Poisson's ratio over two centuries: challenging hypotheses

Abstract

This article explores Poisson's ratio, starting with the controversy concerning its magnitude and uniqueness in the context of the molecular and continuum hypotheses competing in the development of elasticity theory in the nineteenth century, moving on to its place in the development of materials science and engineering in the twentieth century, and concluding with its recent re-emergence as a universal metric for the mechanical performance of materials on any length scale. During these episodes France lost its scientific pre-eminence as paradigms switched from mathematical to observational, and accurate experiments became the prerequisite for scientific advance. The emergence of the engineering of metals followed, and subsequently the invention of composites—both somewhat separated from the discovery of quantum mechanics and crystallography, and illustrating the bifurcation of technology and science. Nowadays disciplines are reconnecting in the face of new scientific demands. During the past two centuries, though, the shape versus volume concept embedded in Poisson's ratio has remained invariant, but its application has exploded from its origins in describing the elastic response of solids and liquids, into areas such as materials with negative Poisson's ratio, brittleness, glass formation, and a re-evaluation of traditional materials. Moreover, the two contentious hypotheses have been reconciled in their complementarity within the hierarchical structure of materials and through computational modelling.

Introduction

Most physical parameters in classical physics relate to laws that emerged between the seventeenth and nineteenth centuries. Where other physicists are honoured in the names of physical units used to measure these parameters—Pascal, Newton, Coulomb, Ampère, Volta, Dalton, Faraday, Gauss, Hertz, Kelvin—Siméon Denis Poisson (1787–1840) is particularly remembered for a ratio,¹ a dimensionless quantity ν, which today has surprisingly acquired ubiquitous physical significance.

The phenomenon that Poisson's ratio is based on was well expressed by Thomas Young (1773–1829) in his remarkable 1807 Lectures on Natural Philosophy and the Mechanical Arts: ‘We may easily observe that if we compress a piece of elastic gum in any direction, it extends itself in other directions; and if we extend it in length, its breadth and thickness are diminished’.² This was often called the ‘squeeze–stretch ratio’ but was not specifically defined by Young (figure 1). In the same lecture, though, he described the concepts of stress σ_l, the external pressure, and strain e_l, the resulting fractional distortion (ΔL/L); the stress–strain curve, starting with the linear elastic region from which Young's modulus E = σ_l/e_l became defined; the critical yield point σ_c, beyond which elasticity ceases and plastic flow starts; and the eventual mechanical failure in which a body is quite simply broken ‘by tearing it asunder’.³ This overarching description of the mechanics of solids, liquids and gases came largely from observation,^2,3 but it took 20 years for a mathematical theory to emerge to handle even the elastic regime, first in France in the hands of Navier (1785–1836)⁴ and then Cauchy (1789–1857),⁵ but rather later in Britain, notably by Stokes (1819–1903).⁶

Figure 1.

Defining Poisson's ratio from the negative ratio of lateral and longitudinal stress −e_t/e_l for a rod under tension. Originally called the squeeze–stretch ratio,² the negative sign ensures that ν is a positive number for common materials such as glass, metals and rubber. When Poisson's ratio is negative, e_t/e_l is positive. These materials swell when they are stretched, and narrow when they are compressed.^29,76

In the first two decades of the nineteenth century French science was enjoying a golden epoch.⁷ The major centre for mathematical physics was Laplace's Société d'Arcueil,⁸ in which elasticity was a central theme. Poisson entered the École Polytechnique in 1798 and by 1812 had been elected to the First Class (the scientific class) of the Institut de France and had become a member of the Arcueil circle, loyal to Laplace and his devotion to the Newtonian view of the Universe. In particular Newton's corpuscular hypothesis of interacting molecules was an article of faith.⁹ In Laplace's words, ‘all attractive and repulsive forces in nature reduce, in the final analysis, to similar forces acting between one molecule and another … attractions that are sensible only at imperceptible distances.’¹⁰ This gave rise to the notion of imponderable elastic materials, which became hardwired into Poisson's physics, starting with his early Traité de Mécanique.¹¹ The argument has been made¹² that this ‘molecular mentality’, for which there was as then no direct evidence, complicated the development of elasticity theory, as well as that of optics and heat.¹³ It was only after this tenet was overtaken by the precept of an infinitely divisible continuum in which stress and strain were coupled linearly through many elastic moduli^5,14 that the rich phenomenology of elasticity acquired the mathematical framework of an effective medium that sustained nearly two centuries of mechanics. When Poisson published his Note¹ in 1827 describing the ratio ν, however, he was still using the molecular hypothesis simplified to a single constant theory. That resulted in a unique number for ν, which happened to agree with novel, but as it transpired inaccurate,¹⁵ experiments on brass by Cagniard de la Tour (1777–1859).¹⁶ Later, between 1850 and 1870, as more materials were examined (figure 2) with more reliable measurements (figure 3),¹⁵ Poisson's ratio emerged multi-valued, in line with Cauchy's continuum theory,^5,17 with stress and strain tensor quantities—re-emphasizing the dichotomy between shape and volume in mechanical response.^14,18 My initial objective is therefore to use the squeeze–stretch ratio as an idée fixe to follow the two hypotheses competing to describe elastic media, from 1811 when Poisson's career was at its height, through the next 60 years, by which time both experimentation had become reliable and the theory and ratio were fully understood at the macroscopic level.¹⁴

Figure 2.

Growth of publications involving Poisson's ratio, from 1827 to the present day: white bars, from note 15; black bars, Web of Science; grey bars, average exponential rise expected from 1970, using the scaling given in note 72. Examples of the diverse applications of Poisson's ratio responsible for the huge rise in publications in recent years are listed on the right.

Figure 3.

