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This special section in physica status solidi (b) is the ninth one dedicated to systems exhibiting an anomalous (negative) Poisson's ratio (PR), called auxetics or partial auxetics, as well as to other mechanical metamaterials and related systems [1]. Recently, such materials and models have attracted significant interest due to their superior properties with respect to common systems [1].

The first three papers in this section concern auxetic textiles which have represented a growing class of auxetic materials in recent years.

Glazzard and Breedon [2] report on the design, fabrication and testing of an auxetic weft-knit fabric. The design approach combines subjective consideration of aesthetic and haptic aspects together with the objective consideration of technical issues in fabric production. The developed fabric is a combination of relief structure and transfer stitching to create a folded chevron-like material based on the double arrowhead geometry previously reported to give rise to auxetic behaviour. The authors present an interesting discussion of issues related to transferring auxetic fabrics to commercial applications and to disseminating the technology to non-specialist communities.

Lim [3] proposes a concept for a semi-auxetic yarn which builds on earlier work by Wright and co-workers who developed an auxetic yarn by helically wrapping a stiff thin yarn around a compliant thick one. In Lim's concept the thin stiff yarn is stitched in a zig-zag pattern through the thick compliant yarn in a given plane. In other words, the stiff element is incorporated into the compliant element rather than wrapped around it. Analytical model expressions are derived and used to explore the anisotropic Poisson's ratio response (including conventional and auxetic planes) as functions of zig-zag angle and PR of the compliant component. The concept is then discussed in terms of alternative stiff yarn configurations and practical considerations for the production of semi-auxetic yarns.

Wang and Hu [4] report the development of 3D spacer fabrics having in-plane auxetic behaviour achieved through incorporating auxetic warp knit face skin components. The effects of fabric structure geometry and in-plane loading direction are explored in detail. The fabrics are auxetic when stretched along the major and diagonal in-plane loading directions, and display anisotropy with the largest magnitude of negative PR occurring for stretching along the weft direction. The PR is strain dependent and transforms from negative to positive values at sufficiently high strains. The authors also undertake repeated loading tests to investigate the stability of the auxetic effect and show that the magnitude of the negative PR decreases from the initial (first load cylcle) value, but is approximately constant after the second cycle up to 10 cycles of loading. The auxetic spacer fabric is shown to have improved conformability to a spherical surface compared to a conventional spacer fabric, illustrating the potential to use these materials in apparel where conformability to the doubly-curved human form is required.

The following paper represents an alternative fibrous material to textile fabrics and yarns. Verma, Shofner and Griffin [5] undertake microstructural and mechanical characterisation of a range of commercially available papers. In particular they measure the out-of-plane thickness variations in response to tensile loading along the in-plane paper machine direction. The papers show different thickness responses, including auxetic response in several of the papers tested, indicating the PR is likely to be a complex interplay between fibrous material geometry, fibre length and surface chemistry. Based on microstructural observations, the authors develop a simple idealised geometrical model which provides transverse strain vs axial strain predictions in qualitative agreement with the experimental data. The model provides a starting point in understanding how fibre source and paper processing conditions can influence the production of paper with specific PR response.

The next three studies are related to foams.

The work by Chetcuti, Ellul, Manicaro et al. [6] presents a highly realistic study aimed at assessing how the auxetic potential of real non-perfectly rigid rotating systems decreases with an increase in the flexibility of the rotating units. This study has important implications in the studies related to auxetic foams where the auxeticity arises, at least to some extent, due to rotation of the joints, which essentially behave as rotating semi-rigid units.

Lim, Alderson and Alderson [7] report results from comparative impact testing of auxetic and conventional polyurethane foams. Under the highest impact energy investigated, the auxetic foam showed visible signs of increased foam damage, contrary to the expected outcome based on localised densification arguments. Based on a photomicroscopy examination of foam structure the authors suggest an explanation of the observed impact damage.

The paper by Lisiecki, Kłysz, Błażejewicz and co-workers [8] reports results from tomography and compression testing of polyurethane foam material. Characterisation is performed before (tomography) and after (tomography and compression testing) application of a “hybrid mechanical-chemical-thermal (M-Ch-T) process”. The link between level of compression applied during the conversion process, density of foam structure, and mechanical response is explored, including spatial variations in structure and properties associated with the conversion process.

The following six papers are related to smart structures.

Gatt, Caruana-Gauci, Attard et al. [9] investigate differences between infinite and finite auxetic structures having planar and tubular configurations. They show that although the PRs of the studied infinite and finite systems may be the same, the Young's moduli for finite systems may be overestimated if calculated through analytical models based on infinite systems due to edge effects. Correction factors that are dependent on the size of the systems are derived for the Young's moduli, to take these edge effects into account. These findings may have implications in the design of various real systems which may range from smart filters to oesophageal stents and other biomedical devices.

Mizzi, Attard, Casha and co-workers [10] analyze the potential of honeycomb geometries for stent designs, employing Finite Element modelling. This work highlights some of the challenges faced by scientists in their quest to create an ideal geometric stent design. As shown in this work, although some designs, such as the hybrid honeycomb, may eliminate certain problems such as foreshortening, these advantages may be overshadowed by other undesirable effects, such as dogboning and lack of conformability of the stent.

