Poisson’s ratio: identify the correct definition\n\nIn strength of materials, Poisson’s ratio (ν) relates the lateral contraction to the axial extension in a uniaxial test. Is the statement “Poisson's ratio is the ratio of linear strain to the volumetric strain” correct?

Introduction / Context:
Understanding Poisson’s ratio is fundamental in solid mechanics. It links how a material thins laterally when stretched longitudinally (or bulges when compressed). Confusing it with volumetric strain leads to serious errors in design and analysis.

Given Data / Assumptions:

• Uniaxial loading of a prismatic specimen within the elastic range.
• Lateral strain is measured perpendicular to the loading axis; linear (axial) strain is along the load.
• Small strain, linear elasticity (Hooke’s law) applies.

Concept / Approach:
Poisson’s ratio ν is defined as the magnitude of lateral strain divided by the linear (axial) strain: ν = −(ε_lateral / ε_axial). The minus sign reflects that lateral strain is typically opposite in sign to axial strain (contraction vs. extension). Volumetric strain is a different quantity equal to the sum of principal strains.

Step-by-Step Solution:
Define axial strain: ε_axial = ΔL / L.Define lateral strain: ε_lateral = Δd / d (perpendicular to loading).Poisson’s ratio: ν = − ε_lateral / ε_axial.Volumetric strain (small strain): ε_v = ε1 + ε2 + ε3 (sum of principal strains).Therefore ν does not equal “linear strain / volumetric strain” nor its inverse.

Verification / Alternative check:
For an isotropic linear elastic solid under uniaxial stress σ, the lateral strain equals −ν σ / E, and axial strain equals σ / E. Substituting confirms ν = −ε_lateral / ε_axial, independent of volumetric strain.

Why Other Options Are Wrong:
“True” is incorrect; ν is not defined using volumetric strain. “True only for incompressible materials” is still false; even for ν ≈ 0.5, the definition remains lateral-to-axial. “True only at very small strains” misstates the definition. “It equals bulk modulus divided by Young’s modulus” is incorrect; elastic constants relate as E = 3K(1 − 2ν) and E = 2G(1 + ν).

Common Pitfalls:
Mixing up ν with bulk modulus K or with volumetric strain; forgetting the negative sign convention; applying the definition outside the elastic range.

Final Answer:
False