The value of Poisson’s ratio lies between

The correct option is -1 to 1/2.

Poisson's ratio ($\sigma$) is defined as the ratio of lateral strain to longitudinal strain for a given material under tensile stress. The theoretical limits of Poisson's ratio can be determined using the relations between elastic constants.

The relationship between Young's modulus ($Y$), Bulk modulus ($K$), and Poisson's ratio ($\sigma$) is given by:

$Y=3K(1-2\sigma )$

Similarly, the relationship between Young's modulus ($Y$), Shear modulus ($\eta$), and Poisson's ratio ($\sigma$) is given by:

$Y=2\eta (1+\sigma )$

Since the elastic moduli ($Y$, $K$, and $\eta$) must always be positive values for physical stability, we can analyze the expressions inside the brackets:

1. From the first equation, since $Y$ and $K$ are positive:
$1-2\sigma >0$
$2\sigma <1$
$\sigma <\frac{1}{2}$

2. From the second equation, since $Y$ and $\eta$ are positive:
$1+\sigma >0$
$\sigma >-1$

Combining these two inequalities, the theoretical range for Poisson's ratio ($\sigma$) is:
$-1<\sigma <\frac{1}{2}$

Thus, the value of Poisson's ratio lies between -1 to 1/2.