2403.18854v2)

Published 18 Mar 2024 in math.AP, math-ph, and math.MP

Abstract: When used in bulk applications, mechanical metamaterials set forth a multiscale problem with many orders of magnitude in scale separation between the micro and macro scales. However, mechanical metamaterials fall outside conventional homogenization theory on account of the flexural, or bending, response of their members, including torsion. We show that homogenization theory, based on calculus of variations and notions of Gamma-convergence, can be extended to account for bending. The resulting homogenized metamaterials exhibit intrinsic generalized elasticity in the continuum limit. We illustrate these properties in specific examples including two-dimensional honeycomb and three-dimensional octet-truss metamaterials.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (53)
  1. Materials by design: Using architecture in material design to reach new property spaces. MRS Bulletin, 40(12):1122–1129, 2015.
  2. Three-dimensional architected materials and structures: Design, fabrication, and mechanical behavior. MRS Bulletin, 44(10):750–757, 2019.
  3. Architectural design and additive manufacturing of mechanical metamaterials: A review. Engineering, 17:44–63, 2022.
  4. Micro-architectured materials: Past, present and future. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466(2121):2495–2516, 2010.
  5. Architected cellular materials: A review on their mechanical properties towards fatigue-tolerant design and fabrication. Materials Science and Engineering: R: Reports, 144:100606, 2021.
  6. A brief review of dynamic mechanical metamaterials for mechanical energy manipulation. Materials Today, 44:168–193, 2021.
  7. Responsive materials architected in space and time. Nature Reviews Materials, 7(9):683–701, 2022.
  8. H. Jin and H.D. Espinosa. Mechanical metamaterials fabricated from self-assembly: A perspective. Journal of Applied Mechanics, Transactions ASME, 91(4), 2024.
  9. 3d-printed highly stretchable curvy sandwich metamaterials with superior fracture resistance and energy absorption. International Journal of Solids and Structures, 286-287:112570, 2024.
  10. M.F. Ashby. The properties of foams and lattices. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 364(1838):15–30, 2006.
  11. Periodic bicontinuous composites for high specific energy absorption. Nano Lett, 12:4392–4396, 2012.
  12. Fabrication and deformation of three-dimensional hollow ceramic nanostructures. Nature Mater, 12:893–898, 2013.
  13. High-strength cellular ceramic composites with 3D microarchitecture. Proc. Natl. Acad. Sci. USA, 111:2453–2458, 2014.
  14. Strong, lightweight, and recoverable threedimensional ceramic nanolattices. Science, 345:1322–1326, 2014.
  15. Fabrication and deformation of metallic glass micro-lattices. Adv. Eng. Mater, 16:889–896, 2014.
  16. Ultralight, ultrastiff mechanical metamaterials. Science, 344:1373–1377, 2014.
  17. Resilient 3d hierarchical architected metamaterials. Proc. Natl. Acad. Sci. USA, 112:11502–11507, 2015.
  18. Ultra-strong architected Cu meso-lattices. Extreme Mech. Lett, 2:7–14, 2015.
  19. Push-to-pull tensile testing of ultra-strong nanoscale ceramic–polymer composites made by additive manufacturing. Extreme Mech. Lett, 3:105–112, 2015.
  20. Approaching theoretical strength in glassy carbon nanolattices. Nature Mater, 15:438–443, 2016.
  21. Effective properties of the octet-truss lattice material. J. Mech. Phys. Solids, 49:1747–1769, 2001.
  22. Selfassembled ultra high strength, ultra stiff mechanical metamaterials based on inverse opals. Adv. Eng. Mater, 17:1420–1424, 2015.
  23. The stiffness and strength of metamaterials based on the inverse opal architecture. Extreme Mechanics Letters, 12:86–96, 2017.
  24. Foam topology: bending versus stretching dominated architectures. Acta Mater, 49:1035–1040, 2001.
  25. Fracture analysis of cellular materials: A strain gradient model. Journal of the Mechanics and Physics of Solids, 46(5):789–828, 1998.
