Mechanical Properties of Solids— Notes, Formulas & Revision

A typical stress-strain curve for a ductile material (e.g., mild steel) has several distinct regions.

1. Proportional region (linear) — stress ∝ strain; Hooke's law holds; modulus is the slope.

2. Elastic limit (just beyond proportional limit) — still recoverable, but stress and strain not strictly proportional.

3. Yield point — material begins permanent deformation; visible necking starts.

4. Plastic region — large strain for small stress increase; ductile materials stretch significantly.

5. Ultimate tensile strength — maximum stress before necking dominates.

6. Fracture point — material breaks.

Ductile materials (steel, copper) show large plastic region. Brittle materials (glass, ceramic) break almost immediately past elastic limit.

Area under the curve = energy absorbed per unit volume (toughness).

For small deformations, the stress in a solid is DIRECTLY proportional to the strain: stress = E × strain. This is Hooke's law.

The proportionality constant E is the MODULUS OF ELASTICITY — material-specific, geometry-independent.

Stress = force per unit area (N/m² = Pa). Strain = fractional deformation (dimensionless).

Three types of stress/strain: tensile/compressive (longitudinal), volumetric (bulk), shear (tangential).

Each has its own modulus: Young's (Y) for longitudinal, Bulk (B) for volume, Shear (G) for shape.

Linear region: Hooke's law holds. Beyond the proportional limit, the curve deviates.

Spring force F = −kx is a special case of Hooke's law applied to a spring; k = stiffness.

Elastic deformations are REVERSIBLE — strain disappears when stress is removed. Beyond the elastic limit, deformation is permanent (plastic).

Young's modulus Y measures a material's stiffness under TENSILE/COMPRESSIVE stress: Y = stress/strain = (F/A)/(ΔL/L).

Units: Pa (N/m²). Reported in GPa for typical solids.

Y values: steel ≈ 200 GPa, copper ≈ 110 GPa, aluminum ≈ 70 GPa, glass ≈ 70 GPa, rubber ≈ 0.01-0.1 GPa, wood (along grain) ≈ 12 GPa.

Wire of length L, cross-section A: extension ΔL = FL/(YA). Stiffer (higher Y) ⇒ smaller ΔL for given F.

Force constant of a wire viewed as a spring: k = YA/L. Long, thin wires are softer; short, thick ones are stiffer.

Used in structural engineering, materials selection, vibration analysis, and acoustics.

Y is approximately constant for moderate strains; varies slightly with temperature.

For composite or anisotropic materials (wood, carbon fibre), Y depends on direction.

Bulk modulus B (sometimes K) measures resistance to VOLUME change under hydrostatic pressure.

Definition: B = −P/(ΔV/V). Negative sign because volume DECREASES as pressure INCREASES.

Units: Pa, same as pressure.

Typical values: steel ≈ 160 GPa, water ≈ 2.2 GPa, air ≈ 0.0001 GPa (at 1 atm). Solids are nearly incompressible; gases very compressible.

Compressibility κ = 1/B — fractional volume change per unit pressure.

Bulk modulus determines the speed of sound in fluids and solids: v = √(B/ρ) for longitudinal waves.

For ideal gas: isothermal B_T = P (= 10⁵ Pa at 1 atm); adiabatic B_S = γP.

Solids' bulk modulus is HIGH (~10² GPa) ⇒ they barely change volume even under enormous pressure.

Shear modulus G (also called rigidity modulus η or μ) measures resistance to SHAPE change at constant volume.

Apply tangential force F to top face of a solid block (area A) while bottom is fixed: produces angular shear θ.

Shear stress = F/A; shear strain = θ ≈ Δx/L (for small angles). G = shear stress / shear strain.

Units: Pa. Typical values: steel ≈ 80 GPa, copper ≈ 45 GPa, rubber ~ MPa.

Liquids and gases have G = 0 — they have no rigidity and cannot sustain shear stress at equilibrium.

Shear modulus governs torsion of rods, propagation of TRANSVERSE waves in solids, etc.

Torsion of a wire: torque τ = (πGr⁴/2L)·θ, where r = radius, L = length, θ = twist angle.

Speed of transverse (shear) waves in solid: v_s = √(G/ρ).

Poisson's ratio σ_p (often ν) measures lateral contraction during axial stretching: σ_p = −(lateral strain)/(longitudinal strain).

When you stretch a rod axially, it gets THINNER laterally — both effects related by σ_p.

Range: 0 ≤ σ_p ≤ 0.5 for isotropic materials. Most metals: 0.25-0.35. Cork: ~0. Rubber: ~0.5 (essentially incompressible).

σ_p = 0.5: material is incompressible (volume unchanged when stretched). Rubber, soft tissues approach this.

σ_p = 0: no lateral contraction at all. Cork has ~0.04 — useful for bottle stoppers.

Some special materials (auxetics) have NEGATIVE Poisson's ratio — they expand laterally when stretched.

Links Young's modulus, bulk modulus, and shear modulus: B = Y/[3(1−2σ_p)], G = Y/[2(1+σ_p)].

Dimensionless quantity — pure number.

Elastic potential energy is the energy stored in a deformed elastic body — recoverable when the body returns to its natural shape.

For a spring or wire obeying Hooke's law: U = ½kx² (spring) or U = ½·F·ΔL (wire under tension F, extension ΔL).

Energy per unit volume (energy density): u = ½·σ·ε = ½·Y·ε² = σ²/(2Y).

All this is recovered when stress is released — provided you stay in the elastic region.

Beyond the elastic limit, some energy goes into permanent (plastic) deformation, heat, or fracture.

Springs in series and parallel: in series, k decreases (1/k_eq = Σ 1/k_i); in parallel, k increases (k_eq = Σ k_i).

Application: bow-and-arrow stores elastic energy in the bow, transferred to KE of arrow.

Earthquake faults store enormous elastic energy in stressed rocks — released suddenly during slip.