Hooke’s Law in terms of Stress and Strain

Hooke’s Law is a law of physics used to solve problems in the field of elasticity. Hooke’s Law equations prove that the force (F) required to expand or contract a spring by some distance has a linear measure to the distance.

The basic Hooke’s Law equation is Fs = kx, where k is a constant element of the spring and x is small compared to the distortion of the spring. An elastic material for which Hooke’s Law equations can be applied is called linear-elastic or Hookean. This Law is a precise approximation for most solid bodies. The condition is that the forces and deformations should be small. 

Definitions and Types of Stress and Strain 

• Stress

Stress is defined as force per unit area. It is the ratio of applied force to a cross-sectional area. There are three types of stress. 

• Tensile stress

Tensile stress stretches the material and acts parallel to the stressed region. 

Formula: σ = Fn / A where σ is the normal stress, Fn is the force, and A is the area. 

• Compressive stress

Compressive stress compresses the material and acts parallel to the stressed region. The formula is the same as tensile stress. 

• Shearing stress

Shearing stress cuts the material and acts in the plane to the stressed area. Shearing stress is perpendicular to compressive and tensile stresses. 

Formula : T = Fp / A

• Strain 

Strain is the deformation or distortion of a solid because of stress. There are two types of strain.

• Normal strain

This type of strain elongates or contracts the line segment.

Formula: e = dl/l0, where e is the strain, dl is the change in length and l0 is the initial length. 

• Shear strain

This type of strain changes the angle between two line segments to 

right angle. 

Hooke’s Law  for linear springs 

Before finding out about the equation, you must know linear springs’ meaning. Spring with the same diameter throughout its length is called a linear spring. This constant diameter gives a continuous spring rate. The spring rate does not change when a load acts on the spring. The deflection/displacement of the spring is proportional to the force applied. The following is the equation for linear springs.  

1. F = -kx
2. U = ½ kx²

Linear springs are coil springs that are helical in shape. They can be compressed and extended. These springs undergo a constant deflection per unit force. Their load versus deflection curve can be modified by changing the diameter of the coils. 

Derivation of Hooke’s Law for linear springs

Consider a normal helical spring. One end of the spring is attached to a fixed object and another end is pulled by a force. The magnitude of this force is Fs. Assume that the spring has reached the state of equilibrium, where its length is not varying. Let x be the quantity by which the free end of the spring was extended from its relaxed position. The relaxed position is the position where the spring is not stretched. 

Hooke’s Law equation states that Fs = kx or x = Fs / k, where k is a positive real number. This formula is the same for both expansion and contraction of spring. This causes Fs and x both to be negative in that case. According to Hooke’s Law equation, the graph of force (Fs) as a function of displacement (x) will be a straight line. This straight line will pass through the origin. The slope of this line will be k. 

Fs is the restoring force exerted by the spring on the thing that is pulling its free end. 

In this case, Hooke’s Law equation becomes 

Fs = -kx

This is because the direction of the restoring force is opposite to the displacement.  

Applications of Hooke’s Law equations

1. Hooke’s Law equations are applied in strings due to their use in elasticity. 
2. They are used in engineering, medical sciences, etc. 
3. They are used as a fundamental concept in the manometer, balance wheel on a clock, and a spring scale.
4. These equations are the base of acoustics, molecular mechanics, and seismology. 

Conclusion 

Hooke’s Law for linear springs in terms of stress and strain states that the force required to expand or contract a spring by some distance estimates linearly to the distance. Hooke’s Law equations prove that the force (F) required to expand or contract a spring by some distance has a linear measure to the distance. The basic Hooke’s Law equation is Fs = kx. The spring with the same diameter throughout its length is called a linear spring. This constant diameter gives a continuous spring rate. Linear springs are coil springs that are helical in shape.