Auxetic Materials and Structures

Auxetic materials are solids that possess negative Poisson’s ratio. This chapter introduces the reader to the definition of Poisson’s ratio and its historical development. Thereafter the definition and historical development of auxetic materials are given. Both naturally occurring as well as man-made auxetic materials are introduced—the former in terms of α-cristobalite and the latter in terms of foams and yarns.

This chapter provides a survey of micromechanical models that seek to predict and explain auxetic behavior, based on re-entrant microstructures, nodule-fibril microstructure, 3D tethered-nodule model, rotating squares, rectangles, triangles and tetrahedrals models, hard cyclic hexamers model, missing rib models, chiral and anti-chiral models, interlocking hexagon model, and the “egg rack” model. All the micromechanical models exhibit a common trait—auxeticity is highly dependent on the microstructural geometry. In some of the micromechanical geometries, comparisons between analytical results have been made with experimental or computational results.

Fundamental behavior of auxetic solids is laid down in terms of linear anisotropic constitutive relationship, followed by the derivation of Poisson’s ratio bounds for isotopic solids in 3D and 2D cases. Increasing simplifications are then imposed on the compliance matrices of the complete anisotropic solid until linear isotropic case is obtained, whereby special trends are observed for Poisson’s ratio of −1, −2/3, −1/2 and 0, followed by distinct moduli ratio that separates auxetic solids from conventional ones. Thereafter the chapter explores large elastic deformation, anisotropic crystals, elastoplasticity and viscoelasticity of auxetic media.

This chapter considers the damage properties of auxetic solids. In the study of stress concentration factors in auxetic solids and plates arising from cavities and rigid inclusions, most cases exhibit minimum stress concentration when the solids possess negative Poisson’s ratio. In discussing the three modes of fracture in auxetic solids in dimensionless terms, most plots exhibit a clear demarcation between auxetic and conventional regions. The consideration of damage criteria based on thermodynamic analysis by Lemaitre and Baptiste (NSF workshop on mechanics of damage and fracture, 1982) shows that as an isotropic solid changes from conventional to auxetic, the damage criterion shifts from being highly dependent on the von Mises equivalent stress to being highly dependent on the hydrostatic stress. Progress on fatigue failure of auxetic materials is then given.

Contact mechanics and its accompanying description of stress and displacement fields is of great practical importance. This chapter begins by identifying the uniqueness of line and point contacts on auxetic half-space in comparison to conventional ones in terms of displacement and stress fields. This is then followed by a study on the influence of indenter shape on auxetic materials. In the contact between two isotropic elastic spheres, the tangential-to-normal compliance ratio is least when both spheres possess Poisson’s ratio of −1. A summary of works on contact of auxetic composites and indentation of auxetic foams are then furnished.

The uniqueness of auxetic beams, in comparison to conventional ones, is explored in this chapter in terms of axial deformation of prismatic bars, cantilever bending of beams, and torsion of rods. In axial deformation of prismatic bars, it is herein shown that the change in density is more drastic in auxetic solids than in conventional ones. The cantilever bending is herein considered for circular cross sections, as well as rectangular cross sections of various aspect ratios. Emphasis is given to the loci of conventional, moderately auxetic and highly auxetic regions in the analyses. For the special case of twisting of a beam in which one cross section remains plane, there is no diminution of the beam length when the Poisson’s ratio of the beam material is −1.

