The modulus of resilience is characterised by the area located under the stress-strain curve up to the:

Explanation:

Modulus of Resilience

• The modulus of resilience is a property of materials that measures the maximum energy a material can absorb per unit volume without undergoing permanent deformation. It represents the area under the stress-strain curve up to the elastic limit (or proportional limit), where the material behavior remains elastic. This energy is recoverable after the removal of the applied load. The modulus of resilience is an important parameter in designing components subjected to dynamic and impact loads, as it provides insight into a material's ability to absorb energy without yielding.

The modulus of resilience can be mathematically expressed as:

U_r = (σ_y)² / 2E

Where:

• U_r = Modulus of resilience (energy per unit volume)
• σ_y = Yield strength of the material
• E = Modulus of elasticity of the material

The modulus of resilience is essentially the area of the triangle formed under the stress-strain curve up to the proportional limit (elastic region). The proportional limit is the point on the stress-strain curve up to which Hooke's Law is valid, and the relationship between stress and strain is linear.

Correct Option Analysis:

The correct option is:

Option 1: Proportional limit

This is the correct answer because the modulus of resilience is characterized by the area under the stress-strain curve up to the proportional limit. Within this region, the material behaves elastically, and all the energy absorbed during deformation can be recovered when the load is removed. Beyond the proportional limit, the material enters the plastic deformation region, where permanent deformation occurs, and the energy absorbed in this region is not recoverable. Hence, only the elastic region (up to the proportional limit) is used to calculate the modulus of resilience.