Mechanical metamaterials are artificial structures with mechanical properties defined by their structure rather than their composition. They can be seen as a counterpart to the rather well-known family of optical metamaterials and include acoustic metamaterials as a special case of vanishing shear. Their mechanical properties can be designed to have values which cannot be found in nature.
Examples of mechanical metamaterials
Acoustic / phononic metamaterials
Acoustic or phononic metamaterials can exhibit acoustic properties not found in nature, such as negative effective bulk modulus, negative effective mass density, or double negativity. They find use in (mostly still purely scientific) applications like acoustic subwavelength imaging, superlensing, negative refraction  or transformation acoustics.
Materials with negative Poisson's ratio (auxetics)
Poisson's ratio defines how a material expands (or contracts) transversely when being compressed longitudinally. While basically all known natural materials have a positive Poisson's ratio (coinciding with our intuitive idea that by compressing a material it must expand in the orthogonal direction), metamaterials with a Poisson's ratio below zero have been fabricated in 2D as well as in 3D. Simple designs of composites possessing negative Poisson's ratio (inverted hexagonal periodicity cell) were published in 1985. 
Metamaterials with negative longitudinal and volume compressibility transitions
In a closed thermodynamic system in equilibrium, both the longitudinal and volumetric compressibility are necessarily non-negative because of stability constraints. For this reason, when tensioned, ordinary materials expand along the direction of the applied force. It has been shown, however, that metamaterials can be designed to exhibit negative compressibility transitions, during which the material undergoes contraction when tensioned (or expansion when pressured). When subjected to isotropic stresses, these metamaterials also exhibit negative volumetric compressibility transitions. In this class of metamaterials, the negative response is along the direction of the applied force, which distinguishes these materials from those that exhibit negative transversal response (such as in the study of negative Poisson's ratio).
Pentamode metamaterials or meta-fluids
A pentamode metamaterial is an artificial three-dimensional structure which, despite being a solid, ideally behaves like a fluid. Thus, it has a finite bulk but vanishing shear modulus, or in other words it is hard to compress yet easy to deform. Speaking in a more mathematical way, pentamode metamaterials have an elasticity tensor with only one non-zero eigenvalue and five (penta) vanishing eigenvalues.
Pentamode structures have been proposed theoretically by G. W. Milton in 1995  but have not been fabricated until early 2012. According to theory, pentamode metamaterials can be used as the building blocks for materials with completely arbitrary elastic properties. Anisotropic versions of pentamode structures are a candidate for transformation elastodynamics and elastodynamic cloaking.
- Lee, Sam Hyeon; Park, Choon Mahn; Seo, Yong Mun; Wang, Zhi Guo; Kim, Chul Koo (29 April 2009). "Acoustic metamaterial with negative modulus". Journal of Physics: Condensed Matter 21 (17): 175704. arXiv:0812.2952. Bibcode:2009JPCM...21q5704L. doi:10.1088/0953-8984/21/17/175704.
- Lee, Sam Hyeon; Park, Choon Mahn; Seo, Yong Mun; Wang, Zhi Guo; Kim, Chul Koo (1 December 2009). "Acoustic metamaterial with negative density". Physics Letters A 373 (48): 4464–4469. Bibcode:2009PhLA..373.4464L. doi:10.1016/j.physleta.2009.10.013.
- Yang, Z.; Mei, Jun; Yang, Min; Chan, N.; Sheng, Ping (1 November 2008). "Membrane-Type Acoustic Metamaterial with Negative Dynamic Mass". Physical Review Letters 101 (20). Bibcode:2008PhRvL.101t4301Y. doi:10.1103/PhysRevLett.101.204301.
- Ding, Yiqun; Liu, Zhengyou; Qiu, Chunyin; Shi, Jing. "Metamaterial with Simultaneously Negative Bulk Modulus and Mass Density". Physical Review Letters 99 (9): 093904. Bibcode:2007PhRvL..99i3904D. doi:10.1103/PhysRevLett.99.093904. PMID 17931008.
- Lee, Sam Hyeon; Park, Choon Mahn; Seo, Yong Mun; Wang, Zhi Guo; Kim, Chul Koo (1 February 2010). "Composite Acoustic Medium with Simultaneously Negative Density and Modulus". Physical Review Letters 104 (5). arXiv:0901.2772. Bibcode:2010PhRvL.104e4301L. doi:10.1103/PhysRevLett.104.054301.
