Moduli of elasticity

Within the elastic limit, since stress is proportional to strain, the ratio $\dfrac{\text{stress}}{\text{strain}}$ is a constant for a given material -- this constant is called the MODULUS OF ELASTICITY of that material. Its SI unit is N m$^{-2}$ (or pascal), and its dimensional formula is $[ML^{-1}T^{-2}]$, exactly the same as stress (since strain is dimensionless). The chapter defines three distinct elastic moduli, one for each kind of stress-strain pair. (a) YOUNG'S MODULUS, $Y=\dfrac{\sigma_t}{\varepsilon_t}$ (or $\sigma_c/\varepsilon_c$ for compression), is the ratio of tensile (or compressive) stress to tensile (or compressive) strain -- it measures a solid's resistance to a change in its LENGTH under a stretching or compressing force. (b) BULK MODULUS, $K=-\dfrac{\sigma_n}{\varepsilon_v}=-\dfrac{P}{\Delta V/V}$, is the ratio of the (normal) volume stress -- equal to the applied pressure P -- to the volume strain; the negative sign reflects that volume DECREASES when pressure INCREASES. Bulk modulus measures a material's resistance to a change in its VOLUME: a small K means the material is easily compressed (gases have a much smaller K than solids, which is why gases compress far more easily). The RECIPROCAL of the bulk modulus is called COMPRESSIBILITY, $C=\dfrac{1}{K}=-\dfrac{\Delta V/V}{P}$, the fractional change in volume per unit increase in pressure; since gases have small K, their compressibility is correspondingly very high. (c) RIGIDITY MODULUS or SHEAR MODULUS, $\eta_R=\dfrac{\sigma_s}{\varepsilon_s}=\dfrac{F/A}{x/h}$, is the ratio of shearing stress to shearing strain -- it measures a material's resistance to being TWISTED or sheared: a small $\eta_R$ means the material twists easily under a given torque, since a restoring torque $\tau\propto\theta$ (angle of shear) developed in a twisted wire is smaller for a given theta when $\eta_R$ is small. For a solid cylindrical rod of length L and radius r rigidly fixed at one end, a tangential (twisting) force F applied at the rim of the free end -- producing a torque $\tau=Fr$ -- twists that end through an angle $\theta=\dfrac{2\tau L}{\pi r^{4}\eta_R}$ (equivalently, the couple per unit twist of the rod is $C=\pi\eta_R r^{4}/2L$); this torsion relation, applying the rigidity modulus to a solid rod rather than a simple sheared cube, is what is used to compute the angle through which a shaft or rod tw …