The linear-elastic material constant G (modulus of rigidity) defined as the ratio of shear (tangential) stress to engineering shear strain within the material's elastic regime (τ = G·γ); units: pascal (Pa).

Principle

In linear elastic shear, shear stress and shear strain are proportional; G quantifies resistance to shape distortion (change of angle) under shear loads and, together with Poisson's ratio, links to Young's modulus for isotropic materials by E = 2G(1+ν).

Demonstration

Illustrative scenario — torsion test on a circular shaft: measure torque and angle of twist per unit length; convert torque to shear stress and twist to shear strain; the initial slope of shear stress versus shear strain gives G; predict elastic twist under service torque using τ = G·γ.

Misapplication

Applying the isotropic relation E = 2G(1+ν) without verifying isotropy or using G measured at small strains to predict shear behaviour in plastic, rate-dependent, or anisotropic regimes; confusing shear modulus (stiffness) with shear strength (failure stress in shear).

Consequence

Correct use of G yields predictions of elastic shear deformation, torsional stiffness and natural frequencies involving shear deformation; misuse leads to incorrect shape-deflection, resonance estimates or inadequate shear-critical design.

Reversal

For anisotropic materials (laminates, single crystals) the scalar G is replaced by directional shear moduli or a shear-compliance matrix; in materials with significant shear yielding or shear banding, linear-elastic G has limited applicability and nonlinear constitutive descriptions are required.

Boundary

Clearly within: homogeneous isotropic or effectively isotropic solids under small elastic shear strains. Boundary case: orthotropic laminates where in-plane shear modulus differs from out-of-plane. Clearly outside: plastic shear flow, large rotations where engineering shear strain definitions differ from true measures, or multi-axial coupled shear–normal states requiring full constitutive representation.

Semantic Tension

Shear stiffness (G) versus volumetric stiffness (bulk modulus K) and versus Young's modulus (E): design must balance resistance to shape change with resistance to volume change and axial deformation, especially in dynamic or multi-axial loading.

Synthesis

Shear modulus is the elastic parameter that controls shape distortion under tangential loads; it complements Young's modulus by governing angular change and torsional response rather than axial extension, so both must be used appropriately in structural and modal analyses.