Materials and structures with negative Poisson’s ratio exhibit a counter-intuitive behaviour. Under uniaxial compression (tension), these materials and structures contract (expand) transversely. The materials and structures that possess this feature are also termed as ‘auxetics’. Many desirable properties resulting from this uncommon behaviour are reported. These superior properties offer auxetics broad potential applications in the fields of smart filters, sensors, medical devices and protective equipment. However, there are still challenging problems which impede a wider application of auxetic materials. This review paper mainly focuses on the relationships among structures, materials, properties and applications of auxetic metamaterials and structures. The previous works of auxetics are extensively reviewed, including different auxetic cellular models, naturally observed auxetic behaviour, different desirable properties of auxetics, and potential applications. In particular, metallic auxetic materials and a methodology for generating 3D metallic auxetic materials are reviewed in details. Although most of the literature mentions that auxetic materials possess superior properties, very few types of auxetic materials have been fabricated and implemented for practical applications. Here, the challenges and future work on the topic of auxetics are also presented to inspire prospective research work. This review article covers the most recent progress of auxetic metamaterials and auxetic structures. More importantly, several drawbacks of auxetics are also presented to caution researchers in the future study.

Material properties can be tailored through modification of their geometry or architecture. With this concept, a lot of smart materials, metamaterials, and smart structures have been developed. Auxetic materials and structures are a novel class of materials which exhibit an interesting property of negative Poisson's ratio. By virtue of the auxetic behavior, mechanical properties such as fracture toughness, indentation resistance, etc., can be improved. In order to exploit the interesting properties of auxetic materials, several potential applications of auxetic materials have been explored in medical, sports, automobile, defense, etc. Design and modeling of novel auxetic materials and structures is still on the way. Here, the article focuses upon the different aspects of auxetic materials and structures. A comprehensive updated review of auxetic materials, their types and properties, and applications has been presented. This paper also discusses the design and modeling approaches of auxetic structures.

We present results from an experiment similar to one performed by Packard 23], in which a genetic algorithm is used to evolve cellular automata (CA) to perform a particular computational task. Packard examined the frequency of evolved CA rules as a function of Langton's parameter 16], and interpreted the results of his experiment as giving evidence for the following two hypotheses: (1) CA rules able to perform complex computations are most likely to be found near \critical" values, which have been claimed to correlate with a phase transition between ordered and chaotic behavioral regimes for CA; (2) When CA rules are evolved to perform a complex computation, evolution will tend to select rules with values close to the critical values. Our experiment produced very di erent results, and we suggest that the interpretation of the original results is not correct. We also review and discuss issues related to , dynamical-behavior classes, and computation in CA.The main constructive results of our study are identifying the emergence and competition of computational strategies and analyzing the central role of symmetries in an evolutionary system. In particular, we demonstrate how symmetry breaking can impede the evolution toward higher computational capability. are referred to as the \output bits" of the rule table. To run the CA, this look-up table is applied to each neighborhood in the current lattice con guration, respecting the choice of boundary conditions, to produce the con guration at the next time step.A common method for examining the behavior of a two-state one-dimensional CA is to display its space-time diagram, a two-dimensional picture that vertically strings together the one-dimensional CA lattice con gurations at each successive time step, with white squares corresponding to cells in state 0, and black squares corresponding to cells in state 1. Two such space-time diagrams are displayed in Figure 2. These show the actions of the Gacs-Kurdyumov-Levin (GKL) binary-state CA on two random initial con gurations of di erent densities of 1's 5, 7]. In both cases, over time the CA relaxes to a xed pattern|in one case, all 0's, and in the other case, all 1's. These patterns are, in fact, xed points of the GKL CA. That is, once reached, further applications of the CA do not change the pattern. The GKL CA will be discussed further below.CA are of interest as models of physical processes because, like many physical systems, they consist of a large number of simple components (cells) which are modi ed only by local interactions, but which acting together can produce global complex behavior. Like the class of dissipative dynamical systems, even the class of elementary one-dimensional CA exhibit the full spectrum of dynamical behavior: from xed points, as seen in Figure 2, to limit cycles (periodic behavior) to unpredictable (\chaotic") behavior. Wolfram considered a coarse classi cation of CA behavior in terms of these categories. He proposed the following four classes with the intention of capturing all possible CA be...

Please cite this article as: Chen, Y., Das, R., Battley, M., Effects of cell size and cell wall thickness variations on the stiffness of closed-cell foams, International Journal of Solids and Structures (2014), doi: http://dx.Abstract This paper concerns with the micromechanical modelling of closed-cell polymeric foams (M130) using Laguerre tessellation models incorporated with realistic foam cell size and cell wall thickness distributions. The cell size and cell wall thickness distributions of the foam were measured from microscope images. The Young's modulus of cell wall material of the foam was characterized by nanoindentation tests. It is found that when the cell size and cell wall thickness are assumed to be uniform in the models, the Kelvin, Weaire-Phelan and Laguerre models overpredict the stiffness of the foam. However, the Young's modulus and shear modulus predicted by the Laguerre models incorporating measured foam cell size and cell wall thickness distributions agree well with the experimental data. This emphasizes the fact that the integration of realistic cell wall and cell size variations is vital for foam modelling. Subsequently the effects of cell size and cell wall thickness variations on the stiffness of closed-cell foams were investigated using Laguerre models. It is found that the Young's modulus and shear modulus decrease with increasing cell size and cell wall thickness variations. The degree of stiffness variation of closed-cell foams resulting from the cell size dispersion and cell wall thickness dispersion are comparable. There is little interaction between the cell size variation and cell wall thickness variation as far as their effects on foam moduli are concerned. Based on the simulation results, expressions incorporating cell size and cell wall thickness variations were formulated for predicting the stiffness of closed-cell foams. Lastly, a simple spring system model was proposed to explain the effects of cell size and cell wall thickness variations on the stiffness of cellular structures.

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