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The toughness of mechanical metamaterials

Nature Materials volume 21, pages 297–304 (2022)Cite this article

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A Publisher Correction to this article was published on 16 February 2022

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Abstract

Rapid progress in additive manufacturing methods has created a new class of ultralight mechanical metamaterials with extreme functional properties. Their application is ultimately limited by their tolerance to damage and defects, but an understanding of this sensitivity has remained elusive. Using metamaterial specimens consisting of millions of unit cells, we show that not only is the stress intensity factor, as used in conventional elastic fracture mechanics, insufficient to characterize fracture, but also that conventional fracture testing protocols are inadequate. Via a combination of numerical and asymptotic analysis, we extend the ideas of elastic fracture mechanics to truss-based metamaterials and develop a general test and design protocol. This framework can form the basis for fracture characterization in other discrete elastic-brittle solids where the notion of fracture toughness is known to break down.

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Fig. 1: Failure under uniaxial tensile loading.
Fig. 2: Failure under multiaxial loading.
Fig. 3: Fracture mechanism map.
Fig. 4: Through-thickness cracks.

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All data are available in the main text or the supplementary materials.

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References

  1. Valdevit, L., Pantano, A., Stone, H. A. & Evans, A. G. Optimal active cooling performance of metallic sandwich panels with prismatic cores. Int. J. Heat Mass Transf. 49, 3819–3830 (2006).

  2. Chung, D. D. L. Composite Materials for Electrical Applications. Composite Materials: Science and Applications (Springer, 2003).

  3. Tancogne-Dejean, T., Spierings, A. B. & Mohr, D. Additively-manufactured metallic micro-lattice materials for high specific energy absorption under static and dynamic loading. Acta Mater. 116, 14–28 (2016).

  4. Zheng, X. et al. Ultralight, ultrastiff mechanical metamaterials. Science 344, 1373–1377 (2014).

    Article  CAS  Google Scholar 

  5. Zheng, X. et al. Multiscale metallic metamaterials. Nat. Mater. 15, 1100–1106 (2016).

    Article  CAS  Google Scholar 

  6. Bauer, J., Schroer, A., Schwaiger, R. & Kraft, O. Approaching theoretical strength in glassy carbon nanolattices. Nat. Mater. 15, 438–443 (2016).

    Article  CAS  Google Scholar 

  7. Zheng, X. et al. Design and optimization of a light-emitting diode projection micro-stereolithography three-dimensional manufacturing system. Rev. Sci. Instrum. 83, 125001 (2012).

    Article  Google Scholar 

  8. Cabrini, S. & Kawata, S. (eds) Nanofabrication Handbook (CRC Press, 2012).

  9. Eckel, Z. C. et al. Additive manufacturing of polymer-derived ceramics. Science 351, 58–62 (2016).

    Article  CAS  Google Scholar 

  10. Meza, L. R. et al. Resilient 3D hierarchical architected metamaterials. Proc. Natl Acad. Sci. USA 112, 11502–11507 (2015).

  11. Kooistra, G. W., Deshpande, V. S. & Wadley, H. N. G. Hierarchical corrugated core sandwich panel concepts. J. Appl. Mech. 74, 259–268 (2007).

    Article  CAS  Google Scholar 

  12. Deshpande, V. S., Ashby, M. F. & Fleck, N. A. Foam topology: bending versus stretching dominated architectures. Acta Mater. 49, 1035–1040 (2001).

    Article  CAS  Google Scholar 

  13. Deshpande, V. S., Fleck, N. A. & Ashby, M. F. Effective properties of the octet-truss lattice material. J. Mech. Phys. Solids 49, 1747–1769 (2001).

    Article  CAS  Google Scholar 

  14. Montemayor, L. C., Wong, W. H., Zhang, Y. W. & Greer, J. R. Insensitivity to flaws leads to damage tolerance in brittle architected meta-materials. Sci. Rep. 6, 20570 (2016).

    Article  CAS  Google Scholar 

  15. Gu, H. et al. Fracture of three-dimensional lattices manufactured by selective laser melting. Int. J. Solids Struct. 180, 147–159 (2019).

    Article  Google Scholar 

  16. O’Masta, M. R., Dong, L., St-Pierre, L., Wadley, H. N. G. & Deshpande, V. S. The fracture toughness of octet-truss lattices. J. Mech. Phys. Solids 98, 271–289 (2017).

    Article  Google Scholar 

  17. Liu, Y., St-Pierre, L., Fleck, N. A., Deshpande, V. S. & Srivastava, A. High fracture toughness micro-architectured materials. J. Mech. Phys. Solids 143, 104060 (2020).

    Article  Google Scholar 

  18. Anderson, T. L. Fracture Mechanics: Fundamentals and Applications 4th edn (CRC Press, 2017).

  19. Tankasala, H. C., Deshpande, V. S. & Fleck, N. A. 2013 Koiter Medal Paper. Crack-tip fields and toughness of two-dimensional elastoplastic lattices. J. Appl. Mech. 82, 091004 (2015).

