Abstract
A band gap region, or simply a band gap, is a range of frequencies where vibrations of certain frequency ranges are isolated. In the present paper, such ranges are sought through the study of different cases for the shape of the unit cells of a lattice, i.e., of an assembly of classical structural elements, such as beams and plates. A lattice with a specific, special designed microstructure is considered in the present investigation. Each particular cell of the examined lattice is studied as a classical composite material consisting of a matrix and the reinforcing core (e.g., matrix-fiber composite), and it is discretized by using two-dimensional plane stress finite elements. The form of the core of the unit cells can be of several shapes, e.g., quadratic, circular, and star. Some of these shapes provide the whole lattice with auxetic behavior, with negative Poisson’s ratio at the homogenized properties. The shape and the microstructure of the lattice is optimized in order to achieve isolation of the desired frequencies. A first attempt on the optimization of star-shaped microstructures is also presented. The optimization is carried out using powerful global optimization methods, such as the genetic algorithms. Results indicate that band gaps may appear in both conventional and auxetic microstructures. Moreover, the appearance and the size of the band gaps depend on the selected microstructure.
Similar content being viewed by others
References
Jianbao, L., Yue-Sheng, W., Chuanzeng, Z.: Finite element method for analysis of band structures of 2D phononic crystals with archimedean-like tilings. AIP Conf. Proc. 1233, 131–136 (2010). https://doi.org/10.1063/1.3452095
Spadoni, A., Ruzzene, M., Gonella, S., Scarpa, F.: Phononic properties of hexagonal chiral lattices. Wave Motion 46, 435–450 (2009)
Phani, A.S., Woodhouse, J., Fleck, N.A.: Wave propagation in two-dimensional periodic lattices. J. Acoust. Soc. Am. 119, 1995–2005 (2006)
Brillouin, L.: Wave Propagation in Periodic Structures, 2nd edn. Dover, New York (1953)
Kittel, C.: Elementary Solid-State Physics: A Short Course, 1st edn. Wiley, New York (1962)
Mace, B.R., Manconi, E.: Modelling wave propagation in two-dimensional structures using finite element analysis. J. Sound Vib. 318, 884–902 (2008)
Ma, Y., Scarpa, F., Zhang, D., Zhu, B., Chen, L., Hong, J.: A nonlinear auxetic structural vibration damper with metal rubber particles. Smart Mater. Struct. 22, 084012 (2013)
Meng, J., Deng, Z., Zhang, K., Xu, X., Wen, F.: Band gap analysis of star-shaped honeycombs with varied Poisson’s ratio. Smart Mater. Struct. 24, 095011 (2015)
Duncan, O., Shepherd, T., Moroney, C., Foster, L., Venkatraman, P.D., Winwood, K., Allen, T., Alderson, A.: Review of auxetic materials for sports applications: expanding options in comfort and protection. Appl. Sci. 8, 941 (2018)
Lim, T.C.: Auxetic Materials and Structures. Springer, Singapore (2015)
Lucas, V.B., Claus, C., Elke, D., Wim, D.: On the impact of damping on the dispersion curves of a locally resonant metamaterial: modelling and experimental validation. J. Sound Vib. 409, 1–23 (2017)
El Sherbiny, M.G., Placidi, L.: Discrete and continuous aspects of some metamaterial elastic structures with band gaps. Arch. Appl. Mech. 88, 1725–1742 (2018)
Sang, M.J., Massimo, R.: Analysis of vibration and wave propagation in cylindrical grid-like structures. Shock Vib. 11, 311–331 (2004)
Parthkumar, G.D., Elisabetta, M., Marcello, V., Lars, V.A., Andrea, R.: Numerical and experimental investigation of stop-bands in finite and infinite periodic one-dimensional structures. J. Vib. Control 22, 1–12 (2014)
Elif, D., Pascal, L.: Finite element method analysis of band gap and transmission of two-dimensional metallic photonic crystals at terahertz frequencies. Appl. Opt. 52, 7367–7375 (2013)
