Stress and Strain

One may have found that some things can stretch quickly, but stretching an iron rod seems difficult, doesn’t it? This article helps to understand why certain things are more malleable than others. Here topics like stress-strain curves are discussed because they’re useful for determining the tensile strength of a given material. When a deforming force is applied to a body its length decreases, as a result of this the molecules of the body go far apart. As a result of this, the spring is slightly compressed and the intermolecular forces get changed. The restoring forces developed in the spring bring these displaced molecules to their respective equilibrium potions. In other words, the body regains its original shape or length on the removal of deforming force. This property of the body is known as elasticity. 

What is Elasticity?

Atoms and molecules in a solid are structured in such a manner that neighboring molecules exert a force on each other are known as the intermolecular forces. After the deforming forces are removed, the body returns to its original structure (length, shape, or volume). This is an example of elasticity.

Elasticity is defined as the property of a solid body by virtue of which it regains its original configuration (shape and size) when the external deforming force and on it is removed is called elasticity. 

Restoring Force

When a deforming force is removed from a perfectly elastic body, it automatically and totally returns to its original state. Quartz and phosphor bronze are examples of almost perfectly elastic materials. A body is said to be perfectly elastic if it completely regains its original form when the deforming force acting on it is removed. There is no such material that can regain completely its original form. In other words, the concept of a perfectly elastic body is only an ideal concept. The nearest approach to the perfectly elastic body is quartz fiber.

And after the deforming force is removed, a plastic body is unable to revert to the original size and shape. Perfectly plastic body are those bodies which do not regain its original form, even slightly, when the deforming force is removed. When the deforming force is removed every material partially regains its original form. So the concept of a perfectly plastic body is also an ideal concept. Paraffin wax, wet clay is the nearest approach to perfect plastic bodies. Thus, plasticity is the property of the material body by virtue of which it does not regain its original configuration when the external force acting on it is removed.

Stress

It is known that when a deforming force is applied to a body then the restoring forces are developed inside the body. Therefore, the restoring force per unit area of a body is called stress. 

The restoring force is equal and the opposite to the deforming force applied to the body. It can also be defined as the deforming force per unit area of the body.

Stress = Deforming force (F) / Area of the body (A)

or 

σ = F / A  

where, σ is the stress applied, F is the force applied and A is the area of force application

In SI, the unit of stress is N/m² or Nm^-2, another unit is Pascal (Pa).

The Dimensional formula of stress is [ML^-1T^-2].

Stress is a scalar quantity i.e. it has only magnitude. 

Different Types of Stress

Stress applied to a material can be of five types. They are:

(1) Normal stress: Normal Stress is defined as the restoring force per unit area perpendicular to the surface of the body. It is further of two types: Tensile stress and Compressive stress. 

The length of a solid can be changed in two ways:

• When two equal and opposite forces are applied at the end of a rod as shown in the figure.
• When two equal and opposite forces are applied on two ends of the rod as shown in the figure.

1. Tensile stress: When two equal and opposite forces are applied on a circular rod to increase its length, a restoring force equal to be applied force F normal to the cross-sectional area of the rod comes cross-section is known as tensile stress. Thus, tensile stress is defined as the restoring force or deforming force acting per unit area perpendicular to the cross-section of the body.
2. Compressive Stress When two equal and opposite forces are applied at the ends of a rod as shown in figure (h) to decrease its length or to of the rod is known as compressive stress. This compressive stress is defined as the restoring force or deforming force acting per unit area perpendicular to the archon of the body. That is Compressive stress A in Under tensile stress or compressive stress, the net force acting on an object is zero, but the object is deformed. Tensile stress or compressive stress and also termed longitudinal stress. 

(2) Tangential stress or shearing stress: When two equal and opposite forces act along the tangents to the surfaces of de opposite faces of an object, then one face of the object is displaced with respect to the other face as shown in the figure. In this case, the object is under the stress known as tangential stress or shearing stress. Thus, tangential stress or shearing stress is defined as the ratio of the force tangent to the surface to the area of the surface. 

(3) Bulk stress or volume stress or hydraulic stress-: When an object is immersed in a fluid (liquid or gas), the fluid exerts a force on the surfaces of the object as shown in the figure. As a result of this, the volume of the object decreases, and the object is under a stress known as Bulk stress or hydraulic stress. 

Strain

The ratio of the change in the configuration (i.e. shape, length, or volume) to the original configuration of the body is called strain.

The strain clearly says it as the amount of deformation experienced by the body in the direction of force applied, separated by the original proportions of the body. The relationship for deformation in terms of a solid’s length is given below:

Strain (ϵ) = Change in the configuration (δl) / Original configuration (L)

or

ϵ = δl / L

where ϵ is the strain due to stress applied, δl is the change in length and L is the original length of the material.

The strain is a dimensionless quantity as it just defines the relative change in shape.

Different Types of Strain

There are three types of strain: 

(1) Longitudinal Strain: This type of strain is produced when the body is under tensile stress or compress stress. It is defined as the ratio of the change in length to the original length of the body. Consider a rod of length L. When the rod is under tensile stress or compressive stress, the change in its length is △L. 

Longitudinal strain = Change in length (△L) / Original length (L)

(2) Volume strain: This type of strain is produced when the body is under bulk stress or hydraulic stress Longitudinal strain original length or Longitudinal strain defined as the rate of the change in volume to the original volume of the body. 

If △V is the change in volume or V₀ – V, where V₀ is the original volume and V is the volume of the body under bulk stress.

Volume strain = – △V / V

A negative sign shows that volume decreases when the body is under bulk stress.

(3) Shear Strain: This type of strain is produced when the body is under tangential stress or shearing stress. is defined as the angle (θ) through which the face of a body originally perpendicular to the fixed face is turned when it is under the shearing stress.

tanθ = x / L

Sample Problems

Problem 1: A body is under tensile stress, its original length was L m, after applying tensile stress its length become  L/4 m. Calculate the tensile strain applied to the body.

Solution:

Given that, 

The original length is L m. 

The change in the length = L – L/4 = 3L/4

Since, the Longitudinal strain = change in length/original length =△L/L

                                               = (3L/4)/L

                                               = 0.75

Problem 2: A copper wire of length 2.5m has a percentage strain of 0.012 % under a tensile force. Calculate the extension in the wire.

Solution:

Given that, The original length is 2.5 m

The Strain = △L/L = 0.012 %

                             =0.012/100

                      △L = (0.012/100) x 2.5

                            = 0.3 m

Problem 3: Given the deforming force as 150 N applied on a body of area of cross-section as 10 m². Calculate the stress in the body.

Solution:

Given that,

Stress = Deforming force / Area of the body

         = F/A

        = 150/10

       = 15 N/m²

Problem 4: Why are the bridges declared unsafe after a long time of use?

Solution:

Due to the repeated stress and strain , the material used in bridges loses elastic strength and ultimately may be collapsed. That is why bridges are declared unsafe after long time of use.

Example 5: Write down the cause of restoring stress in a stretched wire compressed wire.

Solution:

The restoring stress is caused by the interatomic attraction in a stretched wire and by inter atomic repulsion in a compressed wire.