Review Article

Exploiting Microstructural Instabilities in Solids and Structures: From Metamaterials to Structural Transitions

[+] Author and Article Information
Dennis M. Kochmann

Department of Mechanical
and Process Engineering,
ETH Zürich,
Leonhardstr. 21,
Zürich 8092, Switzerland;
Division of Engineering and Applied Science,
California Institute of Technology,
Pasadena, CA 91125
e-mail: dmk@ethz.ch

Katia Bertoldi

John A. Paulson School of Engineering and
Applied Sciences,
Harvard University,
Cambridge, MA 02138;
Kavli Institute,
Harvard University,
Cambridge, MA 02138
e-mail: bertoldi@seas.harvard.edu

Manuscript received May 4, 2017; final manuscript received September 13, 2017; published online October 17, 2017. Editor: Harry Dankowicz.

Appl. Mech. Rev 69(5), 050801 (Oct 17, 2017) (24 pages) Paper No: AMR-17-1031; doi: 10.1115/1.4037966 History: Received May 04, 2017; Revised September 13, 2017

Instabilities in solids and structures are ubiquitous across all length and time scales, and engineering design principles have commonly aimed at preventing instability. However, over the past two decades, engineering mechanics has undergone a paradigm shift, away from avoiding instability and toward taking advantage thereof. At the core of all instabilities—both at the microstructural scale in materials and at the macroscopic, structural level—lies a nonconvex potential energy landscape which is responsible, e.g., for phase transitions and domain switching, localization, pattern formation, or structural buckling and snapping. Deliberately driving a system close to, into, and beyond the unstable regime has been exploited to create new materials systems with superior, interesting, or extreme physical properties. Here, we review the state-of-the-art in utilizing mechanical instabilities in solids and structures at the microstructural level in order to control macroscopic (meta)material performance. After a brief theoretical review, we discuss examples of utilizing material instabilities (from phase transitions and ferroelectric switching to extreme composites) as well as examples of exploiting structural instabilities in acoustic and mechanical metamaterials.

