Skip to main content
SpringerLink
Log in

Negative extensibility metamaterials: phase diagram calculation

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

Negative extensibility metamaterials are able to contract against the line of increasing external tension. A bistable unit cell exhibits several nonlinear mechanical behaviors including the negative extensibility response. Here, an exact form of the total mechanical potential is used based on engineering strain measure. The mechanical response is a function of the system parameters that specify unit cell dimensions and member stiffnesses. A phase diagram is calculated, which maps the response to regions in the diagram using the system parameters as the coordinate axes. Boundary lines pinpoint the onset of a particular mechanical response. Contour lines allow various material properties to be fine-tuned. Analogous to thermodynamic phase diagrams, there exist singular “triple points” which simultaneously satisfy conditions for three response types. The discussion ends with a brief statement about how thermodynamic phase diagrams differ from the phase diagram in this paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Klatt T, Haberman MR (2013) A nonlinear negative stiffness metamaterial unit cell and small-on-large multiscale material model. J Appl Phys 114(3):033503

    Article  Google Scholar 

  2. Wu Y, Lai Y, Zhang ZQ (2011) Elastic metamaterials with simultaneously negative effective shear modulus and mass density. Phys Rev Lett 107(10):105506

    Article  Google Scholar 

  3. Qu J, Kadic M, Wegener M (2017) Poroelastic metamaterials with negative effective static compressibility. Appl Phys Lett 110(17):171901

    Article  Google Scholar 

  4. Babaee S, Shim J, Weaver JC, Chen ER, Patel N, Bertoldi K (2013) 3D Soft metamaterials with negative Poisson’s ratio. Adv Mater 25(36):5044

    Article  Google Scholar 

  5. Bückmann T, Stenger N, Kadic M, Kaschke J, Frölich A, Kennerknecht T, Eberl C, Thiel M, Wegener M (2012) Tailored 3D mechanical metamaterials made by dip-in direct-laser-writing optical lithography. Adv Mater 24(20):2710

    Article  Google Scholar 

  6. Bückmann T, Thiel M, Kadic M, Schittny R, Wegener M (2014) An elasto-mechanical unfeelability cloak made of pentamode metamaterials. Nat Commun 5:4130

    Article  Google Scholar 

  7. Wang K, Zhao Y, Chang YH, Qian Z, Zhang C, Wang B, Vannan MA, Wang MJ (2016) Controlling the mechanical behavior of dual-material 3D printed meta-materials for patient-specific tissue-mimicking phantoms. Mater Des 90:704

    Article  Google Scholar 

  8. Coulais C, Sounas D, Alù A (2017) Static non-reciprocity in mechanical metamaterials. Nature 542:461

    Article  Google Scholar 

  9. Lakes R (2001) Extreme damping in compliant composites with a negative-stiffness phase. Philos Mag Lett 81(2):95

    Article  Google Scholar 

  10. Lakes R, Rosakis P, Ruina A (1993) Microbuckling instability in elastomeric cellular solids. J Mater Sci 28(17):4667

    Article  Google Scholar 

  11. Moore B, Jaglinski T, Stone D, Lakes R (2006) Negative incremental bulk modulus in foams. Philos Mag Lett 86(10):651

    Article  Google Scholar 

  12. Nicolaou ZG, Motter AE (2012) Mechanical metamaterials with negative compressibility transitions. Nat Mater 11(7):608

    Article  Google Scholar 

  13. Jaglinski T, Kochmann D, Stone D, Lakes R (2007) Composite materials with viscoelastic stiffness greater than diamond. Science 315(5812):620

    Article  Google Scholar 

  14. Lakes R, Wojciechowski K (2008) Negative compressibility, negative Poisson’s ratio, and stability. Physica Status Solidi (b) 245(3):545

    Article  Google Scholar 

  15. Dyskin AV, Pasternak E (2012) Elastic composite with negative stiffness inclusions in antiplane strain. Int J Eng Sci 58:45

    Article  MATH  Google Scholar 

  16. Silverberg JL, Evans AA, McLeod L, Hayward RC, Hull T, Santangelo CD, Cohen I (2014) Using origami design principles to fold reprogrammable mechanical metamaterials. Science 345(6197):647