    Article  Google Scholar 

  20. Huang, J. S. & Lin, J. Y. Mixed-mode fracture of brittle cellular materials. J. Mater. Sci. 31, 2647–2652 (1996).

    Article  CAS  Google Scholar 

  21. Huang, J. S. & Gibson, L. J. Fracture toughness of brittle honeycombs. Acta Metall. Mater. 39, 1617–1626 (1991).

    Article  Google Scholar 

  22. Hsieh, M. T., Deshpande, V. S. & Valdevit, L. A versatile numerical approach for calculating the fracture toughness and R-curves of cellular materials. J. Mech. Phys. Solids 138, 103925 (2020).

    Article  CAS  Google Scholar 

  23. ASTM E1820 Standard Test Method for Measurement of Fracture Toughness (ASTM International, 2009).

  24. Fuller R. B. Octet truss. US Patent 2,986,241 (1961).

  25. Fleck, N. A., Deshpande, V. S. & Ashby, M. F. Micro-architectured materials: past, present and future. Proc. R. Soc. A 466, 2495–2516 (2010).

  26. Fleck, N. A. & Qiu, X. The damage tolerance of elastic–brittle, two-dimensional isotropic lattices. J. Mech. Phys. Solids 55, 562–588 (2007).

    Article  CAS  Google Scholar 

  27. ASTM E399-74 Standard Method of Test for Plane-Strain Fracture Toughness of Metallic Materials (ASTM International, 1997).

  28. Bažant, Z. P. & Planas, J. Fracture and Size Effect in Concrete and Other Quasibrittle Materials (Routledge, 2019).

  29. Nguyen, H. et al. New perspective of fracture mechanics inspired by gap test with crack-parallel compression. Proc. Natl Acad. Sci. USA 117, 14015–14020 (2020).

  30. O’Dowd, N. P. & Shih, C. F. Family of crack-tip fields characterized by a triaxiality parameter—I. Structure of fields. J. Mech. Phys. Solids 39, 989–1015 (1991).

    Article  Google Scholar 

  31. O’Dowd, N. P. & Shih, C. F. Family of crack-tip fields characterized by a triaxiality parameter—II. Fracture applications. J. Mech. Phys. Solids 40, 939–963 (1992).

    Article  Google Scholar 

  32. Betegoǹ, C. & Hancock, J. W. Two-parameter characterization of elastic-plastic crack-tip fields. J. Appl. Mech. 58, 104–110 (1991).

    Article  Google Scholar 

Download references

Acknowledgements

We acknowledge funding from the Office of Naval Research (N00014-18-1-2658 and N00014-20-1-2504:P00001). H.C. and X.Z. also thank the NSF (2001677) and Air Force Office of Scientific Research (FA9550-18-1-0299) for funding support. A.J.D.S. is supported by the Cambridge-India Ramanujan scholarship from the Cambridge Trust and the SERB (Government of India). A.J.D.S. and V.S.D. thank S. R. Marshall and G. Smith for their help and support in the laboratory.

Author information

Author notes

  1. These authors contributed equally: Angkur Jyoti Dipanka Shaikeea, Huachen Cui.

Authors and Affiliations

  1. Department of Engineering, University of Cambridge, Cambridge, UK

    Angkur Jyoti Dipanka Shaikeea, Mark O’Masta & Vikram Sudhir Deshpande

  2. Advanced Manufacturing and Metamaterials Laboratory, Department of Mechanical Engineering, Virginia Polytechnical Institute and State University, Blacksburg, VA, USA

    Huachen Cui & Xiaoyu Rayne Zheng

  3. Advanced Manufacturing and Metamaterials Laboratory, Civil and Environmental Engineering, University of California, Los Angeles, CA, USA

    Huachen Cui & Xiaoyu Rayne Zheng

  4. HRL Laboratories, Malibu, CA, USA

    Mark O’Masta

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  1. Angkur Jyoti Dipanka Shaikeea
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Contributions

V.S.D., X.Z. and M.O’.M. conceived and designed the research. H.C. developed the 3D printing platform, fabricated the octet-truss specimens and performed experiments. V.S.D. and A.J.D.S. designed the experiments (in situ XCT with multiaxial loading for fracture toughness measurements, digital volume correlation and imperfection analysis from XCT dataset) and the analytical formulations. A.J.D.S. performed these experiments, the numerical calculations and data analysis. V.S.D. and A.J.D.S. wrote the first draft of the manuscript. All authors participated in revising the manuscript, discussion and interpretation of the data.

Corresponding authors

Correspondence to Xiaoyu Rayne Zheng or Vikram Sudhir Deshpande.

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The authors declare no competing interests.

Peer review

Peer review information

Nature Materials thanks Bin Liu, Yang Lu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Imperfection analysis.

(a) CAD model for the \(\bar \rho\) = 0.1 specimen with strut diameter 2r0=65 μm along with (b) an optical image of the printed specimen. (c) XCT image of the entire specimen comprising ~64,000 unit cells with the inset showing a more detailed view of a single unit cell within the specimen. (d) A magnified view of a larger region in the specimen in (c) with each location within the solid material shaded by the diameter D of the best-fit sphere at that location. (e) Probability density of diameters D within the entire specimen. The two modes correspond to best-fit spheres within the struts and node regions as marked. (f) The printed specimen was divided into 27 sub-cubes as shown, (g) cut and (h) then each individual cube was separately imaged via XCT to determine the distribution of the diameter D of the best-fit spheres as before. (i) The measured probability density function of D over the 27 sub-cubes is shown by the shaded zone with the solid line giving the mean over the 27 separate sub-cubes.

