Question

The modulus of resilience is equal to:

• A. Area under elastic region only
• B. Area under entire stress-strain curve
• C. Maximum stress \(\times\) strain
• D. Plastic work

Hint:

Be clear on these area-under-curve energy terms:
- Area under Elastic Region \(\rightarrow\) Modulus of Resilience (\( \sigma_y^2 / 2E \)).
- Area under Entire Curve (Elastic + Plastic) \(\rightarrow\) Modulus of Toughness.
To maximize resilience (e.g., for springs), select materials with high yield strength and low elastic modulus.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question asks for the mathematical/graphical definition of the "modulus of resilience" of a material from its uniaxial tensile stress-strain diagram.

Step 2: Key Formula or Approach:
Resilience is the capacity of a material to absorb energy when it is deformed elastically and then, upon unloading, to have this energy recovered. The Modulus of Resilience (\( U_r \)) is defined as the strain energy per unit volume absorbed up to the elastic limit (yield point \( \sigma_y \)): \[ U_r = \int_0^{\epsilon_y} \sigma d\epsilon \] For a linear elastic material obeying Hooke's Law (\( \sigma = E\epsilon \)), this integration simplifies to: \[ U_r = \frac{1}{2} \sigma_y \epsilon_y = \frac{\sigma_y^2}{2E} \] where:
\( \sigma_y \) is the yield strength.
\( E \) is the Young's modulus.

Step 3: Detailed Explanation:

• Graphical Representation:
- On a stress-strain plot, the integral \( \int_0^{\epsilon_y} \sigma d\epsilon \) represents the area under the curve.
- Specifically, it is the area of the triangle bounded by the origin, the yield point, and the strain axis (representing the elastic region only).
- This represents the maximum elastic energy storage capacity per unit volume of the material.

• Contrast with Modulus of Toughness:
- The area under the *entire* stress-strain curve up to the point of fracture is called the modulus of toughness. This represents the total energy absorbed before catastrophic failure.
- Plastic work represents the energy consumed in permanent plastic deformation.


Step 4: Final Answer:
The modulus of resilience is equal to the area under the elastic region of the stress-strain curve.
Therefore, the correct choice is option (A).