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Strength of Materials: Hook's Law, Poisson's Ratio, Brittle & Ductile Material

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Strength of Materials: Hook's Law, Poisson's Ratio, Brittle & Ductile Material

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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.

Energy

To deform a part energy is required. The energy is the area under the stress strain curve seen in the figure below.

Modulus of Resilience & Modulus of Toughness

The Modulus of Resilience represents the energy under the elastic region, while the Modulus of Toughness represents the entire curve. To calculate the Modulus of Resilience equation 2 would be used. To calculate the Modulus of Toughness an experimental data set would need to be acquired, so that some form of integration like the trapezoidal rule could be used.

Modulus of Resileince Energy Equation (2)

Poisson's Ratio

So far I have only discussed Young's Modulus. However, Hook's Law also applies to shear, which is known as the Shear Modulus. To relate the two, Poisson's Ratio would have to be used, equation 3.

Poisson's Ration / Calculate Shear Modulus (3)

Ductile & Brittle Material

A material can be considered either ductile or brittle. The main difference between a ductile and a brittle material is a ductile material can deform a lot more than a brittle material, which means it can absorb more energy. However, a brittle materials typically can handle large applied loads before they break.

Temperature can also effect a material. Normally, warmer temperatures will cause a material to become more ductile, while a colder temperature can cause materials to become more brittle.




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