Name of the modulus: ratio of shear stress to shear strain What do we call the material constant defined as (shear stress) / (shear strain)?

Introduction / Context:
Materials exhibit elastic response under small deformations. In shear, the proportionality constant between shear stress and shear strain is fundamental for torsion of shafts, shear deflection of beams, and plate/shell behavior.

Given Data / Assumptions:

• Linear elastic, small-strain behavior (Hooke-type relationship) is assumed.
• Terminology may vary between texts but standard names exist.

Concept / Approach:
The constant G = tau / gamma is called the Shear Modulus or Modulus of Rigidity. It is related to Young's Modulus E and Poisson's ratio ν through G = E / (2*(1 + ν)). Young's Modulus (also called Modulus of Elasticity in tension/compression) pertains to normal stress–strain, not shear.

Step-by-Step Solution:
Identify definition: G = shear stress / shear strain.Recognize synonyms: Modulus of Rigidity = Shear Modulus.Exclude E (Young's Modulus) because it governs normal stress–strain, not shear.

Verification / Alternative check:
If E = 200 GPa and ν = 0.3 for steel, then G = 200 / (2*(1+0.3)) = 76.9 GPa, which matches typical tabulated values for structural steels.

Why Other Options Are Wrong:
(c) and (d) refer to axial elasticity; while some authors loosely call E the 'modulus of elasticity,' in this context the required modulus is specifically G, not E.

Common Pitfalls:
Using E when shear deformation dominates; ignoring the role of Poisson's ratio in relating E and G.

Final Answer:
Both (a) and (b)