Periodic variations in Poisson's ratio for elements replicated for all three major periods of the Periodic Table and attributable to differences in interatomic distance. Taken from the comprehensive 1961 review of Poisson's ratio for metals and alloys by Köster and Franz.¹⁵

By 1870 elasticity theory was being adopted by engineers endeavouring to improve the strength of the materials of the day such as iron and steel in diverse applications.^15,19 With the continuum approximation, this led to advances in metallurgy in both the elastic¹⁴ and plastic regimes.²⁰ By the early twentieth century new materials were being invented,²¹ eventually leading to composites in the 1960s.²² As the science of materials emerged, Poisson's ratio had a meaningful role in characterizing mechanical performance, as can be seen by the steady rise in publications before 1970 (figure 2). At the same time, with the discovery of crystallography²³ revealing for the first time the periodic structure of matter at the atomic level, the physics that was missing from Laplace's molecular hypothesis slowly came to light. Developments in microscopy²⁴ in the 1950s enabled the nanostructural defects that control macroscopic properties to be imaged. Given that quantum mechanics was by then able to predict physical properties at the atomic level, relationships between microscopic structure and macroscopic function gradually became established²⁵—ironically reasserting the molecular hypothesis, but with different crystalline symmetries adding directionality to the centrosymmetric case envisaged by Poisson.^1,17 This was given added momentum when molecular dynamics emerged in the 1960s²⁶ and trajectories of interacting molecules were first simulated computationally by solving Newton's equations of motion—‘pursuing Laplace's vision on modern computers'.²⁷ Accordingly I intend to illustrate through Poisson's ratio how differences between continuum and molecular approaches to materials science were perpetuated between 1870 and 1970, sometimes resulting in divisions between physics and technology, as specialization took hold. I shall also illustrate how,when the two came together, extraordinary progress was achieved.²⁸

After 1970 there was a surprising revival in the exploitation of Poisson's ratio across the applied sciences (figure 2). New types of material, such as those with negative values, emerged (figure 4),²⁹ and, with Poisson's ratio for all isotropic materials falling between −1 and ½, a universal metric opened up to quantify mechanical performance (figure 5).³⁰ Links were established between the mechanical responses of novel systems sharing the same architecture but on colossally different length scales (figure 6).³¹ New light was shone on the elastic properties of the earliest known materials such as cork (figure 7) and glass, and also the circumstances of mechanical failure.³² This rebirth in Poisson's ratio revealed its utility for unifying the mechanical properties of manifold systems, whether they were molecular in origin or better considered a continuum. By continuing this history through to the present time, a final objective is to show how this fundamental concept has proved fertile in completely different areas of science in the twenty-first century, encouraged by the gathering culture of interdisciplinarity, in which concepts and themes from the physical sciences are being creatively regenerated in the applied sciences. Importantly, during the past two centuries the competing hypotheses of elasticity led first to disconnections between physics and engineering, but then to reconnections between the physical and applied sciences. Progress there has been, however unplanned and unpredictable.

Figure 4.

Anisotropic variations in Poisson's ratio for a fibre composite ply laminate, varying as the fibre angle θ changes with respect to the direction of the laminate and its normal. This behaviour was first predicted by Saint-Venant⁵⁴ and analysed by Voigt⁶¹. (Courtesy of A. Kelly.)

Figure 5.

Universal relationship between Poisson's ratio ν and the ratio of bulk to shear moduli B/G.³⁰ All isotropic materials have ν values somewhere within the number window −1 to ½. Materials with negative Poisson's ratio have been called auxetic.⁷⁶ In going from negative to positive values, the shapes of materials are sacrificed for incompressibility.

Figure 6.

Vast sizes and atomistic length scales over which Poisson's ratio operates.³¹ (a) A laser-cooled sparse crystal comprising ⁹Be⁺ ions held at in a potential trap ultralow temperatures with an atomic periodicity of micrometres. (Reproduced from note 93 with permission. Copyright © 1999 American Institute of Physics.) (b) The butterfly nebula, a planetary nebula containing a white dwarf at its centre. White dwarfs are predicted to have body-centred structures similar to sparse crystals but with extremely narrow atomic spacings of picometres. (Reproduced from note 94. Credit: the NASA, ESA and the Hubble SM4 ERO Team.) (Online version of (b) in colour.)

Figure 7.

Similar images of the microstructure of cork separated by 350 years. (a) Differing microstructures of cork from Hooke's Micrographia, the first publication of the Royal Society;⁸¹ engravings obtained from shavings of cork showing the structures (A) radial and (B) tangential to the direction of growth. (b) Recent scanning electron microscope images for equivalent directions.³⁰ (Courtesy of R. L. Reis.)

Response to prejudice and inaccurate measurement: 1811–70

Poisson published his Traité de Mécanique in 1811. He had already taken over Fourier's chair at the École Polytechnique in 1806, and in 1818 was elected a Foreign Member of the Royal Society.³³ In 1820 he was elected to the Conseil Royal de l'Instruction Publique, becoming the director of mathematics teaching throughout France.³⁴ He was fond of saying, ‘Life is good for only two things: to study mathematics and to teach it.’³⁵ By the time he published his Note describing the ratio in 1827, things were very different. Although he was respected as France's grand géomètre,³⁶ increasingly his originality in physics was not; this was epitomized in Fourier's remarks to Laplace in 1815: ‘Poisson has too much talent to apply it to the work of others.’³⁷ Poisson's at times unscrupulous handling of research presented in confidence to the Académie for review and assessment antagonized not only Fourier—the most notorious case being Poisson's several attempts to undermine the candidature of Sophie Germain (1776–1831) for the prix extraordinaire set by Napoleon to explain analytically the Chladni vibrations of thin elastic plates.¹² Germain's eventual success³⁸ in 1816 proved a high watermark preceding the decline of physics in France,³⁹ Britain taking precedence in electromagnetism and Germany in optics and instrumentation. In 1827 Poisson would therefore have felt somewhat insecure and perhaps saw the experiments on brass by his friend Cagniard de la Tour¹⁶ as an opportunity to restate the molecular hypothesis and his own credentials in elasticity theory. In his 1827 Note¹ Poisson proposes that if a is the equilibrium length of a rod and b its cross-sectional area, then if a increases to a(1 + α) and b decreases to b(1 − β) under tensile stress, the volume increases to a × b(1 + α − β). He says that he has deduced from the theory of molecular interactions that, for a solid body composed of molecules simply held together by central forces on a crystalline lattice, β = ½α. Poisson clearly regarded the value of β/α = ½ as a constant, claiming ‘This result agrees perfectly with an experiment that M. Cagniard-Latour [Cagniard de la Tour] has recently communicated to the Académie’,¹ which he then went on to describe¹⁶—evidently a success for the Laplacian school.

