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A mechanical reduced order model for elastomeric 3D printed architectures

  • Todd H. Weisgraber (a1), Thomas Metz (a1), Christopher M. Spadaccini (a1), Eric B. Duoss (a1), Ward Small (a2), Jeremy M. Lenhardt (a2), Robert S. Maxwell (a2) and Thomas S. Wilson (a2)...


Direct ink writing of silicone elastomers enables printing with precise control of porosity and mechanical properties of ordered cellular solids, suitable for shock absorption and stress mitigation applications. With the ability to manipulate structure and feedstock stiffness, the design space becomes challenging to parse to obtain a solution producing a desired mechanical response. Here, we derive an analytical design approach for a specific architecture. Results from finite element simulations and quasi-static mechanical tests of two different parallel strand architectures were analyzed to understand the structure-property relationships under uniaxial compression. Combining effective stiffness-density scaling with least squares optimization of the stress responses yielded general response curves parameterized by resin modulus and strand spacing. An analytical expression of these curves serves as a reduced order model, which, when optimized, provides a rapid design capability for filament-based 3D printed structures. As a demonstration, the optimal design of a face-centered tetragonal architecture is computed that satisfies prescribed minimum and maximum load constraints.


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1. Truby, R.L. and Lewis, J.A.: Printing soft matter in three dimensions. Nature 540, 371 (2016).
2. King, W.E., Anderson, A.T., Ferencz, R.M., Hodge, N.E., Kamath, C., Khairallah, S.A., and Rubenchik, A.M.: Laser powder bed fusion additive manufacturing of metals; physics, computational, and materials challenges. Appl. Phys. Rev. 2, 041304 (2015).
3. Lewis, J.A.: Direct ink writing of 3D functional materials. Adv. Funct. Mater. 16, 2193 (2006).
4. Duoss, E.B., Weisgraber, T.H., Hearon, K., Zhu, C., Small, W., Metz, T.R., Vericella, J.J., Barth, H.D., Kuntz, J.D., Maxwell, R.S., Spadaccini, C.M., and Wilson, T.S.: Three-dimensional printing of elastomeric, cellular architectures with negative stiffness. Adv. Funct. Mater. 24, 4905 (2014).
5. Maiti, A., Small, W., Lewicki, J.P., Weisgraber, T.H., Duoss, E.B., Chinn, S.C., Pearson, M.A., Spadaccini, C.M., Maxwell, R.S., and Wilson, T.S.: 3D printed cellular solid outperforms traditional stochastic foam in long-term mechanical response. Sci. Rep. 6, 24871 (2016).
6. 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).
7. Meza, L.R., Zelhofer, A.J., Clarke, N., Mateos, A.J., Kochmann, D.M., and Greer, J.R.: Resilient 3D hierarchical architected metamaterials. Proc. Natl. Acad. Sci. U. S. A. 112, 11502 (2015).
8. Wang, Q., Jackson, J.A., Ge, Q., Hopkins, J.B., Spadaccini, C.M., and Fang, N.X.: Lightweight mechanical metamaterials with tunable negative thermal expansion. Phys. Rev. Lett. 117, 175901 (2016).
9. Liu, J., Gu, T., Shan, S., Kang, S.H., Weaver, J.C., and Bertoldi, K.: Harnessing buckling to design architected materials that exhibit effective negative swelling. Adv. Mater. 28, 6619 (2016).
10. Gladman, A.S., Matsumoto, E.A., Nuzzo, R.G., Mahadevan, L., and Lewis, J.A.: Biomimetic 4D printing. Nat. Mater. 15, 413 (2016).
11. Wu, A.S., Small, W. IV, Bryson, T.M., Cheng, E., Metz, T.R., Schulze, S.E., Duoss, E.B., and Wilson, T.S.: 3D printed silicones with shape memory. Sci. Rep. 7, 4664 (2017).
12. Lode, A., Meyer, M., Brüggemeier, S., Paul, B., Baltzer, H., Schropfer, M., Winkelmann, C., Sonntag, F., and Gelinsky, M.: Additive manufacturing of collagen scaffolds by three-dimensional plotting of highly viscous dispersions. Biofabrication 8, 015015 (2016).
13. Mrozek, R.A., Gold, C.S., Leighliter, B., Sietins, J.M., and Lenhart, J.L.: Open pore, elastomeric scaffolds through frustrated particle collapse. J. Mater. Sci. 51, 10761 (2016).
14. Mac Murray, B.C., An, X., Robinson, S.S., van Meerbeek, I.M., O’Brien, K.W., Zhao, H., and Shepherd, R.F.: Poroelastic foams for simple fabrication of complex soft robots. Adv. Mater. 27, 6334 (2015).
15. Bendsoe, M.P. and Sigmund, O.: Topology Optimization (Springer Science & Business Media, Berlin, Heidelberg, 2013).
16. 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).
17. Messner, M.C.: Optimal lattice-structured materials. J. Mech. Phys. Solids 96, 162 (2016).
18. Watts, S. and Tortorelli, D.A.: A geometric projection method for designing three-dimensional open lattices with inverse homogenization. Int. J. Numer. Meth. Eng. 60, 351 (2017).
19. Zheng, Q., Jiang, D., Huang, C., Shang, X., and Ju, S.: Analysis of failure loads and optimal design of composite lattice cylinder under axial compression. Compos. Struct. 131, 885 (2015).
20. Weickum, G., Eldred, M.S., and Maute, K.: A multi-point reduced-order modeling approach of transient structural dynamics with application to robust design optimization. Struct. Multidiscipl. Optim. 38, 599 (2008).
21. Puso, M.A.: NIKE3D: A nonlinear, implicit, three-dimensional finite element code for solid and structural mechanics - User's Manual (Lawrence Livermore National Laboratory, Livermore, California, 2012).
22. Gibson, L.J. and Ashby, M.F.: Cellular Solids, Structure and Properties (Cambridge University Press, Cambridge, U. K., 1999).
23. Worsley, M.A., Kucheyev, S.O., Satcher, J.H. Jr., Hamza, A.V., and Baumann, T.F.: Mechanically robust and electrically conductive carbon nanotube foams. Appl. Phys. Lett. 94, 073115 (2009).
24. Ogden, R.W.: Recent advances in the phenomenological theory of rubber elasticity. Rubber Chem. Technol. 59, 361 (1986).
25. Hill, R.: Advances in Applied Mechanics, Vol. 18 (Elsevier, Cambridge, Massachusetts, 1979); pp. 175.
26. Finkel, D.E. and Kelley, C.: Convergence analysis of the direct algorithm. Optim. Online 14, 1 (2004).



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