Poisson's ratio is the negative ratio of transverse to axial strain in a body under uniaxial stress. For nearly all common solids it is positive: a stretched rubber band gets thinner. A negative value means the opposite, lateral expansion in tension and lateral contraction in compression. Such behaviour was known in a few crystals and discussed in theory, but it became a subject of engineering interest when Lakes (1987) converted an ordinary polymer foam into one with a negative Poisson's ratio by compressing it triaxially and heat-setting the buckled cell ribs. The adjective auxetic, from a Greek root meaning that which tends to increase, was introduced a few years later by Evans et al. (1991).

Since then the subject has grown along two lines. One concerns naturally occurring or processed auxetic materials: foams, microporous polymers, certain crystals. The other concerns mechanical metamaterials, in which a designed cell, repeated periodically and usually made by additive manufacturing, produces the effect through its kinematics (Bertoldi et al., 2017). Several broad reviews, from the early survey of Evans and Alderson (2000) onward, catalogue the architectures and their reported properties (Saxena et al., 2016; Kolken and Zadpoor, 2017; Ren et al., 2018).

Among the advantages claimed for auxetic structures, energy absorption attracts the most applied interest, with protective padding, crash structures and blast panels as intended uses. The argument is intuitive: under a compressive blow the material draws inward and densifies beneath the load. The present review has a narrower aim than the catalogues cited above. It asks which design rules reliably yield a negative Poisson's ratio together with useful stiffness and strain range, and what bounds the energy an auxetic metamaterial can absorb in comparison with a conventional cellular solid of the same mass.

The article is a narrative review of the published literature. Sources were chosen for their standing in the field: the papers that introduced each family of architecture, the analytical models in general use, representative experimental and numerical studies of crushing, and earlier reviews. The treatment is restricted to passive, periodic structures in the quasi-static and low-to-moderate velocity impact regimes. Active or programmable metamaterials, auxetic textiles and laminates made auxetic by their stacking sequence are left out. No new data are presented, and comparisons between studies are qualitative, because specimens, base materials and test conditions differ too much for pooled figures to mean anything.

For an isotropic linear elastic solid, positive definiteness of the strain energy requires Poisson's ratio to lie between -1 and 0.5. The upper limit corresponds to incompressibility; the lower limit corresponds to a material that keeps its shape and changes only its size, with a shear modulus far exceeding its bulk modulus (Greaves et al., 2011). Nothing in thermodynamics forbids the negative half of this interval, and Milton (1992) showed that composites of ordinary phases can be constructed with a Poisson's ratio arbitrarily close to -1. For anisotropic materials the ratio between a given pair of directions has no bounds at all (Ting and Chen, 2005), which is why strongly directional cells can show values well beyond the isotropic range.

Two relations from isotropic elasticity explain most of the benefits attributed to auxetic behaviour. The shear modulus equals Young's modulus divided by 2(1 + ν), so it grows without limit relative to Young's modulus as ν approaches -1. The contact stiffness in indentation scales with Young's modulus divided by (1 - ν2), which grows in the same way. Both statements hold at fixed Young's modulus. That qualification is the source of much confusion, because the geometric changes that make a cell auxetic usually lower its Young's modulus as well, and a fair comparison has to be made at equal density or equal mass.

Auxetic architectures can be sorted by the mechanism that produces lateral contraction under compression. Figure 1 shows the four families considered here, with common examples of each.

A good energy absorber crushes at a nearly constant stress over a long stroke. The absorbed energy per unit volume is the area under the stress-strain curve up to densification, and the plateau stress must stay below the level that would injure the occupant or damage the protected structure (Lu and Yu, 2003). A conventional honeycomb or foam meets this description well. An auxetic design can alter three things: the plateau stress, the densification strain, and the way the load spreads.

The clearest benefit is in localised loading. When a small indenter or striker presses into an auxetic material, the lateral contraction draws material in from the sides, so the region under the load becomes denser and harder as it is compressed. Lakes and Elms (1993) compared conventional and re-entrant foams under indentation and reported greater resistance to indentation and higher energy absorption for the re-entrant material. These are foams, with their own cell-rib mechanics, but the densification mechanism does not depend on the particular microstructure and carries over to designed lattices.

For uniform crushing between flat platens the picture is less simple. Finite element studies of re-entrant honeycombs under in-plane impact show the cells collapsing inward into a compact band, with a plateau stress that rises with crushing velocity (Liu et al., 2016), as it does for hexagonal honeycombs (Ruan et al., 2003). Qiao and Chen (2015) derived expressions for the plateau stress of double arrowhead honeycombs and examined graded versions, in which wall thickness varies through the depth so that collapse proceeds layer by layer. Numerical work on sandwich panels with auxetic cores under blast loading suggests that the inward flow of core material towards the loaded zone can reduce back-face deflection in comparison with a monolithic plate of equal mass (Imbalzano et al., 2016). Much of this evidence is computational, and it compares auxetic designs with a limited set of conventional ones.

Four limits recur in the studies reviewed. They concern the strength available at a given mass, the lateral boundary conditions, the usable strain range and the rate of loading.

Three gaps recur in the literature. Comparisons are often made against a single conventional baseline chosen by the authors, and seldom at equal mass, equal base material and equal envelope together. A common protocol, stating relative density, cell count, boundary conditions and strain rate, would make results cumulative. Second, most dynamic studies are numerical, with idealised geometry and elastic, perfectly plastic walls. Printed metal and polymer cells have rough surfaces, residual porosity and limited ductility, and their walls may fracture before the mechanism has run its course. Experiments that follow cell collapse at realistic impact speeds are needed to test the simulations.

Third, the more promising uses seem to lie in combining auxetic behaviour with something else. Graded or hierarchical cores, auxetic layers placed only where loading is local, and hybrid cells that mix stretching-dominated members with a re-entrant mechanism all aim to recover strength without giving up lateral contraction. Fracture and fatigue of hinge and ligament regions under repeated impact have received little attention, although protective equipment is rarely used only once. Recoverable designs based on elastic instability address reuse, at the price of low absorbed energy per unit volume in soft base materials.

A negative Poisson's ratio is permitted by elasticity and can be engineered by a small number of well-understood mechanisms: re-entrant walls, rotating units, chiral ligaments and controlled buckling. The design rules linking geometry to Poisson's ratio are mature. The case for energy absorption is narrower than is often stated. Auxetic metamaterials offer a demonstrable advantage under indentation and localised impact, where inward flow of material stiffens the loaded region, and they can be graded to shape the crushing sequence. Against this stand low strength per unit mass, dependence on lateral freedom, a shortened stroke and a diminishing role of topology as impact speed rises. Design should therefore begin from the loading case and ask whether lateral contraction helps it, and should not begin from the auxetic property itself.

To be declared by the author(s) before publication.

To be declared by the author(s) before publication.