By the 1850s Poisson's ratio had become redefined as the strain across a bar ΔD/D divided by the strain lengthways ΔL/L, namely ν = −(ΔD/D)/(ΔL/L), the minus sign relating to tension. In terms of figure 1, ΔL/L = α, b = πD²/4 and therefore ΔD/D = ½β and so, if β = ½α, ν = ¼. In choosing not to address any variability in his ratio, Poisson ignored the case for which there might be no change in volume under stress, which would have led him to a value for β/α of 1 and for ν of ½, the value later reported for all liquids, including elastic gum.^2,18 Because of their almost complete compliance with shape change and virtual incompressibility, liquids were already well known to be the perfect pressure-transmitting medium, as Young had summarized in 1807.³ By contrast, considering cork, Poisson would have known that there is little change in the cross-sectional area of stoppers when wine bottles are opened, so if β approximates to zero so then should his ratio. A perhaps less obvious observation, but still one familiar to someone brought up in a poor family outside Paris, Poisson as a boy might just have watched cows being milked, teats being squeezed and contracting while swelling on elongation, for which β would be negative and so also would ν. The paradigm of positivism, in which observation was paramount, was spreading in France after the Bourbon Restoration,⁴⁰ and might have encouraged Poisson in the late 1820s to make some of these commonplace connections; however, his allegiance to the elite traditions of Arcueil and the Académie⁴¹ clearly prevailed. This conservatism, combined with the difficulty of Cagniard de la Tour's experiments,^15,16 left the ratio in limbo for more than a decade (figure 2).

Cagniard de la Tour's technique did not measure ν directly^1,15,16 but detected changes in the length and volume of a wire by fixing one end to the bottom of a tube filled with water, stretching the other end, and recording the change in volume from the change in water level. When Wertheim adapted Cagniard de la Tour's method later by stressing water-filled tubes⁴² he reported ν = 1/3 for brass and also for glass. He ‘wished’ it to be 1/3 rather than 1/4, but all Cagniard de la Tour-type measurements were prone to serious errors, even if the testpiece was exceptionally long.¹⁵ By 1859 Kirchhoff⁴³ had measured several metals by using what William Thomson (1824–1907) considered ‘a well-devised experimental method’,¹⁸ measuring the shear modulus G and Young's modulus E independently, from which ν = [(E/2G) − 1]. Later, Everett used the same technique for glass,⁴⁴ which was reviewed by James Clerk Maxwell (1831–79) in 1867.¹⁷ For these far more accurate measurements,¹⁵ ν for brass equalled 0.387,⁴³ not Cagniard de la Tour's 1/4,¹⁶ and for glass the value was 0.258,⁴⁴ not Wertheim's 1/3.⁴² Moreover, iron measured 0.294 and values for copper ranged from 0.226 to 0.441.⁴³ Poisson's ratio was clearly not a constant, but ‘varies from substance to substance’,⁴³ and could also vary for the same material fabricated in different ways. Theodor Kupffer (1799–1865), writing on experimental research in St Petersburg in 1860,⁴⁵ quoted experiments by Franz Ernst Neumann (1798–1895) in Königsberg reporting the volume of a wire increasing under traction up to the elastic limit σ_c, beyond which it remained constant; that is, ν increasing to 1/2. This was evocatively referred to as being in ‘the state of ease’.⁴⁵

In the first half of the nineteenth century the saga disproving the fixed value of 1/4 was concurrent with an often vitriolic tussle between the hypotheses competing to describe elasticity. Stress and strain, which Young had described in 1807,² were not included in the assumption of interacting molecules, even in Navier's important Mémoire.⁴ This was a one-constant theory for isotropic media, resulting in Poisson's ratio being fixed. The continuum hypothesis stemming from Germain's work on vibrating plates,³⁸ which Cauchy was later to formalize,⁵ led to the system of stress–strain ‘Cauchy relationships’,¹⁴ now often considered ‘the fundamental mathematical apparatus of elasticity theory’.⁴⁶ Stokes, initially unaware of the French work when he turned his attention to the mathematics of elasticity in 1845,⁶ introduced two independent constants for isotropic solids. These were called ‘resistance to compression’ and ‘rigidity’;⁴⁷ these explicitly distinguished changes in volume and shape, respectively, thereby admitting many values for ν.⁴⁸

By the middle of the nineteenth century, amid all of this controversy, Maxwell¹⁷ returned to Poisson's centrosymmetric molecular interaction theory for which ν = 1/4:¹

This relation … is of great importance in the theory of Elastic Solids, since if we assume elasticity to arise from the direct action of molecules on one another by attractive or repulsive forces acting upon the line of centres, this ratio can be shown to be 1/4 (as stated by Siméon Denis Poisson), provided the displacement of each molecule is a function of its position in the body. If however the displacement of each molecule is a function of a different form from that of the others, the ratio may be different.¹⁷

