This special section in physica status solidi (b) is the eighth one [1-7] dedicated to auxetics (defined as materials showing negative Poisson's ratio (NPR) in all directions), partial auxetics (for which Poisson's ratio is negative in some directions only), mechanical metamaterials and related systems. Due to the various superior properties exhibited by these systems, which show counter-intuitive so-called ‘negative’ behaviour, their study and characterisation has attracted significant interest reflected in an increasing number of publications and patents in these areas, particularly in the recent few years.
As in the previous special issues [1-7] the main emphasis in this issue is put on systems exhibiting NPR. They are discussed in the first ten papers [8-17]. (To remind, NPR, usually denoted as ναβ in a general case of an anisotropic material, means that when stretched in direction α the material becomes fatter (not thinner) in direction β (transverse to α) in which the response is measured.) The remaining six papers concern negative thermal expansion (NTE) [18-20], negative stiffness (NS) , ‘negative’ magnetic properties , and graphene .
The thematic issue starts with the Feature Article by Robert Critchley et al.  who review auxetic foams. As they write, this is due to the low price, easy availability, desirable mechanical properties and enhanced properties, such as density, stiffness, fracture toughness and dampening of such materials. Auxetic foams constitute a modern class of materials fabricated by altering the material microstructure. The key areas discussed include the fabrication method and the effects played by different parameters (temperature, heating time, cell shape and size and volumetric compression ratio), microstructural models, mechanical properties and potential applications. Auxetic foams are also the subject of two following papers. The first of them concerns experiments performed on Cu foams. By applying a sequence of permanent tri-axial compression deformations Dong Li et al.  accomplished transformation of such foams into the re-entrant (auxetic) ones. Poisson's ratios of the re-entrant foams were then studied by resonant ultrasound spectroscopy. It has been found that Poisson's ratio first decreases and then increases with the increasing compression strain. A minimum in Poisson's ratio of approximately –0.7 was achieved for all initial densities. However, foams with higher values of the initial density attained minima in Poisson's ratio at lower permanent volumetric compression. Studies of the shear modulus showed that it increased with the increasing volumetric compression ratio, revealing a small hump near the point corresponding to the Poisson's ratio minimum. In the second paper, also related to an experiment, J. Lisiecki et al.  describe the manufacturing and testing of auxetic polyurethane foams for potential application as seats in military helicopters. In the first part of their paper they sketch the manufacturing process which they developed to convert conventional foams into auxetic ones. In the second part of the paper, some static and dynamic tests are described which were used to demonstrate advantages of the obtained auxetic specimens with respect to conventional foams available on the market and used in production of standard seats.
The next three papers concern macroscopic structures exhibiting NPR. Zhengkai Zhang et al.  propose a partially auxetic structure made of tubes and corrugated sheets using a conventional technology. They investigate auxetic properties of the structure by both geometrical analysis and the Finite Element (FE) method. For validating the analyses, a real auxetic structure is made by them of aluminium and tested under compression conditions. They show that the investigated structure has a very significant NPR effect under compression. Zhang et al. expect that, because of low production costs, their structure will find a wide range of applications in industrial fields. Reuben Cauchi et al.  investigate centro-symmetric cellular honeycombs at the transition between the convex and re-entrant (auxetic) form, i.e. when the joints form a T-shaped junction. They re-examine the mechanical properties of such honeycombs using FE simulations. The simulations show that, contrary to the current understanding, such honeycombs may deform in a manner which results in bending of the horizontal ligaments upon loading in the horizontal direction, with the result that such honeycombs may exhibit non-zero Poisson's ratio values. This effect may be employed to design honeycombs with negative Poisson's ratios. Analytical and FE methods are applied by Ruben Gatt et al.  to study chiral systems. Such systems may exhibit an anomalous (negative) Poisson's ratio. Recently, these structures have aroused interest due to their remarkable mechanical properties and numerous potential applications. The obtained results suggest that the Poisson's ratio is highly dependent on the structure geometry and material properties of the constituent materials. It is also shown that the rigidity of an anti-tetrachiral system can be changed without altering the Poisson's ratio.
Microscopic modelling of materials by computer simulations is presented in the following two papers. Elastic properties of fcc phases of soft polydisperse spheres interacting through n-inverse-power-potential are studied by Konstantin V. Tretiakov and Krzysztof W. Wojciechowski . Monte Carlo simulations show that both the size polydispersity δ and temperature T strongly influence the elastic properties of the studied system. Poisson's ratio is found to be negative and decreasing with an increase in the temperature and size polydispersity in the  direction. The obtained results show that a wide range of Poisson's ratio ναβ values (from ν ≈ –0.3 to ν ≈ 1.1) can be obtained by changing the system polydispersity, potential softness and sample orientation. Joseph N. Grima and Christine Zerafa  describe numerical studies of the effect of solvent molecules on structural and mechanical properties of porous polyphenylacetylene auxetic organic networked polymers. The simulations indicate that the presence of solvent molecules within the molecular framework of the networked polymers has a significant effect on their density and stiffness but does not influence the Poisson's ratio of the material. The results of this paper may have important implications in future synthesis of such systems.
