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Physics · Mechanical Properties of Solids

Stress-Strain Curve

Load a metallic wire in steps, record the stress and the strain, and plot one against the other. The resulting stress-strain curve is a fingerprint of the material — it marks where Hooke's law fails, where recovery ends, where the material can carry the most stress, and where it finally breaks. NCERT §8.4 walks the curve point by point. This deep-dive maps every region of the metallic-wire curve, separates the proportional limit from the elastic limit, contrasts ductile with brittle, and shows why rubber traces a hysteresis loop while steel — the less stretchy of the two — is in fact the more elastic.

The tensile test and the curve

NCERT describes a standard tensile test. A test cylinder or wire is clamped and stretched by an applied force that is increased in steps. At each step the fractional change in length — the strain — is recorded against the stress, which equals the applied force per unit cross-sectional area. Plotting stress on the vertical axis and strain on the horizontal axis produces the stress-strain curve. A typical curve for a metallic wire has a characteristic shape with five landmarks worth naming.

The curve is read left to right as the load grows. It begins as a straight line through the origin, bends as the material yields, climbs to a peak, then falls before the wire snaps. Each segment tells you what the wire is doing internally — recovering elastically, flowing plastically, or about to fail.

Stress-strain curve for a metallic wire A curve rising linearly from the origin O to the proportional limit P, bending to the elastic/yield point E, flowing through the plastic region to the ultimate tensile strength U, then falling to the fracture point B. A dashed unloading line from a point C drops to a permanent set on the strain axis. Strain Stress ✕ C permanent set O P E U B elastic region plastic region yield point
Figure 1. The metallic-wire stress-strain curve. O→P linear (Hooke's law obeyed); P proportional limit; E elastic limit / yield point; E→U plastic region; U ultimate tensile strength (the peak); B fracture point. The dashed line from C shows unloading from the plastic region, leaving a permanent set.

The five landmarks on the curve

Five named points organise the entire curve. Memorise their order along the strain axis and what physically changes at each.

PointNameWhat it marks
PProportional limitEnd of the straight line; Hooke's law (stress ∝ strain) holds only up to here
EElastic limit / yield pointMaximum stress for which the wire still returns to its original length on unloading; beyond it deformation becomes plastic
—Plastic regionStrain rises rapidly for small stress increases; unloading now leaves a permanent set
UUltimate tensile strength \(\sigma_u\)The highest point — the maximum stress the material can sustain
BFracture pointThe wire actually breaks; the corresponding stress is the breaking stress

The region from O to P is linear, and here the wire behaves as a perfectly elastic body — remove the load and it springs back exactly. The yield strength \(\sigma_y\) is the stress at the elastic limit E, and the ultimate tensile strength \(\sigma_u\) is the stress at the peak U. Note that NCERT's "yield point" coincides with the elastic limit on the school-level curve; both mark the boundary between recoverable and permanent deformation.

Proportional limit vs elastic limit

These two limits are the most commonly confused pair in the whole chapter, and NEET exploits that. They are not the same point, and they mark different physical events.

Proportional limit (P)

  • End of the straight-line portion of the curve.
  • Up to here, stress and strain are in constant ratio: Hooke's law holds.
  • The slope of OP is Young's modulus \(Y\).
  • Comes first along the curve.

Elastic limit (E)

  • Maximum stress for which the wire fully recovers on unloading.
  • The curve is already non-linear here — Hooke's law has failed, yet recovery is still complete.
  • Coincides with the yield point; stress here is the yield strength \(\sigma_y\).
  • Comes after the proportional limit.

Between P and E lies a narrow band where the curve has bent away from the straight line but the deformation is still completely recoverable. In other words, the material can stop obeying Hooke's law and still remain elastic. Linearity ends at P; recoverability ends at E.

Plastic region and permanent set

Push the load past the elastic limit and the wire enters the plastic region. Here the strain grows rapidly even for a small increase in stress — the curve flattens out and rises gently. The crucial change is in what happens on unloading.

Suppose the load is removed at some point C in the plastic region. The wire does not retrace its path back to the origin; it follows a new line roughly parallel to the original elastic line, and lands at a non-zero strain even when the stress has returned to zero. This left-over strain is the permanent set, and the deformation that produced it is plastic deformation. The wire is permanently longer than it started.

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Foundation check

The straight portion OP is exactly the regime where Hooke's law defines the elastic moduli. Revisit it if "stress ∝ strain" feels shaky.

Ultimate strength vs fracture point

The peak of the curve is the ultimate tensile strength U — the largest stress the material can bear. It is tempting to think the wire breaks at this peak, but it does not. Beyond U the curve turns downward: the wire begins to neck (thin locally), and fracture finally occurs at point B, at a stress lower than the ultimate strength. The wire breaks past the highest point of the curve, not at it.

