Stress-strain behaviour and mechanical properties

Why it matters

The tensile test is the single most important source of design data: Young's modulus for deflection, yield strength for allowable stress, ultimate strength and elongation for forming and for judging ductility. Reading the stress-strain curve correctly, including the difference between engineering and true values, is needed in machine design, metal forming and every materials-selection decision.

Key ideas

The tensile test. A standard specimen of original gauge length L0 and cross-section A0 is pulled at a slow, constant rate; load F and elongation ΔL are recorded. Engineering stress and strain are based on the original dimensions, so they describe the specimen, not the instantaneous state of the material.

Regions of a ductile metal's curve.

1. Linear elastic: stress proportional to strain (Hooke's law) up to the proportional limit. Unloading returns the specimen to zero strain. The elastic limit is the highest stress with no permanent set; it is very close to the proportional limit.
2. Yielding: plastic flow starts. Some low-carbon steels show a sharp upper yield point, a drop to a lower yield point and a flat yield-point elongation (Lüders bands), caused by carbon and nitrogen atoms pinning dislocations. Metals without a distinct yield point are given a 0.2 % offset (proof) stress: draw a line parallel to the elastic slope from strain 0.002 and read where it meets the curve.
3. Uniform plastic deformation with strain hardening: the load keeps rising because the metal hardens faster than its area shrinks.
4. Ultimate tensile strength (UTS): the maximum engineering stress, F_max/A0. Beyond it, deformation localises in a neck.
5. Necking to fracture: the load falls on the engineering curve even though the true stress in the neck keeps rising.

Elastic properties. E (Young's modulus) measures stiffness and depends on bonding, so heat treatment and alloying hardly change it (all steels are about 200–210 GPa). Poisson's ratio ν = −lateral strain / axial strain, about 0.3 for metals; G = E / [2(1 + ν)] for isotropic materials.

Ductility. Percent elongation (depends on gauge length, so always quote it) and percent reduction in area (independent of gauge length, a better measure of the neck's ductility). A material with less than about 5 % elongation is usually called brittle.

Energy measures. Resilience is the elastic strain energy per unit volume up to yield (area under the elastic part). Tensile toughness is the total area under the curve to fracture; it needs both strength and ductility.

True stress and true strain. True stress σT = F/Ai uses the instantaneous area; true (logarithmic) strain εT = ln(Li/L0) is additive over successive steps. Assuming constant volume (plastic flow is volume-conserving) and uniform deformation (valid only up to UTS), they convert from engineering values. Beyond UTS, use the measured neck area.

Strain hardening. In the uniform plastic region many metals follow Hollomon's law σT = K·εTⁿ. The exponent n (0.1–0.5) measures how strongly the metal hardens; necking starts when εT = n (Considère criterion), so high-n metals stretch-form well.

Brittle and polymeric behaviour. Ceramics and grey cast iron fracture in the elastic region with almost no plasticity and are much stronger in compression than tension. Polymers are viscoelastic: their curves depend strongly on strain rate and temperature.

Effect of temperature and rate. Raising temperature lowers E, yield strength and UTS and usually raises ductility; raising strain rate does the reverse. BCC steels become brittle at low temperature (ductile-to-brittle transition).

Formulas

σ = F / A0 and ε = ΔL / L0 Engineering stress (Pa) and strain (dimensionless); F in N, A0 in m², L0 in m.

σ = E·ε Hooke's law, linear elastic region only; E in Pa.

ν = −ε_lateral / ε_axial and G = E / (2(1 + ν)) Poisson's ratio (dimensionless); shear modulus G in Pa, isotropic materials.

%EL = (Lf − L0) / L0 × 100 and %RA = (A0 − Af) / A0 × 100 Lf, Af = gauge length and minimum area after fracture.

Ur = σy² / (2E) Modulus of resilience (J/m³), assumes linear elasticity up to yield σy.

σT = σ·(1 + ε) and εT = ln(1 + ε) True stress (Pa) and true strain; valid up to the onset of necking.

σT = K·εTⁿ Hollomon law; K = strength coefficient (Pa), n = strain-hardening exponent. At necking, εT = n.

Worked examples

Example 1 (standard): reducing tensile-test data. Given: steel specimen, d0 = 12.5 mm, L0 = 50 mm; load at yield 40 kN; maximum load 62 kN; after fracture Lf = 62 mm and neck diameter df = 9.0 mm; E = 200 GPa.

1. A0 = π/4 × 12.5² = 122.7 mm².
2. σy = 40 000 / 122.7 = 326 MPa.
3. UTS = 62 000 / 122.7 = 505 MPa.
4. %EL = (62 − 50)/50 × 100 = 24 %.
5. Af = π/4 × 9.0² = 63.6 mm²; %RA = (122.7 − 63.6)/122.7 × 100 = 48.2 %.
6. Ur = σy²/(2E) = 326²/(2 × 200 000) = 0.266 MPa = 0.266 MJ/m³.

Example 2 (GATE level): true values at necking and the Hollomon law. Given: an annealed metal reaches its UTS of 400 MPa at an engineering strain of 0.20. Assume Hollomon behaviour.

1. True stress at UTS: σT = 400 × (1 + 0.20) = 480 MPa.
2. True strain at UTS: εT = ln 1.20 = 0.182.
3. Since necking starts at εT = n, n = 0.182.
4. K = σT / εTⁿ = 480 / 0.182^0.182 = 480 / 0.733 = 655 MPa.

Example 3 (elastic): a 10 mm diameter, 2 m long steel rod carries 20 kN (E = 200 GPa, ν = 0.3).

1. σ = 20 000 / (π/4 × 10²) = 254.6 MPa.
2. ε = 254.6 / 200 000 = 1.273 × 10⁻³; ΔL = 1.273 × 10⁻³ × 2000 mm = 2.55 mm.
3. Diameter change = −ν·ε·d = −0.3 × 1.273 × 10⁻³ × 10 = −0.0038 mm.

Common mistakes

• Mixing MPa, N/mm² and Pa: 1 MPa = 1 N/mm², which makes hand calculations in mm easy.
• Applying the true-stress conversions after necking has started.
• Quoting %EL without the gauge length, or comparing elongations from different gauge lengths.
• Thinking heat treatment raises E. It raises yield strength and hardness, not stiffness.
• Reading the 0.2 % offset as 0.2 (instead of 0.002) strain.
• Assuming the drop in the engineering curve after UTS means the material is getting weaker; it is the neck's area that is falling.

For GATE PI

Expect direct numericals on stress, strain, elongation, Hooke's law and Poisson's ratio; conversions between engineering and true stress/strain; the necking criterion εT = n with Hollomon's law (a favourite in metal-forming questions); and resilience or toughness as area under the curve. Conceptual one-markers ask which property measures stiffness, ductility or toughness and what the 0.2 % proof stress is.

Quick check

1. A specimen of 100 mm gauge length breaks at 122 mm. What is %EL?
2. Engineering strain is 0.10. What is the true strain?
3. Which property is unchanged by quenching and tempering a steel: yield strength, hardness or Young's modulus?
4. For a Hollomon material with n = 0.25, at what true strain does necking begin?
5. E = 200 GPa and ν = 0.25. What is G?

Answers: 1. 22 %; 2. 0.0953; 3. Young's modulus; 4. 0.25; 5. 80 GPa.