Auxeticity of cubic materials under pressure

The global maximum and global minimum Poisson's ratio (PR) surfaces, regions of different auxetic behaviour and the domains of the different extreme directions of cubic materials are shown and discussed in terms of the elastic moduli ratios, $X = G/K,Y = G/W$, where K is the bulk modulus and $G,W$ are shear moduli. A straightforward way is given to classify any cubic material as being auxetic, nonauxetic and partially auxetic, as well as calculating its extreme PR values. A decomposition of the XY plane with the elastic constant ratio ${\lambda} = C_{12} /C_{44} $ is introduced to facilitate the interpretation of cubic materials under hydrostatic pressure, $P> 0$, and isotropic tension, $P< 0$, which are two qualitatively different situations. Using this representation it is demonstrated that cubic materials with ${\lambda} > 0$ can be auxetic only under ‘negative’ pressure ($P< 0$). It is shown that the influence of pressure on the auxetic behaviour is different for the positive and negative ${\lambda} $ cases. In particular, a cubic material with $0> {\lambda} > {-} 1$ may be auxetic at negative as well as positive pressure. The work demonstrates the crucial role of P in obtaining desired auxetic behaviour. Microscopic mechanisms which tune the cubic system to targeted regions in the XY plane are investigated with a fcc crystal of particles interacting via the pairwise spherically symmetric potential, ${\phi} (r)$. It is found that all fcc static models in which the range of interaction is only between nearest neighbour particles are placed on a universal curve in the XY plane. It is shown that auxetic behaviours can be achieved with simple static ${\phi} (r)$ analytic forms. The influence of the next neighbour interactions on the universal curve is also considered. In order to assess the role of thermal fluctuations, molecular dynamic simulations for two different ${\phi} (r)$ (the Lennard–Jones (LJ) and the tethered particle potentials) were performed. At low temperatures the studied systems are well represented by the universal curve and on increasing temperature a systematic departure from it is observed.