Define the modulus of rigidity (shear modulus, G). Select the correct ratio that defines G in terms of basic stress–strain quantities.

Introduction:
The modulus of rigidity, also known as the shear modulus G, quantifies a material's resistance to shear deformation and is one of the three fundamental elastic constants alongside E and K (or ν).

Given Data / Assumptions:

• Small strains; linear elastic behavior.
• Uniform shear state in the considered element.

Concept / Approach:
By definition, an elastic modulus is a stress–strain ratio. For shear, the relevant quantities are shear stress tau and shear strain gamma. Thus G = tau / gamma.

Step-by-Step Solution:
1) Apply a pure shear tau to a small element.2) Measure resulting shear strain gamma (angular distortion).3) Define G = tau / gamma.4) Relate to other constants when needed: E = 2 * G * (1 + ν).

Verification / Alternative check:
Check units: tau in N/m^2; gamma is dimensionless; hence G has units of N/m^2, consistent with other elastic moduli.

Why Other Options Are Wrong:
Linear stress to lateral strain and lateral to linear strain: unrelated ratios for G.

Linear stress to linear strain: that defines Young's modulus E, not G.

Common Pitfalls:
Confusing shear modulus G with bulk modulus K or Young's modulus E; also mixing up lateral strain definitions used with Poisson's ratio ν.

Final Answer:
Shear stress to shear strain