Ductile vs brittle materials

The shape of the curve between the ultimate strength U and the fracture point B classifies the material. The decisive feature is how much plastic deformation happens before the wire breaks — measured by the gap between U and B along the strain axis.

Ductile material

  • U and B are far apart.
  • Large plastic deformation before fracture — the wire stretches, necks, and elongates substantially.
  • Long, gentle plastic region.
  • Examples: copper, mild steel, aluminium.

Brittle material

  • U and B are close together.
  • Breaks soon after the elastic limit is crossed, with little plastic flow.
  • Short or almost absent plastic region.
  • Examples: glass, cast iron.

A ductile material gives visible warning before failure — it deforms a lot first — which is why ductile metals are favoured for structural members. A brittle material such as glass fractures abruptly with almost no plastic stretching to warn of impending failure.

Elastomers and the hysteresis loop

Not all materials follow the metallic-wire pattern. Rubber, and biological tissue such as the elastic tissue of the aorta, can be stretched to several times their natural length and still return to the original shape. NCERT calls such materials elastomers. Their stress-strain curve is qualitatively different from a metal's in two key ways.

Stress-strain curve for rubber showing a hysteresis loop A loading curve rising with increasing slope from the origin, and a separate, lower unloading curve returning to the origin, the two enclosing a loop whose area represents energy dissipated as heat. Strain Stress loading ↗ unloading ↘ loop area = heat O
Figure 2. Rubber's stress-strain curve. There is no straight (proportional) region. On unloading, the curve (purple) does not retrace the loading curve (teal); the two enclose a hysteresis loop. The wire returns to its natural length, but along a different path, and the loop area is energy dissipated as heat.

First, there is no region of proportionality — the curve is non-linear from the very start, so an elastomer does not obey Hooke's law over most of its range, even though its elastic region is enormous. Second, when the deforming force is gradually reduced, the unloading curve does not retrace the loading curve. The sample does end up back at its natural length, but it travels there along a lower path.

The work done by the material in returning is less than the work done on it during stretching. That energy difference is absorbed by the material and reappears as heat — you can feel a stretched-and-released rubber band warm against your lips. This effect is elastic hysteresis, and the enclosed loop area equals the energy dissipated per cycle. It is precisely this dissipation that makes elastomers ideal for shock absorbers: the absorber retains part of the energy delivered by a jolt and transmits only a small fraction onward.

Why steel is more elastic than rubber

Everyday language calls rubber "more elastic" because it stretches so much. Physics reverses this. A body is more elastic when it resists deformation more strongly — when a given stress produces a smaller strain. That is exactly the body with the larger Young's modulus.

Take identical steel and rubber wires under the same deforming force. The steel extends far less than the rubber. Equivalently, to produce the same strain in both, the steel needs a much larger stress, which means the internal restoring force developed inside steel is much greater. Larger restoring force for a given strain is precisely what "more elastic" means in physics.

CriterionSteelRubber
Strain for a given stressVery smallVery large
Young's modulus \(Y\)Large (\(\sim 2\times10^{11}\) Pa)Small
Internal restoring forceLargeSmall
More elastic (physics sense)?YesNo
Quick recap

The curve in one breath

  • Five landmarks left to right: O (origin) → P (proportional limit) → E (elastic limit / yield) → U (ultimate tensile strength, the peak) → B (fracture).
  • OP is linear and obeys Hooke's law; its slope is Young's modulus. Recoverability extends a little beyond, up to E.
  • Proportional limit ends linearity; elastic limit ends recoverability — they are different points.
  • Past E, deformation is plastic; unloading leaves a permanent set (non-zero strain at zero stress).
  • U is the maximum stress; the wire breaks later at B, at a lower stress. Ultimate strength ≠ fracture point.
  • U and B far apart → ductile (copper, mild steel); close together → brittle (glass).
  • Elastomers (rubber, aorta tissue): no proportional region; loading and unloading differ → hysteresis loop; loop area = heat. Recovers shape but dissipates energy.
  • Steel is more elastic than rubber: smaller strain per stress, larger Young's modulus, larger restoring force.

NEET PYQ Snapshot — Stress-Strain Curve

Curve-reading and elastic-limit questions NEET has actually asked. Anchor each to a region of the graph.