The work by Ribeiro Filho, Silva, Brandão et al. [11] concerns a novel solid composite based on recycled rubber. The tensile and failure behaviour of multi re-entrant honeycomb topologies consisting of width, thickness and angle variations was studied experimentally. Finite element (FE) models were also developed to simulate tensile tests and failure behaviour of the cellular structures. The experimental and FE results showed a good correlation, validating the numerical model which was used to perform a parametric analysis on the influence of the cell geometry over the failure behaviour of the structures. It was found that the structures with lower PR values were able to absorb more energy, and consequently, exhibiting less points of failure. It was also found that both the thickness and the width have a significant effect on the tensile properties and failure behaviour.

In the paper by Rodriguez, Kalathur and Lakes [12] lattice structures based on bimorph rib elements are fabricated and studied experimentally. The authors find that the effective piezoelectric sensitivity is much larger, by a factor of at least 10,000, in magnitude than that of material comprising the lattice ribs. This interesting result of very large sensitivity is explained by bending of the ribs in response to input voltage.

Strek, Jopek, Maruszewski et al. [13] have analysed stiffness of sandwich-structured composites consisting of two different materials: auxetic and structural steel. The authors conclude that maximally stiff structures have, in general, complicated geometries while minimum value of stiffness is obtained when the structure of the composite is a flat laminate.

Influence of disorder on PR is an important subject not only from the point of view of basic research but also for applications. Using Finite Element computer simulations, Pozniak and Wojciechowski [14] studied PR for anti-chiral structures built on rectangular lattices with disorder introduced by stochastic distributions of circular node sizes. Such structures are of interest as, in the case of sufficiently large anisotropy, they show extremely negative PR. The investigated models were parameterized by the lattice anisotropy, the rib and node thickness, and the radii distribution of circular nodes. The simulations have shown that thin ribs and thin-walled circular nodes favour low values of PR. In the case of thick ribs and thick-walled circular nodes PR is essentially higher. In both cases the dispersion of the values of circular nodes radii has a minor effect on the lowest values of PR. Comparison of an exact approach, using only planar elements, with Timoshenko beam based approximations, showed that the latter are valid only in the very thin rib limit, with the approximation error growing with increasing thickness.

Two further papers concern microscopic mechanisms of auxeticity.

The work by Grima, Zerafa and Brincat [15] discusses frameworks which may exhibit auxetic behaviour through a mechanism involving rotating semi-rigid triangular units, where the magnitude of the PR may be controlled through the shape of the network.

Tretiakov and Wojciechowski [16] used Monte Carlo simulations to investigate elastic properties of the face centered cubic (fcc) phase of particles, interacting through hard-core repulsive Yukawa potential. Effects of the Debye screening length and contact value of the potential on elastic properties of the system were studied. The authors have found that the hard-core repulsive Yukawa system is partially auxetic, i.e. exhibits negative PR in some directions.

This section is closed by two papers discussing dynamic properties of materials exhibiting anomalous mechanical properties.

The potential to optimise control of elastic wave propagation through tailoring the PR of isotropic materials is explored by Lim, Cheang and Scarpa [17]. Expressions relating wave velocity to elastic properties and density for isotropic solids are developed for longitudinal waves in prismatic bars, plane waves of dilatation, plane waves of distortion, and surface waves. The velocity of plane waves of dilatation is the highest over all allowable values of PR, and the velocity of plane waves of distortion always exceeds the velocity of surface waves. The velocity of longitudinal waves in prismatic bars shows the most dramatic variation with PR, equalling the velocity of plane waves of dilatation when ν = 0, and being second highest for all other values of PR when –0.5 < ν < +0.5; dropping to third highest when –0.733 < ν < –0.5, and being lowest when –1 < ν < –0.733.

A theoretical investigation into the stability of the static and dynamic bulk moduli of a simple two-phase system of a spherical core surrounded by an outer shell layer is reported by Wojnar and Kochmann [18]. This simple system is suggested to provide insight into whether or not extreme effective moduli due to the existence of a negative stiffness phase, previously found experimentally in dynamic tests on such systems, can be stabilised in the static regime. The results show that stable extreme values of the static effective bulk modulus cannot be achieved for this system, but that extreme values of the dynamic effective bulk modulus can be. Further the resonant-like extreme dynamic stiffnesses are shown to vary with the mechanical, mass and geometrical parameters of the constituents and to occur at frequencies significantly lower than those of positive stiffness constituent combinations.

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 Sheffield Hallam University, the University of Malta, the Malta Council for Science and Technology, the Institute of Molecular Physics of the Polish Academy of Sciences in Poznan, and the Poznan Supercomputing and Networking Center.

  • Kim L. Alderson1, Andrew Alderson2, Joseph N. Grima3, Krzysztof W. Wojciechowski4

  • 1 The Open University, UK

  • 2 Materials and Engineering Research Institute, Sheffield Hallam University, UK

  • 3 Faculty of Science, University of Malta

  • 4 Institute of Molecular Physics, Polish Academy of Sciences, Poznan

  • Kim L. Alderson (The Open University, UK)

  • Andrew Alderson (Materials and Engineering Research Institute, Sheffield Hallam University, UK)

  • Joseph N. Grima (Faculty of Science, University of Malta)

  • Krzysztof W. Wojciechowski (Institute of Molecular Physics, Polish Academy of Sciences, Poznan)

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