To understand the appeal of the molecular hypothesis and its possible modification it is helpful to appreciate that before the advent of X-ray crystallography²³ and the ability to deduce the position of atoms in solids, careful measurement of elastic deformation was one of the few ways for understanding the cohesive forces operating in solids. However, crystallography goes back much further, notably to René Just Haüy (1743–1822),⁴⁹ who in the 1780s was inferring atomic structure from the faces of minerals in the stacking of blocks of the same shape and size. These unit cells were later systematically indexed by William Miller (1801–80) by the three integers (h,k,l) that are now used to identify atomic planes.⁵⁰ Unit cells were first rationalized by Auguste Bravais (1811–63) into a handful of distinct lattices, decreasing in symmetry and increasing in anisotropy from cubic to triclinic.⁵⁰ It has been argued that the fact that Laplace's circle did not embrace crystallography in supporting the molecular hypothesis was ‘a missed opportunity’.⁵¹

Notwithstanding this, in comparison with Maxwell's positive response to Poisson's molecular reasoning, Thomson viewed the theory far less charitably,¹⁸ considering it to be

fallacious Mathematics … that the ratio of lateral contraction to elongation by pull without transverse force is 1/4. This would require the rigidity to be 3/5 of the resistance to compression for all solids;⁵² which was first shown to be false by Stokes⁶.

In many ways this was unfair, because Poisson had already embraced Cauchy's multi-constant continuum theory⁵ by 1829, admittedly with reservations.⁵³ Indeed, Poisson had encouraged Cauchy to admit linear dependences between stress and strain for anisotropic media, invoking 36 elastic parameters⁴⁶ that Green later showed could be further reduced to 21 independent parameters.^14,46 A further twist, though, had come earlier in 1828 when Cauchy, pursuing Navier's tenet of interacting molecules,⁴⁶ reduced the elastic parameters to 15 for anisotropic systems and 1 for isotropic systems, for which ν = 1/4. The fact that experiments showed that Poisson's ratio was not a constant meant that elasticity theory had to be multi-constant—up to 21 if the system was anisotropic but 2 if it was isotropic. The Cauchy relations were restated by Jean Claude Barré de Saint-Venant (1797–1886) in 1848,⁵⁴ who also seems to have been first to propose that Poisson's ratio might be negative in anisotropic media and might also have values greater than 1/2, ideas to be returned to below. In the face of these ensuing complications one can only be impressed by Poisson's prejudice in asserting his value of 1/4. Maxwell¹⁷ must have recognized that this was very close to Everett's value for glass,⁴⁴ whose silicate structures are now known to comprise networks of centrosymmetric tetrahedral polyhedra.⁵⁵ Indeed, ν averages out at almost 1/4 for the Earth's crust, which is nowadays referred to as a ‘Cauchy solid’,⁵⁶ flying in the face of Thomson's view that Navier's and Poisson's theory ‘may be regarded as the proper theory for an ideal perfect solid, and as indicating an amount of rigidity not quite reached by any real substance’¹⁸!

A history of the ordinary during extraordinary times: ca. 1870 to ca. 1970

The genealogy of a scientific idea becomes difficult to trace once experiments become ordinary measurements that may also be performed in a different field. With the theory of elasticity ‘settled’ by 1870 in favour of the continuum hypothesis and with Poisson's ratio able to be measured with high precision, the controversy over the previous 40 years moved out of the scientific limelight. Over the next 100 years, an extraordinary period that embraced the second scientific revolution which incorporated the transition to modern physics culminating in the establishment of modern technology,⁵⁷ Poisson's ratio continued to engage, but mainly with the engineering profession, for whom macroscopic properties were sovereign. Nevertheless, lineaments emerged at the atomic level in the physics of condensed matter, not least in the velocities of sound, which relate Poisson's ratio to the phonon properties of solids.⁵⁸ The rapid growth of scientific knowledge demanded empirical reproducibility, symbolized by the establishment of the Bureau International des Poids et Mesures at Sèvres in 1875 followed by associated institutes worldwide such as the National Physical Laboratory in 1900.⁵⁹ Although Poisson's ratio is unitless, it can be determined from accurate measurements of elastic moduli, as Kirchhoff showed,^18,43 or directly from ΔL/L and ΔD/D by using very precise interferometry, such as Grüneison developed in the early 1900s.⁶⁰ By 1891 Poisson's fixed value of 1/4 for ν had been revisited in Germany as many more metals were measured, and Cauchy's 15-parameter theory was again deemed insufficient.^15,61

With standardization, modern technology quickly identified the need for high-quality reproducible materials. Kupffer reflected this in his 1860 review of research in Russia, recommending ‘a special institution, dedicated to experiments on the strength of materials between and beyond the limits of elasticity, where you can test the metal production of all factories in the country’.⁴⁵ When John Anderson (1814–86), superintendent engineer at the Arsenal at Woolwich, introduced ‘a machine for testing the strength and elasticity of materials’²⁰ to identify the best iron from different countries for the manufacture of ordnance, he observed that the tenacity of US cast iron was more than twice that produced in Britain. Tenacity is now called critical yield stress, σ_c, where the shear modulus G drops sharply, ν rises to 1/2 and plastic flow commences, as Neumann had first observed 10 years earlier.⁴⁵ For a perfect crystalline array, σ_c should scale approximately with the shear modulus as G/6.⁶² Anderson's values, however, were typically thousands of times less, pointing to imperfections in crystallinity when metals were tested beyond their elastic limits.

