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Strength of Materials: Hook's Law

Stress and strain can be related towards one another graphically which can be seen in the stress strain plot below.

Stress Strain Curve

Elastic Region

The stress and strain relation can happen in the elastic region which exhibits elastic behavior, or the plastic region. The elastic region is the area of the stress strain curve that engineers are most interested in because it exhibits a linear relationship which is known as Hooke's Law, equation 1. Due to this linear relationship a constant known as the Young's Modulus can be derived to relate stress and strain.

Hook's Law Equation (1)

Plastic Region

Plastic Region Consists of different parts, which are Yielding, Strain Hardening, and Necking.

In the Yielding section the part under load will start to deform. However, the load will stay the same, which means the stress on the part will be the same even though strain is increasing. The next section, Strain Hardening, additional loading will occur on the object, however stress and strain will no long be linearly related. During strain hardening, if the part is unloaded, the Young's Modulus (Elastic Region) will shift. Due to this shift a larger force on the object will be required to cause the part to continue to deform.

Effect of Strain Hardening

Finally, necking will occur after the ultimate stress has been reached. Necking will continue until the part reaches its fracture stress and breaks.

Necking due to deformation of a part

Hooke's Law no longer applies to this region due to the fact that there is no longer a linear relationship between stress and strain.

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