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Euler, L. , 1759, “ Sur la force des colonnes,” Mem. Acad. Berlin, 13, pp. 252–282.
Euler, L. , 1778, “ De altitudine colomnarum sub proprio pondere corruentium,” Acta Acad. Sci. Petropolitana, 1, pp. 191–193.
Euler, L. , 1780, “ Determinatio onerum quae columnae gestare valent,” Acta Acad. Sci. Petropolitana, 2, pp. 163–193.
Kirchhoff, G. , 1859, “ Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes,” J. Reine Angew. Math., 1859(56), pp. 285–313. [CrossRef]
Love, A. , 1893, Mathematical Theory of Elasticity, Cambridge Press, Cambridge, MA.
Thomson, W. L. K. , 1875, “ Elasticity,” Encyclopedia Britannica, Vol. 7, Encyclopædia Britannica, Inc., London.
Lord Rayleigh, F. R. S. , 1878, “ On the Instability of Jets,” Proc. London Math. Soc., s1-10(1), pp. 4–13. [CrossRef]
Southwell, R. V. , 1914, “ On the General Theory of Elastic Stability,” Philos. Trans. R. Soc. London A, 213(497–508), pp. 187–244. [CrossRef]
Timoshenko, S. , and Gere, J. , 1961, Theory of Elastic Stability, McGraw-Hill, New York.
Leipholz, H. , 1970, Stability Theory, Academic Press, New York.
Thompson, J. , and Hunt, G. , 1973, A General Theory of Elastic Stability, Wiley, New York.
Iooss, G. , and Joseph, D. , 1980, Elementary Bifurcation and Stability Theory, Springer, New York. [CrossRef]
Geymonat, G. , Müller, S. , and Triantafyllidis, N. , 1993, “ Homogenization of Nonlinearly Elastic Materials, Microscopic Bifurcation and Macroscopic Loss of Rank-One Convexity,” Arch. Ration. Mech. Anal., 122(3), pp. 231–290. [CrossRef]
Como, M. , and Grimaldi, A. , 1995, Theory of Stability of Continuous Elastic Structures, CRC Press, Boca Raton, FL.
Nguyen, Q. , 2000, Stability and Nonlinear Solid Mechanics, Wiley, New York.
Ziegler, H. , 1968, Principles of Structural Stability, Blaisdell, New York.
Simitses, G. J. , and Hodges, D. H. , 2006, Fundamentals of Structural Stability, Butterworth-Heinemann, Burlington, MA.
Knops, R. , and Wilkes, W. , 1973, “ Theory of Elastic Stability,” Handbook of Physics, Vol. VIa, Springer, Berlin, p. 125302. [CrossRef]
Budiansky, B. , 1974, “ Theory of Buckling and Post-Buckling Behavior in Elastic Structures,” Advances of Applied Mechanics, Elsevier, Amsterdam, The Netherlands, pp. 1–65. [CrossRef]
Goncu, F. , Willshaw, S. , Shim, J. , Cusack, J. , Luding, S. , Mullin, T. , and Bertoldi, K. , 2011, “ Deformation Induced Pattern Transformation in a Soft Granular Crystal,” Soft Matter, 7(6), pp. 2321–2324. [CrossRef]
Overvelde, J. , Shan, S. , and Bertoldi, K. , 2012, “ Compaction Through Buckling in 2D Periodic, Soft and Porous Structures: Effect of Pore Shape,” Adv. Mater., 24(17), pp. 2337–2342. [CrossRef] [PubMed]
Shim, J. , Perdigou, C. , Chen, E. , Bertoldi, K. , and Reis, P. , 2012, “ Buckling-Induced Encapsulation of Structured Elastic Shells Under Pressure,” Proc. Natl. Acad. Sci., 109(16), pp. 5978–5983. [CrossRef]
Raney, J. R. , Nadkarni, N. , Daraio, C. , Kochmann, D. M. , Lewis, J. A. , and Bertoldi, K. , 2016, “ Stable Propagation of Mechanical Signals in Soft Media Using Stored Elastic Energy,” Proc. Natl. Acad. Sci., 113(35), pp. 9722–9727. [CrossRef]
Tavakol, B. , Bozlar, M. , Punckt, C. , Froehlicher, G. , Stone, H. A. , Aksay, I. A. , and Holmes, D. P. , 2014, “ Buckling of Dielectric Elastomeric Plates for Soft, Electrically Active Microfluidic Pumps,” Soft Matter, 10(27), pp. 4789–4794. [CrossRef] [PubMed]
Meza, L. R. , Das, S. , and Greer, J. R. , 2014, “ Strong, Lightweight, and Recoverable Three-Dimensional Ceramic Nanolattices,” Science, 345(6202), pp. 1322–1326. [CrossRef] [PubMed]
Meza, L. R. , Zelhofer, A. J. , Clarke, N. , Mateos, A. J. , Kochmann, D. M. , and Greer, J. R. , 2015, “ Resilient 3D Hierarchical Architected Metamaterials,” Proc. Natl. Acad. Sci., 112(37), pp. 11502–11507. [CrossRef]
Lee, J.-H. , Ro, H. W. , Huang, R. , Lemaillet, P. , Germer, T. A. , Soles, C. L. , and Stafford, C. M. , 2012, “ Anisotropic, Hierarchical Surface Patterns Via Surface Wrinkling of Nanopatterned Polymer Films,” Nano Lett. 12(11), pp. 5995–5999.
Pezzulla, M. , Shillig, S. A. , Nardinocchi, P. , and Holmes, D. P. , 2015, “ Morphing of Geometric Composites Via Residual Swelling,” Soft Matter, 11(29), pp. 5812–5820. [CrossRef] [PubMed]
Platus, D. L. , 1999, “ Negative-Stiffness-Mechanism Vibration Isolation Systems,” Proc. SPIE, 3786, pp. 98–105.
Yap, H. W. , Lakes, R. S. , and Carpick, R. W. , 2008, “ Negative Stiffness and Enhanced Damping of Individual Multiwalled Carbon Nanotubes,” Phys. Rev. B, 77(4), p. 045423. [CrossRef]
Mizuno, T. , 2010, “ Vibration Isolation System Using Negative Stiffness,” Vibration Control, Vol. 3786, M. Lallart , ed., InTech, Rijeka, Croatia.
Lee, C.-M. , and Goverdovskiy, V. , 2012, “ A Multi-Stage High-Speed Railroad Vibration Isolation System With Negative Stiffness,” J. Sound Vib., 331(4), pp. 914–921. [CrossRef]
Goncu, F. , Luding, S. , and Bertoldi, K. , 2012, “ Exploiting Pattern Transformation to Tune Phononic Band Gaps in a Two-Dimensional Granular Crystal,” J. Acoust. Soc. Am., 131(6), pp. EL475–EL480. [CrossRef] [PubMed]
Wang, P. , Casadei, F. , Shan, S. , Weaver, J. , and Bertoldi, K. , 2014, “ Harnessing Buckling to Design Tunable Locally Resonant Acoustic Metamaterials,” Phys. Rev. Lett., 113, p. 014301. [CrossRef] [PubMed]
Rudykh, S. , and Boyce, M. , 2014, “ Transforming Wave Propagation in Layered Media Via Instability-Induced Wrinkling Interfacial Layer,” Phys. Rev. Lett., 112(3), p. 034301. [CrossRef] [PubMed]
Xin, F. , and Lu, T. , 2016, “ Tensional Acoustomechanical Soft Metamaterials,” Sci. Rep., 6, p. 27432. [CrossRef] [PubMed]
Lakes, R. S. , 2001, “ Extreme Damping in Compliant Composites With a Negative-Stiffness Phase,” Philos. Mag. Lett., 81(2), pp. 95–100. [CrossRef]
Lakes, R. S. , 2001, “ Extreme Damping in Composite Materials With a Negative Stiffness Phase,” Phys. Rev. Lett., 86(13), pp. 2897–2900. [CrossRef] [PubMed]
Lakes, R. S. , Lee, T. , Bersie, A. , and Wang, Y. C. , 2001, “ Extreme Damping in Composite Materials With Negative-Stiffness Inclusions,” Nature, 410(6828), pp. 565–567. [CrossRef] [PubMed]
Jaglinski, T. , Kochmann, D. , Stone, D. , and Lakes, R. S. , 2007, “ Composite Materials With Viscoelastic Stiffness Greater Than Diamond,” Science, 315(5812), pp. 620–622. [CrossRef] [PubMed]
Popa, B.-I. , Zigoneanu, L. , and Cummer, S. A. , 2013, “ Tunable Active Acoustic Metamaterials,” Phys. Rev. B, 88(2), p. 024303. [CrossRef]
Wojnar, C. S. , le Graverend, J.-B. , and Kochmann, D. M. , 2014, “ Broadband Control of the Viscoelasticity of Ferroelectrics Via Domain Switching,” Appl. Phys. Lett., 105(16), p. 162912.
Reis, P. M. , 2015, “ A Perspective on the Revival of Structural (In)Stability With Novel Opportunities for Function: From Buckliphobia to Buckliphilia,” ASME J. Appl. Mech., 82(11), p. 111001. [CrossRef]
Morrey, C. B. , 1952, “ Quasi-Convexity and the Lower Semicontinuity of Multiple Integrals,” Pac. J. Math., 2(1), p. 25–53.
Knops, R. J. , and Stuart, C. A. , 1986, Quasiconvexity and Uniqueness of Equilibrium Solutions in Nonlinear Elasticity, Springer, Berlin, pp. 473–489.
Landau, L. , 1937, “ On the Theory of Phase Transitions,” Zh. Eksp. Teor. Fiz., 7, pp. 19–32 (in Russian).
Devonshire, A. , 1949, “ XCVI. Theory of Barium Titanate,” Philos. Mag., 40(309), pp. 1040–1063. [CrossRef]
Devonshire, A. , 1951, “ CIX. Theory of Barium Titanate: Part II,” Philos. Mag., 42(333), pp. 1065–1079. [CrossRef]
Ball, J. , and James, R. , 1987, “ Fine Phase Mixtures as Minimizers of Energy,” Arch. Ration. Mech. Anal., 100(1), pp. 13–52. [CrossRef]
James, R. , 1981, “ Finite Deformation by Mechanical Twinning,” Arch. Ration. Mech. Anal., 77(2), pp. 143–176. [CrossRef]
Christian, J. , and Mahajan, S. , 1995, “ Deformation Twinning,” Prog. Mater. Sci., 39(1–2), pp. 1–157. [CrossRef]
Carstensen, C. , Hackl, K. , and Mielke, A. , 2002, “ Non-Convex Potentials and Microstructures in Finite-Strain Plasticity,” Proc. R. Soc. London, Ser. A, 458(2018), pp. 299–317. [CrossRef]
Ortiz, M. , and Repettto, E. A. , 1999, “ Nonconvex Energy Minimization and Dislocation Structures in Ductile Single Crystals,” J. Mech. Phys. Solids, 47(2), pp. 397–462. [CrossRef]
Govindjee, S. , Mielke, A. , and Hall, G. J. , 2003, “ The Free Energy of Mixing for n-Variant Martensitic Phase Transformations Using Quasi-Convex Analysis,” J. Mech. Phys. Solids, 51(4), pp. I–XXVI. [CrossRef]
Miehe, C. , Lambrecht, M. , and Gurses, E. , 2004, “ Analysis of Material Instabilities in Inelastic Solids by Incremental Energy Minimization and Relaxation Methods: Evolving Deformation Microstructures in Finite Plasticity,” J. Mech. Phys. Solids, 52(12), pp. 2725–2769. [CrossRef]