NEET 2024

The maximum elongation of a steel wire of 1 m length, if the elastic limit of steel and its Young's modulus are \(8\times10^{8}~\text{N m}^{-2}\) and \(2\times10^{11}~\text{N m}^{-2}\) respectively, is:

  1. 4 mm
  2. 0.4 mm
  3. 40 mm
  4. 8 mm
Answer: (1) 4 mm

Curve relevance. The maximum recoverable elongation occurs at the elastic limit. From \(Y=\dfrac{\sigma}{\varepsilon}\), the limiting strain is \(\varepsilon=\dfrac{\sigma_{\text{el}}}{Y}=\dfrac{8\times10^{8}}{2\times10^{11}}=4\times10^{-3}\). With \(L=1~\text{m}\), \(\Delta L=\varepsilon L=4\times10^{-3}~\text{m}=4~\text{mm}\).

NEET 2022

Assertion (A): The stretching of a spring is determined by the shear modulus of the material of the spring. Reason (R): A coil spring of copper has more tensile strength than a steel spring of the same dimensions.

  1. Both (A) and (R) true, (R) is the correct explanation
  2. Both (A) and (R) true, (R) is not the correct explanation
  3. (A) true but (R) false
  4. (A) false but (R) true
Answer: (3) (A) true, (R) false

Curve relevance. Stretching a coil changes its shape, not its length or volume — so it is governed by shear modulus: (A) is true. But on the strength comparison, steel has higher tensile strength than copper, so (R) is false. The "steel is stronger / more elastic than copper" theme of the curve carries the answer.

NEET 2023

A wire is suspended from the ceiling and stretched by a weight \(W\) attached at its free end. The longitudinal stress at any point of cross-sectional area \(A\) of the wire is:

  1. Zero
  2. \(2W/A\)
  3. \(W/A\)
  4. \(W/2A\)
Answer: (3) W/A

Curve relevance. The tension at any cross-section equals the supported weight \(W\) (not \(2W\), despite the equal-and-opposite ceiling reaction). The horizontal axis stress used to plot the curve is exactly this \(F/A=W/A\). Getting the stress right is the first step in reading any stress-strain graph.

FAQs — Stress-Strain Curve

Short answers to the curve-reading questions NEET aspirants get wrong most often.

What is the difference between the proportional limit and the elastic limit?
The proportional limit (point P) is where Hooke's law stops holding — stress and strain are no longer in a straight-line ratio. The elastic limit, which lies at the yield point (point E), is the maximum stress up to which the wire still returns to its original length on unloading. Between P and E the curve is no longer straight, yet the deformation is still fully recoverable. So the proportional limit comes first and is the end of linearity; the elastic limit comes a little after and is the end of recoverability.
Is the ultimate tensile strength the point where the wire breaks?
No. The ultimate tensile strength (point U) is the maximum stress the material can carry — it is the highest point on the curve. The wire does not break here. Beyond U the stress the material can bear actually falls, the wire necks down, and fracture finally occurs at a later point B. Confusing U with the fracture point is a standard NEET trap.
What is a permanent set?
If the wire is loaded past the elastic limit into the plastic region and then unloaded, it does not return to its original length. A residual strain remains even when the stress is zero. This left-over, non-recoverable strain is called the permanent set, and the deformation that produced it is plastic deformation.
Why does the rubber stress-strain curve form a loop?
When rubber is stretched and then released, the unloading curve does not retrace the loading curve, so the two paths enclose a loop. The wire returns to its natural length, but along a different path. The area of the loop is the energy dissipated as heat in each loading-unloading cycle. This effect is called elastic hysteresis and is the basis of shock absorbers.
Why is steel said to be more elastic than rubber when rubber stretches far more?
In physics, the more elastic material is the one that resists deformation more strongly — it produces a smaller strain for a given stress, i.e. it has a larger Young's modulus. To produce the same strain, steel needs far more stress than rubber, which means its internal restoring forces are larger. So steel is more elastic than rubber, even though rubber visibly stretches much further.
How does the stress-strain curve tell ductile and brittle materials apart?
Look at the gap between the ultimate tensile strength (U) and the fracture point (B). If the two are far apart, the material undergoes large plastic deformation before breaking and is ductile (for example, copper or mild steel). If U and B are close together, the material breaks almost as soon as the elastic limit is crossed, with little plastic flow — it is brittle (for example, glass).
Related Topics
Mechanical Properties of Solids — Parent chapter Full chapter map: stress, strain, Hooke's law, elastic moduli, applications. Stress and Strain Tensile, shearing and hydraulic stress; the three strains; SI units. Hooke's Law Stress ∝ strain in the linear region; modulus of elasticity. Elastic Moduli Young's, shear and bulk modulus; Poisson's ratio; elastic energy. Applications of Elastic Behaviour Crane ropes, I-section beams, pillars and bridge design.