  26. The toughness of mechanical metamaterials. Nature Materials, 21(3):297–304, 2022.
  27. Fracture resistance of 3d nano-architected lattice materials. Extreme Mechanics Letters, 56, 2022.
  28. Energy-based fracture mechanics of brittle lattice materials. Journal of the Mechanics and Physics of Solids, 169, 2022.
  29. Grid octet truss lattice materials for energy absorption. International Journal of Mechanical Sciences, 259:108616, 2023.
  30. A quasicontinuum theory for the nonlinear mechanical response of general periodic truss lattices. Journal of the Mechanics and Physics of Solids, 124:758–780, 2019.
  31. Multiscale modeling and optimization of the mechanics of hierarchical metamaterials. MRS Bulletin, 44(10):773–781, 2019.
  32. Quasicontinuum analysis of defects in solids. Philosophical Magazine a-Physics of Condensed Matter Structure Defects and Mechanical Properties, 73(6):1529–1563, 1996.
  33. Mixed atomistic and continuum models of deformation in solids. Langmuir, 12(19):4529–4534, 1996.
  34. A ΓΓ\Gammaroman_Γ-convergence analysis of the quasicontinuum method. SIAM Multiscale Modeling & Simulation, 11(3):766–794, 2013.
  35. Data-driven finite element computation of open-cell foam structures. Computer Methods in Applied Mechanics and Engineering, 400:115487, 2022.
  36. Kerstin Weinberg. Data-driven finite element computation of microstructured materials. PAMM, 23(4):e202300285, 2023.
  37. Mechanical metamaterials and beyond. Nature Communications, 14(6004), 2023.
  38. K. Karapiperis and D.M. Kochmann. Prediction and control of fracture paths in disordered architected materials using graph neural networks. Communications Engineering, 2:32, 2023.
  39. D. Cioranescu and P. Donato. An introduction to homogenization, volume 17 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York, 1999.
  40. Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers. Networks and Heterogeneous Media, 4(4):667–708, 2009.
  41. From discrete systems to continuous variational problems: an introduction. In Topics on concentration phenomena and problems with multiple scales, volume 2 of Lect. Notes Unione Mat. Ital., pages 3–77. Springer, Berlin, 2006.
  42. A.C. Eringen. Linear theory of micropolar elasticity. Journal of Mathematics and Mechanics, 15(6):909–923, 1966.
  43. H. Askes and E.C. Aifantis. Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results. International Journal of Solids and Structures, 48:1962–1990, 2011.
  44. Theory of Structures. McGraw-Hill Book Company, New York, 2nd edition, 1965.
  45. G. dal Maso. An introduction to Γnormal-Γ\Gammaroman_Γ-convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston Inc., Boston, MA, 1993.
  46. R. Alicandro and M. Cicalese. A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM Journal on Mathematical Analysis, 36(1):1–37, 2004.
  47. S. Mallat. A wavelet tour of signal processing. Elsevier/Academic Press, Amsterdam, third edition, 2009. The sparse way, With contributions from Gabriel Peyré.
  48. M.P. Ariza and M. Ortiz. Discrete crystal elasticity and discrete dislocations in crystals. Archive for Rational Mechanics and Analysis, 178(2):149–226, 2005.
  49. J. H. Weiner. Statistical mechanics of elasticity. Dover Publications, Mineola, N.Y., 2nd edition, 2002.
  50. M. P. Ariza and M. Ortiz. Discrete dislocations in graphene. Journal of the Mechanics and Physics of Solids, 58(5):710–734, 2010.
  51. Reversibly assembled cellular composite materials. Science, 341(6151):1219–1221, 2013.
  52. Ultralight, strong, and self-reprogrammable mechanical metamaterials. Science Robotics, 9(86):eadi2746, 2024.
  53. A. Braides and L. Truskinovsky. Asymptotic expansions by ΓΓ\Gammaroman_Γ-convergence. Continuum Mechanics and Thermodynamics, 20:21–62, 2008.
Citations (3)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
Lightbulb Streamline Icon: https://streamlinehq.com

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets

This paper has been mentioned in 1 post and received 1 like.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up to View