Homogenization and equivalent in-plane properties of two-dimensional periodic lattices

Abstract The equivalent in-plane properties for hexagonal and re-entrant (auxetic) lattices are investigated through the analysis of partial differential equations associated with their homogenized continuum models. The adopted homogenization technique interprets the discrete lattice equations according to a finite differences formalism, and it is applied in conjunction with the finite element description of the lattice unit cell. It therefore allows handling of structures with different levels of complexity and internal geometry within a general and compact framework, which can be easily implemented. The estimation of the mechanical properties is carried out through a comparison between the derived differential equations and appropriate elasticity models. Equivalent Young’s moduli, Poisson’s ratios and relative density are estimated and compared with analytical formulae available in the literature. In-plane wave propagation characteristics of honeycombs are also investigated to evaluate phase velocity variation in terms of frequency and direction of propagation. Comparisons are performed with the values obtained through the application of Bloch theorem for two-dimensional periodic structures, to show the accuracy of the technique and highlight limitations introduced by the long wavelength approximation associated with the homogenization technique.

[1]  Equivalent continuum models of large platelike lattice structures , 1994 .

[2]  A. Maewal.  Construction of models of dispersive elastodynamic behavior of periodic composites: A computational approach , 1986 .

[3]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[4]  J. Achenbach.  Wave propagation in elastic solids , 1962 .

[5]  Martin Ostoja-Starzewski,et al.  Lattice models in micromechanics , 2002 .

[6]  J. B. Park,et al.  Negative Poisson's ratio polymeric and metallic foams , 1988 .

[7]  Andrei V. Metrikine,et al.  Comparison of wave propagation characteristics of the Cosserat continuum model and corresponding discrete lattice models , 2001 .

[8]  Tadeusz Paszkiewicz,et al.  Physics of Phonons , 1987 .

[9]  N. Kikuchi,et al.  Plane harmonic wave propagation in three-dimensional composite media , 1999 .

[10]  J. Tichy,et al.  Dependence of ultrasonic propagation velocities and transit times on an electric biasing field in alpha quartz , 1983 .

[11]  P. Martinsson,et al.  Fast multiscale methods for lattice equations , 2002 .

[12]  Per-Gunnar Martinsson,et al.  VIBRATIONS OF LATTICE STRUCTURES AND PHONONIC BAND GAPS , 2003 .

[13]  G. P. Srivastava,et al.  The Physics of Phonons , 2019 .

[14]  Y. Ishibashi,et al.  Elastic properties of a two-dimensional model of crystals containing particles with rotational degrees of freedom , 2002 .

[15]  Jacob Fish,et al.  Higher-Order Homogenization of Initial/Boundary-Value Problem , 2001 .

[16]  Massimo Ruzzene,et al.  Analysis of in-plane wave propagation in hexagonal and re-entrant lattices , 2008 .

[17]  D. McDowell,et al.  Generalized continuum modeling of 2-D periodic cellular solids , 2004 .

[18]  Jaan Kiusalaas,et al.  Numerical Methods in Engineering , 2010 .

[19]  R. Cook,et al.  Concepts and Applications of Finite Element Analysis , 1974 .

[20]  Wen Chen,et al.  Nonlocal Dispersive Model For Wave Propagation In Heterogeneous Media . Part 1 : One-Dimensional Case , 2022 .

[21]  Richard M. Christensen,et al.  Mechanics of cellular and other low-density materials , 2000 .

[22]  Ole Sigmund,et al.  Systematic design of phononic band–gap materials and structures by topology optimization , 2003, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[23]  Massimo Ruzzene,et al.  Wave beaming effects in two-dimensional cellular structures , 2003 .

[24]  R. Lakes,et al.  Properties of a chiral honeycomb with a poisson's ratio of — 1 , 1997 .

[25]  Wen Chen,et al.  Nonlocal Dispersive Model For Wave Propagation In Heterogeneous Media . Part 2 : Multi-Dimensional Case , 2001 .

[26]  P. Cartraud,et al.  Continuum modeling of beamlike lattice trusses using averaging methods , 1999 .

[27]  M. G. Salvadori,et al.  Numerical methods in engineering , 1955 .

[28]  M. Ashby,et al.  Cellular solids: Structure & properties , 1988 .

[29]  Ahmed K. Noor,et al.  Continuum Modeling for Repetitive Lattice Structures , 1988 .

[30]  L. Brillouin.  Wave propagation in periodic structures : electric filters and crystal lattices , 1953 .

[31]  N. Fleck,et al.  Wave propagation in two-dimensional periodic lattices. , 2006, The Journal of the Acoustical Society of America.

[32]  K. P. Byrne,et al.  Determining the response of infinite, one-dimensional, non-uniform periodic structures by substructuring using waveshape coordinates , 2005 .

[33]  Gregory M. Hulbert,et al.  Band-Gap Engineering of Elastic Waveguides Using Periodic Materials , 2003 .

[34]  N. Kikuchi,et al.  An assumed strain method for the dispersive global–local modeling of periodic structures , 2001 .

[35]  M. Ruzzene,et al.  Analysis of Vibration and Wave Propagation in Cylindrical Grid-Like Structures , 2003 .

[36]  Jacob Fish,et al.  Non‐local dispersive model for wave propagation in heterogeneous media: one‐dimensional case , 2002 .