Poisson's Ratio Explained: How Materials Deform Perpendicular to Load

Poisson’s ratio is a fundamental material property that tells us how a material deforms in directions perpendicular to an applied load. This video introduces the concept with intuitive examples such as a stretching rubber band and a compressive cuboid, then defines the ratio between longitudinal and lateral strains. It explains the sign convention, the reason for the minus sign in the equation, and why the concept is valid for isotropic, elastic materials. The talk moves from simple uniaxial loading to tri-axial loading, highlights the range of possible values (-1 to 0.5), and shows common materials with typical values. It also covers special cases like cork with near-zero Poisson’s ratio and auxetic materials with negative values, and connects Poisson’s ratio to volumetric strain and generalized Hooke’s law.

Introduction to Poisson's ratio

Poisson's ratio is a dimensionless material property that describes how a solid deforms laterally when stretched or compressed along a single direction. The video uses simple demonstrations, starting with a rubber band that lengthens in tension and thins laterally, and a small orange cuboid under compressive or tensile loading to illustrate the same idea in three dimensions. The key point is that axial loading in one direction induces deformation in the perpendicular directions, and Poisson's ratio quantifies how large that lateral deformation is relative to the axial deformation.

Definition and sign convention

When a tensile force is applied along the longitudinal x-direction, the longitudinal strain is εx = ΔLx/Lx, and the lateral strains are εy and εz. By convention, tensile strains are positive and compressive strains negative. Poisson's ratio ν is defined as the negative ratio of the lateral strain to the longitudinal strain, ν = -εy/εx = -εz/εx. The minus sign ensures a positive ν for the usual case where lateral strains oppose the sign of the longitudinal strain. This definition is valid for isotropic, elastic materials; plastic deformation and anisotropy complicate the picture.

Isotropy and elastic regime

The discussion assumes isotropic materials, meaning identical properties in all directions, and deformations within the elastic region where Hooke’s law applies. If plastic deformation occurs, the simple Poisson’s ratio description no longer fully captures the material behavior. The video emphasizes these assumptions to delineate when the described relationships hold.

Relation to Hooke's law and generalized forms

Under uniaxial stress, the longitudinal strain is given by Hooke’s law as εx = σx/E, where E is Young's modulus. The lateral strains follow from Poisson’s ratio: εy = ν εx and εz = ν εx. For more complex loading, where stresses exist in all three directions, the simple uniaxial formula no longer suffices. The video derives the generalized Hooke’s law by combining Hooke’s law with Poisson’s ratio and the principle of superposition to determine strains in the X, Y, and Z directions. This provides a practical framework for predicting deformations under triaxial stress states.

Volumetric strain and incompressibility

Volumetric strain is the sum of the principal strains: εvol = εx + εy + εz. Substituting the relations from generalized Hooke’s law, one finds a key result: when ν = 0.5, the volumetric strain becomes zero, meaning the material is incompressible and its volume remains constant under deformation. Rubber is given as a classic example of an incompressible material, illustrating how large shape changes can occur without volume change.

Range of Poisson's ratio and material examples

The theoretical range for ν is from -1 to 0.5. In practice, most real materials have 0 ≤ ν ≤ 0.5, with many metals around 0.3. The video highlights cork as a near-zero ν material, which explains why cork easily fits into bottle necks without expanding laterally when compressed. It also introduces auxetic materials, a class of engineered materials with negative Poisson’s ratios that expand laterally when stretched and contract when compressed, a counterintuitive but increasingly exploited behavior in advanced applications.

Applications and significance

Poisson's ratio is essential in continuum mechanics for predicting how bodies deform under applied stresses. It feeds into the analysis of uniaxial and tri-axial loading, informs design decisions in mechanical and structural engineering, and helps in understanding material behavior during deformation. The video connects ν to practical concepts like volumetric strain and the generalized Hooke’s law, reinforcing its role in determining how a material responds to complex loading scenarios.

Summary of key takeaways

Poisson's ratio is a dimensionless parameter linking longitudinal and lateral deformations in isotropic, elastic materials. ν is defined as ν = -εlateral/εlongitudinal, with ν ranging theoretically from -1 to 0.5. Under uniaxial loading, lateral strains are proportional to the longitudinal strain, and the generalized Hooke’s law extends these ideas to tri-axial stress states. Volumetric strain reveals a special case: ν = 0.5 yields no volume change, characterizing incompressible materials like rubber. Real materials typically have ν between 0 and 0.5, though special engineered materials with negative ν (auxetics) open new possibilities in design. The concepts rely on isotropy and the elastic regime, forming a foundation for understanding material deformation under load.