Hooke’s Law

In material science and engineering work, we deal with various types of materials. In engineering structures, we use different materials to build different components. But how do engineers choose the material for a component? There are several parameters for choosing a material, but the most important one is to know about the load-bearing capacity, which means how much a material or a body elongates for a certain magnitude of applied force, which we can find by using Hooke’s law. So, let us discuss Hooke’s law. 

Hooke’s law

“Within the proportional limit, the stress in a body is directly proportional to the corresponding strain.”

We know stress =FA

Where F is restoring force and A is the area of cross section.

And strain = ll

Where l is the change in length.

And l is the original length. 

According to Hooke’s law, stress ∝ strain 

Hence, mathematically, FA∝ll

If we replace the proportionality symbol, then it will be

FA=Ell

Here E is the proportionality constant called Young’s modulus. 

Elastic modulus 

Above, we mention a proportionality constant, Young’s modulus, in which the ratio of tensile and tensile strain is symbolised by Y.

Mathematically  Y= Tensile stressTensile strain

                      Or

Y=FAll

Modulus of rigidity: Another type of Elastic constant is called modulus of rigidity, defined as the ratio of shear stress and shear strain. So, it is also called shear modulus. 

It is symbolised by η

Mathematically, η=FAxh= FhxA

Bulk modulus:

Bulk modulus is defined as the ratio of volume stress and volume strain. 

Mathematically, B=-PVV

Here the negative sign refers to volume decreasing as pressure increases. 

Sometimes, the change in pressure is given as a gradient of volume; that time, we can write Bulk modulus for a slight change in pressure as: 

B=-VdPdV

Compressibility: Compressibility is defined as the reciprocal of Bulk modulus. 

Mathematically, K=1B=-1vdVdP 

Hooke’s law in spring force

When we elongate a spring, it exerts a force known as spring force. If we apply Hooke’s law in a spring, then: 

FA=Ell

Where F is force delivered by spring 

A is the area of cross section of spring

l is the natural length of the spring

l is the change in length of spring.

Now we can arrange the equation as F=(AEl)l

Or F=kx

Where k is a constant, known as the spring constant, 

 k=AEl 

And x=l change in length 

For a given spring, A, E, and l are constant, so k is the spring constant.

As force delivered by spring is opposite to the direction of elongation of spring, a negative sign can be applied. 

 So F=-kx

Experimental proof of Hooke’s law 

Take a length of spring. Measure the initial length and cross section of the spring. Now increase the load gradually and measure the elongation. Note down the applied load and its corresponding elongation. From the elongation, calculate the strain, and from the applied load, calculate the stress. 

Now make a stress-vs-strain graph. We can observe that up to a certain limit, stress is proportional to the strain. This limit is called the proportionality limit, and the graph is a straight line up to the proportionality limit. The slope of this straight line gives Young’s modulus. 

Let’s discuss the stress-vs-strain graph.

This is the stress-vs-strain graph for mild steel. 

When a tensile force is applied to a body, atoms in the body get displaced from their equilibrium position. But the molecular attraction force tries to back the molecule to its initial position. When the load is removed due to molecular attraction force, molecules return to their initial positions. This is called the elasticity of the material. But after a certain limit, molecular bonds between molecules are broken, so they cannot return to their initial position and cause permanent deformation. This deformation is called plastic deformation. There are several limit points in the stress-strain curve. Let us briefly discuss these points.

1. Proportionality limit: It is the limit up to which Hooke’s law is valid. This means that up to this point, stress is directly proportional to the strain. 

2. Elastic limit: After the elastic limit, plastic deformation is formed. Elasticity is valid up to that point. After the elastic limit, plastic deformation is formed. 

3. Yield point: At the yield point, the molecular bond between molecules is broken; therefore, the curve is a little bit down here. 

4. Ultimate tensile stress: At the UTS point, maximum stress is observed, and after that point, the strain goes downward. 

5. Breaking point: At that point, the material finally breaks into two parts. 

Some features of Hooke’s law

• For spring force, manometer, and wheel balance, Hooke’s law is used as the fundamental law.

• It can also explain the molecular behaviour of the material.

• Hooke’s law only used up to a certain limit, called proportional limit, and after that, it will fail.

• It is accurate for a solid body. In the case of gases and liquids, it does not work properly.

Conclusion

In material engineering and science, there are various studies and applications of the elastic properties of materials. Hooke’s law is one of the fundamental laws which explain the mechanical properties of the material. Hooke’s law only used up to a certain limit, called proportional limit, and after that, it will fail. This study material contains notes on Hooke’s law, elastic constants, stress-strain characteristics, etc.