Relation between G and E For most engineering materials, how does shear modulus G compare with half the modulus of elasticity E?

Introduction:
Knowing how elastic constants relate allows engineers to estimate one property from another. The ratio between shear modulus G and modulus of elasticity E depends on Poisson ratio v for isotropic materials.

Given Data / Assumptions:

• Isotropic, homogeneous, linear elastic materials.
• Typical Poisson ratio v between about 0.25 and 0.35 for metals.

Concept / Approach:
The fundamental relation is G = E / (2 * (1 + v)). Because v is positive for stable materials, the denominator exceeds 2. Hence G is strictly less than E / 2 for ordinary materials (v > 0). If v were zero, G would equal E / 2, but that is uncommon in practice.

Step-by-Step Solution:
Start with isotropic relation: G = E / (2 * (1 + v)).For v = 0.3 (typical steel): G = E / (2 * 1.3) = E / 2.6 which is less than E / 2.Thus, for most materials with v between 0.25 and 0.35, G < E / 2.

Verification / Alternative check:
Cross-check with measured values: for mild steel E ≈ 200 GPa, G ≈ 80 GPa; 80 is less than 100 which equals E / 2.

Why Other Options Are Wrong:
equal to half: only true if v = 0, rarely the case.more than half: impossible for v > 0 because 1 + v > 1 makes denominator > 2.none of these: incorrect since the correct qualitative relation is known.

Common Pitfalls:
Forgetting the role of v and assuming a fixed ratio independent of material.

Final Answer:
less than half