JEE Physics Mechanical Properties Of Solids Notes Set 01

Elasticity, Plasticity, Stress and Strain

A body that regains its original size and shape when the deforming force is removed is elastic; one that stays deformed is plastic. Quartz fibre is the nearest real example of a perfectly elastic body and wet clay or wax of a perfectly plastic one. The restoring force per unit area developed inside the body is the stress \( \sigma=\frac{F}{A} \) (unit N/m² or Pa). Stress may be normal (tensile or compressive), tangential (shear) or a uniform pressure (bulk stress).
Strain is the fractional change of configuration and has no unit. Longitudinal strain is \( \frac{\Delta L}{L} \), volume strain is \( \frac{\Delta V}{V} \), and shear strain is the angle \( \phi \) through which a face turns, \( \tan\phi=\frac{x}{L} \).
If a force F acts on a face of area A at an angle to the face, only its normal part produces tensile stress and its tangential part produces shear stress. For a bar pulled by F, a plane whose normal makes angle \( \theta \) with the axis carries tensile stress \( \frac{F\cos^2\theta}{A} \) and shear stress \( \frac{F\sin2\theta}{2A} \); the tensile stress is largest at \( \theta=0 \) and the shear stress is largest at \( \theta=45^\circ \).

Hooke's Law and the Three Moduli

Within the elastic limit stress is proportional to strain, and the ratio is the modulus of elasticity, which depends only on the material and not on the shape. There are three moduli.
Young's modulus: \( Y=\frac{F/A}{\Delta L/L}=\frac{FL}{A\,\Delta L} \). Bulk modulus: \( K=-\frac{\Delta P}{\Delta V/V} \), with compressibility \( \frac1K \); for a gas the isothermal bulk modulus equals P and the adiabatic one equals \( \gamma P \). Modulus of rigidity: \( \eta=\frac{F/A}{\phi} \). In general \( K_{solids}>K_{liquids}>K_{gases} \).
Worked example: a rubber cube of side 5 cm with one face fixed is pushed tangentially by 1800 N on the opposite face. With \( \eta=2.4\times10^6 \) N/m², the shear strain is \( \frac{1800}{25\times10^{-4}\times2.4\times10^6}=0.3 \) rad and the face shifts by 1.5 cm.
Series and parallel wires: wires of equal length and area hung side by side share the load, so the effective Young's modulus is the average \( \frac{Y_1+Y_2}{2} \). In series (end to end) the extensions add.

Stress-Strain Curve and Breaking

On a stress-strain graph the straight part ends at the limit of proportionality, and the elastic limit follows slightly later. Beyond the yield point the wire flows with a large strain for a small stress increase (plastic region). The maximum stress before fracture is the breaking (ultimate) stress or tensile strength. A ductile material has a long plastic range, a brittle one breaks soon after the elastic limit.
Breaking stress is a property of the material, so the breaking force is proportional to the cross-sectional area (a wire of twice the radius needs four times the force). The safety factor is breaking stress divided by working stress.
Elastic after-effect is the delay in recovery after the load is removed (very small for quartz and phosphor bronze, which are used for galvanometer suspensions). Elastic fatigue is the loss of strength after repeated straining.

Elongation of Rods in Different Situations

A rod hanging under its own weight W has tension \( \frac{Wx}{L} \) at a distance x from the bottom, giving a total elongation \( \frac{WL}{2AY} \), which is half the value obtained by hanging the whole weight at the end.
A rod rotating about one end with angular velocity \( \omega \) has tension decreasing from the axis to the tip, and its extension is \( \frac{\rho\omega^2L^3}{3Y} \). A bar of mass m pulled by a force F on a smooth surface has tension \( \frac{Fx}{L} \) at distance x from the free end and a total elongation \( \frac{FL}{2AY} \). A thin ring of density \( \rho \) spinning at \( \omega \) stretches so that its radius increases by \( \frac{\rho\omega^2R^3}{Y} \).
A rod or cable behaves as a spring of constant \( k=\frac{AY}{L} \). Rods in series add flexibilities: \( \frac1k=\frac1{k_1}+\frac1{k_2} \). A mass m hung from it oscillates with \( \omega=\sqrt{k/m} \). If a body falls on a collar attached to the rod, the maximum extension follows from energy conservation: \( mg(h+x)=\tfrac12kx^2 \).

Elastic Potential Energy

The work done in stretching a wire is stored as elastic potential energy: \( U=\tfrac12F\,\Delta L=\frac12\frac{AY}{L}(\Delta L)^2=\frac{F^2L}{2AY} \). The energy per unit volume is \( \tfrac12\times\text{stress}\times\text{strain}=\frac{\sigma^2}{2Y}=\tfrac12Y\varepsilon^2 \).
When a mass M is attached to a vertical wire and allowed to settle, the loss of gravitational potential energy is \( Mg\,\delta \), the elastic energy stored is \( \tfrac12Mg\,\delta \), and the remaining half appears as heat. If instead the load is applied very slowly, no heat is produced and the work done equals the stored energy.
Thermal stress: a rod prevented from expanding or contracting when its temperature changes by \( \Delta T \) develops a stress \( Y\alpha\,\Delta T \), so the force is \( YA\alpha\,\Delta T \).
Worked example: a wire stretched by 1 mm needs 2 J of work. A wire of the same material with double the radius and half the length has 8 times the spring constant, so it needs 16 J for the same extension.

Torsion of a Wire

When one end of a wire is twisted through an angle \( \theta \) the restoring torque is \( \tau=C\theta \), where the torsion constant is \( C=\frac{\pi\eta r^4}{2L} \) for a wire of radius r, length L and modulus of rigidity \( \eta \). Doubling the radius makes the wire 16 times harder to twist. The energy stored is \( \tfrac12C\theta^2 \).