Consider a linear rectangular thin sheet of metal, subjected to uniform uniaxial tensile stress of 100 MPa along the length direction. Assume plane stress conditions in the plane normal to the thickness. The Young’s modulus E = 200 MPa and Poisson’s ratio v = 0.3 are given. The principal strains in the plane of the sheet are

The correct option is (0.5, -0.15).

Let us break down the logical reasoning and mathematical derivation step-by-step to understand why this option is correct.

1. Understanding the Given Data:
We are given a linear rectangular thin sheet of metal subjected to a uniform uniaxial tensile stress along its length direction (let this be the x-direction).
- Uniaxial tensile stress along the length: ${\sigma }_x=100\text{MPa}$
- Since the loading is uniaxial, the stress in the perpendicular in-plane direction (y-direction) is: ${\sigma }_y=0$
- Young’s modulus: $E=200\text{MPa}$
- Poisson’s ratio: $\nu =0.3$ (represented as v in the question description)
- The sheet is thin and we assume plane stress conditions in the plane of the sheet (x-y plane).

2. Formulas for Principal Strains under Plane Stress:
Using Hooke's Law for generalized plane stress, the principal strains in the plane of the sheet (${\epsilon }_x$ and ${\epsilon }_y$) are related to the stresses by the following equations:

${\epsilon }_x=\frac{1}{E}({\sigma }_x-\nu {\sigma }_y)$

${\epsilon }_y=\frac{1}{E}({\sigma }_y-\nu {\sigma }_x)$

3. Step-by-Step Calculation:

First, let us calculate the principal strain along the direction of the applied load (${\epsilon }_x$):
Substituting ${\sigma }_x=100\text{MPa}$, ${\sigma }_y=0$, and $E=200\text{MPa}$ into the equation for ${\epsilon }_x$:

${\epsilon }_x=\frac{1}{200}(100-0.3\times 0)$

${\epsilon }_x=\frac{100}{200}=0.5$

Next, let us calculate the principal strain in the lateral direction (${\epsilon }_y$) due to the Poisson effect:
Substituting the given values into the equation for ${\epsilon }_y$:

${\epsilon }_y=\frac{1}{200}(0-0.3\times 100)$

${\epsilon }_y=\frac{-30}{200}=-0.15$

4. Conclusion:
The principal strains in the plane of the sheet are $({\epsilon }_x,{\epsilon }_y)=(0.5,-0.15)$, which matches the correct option.