Optical microscopes provided clues for identifying the mechanical weakness of metals. Capable of resolving a fraction of a micrometre, they revealed a myriad of submillimetre polycrystalline grains in polished specimens of cast iron. In particular, slip within crystals was observed at the National Physical Laboratory by Robert Rosenhain (1875–1934),⁶³ who also detected the grain boundaries between metallic polycrystals that seemed to prevent slip from extending. Importantly, he considered⁶⁴ that experimental yield stresses resulted from a balance between the inter-grain cohesion and intra-grain strength; the former, he judged, resulted from grains being cemented by a thin layer of amorphous material ‘analogous to the condition of a greatly supercooled liquid’. Twenty years later Taylor, Polanyi and Orowan described how grain boundary motion was governed by dislocations, as adjacent planes of atoms rolled against one another. Dislocations themselves could only be understood⁶⁵ once atomic arrangements in perfect single crystals were known,²³ eventually being imaged in the 1950s with new electron microscopes able to resolve down to 1 nm.²⁴ The physics of dislocation formation was also worked out at that time, in which the interfacial energy was found to be strongly influenced by orthogonal strains between crystalline planes and to be governed not just by the shear modulus G but also by Poisson's ratio ν.⁶⁶

The combined effects of microstructure and flow in metals bedevilled specimen integrity and limited the accuracy of measurements of Poisson's ratio to about 1%.¹⁵ Nevertheless, by the 1960s Köster and Franz were able to assemble the most reliable values for pure metals (figure 3), finding fluctuations across the three long periods of the Periodic Table that showed remarkable similarity and indicated ‘a marked effect of the configuration of the outer electrons’.¹⁵ They concluded, ‘Poisson's ratio depends to a much greater extent on the conditions of bonding than do the other elastic coefficients’, reasserting Poisson's prejudice for the hypothesis of intermolecular forces.

The interplay between science and technology was powerful if disjointed in the twentieth century, but it led to the development of new materials quite distinct from nineteenth-century materials such as iron and glass. The first was Bakelite (1907), talismanic for the polymer chemistry that advanced later between World War I and World War II²¹ and leading to the development of composites and glass-reinforced products in the 1960s.⁵⁹ Subsequently, with the invention of stiff carbon fibres virtually devoid of internal atomic defects and with high yield stresses, carbon fibre composites came to the fore as the tough lightweight material of choice.⁵⁹ Starting with Brian Lempriere's work on orthotropic composites⁶⁷ employing the Cauchy relations,^5,14 laminate composites were a natural testbed for directional Poisson's ratios, each of which might fluctuate widely between positive and negative values (figure 4). However, the huge increase in the commercial importance of composites²² generated only sporadic academic interest at the time. Their haphazard acceptance resulted partly from incomplete scientific dissemination but also from perennial cultural demarcations between scientists and engineers.

There are exceptions to every rule, the most spectacular being the Manhattan Project (1943–45), combining science at the most fundamental level with the most sophisticated technology available at the time. Led by General Leslie Groves and directed by Robert Oppenheimer, the project delivered the first atomic bomb under the most extraordinary circumstances.²⁸ At the heart of the detonation of nuclear fission was the implosion of plutonium, requiring huge temperatures and pressures for liquefaction. Spherical lenses reflected the shock wave inwards, achieving sufficient density for criticality as well as mechanically supporting the liquid charge momentarily. The implosion geometry followed from the physics of liquids. With a Poisson's ratio of 1/2, liquids are shapeless and hard to compress. A fascinating anecdote from Los Alamos concerns one of the laboratory visits made by General Groves when he happened to trample on rubber hoses carrying water. If pipes swell they shorten—e_t/e_l is negative—and are therefore liable to disconnect, which abruptly happened in this case, instantly shooting hot water everywhere, soaking the general and prompting the informative retort from Oppenheimer, ‘Well, it just goes to show the incompressibility of water.’²⁸

After World War II came a further large expansion in scientific activity with the creation of national funding agencies and the building of central facilities—synchrotrons, collider accelerators and neutron sources.⁶⁸ Often adopting the Manhattan template, civil science programmes were based on the paradigm that ‘blue skies’ research would deliver world-beating technology. This did indeed happen, with major advancements in nuclear reactors, electronics, telecommunications, lasers and also computers. In the present context the birth of computational molecular dynamics in the 1950s is apposite. As Adler and Wainwright wrote then:⁶⁹ ‘the behaviour of systems of many interacting particles cannot, in general, be dealt with theoretically in an exact way. … Since these difficulties are not conceptual but mathematical, high-speed computers are well suited to deal with them.’ From that moment on, Laplace's molecular hypothesis began to regain lost ground. Complementary to this, but quite independently, the finite-element method, based on Cauchy's continuum hypothesis and exploiting Lord Rayleigh's variational methods, was pioneered in the 1950s by the engineer Olgierd Zienkiewicz (1921–2009) to analyse the distribution of stress in large structures.⁷⁰ The finite-element method and molecular dynamics were to feed back jointly into solid mechanics as computational science advanced towards the close of the twentieth century.⁷¹

Renaissance through interdisciplinarity: ca. 1970 to the present day

During the twentieth century the number of scientific publications rose exponentially, ‘growing by a factor of 10 every 50 years’.⁷² Over the past 40 years, however, papers incorporating Poisson's ratio across the applied sciences have exceeded this baseline 15-fold (figure 2). The appeal is probably attributable to the shape versus volume concept³⁰ embedded in the ratio of bulk to shear modulus B/G^47,52 that caught Thomson's eye.¹⁸ This notion has illuminated the understanding of the narrowing of arteries during hypertension, the resilience of bones and medical implants, the rheology of liquid crystals, the shaping of ocean floors, the oblateness of the Earth, and planetary seismology after meteor impact—all extraordinary examples, but with their explanations ordinarily the same (figure 2). Whereas such research had previously followed rules of thumb, in the context of Poisson's ratio it gathered fresh meaning. Another attraction would have been the realization that, in contrast to Young's modulus and the other elastic moduli that differ hugely between soft and stiff materials, compliant and incompliant, Poisson's ratio was contained within narrow numerical bounds.³⁰ For materials in isotropic form this meant that −1 ≥ ν ≥ 1/2. Accordingly the uniqueness of the physical number that Poisson had proposed,¹ and which caused so much controversy in the nineteenth century (see above), became the uniqueness of a number window. The idea that this embraced the elastic response of every isotropic material in the Universe had to be very seductive,³⁰ and proved to be so (figure 2).

