Strength of materials – Resilience:\nIs the modulus of resilience defined as the proof resilience (i.e., maximum elastic strain energy) stored per unit volume of a material?

Introduction / Context:
Modulus of resilience is a key property in strength of materials that quantifies the amount of elastic strain energy a material can absorb per unit volume right up to its elastic limit (also called proof stress for design). This concept is crucial when selecting materials for components that must endure impact or cyclic loading without permanent deformation, such as springs, couplings, and energy-absorbing devices.

Given Data / Assumptions:

• The discussion is limited to linear elastic behavior up to the elastic limit.
• Stress–strain relationship is taken as linear with slope E (Young's modulus).
• Proof resilience is the total elastic strain energy stored at the elastic limit.

Concept / Approach:
Strain energy density (energy per unit volume) in uniaxial linear elasticity is the area under the stress–strain curve up to a specified stress. For a straight line up to the elastic limit, this area is a right triangle. Therefore, the modulus of resilience equals the triangular area up to proof stress.

Step-by-Step Solution:

Verification / Alternative check:
This same result can be written using resilience terminology: modulus of resilience = proof resilience per unit volume = sigma_p^2 / (2 * E). Units check: (stress^2)/(modulus) gives energy per volume, as required.

Why Other Options Are Wrong:

• Incorrect: contradicts the standard definition used in design handbooks.
• Not applicable: the definition is broadly applicable to linear elastic materials.
• Depends on loading type: while multi-axial states need corresponding forms, the uniaxial definition remains per unit volume up to elastic limit.

Common Pitfalls:
Confusing modulus of resilience with toughness (the latter is the entire area under the stress–strain curve to fracture). Another mistake is using yield strength beyond proportional limit without clarifying the definition of proof stress.

Final Answer:
Correct