Using a single dimensionless complex parameter, expressed by the elastic compliances of cubic crystals, Robert V. Goldstein et al.  propose a new classification of crystalline cubic auxetics. After a brief analysis of the equivalence of their approach with other known classifications, they present some examples of the surface separating regions with negative and positive Poisson's ratio for partially auxetic materials. They also show how the character of this surface changes with changing the dimensionless complex parameter and find a critical value of this parameter for which a topological rearrangement of the ‘separating surface’ occurs.
The ‘auxetic part’ of this special section is closed by the paper by B. T. Maruszewski et al.  who deal with a nonlinear elastic solid of NPR which dissipates energy during thermoelastic damping. Numerical simulations of a model based, inter alia, on Murnaghan's free energy show some unconventional effects occurring in such materials.
Three further papers describe models of negative thermal expansion (NTE). Joseph N. Grima et al.  present the concept of reducing the thermal expansion of composites by exploiting the Poisson's effect. The FE simulation was used by them to show that systems made of needle, cylindrical or coin shaped inclusions embedded in a soft matrix may exhibit NTE under certain conditions. The conditions concern both the matrix, which should have a highly positive Poisson's ratio and a low coefficient of thermal expansion, and inclusions, which should be hard and exhibit a highly positive thermal expansion coefficient. In the case of coin-shaped inclusions, the NTE is at a maximum in the direction orthogonal to the surface of the coins. For systems with needle-shaped inclusions, the NTE is at a maximum in the direction orthogonal to the length of the needles. The subject of negative NTE is also the topic of the paper by Brian Ellul and Joseph N. Grima  who present a detailed analytical model of a composite system which enables control of the thermal expansion through the use of the earlier mentioned Poisson's effect. It follows from the presented analysis that the model can predict the thermal expansion of such systems to an extent comparable to more complex FE simulations. The proposed model also permits optimisation of the system to exhibit maximum negative thermal expansion, or, thermal expansion of a desired magnitude. NTE trusses studied in the paper by Teik-Cheng Lim  exhibit a special case of anisotropy known as transverse isotropy. Conditions for attaining negative volumetric strain and negative coefficients of volumetric expansion due to change in temperature are established by him for these models.
Stability of viscoelastic composites with negative stiffness (NS) inclusions and their associated extreme mechanical properties are studied by Yun-Che Wang and Chih-Chin Ko  with time-domain Finite Element (FE) analysis and composite theory. Effective stiffness and damping anomalies are observed in two-dimensional (2D) and three-dimensional (3D) models of two-phase composites, and the stability of the composites is evaluated. The analysis shows that the extreme mechanical properties for both of the 2D and 3D cases are located near the strong ellipticity boundary. When the matrix modulus is suitably chosen, the effective damping enhancement may be reached in the stability range.
‘Negative’ magnetic properties are the subject of the following paper in which L. M. Thu et al.  study the orbital diamagnetic response of three-dimensional (3D) arrays of embedded InAs/GaAs wobbled nano-rings. Using the effective one-band Hamiltonian and smooth 3D confinement potential that is mapping the actual strain and material content inside the rings they obtain magnetic susceptibility of an individual nano-ring. Then, using the Clausius–Mossotti relation they estimate the effective susceptibility of 3D arrays of the rings. It follows from their study that conventionally diamagnetic InAs/GaAs ring structures under certain conditions can demonstrate a positive peak of the effective magnetic susceptibility of the arrays, which they call ‘negative’-diamagnetic response. That ‘negative’-diamagnetic peak remains Lorentz-like shaped and gradually disappears when the rings' concentration in the arrays decreases.
The last paper in this special section concerns properties of graphene. Using a molecular mechanics approach based on an atomistic FE formulation, Kenneth Kam et al.  studied the out-of-plane mechanical bending properties of single layer graphene sheets (SLGS). Force/displacement curves for different rectangular SLGS with different aspect ratios were obtained by them for distributed (uniform pressure) and concentrated central loadings. They show that membrane and bending deformations scale differently based on the type of load, as well as geometry of the graphene sensor films.
We thank all the contributors of this thematic issue for submitting their papers and some of them for their patience in waiting until this issue has been completed. We are grateful to all the reviewers for valuable comments. Finally, we acknowledge the support from the Institute of Molecular Physics of the Polish Academy of Sciences in Poznań, the Poznań Supercomputing and Networking Center, PWSZ im. Prezydenta Stanisława Wojciechowskiego in Kalisz, the University of Malta, and the Malta Council for Science and Technology.
Krzysztof W. Wojciechowski,
Institute of Molecular Physics, Polish Academy of Sciences, Poznań, Poland
Joseph N. Grima,
Faculty of Science, University of Malta, Msida, Malta
Kim L. Alderson,
University of Bolton, Bolton, UK
Gdańsk University of Technology, Gdańsk, Poland