Mechanics of materials – resilience: In a standard stress–strain diagram for a ductile metal, the area under the curve up to which point is defined as the modulus of resilience (elastic strain energy per unit volume)?

Introduction / Context:
The modulus of resilience quantifies the energy a material can absorb elastically and release upon unloading. It is critical in applications involving impact or spring-like behavior, where permanent deformation must be avoided.

Given Data / Assumptions:

• Conventional engineering stress–strain curve for a ductile metal.
• Elastic region followed by yielding and plastic deformation.
• Elastic limit approximates the end of purely elastic behavior (for many metals it is close to the proportional limit and lower yield point).

Concept / Approach:
Modulus of resilience is the area under the stress–strain curve from zero strain up to the elastic limit. Within this region, stress is recoverable and energy is stored elastically. For linear elasticity, an approximate formula is U_r = σ_y^2 / (2E) when yield and elastic limits coincide; otherwise, the integral is taken to the elastic limit.

Step-by-Step Solution:
Identify the elastic region of the curve ending at the elastic limit.Define modulus of resilience as the elastic strain energy density up to that limit.For linear metals: U_r ≈ (1/2) * σ_y * ε_y = σ_y^2 / (2E) when σ_y corresponds to elastic limit/yield.Hence, the reference point is the elastic limit.

Verification / Alternative check:
Materials handbooks define resilience with respect to the elastic limit; toughness, by contrast, is the area up to fracture, far beyond yield.

Why Other Options Are Wrong:
Yield strength: often close, but resilience is formally referenced to the elastic limit, not necessarily the engineering yield point in all materials.

Proportional limit: sometimes near the elastic limit, but the accepted definition uses the elastic limit where unloading is fully recoverable.

Maximum point (UTS): relates to necking and toughness, not resilience.

Common Pitfalls:

• Mixing up resilience (elastic energy) with toughness (total energy to fracture).
• Using σ_y without checking if the material exhibits a clear yield plateau.

Final Answer:
Elastic limit