Bhattacharya, K. , 2004, Microstructure of Martensite—Why It Forms and How It Gives Rise to the Shape-Memory Effect, Oxford University Press, Oxford, UK.
Lee, J.-H. , Singer, J. P. , and Thomas, E. L. , 2012, “ Micro/Nanostructured Mechanical Metamaterials,” Adv. Mater., 24(36), pp. 4782–4810. [CrossRef] [PubMed]
Spadaccini, C. M. , 2015, “ Mechanical Metamaterials: Design, Fabrication, and Performance,” Winter, 45(4), pp. 28–36.
Li, X. , and Gao, H. , 2016, “ Mechanical Metamaterials: Smaller and Stronger,” Nat. Mater., 15, pp. 373–374.
Papanicolaou, G. , Bensoussan, A. , and Lions, J. , 1978, Asymptotic Analysis for Periodic Structures. Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands.
Sánchez-Palencia, E. , 1980, Non-Homogeneous Media and Vibration Theory (Lecture Notes in Physics), Springer, New York.
Schröder, J. , 2000, “ Homogenisierungsmethoden der nichtlinearen kontinuumsmechanik unter beachtung von stabilitätsproblemen,” Ph.D. thesis, Universität Stuttgart, Stuttgart, Germany.
Terada, K. , and Kikuchi, N. , 2001, “ A Class of General Algorithms for Multi-Scale Analyses of Heterogeneous Media,” Comput. Methods Appl. Mech. Eng., 190(4041), pp. 5427–5464. [CrossRef]
Miehe, C. , Schröder, J. , and Becker, M. , 2002, “ Computational Homogenization Analysis in Finite Elasticity: Material and Structural Instabilities on the Micro- and Macro-Scales of Periodic Composites and Their Interaction,” Comput. Methods Appl. Mech. Eng., 191(44), pp. 4971–5005. [CrossRef]
Kouznetsova, V. , Brekelmans, W. A. M. , and Baaijens, F. P. T. , 2001, “ An Approach to Micro-Macro Modeling of Heterogeneous Materials,” Comput. Mech., 27(1), pp. 37–48. [CrossRef]
Milton, G. , 2001, Theory of Composites, Cambridge University Press, Cambridge, UK.
Floquet, G. , 1883, “ Sur les équations différentielles linéaires à coefficients périodiques,” Ann. Ec. Norm. Supér., 12, p. 4788.
Bloch, F. , 1928, “ Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys., 52, pp. 550–600.
Krödel, S. , Delpero, T. , Bergamini, A. , Ermanni, P. , and Kochmann, D. M. , 2014, “ 3D Auxetic Microlattices With Independently-Controllable Acoustic Band Gaps and Quasi-Static Elastic Moduli,” Adv. Eng. Mater., 16(4), pp. 357–363. [CrossRef]
Willis, J. , 1981, “ Variational and Related Methods for the Overall Properties of Composites,” Adv. Appl. Mech., 21, pp. 1–78.
Willis, J. , 1981, “ Variational Principles for Dynamic Problems for Inhomogeneous Elastic Media,” Wave Motion, 3(1), pp. 1–11. [CrossRef]
Willis, J. , 1997, “ Dynamics of Composites,” Continuum Micromechanics, P. Suquet , ed., Springer, New York, pp. 265–290. [CrossRef]
Milton, G. , and Willis, J. , 2007, “ On Modifications of Newton's Second Law and Linear Continuum Elastodynamics,” Proc. R. Soc. A, 463(2079), pp. 855–880. [CrossRef]
Willis, J. , 2009, “ Exact Effective Relations for Dynamics of a Laminated Body,” Mech. Mater., 41(4), pp. 385–393. [CrossRef]
Willis, J. , 2011, “ Effective Constitutive Relations for Waves in Composites and Metamaterials,” Proc. R. Soc. A, 467(2131), pp. 1865–1879.
Norris, A. , Shuvalov, A. , and Kutsenko, A. , 2012, “ Analytical Formulation of Three-Dimensional Dynamic Homogenization for Periodic Elastic Systems,” Proc. R. Soc. A, 468(2142), pp. 1629–1651. [CrossRef]
Pham, K. , Kouznetsova, V. , and Geers, M. , 2013, “ Transient Computational Homogenization for Heterogeneous Materials Under Dynamic Excitation,” J. Mech. Phys. Solids, 61(11), pp. 2125–2146. [CrossRef]
Liu, C. , and Reina, C. , 2015, “ Computational Homogenization of Heterogeneous Media Under Dynamic Loading,” preprint arXiv:1510.02310.
Hill, R. , 1957, “ On Uniqueness and Stability in the Theory of Finite Elastic Strain,” J. Mech. Phys. Solids, 5(4), pp. 229–241. [CrossRef]
Hill, R. , 1961, “ Uniqueness in General Boundary-Value Problems for Elastic or Inelastic Solids,” J. Mech. Phys. Solids, 9(2), pp. 114–130. [CrossRef]
Ball, J. M. , 1976, “ Convexity Conditions and Existence Theorems in Nonlinear Elasticity,” Arch. Ration. Mech. Anal., 63(4), pp. 337–403. [CrossRef]
Lyapunov, A. , 1892, “ The General Problem of the Stability of Motion,” Ph.D. thesis, University of Kharkov, Kharkov, Russia (in Russian).
Lyapunov, A. , 1966, The General Problem of the Stability of Motion, Academic Press, New York.
Koiter, W. , 1965, “ The Energy Criterion of Stability for Continuous Elastic Bodies,” Proc. K. Ned. Acad. Wet. B, 868, pp. 178–202.
Kochmann, D. M. , and Drugan, W. J. , 2011, “ Infinitely Stiff Composite Via a Rotation-Stabilized Negative-Stiffness Phase,” Appl. Phys. Lett., 99(1), p. 011909.
Ziegler, H. , 1953, “ Linear Elastic Stability,” Z. Angew. Math. Phys., 4(3), pp. 167–185. [CrossRef]
Ortiz, M. , and Stainier, L. , 1999, “ The Variational Formulation of Viscoplastic Constitutive Updates,” Comput. Methods Appl. Mech. Eng., 171(34), pp. 419–444. [CrossRef]
Bazant, Z. P. , 2000, “ Structural Stability,” Int. J. Solids Struct., 37(12), pp. 55–67. [CrossRef]
Ning, X. , and Pellegrino, S. , 2015, “ Imperfection-Insensitive Axially Loaded Thin Cylindrical Shells,” Int. J. Solids Struct., 62, pp. 39–51. [CrossRef]
Thomson, W. L. K. , 1856, “ Elements of a Mathematical Theory of Elasticity,” Philos. Trans. R. Soc. London, 146, pp. 481–498. [CrossRef]
Van Hove, L. , 1947, “ Sur l'extension de la condition de legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues,” Proc. Ned. Akad. Wet., 50, p. 1823.
Mandel, J. , 1962, “ Ondes plastique dans un milieu indéfini à trois dimensions,” J. Mech., 1, pp. 3–30.
Mandel, J. , 1966, “ Conditions de stabilité et postulat de drucker,” Rheology and Solid Mechanics, J. Kravtchenko and M. Sirieys , eds., Springer, Berlin, pp. 58–68.
Kochmann, D. M. , 2012, “ Stability Criteria for Continuous and Discrete Elastic Composites and the Influence of Geometry on the Stability of a Negative-Stiffness Phase,” Phys. Status Solidi B, 249(7), pp. 1399–1411. [CrossRef]
Pearson, C. E. , 1956, “ General Theory of Elastic Stability,” Q. Appl. Math., 14, pp. 133–144. [CrossRef]
Kochmann, D. M. , and Drugan, W. J. , 2012, “ Analytical Stability Conditions for Elastic Composite Materials With a Non-Positive-Definite Phase,” Proc. R. Soc. A, 468(2144), pp. 2230–2254. [CrossRef]
Triantafyllidis, N. , and Maker, B. , 1985, “ On the Comparison Between Microscopic and Macroscopic Instability Mechanisms in a Class of Fiber-Reinforced Composites,” ASME J. Appl. Mech., 52(4), pp. 794–800. [CrossRef]
Kochmann, D. , and Drugan, W. , 2009, “ Dynamic Stability Analysis of an Elastic Composite Material Having a Negative-Stiffness Phase,” J. Mech. Phys. Solids, 57(7), pp. 1122–1138. [CrossRef]
Kochmann, D. M. , and Milton, G. W. , 2014, “ Rigorous Bounds on the Effective Moduli of Composites and Inhomogeneous Bodies With Negative-Stiffness Phases,” J. Mech. Phys. Solids, 71, pp. 46–63. [CrossRef]
Lakes, R. , 1999, Viscoelastic Solids (CRC Mechanical Engineering Series), CRC Press, Boca Raton, FL.
Fritzen, F. , and Kochmann, D. M. , 2014, “ Material Instability-Induced Extreme Damping in Composites: A Computational Study,” Int. J. Solids Struct., 51(23–24), pp. 4101–4112. [CrossRef]
Lakes, R. , 1993, “ Materials With Structural Hierarchy,” Nature, 361(6412), pp. 511–515. [CrossRef]
Kim, H. , Swan, C. , and Lakes, R. , 2002, “ Computational Studies on High-Stiffness, High-Damping SiC–InSn Particulate Reinforced Composites,” Int. J. Solids Struct., 39(23), pp. 5799–5812. [CrossRef]
Meaud, J. , Sain, T. , Hulbert, G. , and Waas, A. , 2013, “ Analysis and Optimal Design of Layered Composites With High Stiffness and High Damping,” Int. J. Solids Struct., 50(9), pp. 1342–1353. [CrossRef]
Meaud, J. , Sain, T. , Yeom, B. , Park, S. J. , Shoultz, A. B. , Hulbert, G. , Ma, Z.-D. , Kotov, N. A. , Hart, A. J. , Arruda, E. M. , and Waas, A. M. , 2014, “ Simultaneously High Stiffness and Damping in Nanoengineered Microtruss Composites,” ACS Nano, 8(4), pp. 3468–3475. [CrossRef] [PubMed]
Wang, Y. C. , and Lakes, R. S. , 2004, “ Extreme Stiffness Systems Due to Negative Stiffness Elements,” Am. J. Phys., 72(1), pp. 40–50. [CrossRef]
Wojnar, C. S. , and Kochmann, D. M. , 2014, “ A Negative-Stiffness Phase in Elastic Composites Can Produce Stable Extreme Effective Dynamic But Not Static Stiffness,” Philos. Mag., 94(6), pp. 532–555. [CrossRef]
Liu, Z. , Zhang, X. , Mao, Y. , Zhu, Y. , Yang, Z. , Chan, C. , and Sheng, P. , 2000, “ Locally Resonant Sonic Materials,” Science, 289(5485), pp. 1734–1736. [CrossRef] [PubMed]
