Stress-Strain Curve: Stress, Strain, Hook’s Law & Graphs

Stress-Strain Curve: When we stretch a spring by applying load to its ends, the length of the spring increases slightly. When you remove the load at the ends, it regains its original size and shape. Similarly, rubber can be pulled to several times its original length and still return to its original shape. But after crossing a certain limit of the load, we will observe that cracks are generated, and then the rubber breaks. The same thing will happen with spring or any other material. This phenomenon also occurs in compression. After a certain limit of compressive force, the material fails.

What is Stress-Strain Curve?

When a deforming force is applied to a material such that the material is in static equilibrium, a resistive force is developed inside the material to oppose the external force. The resistive force per unit area is called stress \((\rm{Force/Area})\). Due to the application of deforming force, there will be a change in the length of the material. The change in length of material per unit length is called strain ( \(\Delta L/L\) ). On plotting the stress and its corresponding strain on the graph, we get a curve, and this curve is called the stress-strain curve or stress-strain diagram. Generally, we take the strain (no units) on the \(x-\)axis and stress (\(\rm{Pa}\) or \(\rm{MPa}\)) on the \(y-\)axis. This stress strain helps us to understand how a given material behaves with increasing loads. The stress-strain behaviour varies from material to material. For the same material, the curve changes with the environment temperature and loading condition of the material.

How to Draw Stress-Strain Curve/Diagram?

The relation between the stress and the strain for a given material under tensile stress can be found experimentally using a universal testing machine (UTM). In this standard test, a test cylindrical rod or a wire is stretched on the application of force (tensile). The strain and the stress due to the applied force needed to cause the strain are recorded. The applied force is gradually increased in steps, and the change in length is noted. Then a graph is plotted between the stress and the strain produced. A typical graph for metal is shown in the figure below.

Stress-Strain Curve Explanation

The stress-strain curve has different regions and points.

1. Point \(A\) is the elastic limit or proportional limit.
2. Point \(B\) is the elastic limit or upper yield point.
3. Point \(C\) is the lower yield point.
4. Point \(D\) represents ultimate strength.
5. Point \(E\) represents the fracture point.
6. The area under the curve \(O\) to \(B\) is the elastic region, and the area under the curve \(B\) to \(E\) is the plastic region.

Observations From the Stress-Strain Curve

From the graph, we can see that in the region between \(O\) to \(A\), the curve is linear (straight line), so we can say that the stress is directly proportional to strain and obeys Hooke’s law. On unloading, the material regains its original dimensions. In this region, the material behaves as an elastic body. The slope of the line \(OA\) gives Young’s modulus and is denoted by the symbol \(Y\). It has the same units as stress, i.e \(\rm{N/m}^2\) or \(\rm{Pa}.\)
\(Y = {\text{Slope}} = \frac{{{\text{Stress}}}}{{{\text{Strain}}}}\)
In the region from \(A\) to \(B\), stress and strain are not proportional. Nevertheless, the body still returns to its original dimension when the load is removed. The point \(B\) in the curve is known as yield point (also known as elastic limit), and the corresponding stress is known as the yield strength of the material.
The portion of the curve between \(B\) and \(D\) shows, if the load is increased further, the stress developed exceeds the yield strength, and strain increases rapidly even for a small change in the stress. From point \(D\), material starts to fail.

When the load is removed, say at some point \(C\) between \(B\) and \(D\), the body does not regain its original dimension. In this case, even when the stress is zero, the strain is not Zero, i.e. the material reaches point \(P\) from \(C\). The material is said to have a permanent deformation. The deformation is said to be plastic deformation.
Beyond point \(D\), additional strain is produced even by a reduced applied force, and fracture occurs at point \(E.\) The material breaks into pieces. If the ultimate strength and fracture points \(D\) and \(E\) are close, the material is said to be brittle. But if they are far apart, the material will be ductile.

