Elastic Constants — Shear Modulus (Modulus of Rigidity) What is shear modulus defined as, in terms of fundamental stress–strain ratios for small deformations?

Introduction:
Shear modulus, also called modulus of rigidity and denoted by G, quantifies how a material resists shape change under tangential loading. It complements the modulus of elasticity E (normal loading) and bulk modulus K (volumetric loading), forming the core elastic constants used in strength of materials and machine design.

Given Data / Assumptions:

• Small strains and linear elastic behavior.
• Homogeneous, isotropic material so standard relationships among E, G, K, and Poisson ratio v hold.
• Pure shear loading leading to angular distortion without significant volume change.

Concept / Approach:
By definition, shear modulus G is the ratio of shear stress tau to shear strain gamma for small deformations: G = tau / gamma. For isotropic linear elastic solids, G relates to E and v through G = E / (2 * (1 + v)), showing why G is typically a little less than E/2 when v is between 0.25 and 0.35.

Step-by-Step Solution:
Identify loading mode: tangential force causes layers to slide relative to each other.Measure shear stress: tau = V / A on the considered plane.Measure shear strain: gamma is the change in right angle between orthogonal lines in radians.Apply definition: G = tau / gamma.

Verification / Alternative check:
Use isotropic relation G = E / (2 * (1 + v)) to cross-check measured values of E and v from tests; the computed G should match tau / gamma from shear tests.

Why Other Options Are Wrong:
linear stress to linear strain: that is modulus of elasticity E in tension or compression.linear stress to lateral strain: this does not represent a standard elastic constant.volumetric strain to linear strain: this is a nonsensical ratio for defining a basic modulus.

Common Pitfalls:
Confusing shear modulus with E; note that shear involves angular change, not normal extension.Using large-strain values where linear definitions no longer apply.

Final Answer:
shear stress to shear strain