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Analytical models of the geometric properties of solid and hollow architected lattice cellular materials

  • Christopher J. Ro (a1) and Christopher S. Roper (a1)

Abstract

New closed-form analytical equations for volume fractions and surface-area-to-volume ratios for architected lattice cellular materials are derived. Prior approximate equations which erroneously over count overlapping volumes and the associated surface area are commonly used in the literature. These equations are found to have up to 184% error for volume fraction calculations for hollow lattices and 211% error for surface-area-to-volume ratio calculations, thus necessitating computational methods to arrive at accurate geometric properties for cellular lattice materials. This work derives new equations which are accurate to better than 1% for both volume fraction and surface-area-to-volume ratio as compared to the computational models. These new equations for cellular lattice materials are applicable to both pyramidal and tetrahedral unit cells as well as to both hollow and solid lattice members. By eliminating the need for numerical models to compute accurate volume fractions and surface-area-to-volume ratios of architected cellular materials, these new analytical equations will enable accurate yet computationally efficient optimization of the physical properties of architected cellular materials.

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a) Address all correspondence to this author. e-mail: csroper@hrl.com

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Contributing Editor: Lorenzo Valdevit

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References

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1. Evans, A.G., Hutchinson, J.W., Fleck, N.A., Ashby, M.F., and Wadley, H.N.G.: The topological design of multifunctional cellular metals. Prog. Mater. Sci. 46, 309 (2001).
2. Chiras, S., Mumm, D.R., Evans, A.G., Wicks, N., Hutchinson, J.W., Dharmasena, K., Wadley, H.N.G., and Fichter, S.: The structural performance of near-optimized truss core panels. Int. J. Solids Struct. 39, 4093 (2002).
3. Wicks, N. and Hutchinson, J.W.: Performance of sandwich plates with truss cores. Mech. Mater. 36, 739 (2004).
4. Schaedler, T.A., Ro, C.J., Sorensen, A.E., Eckel, Z., Yang, S.S., Carter, W.B., and Jacobsen, A.J.: Designing metallic microlattices for energy absorber applications. Adv. Eng. Mater. 16, 276 (2014).
5. Evans, A.G., He, M.Y., Deshpande, V.S., Hutchinson, J.W., Jacobsen, A.J., and Carter, W.B.: Concepts for enhanced energy absorption using hollow micro-lattices. Int. J. Impact Eng. 37, 947 (2010).
6. Lu, T.J., Stone, H.A., and Ashby, M.F.: Heat transfer in open-cell metal foams. Acta Mater. 46, 3619 (1998).
7. Maloney, K.J., Fink, K.D., Schaedler, T.A., Kolodziejska, J.A., Jacobsen, A.J., and Roper, C.S.: Multifunctional heat exchangers derived from three-dimensional micro-lattice structures. Int. J. Heat Mass Transfer 55, 2486 (2012).
8. Roper, C.S., Schubert, R.C., Maloney, K.J., Page, D., Ro, C.J., Yang, S.S., and Jacobsen, A.J.: Scalable 3D bicontinuous fluid networks: Polymer heat exchangers toward artificial organs. Adv. Mater. 27, 24792484 (2015).
9. Jones, J.R., Lin, S., Yue, S., Lee, P.D., Hanna, J.V., Smith, M.E., and Newport, R.J.: Bioactive glass scaffolds for bone regeneration and their hierarchical characterisation. Proc. Inst. Mech. Eng., Part H 224, 1373 (2010).
10. Lu, T.J., Hess, A., and Ashby, M.F.: Sound absorption in metallic foams. J. Appl. Phys. 85, 7528 (1999).
11. Ashby, M.F.: The properties of foams and lattices. Philos. Trans. R. Soc., A 364, 15 (2006).
12. Wadley, H.N.G.: Cellular metals manufacturing. Adv. Eng. Mater. 4, 726 (2002).
13. Finnegan, K., Kooistra, G., Wadley, H.N.G., and Deshpande, V.S.: The compressive response of carbon fiber composite pyramidal truss sandwich cores. Int. J. Mater. Res. 98, 1264 (2007).
14. Gibson, L.J. and Ashby, M.F.: The mechanics of three-dimensional cellular materials. Philos. Trans. R. Soc., A 382, 43 (1982).
15. Gibson, L. and Ashby, M.F.: Cellular Solids: Structure and Properties, 2nd ed. (Cambridge University Press, Cambridge, U.K., 1997).
16. Gibson, L.J., Ashby, M.F., and Harley, B.A.: Cellular Materials in Nature and Medicine (Cambridge University Press, Cambridge, U.K., 2010).
17. Deshpande, V.S., Ashby, M.F., and Fleck, N.A.: Foam topology: Bending versus stretching dominated architectures. Acta Mater. 49, 1035 (2001).