Istvan, A.V., Thomas, B., Osamu, M.: Analysis by the finite element method. J. Appl. Phys. 114, 083519 (2013)
Hsiang-Wen, T., Wei-Di, C., Lien-Wen, C.: Wave propagation in the polymer-filled star-shaped honeycomb periodic structure. Appl. Phys. A 123, 523 (2017)
Meng, J., Deng, Z., Zhang, K., Xu, X., Wen, F.: Band gap analysis of star-shaped honeycombs with varied Poisson’s ratio. Smart Mater. Struct. 24, 095011 (2015)
Chen, W., Tian, X., Gao, R., Liu, Sh: A low porosity perforated mechanical metamaterial with negative Poisson’s ratio and band gaps. Smart Mater. Struct. 27, 115010 (2018)
Bacigalupo, A., De Belis, M.L.: Auxetic anti-tetrachiral materials: equivalent elastic properties and frequency band-gaps. Compos. Struct. 131, 530–544 (2015)
Mukherjee, S., Scarpa, F., Gopalakrishnan, S.: Phononic band gap design in honeycomb lattice with combinations of auxetic and conventional core. Smart Mater. Struct. 25, 054011 (2016)
Jian, L., Viacheslav, S., Stephan, R.: Auxetic multiphase soft composite material design through instabilities with application for acoustic metamaterials. Soft Matter 14, 6171 (2018)
Aage, N., Gersborg, A.R., Sigmund, O.: Topology optimization of optical band gap effects in slab structures modulated by periodic rayleigh waves. In: XXII ICTAM, Adelaide, Australia, 25–29 August (2008)
Vatanabe, S.L., Paulino, G.H., Silva, E.C.: Maximizing phononic band gaps in piezocomposite materials by means of topology optimization. J. Acoust. Soc. Am. 136, 494–501 (2014)
Halkjær, S., Sigmund, O., Jensen, J.S.: Maximizing band gaps in plate structures. Struct. Multidiscip. Optim. 32, 263–275 (2006)
Zhang, G.H., Gao, X.L., Ding, S.R.: Band gaps for wave propagation in 2-D periodic composite structures incorporating microstructure effects. Acta Mech. 229, 4199–4214 (2018)
Wagner, P.R., Dertimanis, V.K., Antoniadis, I.A., Chatzi, E.N.: On the feasibility of structural metamaterials for seismic-induced vibration mitigation. Int. J. Earthq. Impact Eng. 1, 20–56 (2016)
Bloch, F.: Über die quantenmechanik der elektronen in kristallgittern. Z. Phys. 52, 555–600 (1928)
Floquet, G.: Sur les équations différentielles linéaires à coefficients périodiques. Ann. Sci. Norm. Supérieure 12, 47–88 (1883)
Theocaris, P.S., Stavroulakis, G.E., Panagiotopoulos, P.D.: Negative Poisson’s ratios in composites with star-shaped inclusions: a numerical homogenization approach. Arch. Appl. Mech. 67, 274–286 (1997)
Grima, J.N., Cauchi, R., Gatt, R., Attard, D.: Honeycomb composites with auxetic out-of-plane characteristics. Compos. Struct. 106, 150–159 (2013)
Kaminakis, N.T., Stavroulakis, G.E.: Topology optimization for compliant mechanisms, using evolutionary-hybrid algorithms and application to the design of auxetic materials. Compos. Part B Eng. 43, 2655–2668 (2012)
Kaminakis, N.T., Drosopoulos, G.A., Stavroulakis, G.E.: Design and verification of auxetic microstructures using topology optimization and homogenization. Arch. Appl. Mech. 85, 1289–1306 (2015)
Koutsianitis, P., Drosopoulos, G., Tairidis, G.K., Stavroulakis, G.E.: Shape optimization of unit cells for vibration isolation using auxetic materials. In: 13th World Congress in Computational Mechanics WCCM 2018, New York City, USA, 22–27 July (2018)
Bendsoe, M.P., Sigmund, O.: Topology Optimization: Theory, Methods and Applications. Springer, Berlin (2003)
Acknowledgements
The research work was supported by the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), under the HFRI PhD Fellowship Grant (GA. no. 34254).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Koutsianitis, P.I., Tairidis, G.K., Drosopoulos, G.A. et al. Conventional and star-shaped auxetic materials for the creation of band gaps. Arch Appl Mech 89, 2545–2562 (2019). https://doi.org/10.1007/s00419-019-01594-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-019-01594-1