The length scale originally conceptualized by Navier⁴ and Poisson^1,53 was defined by the distances between molecules. The connection between the sensible forces at insensible distances started to be re-established with atomistic computer simulation;²⁶ in the past few years, by employing ab initio quantum mechanical methods, elastic constants have been accurately predicted,⁷³ including Poisson's ratio.⁷⁴ By contrast, in experimental science, much of the novelty in revisiting Poisson's ratio has come through the discovery of new materials whose functionality relates to length scales larger than atomistic and to which continuum mechanics applies, affirming what Roderick Lakes has argued since 1987: that ‘the classical theory of elasticity has no length scale.’^29,75 Poisson's ratio therefore applies to architectural frameworks just as it does to colloids, for example, or to the atomic architecture of the mechanically active components that all materials are constructed from. This awareness has become of great practical significance, offering scope for reverse engineering from the macro to the micro scale,^76,77 bridging the gap between continuum and atomistic topologies.

Laying out the elastic properties of diverse materials in the frame of Poisson's ratio and B/G (figure 5) reveals how the inherent universality applies over vastly different length scales,³⁰ ranging from 0.5 mm for re-entrant polymer foams to 50 µm for cork aggregate. For rubber the scale is 0.5 µm, for many liquids it is nanometres, and for glass, steel and lead it is the atomic scale of 0.1 nm. Moreover, in ways in which Poisson seemed not to consider with his fixed value of 1/4 (see above), the elastic response of different materials modulates from negative to positive values: negative for those that resist shape change but are compressible (G ≫ B), and positive for those that resist compression in favour of changing shape (B ≫ G). With all of the data accumulated in previous years, trends could be deciphered between Poisson's ratio and atomic geometry and density^30,78 that would have impressed the nineteenth-century engineers engaged in optimizing constraints for the new framework buildings of the time.¹⁹ For example, for glassy materials Poisson's ratio was found to decrease as atomic geometry became better connected, resulting in glasses that were more rigid but also prone to breaking;⁷⁸ conversely, Poisson's ratio increased as atomistic structures became better packed, but at the same time had the tendency to distort plastically. As Tanguy Rouxel has put it: ‘One can hence produce glasses with Poisson's ratio à la carte.’⁷⁸ He has also drawn attention to crystalline elements from the same column in the Periodic Table that often all share the same structure, such as the semiconductors germanium, silicon and diamond or the face-centred cubic metals gold, silver and copper (figure 3), but whose Poisson's ratio decreases with the interatomic distance, demonstrating how ‘the valence electron density plays a key role’.³⁰ It is therefore not only the architecture but also the strength of the framework members that affects the size of ν.

The possibility that Poisson's ratio might take negative values, foreseen in 1848 by Saint-Venant for anisotropic continua,⁵⁴ resulted in materials that on the atomic scale, as Ray Baughman has noted,³¹ ‘have the counterintuitive property of expanding laterally when stretched’. This property was studied much earlier when the elastic moduli of single crystals were measured,^61,79 in which anisotropy was noted to be virtually always present, even in cubic materials, cubic structures requiring three independent elastic constants. Directionally negative values, however, largely went unnoticed because ν usually averages out positive for the randomized microstructure of polycrystalline materials.^14,61 Against this background the paper by Roderick Lakes in 1987,²⁹ announcing the fabrication of materials of negative ν, was particularly novel because these materials were isotropic. Lakes refabricated conventional polymer foams with positive Poisson's ratio by heating them under pressure to create novel re-entrant structures on the submillimetre scale. These reprocessed foams, Lakes demonstrated, created a new hierarchical structure with negative values of ν as low as −0.8. Topologically the submillimetre structure that delivers negative ν from a polymer with positive ν comprises voids that are convex rather than concave. Since then, numerous mechanical models have been devised that mimic re-entrant geometry and, in engineering terms, could operate on any length scale.^76,77 Furthermore, ‘Work on these strange materials is not meant just for our entertainment; negative Poisson's ratio materials may be useful for a variety of reasons.’⁷⁵

By 1991 the unusual mechanical behaviour of expansion under tension was attracting wider attention and was coined ‘auxetic’⁷⁶ by Ken Evans, basing this on the Greek noun for ‘growth’ (αűξεσις, or auxesis). Both Evans and Lakes have stressed how materials with a negative Poisson's ratio have a natural tendency to form dome-shaped surfaces—synclastic curvature in place of the anticlastic bend of conventional materials, first mooted in the vibration of plates.^12,38 Despite the fact that auxetics are often very compliant materials, they are also very tough and difficult to indent, absorbing shock far more efficiently than materials with positive Poisson's ratios—genuinely a new branch of materials. Since then, and rather spectacularly, auxeticity has been predicted for extreme states of matter, from sparse crystals in the laboratory to the cores of white dwarf stars (figure 6).³¹ For these body-centred cubic systems Baughman found that negative directional Poisson's ratios were virtually independent of density, operating for crystalline arrays with densities from 10⁻¹⁵ g cm⁻³ to about 10⁶ g cm⁻³ for white dwarfs at the centres of planetary nebulae (figure 6). ‘It is remarkable that negative Poisson's ratios, a property once considered rare, can be expected from Coulomb interactions over such a broad range of densities.’³¹

