The relation between the stress and the strain for a given material under tensile stress can be found experimentally. In a standard test of tensile properties, a test cylinder or a wire is stretched by an applied force. The fractional change in length (Strain = \(\Delta L / L\)) and 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. A graph is plotted between the stress (which is equal in magnitude to the applied force per unit area) and the strain produced. A typical graph for a metal is shown in the Figure below. The stress-strain curves vary from material to material. These curves help us to understand how a given material deforms with increasing loads.

The stress-strain curve has different regions and points.

From the graph, we can see that in the region between O to A, the curve is linear. In this region, Hooke’s law is obeyed. The body regains its original dimensions when the applied force is removed. In this region, the solid behaves as an elastic body.
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. Point B in the curve is known as the yield point (also known as elastic limit) and the corresponding stress is known as yield strength \(\left(\sigma_y\right)\) of the material.
In the region from B to D, when the load is increased further, the stress developed exceeds the yield strength, and strain increases rapidly even for a small change in the stress. 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. The material is said to have a permanent set. The deformation is said to be plastic deformation. Point D on the graph is the ultimate tensile strength \(\left(\sigma_u\right)\) of the material.
Beyond this point, additional strain is produced even by a reduced applied force and fracture occurs at point E. If the ultimate strength and fracture points D and E are close, the material is said to be brittle. If they are far apart, the material is said to be ductile.

As stated earlier, the stress-strain behaviour varies from material to material. For example, rubber can be pulled to several times its original length and still returns to its original shape. The figure below shows the stress-strain curve for the elastic tissue of aorta, present in the heart. Note that although elastic region is very large, the material does not obey Hooke’s law over most of the region. Substances like tissue of aorta, rubber, etc. which can be stretched to cause large strains are called elastomers.

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