Calculating Lateral and Longitudinal Strains

Understanding how materials deform under load is fundamental in mechanics. When a tensile load is applied to a metal bar, it stretches in the direction of the load (longitudinally) and shrinks perpendicular to the load (laterally).

Understanding Strain

Strain is a measure of deformation. It is defined as the change in dimension divided by the original dimension. Strain is a dimensionless quantity, although sometimes it is expressed with units like mm/mm or as a percentage or in scientific notation like $10^{-4}$.

• Longitudinal Strain: This is the strain in the direction of the applied load.
• Lateral Strain: This is the strain perpendicular to the applied load.

Calculating Longitudinal Strain

The longitudinal strain ($\varepsilon_L$) is calculated using the change in length ($\Delta L$) and the original gauge length ($L_0$).

Given:

• Original gauge length, $L_0 = 200$ mm
• Extension (change in length), $\Delta L = 0.2$ mm

The formula for longitudinal strain is:

$\varepsilon_L = \frac{\Delta L}{L_0}$

Substituting the given values:

$\varepsilon_L = \frac{0.2 \text{ mm}}{200 \text{ mm}}$

$\varepsilon_L = \frac{0.2}{200} = \frac{2 \times 10^{-1}}{2 \times 10^2} = 1 \times 10^{-1} \times 10^{-2} = 1 \times 10^{-3}$

In scientific notation asked in options, $1 \times 10^{-3}$ can also be written as $10 \times 10^{-4}$.

So, longitudinal strain is $10 \times 10^{-4}$.

Calculating Lateral Strain

The lateral strain ($\varepsilon_{Lat}$) is calculated using the change in thickness or width ($\Delta t$) and the original thickness or width ($t_0$).

Given:

• Original thickness, $t_0 = 40$ mm (since the section is 40 mm x 40 mm)
• Decrease in thickness (change in thickness), $\Delta t = 0.012$ mm

The formula for lateral strain is:

$\varepsilon_{Lat} = \frac{\Delta t}{t_0}$

Substituting the given values:

$\varepsilon_{Lat} = \frac{0.012 \text{ mm}}{40 \text{ mm}}$

$\varepsilon_{Lat} = \frac{0.012}{40} = \frac{12 \times 10^{-3}}{40} = \frac{12}{40} \times 10^{-3} = 0.3 \times 10^{-3}$

In scientific notation, $0.3 \times 10^{-3}$ can also be written as $3 \times 10^{-4}$.

So, lateral strain is $3 \times 10^{-4}$.

Summary of Strains

The question asks for the lateral and longitudinal strains respectively.

Lateral strain $= 3 \times 10^{-4}$

Longitudinal strain $= 10 \times 10^{-4}$

Therefore, the lateral and longitudinal strains are $3 \times 10^{-4}$ and $10 \times 10^{-4}$ respectively.