Poisson's ratio
Poisson's ratio, denoted by the symbol \nu, is a dimensionless mechanical property that describes the Poisson effect in materials, defined as the negative ratio of the transverse (lateral) strain to the axial strain under uniaxial loading.[1] This ratio quantifies how a material deforms perpendicular to the applied stress, typically contracting laterally when stretched axially in most common substances.[2] Named after the French mathematician and physicist Siméon Denis Poisson, who derived it theoretically in 1827 as part of his work on the molecular-continuum theories of elasticity, the concept challenged prevailing hypotheses about material isotropy and deformation.[3] In isotropic linear elastic materials, Poisson's ratio ranges from -1 to 0.5, with values approaching 0.5 indicating near-incompressibility, as seen in rubbers and liquids where volume change is minimal under deformation.[4] For most engineering metals like steel and aluminum, \nu is approximately 0.3, reflecting moderate lateral contraction during axial extension.[5] Materials exhibiting negative Poisson's ratios, known as auxetics, expand laterally when stretched and were first realized in engineered foams in the late 1980s, offering unique properties for applications in vibration damping and biomedical devices.[6] Poisson's ratio plays a crucial role in continuum mechanics, linking to other elastic constants such as Young's modulus and the shear modulus through relations like \nu = \frac{E}{2G} - 1, where E is Young's modulus and G is the shear modulus, enabling predictions of material behavior under complex loading in fields like structural engineering and geomechanics.[7] Its measurement and variation with factors like temperature, strain rate, and microstructure are essential for designing composites, polymers, and metamaterials with tailored deformation responses.[8]Definition and Origin
Historical Development
The phenomenon of lateral contraction accompanying longitudinal extension in elastic materials was first systematically observed and quantified by Thomas Young in his 1807 A Course of Lectures on Natural Philosophy and the Mechanical Arts. Young described the effect for various substances, noting the proportional relationship between lateral contraction and longitudinal extension.[3] Building on such empirical observations, Siméon Denis Poisson introduced the theoretical framework for what became known as Poisson's ratio in his 1829 memoir Mémoire sur l'équilibre et le mouvement des corps élastiques homogènes. Drawing from a molecular-kinetic theory of elasticity, Poisson derived that, for isotropic materials, the negative ratio of transverse strain to axial strain under uniaxial stress is universally 1/4, a prediction rooted in assumptions about inter-molecular forces and equilibrium.[3] This work marked a pivotal shift from descriptive phenomenology to a predictive model within continuum mechanics.[9] In the broader 19th-century development of elastic theory, Augustin-Louis Cauchy contributed foundational mathematical formulations between 1822 and 1828, establishing the stress-strain relations that incorporated the lateral contraction effect as an inherent property of elastic solids.[10] Similarly, George Gabriel Stokes advanced the understanding in 1845 through his analysis of elastic vibrations and deformations, integrating Poisson's ratio into the general relations among elastic constants and highlighting its implications for wave propagation in solids.[11] Early experimental verification came with Wertheim's measurements in 1848, who reported values around 1/3 for brass and glass, and Ferdinand Redtenbacher's precise measurements on metals in 1858, conducted at the Polytechnic Institute in Karlsruhe, which provided quantitative data on Poisson's ratio values deviating from Poisson's predicted constant, thus prompting further scrutiny of the molecular assumptions underlying the theory.[12][13]Basic Concept
Poisson's ratio, denoted by the symbol ν, is a fundamental measure of a material's lateral response to axial deformation, defined as the negative ratio of the transverse strain to the axial strain under uniaxial loading. This ratio quantifies how much a material contracts (or expands) in directions perpendicular to the applied stretch, capturing the coupled nature of deformation in elastic solids.[14] Physically, this phenomenon arises in a uniaxial tension test, where a prismatic specimen—such as a cylindrical rod—is subjected to pulling forces along its longitudinal axis, causing it to elongate while typically narrowing in the transverse directions. The observed contraction reflects the material's internal atomic or molecular rearrangements that resist changes in volume, particularly in nearly incompressible substances like rubber, where the lateral shrinkage balances the axial extension to preserve overall volume. For such incompressible materials, ν approaches 0.5, ensuring exact volume conservation during small elastic deformations.[15][16] For most engineering materials, Poisson's ratio falls within the range 0 < ν < 0.5, indicating positive lateral contraction without full incompressibility, though values near zero are rare and imply minimal transverse effect. This range stems from thermodynamic stability constraints in isotropic linear elasticity, highlighting ν's role as a key indicator of a material's compressibility and deformation behavior.[17]Mathematical Description
