Axial deformation and Young’s Modulus for a prismatic rod A straight rod of uniform cross-sectional area A and length L is subjected to an axial force P and undergoes an elastic deformation δ. What is the Young’s Modulus E of the material?

Introduction / Context:
In strength of materials, axial deformation of a prismatic (uniform) rod under axial load provides a direct route to determine Young’s Modulus E when the load, geometry, and measured elongation are known. This test principle underlies tensile testing and field evaluations of stiffness.

Given Data / Assumptions:

• Rod length = L; uniform area = A.
• Axial force = P (within elastic limit).
• Measured elastic extension = δ.
• Homogeneous, isotropic, linear-elastic behaviour is assumed.

Concept / Approach:
Axial stress σ = P / A. Axial strain ε = δ / L. By definition, Young’s Modulus E = σ / ε. Substituting yields E = (P / A) / (δ / L) = (P * L) / (A * δ). This relation is foundational for elastic analysis of bars and for calibration of material properties.

Step-by-Step Solution:
Compute stress: σ = P / A.Compute strain: ε = δ / L.Apply definition: E = σ / ε = (P / A) / (δ / L).Simplify: E = (P * L) / (A * δ).

Verification / Alternative check:
Dimension check: [E] = force/area divided by deformation/length = (N/m^2) / (m/m) = N/m^2, consistent with modulus units (Pa).

Why Other Options Are Wrong:
(A * δ) / (P * L) and others: These invert the correct relationship or scramble variables; they would not pass a simple dimensional or limiting-case check (e.g., larger P should increase E, not decrease it).

Common Pitfalls:

• Confusing engineering strain (δ/L) with percentage elongation (100 * δ/L).
• Using plastic deformation values; the formula is valid only in the elastic range.

Final Answer:
E = (P * L) / (A * δ)