Generalized Hooke’s law in shear Within the elastic limit, shear stress is __________ shear strain.

Introduction / Context:
Hooke’s law extends to shear: in the elastic range, material response is linear, and shear stress is proportional to shear strain. This underpins torsion formulas, beam shear relations, and finite element constitutive models.

Given Data / Assumptions:

• Material is homogeneous and isotropic.
• Deformations are small; elastic limit not exceeded.
• Shear modulus G (also written as C) is constant in this range.

Concept / Approach:
Generalized Hooke’s law in shear states τ = G * γ, where τ is shear stress and γ is engineering shear strain. The direct proportionality is characterized by the slope G of the τ–γ line in the elastic region.

Step-by-Step Reasoning:
Identify the elastic region (no permanent deformation).Relate stress and strain: τ = G * γ implies linear proportionality.Therefore, within elastic limit, τ varies directly with γ.

Verification / Alternative check:
For torsion of circular shafts: τ = T r / J and γ = r θ / L. Eliminating r shows τ ∝ γ with slope G when Torsion equation G J θ / L = T is used, reaffirming linearity.

Why Other Options Are Wrong:
“Equal to” ignores units (stress vs strain). “Less than” or “inversely proportional” contradicts basic elasticity. “Independent of” is inconsistent with elastic behavior.

Common Pitfalls:
Confusing proportionality (τ ∝ γ) with equality; mixing G with E (Young’s modulus) or K (bulk modulus).

Final Answer:
directly proportional to