Explain the different types of modulus of elasticity.

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Q. Explain the different types of modulus of elasticity.

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Concept understanding — Rigidity Modulus

The RIGIDITY MODULUS (or SHEAR MODULUS), $\eta_R$ , is one of the three elastic moduli of a material, defined as the ratio of shearing stress to shearing strain: $\eta_R=\dfrac{\sigma_s}{\varepsilon_s}=\dfrac{F/A}{x/h}$ , where F is the tangential force applied over area A, and x/h is the resulting shear (the tangent of the angle of shear, which for a small angle equals the angle itself, $\theta$ ). Its SI unit is N m $^{-2}$ (pascal), matching the other elastic moduli, since strain is dimensionless. Physically, the rigidity modulus measures how strongly a material resists being TWISTED or sheared: a SMALL $\eta_R$ means a material twists easily for a given applied torque, since the restoring torque $\tau$ developed in a twisted wire or rod is proportional to the angle of twist $\theta$ ( $\tau\propto\theta$ ), and a small rigidity modulus means a large twist angle results from even a modest torque. TWIST OF A SOLID CYLINDRICAL ROD. For a solid cylindrical rod of length L and radius r, rigidly fixed at one end, applying a tangential force F at the rim of the free end (producing a torque $\tau=Fr$ about the rod's axis) twists that free end through an angle $\theta=\dfrac{2\tau L}{\pi r^4\eta_R}$ -- equivalently, the couple needed per unit angle of twist (the torsional rigidity) of the rod is $C=\dfrac{\pi\eta_R r^4}{2L}$ . This torsion relation extends the simple cube-shear picture of rigidity modulus to the practically important case of a shaft or rod twisted about its own axis, and is the formula used whenever a problem asks for the angle through which a rod, cylinder, or shaft twists under an applied tangential force or torque. Table 7.1 of the chapter lists measured rigidity-modulus values (in units of $10^{10}$ N m $^{-2}$ ) for common materials: steel (8.0), aluminium (2.5), copper (4.0), iron (6.4), and glass (2.6) -- steel again has the highest value among these, meaning it resists twisting more strongly than any of the other four materials listed. The three elastic moduli of a material are not independent of each other: they are tied together, along with Poisson's ratio $\mu$ , by the standard relations $Y=2\eta_R(1+\mu)=3K(1-2\mu)$ , which let any one of Y, K, $\eta_R$ , or $\mu$ be computed from the other two.

Step 1. Young's modulus. $Y=\dfrac{\sigma_t}{\varepsilon_t}$ (tensile or compressive stress over the corresponding strain) measures a solid's resistance to a change in its LENGTH. For a wire of length L, area A, stretched by force F to extension $\Delta L$ : $Y=\dfrac{FL}{A\,\Delta L}$ . A higher Y means less strain (elongation) for a given stress -- steel (Y about $20\times10^{10}$ N m $^{-2}$ ) is far stiffer in this sense than rubber.

Step 2. Bulk modulus. $K=-\dfrac{\sigma_n}{\varepsilon_v}=-\dfrac{P}{\Delta V/V}$ measures a material's resistance to a change in its overall VOLUME under a uniform pressure P; the negative sign makes K positive, since volume decreases as pressure increases. Its reciprocal, $C=1/K$ , is the compressibility. Gases have a much smaller K (and hence much larger compressibility) than solids or liquids, which is why gases compress so much more readily.

Step 3. Rigidity (shear) modulus. $\eta_R=\dfrac{\sigma_s}{\varepsilon_s}=\dfrac{F/A}{x/h}$ measures a material's resistance to a change in its SHAPE (a twisting or shearing deformation) at constant volume -- for a cuboid of height h whose top face is displaced sideways by x under a tangential force F over area A. A small $\eta_R$ means the material twists easily under a given torque.

Step 4. Relation. All three moduli, together with Poisson's ratio $\mu$ , are linked by $Y=2\eta_R(1+\mu)=3K(1-2\mu)$ , so any one can be computed from the other two.

Step 5. Comparison. Table 7.1 tabulates Y, K, and $\eta_R$ (all $\times10^{10}$ N m $^{-2}$ ) for steel, aluminium, copper, iron and glass; steel has the highest value of all three moduli among these materials, which is why it is preferred for heavy-duty, high-stress engineering applications.

✓Final answer

Y=\sigma_t/\varepsilon_t

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Q.Explain the different types of modulus of elasticity.

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