    Article  Google Scholar 

  17. Boatti E, Vasios N, Bertoldi K (2017) Origami metamaterials for tunable thermal expansion. Adv Mater 29:1700360

    Article  Google Scholar 

  18. Chen Y, Li T, Scarpa F, Wang L (2017) Lattice metamaterials with mechanically Tunable Poisson’s ratio for vibration control. Phys Rev Appl 7(2):024012

    Article  Google Scholar 

  19. Chen M, Karpov E (2014) Bistability and thermal coupling in elastic metamaterials with negative compressibility. Phys Rev E 90(3):033201

    Article  Google Scholar 

  20. Harne RL, Wu Z, Wang KW (2016) Designing and harnessing the metastable states of a modular Metastructure for programmable mechanical properties adaptation. J Mech Des 138(2):021402

    Article  Google Scholar 

  21. Che K, Yuan C, Wu J, Qi HJ, Meaud J (2017) Three-dimensional-printed multistable mechanical metamaterials with a deterministic deformation sequence. J Appl Mech 84(1):011004

    Article  Google Scholar 

  22. Liu A, Zhu W, Tsai D, Zheludev NI (2012) Micromachined tunable metamaterials: a review. J Opt 14(11):114009

    Article  Google Scholar 

  23. Rafsanjani A, Akbarzadeh A, Pasini D (2015) Snapping mechanical metamaterials under tension. Adv Mater 27(39):5931

    Article  Google Scholar 

  24. Karpov EG, Danso LA, Klein JT (2017) Negative extensibility metamaterials: occurrence and design-space topology. Phys Rev E 96(2):023002

    Article  Google Scholar 

  25. Otsuka K, Wayman CM (1999) Shape memory materials. Cambridge University Press, Cambridge

    Google Scholar 

  26. Dong J, Cai C, Okeil AM (2010) Overview of potential and existing applications of shape memory alloys in bridges. J Bridge Eng 16(2):305

    Article  Google Scholar 

  27. Zeeman EC (1977) Catastrophe theory: selected papers, 1972–1977. Addison-Wesley, Boston

    MATH  Google Scholar 

  28. Saunders PT (1980) An introduction to catastrophe theory. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  29. Gilmore R (1993) Catastrophe theory for scientists and engineers. Courier Corporation, North Chelmsford

    MATH  Google Scholar 

  30. Liu WK, Karpov E, Zhang S, Park H (2004) An introduction to computational nanomechanics and materials. Comput Methods Appl Mech Eng 193(17):1529

    Article  MathSciNet  MATH  Google Scholar 

  31. Kan Q, Yu C, Kang G, Li J, Yan W (2016) Experimental observations on rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy. Mech Mater 97:48

    Article  Google Scholar 

  32. Ortın J, Delaey L (2002) Hysteresis in shape-memory alloys. Int J Non-Linear Mech 37(8):1275

    Article  MATH  Google Scholar 

  33. Dolce M, Cardone D (2001) Mechanical behaviour of shape memory alloys for seismic applications 1. Martensite and austenite NiTi bars subjected to torsion. Int J Mech Sci 43(11):2631

    Article  Google Scholar 

  34. Saadat S, Salichs J, Noori M, Hou Z, Davoodi H, Bar-On I, Suzuki Y, Masuda A (2002) An overview of vibration and seismic applications of NiTi shape memory alloy. Smart Mater Struct 11(2):218

    Article  Google Scholar 

  35. Liu WK, Karpov EG, Park HS (2009) Nano mechanics and materials: theory, multiscale methods and applications. Wiley, Hoboken

    Google Scholar 

  36. Lukas HL, Fries SG, Sundman B (2007) Computational thermodynamics: the Calphad method. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil and Materials Engineering, University of Illinois at Chicago, 842 W Taylor St, Chicago, IL, 60607, USA

    John T. Klein & Eduard G. Karpov

Authors
  1. John T. Klein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eduard G. Karpov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John T. Klein.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Klein, J.T., Karpov, E.G. Negative extensibility metamaterials: phase diagram calculation. Comput Mech 62, 669–683 (2018). https://doi.org/10.1007/s00466-017-1520-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-017-1520-2

Keywords

Search

Navigation