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How is Young’s modulus calculated from the stretching of a metal bar?

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mechanics of solids Figure 1: The position vector x and the velocity vector v of a material point, the body force fdV acting on an element dV of volume, and the surface force TdS acting on an element dS of surface in a Cartesian coordinate system 1, 2, 3 (see text).

Young’s modulus, also known as the modulus of elasticity, is a numerical constant that measures a material's ability to withstand changes in length when subjected to lengthwise tension or compression. It is calculated as the ratio of longitudinal stress to strain.

For a metal bar with an original length $L_{0}$ and a cross-sectional area $A$ , subjected to a tensile force $F$ that stretches it to a new length $L_{n}$ , the calculation is as follows:

Stress: The tensile force divided by the cross-sectional area, expressed as $F / A$ .
Strain: The relative deformation, calculated as the change in length divided by the original length: $(L_{n} - L_{0}) / L_{0}$ .

Mathematically, Young’s modulus ( $Y$ ) is expressed as:

$Y = \frac{stress}{strain} = \frac{F L_{0}}{A (L_{n} - L_{0})}$

This relationship is a specific application of Hooke’s law and is valid only within the range where the material exhibits elastic behavior—meaning the stress is proportional to the strain and the bar returns to its original dimensions once the force is removed.1