Stress-Strain Relationship

Ever stretched a rubber band till it breaks and hits you hard? Just like that rubber band, every material has resistive properties and there is a limit to which these materials can be stretched. The stress-strain relationship can be plotted on a graph in the form of a curve that tells us the limit to which a material can be stretched before it fractures.

Stress

Let’s first understand some terms before jumping into the stress-strain relationship.

To put it simply, when an external force is applied to a material, stress is the measure of what it feels. So, when you stretch a rubber band, the force you apply with your thumbs divided by the cross-sectional area of the rubber band on which the force is applied is called stress.

Hence, stress has units of force per area:

N/m2 (SI)

Stress is usually denoted by the symbol σ.

The formula for stress is as follows:

σ = F/A

Where,

σ: Stress

F: Force applied

A: Area on which the force is applied

Types of stress

A material can experience two types of stress:

• Normal stress

• Shear stress

Normal stress

A material experiences normal stress when the external force applied is perpendicular to the cross-sectional area (or the surface of the material).

Shear stress

On the other hand, a material experiences shear stress when the external force applied is parallel to the cross-sectional area (or the surface).

Strain

Let’s try to understand what strain is now by taking the example of stretching a rubber band again. If stress is the measure of what the rubber band feels when force is applied to stretch it, then strain is the elongation of the rubber band divided by the original length.

Note that the elongation is the change in length caused by the application of the force. Since strain is the ratio of elongation to the original length, it is a dimensionless quantity.

The formula for strain is as follows:

ϵ = ΔL / L

Where,

ϵ: Strain

ΔL: Elongation

L: The original length

Types of strain

Similar to stress, there are two types of strain:

• Normal strain

• Shear strain

Normal strain

When the elongation in a material is caused by normal stress, then it is a normal strain.

Shear strain

When the change in length of the material is caused by shear stress, then it is a shear strain.

Hooke’s law

Hooke’s law simply states that as long as the material is within its elastic limit, the strain is directly proportional to the stress applied.

Stress ∝ Strain

Thus,

σ ∝ ϵ

σ = Y×ϵ

F/A = Y×(ΔL / L)

Where,

Y is a constant of proportionality called Young’s Modulus or Elastic Modulus.

F is the applied force,

A is the area on which force F is applied,

L is the original length of the material,

ΔL is the change in the length.

Since strain is dimensionless, Young’s Modulus has the same dimensions as stress.

Stress-strain curve

Elasticity is a property that materials possess which allows them to go back to the original shape once the external force that was applied to the material is removed. And a simple way to study the elastic properties of materials is to study their stress-strain curve.

Let’s understand the stress-strain relationship now.

Let’s say a material with high ductility and with a uniform cross-sectional area is applied an axial force that is gradually increased. This is done until the material reaches a point where it breaks. The applied force is noted with the corresponding elongation, and a curve that is plotted looks as follows:

Segment OA: This segment is where the material obeys Hooke’s law, i.e., stress is directly proportional to stress. 

Segment AB: This segment is not exactly a straight line, and thus Hooke’s law is disobeyed. However, the material still appears to be elastic and the last point i.e., point B is the elastic limit.

Segment BC: The material starts losing its elasticity where there is strain, even if there is no increase in stress. In this segment, when the external force is removed, the strain or the elongation still remains, meaning the material does not come back to its original position.

Segment CD: It is here where Yielding begins. Point C is called the lower yield point, while Point D is called the upper yield point. 

Segment DE: Once yielding takes place at Point D, the material strains even more. As it reaches Point E, local necking begins. Point E is called the ultimate tensile stress point.

Segment EF: As the local necking proceeds, and by the time we reach Point F, fracture takes place. Point F is called fracture/breaking point and the corresponding stress is called breaking stress.

Conclusion

A stress-strain curve assesses the elastic properties of a material that helps in its engineering applications. It helps in calculating important parameters like strength, yielding point, elongation, fracture stresses, toughness, and more.