Sheng, P. , Zhang, X. , Liu, Z. , and Chan, C. , 2003, “ Locally Resonant Sonic Materials,” Physica B, 338(1–4), pp. 201–205. [CrossRef]
Fang, N. , Xi, D. , Xu, J. , Ambati, M. , Srituravanich, W. , Sun, C. , and Zhang, X. , 2006, “ Ultrasonic Metamaterials With Negative Modulus,” Nat. Mater., 5(6), pp. 452–456. [CrossRef] [PubMed]
Platus, D. L. , 1992, “ Negative Stiffness-Mechanism Vibration Isolation System,” Proc. SPIE, 1619, p. 44–54.
Platus, D. L. , and Ferry, D. K. , 2007, “ Negative-Stiffness Vibration Isolation Improves Reliability of Nanoinstrumentation,” Laser Focus World, 43(10), pp. 107–109.
Lee, C.-M. , Goverdovskiy, V. , and Temnikov, A. , 2007, “ Design of Springs With Negative Stiffness to Improve Vehicle Driver Vibration Isolation,” J. Sound Vib., 302(4–5), pp. 865–874. [CrossRef]
Sarlis, A. A. , Pasala, D. T. R. , Constantinou, M. C. , Reinhorn, A. M. , Nagarajaiah, S. , and Taylor, D. P. , 2016, “ Negative Stiffness Device for Seismic Protection of Structures: Shake Table Testing of a Seismically Isolated Structure,” J. Struct. Eng., 142(5), p. 04016005.
Moore, B. , Jaglinski, T. , Stone, D. S. , and Lakes, R. S. , 2006, “ Negative Incremental Bulk Modulus in Foams,” Philos. Mag. Lett., 86(10), pp. 651–659. [CrossRef]
Estrin, Y. , Dyskin, A. V. , Pasternak, E. , Schaare, S. , Stanchits, S. , and Kanel-Belov, A. J. , 2004, “ Negative Stiffness of a Layer With Topologically Interlocked Elements,” Scr. Mater., 50(2), pp. 291–294. [CrossRef]
Cherkaev, A. , Cherkaev, E. , and Slepyan, L. , 2005, “ Transition Waves in Bistable Structures. I. Delocalization of Damage,” J. Mech. Phys. Solids, 53(2), pp. 383–405. [CrossRef]
Kashdan, L. , Seepersad, C. C. , Haberman, M. , and Wilson, P. S. , 2012, “ Design, Fabrication, and Evaluation of Negative Stiffness Elements Using SLS,” Rapid Prototyping J., 18(3), pp. 194–200. [CrossRef]
Klatt, T. , and Haberman, M. R. , 2013, “ A Nonlinear Negative Stiffness Metamaterial Unit Cell and Small-on-Large Multiscale Material Model,” J. Appl. Phys., 114(3), p. 033503.
Matthews, J. , Klatt, T. , Morris, C. , Seepersad, C. C. , Haberman, M. , and Shahan, D. , 2016, “ Hierarchical Design of Negative Stiffness Metamaterials Using a Bayesian Network Classifier,” ASME J. Mech. Des., 138(4), p. 041404.
Konarski, S. G. , Hamilton, M. F. , and Haberman, M. R. , 2014, “ Elastic Nonlinearities and Wave Distortion in Heterogeneous Materials Containing Constrained Negative Stiffness Inclusions,” Eighth International Congress on Advanced Electromagnetic Materials in Microwaves and Optics (METAMATERIALS), Lyngby, Denmark, Aug. 25–28, pp. 130–132.
Novak, I. , and Truskinovsky, L. , 2015, “ Nonaffine Response of Skeletal Muscles on the ‘Descending Limb’,” Math. Mech. Solids, 20(6), pp. 697–720. [CrossRef]
Caruel, M. , Allain, J.-M. , and Truskinovsky, L. , 2013, “ Muscle as a Metamaterial Operating Near a Critical Point,” Phys. Rev. Lett., 110, p. 248103. [CrossRef] [PubMed]
Snyder, K. V. , Brownell, W. E. , and Sachs, F. , 1999, “ Negative Stiffness of the Outer Hair Cell Lateral Wall,” Biophys. J., 76(1), p. A60.
Hashin, Z. , and Shtrikman, S. , 1963, “ A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials,” J. Mech. Phys. Solids, 11(2), pp. 127–140. [CrossRef]
Novikov, V. V. , and Wojciechowski, K. W. , 2005, “ Extreme Viscoelastic Properties of Composites of Strongly Inhomogeneous Structures Due to Negative Stiffness Phases,” Phys. Status Solidi B, 242(3), pp. 645–652. [CrossRef]
Wang, Y. C. , and Lakes, R. S. , 2001, “ Extreme Thermal Expansion, Piezoelectricity, and Other Coupled Field Properties in Composites With a Negative Stiffness Phase,” J. Appl. Phys., 90(12), pp. 6458–6465. [CrossRef]
Wang, Y.-C. , Ko, C.-C. , and Chang, K.-W. , 2015, “ Anomalous Effective Viscoelastic, Thermoelastic, Dielectric, and Piezoelectric Properties of Negative-Stiffness Composites and Their Stability,” Phys. Status Solidi B, 252(7), pp. 1640–1655. [CrossRef]
Chronopoulos, D. , Antoniadis, I. , Collet, M. , and Ichchou, M. , 2015, “ Enhancement of Wave Damping Within Metamaterials Having Embedded Negative Stiffness Inclusions,” Wave Motion, 58, pp. 165–179. [CrossRef]
Jeeva, L. L. , Choi, J. B. , and Lee, T. , 2014, “ Improvement of Viscoelastic Damping by Using Manganese Bronze With Indium,” Mech. Time-Depend. Mater., 18(1), pp. 217–227. [CrossRef]
Wang, Y. C. , Ludwigson, M. , and Lakes, R. S. , 2004, “ Deformation of Extreme Viscoelastic Metals and Composites,” Mater. Sci. Eng. A, 370(1–2), pp. 41–49. [CrossRef]
Prasad, J. , and Diaz, A. R. , 2009, “ Viscoelastic Material Design With Negative Stiffness Components Using Topology Optimization,” Struct. Multidiscip. Optim., 38(6), pp. 583–597. [CrossRef]
Lakes, R. S. , and Drugan, W. J. , 2002, “ Dramatically Stiffer Elastic Composite Materials Due to a Negative Stiffness phase?,” J. Mech. Phys. Solids, 50(5), pp. 979–1009. [CrossRef]
Drugan, W. J. , 2007, “ Elastic Composite Materials Having a Negative Stiffness Phase Can Be Stable,” Phys. Rev. Lett., 98(5), p. 055502.
Hoang, T. M. , and Drugan, W. J. , 2016, “ Tailored Heterogeneity Increases Overall Stability Regime of Composites Having a Negative-Stiffness Inclusion,” J. Mech. Phys. Solids, 88, pp. 123–149. [CrossRef]
Wojnar, C. S. , and Kochmann, D. M. , 2014, “ Stability of Extreme Static and Dynamic Bulk Moduli of an Elastic Two-Phase Composite Due to a Non-Positive-Definite Phase,” Phys. Status Solidi B, 251(2), pp. 397–405. [CrossRef]
Wang, Y. C. , and Lakes, R. , 2004, “ Negative Stiffness-Induced Extreme Viscoelastic Mechanical Properties: Stability and Dynamics,” Philos. Mag., 84(35), pp. 3785–3801. [CrossRef]
Wang, Y. C. , and Lakes, R. , 2005, “ Stability of Negative Stiffness Viscoelastic Systems,” Q. Appl. Math., 63(1), pp. 34–55. [CrossRef]
Wang, Y. C. , Swadener, J. G. , and Lakes, R. S. , 2006, “ Two-Dimensional Viscoelastic Discrete Triangular System With Negative-Stiffness Components,” Philos. Mag. Lett., 86(2), pp. 99–112. [CrossRef]
Wang, Y. C. , 2007, “ Influences of Negative Stiffness on a Two-Dimensional Hexagonal Lattice Cell,” Philos. Mag., 87(24), pp. 3671–3688. [CrossRef]
Wang, Y.-C. , Swadener, J. G. , and Lakes, R. S. , 2007, “ Anomalies in Stiffness and Damping of a 2D Discrete Viscoelastic System Due to Negative Stiffness Components,” Thin Solid Films, 515(6), pp. 3171–3178. [CrossRef]
Kochmann, D. M. , 2014, “ Stable Extreme Damping in Viscoelastic Two-Phase Composites With Non-Positive-Definite Phases Close to the Loss of Stability,” Mech. Res. Commun., 58, pp. 36–45. [CrossRef]
Wang, Y. C. , and Lakes, R. S. , 2004, “ Stable Extremely-High-Damping Discrete Viscoelastic Systems Due to Negative Stiffness Elements,” Appl. Phys. Lett., 84(22), pp. 4451–4453. [CrossRef]
Wang, Y.-C. , and Ko, C.-C. , 2010, “ Stability of Viscoelastic Continuum With Negative-Stiffness Inclusions in the Low-Frequency Range,” Phys. Status Solidi B, 250(10), pp. 2070–2079.
Junker, P. , and Kochmann, D. M. , 2017, “ Damage-Induced Mechanical Damping in Phase-Transforming Composites Materials,” Int. J. Solids Struct., 113–114, pp. 132–146. [CrossRef]
Kapitza, P. L. , 1951, “ Dynamic Stability of a Pendulum When Its Point of Suspension Vibrates,” Sov. Phys. JETP, 21, pp. 588–592.
Kapitza, P. L. , 1951, “ Pendulum With a Vibrating Suspension,” Usp. Fiz. Nauk, 44, pp. 7–15. [CrossRef]
Kochmann, D. M. , and Drugan, W. J. , 2016, “ An Infinitely-Stiff Elastic System Via a Tuned Negative-Stiffness Component Stabilized by Rotation-Produced Gyroscopic Forces,” Appl. Phys. Lett., 108(26), p. 261904. [CrossRef]
Lakes, R. S. , 2012, “ Stable Singular or Negative Stiffness Systems in the Presence of Energy Flux,” Philos. Mag. Lett., 92(5), pp. 226–234. [CrossRef]
Jaglinski, T. , and Lakes, R. S. , 2004, “ Anelastic Instability in Composites With Negative Stiffness Inclusions,” Philos. Mag. Lett., 84(12), pp. 803–810. [CrossRef]
Jaglinski, T. , Stone, D. , and Lakes, R. S. , 2005, “ Internal Friction Study of a Composite With a Negative Stiffness Constituent,” J. Mater. Res., 20(9), pp. 2523–2533. [CrossRef]
Jaglinski, T. , Frascone, P. , Moore, B. , Stone, D. S. , and Lakes, R. S. , 2006, “ Internal Friction Due to Negative Stiffness in the Indium-Thallium Martensitic Phase Transformation,” Philos. Mag., 86(27), pp. 4285–4303. [CrossRef]
Dong, L. , Stone, D. , and Lakes, R. , 2011, “ Giant Anelastic Responses in (BaZrO3-ZnO)-BaTiO3 Composite Materials,” EPL, 93(6), p. 66003.
Dong, L. , Stone, D. S. , and Lakes, R. S. , 2011, “ Extreme Anelastic Responses in Zn80Al20 Matrix Composite Materials Containing BaTiO3 Inclusion,” Scr. Mater., 65(4), pp. 288–291. [CrossRef]