- Zhu, J.; Christensen, J.; Jung, J.; Martin-Moreno, L.; Yin, X.; Fok, L.; Zhang, X.; Garcia-Vidal, F. J. (2011). "A holey-structured metamaterial for acoustic deep-subwavelength imaging". Nature Physics 7 (1): 52–55. Bibcode:2011NatPh...7...52Z. doi:10.1038/nphys1804.
- Li, Jensen; Fok, Lee; Yin, Xiaobo; Bartal, Guy; Zhang, Xiang (2009). "Experimental demonstration of an acoustic magnifying hyperlens". Nature Materials 8 (12): 931–934. Bibcode:2009NatMa...8..931L. doi:10.1038/nmat2561. PMID 19855382.
- Christensen, Johan; de Abajo, F. (2012). "Anisotropic Metamaterials for Full Control of Acoustic Waves". Physical Review Letters 108 (12). Bibcode:2012PhRvL.108l4301C. doi:10.1103/PhysRevLett.108.124301.
- Farhat, M.; Enoch, S.; Guenneau, S.; Movchan, A. (2008). "Broadband Cylindrical Acoustic Cloak for Linear Surface Waves in a Fluid". Physical Review Letters 101 (13). Bibcode:2008PhRvL.101m4501F. doi:10.1103/PhysRevLett.101.134501.
- Cummer, Steven A; Schurig, David (2007). "One path to acoustic cloaking". New Journal of Physics 9 (3): 45–45. Bibcode:2007NJPh....9...45C. doi:10.1088/1367-2630/9/3/045.
- Xu, B.; Arias, F.; Brittain, S. T.; Zhao, X.-M.; Grzybowski, B.; Torquato, S.; Whitesides, G. M. (1999). "Making Negative Poisson's Ratio Microstructures by Soft Lithography". Advanced Materials 11 (14): 1186–1189. doi:10.1002/(SICI)1521-4095(199910)11:14<1186::AID-ADMA1186>3.0.CO;2-K.
- Bückmann, Tiemo; Stenger, Nicolas; Kadic, Muamer; Kaschke, Johannes; Frölich, Andreas; Kennerknecht, Tobias; Eberl, Christoph; Thiel, Michael; Wegener, Martin (22 May 2012). "Tailored 3D Mechanical Metamaterials Made by Dip-in Direct-Laser-Writing Optical Lithography". Advanced Materials 24 (20): 2710–2714. doi:10.1002/adma.201200584. PMID 22495906.
- A.G. Kolpakovs "Determination of the average characteristics of elastic frameworks" Journal of Applied Mathematics and Mechanics (Elsevier http://www.sciencedirect.com/science/article/pii/0021892885900115), 49(6), 1985, pp.739-745
- R.F. Almgren "An isotropic three-dimensional structure with Poisson's ratio=-1", J. Elasticity 15 (1985), 427-430.
- P.S. Theocaris, G.E. Stavroulakis, P.D. Panagiotopoulos: Negative Poisson's ratio in composites with star-shaped inclusions: a numerical homogenization approach . Archive of Applied Mechanics (former Ingenieur Archiv) 67(4), 274-286, 1997.
- P.S. Theocaris, G.E. Stavroulakis: The homogenization method for the study of variation of Poisson's ratio in fiber composites. Archive of Applied Mechanics (former Ingenieur Archiv), 69(3-4), 281-295, 1998.
- G.E. Stavroulakis: Auxetic behaviour: Appearance and engineering applications. Physica Status Solidi (b), 242(3), 710-720, 2005.
- N.T. Kaminakis, G.E. Stavroulakis: Topology optimization for compliant mechanisms, using evolutionary-hybrid algorithms and application to the design of auxetic materials. Composites Part B: Engineering, 43(6), 2655-2668, 2012.
- Nicolaou Z. G. and Motter A. E., Mechanical metamaterials with negative compressibility transitions, Nature Materials 11, 608-613 (2012).
- Milton, Graeme W.; Cherkaev, Andrej V. (1 January 1995). "Which Elasticity Tensors are Realizable?". Journal of Engineering Materials and Technology 117 (4): 483. doi:10.1115/1.2804743.
- Kadic, Muamer; Bückmann, Tiemo; Stenger, Nicolas; Thiel, Michael; Wegener, Martin (1 January 2012). "On the practicability of pentamode mechanical metamaterials". Applied Physics Letters 100 (19): 191901. arXiv:1203.1481. Bibcode:2012ApPhL.100s1901K. doi:10.1063/1.4709436.