Strain-Based Derivation
In the context of uniaxial loading along the x-direction, the axial strain \varepsilon_x is defined as the relative change in length: \varepsilon_x = \frac{\Delta L}{L} where \Delta L is the elongation and L is the original length of the specimen.[18] The transverse strain \varepsilon_y in a perpendicular direction, such as the y-direction, is similarly the relative change in width: \varepsilon_y = \frac{\Delta w}{w} where \Delta w is the lateral contraction and w is the original width.[18] Poisson's ratio \nu is then defined as the negative ratio of the transverse strain to the axial strain: \nu = -\frac{\varepsilon_y}{\varepsilon_x} This expression quantifies the proportional lateral contraction that accompanies axial extension under uniaxial stress conditions, assuming small deformations where strains are linear.[18] Extending this to three dimensions requires the strain tensor \varepsilon_{ij}, which captures the full deformation state through its components, with normal strains along the principal axes. For an isotropic material subjected to uniaxial stress along the x-direction, the relevant strain tensor components are \varepsilon_{xx} (axial), \varepsilon_{yy}, and \varepsilon_{zz} (transverse), while shear components \varepsilon_{ij} (for i \neq j) vanish in the principal coordinate system. Poisson's ratio is expressed as \nu = -\frac{\varepsilon_{ii}}{\varepsilon_{jj}} (no sum over indices, i \neq j), where \varepsilon_{jj} is the axial strain and \varepsilon_{ii} are the equal transverse strains under this loading.[19] Geometrically, these strains manifest in the deformation of a prismatic volume element. For infinitesimal strains, the relative volume change \frac{\Delta V}{V} of such an element is the first invariant of the strain tensor, approximated as the sum of the principal normal strains: \frac{\Delta V}{V} \approx \varepsilon_x + \varepsilon_y + \varepsilon_z Under uniaxial loading, substituting \varepsilon_y = \varepsilon_z = -\nu \varepsilon_x yields \frac{\Delta V}{V} \approx \varepsilon_x (1 - 2\nu), highlighting Poisson's ratio's role in determining whether the material exhibits volume conservation (as \nu approaches 0.5) or expansion/contraction.[5]Relation to Elastic Constants
In isotropic linear elasticity, the generalized Hooke's law relates the stress tensor \sigma_{ij} to the strain tensor \epsilon_{ij} through the Lamé constants \lambda and \mu as \sigma_{ij} = \lambda \delta_{ij} \epsilon_{kk} + 2\mu \epsilon_{ij}, where \delta_{ij} is the Kronecker delta and \epsilon_{kk} is the trace of the strain tensor (summation over repeated indices implied). This form captures both volumetric and deviatoric responses, with \mu representing the shear response and \lambda influencing the volumetric change.[20] Poisson's ratio \nu emerges from this relation under uniaxial loading, where the lateral strain is -\nu times the axial strain. By applying the constitutive equation to a uniaxial stress state \sigma_{11} = \sigma, \sigma_{22} = \sigma_{33} = 0, the resulting strains yield \nu = \frac{\lambda}{2(\lambda + \mu)}. This expression links \nu directly to the Lamé constants, highlighting its role in describing transverse contraction relative to longitudinal extension. The shear modulus G is identical to \mu, providing a direct connection to shear deformation.[20][21] Further relations connect \nu to other fundamental elastic moduli. Young's modulus E, which measures stiffness under uniaxial tension, relates via \nu = \frac{E - 2G}{2G}, or equivalently E = \frac{G(3\lambda + 2\mu)}{\lambda + \mu}. The bulk modulus K, governing resistance to uniform compression, is K = \lambda + \frac{2}{3}\mu, leading to \nu = \frac{3K - E}{6K}. These interrelations allow \nu to be expressed or computed from any two independent moduli, facilitating material characterization across experimental methods. The inverse forms for the Lamé constants are \lambda = \frac{E \nu}{(1 + \nu)(1 - 2\nu)} and \mu = \frac{E}{2(1 + \nu)}, enabling practical computation from measured E and \nu.[22][23] Thermodynamic stability for isotropic materials imposes bounds on \nu, requiring the stiffness matrix to be positive definite: \mu > 0 and $3\lambda + 2\mu > 0, which translates to -1 < \nu < 0.5. The lower bound \nu > -1 ensures resistance to excessive expansion under tension, while the upper bound \nu < 0.5 prevents instability under compression.[24]Material Classifications
Isotropic Case
In isotropic materials, Poisson's ratio is defined as a single scalar value, denoted by \nu, that quantifies the uniform lateral contraction (or expansion) relative to axial extension (or compression) in all transverse directions perpendicular to the applied load. This uniformity arises because the material exhibits identical mechanical properties regardless of the direction of measurement or loading, making \nu a material constant that applies equally across all orientations. For such solids under linear elastic conditions, the transverse strain \epsilon_t is related to the axial strain \epsilon_a by \nu = -\frac{\epsilon_t}{\epsilon_a}, ensuring symmetric deformation responses. The isotropic case assumes full rotational symmetry in the material's elastic properties, which reduces the number of independent elastic constants to just two, such as Young's modulus E and Poisson's