Stress-Strain Curve for Various Materials

As we have discussed the nature of stress-strain cure varies from material to material. For example, rubber can be pulled several times its original length but still return to its original shape. This does not happen with metals, the limit of elastic deformation of metals is very small.

Rubber

A distinctly different stress-strain relation exists for vulcanized rubber. The behaviour is qualitatively shown in the figure.

Rubber remains elastic even when it is stretched to over several times its original length. In the case shown in the figure, the length is increased to \(8\) times its natural length. Even then, if the stretching forces are removed, it will come back to its original length. In this respect, rubber is more elastic than ductile metal like steel.

However, the magnitude of stress for a given strain is much more in steel than in rubber. This means large internal forces appear if the steel wire is deformed. In this sense, steel is more elastic than rubber. There are two important phenomena to note from the figure. There is almost no region of proportionality. Secondly, when the applied load is removed, the original curve is not retraced. Although the material finally acquires its original length.

The work done (area under the curve) by the material in returning to its original shape is less than the work done by the deforming force when it was deformed. A particular amount of energy is absorbed by the material in the cycle. This absorbed energy appears as heat. This phenomenon is called elastic hysteresis. Substances like tissue of aorta, rubber, etc., that can be stretched to cause large deformation are called elastomers.

Stress and Strain Curve- Sample Problems

Q.1. The figure shows the strain-stress curve for a given material. What are,
(a) Young’s modulus and
(b) approximate yield strength for this material?

Ans: (a) We have to find the slope of the curve to find out Young’s modulus. For finding slope, take two points lying on the line. For example, take origin \((0,\,0)\) and \((150 × 10^6,\,0.002)\).
Since it is passing through the origin \((y = mx)\) we can directly write,
Young’s modulus, \(Y = {\text{Slope}} = \frac{{{\text{Stress}}}}{{{\text{Strain}}}}\)
\( \Rightarrow Y = \frac{{150 \times {{10}^6}}}{{0.002}}\)
\( \Rightarrow Y = 75 \times {10^6} \times {10^3}\)
\( \Rightarrow Y = 75 \times {10^9}\,{\text{N}}\,{{\text{m}}^{ – 2}}\)
(b) Yield strength of a material is defined as the maximum stress it can withstand before failure. Here there is no plastic region, the yield strength and Ultimate tensile strength are almost the same. From the graph, the approximate yield strength of the given material, \({U_T} = 300 \times {10^6}\,{\text{N}}\;{{\text{m}}^{ – 2}}.\)

Summary

The stress-strain curve is the relationship between stress and corresponding strain. Up to a certain limit only, the stress is directly proportional to strain. Hooke’s law is valid only in this linear part of the stress-strain curve. The slope of the linear curve gives Young’s modulus of the material. A material with a large value of Young’s modulus requires a large force to produce small changes in its length.
In daily life, we feel that a material that stretches more is more elastic, but it is a misnomer. In fact, material that stretches to a lesser extent even after the application of a large load is considered to be more elastic. Steel is more elastic than rubber.

FAQs on Stress-Strain Curve

Q.1. Why are steel and brass are preferred over copper and aluminum in heavy machines?
Ans: Steel and brass are preferred over copper and aluminum in heavy machines because steel and brass have high Young’s modulus and required a large force to deform.

Q.2. What are elastomers?
Ans: Many materials like rubber, tissue of aorta, etc., can have large elastic deformation. They can be stretched much larger than metals without plastic deformation.

Q.3. What is the relationship between stress and strain?
Ans: Under elastic limit, stress is directly proportional to strain. According to Hook’s law,
Young’s modulus, \(Y = \frac{{{\text{Stress}}}}{{{\text{Strain}}}}.\)

Q.4. What is the use of a stress-strain diagram?
Ans: The stress-strain diagram provides an estimate of the strength and elasticity of the material. It also tells about the behaviour of the materials with the application of load.

Q.5. What is the elastic limit?
Ans: It is the point on the stress-strain curve after which material deforms plastically. Up to this point, the material returns to its original size after the load is released.

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