Minimum and maximum values of Poisson’s ratio for a metal lies between

The correct option is 0 to 0.5.

Step-by-Step Explanation:

1. Definition of Poisson's Ratio:
Poisson's ratio (denoted by the Greek letter $\nu$ or $\mu$) is a measure of the Poisson effect, which describes the expansion or contraction of a material in directions perpendicular to the direction of loading. Mathematically, it is defined as the ratio of lateral strain to longitudinal strain under uniaxial tensile stress:

$\nu =-\frac{{\epsilon }_{\text{lateral}}}{{\epsilon }_{\text{longitudinal}}}$

2. Theoretical Limits from Elasticity Theory:
From the theory of isotropic linear elasticity, the relations between elastic constants (Young's modulus $E$, Shear modulus $G$, and Bulk modulus $K$) must yield positive values for these moduli to ensure stable material behavior under load. The relationships are given by:

$G=\frac{E}{2(1+\nu )}$

and

$K=\frac{E}{3(1-2\nu )}$

For the Shear modulus $G$ and Bulk modulus $K$ to be positive (assuming Young's modulus $E$ is positive):
From $1+\nu >0$, we get $\nu >-1$.
From $1-2\nu >0$, we get $\nu <0.5$.
Thus, the theoretical limits of Poisson's ratio for a stable, isotropic elastic material are:

$-1<\nu <0.5$

3. Practical Limits for Metals:
For most common physical materials, particularly metals, stretching a material in one direction causes it to contract in the transverse directions. This lateral contraction corresponds to a positive Poisson's ratio ($\nu >0$).
Metals do not exhibit auxetic (negative Poisson's ratio) behavior under ordinary conditions. Therefore, the minimum practical value for metals starts at 0 (representing no lateral contraction, such as cork, though most metals have values around 0.25 to 0.35). The maximum theoretical limit is 0.5 (representing a perfectly incompressible material where volume remains constant during deformation).
Consequently, for metals, the Poisson's ratio practically lies between 0 and 0.5.