In between the countless materials so far measured (figure 5), some with positive and some negative ν, have come those with values close to zero, of which cork is the archetype. Cork, with its remarkable physical, chemical, thermal and elastic properties, has an illustrious history from Greek and Roman times.⁸⁰ Mechanically it retains its cross-section well, despite being stretched or squeezed along its length, which Poisson would have been aware of had he entertained the possibility that β could approach 0 (see above). The complex technology of cork, together with its special elastic properties, predates the arrival of Poisson's ratio¹ by more than 22 centuries, and yet precise elastic measurements and microscopy have been accomplished only in the past 30 years.⁸⁰

A remarkable foretaste of the complex structure of cork came with the amazing engravings that Hooke made of shavings of cork while peering down his seventeenth-century optical microscope (figure 7a). Micrographia was the first book to be published by the Royal Society,⁸¹ and these flawless images of cork appeared alongside those of the intimidating human mite and other fantastic biological and botanical specimens. Hooke was the first to use the word ‘cell’, actually in describing the microstructure of cork. More particularly he brilliantly identified its hierarchical structure and inherent anisotropy, distinguishing the honeycomb structure in the direction of growth of the bark from the tangential brick-like structure (figure 7a). His anticipation is palpable: ‘but me thought I had with the discovery of them, presently hinted to me the true and intelligible reason of the Phenomena of Cork.’ Comparing Hooke's engravings with scanning electron microscope images of cork today³⁰ (figure 7b) one can only marvel at his natural philosopher's objectivity in drawing with ‘a sincere Hand, and faithful Eye, to examine, and record, the things themselves as they appear.’⁸¹

What Hooke could not have seen were the corrugations in cork cells, with their convex protrusions (figure 7b). Michael Ashby's Cambridge engineering group observed this for the first time, finding that ‘When cork deforms, the cell walls bend and buckle’,⁸⁰ leading to positive and negative values for ν in the radial and tangential directions, respectively. Ashby also managed to measure the cell wall density, finding it to be 15% greater than water, together with the pressure at which the porous structure of cork collapses: ‘We think that this is because the cooperative buckling of neighbouring cells … can take place.’ The cellular structure caves in at pressures close to a 1 MPa, approximately the value that Thomson had reported a century earlier in his celebrated observation of cork sinking in water under pressure.⁸²

In comparison with cork, perhaps the most complex of the oldest materials is glass, with a history extending over five millennia, yet even in 1995 the noble laureate Phil Anderson was claiming⁸³ that ‘The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of glass and the glass transition.’ Glass fabricated by cooling from the molten state is popularly recognized as a frozen liquid,⁸⁴ ν decreasing from 1/2 in the melt to close to Poisson's chosen value of 1/4 when glass is formed. Could the elastic properties for a glass, epitomized in Poisson's ratio, somehow be connected with the viscosity of the liquid from which it is formed⁸⁴ or, as Tulio Scorpigno tantalizingly asked in 2003,⁸⁵ ‘Is the fragility of a liquid embedded in the properties of its glass?’ This controversial idea was taken one step further a year later by Academician Victor Novikov⁸⁶ by correlating ν for different glasses with the fragility⁸⁷ m of the melt from which they were quenched. Such an empirical relationship implies that the elastic timescale embodied in Poisson's ratio might somehow be correlated with the enormously slower timescale of the glass as it sets.^30,88 Because of the vast difference in timescales, trying to predict glass formation from Poisson's ratio ‘is akin to trying to predict the rate of global climate variations over millions of years from observations of the world's weather collected over a single minute’⁸⁸—a new mystery implicating Poisson's ratio that has yet to be solved.

One of the conundrums of glass is that, although it ‘constitutes the intrinsically strongest man-made material that can be produced on a large scale … these intrinsic strengths are compromised by the material's low resistance to surface damage.’⁸⁹ Embrittlement is related to resistance to shear, so could this weakness be manifested elastically in small values of B/G, and therefore of ν? In contrast, when crystalline materials flow plastically, which limits the strength of metals^20,62–66, Poisson's ratio approaches 1/2,⁵² so B/G becomes large and resistance to shear low. Might Poisson's ratio obtained from elastic properties differentiate brittle from plastic behaviour when materials are stressed beyond σ_c?³⁰ The antecedents for this idea date back to the 1890s, when relationships for crystalline metals suggested that low values of Poisson's ratio might be related to hardness and also to high melting points.¹⁵ Despite much subsequent controversy concerning how general this might be, it now transpires that the fracture energy for metallic glasses actually does increase markedly within narrow ranges of Poisson's ratio for different materials.^30,90 There is no definitive answer as to why this critical change in mechanical behaviour should be so sharply defined by Poisson's ratio, but it does seem to be empirically correlated with the fragility of the liquid antecedent.³⁰ In so doing it draws a connection between the speculations of German¹⁵ and British⁶⁴ metallurgists around the start of the twentieth century and current ideas about the strength of modern materials being related to the dynamics of the molten state.^30,85,86

Conclusion

Consideration of the history of Poisson's ratio over two centuries reveals the cumulative career of this ubiquitous physical number. The three periods chosen—1811–70, 1870–1970, and 1970 to the present day—also chart the progress and diversification of science from classical mathematical physics to the applied sciences.

