Question

A cylindrical sample of granite (diameter = 54.7 mm; length = 137 mm) shows a linear relationship between axial stress and axial strain under uniaxial compression up to the peak stress level at which the specimen fails. If the uniaxial compressive strength of this sample is 200 MPa and the axial strain corresponding to this peak stress is 0.005, the Young's modulus of the sample in GPa is _______ (in integer).

Hint:

Always remember: For linear elastic behavior, \(E = \sigma / \varepsilon\). Check the units carefully — stress in MPa and strain as dimensionless — then convert MPa to GPa if required.

Answer 1

Step 1: Recall the definition of Young's modulus.
Young's modulus \(E\) is the ratio of stress to strain: \[ E = \frac{\sigma}{\varepsilon} \] where,
\(\sigma =\) axial stress,
\(\varepsilon =\) axial strain. Step 2: Substitute given values.
Given uniaxial compressive strength (peak stress): \[ \sigma = 200 \, \text{MPa} \] Axial strain at failure: \[ \varepsilon = 0.005 \] Step 3: Calculate Young's modulus.
\[ E = \frac{\sigma}{\varepsilon} = \frac{200 \, \text{MPa}}{0.005} \] \[ E = 40,000 \, \text{MPa} \] Step 4: Convert to GPa.
Since \(1 \, \text{GPa} = 1000 \, \text{MPa}\): \[ E = \frac{40,000}{1000} = 40 \, \text{GPa} \] Final Answer: \[ \boxed{40 \, \text{GPa}} \]