Mechanical Properties of Solids

Everything you have solved so far assumed rigid bodies. Nothing is rigid. This chapter measures how much real materials give, and how engineers turn that measurement into cables that hold and bridges that stand. It begins with elastic behaviour and the atomic picture of a solid as a lattice of tiny springs, then defines the two quantities that make deformation comparable across objects of any size — stress, the restoring force per unit area, and strain, the fractional change in dimension — in the three pairs that run through the whole subject: longitudinal, shearing and hydraulic. Hooke's law makes stress proportional to strain for small deformations, and the stress-strain curve shows exactly where that proportionality dies, marking the elastic limit, the plastic region, the ultimate tensile strength and fracture. Three moduli follow: Young's modulus $Y = \frac{FL}{A\,\Delta L}$ for stretching, the shear modulus $G$ for sliding and twisting, and the bulk modulus $B = -\frac{\Delta p}{\Delta V/V}$ for squeezing from every side, with compressibility as its reciprocal. Poisson's ratio links the sideways contraction to the lengthways stretch, and the relations $Y = 3B(1 - 2\sigma) = 2G(1 + \sigma)$ tie all four constants together. The elastic potential energy $U = \frac{1}{2}F\,\Delta L$ and the strain energy density $u = \frac{1}{2}\sigma\varepsilon$ account for the work stored, and the chapter closes on applications — sizing crane ropes, why an I-section girder is shaped as it is, the sag of a loaded beam, and why no mountain on Earth can be much taller than 10 km. Topics the rationalised syllabus trimmed — thermal stress, elastic hysteresis and fatigue, and the relations between the elastic constants — are restored in full, because Boards, JEE Main and NEET ask about them every year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.