The shear strain energy stored in a solid body under a state of pure shear is to be expressed in terms of shear stress τ, shear modulus C (modulus of rigidity, G), and volume V. Select the correct expression for the total strain energy due to shear.

Introduction:
Strain energy represents recoverable elastic energy stored in a body under loading. Under pure shear, this energy depends on shear stress, shear strain, and the material's shear modulus. The correct expression is essential for impact, resilience, and torsion problems.

Given Data / Assumptions:

• Uniform shear stress τ in the body.
• Linear elastic behavior with shear modulus C (G).
• Body volume V.

Concept / Approach:
For linear elasticity, energy density u = 0.5 * stress * strain. In shear: u = 0.5 * τ * γ. Since γ = τ / C, substitute to get u = 0.5 * τ * (τ / C) = τ^2 / (2C). Total energy U = u * V = (τ^2 * V) / (2C).

Step-by-Step Solution:
1) Start with u = 0.5 * τ * γ.2) Use γ = τ / C for linear shear.3) Obtain u = τ^2 / (2C).4) Multiply by volume: U = u * V = (τ^2 * V) / (2 * C).

Verification / Alternative check:
Dimensional check: τ in N/m^2, C in N/m^2, so τ^2/C has units of N/m^2; multiplying by V (m^3) gives N·m (energy), which is consistent.

Why Other Options Are Wrong:
(τ * V) / C and (τ * V) / (2C): miss the quadratic dependence on τ for energy.

(τ^2 * V) / C: missing the 1/2 factor required by energy derivation.

(τ^2) / (2 C V): incorrect placement of V in the denominator.

Common Pitfalls:
Forgetting that energy scales with the square of stress in linear elasticity; omitting the 1/2 factor.

Final Answer:
U = (τ^2 * V) / (2 * C)