Physics · Ch 6 — Mechanical Properties of Solids
Modulus of rigidity (η)
Modulus of rigidity (η)
The modulus of elasticity associated with a change in the shape of an object — with no accompanying change in its size — is called the modulus of rigidity, or shear modulus, usually denoted . Like Young's modulus, it is a property of solids only, since only solids possess a definite shape to begin with; liquids and gases have no fixed shape and therefore cannot resist a shearing deformation at all.
Consider a block of a uniform, isotropic material, of uniform cross-sectional area and height (Fig. 6.5). Two equal and opposite forces, each of magnitude , are applied along the top and bottom surfaces of the block, parallel to those surfaces — together these two forces constitute a couple. (The same effect results if the bottom surface is simply held fixed while only the top surface is pushed.) Because the forces and act parallel to, rather than normal to, the block's cross section, this is fundamentally different from the tensile-stress geometry of section 6.5.1, where the force acts normal to the cross section.
As a result of this couple, the top surface of the block is displaced sideways relative to the bottom surface by a small distance , and the block's side profile tilts through a small angle . The shear stress produced is defined, exactly as in section 6.3, as
and the corresponding shear strain, expressed in radians, is
The modulus of rigidity (shear modulus) is then the ratio of shear stress to shear strain, within the elastic limit: …
What this figure shows. A rectangular block of uniform cross-sectional area A and height l is shown with two equal and opposite forces of magnitude F applied along its top and bottom surfaces, in opposite horizontal directions, forming a couple (as if the bottom is pushed one way and the top pushed the opposite way, or equivalently the bottom is fixed and only the top is pushed). Because of this couple, the top surface is shown displaced sideways relative to the bottom surface by a small horizontal distance Δl, so the block's originally rectangular side profile becomes a slanted parallelogram. The small angle θ = Δl/l formed at the corner between the original vertical edge and the new slanted edge is marked and labelled as the shear strain. The figure is the general version of the shearing-force setup already introduced in Fig. 6.3, now showing both forces of the couple explicitly …
Material | Rigidity modulus η ×10^10 Pa (N/m2)
Lead | 0.6
Aluminium | 2.5
Glass (crown) | 2.5
Silver | 2.7
Gold | 2.9
Brass | 3.5 …
Worked out. Worked example finding the modulus of rigidity of a metal, given a metal cube of side l = 40 cm = 0.40 m subjected to a shearing force F = 2000 N applied on its upper surface, which is displaced by Δl = 0.5 cm = 0.005 m with respect to the (fixed) bottom surface. The method first computes the cube's cross-sectional area A = l² = 0.16 m² and the shear strain θ = Δl/l = 0.0125, then substitutes F, A and θ into η = (F/A)/θ, obtaining η = 1.0×10^6 N/m² — a direct application of the modulus-of-rigidity formula to the exact couple-and-shear-angle geometry shown in Fig. 6.5, with l here playi …