Definition of Young’s modulus (modulus of elasticity) Choose the correct definition of Young’s modulus for linear elastic, uniaxial loading.

Introduction / Context:
Young’s modulus quantifies the axial stiffness of a material under uniaxial loading. It is fundamental for predicting elongation of bars, axial stiffness of machine elements, and the bending stiffness of beams through EI.

Given Data / Assumptions:

• Small strains and linear elastic behavior per Hooke’s law.
• Uniaxial loading (tension or compression) with uniform stress and strain.

Concept / Approach:
Young’s modulus E is defined by E = σ / ε, where σ is normal (linear) stress along the loading direction, and ε is corresponding normal (linear) strain along the same direction.

Step-by-Step Solution:
Normal stress: σ = P / A.Engineering strain: ε = ΔL / L.Hooke’s relation: σ = E ε → E = σ / ε.

Verification / Alternative check:
Other elastic constants include shear modulus G = τ / γ and bulk modulus K = hydrostatic stress / volumetric strain. These confirm that option (d) corresponds to shear modulus, not E.

Why Other Options Are Wrong:
Linear stress to lateral strain mixes orthogonal components; lateral-to-linear strain is Poisson’s ratio; shear stress to shear strain defines G; bulk stress to volumetric strain defines K.

Common Pitfalls:
Confusing E with G or K; mixing axial and lateral responses; using true strain outside the linear regime where the definition shifts to tangent modulus.

Final Answer:
linear stress to linear strain