Dong, L. , Stone, D. S. , and Lakes, R. S. , 2011, “ Viscoelastic Sigmoid Anomalies in BaZrO3-BaTiO3 Near Phase Transformations Due to Negative Stiffness Heterogeneity,” J. Mater. Res., 26(11), pp. 1446–1452. [CrossRef]
Skandani, A. A. , Ctvrtlik, R. , and Al-Haik, M. , 2014, “ Nanocharacterization of the Negative Stiffness of Ferroelectric Materials,” Appl. Phys. Lett., 105(8), p. 082906.
Romao, C. P. , and White, M. A. , 2016, “ Negative Stiffness in ZrW2O8 Inclusions as a Result of Thermal Stress,” Appl. Phys. Lett., 109(3), p. 031902.
le Graverend, J.-B. , Wojnar, C. S. , and Kochmann, D. M. , 2015, “ Broadband Electromechanical Spectroscopy: Characterizing the Dynamic Mechanical Response of Viscoelastic Materials Under Temperature and Electric Field Control in a Vacuum Environment,” J. Mater. Sci., 50(10), pp. 3656–3685. [CrossRef]
Mullin, T. , Deschanel, S. , Bertoldi, K. , and Boyce, M. C. , 2007, “ Pattern Transformation Triggered by Deformation,” Phys. Rev. Lett., 99(8), p. 084301. [CrossRef] [PubMed]
Zhang, Y. , Matsumoto, E. A. , Peter, A. , Lin, P. C. , Kamien, R. D. , and Yang, S. , 2008, “ One-Step Nanoscale Assembly of Complex Structures Via Harnessing of an Elastic Instability,” Nano Letters, 8(4), pp. 1192–1196. [CrossRef] [PubMed]
Michel, J. , Lopez-Pamies, O. , Castañeda, P. P. , and Triantafyllidis, N. , 2007, “ Microscopic and Macroscopic Instabilities in Finitely Strained Porous Elastomers,” J. Mech. Phys. Solids, 55(5), pp. 900–938. [CrossRef]
Triantafyllidis, N. , Nestorovic, M. D. , and Schraad, M. W. , 2006, “ Failure Surfaces for Finitely Strained Two-Phased Periodic Solids Under General In-Plane Loading,” ASME J. Appl. Mech., 73(3), pp. 505–515. [CrossRef]
Bertoldi, K. , Reis, P. M. , Willshaw, S. , and Mullin, T. , 2010, “ Negative Poisson's Ratio Behavior Induced by an Elastic Instability,” Adv. Mater., 22(3), pp. 361–366. [CrossRef] [PubMed]
Babaee, S. , Shim, J. , Weaver, J. , Patel, N. , and Bertoldi, K. , 2013, “ 3D Soft Metamaterials With Negative Poisson's Ratio,” Adv. Mater., 25(36), pp. 5044–5049. [CrossRef] [PubMed]
Liu, J. , Gu, T. , Shan, S. , Kang, S. , Weaver, J. , and Bertoldi, K. , 2016, “ Harnessing Buckling to Design Architected Materials That Exhibit Effective Negative Swelling,” Adv. Mater., 28(31), pp. 6619–6624.
Kang, S. , Shan, S. , Noorduin, W. , Khan, M. , Aizenberg, J. , and Bertoldi, K. , 2013, “ Buckling-Induced Reversible Symmetry Breaking and Amplification of Chirality Using Supported Cellular Structures,” Adv. Mater., 25(24), pp. 3380–3385. [CrossRef] [PubMed]
Cao, B. , Wu, G. , Xia, Y. , and Yang, S. , 2016, “ Buckling Into Single-Handed Chiral Structures From pH-Sensitive Hydrogel Membranes,” Extreme Mech. Lett., 7, pp. 49–54. [CrossRef]
Wu, G. , Xia, Y. , and Yang, S. , 2014, “ Buckling, Symmetry Breaking, and Cavitation in Periodically Micro-Structured Hydrogel Membranes,” Soft Matter, 10(9), pp. 1392–1399. [CrossRef] [PubMed]
Lazarus, A. , and Reis, P. , 2015, “ Soft Actuation of Structured Cylinders Through Auxetic Behavior,” Adv. Eng. Mater., 17(6), pp. 815–820. [CrossRef]
Yang, D. , Mosadegh, B. , Ainla, A. , Lee, B. , Khashai, F. , Suo, Z. , Bertoldi, K. , and Whitesides, G. , 2015, “ Buckling of Elastomeric Beams Enables Actuation of Soft Machines,” Adv. Mater., 27(41), pp. 6323–6327. [CrossRef] [PubMed]
Li, J. , Shim, J. , Deng, J. , Overvelde, J. , Zhu, X. B. K. , and Yang, S. , 2012, “ Switching Periodic Membranes Via Pattern Transformation and Shape Memory Effect,” Soft Matter, 8(40), pp. 10322–10328. [CrossRef]
Zhu, X. , Wu, G. , Dong, R. , Chen, C. , and Yang, S. , 2012, “ Capillarity Induced Instability in Responsive Hydrogel Membranes With Periodic Hole Array,” Soft Matter, 8(31), pp. 8088–8093. [CrossRef]
Bertoldi, K. , and Boyce, M. C. , 2008, “ Mechanically-Triggered Transformations of Phononic Band Gaps in Periodic Elastomeric Structures,” Phys. Rev. B, 77(5), p. 052105. [CrossRef]
Shan, S. , Kang, S. , Wang, P. , Qu, C. , Shian, S. , Chen, E. , and Bertoldi, K. , 2014, “ Harnessing Multiple Folding Mechanisms in Soft Periodic Structures for Tunable Control of Elastic Waves,” Adv. Funct. Mater., 24(31) , p. 4935. [CrossRef]
Raney, J. , and Lewis, J. , 2015, “ Printing Mesoscale Architectures,” MRS Bull., 40(11), pp. 943–950. [CrossRef]
Williams, F. , and Anderson, M. , 1983, “ Incorporation of Lagrangian Multipliers Into an Algorithm for Finding Exact Natural Frequencies or Critical Buckling Loads,” Int. J. Mech. Sci., 25(8), pp. 579–584. [CrossRef]
Triantafyllidis, N. , and Schnaidt, W. , 1993, “ Comparison of Microscopic and Macroscopic Instabilities in a Class of Two-Dimensional Periodic Composites,” J. Mech. Phys. Solids, 41(9), pp. 1533–1565. [CrossRef]
Bertoldi, K. , Boyce, M. C. , Deschanel, S. , Prange, S. M. , and Mullin, T. , 2008, “ Mechanics of Deformation-Triggered Pattern Transformations and Superelastic Behavior in Periodic Elastomeric Structures,” J. Mech. Phys. Solids, 56(8), pp. 2642–2668. [CrossRef]
Ning, X. , and Pellegrino, S. , 2015, “ Buckling Analysis of Axially Loaded Corrugated Cylindrical Shells,” AIAA Paper No. 2015-1435.
Shim, J. , Shan, S. , Kosmrlj, A. , Kang, S. , Chen, E. , Weaver, J. , and Bertoldi, K. , 2013, “ Harnessing Instabilities for Design of Soft Reconfigurable Auxetic/Chiral Materials,” Soft Matter, 9(34), pp. 8198–8202. [CrossRef]
Kang, S. H. , Shan, S. , Košmrlj, A. , Noorduin, W. L. , Shian, S. , Weaver, J. C. , Clarke, D. R. , and Bertoldi, K. , 2014, “ Complex Ordered Patterns in Mechanical Instability Induced Geometrically Frustrated Triangular Cellular Structures,” Phys. Rev. Lett., 112(9), p. 098701. [CrossRef] [PubMed]
Javid, F. , Liu, J. , Shim, J. , Weaver, J. C. , Shanian, A. , and Bertoldi, K. , 2016, “ Mechanics of Instability-Induced Pattern Transformations in Elastomeric Porous Cylinders,” J. Mech. Phy. Solids, 96, pp. 1–17.
Hong, W. , Liu, Z. , and Suo, Z. , 2009, “ Inhomogeneous Swelling of a Gel in Equilibrium With a Solvent and Mechanical Load,” Int. J. Solids Struct., 46(17), pp. 3282–3289. [CrossRef]
Tipton, C. , Han, E. , and Mullin, T. , 2012, “ Magneto-Elastic Buckling of a Soft Cellular Solid,” Soft Matter, 8(26), pp. 6880–6883. [CrossRef]
Zhu, X. , 2011, “ Design and Fabrication of Photonic Microstructures by Holographic Lithography and Pattern Transformation,” Ph.D. thesis, University of Pennsylvania, Philadelphia, PA.
Rivlin, R. S. , 1948, “ Large Elastic Deformations of Isotropic Materials. II. Some Uniqueness Theorems for Pure, Homogeneous Deformation,” Philos. Trans. R. Soc. London A, 240(822), pp. 491–508. [CrossRef]
Overvelde, J. T. B. , Dykstra, D. M. J. , de Rooij, R. , Weaver, J. , and Bertoldi, K. , 2016, “ Tensile Instability in a Thick Elastic Body,” Phys. Rev. Lett., 117(9), p. 094301. [CrossRef] [PubMed]
Bertoldi, K. , and Boyce, M. C. , 2008, “ Wave Propagation and Instabilities in Monolithic and Periodically Structured Elastomeric Materials Undergoing Large Deformations,” Phys. Rev. B, 78(18), p. 184107. [CrossRef]
Rafsanjani, A. , Akbarzadeh, A. , and Pasini, D. , 2015, “ Snapping Mechanical Metamaterials Under Tension,” Adv. Mater., 27(39), pp. 5931–5935. [CrossRef] [PubMed]
Rafsanjani, A. , and Pasini, D. , 2016, “ Bistable Auxetic Mechanical Metamaterials Inspired by Ancient Geometric Motifs,” Extreme Mech. Lett., 9(Pt. 2), pp. 291–296. [CrossRef]
Zhang, Y. , Yan, Z. , Nan, K. , Xiao, D. , Liu, Y. , Luan, H. , Fu, H. , Wang, X. , Yang, Q. , Wang, J. , Ren, W. , Si, H. , Liu, F. , Yang, L. , Li, H. , Wang, J. , Guo, X. , Luo, H. , Wang, L. , Huang, Y. , and Rogers, J. A. , 2015, “ A Mechanically Driven Form of Kirigami as a Route to 3D Mesostructures in Micro/Nanomembranes,” Proc. Natl. Acad. Sci. U.S.A., 112(38), p. 11757. [CrossRef] [PubMed]
Rafsanjani, A. , and Bertoldi, K. , 2017, “ Buckling-Induced Kirigami,” Phys. Rev. Lett., 118(8), p. 084301. [CrossRef] [PubMed]
Song, Z. , Wang, X. , Lv, C. , An, Y. , Liang, M. , Ma, T. , He, D. , Zheng, Y.-J. , Huang, S.-Q. , Yu, H. , and Jiang, H. , 2015, “ Kirigami-Based Stretchable Lithium-Ion Batteries,” Sci. Rep., 5(1), p. 10988. [CrossRef] [PubMed]
Shyu, T. C. , Damasceno, P. F. , Dodd, P. M. , Lamoureux, A. , Xu, L. , Shlian, M. , Shtein, M. , Glotzer, S. C. , and Kotov, N. A. , 2015, “ A Kirigami Approach to Engineering Elasticity in Nanocomposites Through Patterned Defects,” Nat. Mater., 14, p. 785. [CrossRef] [PubMed]
Blees, K. , Barnard, A. W. , Rose, P. A. , Roberts, S. P. , McGill, K. L. , Huang, P. Y. , Ruyack, A. R. , Kevek, J. W. , Kobrin, B. , Muller, D. A. , and McEuen, P. L. , 2015, “ Graphene Kirigami,” Nature, 524(7564), p. 204. [CrossRef] [PubMed]
