Summary

This chapter developed the mechanics of elastic solids from first principles, along exactly the topics WBCHSE's Unit 7 SUB TOPIC 1 lists:

• Elastic and plastic behaviour (Section 8.1): a body is elastic if it fully recovers its original shape once a deforming force is removed, and plastic if it does not.
• Stress and strain (Section 8.2): stress is the internal restoring force per unit area, $F/A$ (SI unit $\text{Pa}$); strain is the resulting fractional deformation (dimensionless). Three kinds of each were distinguished: tensile/compressive (longitudinal), shearing (tangential), and hydraulic (volume).
• Hooke's law (Section 8.3): within the proportional limit, stress $\propto$ strain, with the constant of proportionality called the modulus of elasticity.
• The stress-strain curve (Section 8.4): read for a ductile metal from the proportional limit, through the elastic limit and yield point, up to the ultimate tensile strength and finally the fracture point; contrasted with a brittle material's short, steep curve.
• The three elastic moduli: Young's modulus $Y = \dfrac{FL}{A\,\Delta L}$ for longitudinal deformation (Section 8.5), the shear modulus of rigidity $\eta = \dfrac{F/A}{\theta}$ for shape-changing deformation, with $\eta = 0$ identically for any fluid (Section 8.6), and the bulk modulus $K = -\dfrac{\Delta P}{\Delta V/V}$ for volume-changing deformation, with compressibility $= 1/K$ (Section 8.7).
• Poisson's ratio $\sigma = -\dfrac{\text{lateral strain}}{\text{longitudinal strain}}$ (Section 8.8), a dimensionless number typically between $0.2$ and $0.4$ for ordinary solids, with a theoretical upper limit of $0.5$. …