This chapter considers auxetic solids in such a form that is best analyzed using polar or spherical coordinate system. Specifically, this chapter considers the effect of auxeticity on stresses in rotating disks—both thin and thick—as well as the stresses in thick-walled cylinders and thick-walled spheres arising from internal and external pressure. Plotted results suggest that auxetic materials are advantageous over conventional ones for use as internally pressurized thick-walled cylinders, but inferior in comparison to conventional ones for application as thick-walled cylinders under external pressure. In the case of rotating thin disks with Poisson’s ratio of −1/3, the circumferential stress is independent of the radial distance, i.e. uniform throughout the entire disk, but not so for rotating disks with central hole. The maximum stress in a thin solid disk, as well as both the maximum radial and circumferential stresses in a thin disk with a central hole, decreases linearly as the Poisson’s ratio becomes more negative. Similar to thin rotating disks, the circumferential stress in thick rotating disks is independent from the radial distance when the Poisson’s ratio is −1/3; unlike thin disks, this circumferential stress is not uniform throughout the entire disk, as it varies along the disk thickness. In addition, the radial and circumferential stresses in a thick rotating disk is independent from the through thickness direction when the Poisson’s ratio is either 0 or −1. In the case of thick-walled spheres, the radial displacement is inversely proportional to the square of the radial distance if the Poisson’s ratio is 0.5, but varies linearly to the radial distance if the Poisson’s ratio is −1.

This chapter opens with a discussion on flexural rigidity of auxetic plates vis-à-vis conventional ones, followed by an analysis of circular auxetic plates. Bending moment result from uniform loading of circular plates suggests that the optimal Poisson’s ratio is −1/3 if the plate is simply-supported at the edge. Based on bending and twisting moment minimization on a rectangular plate under sinusoidal load, the optimal Poisson’s ratio for a square plate is 0, and this value reduces until −1 for a rectangular plate with aspect ratio 1 + √2. Auxetic materials are not suitable for uniformly loaded and simply supported square plates, as moment minimization study suggests an optimal Poisson’s ratio of 0.115, but are highly suitable for central point loaded and simply supported square plates, as moment minimization study suggests an optimal Poisson’s ratio of −1. In the study of auxetic plates on auxetic foundation, the plotted results suggest that, in addition to selecting materials of sufficient strength and mechanical designing of plate for reduced stressed concentration, the use of plate and/or foundation materials with negative Poisson’s ratio is useful for designing against failure. The investigations on width-constrained plates under uniaxial in-plane pressure by Strek et al. (J Non-Cryst Solids 354(35–39):4475–4480, 2008) and Pozniak et al. (Rev Adv Mat Sci 23(2):169–174, 2010) exhibit a remarkable and surprising result—at extreme negative Poisson’s ratios the displacement vector has components which are anti-parallel to the direction of loading. In the study of spherical shells under uniform load, the use of auxetic material reduces the ratio of maximum bending stress to the membrane stress, thereby implying that if the shell material possesses a Poisson’s ratio that is sufficiently negative, such as −1, and the boundary condition permits free rotation and lateral displacement, then the use of membrane theory of shell is sufficient even though the shell thickness is significant. Results also recommend the use of auxetic material for spherical shells with simple supports because the bending stress is significantly reduced. However the use of auxetic material as spherical shells, with built-in edge, is not recommended due to the sharp increase in the bending stress as the Poisson’s ratio of the shell material becomes more negative.

This chapter begins with some general considerations on the thermoelasticity of auxetic solids, followed by the thermal elasticity of 3D auxetic solids with geometrical constraints. Thereafter, the thermoelasticity of beams and plates arising from a given temperature profile is furnished. Based on a set of dimensionless thermal stresses for application in auxetic plates and shells, it was found that thermal stresses reduce as the material becomes more auxetic at constant Young’s modulus (E), and constant shear modulus (G), but the thermal stresses increase as the material becomes more auxetic at constant bulk modulus (K). In the case of constant product of EGK, the thermal stress is maximum at Poisson’s ratio of 0.303, but diminishes at Poisson’s ratios of −1 and 0.5. In most cases of solids considered in this chapter, the thermal stresses are minimized in the auxetic region. Finally a summary of thermal conductivity study in multi-re-entrant honeycombs by Innocenti and Scarpa (J Compos Mater 43(21):2419–2439, 2009) is given, in which the results suggest that auxetic honeycomb configurations exhibit higher out-of-plane conductivity, strong in-plane thermal anisotropy, and the lowest peak temperatures during heat transfer between the bottom and top faces of honeycomb panels.

