Stress Strain Relationship

The stress-strain relationship is a fundamental concept in material science that explains how materials deform under various forces. Stress refers to the internal force per unit area exerted within a material, while strain measures the material's deformation in response to that stress. This relationship helps engineers and scientists understand how different materials will behave when subjected to tension, compression, or shear forces, which is critical for designing structures and products.

In real life, the stress-strain relationship can be seen in the stretching of a rubber band. Initially, the rubber band stretches easily (elastic region), but if pulled too far, it can reach a point where it either becomes permanently deformed (plastic region) or breaks (ultimate tensile strength). This example demonstrates how understanding stress and strain is vital in applications ranging from bridge construction to manufacturing everyday objects.

Stress-Strain Relationship

The relation between the stress and the strain of a given material under tensile stress can be plotted on a graph called strain stress curve.

Fig: typical stress-strain curve for a metal.

The stress-strain curves vary from material to material. These curves help us to understand how a given material deforms with increasing loads.

When the strain is small (i.e., in region OA) stress is proportional to strain. This is the region where the Hooke’s law is obeyed. The point A is called the proportional limit and the slope of line OA gives Young’s modulus (Y) of the material of the wire.

If the strain is increased a little bit, i.e., in the region AB, the stress is not proportional to strain. However, the wire still regains its original length after the removal of the stretching force. This behaviour is shown up to point B known as the elastic limit or yield-point. The region OAB represents the elastic behaviour of the material of the wire.

If the wire is stretched beyond the elastic limit B, i.e., between BC, the strain increases much more rapidly and if the stretching force is removed the wire does not come back to its natural length. Some permanent increase in length takes place.

If the stress is increased further by a very small amount, a very large increase in strain is produced (region CD) and after reaching point D, the strain increases even if the wire is unloaded and ruptures at E. In the region DE, the wire literally flows. The maximum stress corresponding to D after which the wire begins to flow and breaks is called breaking or tensile strength. The region BCDE represents the plastic behaviour of the material of wire.

Types of Materials

Ductile material: If a large deformation in the material takes place between the elastic limit and fracture point (or) if the material has a large plastic region, then that material is called ductile material.

Brittle material: If the material breaks down soon after the elastic limit is crossed, it is called as brittle material.

Elastomers:- These materials only have an elastic region (i.e., no plastic region). For example:- rubber

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Solved Examples Based On Stress-Strain Relationship

Example 1:Which of the following material is an elastomer

1) Steel

2) Aluminium

3) Rubber

4) Plastic

Solution:

Elastomers - The Materials can be elastically stretched to large values of strain.

wherein

Hence, the answer is the option (3).

Example 2: A graph is shown between stress and strain for a metal. The part in which Hook's law holds good is

1) OA

2) AB

3) BC

4) CD

Solution:

Graph of stress vs strain

straight line

wherein

During OA, stress $\propto$ strain

Hook's law holds good

Hence, the answer is the option (1).

Example 3: A graph is shown b/w stress and strain. In this graph point B indicates:

1) Breaking point

2) limiting point

3) yield point

4) none of the above

Solution:

Stress-strain curve

Gradually increasing the load on a vertically suspended metal wire.

wherein

Point B indicates the yield point

Hence, the answer is the option (3).

Example 4: In the given figure, if the dimensions of the wires are the same, and the materials are different, Young's modulus is more for-

1) A

2) B

3) Both

4) None of these

Solution:

Use, $F=\frac{Y_A}{l} \cdot \Delta l$
i.e. F- $\Delta I$ graph is a straight line with slope $\frac{Y_A}{l}$ or slope proportional to $Y$.
$(\text { Slope })_A>(\text { Slope })_B$

$
\therefore Y_A>Y_B
$

Hence, the answer is the option (1).

Example 5: The stress versus strain graphs for wires of two materials $A$ and $B$ are shown in the figure. If $Y_A$ and $Y_B$ are Young's modulus of the material then

1) $y_B=2 y_A$
2) $y_A=y_B$
3) $y_B=3 y_A$
4) $y_A=3 y_B$

Solution:

Use, $y_A=\tan \theta_A, y_B=\tan \theta_B$

$
\begin{aligned}
& \frac{y_A}{y_B}=\frac{\tan \theta_A}{\tan \theta_B}=\frac{\tan 60^{\circ}}{\tan 30^{\circ}} \\
& \frac{y_A}{y_B}=\frac{\sqrt{3}}{1 / \sqrt{3}}=3 \\
& y_A=3 y_B
\end{aligned}
$

Hence, the answer is the option (4).

Summary

The stress-strain relationship defines the deformation of materials under the applied forces. Stress is the force applied per unit area to a material whose subsequent deformation or change in shape is referred to as strain. The relationship can commonly be expressed in a stress-strain curve, describing how a material reacts to increased stress. This would include information such as the elastic limit, indicating where permanent deformation occurs, ultimate strength, and the maximum stress a material will support without breaking. The understanding of this relationship helps in the appropriate choice of material for a particular application that ensures safety, durability.