Chapter 8 — Mechanical Properties of Solids
No solid is truly rigid. Stress, strain and Hooke's law describe how solids stretch, shear and squeeze; the elastic moduli measure their stiffness, and the stress–strain curve tells when they give way.
Every morning, as traffic pours onto a great suspension bridge, its steel cables stretch by several centimetres and the deck sags a little; at night they spring back. A lift cable lengthens slightly each time a crowd steps in. Your thigh bone shortens by a few hundredths of a millimetre every time you stand on one leg, and the elastic wall of your aorta swells and recoils with every heartbeat. In the earlier chapters bodies were treated as rigid — a fine approximation for working out how they move, but not for deciding whether a cable will hold, a beam will sag or a bone will break.
This chapter looks inside solids. A solid is a lattice of atoms held by forces that act like tiny springs, so it deforms when it is loaded and pulls back when the load is removed. You will measure loading by stress (force per unit area) and deformation by strain (fractional change), meet Hooke's law — stress is proportional to strain for small deformations — and read the full stress–strain curve from the first gentle stretch to fracture, which separates ductile metals from brittle glass and rubbery elastomers. Then each kind of deformation gets its own measure of stiffness: Young's modulus for stretching, the shear modulus for changes of shape, the bulk modulus for changes of volume, and Poisson's ratio for the narrowing that goes with a stretch.
- explain elastic and plastic behaviour with the ball-and-spring model of a solid;
- calculate tensile, compressive, shearing and hydraulic stress, and longitudinal, shearing and volume strain;
- state Hooke's law and use a modulus of elasticity, including the "wire as a spring" constant $k=YA/L$;
- read a stress–strain curve — proportional limit, elastic limit, permanent set, ultimate tensile strength, fracture — and classify materials as ductile, brittle or elastomeric;
- find breaking loads and design with a factor of safety, and explain applications of elasticity — crane ropes, beams, the greatest height of a mountain;
- use $Y=\dfrac{FL}{A\,\Delta L}$ for single wires and wires joined in series or side by side, and find $Y$ from a load–extension graph (Searle's method);
- use the shear modulus, the bulk modulus and compressibility, and Poisson's ratio with $\dfrac{\Delta V}{V}=\varepsilon(1-2\nu)$;
- find the elastic potential energy of a stretched wire, $U=\tfrac12F\,\Delta L$, and the energy per unit volume $\tfrac12\sigma\varepsilon$;
- tackle Board, JEE Main, JEE Advanced and NEET-style problems on the elastic properties of solids.
Sections in this chapter
Stress, Strain and Hooke's Law
Elastic and plastic behaviour, the three kinds of stress and strain, Hooke's law and the stress–strain curve from proportional limit to fracture.
Elastic Moduli and Poisson's Ratio
Young's modulus and its measurement, the shear and bulk moduli, compressibility and Poisson's ratio.