ratio \nu, or alternatively the Lamé parameters \lambda and \mu. This symmetry implies that the stress-strain relationship can be fully described by the generalized Hooke's law without directional dependencies, valid for small deformations within the elastic regime. These assumptions hold for homogeneous materials like many metals and polymers, where microstructural isotropy at the macroscopic scale dominates. Representative examples of isotropic materials include metals such as steel, where \nu \approx 0.3 leads to moderate lateral contraction during uniaxial tension, and elastomers like rubber, with \nu \approx 0.5 indicating near-incompressible behavior and minimal volume change. In both cases, the uniform strain response ensures that the material deforms predictably without preferred directions, facilitating straightforward engineering applications like structural design. These values reflect the material's inherent isotropy, where the Poisson effect is consistent across loading axes. Measurement of Poisson's ratio in isotropic materials typically involves uniaxial tensile or compressive tests, where strain gauges or digital image correlation capture both axial and transverse displacements, yielding a consistent \nu value independent of the loading direction due to the material's symmetry. Such tests confirm the scalar nature of \nu by averaging results from multiple orientations, with standard deviations often below 5% for well-characterized samples. This directional invariance distinguishes isotropic measurements from those in textured materials.Anisotropic Case
In anisotropic materials, Poisson's ratio exhibits direction dependence, varying according to the orientation of applied stress and the resulting transverse strain, unlike the single uniform value characteristic of isotropic materials. This directional variability arises from the inherent lack of symmetry in the material's elastic response, where the lateral contraction in one direction may differ significantly from that in another under the same axial loading.[25] The tensorial representation of Poisson's ratio is derived from the fourth-rank compliance tensor S_{ijkl}, which relates strains to stresses via Hooke's law in its general form: \epsilon_{ij} = S_{ijkl} \sigma_{kl}. For a uniaxial stress along direction i, the apparent Poisson's ratio \nu_{ij} (with i \neq j) is defined as the negative ratio of the transverse strain in direction j to the axial strain in direction i: \nu_{ij} = -\frac{S_{ij}}{S_{ii}} in Voigt notation (no summation). This formulation generalizes the isotropic scalar Poisson's ratio, which emerges as a special case when all relevant compliance components are equal. In the most general anisotropic case, such as triclinic crystals lacking any symmetry elements, the stiffness or compliance tensor possesses 21 independent elastic constants, allowing for a rich set of direction-dependent Poisson's ratios that must be characterized fully to predict material behavior.[26] A representative example is single-crystal quartz (\alpha-quartz), a trigonal material where Poisson's ratios vary markedly with crystallographic orientation due to its anisotropic stiffness. Measurements indicate values such as \nu_{12} = -S_{12}/S_{11} \approx 0.130 and \nu_{13} = -S_{13}/S_{11} \approx 0.119 in principal directions, with overall values ranging from negative (auxetic behavior in some orientations) to approximately 0.17 depending on the loading axis relative to the crystal lattice.[27][28][29] This variation underscores the need for orientation-specific analysis in applications like piezoelectric devices. Determining the complete stiffness tensor for anisotropic design poses significant challenges, as it requires measuring all independent constants through multiple experimental configurations or advanced techniques such as resonant ultrasound spectroscopy or Brillouin scattering, due to the complexity of decoupling directional couplings and ensuring thermodynamic consistency. Incomplete characterization can lead to inaccurate predictions of deformation, particularly in engineering contexts involving composites or crystals where symmetry is minimal.[30]Specialized Material Behaviors
Orthotropic Materials
Orthotropic materials exhibit three mutually perpendicular planes of symmetry, leading to a simplified form of elastic anisotropy compared to general anisotropic materials. In linear elasticity, such materials are characterized by nine independent elastic constants: three Young's moduli E_x, E_y, E_z along the principal directions, three shear moduli G_{xy}, G_{yz}, G_{zx}, and three independent Poisson's ratios \nu_{xy}, \nu_{yz}, \nu_{zx}.[31] These Poisson's ratios quantify the lateral strain response in one direction due to axial strain in another, and because of the material's symmetry, there are six distinct ratios \nu_{ij} (for i \neq j), such as \nu_{xy} \neq \nu_{xz}, though only three are independent.[32] A key symmetry relation governs these Poisson's ratios in orthotropic materials, ensuring thermodynamic consistency and reciprocity between off-diagonal stiffness terms. Specifically, the ratios satisfy \frac{\nu_{ij}}{E_i} = \frac{\nu_{ji}}{E_j} for i \neq j, where \nu_{ji} is the Poisson's ratio for contraction in the i-direction due to extension in the j-direction.