Lamoureux, A. , Lee, K. , Shlian, M. , Forrest, S. R. , and Shtein, M. , 2015, “ Dynamic Kirigami Structures for Integrated Solar Tracking,” Nat. Comm., 6, p. 8092. [CrossRef]
Wu, C. , Wang, X. , Lin, L. , Guo, H. , and Wang, Z. L. , 2016, “ Paper-Based Triboelectric Nanogenerators Made of Stretchable Interlocking Kirigami Patterns,” ACS Nano, 10(4), p. 4652. [CrossRef] [PubMed]
Isobe, M. , and Okumura, K. , 2016, “ Initial Rigid Response and Softening Transition of Highly Stretchable Kirigami Sheet Materials,” Sci. Rep., 6, p. 24758. [CrossRef] [PubMed]
Yan, Z. , Zhang, F. , Wang, J. , Liu, F. , Guo, X. , Nan, K. , Lin, Q. , Gao, M. , Xiao, D. , Shi, Y. , Qiu, Y. , Luan, H. , Kim, J. H. , Wang, Y. , Luo, H. , Han, M. , Huang, Y. , Zhang, Y. , and Rogers, J. A. , 2016, “ Controlled Mechanical Buckling for Origami-Inspired Construction of 3D Microstructures in Advanced Materials,” Adv. Funct. Mater., 26(16), pp. 2629–2639.
Neville, R. , Scarpa, F. , and Pirrera, A. , 2016, “ Shape Morphing Kirigami Mechanical Metamaterials,” Sci. Rep., 6, p. 31067. [CrossRef] [PubMed]
Coulais, C. , Teomy, E. , de Reus, K. , Shokef, Y. , and van Hecke, M. , 2016, “ Combinatorial Design of Textured Mechanical Metamaterials,” Nature, 535(7653), pp. 529–532. [CrossRef] [PubMed]
Hussein, M. , Leamy, M. , and Ruzzene, M. , 2014, “ Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress and Future Outlook,” Appl. Mech. Rev., 66(4), p. 040802. [CrossRef]
Khelif, A. , Choujaa, A. , Benchabane, S. , Djafari-Rouhani, B. , and Laude, V. , 2004, “ Guiding and Bending of Acoustic Waves in Highly Confined Phononic Crystal Waveguides,” Appl. Phys. Lett., 84(22), pp. 4400–4402. [CrossRef]
Kafesaki, M. , Sigalas, M. M. , and Garcia, N. , 2000, “ Frequency Modulation in the Transmittivity of Wave Guides in Elastic-Wave Band-Gap Materials,” Phys. Rev. Lett., 85(19), pp. 4044–4047. [CrossRef] [PubMed]
Cummer, S. , and Schurig, D. , 2007, “ One Path to Acoustic Cloaking,” New J. Phys., 9, p. 45. [CrossRef]
Elser, D. , Andersen, U. L. , Korn, A. , Glöckl, O. , Lorenz, S. , Marquardt, C. , and Leuchs, G. , 2006, “ Reduction of Guided Acoustic Wave Brillouin Scattering in Photonic Crystal Fibers,” Phys. Rev. Lett., 97(13), p. 133901. [CrossRef] [PubMed]
Elnady, T. , Elsabbagh, A. , Akl, W. , Mohamady, O. , Garcia-Chocano, V. M. , Torrent, D. , Cervera, F. , and Sánchez-Dehesa, J. , 2009, “ Quenching of Acoustic Bandgaps by Flow Noise,” Appl. Phys. Lett., 94(13), p. 134104. [CrossRef]
Casadei, F. , Dozio, L. , Ruzzene, M. , and Cunefare, K. , 2010, “ Periodic Shunted Arrays for the Control of Noise Radiation in an Enclosure,” J. Sound Vib., 329(18), p. 3632. [CrossRef]
Airoldi, L. , and Ruzzene, M. , 2011, “ Design of Tunable Acoustic Metamaterials Through Periodic Arrays of Resonant Shunted Piezos,” New J. Phys., 13(11), p. 113010. [CrossRef]
Casadei, F. , Beck, B. , Cunefare, K. A. , and Ruzzene, M. , 2012, “ Vibration Control of Plates Through Hybrid Configurations of Periodic Piezoelectric Shunts,” J. Intell. Mater. Syst. Struct., 23(10), pp. 1169–1177.
Kittel, C. , 1967, “ Introduction to Solid State Physics,” Am. J. Phys., 35(6), pp. 547–548. [CrossRef]
Kinra, V. K. , and Ker, E. L. , 1983, “ An Experimental Investigation of Pass Bands and Stop Bands in Two Periodic Particulate Composites,” Int. J. Solids Struct., 19(5), pp. 393–410. [CrossRef]
Kafesaki, M. , Sigalas, M. M. , and Economou, E. N. , 1995, “ Elastic Wave Band Gaps in 3-D Periodic Polymer Matrix Composites,” Solid State Commun., 96(5), pp. 285–289. [CrossRef]
Zhang, X. , Liu, Z. , Liu, Y. , and Wu, F. , 2003, “ Elastic Wave Band Gaps for Three-Dimensional Phononic Crystals With Two Structural Units,” Phys. Lett. A, 313(56), pp. 455–460. [CrossRef]
Page, J. , Yang, S. , Cowan, M. , Liu, Z. , Chan, C. , and Sheng, P. , 2003, “ 3D Phononic Crystals,” Wave Scattering in Complex Media: From Theory to Applications (NATO Science Series), Vol. 107, B. van Tiggelen and S. Skipetrov , eds., Springer, Dordrecht, The Netherlands, pp. 282–307. [CrossRef]
Yang, S. X. , Page, J. H. , Liu, Z. Y. , Cowan, M. L. , Chan, C. T. , and Sheng, P. , 2004, “ Focusing of Sound in a 3D Phononic Crystal,” Phys. Rev. Lett., 93(2), p. 024301. [CrossRef] [PubMed]
Sainidou, R. , Djafari-Rouhani, B. , Pennec, Y. , and Vasseur, J. O. , 2006, “ Locally Resonant Phononic Crystals Made of Hollow Spheres or Cylinders,” Phys. Rev. B, 73(2), p. 024302. [CrossRef]
Wang, P. , Shim, J. , and Bertoldi, K. , 2013, “ Effects of Geometric and Material Non-Linearities on the Tunable Response of Phononic Crystals,” Phys. Rev. B, 88(1), p. 014304. [CrossRef]
Mousanezhad, D. , Babaee, S. , Ghosh, R. , Mahdi, E. , Bertoldi, K. , and Vaziri, A. , 2015, “ Honeycomb Phononic Crystals With Self-Similar Hierarchy,” Phys. Rev. B, 92(10), p. 104304. [CrossRef]
Babaee, S. , Wang, P. , and Bertoldi, K. , 2015, “ Three-Dimensional Adaptive Soft Phononic Crystals,” J. Appl. Phys., 117(24), p. 244903. [CrossRef]
Celli, P. , Gonella, S. , Tajeddini, V. , Muliana, A. , Ahmed, S. , and Ounaies, Z. , 2017, “ Wave Control Through Soft Microstructural Curling: Bandgap Shifting, Reconfigurable Anisotropy and Switchable Chirality,” Smart Mater. Struct., 26(3), p. 035001.
Brillouin, L. , 1946, Wave Propagation in Periodic Structures, Dover Publications, Mineola, NY.
Parnell, W. , 2007, “ Effective Wave Propagation in a Prestressed Nonlinear Elastic Composite Bar,” IMA J. Appl. Math., 72(2), pp. 223–244. [CrossRef]
Parnell, W. , 2010, “ Pre-Stressed Viscoelastic Composites: Effective Incremental Moduli and Band-Gap Tuning,” AIP Conf. Proc., 1281(1), pp. 837–840.
Pal, R. , Rimoli, J. , and Ruzzene, M. , 2016, “ Effect of Large Deformation Pre-Loads on the Wave Properties of Hexagonal Lattices,” Smart Mater. Struct., 25, p. 054010. [CrossRef]
Gibson, L. , Ashby, M. , Zhang, J. , and Triantafillou, T. , 1989, “ Failure Surfaces for Cellular Materials Under Multiaxial Loads—I: Modelling,” Int. J. Solids Struct., 31(9), pp. 635–663.
Gibson, L. , and Ashby, M. , 1999, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, UK.
Papka, S. , and Kyriakides, S. , 1999, “ Biaxial Crushing of Honeycombs: Part 1: Experiments,” Int. J. Solids Struct., 36(29), pp. 4367–4396. [CrossRef]
Papka, S. , and Kyriakides, S. , 1999, “ In-Plane Biaxial Crushing of Honeycombs: Part II: Analysis,” Int. J. Solids Struct., 36(29), pp. 4397–4423. [CrossRef]
Chung, J. , and Waas, A. , 2001, “ In-Plane Biaxial Crush Response of Polycarbonate Honeycombs,” J. Eng. Mech., 127(2), pp. 180–193. [CrossRef]
Ohno, N. , Okumura, D. , and Noguchi, H. , 2002, “ Microscopic Symmetric Bifurcation Condition of Cellular Solids Based on a Homogenization Theory of Finite Deformation,” J. Mech. Phys. Solids, 50(5), pp. 1125–1153. [CrossRef]
Combescure, C. , Henry, P. , and Elliott, R. S. , 2016, “ Post-Bifurcation and Stability of a Finitely Strained Hexagonal Honeycomb Subjected to Equi-Biaxial In-Plane Loading,” Int. J. Solids Struct., 88–89, pp. 296–318. [CrossRef]
Drugan, W. , 2017, “ Wave Propagation in Elastic and Damped Structures With Stabilized Negative-Stiffness Components,” J. Mech. Phys. Solids, 106, pp. 34–45.
Puglisi, G. , and Truskinovsky, L. , 2000, “ Mechanics of a Discrete Chain With Bi-Stable Elements,” J. Mech. Phys. Solids, 48(1), pp. 1–27. [CrossRef]
Cherkaev, A. , Kouznetsov, A. , and Panchenko, A. , 2010, “ Still States of Bistable Lattices, Compatibility, and Phase Transition,” Continuum Mech. Thermodyn., 22(6), pp. 421–444. [CrossRef]
Paulose, J. , Meeussen, A. S. , and Vitelli, V. , 2015, “ Selective Buckling Via States of Self-Stress in Topological Metamaterials,” Proc. Natl. Acad. Sci., 112(25), pp. 7639–7644. [CrossRef]
Shan, S. , Kang, S. H. , Raney, J. , Wang, P. , Fang, L. , Candido, F. , Lewis, J. , and Bertoldi, K. , 2015, “ Multistable Architected Materials for Trapping Elastic Strain Energy,” Adv. Mater., 27(29), p. 4296. [CrossRef] [PubMed]
Restrepo, D. , Mankame, N. D. , and Zavattieri, P. D. , 2015, “ Phase Transforming Cellular Materials,” Extreme Mech. Lett., 4, pp. 52–60. [CrossRef]
Fraternali, F. , Blesgen, T. , Amendola, A. , and Daraio, C. , 2011, “ Multiscale Mass-Spring Models of Carbon Nanotube Foams,” J. Mech. Phys. Solids, 59(1), pp. 89–102. [CrossRef]
Schaeffer, M. , and Ruzzene, M. , 2015, “ Wave Propagation in Multistable Magneto-Elastic Lattices,” Int. J. Solids Struct., 56–57, pp. 78–95. [CrossRef]
Betts, D. , Bowen, C. , Kim, H. , Gathercole, N. , Clarke, C. , and Inman, D. , 2013, “ Nonlinear Dynamics of a Bistable Piezoelectric-Composite Energy Harvester for Broadband Application,” Eur. Phys. J. Spec. Top., 222(7), pp. 1553–1562. [CrossRef]