This chapter lays down the foundation for elastic stability of columns, circular plates, rectangular plates, cylindrical shells and spherical shells that possess negative Poisson’s ratio. Results show that the plate Poisson’s ratio has no effect on the elastic stability of rectangular plates under in-plane biaxial loadings when the critical buckling load is expressed in terms of plate flexural rigidity, but the Poisson’s ratio plays a greater role for circular plate buckling. In the elastic stability study of spherical shells, the critical buckling stress is directly proportional to the shell thickness for Poisson’s ratio of 0 and proportional to the square of the shell thickness as the Poisson’s ratio approaches −1. Thereafter a summary of results by Miller et al. (Compos Sci Technol 70:1049–1056, 2010) for flatwise buckling optimization of hexachiral and tetrachiral honeycombs is furnished. Finally examples are given for square plates with array of perforations such that imposition of uniaxial compressive buckling leads to 2D auxeticity.

Vibration study is of great practical importance, as vibration of continuous systems with constraints implies cyclic stresses and the inevitable fatigue damage. This chapter on vibration forms the first part of elastodynamics of auxetic solids, with special emphasis on plates (both circular and rectangular) as well as shells (both cylindrical and spherical). For circular plates with free and simply supported edges, the frequency parameter changes more rapidly in the auxetic region than in the conventional region; consequently the natural vibration frequencies of these plates can be effectively reduced by choosing auxetic materials. For rectangular plates, the effect of negative Poisson’s ratio is evaluated for plates with all four sides and two sides being simply supported, as well as examples of rectangular plates with three sides being simply supported. In the case of cylindrical shells with simply supported edges, the results of frequency study imply that, when expressed in terms of flexural rigidity, the frequency is independent from the cylindrical shell radius at extreme auxetic behavior for isotropic case. In the case of spherical isotropic shells, the natural frequency diminishes as the Poisson’s ratio of the shell material approaches −1 at constant flexural rigidity and at constant shear modulus. Finally, advanced topics on vibration damping, vibration transmissibility and acoustics of auxetic solids and structures are briefly reviewed.

This chapter on wave propagation forms the second part of the elastodynamics of auxetic solids. Special emphasis is placed on the effect of negative Poisson’s ratio towards the velocity of longitudinal waves in prismatic bars c ₀, the velocity of plane waves of dilatation c ₁, the velocities of plane waves of distortion and torsional waves c ₂ and Rayleigh waves c ₃. A set of dimensionless wave velocities is introduced to facilitate the plotting of non-dimensional wave velocity in both the auxetic and conventional regions. As an alternative way of non-dimensionalization, all wave velocities can be normalized against the wave velocity for plane wave of dilatation. It is herein shown that some of the velocities of different types of waves are equal at non-positive Poisson’s ratio, i.e. c ₀ = c ₁ at v = 0, c ₀ = c ₂ at v = −0.5 and c ₀ = c ₃ at v = −0.733. In the case of solitary waves in plates, Kołat et al. (J Non-Cryst Solids 356:2001–2009, 2010) showed that the amplitudes and velocities are approximately related to the magnitude of the Poisson’s ratio, while the width of the initial pulse is related to the number of propagating solitary pulses.

This chapter explores the stress wave transmission and reflection from an incident stress wave, between isotropic solids of different Poisson’s ratio, with special emphasis on systems in which at least one of the solids is auxetic. The dimensionless transmitted stress, in terms of the ratio of transmitted to incident stresses, were investigated for longitudinal stresses in prismatic bars (1D stress), longitudinal stresses in width-constrained plates (2D stress or 2D strain), plane waves of dilatation (1D strain), torsional waves (shear waves), and Rayleigh (surface) waves. Each of these wave transmission study was performed under three special cases, i.e. when the (i) product of density and Young’s modulus for both solids are equal, (ii) product of density and shear modulus for both solids are equal, and (iii) product of density and bulk modulus for both solids are equal. Under these special conditions, results show that the stress transmission is effectively doubled or eliminated when the Poisson’s ratio for the isotropic solids are at their limits.