The first 60 years, as we have seen, started with Poisson's first major publication¹¹ as the zenith of Laplace's influence on French physics approached. The Note of 1827,¹ reacting to Cagniard de la Tour's unusual experiments on the elasticity of brass¹⁶ by predicting a single value for ν, was small by comparison; it may just have been opportune in an effort to shore up the then receding Newtonian agenda for corpuscular forces. Coincidentally, 1827 was the year in which Laplace died, which must have been portentous for Poisson as it was also the centenary of Newton's death. In his last decade Poisson mainly consolidated his research through the writing of books. Recognized abroad—notably by the award of the Copley Medal of the Royal Society in 1832³³—at home he was revered, sometimes feared, with his legacy often misunderstood³⁶. He became increasingly reclusive, routinely working in his home a 10-hour day, inaccessible to visitors.³³ As Guglielmo Libri (1803–69) recalled in his éloge: ‘he (Poisson) especially liked unsolved questions that had been treated by others or areas where there was still work to be done.’⁹¹ Fourier's earlier questioning of Poisson's originality³⁷ was never far away. Antoine Cournot (1801–77), who replaced Poisson as chair of the Jury d'Agrégation in mathematics in 1839, went further:⁹² ‘[Poisson] did not enjoy the rare good fortune of developing one of those completely new and striking conceptions that forever establish the fame of their innovator in the history of science.’ In the event, a ratio already well known at the time² acquired Poisson's name as the result of a controversial prediction that he made,^17,18 and two centuries later it has become a dominant theme in materials science.³⁰ Scientific concepts generally eclipse their originators, whoever they are.

It would seem that there was no follow-up to Poisson's prediction or Cagniard de la Tour's experiments, either in France or elsewhere, until after Poisson's death in 1840 (figure 2). The dispute that led to the establishment of the theory of elasticity based on the continuum hypothesis¹⁴ left discredited the molecular hypothesis, championed by Poisson^1,11 and Navier⁴.^18,61 But that was then. The virtue in taking the long view is that the campaigns by Kirchhoff, Thomson and others^18,42,43,45 against the single value for ν did not affect the subsequent increase in research into this dimensionless modulus (figure 2). Indeed, this must have been fuelled in part by the search for different values for ν among different metals, simply to assert the approach of Cauchy⁵ and Stokes.⁶ Importantly, by championing one hypothesis, attention was diverted from competing views, leading to important questions addressed by the early physics of crystallography^49,61 being ignored and the molecular cohesion of matter^9,17 being overlooked.

Instead, between 1870 and 1970, the continuum theory of elasticity^5,14 formed the foundation for the mechanics of engineering.^19,20 In the first 30 years of that period, the accurate measurement of elastic moduli, including Poisson's ratio, became possible,⁶⁰ and indeed proved necessary. This was primarily to characterize the mechanical performance of existing materials, such as metals^15,63,64 essential in the manufacture of armaments and later alloys developed for aerospace applications. With imperatives more concerned with predicting mechanical failure on the macro scale rather than how this might be precipitated from the dynamics of atoms and molecules, the shift in mechanics was not just away from the new physics of dislocations^65,66 but was also away from elastic behaviour itself, as materials were stressed beyond their critical yield points σ_c.^20,62 Here Poisson's ratio could be informative, the maximum value⁵² of 1/2 coinciding with shapelessness of plastic deformation in metals stressed above σ_c, ⁴⁵ exemplified by the incompressible nature of liquids and rubbers (figure 5). As new materials, such as plastics and glass and carbon fibre, were invented in the twentieth century,^21,59 their elastic properties became far more relevant because they were often attached to one another as composites and laminates,^22,67 creating structures that were generally anisotropic, for which ν could take both negative⁶⁷ and positive values (figure 4) in different directions. Serendipity in Materials Science and Engineering is often intertwined.

Despite the importance of these major developments, which were mostly in technology and were largely disregarded by the physics community,⁵⁹ the course of the effective medium theory of elasticity in general, and Poisson's ratio in particular, was in many ways reciprocally isolated from mainstream physics with the advent of quantum mechanics and crystallography.²³ As things transpired, however, the ensuing physics of condensed matter²⁵ brought with it a fundamental understanding of interatomic bonding, and with this came the eventual rehabilitation of the molecular hypothesis of the Arcueil circle and the Académie des Sciences,^9,10 notably as molecular dynamics emerged.^26,27 Imponderable materials finally become ponderable, a point that can be overlooked in treating the history of elasticity theory as a victory of the continuum hypothesis over the molecular hypothesis.¹²

Indeed, since 1970, understanding the molecular origin of elastic properties of materials has been an expanding theme in the physical sciences,^{15,30,78,84,85} with clear relationships emerging between the strength of intermolecular forces and smaller values for ν (see, for example, figure 3). However, this is only a fraction of the reawakening of interest in Poisson's ratio³⁰ that there has been since 1970. Bibliometrically the size of the renaissance has been huge (figure 2), coming from the geosciences, medicine, biology, materials science, chemistry, engineering and computational science—with little emerging from the mathematical physics that gave birth to elasticity theory. It is important to emphasize that a critical element in this resurgence has been the recognition that elasticity operates over multiple length scales, exemplified by the invention of isotropic materials with negative Poisson's ratio,²⁹ auxetic at the macro level but with positive microscopic values. Concurrently, the hierarchical structure of ancient materials such as cork (figure 7) has been used to explain its unusual mechanical properties, that Hooke first suspected⁸¹ but which were clarified and quantified through the anisotropy of Poisson's ratio.⁸⁰

The challenging of hypotheses is customarily seen as a refining process in the progress of science. For Poisson's ratio, however, the shape versus volume concept explained in different ways at different times has stayed intact for two centuries, once experimentation was sufficient for accurate measurement. The resilience of the ratio today lies in the fact that it applies equally well to continua and to atomic systems. Accordingly, the part that Poisson's ratio has played in the history of science is quite unusual, maybe unique: the integrity of the physical concept surviving controversy in mathematical physics (1811–70), adoption by materials science and engineering (1870–1970), and now revival in the applied sciences (1970 to the present day). Good ideas are always promiscuous.