Yang, K. , Harne, R. L. , Wang, K. W. , and Huang, H. , 2014, “ Dynamic Stabilization of a Bistable Suspension System Attached to a Flexible Host Structure for Operational Safety Enhancement,” J. Sound Vib., 333(24), pp. 6651–6661. [CrossRef]
Harne, R. L. , Thota, M. , and Wang, K. W. , 2013, “ Concise and High-Fidelity Predictive Criteria for Maximizing Performance and Robustness of Bistable Energy Harvesters,” Appl. Phys. Lett., 102(5), p. 053903. [CrossRef]
Wu, Z. Z. , Harne, R. L. , and Wang, K. W. , 2014, “ Energy Harvester Synthesis Via Coupled Linear-Bistable System With Multistable Dynamics,” ASME J. Appl. Mech., 81(6), p. 061005. [CrossRef]
Johnson, D. R. , Harne, R. L. , and Wang, K. W. , 2014, “ A Disturbance Cancellation Perspective on Vibration Control Using a Bistable Snap-Through Attachment,” ASME J. Vib. Acoust., 136(3), p. 031006. [CrossRef]
Bolotin, V. V. , 1965, “ The Dynamic Stability of Elastic Systems. V. V. Bolotin. Translated from the Russian edition (Moscow, 1965) by V. I. Weingarten, L. B. Greszcuzuk, K. N. Trirogoff, and K. D. Gallegos. Holden-Day, San Francisco, CA, 1964. pp. xii + 451,” Science, 148(3670), pp. 627–628.
Feeny, B. F. , and Diaz, A. R. , 2010, “ Twinkling Phenomena in Snap-Through Oscillators,” ASME J. Vib. Acoust., 132(6), p. 061013. [CrossRef]
Nadkarni, N. , Daraio, C. , and Kochmann, D. M. , 2014, “ Dynamics of Periodic Mechanical Structures Containing Bistable Elastic Elements: From Elastic to Solitary Wave Propagation,” Phys. Rev. E, 90(2), p. 023204.
Frenkel, Y. , and Kontorova, T. , 1938, “ On Theory of Plastic Deformation and Twinning,” Phys. Z. Sowjetunion, 13, pp. 1–7.
Braun, O. M. , and Kivshar, Y. S. , 1998, “ Nonlinear Dynamics of the Frenkel Kontorova Model,” Phys. Rep., 306(12), pp. 1–108. [CrossRef]
Prandtl, L. , 1928, “ Ein gedankenmodell zur kinetischen theorie der festen krper,” Z. Angew. Math. Mech., 8(2), pp. 85–106. [CrossRef]
Benichou, I. , and Givli, S. , 2013, “ Structures Undergoing Discrete Phase Transformation,” J. Mech. Phys. Solids, 61(1), pp. 94–113. [CrossRef]
Bour, E. , 1862, “ Théorie de la déformation des surfaces,” J. Ec. Imp. Polytech. 19, pp. 1–48.
Scott, A. C. , 1969, “ A Nonlinear Klein-Gordon Equation,” Am. J. Phys., 37(1), pp. 52–61. [CrossRef]
Fermi, E. , Pasta, J. , and Ulam, S. , 1965, “ Studies of Nonlinear Problems (Los Alamos Report LA-1940),” The Collected Papers of Enrico Fermi, E. Segré , ed., University of Chicago Press, Chicago, IL.
Boussinesq, J. , 1877, “ Essai sur la theorie des eaux courantes, memoires presentes par divers savants,” Acad. Sci. Inst. Nat. France, XXIII, pp. 1–680.
Korteweg, D. D. J. , and de Vries, D. G. , 1895, “ XLI. On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves,” Philos. Mag., 39(240), pp. 422–443. [CrossRef]
Chu, C. , and James, R. D. , 1995, “ Analysis of Microstructures in Cu-14.0%Al-3.9%Ni by Energy Minimization,” J. Phys. IV, 5, pp. C8-143–C8-149.
Jona, F. , and Shirane, G. , 1962, Ferroelectric Crystals, Pergamon Press, New York.
Zhang, W. , and Bhattacharya, K. , 2005, “ A Computational Model of Ferroelectric Domains. Part I: Model Formulation and Domain Switching,” Acta Mater., 53(1), pp. 185–198. [CrossRef]
Su, Y. , and Landis, C. M. , 2007, “ Continuum Thermodynamics of Ferroelectric Domain Evolution: Theory, Finite Element Implementation, and Application to Domain Wall Pinning,” J. Mech. Phys. Solids, 55(2), pp. 280–305. [CrossRef]
Onuki, A. , 2002, Phase Transition Dynamics, Cambridge University Press, Cambridge, UK. [CrossRef]
Bray, A. J. , 2002, “ Theory of Phase-Ordering Kinetics,” Adv. Phys., 51(2), pp. 481–587. [CrossRef]
Dye, D. , 2015, “ Shape Memory Alloys: Towards Practical Actuators,” Nat. Mater., 14, pp. 760–761. [CrossRef] [PubMed]
Fiebig, M. , Lottermoser, T. , Meier, D. , and Trassin, M. , 2016, “ The Evolution of Multiferroics,” Nat. Rev. Mater., 1, p. 16046.
Nadkarni, N. , Daraio, C. , Abeyaratne, R. , and Kochmann, D. M. , 2016, “ Universal Energy Transport Law for Dissipative and Diffusive Phase Transitions,” Phys. Rev. B, 93(10), p. 104109. [CrossRef]
Nadkarni, N. , Arrieta, A. F. , Chong, C. , Kochmann, D. M. , and Daraio, C. , 2016, “ Unidirectional Transition Waves in Bistable Lattices,” Phys. Rev. Lett., 116(24), p. 244501. [CrossRef] [PubMed]
Allen, S. M. , and Cahn, J. W. , 1979, “ A Microscopic Theory for Antiphase Boundary Motion and Its Application to Antiphase Domain Coarsening,” Acta Metall., 27(6), pp. 1085–1095. [CrossRef]
Zhang, W. , and Bhattacharya, K. , 2005, “ A Computational Model of Ferroelectric Domains. Part II: Grain Boundaries and Defect Pinning,” Acta Mater., 53(1), pp. 199–209. [CrossRef]
Bishop, A. R. , and Lewis, W. F. , 1979, “ A Theory of Intrinsic Coercivity in Narrow Magnetic Domain Wall Materials,” J. Phys. C, 12(18), p. 3811. [CrossRef]
Kashimori, Y. , Kikuchi, T. , and Nishimoto, K. , 1982, “ The Solitonic Mechanism for Proton Transport in a Hydrogen Bonded Chain,” J. Chem. Phys., 77(4), pp. 1904–1907. [CrossRef]
Peyrard, M. , and Bishop, A. R. , 1989, “ Statistical Mechanics of a Nonlinear Model for DNA Denaturation,” Phys. Rev. Lett., 62(23), pp. 2755–2758. [CrossRef] [PubMed]
Thevamaran, R. , Fraternali, F. , and Daraio, C. , 2014, “ Multiscale Mass-Spring Model for High-Rate Compression of Vertically Aligned Carbon Nanotube Foams,” ASME J. Appl. Mech., 81(12), p. 121006. [CrossRef]
Stewart, W. C. , 1968, “ Current Voltage Characteristics of Josephson Junctions,” Appl. Phys. Lett., 12(8), pp. 277–280. [CrossRef]
Keizer, J. , and Smith, G. D. , 1998, “ Spark-to-Wave Transition: Saltatory Transmission of Calcium Waves in Cardiac Myocytes,” Biophys. Chem., 72(12), pp. 87–100. [CrossRef] [PubMed]
Scott, A. C. , 1975, “ The Electrophysics of a Nerve Fiber,” Rev. Mod. Phys., 47(2), pp. 487–533. [CrossRef]
Rotermund, H. H. , Jakubith, S. , von Oertzen, A. , and Ertl, G. , 1991, “ Solitons in a Surface Reaction,” Phys. Rev. Lett., 66(23), pp. 3083–3086. [CrossRef] [PubMed]
Comte, J. C. , Marquie, P. , and Remoissenet, M. , 1999, “ Dissipative Lattice Model With Exact Traveling Discrete Kink-Soliton Solutions: Discrete Breather Generation and Reaction Diffusion Regime,” Phys. Rev. E, 60(6), pp. 7484–7489. [CrossRef]
Balk, A. M. , Cherkaev, A. V. , and Slepyan, L. I. , 2001, “ Dynamics of Chains With Non-Monotone Stress Strain Relations. II. Nonlinear Waves and Waves of Phase Transition,” J. Mech. Phys. Solids, 49(1), pp. 149–171. [CrossRef]
Truskinovsky, L. , and Vainchtein, A. , 2005, “ Kinetics of Martensitic Phase Transitions: Lattice model,” SIAM J. Appl. Math., 66(2), pp. 533–553. [CrossRef]
Braun, O. M. , Kivshar, Y. S. , and Zelenskaya, I. I. , 1990, “ Kinks in the Frenkel-Kontorova Model With Long-Range Interparticle Interactions,” Phys. Rev. B, 41(10), pp. 7118–7138. [CrossRef]
Abeyaratne, R. , and Knowles, J. , 1991, “ Kinetic Relations and the Propagation of Phase Boundaries in Solids,” Arch. Ration. Mech. Anal., 114(2), pp. 119–154. [CrossRef]
Remoissenet, M. , and Peyrard, M. , 1984, “ Soliton Dynamics in New Models With Parametrized Periodic Double-Well and Asymmetric Substrate Potentials,” Phys. Rev. B, 29(6), pp. 3153–3166. [CrossRef]
Benichou, I. , Faran, E. , Shilo, D. , and Givli, S. , 2013, “ Application of a Bi-Stable Chain Model for the Analysis of Jerky Twin Boundary Motion in Ni–Mn–Ga,” Appl. Phys. Lett., 102(1), p. 011912. [CrossRef]
Frazier, M. J. , and Kochmann, D. M. , 2017, “ Band Gap Transmission in Periodic Bistable Mechanical Systems,” J. Sound Vib., 388, pp. 315–326. [CrossRef]
Chen, B. G. , Upadhyaya, N. , and Vitelli, V. , 2014, “ Nonlinear Conduction Via Solitons in a Topological Mechanical Insulator,” Proc. Natl. Acad. Sci., 111(36), pp. 13004–13009. [CrossRef]
Frazier, M. J. , and Kochmann, D. M. , 2017, “ Atomimetic Mechanical Structures With Nonlinear Topological Domain Evolution Kinetics,” Adv. Mater., 29(19), p. 1605800.
Zirbel, S. A. , Tolman, K. A. , Trease, B. P. , and Howell, L. L. , 2016, “ Bistable Mechanisms for Space Applications,” PLoS One, 11, p. e0168218. [CrossRef] [PubMed]
Benichou, I. , and Givli, S. , 2015, “ Rate Dependent Response of Nanoscale Structures Having a Multiwell Energy Landscape,” Phys. Rev. Lett., 114(9), p. 095504. [CrossRef] [PubMed]
Cherkaev, A. , and Zhornitskaya, L. , 2005, “ Protective Structures With Waiting Links and Their Damage Evolution,” Multibody Syst. Dyn., 13(1), pp. 53–67. [CrossRef]
Evans, A. , and Cannon, R. , 1986, “ Overview No. 48,” Acta Metall., 34(5), pp. 761–800. [CrossRef]