This chapter considers the effect of Poisson’s ratio on lateral deformation in longitudinal waves in prismatic bars, and the consequent density change. A non-dimensionalization is adopted herein such that the dimensionless velocity of longitudinal wave is constant with Poisson’s ratio. Based on this non-dimensionalization, incorporation of density correction and/or lateral inertia using the strength of materials approach gives a dimensionless velocity that decreases and increases with Poisson’s ratio for tensile and compressive loads, respectively, such that the pivot conditions take place at v = 0.5 considering density correction only, at v = 0 considering lateral inertia only, and at v = 0.25 considering both corrections. An analogy is then extended to the case of plane waves of dilatation, in which only density correction is required. Thereafter, a revisit to the lateral inertia of Love rods provides the combined effect of density, Young’s modulus, Poisson’s ratio, polar radius of gyration and wave number on the velocity of longitudinal waves in Love rods.

This chapter establishes the effect of auxeticity on shear deformation in laterally-loaded thick beams, laterally-loaded thick circular, polygonal and rectangular plates, buckling of thick columns and plates, and vibration of thick plates. Results show that shear deformation reduces as the Poisson’s ratio becomes more negative, thereby implying that geometrically thick beams and plates are mechanically thin beams and plates, respectively, if the Poisson’s ratio is sufficiently negative. In other words, results of deflections in Timoshenko beam and Mindlin plate approximate those by Euler-Bernoulli beam and Kirchhoff plate, respectively, as the Poisson’s ratio approaches −1. In the study of buckling of isotropic columns, it was found that auxeticity increases the buckling load such that the buckling loads of Timoshenko columns approximate those of Euler-Bernoulli columns as \(v \to - 1\). In the case of vibration of thick isotropic plates, it was shown that as a plate’s Poisson’s ratio becomes more negative, the Mindlin-to-Kirchhoff natural frequency ratio increases with diminishing rate. Furthermore, simplifying assumptions such as constant shear correction factor and exclusion of rotary inertia is valid for plates with positive Poisson’s ratio, and that the assumptions of constant shear correction factor and no rotary inertia for auxetic plates give overestimated natural frequency.

Upon defining two types of semi-auxetic solids—directional and positional semi-auxetic solids—this chapter develops the elastic properties of a directional semi-auxetic solid based on a combined hexagonal and re-entrant honeycomb structure and the 3D kinematic characteristics of another directional semi-auxetic solid based on rotating units. This is followed by an analysis and proposed fabrication methods of directional semi-auxetic yarns. An analysis is then given for a functionally-graded beam from positive Poisson’s ratio on one surface to a negative Poisson’s ratio on the opposite surface. Finally analyses are given for the mechanical properties of positional semi-auxetic structures in the form of compound rods and sandwich structures. Results for positional semi-auxetic structures show that the extent of auxeticity depends not only on the auxeticity of the constituent structures and the relative volume fractions, but also on the modes of deformation, i.e. these structures exhibit overall conventional properties under one type of loading mode, but overall auxetic properties under a different type of loading mode.

This chapter begins by establishing the effect of constituents with opposite Poisson’s ratio signs on the effective moduli of composite properties. Results show that the effective Young’s modulus of continuous unidirectional fiber composites in the fiber direction and that for laminates of isotropic laminas in the in-plane direction exceeds the rule of mixture prediction, especially when the difference between Young’s moduli and Poisson’s ratios between the constituents are small and large, respectively. For laminates of isotropic laminas with opposing Poisson’s ratio signs, the effective Young’s modulus in the out-of-plane direction not only exceeds the inverse rule of mixture but also the direct rule of mixture, and this is especially so when the difference between the Young’s modulus of individual laminas is insignificant. The conditions that lead to further counter-intuitive properties whereby the in-plane laminate modulus exceeds the modulus of the stiffer phase is established, followed by an example in which the maximum point of the laminate modulus takes place when the volume fraction of the stiffer phase is lower than the volume fraction of the more compliant phase. Thereafter, investigation on laminates of isotropic laminas with alternating signs of Poisson’s ratio and alternating signs of coefficient of thermal expansion (CTE) gives results of extreme overall CTE. Finally, a review is done for investigation on conventional composites that lead to auxetic properties.