Stress-strain Curve

The Stress-Strain Curve: What It Tells Us

When a solid is subjected to a deforming force, it undergoes a change in shape or size. The relationship between the stress (the internal restoring force per unit area) and the strain (the relative deformation) is not arbitrary — it follows a characteristic pattern that is captured by the stress-strain curve. This curve is obtained experimentally by gradually increasing the load on a specimen (say, a metal wire or a rod) and measuring the corresponding elongation or compression at each stage.

The curve is not a single straight line. It reveals several distinct regions, each corresponding to a different mechanical behaviour of the material. Understanding these regions is essential for predicting how a material will behave under load — whether it will spring back, deform permanently, or eventually break.

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The Four Key Regions of the Curve

The stress-strain curve for a typical ductile material (like mild steel) can be divided into four main parts. We will go through them in the order they appear as stress increases.

1. Proportional Limit (Hooke's Law Region)

At very low stresses, the strain is directly proportional to the stress. This means the graph is a straight line passing through the origin. In this region, the material obeys Hooke's law:

The constant of proportionality is the Young's modulus $Y$ (or the modulus of elasticity in general). So we write:

The point up to which this linear relationship holds is called the proportional limit. Beyond this point, the stress and strain are no longer proportional, and the graph begins to curve.

2. Elastic Limit and Yield Point

After the proportional limit, the curve starts to deviate from a straight line. However, if the load is removed at any point before a certain critical stress, the material returns to its original dimensions. This critical stress is called the elastic limit.

• Elastic limit: The maximum stress that a material can withstand without permanent deformation.
• Yield point: For many materials (especially mild steel), there is a very clear stress at which the strain increases rapidly without any increase in stress. This is the yield point. The stress at this point is called the yield strength ($\sigma_y$).

At the yield point, the material "gives way" — it begins to flow plastically. The strain increases dramatically while the stress remains nearly constant. This flat region on the curve is called the yield plateau.

3. Plastic Region (Work Hardening)

Once the yield point is crossed, the material enters the plastic region. Here, the deformation is permanent — even if the load is removed, the material will not return to its original shape. The strain is now much larger than in the elastic region.

Interestingly, after the yield plateau, the curve begins to rise again. This happens because the material work-hardens (or strain-hardens). As the material is deformed plastically, its internal structure changes — dislocations in the crystal lattice multiply and tangle, making further deformation more difficult. So, to produce additional strain, a higher stress is required. The curve slopes upward, but less steeply than in the elastic region.

This region continues until the stress reaches a maximum value.

4. Ultimate Strength and Fracture

The highest point on the stress-strain curve is called the ultimate tensile strength (or simply ultimate strength, $\sigma_u$). This is the maximum stress the material can withstand before it begins to fail.

After the ultimate strength is reached, a curious thing happens: the material starts to neck. A localised reduction in cross-sectional area occurs at some point along the specimen. Because the area decreases, the true stress in the necked region becomes very high, but the engineering stress (calculated using the original area) actually drops. The curve therefore turns downward.

Finally, at the breaking point (or fracture point), the material ruptures. The stress at which fracture occurs is called the breaking stress ($\sigma_b$). For ductile materials, the breaking stress is lower than the ultimate strength.

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Properties of the Stress-Strain Curve (as listed in the textbook)

The textbook lists several important properties that can be read directly from the curve. Each one is a quantitative measure of the material's mechanical behaviour.

›Proof

Property 1: Young's Modulus ($Y$)

In the linear (Hooke's law) region, the slope of the stress-strain curve gives the Young's modulus.

Let $\sigma$ be the stress and $\epsilon$ be the strain. In the linear region:

$\sigma = Y \epsilon$

Therefore,

$Y = \frac{\sigma}{\epsilon} = \text{slope of the linear portion}$

This is a constant for a given material at a given temperature. A steeper slope means a stiffer material (higher modulus).

›Proof

Property 2: Yield Strength ($\sigma_y$)

The yield strength is the stress at the yield point. On the curve, it is the stress value at which the graph first shows a large increase in strain for a negligible increase in stress (the beginning of the yield plateau).

For materials that do not have a clear yield point (e.g., many aluminium alloys), the yield strength is often defined as the stress required to produce a specific amount of permanent strain, typically 0.2%. This is called the 0.2% offset yield strength.

›Proof

Property 3: Ultimate Tensile Strength ($\sigma_u$)

This is the maximum stress on the curve. It is the highest point of the graph. Mathematically, it is the maximum value of $\sigma$ as a function of $\epsilon$:

$\sigma_u = \max(\sigma(\epsilon))$

It represents the maximum load the material can support per unit original area.

›Proof

Property 4: Percentage Elongation (Ductility)

The total plastic strain at fracture, expressed as a percentage, is a measure of ductility. If the original length of the specimen is $L_0$ and the length at fracture is $L_f$, then:

$\text{Percentage elongation} = \frac{L_f - L_0}{L_0} \times 100\%$

On the stress-strain curve, this corresponds to the total strain at the breaking point (minus the elastic strain, which is recovered upon unloading). A large percentage elongation indicates a ductile material; a small value indicates a brittle material.

›Proof

Property 5: Toughness

The toughness of a material is the total energy absorbed per unit volume before fracture. On the stress-strain curve, this is the area under the entire curve up to the breaking point.

If the curve is given by $\sigma = f(\epsilon)$, then:

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