Extended Data Fig. 2 Uniformity of mechanical properties.

(a) Optical image of the printed specimen and (b) sketches depicting tensile testing along x, y and z− directions where z is the build direction. The measured tensile responses for the (c) \(\bar \rho\) = 0.1 and (d) \(\bar \rho\) = 0.02 specimens in the x, y and z− directions. Predictions based on the measured properties of a single strut (Supplementary Fig. 2) are included in (c) and (d). (e) The specimen cut into 27 sub-cubes and (f) uniaxial tensile tests were conducted on each cube along the (g) z− direction. The measured responses are shown in (h) and (i) for the \(\bar \rho\) = 0.1 specimen with two different choices of unit cell dimensions. The shaded zone depicts the variation over the 27 sub-cubes with the solid line the mean measured response.

Extended Data Fig. 3 Fracture mechanism maps.

Fracture mechanism maps of the stretch-dominated (a) isotropic compound simple cubic (SC) and body centered-cubic (BCC) truss (b) anisotropic compound SC/BCC truss (c) bend-dominated gyroid and (d) a topology selection map indicating which topology maximizes \(\bar K_{IC}/\varepsilon _f\) in different regions of the \(\bar \rho /\varepsilon _f\) versus \(\bar T\)space.

Extended Data Fig. 4 Design with metamaterials.

(a) Octet-truss beam of aspect ratio L/W = 20 with the embedded crack (a/W = 0.2) subjected to four-point bending. (b) Geometry of the continuum anisotropic elastic beam used to determine the calibration factors YI and \(\hat{T}\) for KIand T, respectively. (c) The cross-plotted fracture map from Fig. 3e. (d) Prediction of the normalized failure load P̄f of the \(\bar \rho\) = 0.08 octet-truss over a range of crack sizes and two choices of parent materials. The reference prediction for an assumed \(\bar T\)= 0 is also included. The black markers in (c-d) show examples of the prediction of the normalized failure load P = Pf for a crack with \(a/\ell = 16\) in a \(\bar \rho\) = 0.08 octet-truss metamaterial beam made from parent materials with εf = 0.025 and 0.1. (e-g) Topology selection for maximizing failure load under four-point bending. (e) The four candidate topologies with their orientations labelled in the global beam co-ordinate system (X,Y,Z). (f) Continuum calibration of the geometric constants YI and \(\hat{T}\) and (g) description of the optimal topology and improvement over the next best candidate for a \(\bar \rho\) = 0.08 beam made from a parent material with failure strains εf = 0.025 and 0.1.

Extended Data Fig. 5 Fracture process zone and crack propagation.

(a) Tensile loading of the \(2B = 1\ell\) specimen (\(a/\ell = 10\), \(\bar \rho\) = 0.1) and the corresponding (b) measurement and FE prediction of the tensile stress versus strain response. (c) High-speed photographs and corresponding FE predictions of crack propagation beyond peak stress with time t = 0 corresponding to the instant of the initiation of crack propagation at peak stress. (d) DVC measurements and (e) FE predictions of the distributions of the von-Mises strains εe within struts around the crack-tip. Measurements and predictions are shown at two load levels just prior to peak load marked in (b). The scale bar is applicable for the strains in c, d and e.

Supplementary information

Supplementary Information

Supplementary Sections A–H, Figs. 1–13, Tables 1 and 2 and captions for Videos 1–4.

Supplementary Video 1

The crack flank and front in the as-manufactured \(\bar \rho = 0.03\) and \(a/\ell = 4\) embedded crack specimen.

Supplementary Video 2

Tensile fracture of the crack front struts in the \(\bar \rho = 0.08\) and \(a/\ell = 10\) embedded crack specimen loaded to \(K_{\mathrm{I}} = K_{\mathrm{Ic}}\).

Supplementary Video 3

Buckling of struts over the crack flank in the \(\bar \rho = 0.03\) and \(a/\ell = 4\) embedded crack specimen loaded to \(K_{\mathrm{I}} = K_{\mathrm{Ic}}\).

Supplementary Video 4

The \(2B = 100\,{{{\mathrm{unit}}}}\,{{{\mathrm{cells}}}},\) \(\bar \rho = 0.10\) and \(a/\ell = 10\) specimen with a through-thickness crack loaded to \(K_{\mathrm{I}} = K_{\mathrm{Ic}}\) showing the fracture of surface struts while struts within the specimen remained intact. In the video, we show a slice consisting of the free surface and five layers in thickness instead of the entire 100 layers of thickness.

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Shaikeea, A.J.D., Cui, H., O’Masta, M. et al. The toughness of mechanical metamaterials. Nat. Mater. 21, 297–304 (2022). https://doi.org/10.1038/s41563-021-01182-1

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