[33] This relation links the apparent compliance in coupled directions and reduces the number of free parameters needed to fully describe the material's response under multiaxial loading. Common examples of orthotropic materials include natural substances like wood, which displays unique mechanical properties along its longitudinal (fiber), radial, and tangential directions due to its cellular structure.[34] Engineered fiber-reinforced composites, such as carbon-fiber laminates, also exemplify orthotropy, where aligned fibers in the laminate planes create directionally dependent Poisson effects, often with lower transverse Poisson's ratios perpendicular to the fiber direction.[35] These materials are particularly valuable in applications involving layered structures, like aerospace panels or sporting goods, where orthotropic modeling captures essential behaviors without the complexity of 21 constants required for fully anisotropic cases.[36]Transversely Isotropic Materials
Transversely isotropic materials possess material properties that are isotropic within a designated plane, termed the plane of isotropy, while differing along the perpendicular axis of symmetry. This symmetry class requires only five independent elastic constants: the longitudinal Young's modulus E_L along the unique axis, the transverse Young's modulus E_T in the plane, the longitudinal shear modulus G_{LT}, the transverse shear modulus G_{TT}, and two distinct Poisson's ratios. Specifically, the in-plane Poisson's ratio \nu_{12} equals \nu_{21} due to the planar symmetry, but the perpendicular Poisson's ratio \nu_{13} (or \nu_{31}) generally differs from the in-plane value, reflecting the directional distinction.[37][38] Representative examples include unidirectional fiber-reinforced composites, in which aligned fibers confer the unique axial properties while the matrix provides isotropy in the transverse plane; rolled metal sheets, where manufacturing processes induce equivalent behavior in all in-plane directions; and biological tissues such as skin, which exhibit transverse isotropy from oriented collagen fibers.[37][39][40] Under uniaxial loading along the unique axis, the strain response in transversely isotropic materials manifests as cylindrical deformation patterns, with uniform transverse contraction (or expansion) across the isotropic plane dictated by \nu_{13}, ensuring axisymmetric behavior without preferred directions in that plane.[41] Transversely isotropic behavior emerges as a simplification of orthotropic materials by enforcing equality of in-plane properties, such as setting the two transverse Young's moduli equal and the in-plane Poisson's ratios symmetric (\nu_{12} = \nu_{21}), thereby reducing the nine independent constants of orthotropy to five while preserving the reciprocity relation \nu_{ij}/E_i = \nu_{ji}/E_j.[42]Observed Values and Examples
Typical Materials
Poisson's ratio for typical engineering materials, assuming isotropic behavior, generally falls between 0.2 and 0.5, reflecting the material's resistance to lateral expansion or contraction under uniaxial stress.[43] These values are derived from standard tensile tests and are essential for predicting deformation in structural applications. Variations occur due to factors such as temperature and processing history, which can slightly alter the elastic response. In metals, Poisson's ratios are typically around 0.3, indicating moderate lateral contraction. For aluminum, the value is approximately 0.33, while for mild steel it ranges from 0.27 to 0.30, and for copper it is about 0.34.[44] [43] Processing effects, such as annealing versus cold working, can introduce minor differences; for instance, cold-worked metals may exhibit slightly lower values due to increased dislocation density affecting elastic anisotropy at the microscopic level.[45] Temperature also influences these ratios, with mild steel showing a slight increase (up to 0.01-0.02) as temperature rises from room temperature to 200°C, linked to thermal expansion and modulus changes. Polymers exhibit higher Poisson's ratios, often approaching 0.45-0.5 near incompressibility, due to their molecular chain flexibility. High-density polyethylene has a value of 0.40-0.45, while polystyrene is around 0.33-0.34.[46] [43] These materials' ratios can vary with strain rate and temperature, particularly above the glass transition point, where increased chain mobility reduces lateral contraction resistance. Ceramics, being brittle and rigid, show lower Poisson's ratios, typically 0.2-0.25, owing to their strong ionic or covalent bonding. Soda-lime glass has a value of 0.22-0.25, and alumina (Al₂O₃) is approximately 0.22.[43] [47] Temperature effects are minimal up to 500°C, but processing like sintering can influence porosity, indirectly lowering the ratio by 0.01-0.03 in denser forms.[48] The following table summarizes representative values from engineering handbooks and material databases:| Material Category | Specific Material | Poisson's Ratio | Source |
|---|---|---|---|
| Metals | Aluminum | 0.33 | Engineering ToolBox[43] |
| Metals | Mild Steel | 0.27-0.30 | Stanford Advanced Materials[44] |
| Metals | Copper | 0.34 | Stanford Advanced Materials[44] |
| Polymers | High-Density Polyethylene | 0.40-0.45 | INEOS O&P USA[46] |
| Polymers | Polystyrene | 0.34 | Engineering ToolBox[43] |
| Ceramics | Soda-Lime Glass | 0.22-0.25 | Engineering ToolBox[43] |
| Ceramics | Alumina (Al₂O₃) | 0.22 | AZoM[47] |