Within the elastic limit, the ratio of shearing stress to shearing strain is called what?

Introduction / Context:
Different moduli describe material stiffness under distinct modes of deformation. In shear, the proportionality between shear stress and shear strain in the elastic range defines the shear modulus, a key parameter for torsion, shear deformation, and plate/beam analyses.

Given Data / Assumptions:

• Linear elastic behavior is assumed.
• Shear stress (tau) and shear strain (gamma) are considered.
• Isotropic material relations may link E, G, and nu.

Concept / Approach:
The shear modulus (G), also called modulus of rigidity, is defined by G = tau / gamma. For isotropic materials, G relates to Young’s modulus E and Poisson’s ratio nu through E = 2G(1 + nu). Bulk modulus K is used for volumetric compression under hydrostatic stress, while tangent modulus is a local slope in nonlinear regimes.

Step-by-Step Solution:
Identify the deformation mode: shear.Apply Hooke’s law in shear: tau = G * gamma.Select the modulus corresponding to shear stiffness: G.

Verification / Alternative check:
Typical values: structural steel G ≈ 80,000 MPa when E ≈ 200,000 MPa and nu ≈ 0.3, matching the isotropic relation.

Why Other Options Are Wrong:

• Modulus of elasticity (E): pertains to normal stress–strain.
• Bulk modulus (K): volumetric stiffness under hydrostatic loading.
• Tangent modulus: used for inelastic analysis.
• All the above: only one correct definition applies here.

Common Pitfalls:

• Using E in torsion or shear deflection formulas instead of G, leading to significant error.

Final Answer:
Shear modulus of elasticity.