Grahic Jump Location
Fig. 1

Map of stable regions in the λ–μ plane of a homogeneous, isotropic, linear elastic solid. Dark-gray regions are unstable by violation of ellipticity, whereas the requirement of positive definiteness further restricts the stable region by also making the light-gray region unstable (the three key inequalities from Eqs. (18) and (23) are shown separately).

Grahic Jump Location
Fig. 2

Simple spring examples: (a) bistable mechanical system and (b) composite spring system; the energy and stiffness of system (a) is illustrated in (c); (d) linear spring reduction of system (b) where k1=ka* and (e) its viscoelastic extension; (f) composite system

Grahic Jump Location
Fig. 3

Effective (normalized) stiffness and damping of the linear viscoelastic composite system of Fig. 2(e) versus (normalized) spring stiffness k1 (shown for ηω=0.01 and k2>0)

Grahic Jump Location
Fig. 4

Effective viscoelastic Young's modulus and loss tangent for a Hashin–Shtrikman composite composed of metal matrix (μmat=19.2 GPa, κmat=41.6 GPa, and three values of tan δ=0.01,0.02, and  0.04) and 5 vol % ceramic inclusions (μinc=50 GPa, tan δ=0.001, and varying κinc). The shown effective modulus refers to the absolute value of the complex-valued viscoelastic Young's modulus E*=Ere+iEim.

Grahic Jump Location
Fig. 5

Stiffness (Young's modulus) versus damping (loss tangent) versus mass density for a variety of natural and manmade materials; the shadowed prism highlights the desirable but challenging region of combined high stiffness and high damping

Grahic Jump Location
Fig. 6

Stable and unstable moduli combinations for a coated spherical inclusion (stable and unstable combinations are shown as light and dark gray regions, respectively); both phases are homogeneous, isotropic, linear elastic (with μc,μi>0 for pointwise stability). Under static conditions, an infinite effective bulk modulus of the two-phase body is unstable (shown by the ω = 0 line are all moduli combinations resulting in a positive-infinite bulk modulus). Under dynamic excitation, resonance effects lead to strong stiffness variations which, with increasing excitation frequency ω shift in to the stable region (ω0=μc/ρcb). When admitting negative stiffness κi<0, these become stable at significantly lower frequencies [107].

Grahic Jump Location
Fig. 7

Stability map showing stable (light) and unstable (dark gray) regions of κi/μi, where κi and μi are, respectively, the bulk and shear moduli of a spherical inclusion (radius a) embedded in a concentric coating (outer radius b and moduli κc, μc). The stability limit for κi/μi depends on the applied BCs. Plotted is the effective bulk modulus κ*=pb/3ur(b) for uniform applied pressure p, resulting in a rotational-symmetric expansion of the coated-sphere composite with radial displacement field ur(r). Consequently, under both types of BCs, the solid loses stability before the effective modulus tends to +∞ with decreasing κi/μi. Note that positive-definiteness corresponds to κ/μ≥0, so that the elastic coating expands the stable regime of the inclusion phase.

Grahic Jump Location
Fig. 8

Torsional compliance (inverse stiffness) and loss tangent versus temperature of a composite composed of 1 vol % VO2 particles embedded in a pure tin matrix (damping of pure tin is included for reference). Measurements were conducted well below sample resonance at 100 Hz. Reprinted with permission from Lakes et al. [39]. Copyright 2001 by Nature Publishing Group.

Grahic Jump Location
Fig. 9

Dynamic stiffness and damping variations in viscoelastic Sn–BaTiO3 composites under harmonic loading. Adapted from Ref. [40].

Grahic Jump Location
Fig. 10

(a) Macroscopic (continuous line) and microscopic (dashed line) onset-of-bifurcation surfaces in the principal macroscopic logarithmic strain space for a perfectly periodic neo-Hookean solid (characterized by a bulk to shear moduli ratio equal to 9.8) with a square distribution of circular voids. The eigenmode of the microscopic bifurcation instability is shown on the right. Reprinted with permission from Triantafyllidis et al. [162]. Copyright 2006 by ASME. (b) Experimental images of an elastomeric structure comprising a square array of circular holes for increasing values of the applied deformation. Note that after instability, the lateral boundaries of the sample bend inward, a clear signature of negative Poisson's ratio behavior. Adapted from Ref. [180].

Grahic Jump Location
Fig. 11

(a) Experimental images of an elastomeric structure comprising a triangular array of circular holes when compressed horizontally (top), vertically (center), and equibiaxially (bottom). Three distinct buckling-induced patterns are formed. Adapted from Ref. [174]. (b) Orthogonal side views (onto the y–z and x–z planes) for a cylindrical sample pattern with a square array of circular holes at different levels of deformation. The structure was made watertight by a thin membrane that covered the inner surface of the voids and was then loaded hydraulically. Adapted from Ref. [169]. (c) The Hoberman Twist-O is a commercial toy which comprises a rigid network of struts connected by rotating hinges and can easily collapse into a ball having a fraction of its original size. Adapted from Ref. [22]. (d) The Buckliball is inspired by this popular toy but translates the mechanism design to the structure of an elastic spherical shell—which under pneumatic actuation undergoes buckling-induced folding, opening avenues for a new class of active and reversible encapsulation systems. Adapted from Ref. [22].

Grahic Jump Location
Fig. 12

(a) Experimental snapshots during the swelling process for a square lattice made of plates sandwiched between two thin and stiff layers. Buckling induces an effective negative swelling ratio in this structure. Reprinted with permission from Liu et al. [165]. Copyright 2016 by John Wiley & Sons. (b) An instability is induced by capillary forces during evaporation of water from a swollen hydrogel membrane with micron-sized holes in a square array. Scale bars: 10 μm. Reprinted with permission from Zhu et al. [172]. Copyright 2012 by Royal Society of Chemistry (c) Slow variations in the current through several electromagnetic coils embedded in a soft cellular elastomer induce visible strain and snap-through behavior. Reprinted with permission from Tipton et al. [184]. Copyright 2012 by Royal Society of Chemistry.

Grahic Jump Location
Fig. 13

(a) Buckling-induced reversible pattern formation in a supported microscale honeycomb lattice upon rapid swelling. Depending on the geometry of the plates, buckling induces either an achiral pattern or a chiral pattern. Multiple domains with different chirality are observed, whose boundaries are highlighted by the dashed lines. The insets show magnified images of the buckled patterns within the domains (top) and at the domain boundaries (bottom). The color-coded arrows indicate the handedness of the vertices. Adapted from Ref. [166]. (b) A soft gripper made of a buckling actuator. The claws of the gripper close upon deflation of the buckling actuator and the buckling gripper picks up a piece of chalk. Scale bars: 1 cm. Adapted from Ref. [170]. (c) Buckling-induced pattern transformation in shape-memory polymer membranes comprising a hexagonal array of micron-sized circular holes results in dramatic color switching. Adapted from Ref. [171].

Grahic Jump Location
Fig. 14

(a) Snapshots of a bistable mechanical metamaterial in response to tensile loading. The system comprises an array of double-curved beams which can snap between two stable configurations. Adapted from Ref. [189]. (b) Response of an elastic sheet perforated with a square array of mutually orthogonal cuts under uniaxial tension. In the thick limit, the perforated sheet deforms in-plane and identically to a network of rotating squares (left). For sufficiently small values of thickness, mechanical instabilities triggered under uniaxial tension result in the formation of complex 3D patterns, which are affected by the loading direction (center and right). Scale bars: 6 mm. Adapted from Ref. [192].

Grahic Jump Location
Fig. 15

(a) Analogously to antiferromagnetic systems—in which nearest-neighbor spins cannot align in opposite directions when arranged on a triangle—in triangular frames, the beams cannot buckle into a half sinusoid and at the same time preserve angles at joints. As a result, the system becomes frustrated. (b) Geometric frustration in periodic 2D beam lattices favors the formation of complex buckling-induced ordered patterns. Adapted from Ref. [181].

Grahic Jump Location
Fig. 16

Phononic band structure for a square array of circular voids in an elastic matrix subjected to equibiaxial compression in (a) the undeformed configuration and (b) after buckling. Adapted from Ref. [218].

Grahic Jump Location
Fig. 17

Effect of deformation on the directionality of the propagating waves for a square array of circular voids in an elastic matrix subjected to equibiaxial compression. (a) and (b) Effect of deformation on the directionality of the phase velocity. (c) and (d) Effect of deformation on the directionality of the group velocity. Adapted from Ref. [218].

Grahic Jump Location
Fig. 18

Tunable acoustic metamaterial: (a) the undeformed configuration comprises resonating units dispersed into an elastomeric matrix. Each resonator consists of a metallic mass connected to the matrix through elastic beams, which form a structural coating. The black regions in the picture indicate voids in the structure. The unit cell size is A0=50.0 mm. (b) When a compressive strain ε=−0.10 is applied in the vertical direction, buckling of the beams significantly alters the effective stiffness of the structural coating. (c) Experimentally measured transmittance at different levels of applied deformation. The band gap frequency first decreases linearly as a function of ε and then it completely disappears as ε approaches 0.10. Adapted from Ref. [34].

Grahic Jump Location
Fig. 19

Illustration of domains (constant polarization p within each domain) separated by domain walls in structures and materials. Under excitation, the domain walls move at speed v. The general structure of both systems is schematically shown with an interaction potential V and multistable potential ψ.

Grahic Jump Location
Fig. 20

A chain of laterally compressed, bistable membranes coupled by nonlinear permanent magnets obeys Eq. (47) and displays unidirectional transition wave motion upon impact [266]

Grahic Jump Location
Fig. 21

Stable transition waves in a 3D-printed chain of bistable elements [23]: (a) a compressive transition wave propagates by releasing stored elastic energy, which is dissipated due to material-internal damping mechanisms. (b) A stable transition wave of constant speed v and width w is recorded (after an initial transient period).

Grahic Jump Location
Fig. 22

Periodic array of bistable rotational elements (polarization angle φ) coupled to nearest neighbors by elastic bands [287]. The combination of an excentrically attached linear spring and the action of gravity results in a tunable bistable potential ψ which can be biased as in domain switching (by tilting the whole setup by an angle α) and which can also mimic a second-order phase transition (by adjusting the position fx of the spring anchor points, switching between a bistable potential and a single-well potential).



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