Discussion of an auxetic model possessing a complete bandgap at Auxetics 2007 @ Malta, Mon 24 - Wed 26 September 200

J.N. Grima, A.C. Sparavigna

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[Other] Phonons in a two-dimensional auxetic lattice
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J.N. Grima, A.C. Sparavigna (2007). Phonons in a two-dimensional auxetic lattice. .
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 1 
Auxetics 2007 @ Malta, Mon 24 - Wed 26 September 2007 
 
Phonons in a two-dimensional auxetic lattice 
J.N. Grima
1
 and A. Sparavigna
2,* 
 
1
 Department of Chemistry, University of Malta, Msida MSD 2080, Malta 
2
 Dipartimento di Fisica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy. E-
mail: amelia.sparavigna@polito.it 
 
PACS:  62.20, 63.20 
Auxetics are materials and structures displaying a negative Poisson's ratio, meaning that 
they exhibit a lateral extension, instead of shrinking, when they are stretched. Although in 
the past two decades there has been considerable developments on systems exhibiting 
auxetic behaviour, the studies of the modes of vibrations in auxetic structures is still in its 
infancy. 
Here we aim starting a discussion on models of two-dimensional lattices, in which the 
lattice is represented by a planar network where sites are connected by strings and rigid 
rods, in order to obtain an auxetic structure.  In particular, we discuss a model based on the 
recently proposed ‘rotating squares’ structure [1-3], see Fig. 1(a). We show that since the 
proposed lattice has a basis, translational and rotational mode dispersions are observed 
displaying a complete bandgap when the ratio of masses in the lattice basis is different from 
one (Fig.1(b),(c)). In analogy with the behaviour of crystalline lattices, the acoustic mode 
velocity is strongly reduced when the mass ratio increases.  
 
References: 
 
[1] J.N. Grima, A. Alderson and K.E. Evans, Zeolites with negative Poisson's 
ratios, Paper presented at the 4th Materials Chemistry conference, Dublin (Ireland), 
July 1999 (P81). 
 
[2] J.N. Grima and K.E. Evans, J. Mat. Sci. Lett., 19 (2000) p.1563-1565. 
 
[3] Y. Ishisbashi and M. Iwata, J. Phys. Soc. Jap., 69 (2000) 2702. 
 
(*) Author for correspondence 
 2 
Auxetics 2007 @ Malta, Mon 24 - Wed 26 September 2007 
 
(a) 
 
(b) (c) 
 
Fig. 1(a): The lattice discussed here where masses are connected with strings; (b) the 
reduced mode frequencies as a function of wavenumbers along directions X,Y and D, when 
the units of the lattice basis have the same mass, and (c) the reduced mode frequencies as a 
function of  wavenumbers, when the units of the lattice have different masses (M1=4Mo). 
Note the complete bandgap between translational and rotational modes and the lower group 
velocity of translational (acoustic) modes. 


Phonons in a two-dimensional auxetic lattice

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