Stress-Strain Curve: Material Deformation Explained

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Mild steel: yields at 250 MPa with 25% elongation

A typical mild steel specimen yields at 250 MPa, reaches an ultimate strength of 400 MPa, then necks and fractures at 25% elongation. The area under the curve represents the material's toughness.

Formula

σ = F / A₀ (engineering stress)
ε = ΔL / L₀ (engineering strain)
E = σ / ε (in elastic region — Hooke's law)
Resilience = σ_y² / (2E)

The Universal Mechanical Test

The tensile test is the most fundamental experiment in materials engineering. A specimen is gripped at both ends and pulled apart at a controlled rate while force and displacement are recorded. The resulting stress-strain curve is a fingerprint of the material's mechanical behavior — revealing its stiffness, strength, ductility, and toughness in a single graph.

Elastic Region and Young's Modulus

The initial linear portion of the curve obeys Hooke's law: stress is proportional to strain. The slope is Young's modulus E, a measure of atomic bond stiffness. Metals like steel (E ≈ 200 GPa) are far stiffer than polymers (E ≈ 1-4 GPa) because metallic bonds resist stretching more than the van der Waals forces between polymer chains. All deformation in this region is fully reversible.

Yielding and Plastic Flow

At the yield strength σ_y, dislocations begin to move through the crystal lattice, causing permanent (plastic) deformation. In mild steel, this appears as a sharp yield point followed by a plateau. In aluminum alloys, yielding is gradual and engineers define it using the 0.2% offset method. Beyond yielding, strain hardening increases the stress needed for further deformation as dislocation tangles accumulate.

Necking and Fracture

At the ultimate tensile strength σ_u, the specimen begins to neck — deformation localizes in a narrow region where the cross-section shrinks rapidly. The engineering stress drops because it's calculated from the original area, even though true stress at the neck continues to rise. Eventually the material fractures. The total area under the stress-strain curve represents the material's toughness — its ability to absorb energy before breaking.

FAQ

What does a stress-strain curve show?

A stress-strain curve plots engineering stress (force/original area) versus engineering strain (change in length/original length) as a specimen is pulled in tension. It reveals the elastic modulus, yield strength, ultimate tensile strength, and ductility of a material.

What is the difference between elastic and plastic deformation?

Elastic deformation is reversible — atoms return to their original positions when the load is removed. Plastic deformation is permanent, caused by dislocation movement along slip planes. The yield point marks the transition between the two regimes.

What is Young's modulus?

Young's modulus (E) is the slope of the stress-strain curve in the elastic region. It measures a material's stiffness — how much it resists elastic deformation. Steel has E ≈ 200 GPa while rubber has E ≈ 0.01 GPa.

Why does necking occur before fracture?

After the ultimate tensile strength, strain hardening can no longer compensate for the decreasing cross-sectional area. Deformation localizes in a narrow band (neck), stress concentrates there, and the specimen fractures.

Sources

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