Strain energy under a gradually applied axial load: Which expression is correct for a linearly elastic body (σ = stress, V = volume, E = Young's modulus)?

Introduction / Context:
Strain energy is the elastic energy stored when a body is loaded. For axial loading in the elastic range, the correct formula depends on how the load is applied—suddenly, gradually, or with impact. Here we consider a gradually applied load.

Given Data / Assumptions:

• Axially loaded prismatic member, linearly elastic behaviour.
• Stress at final load level is σ, volume is V, Young’s modulus is E.
• Load is applied gradually from zero to its final value.

Concept / Approach:
For linear elasticity: sigma = E * epsilon. Strain energy per unit volume u equals the area under the stress–strain curve up to σ. With a gradually applied load, the stress–strain path is a straight line from 0 to σ, giving a triangular area.

Step-by-Step Solution:

Verification / Alternative check:
Using force–displacement: U = 1/2 * P * δ. Substituting P = σA and δ = (σL)/E for a bar of area A and length L gives U = (σ^2 A L)/(2E) = (σ^2 V)/(2E).

Why Other Options Are Wrong:
Expressions without the 1/2 correspond to sudden loading or are dimensionally inconsistent.σ * V has wrong dimensions and ignores E.

Common Pitfalls:
Confusing gradually applied with suddenly applied load; forgetting that strain energy density equals the triangular area under the linear σ–ε curve.

Final Answer:

(σ^2 * V) / (2 * E)