18. Jacobsen, A.J., Barvosa-Carter, W., and Nutt, S.: Compression behavior of micro-scale truss structures formed from self-propagating polymer waveguides. Acta Mater. 55, 6724 (2007).
19. Zheng, X., Lee, H., Weisgraber, T.H., Shusteff, M., DeOtte, J., Duoss, E.B., Kuntz, J.D., Biener, M.M., Ge, Q., Jackson, J.A., Kucheyev, S.O., Fang, N.X., and Spadaccini, C.M.: Ultralight, ultrastiff mechanical metamaterials. Science 344, 1373 (2014).
20. Lu, T., Valdevit, L., and Evans, A.: Active cooling by metallic sandwich structures with periodic cores. Prog. Mater. Sci. 50, 789 (2005).
21. Wadley, H.N.G., Fleck, N.A., and Evans, A.G.: Fabrication and structural performance of periodic cellular metal sandwich structures. Compos. Sci. Technol. 63, 2331 (2003).
22. Queheillalt, D.T. and Wadley, H.N.G.: Pyramidal lattice truss structures with hollow trusses. Mater. Sci. Eng., A 397, 132 (2005).
23. Kooistra, G.W., Deshpande, V.S., and Wadley, H.N.G.: Compressive behavior of age hardenable tetrahedral lattice truss structures made from aluminium. Acta Mater. 52, 4229 (2004).
24. Wang, J., Evans, A.G., Dharmasena, K., and Wadley, H.N.G.: On the performance of truss panels with Kagomé cores. Int. J. Solids Struct. 40, 6981 (2003).
25. Jacobsen, A.J., Barvosa-Carter, W., and Nutt, S.: Micro-scale truss structures with three-fold and six-fold symmetry formed from self-propagating polymer waveguides. Acta Mater. 56, 2540 (2008).
26. Maloney, K.J., Roper, C.S., Jacobsen, A.J., Carter, W.B., Valdevit, L., and Schaedler, T.A.: Microlattices as architected thin films: Analysis of mechanical properties and high strain elastic recovery. APL Mater. 1, 022106 (2013).
27. Schaedler, T.A., Jacobsen, A.J., Torrents, A., Sorensen, A.E., Lian, J., Greer, J.R., Valdevit, L., and Carter, W.B.: Ultralight metallic microlattices. Science 334, 962 (2011).
28. Jacobsen, A.J., Barvosa-Carter, W., and Nutt, S.: Micro-scale truss structures formed from self-propagating photopolymer waveguides. Adv. Mater. 19, 3892 (2007).
29. Fink, K.D., Kolodziejska, J.A., Jacobsen, A.J., and Roper, C.S.: Fluid dynamics of flow through microscale lattice structures formed from self-propagating photopolymer waveguides. AIChE J. 57, 2636 (2011).
30. Roper, C.S., Fink, K.D., Lee, S.T., Kolodziejska, J.A., and Jacobsen, A.J.: Anisotropic convective heat transfer in microlattice materials. AIChE J. 59, 622 (2013).
31. Wicks, N. and Hutchinson, J.W.: Optimal truss plates. Int. J. Solids Struct. 38, 5165 (2001).
32. Valdevit, L., Jacobsen, A.J., Greer, J.R., and Carter, W.B.: Protocols for the optimal design of multi-functional cellular structures: From hypersonics to micro-architected materials. J. Am. Ceram. Soc. 94, s15 (2011).
33. Jacobsen, A.J., Barvosa-Carter, W., and Nutt, S.: Shear behavior of polymer micro-scale truss structures formed from self-propagating polymer waveguides. Acta Mater. 56, 1209 (2008).
34. Wadley, H.N.G.: Multifunctional periodic cellular metals. Philos. Trans. R. Soc., A 364, 31 (2006).
35. Deshpande, V.S., Fleck, N.A., and Ashby, M.F.: Effective properties of the octet-truss lattice material. J. Mech. Phys. Solids 49, 1747 (2001).
36. Deshpande, V.S. and Fleck, N.A.: Collapse of truss core sandwich beams in 3-point bending. Int. J. Solids Struct. 38, 6275 (2001).
37. Valdevit, L., Godfrey, S.W., Schaedler, T.A., Jacobsen, A.J., and Carter, W.B.: Compressive strength of hollow microlattices: Experimental characterization, modeling, and optimal design. J. Mater. Res. 28, 2461 (2013).
38. Hammetter, C.I., Rinaldi, R.G., and Zok, F.W.: Pyramidal lattice structures for high strength and energy absorption. J. Appl. Mech. 80, 041014-1041014-11 (2013).
39. Roper, C.S.: Multiobjective optimization for design of multifunctional sandwich panel heat pipes with micro-architected truss cores. Int. J. Heat Fluid Flow 32, 239 (2011).
40. Moreton, M.: Symmetrical intersections of right circular cylinders. Math. Gaz. 58, 181 (1974).
41. Angell, I.O. and Moore, M.: Symmetrical intersections of cylinders. Acta Crystallogr., Sect. A: Found. Crystallogr. 43, 244 (1987).
42. Jacobsen, A.J., Kolodziejska, J.A., Doty, R., Fink, K.D., Zhou, C., Roper, C.S., and Carter, W.B.: Interconnected self-propagating photopolymer waveguides: An alternative to stereolithography for rapid formation of lattice-based open-cellular materials. In 21st Annual International Solid Freeform Fabrication Symposium—An Additive Manufacturing Conference, SFF 2010 (2010); p. 846.

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