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Stress and Strain

Last Updated : 17 May, 2024
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Stress and Strain are the two terms in Physics that describe the forces causing the deformation of objects. Deformation is known as the change of the shape of an object by applications of force. The object experiences it due to external forces; for example, the forces might be like squeezing, squashing, twisting, shearing, ripping, or pulling the object apart.

Let’s learn more about stress, strain, their formula, unit, dimension, relation between stress and strain and curve between them in this article

What is Stress?

It is known that when a deforming force is applied to a body, the restoring forces are developed inside the body. Therefore, the restoring force per unit area of a body is called stress. We define stress using the Greek letter, ‘σ’.

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

Stress Formula

The stress formula is given as:

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

In terms of symbolism, stress and strain can also be expressed as follows:

σ = F / A

where,
σ is the Stress
F is the Force Applied
A is the Area Application of Force

Unit and Dimension of Stress

In SI, the unit of stress is N/m² or Nm-2, and 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. 

Learn more about, Stress

Types of Stress

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

  • Normal Stress
  • Tensile stress
  • Compressive Stress
  • Tangential Stress or Shearing Stress
  • Bulk Stress or Volume Stress or Hydraulic Stress

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. 

If the force is applied perpendicular to the cross-sectional area of the wire or rod its volume changes and the stress generated in the body is called the normal stress.

Tensile Stress

When two equal and opposite forces are applied on a circular rod to increase its length, a restoring force equal to 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.

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 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

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. 

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. 

Various types of stress that we encounter are,

Types of Stress

What is 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. Strain is denoted using the Greek letter ‘ϵ’.

The strain clearly says it is the amount of deformation experienced by the body in the direction of force applied, separated by the original proportions of the body.

Strain Formula

The relationship for deformation in terms of a solid’s length is given below:

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

ϵ = δl / L

where,
ϵ is the Strain due to applied Stress
δl is the Change in Length
L is the Original Length of material

Strain Unit and Dimension

Strain has no unit as it is ratio of same same quantity. Hence, it is a dimensionless quantity

Types of Strain

There are three types of strain, that are

  • Longitudinal Strain
  • Volume Strain
  • Shear Strain

Longitudinal Strain

This type of strain is produced when the body is under tensile stress or compressed 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)

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 is defined as the rate of 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.

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

Stress-Strain Curve

The material’s stress-strain curve gives its stress-strain relationship. In a stress-strain curve, the stress and its corresponding strain values are plotted. When we study solids and their mechanical properties, information regarding their elastic properties is most important. We can learn about the elastic properties of materials by studying the stress-strain relationships, under different loads, in these materials.

An generalized graph of stress-strain curve is given below:

Stress-Strain Curve

Explaining Stress-Strain Graph

The different regions in the stress-strain diagram are:

(i) Proportional Limit

It is the region in the stress-strain curve that obeys Hooke’s Law. In this limit, the stress-strain ratio gives us a proportionality constant known as Young’s modulus. The point OA in the graph represents the proportional limit.

(ii) Elastic Limit

It is the point in the graph up to which the material returns to its original position when the load acting on it is completely removed. Beyond this limit, the material doesn’t return to its original position, and plastic deformation starts to appear in it.

(iii) Yield Point

The yield point is defined as the point at which the material starts to deform plastically. After the yield point is passed, permanent plastic deformation occurs. There are two yield points 

  • Upper Yield Point 
  • Lower Yield Point

(iv) Ultimate Stress Point

It is a point that represents the maximum stress that a material can endure before failure. Beyond this point, failure occurs.

(v) Fracture or Breaking Point

It is the point in the stress-strain curve at which the failure of the material takes place.

What is Elasticity?

We know that in solids the particles are bounded by each other by intermolecular forces and the outer force applied to the solid changes the internal force and the configuration of the solid changes accordingly.

Elasticity is defined as the property of a solid body that allows it to regain its original configuration (shape and size) when the external deforming force on it is removed from it.

When a deforming force is removed from a perfectly elastic body, it automatically returns to its original state. Quartz and Phosphor Bronze are some 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 to the perfectly elastic body is Quartz fiber.

A Plastic Body is a body that is unable to revert to its original size and shape after the deforming force is removed. Perfectly plastic bodies are those bodies that do not regain their 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 and Wet Clay are the nearest 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.

Hooke’s Law

In the 19th century, while studying springs and elasticity, English scientist Robert Hooke noticed that many materials exhibited a similar property when the stress-strain relationship was studied. There was a linear region where the force required to stretch the material was proportional to the extension of the material, known as Hooke’s Law.

Hooke’s Law states that the strain of the material is proportional to the applied stress within the elastic limit of that material.

Mathematically, Hooke’s law is commonly expressed as:

F = –k.x

where
F is the force, 
x is the extension in length
k is the constant of proportionality known as the spring constant in N/m.

What is Elastic Modulus?

Elastic modulus, commonly known as Young’s modulus, measures a material’s stiffness or rigidity. It quantifies a material’s ability to deform under stress and then return to its original shape. In other words, it specifies how much a material expands or compresses in response to a given force. Elastic Modulus is given as

Elastic Modulus = Stress/Strain

Elastic Moduli of Materials

The following table lists Young’s modulus, shear modulus, and bulk modulus for common materials.

Material

Young’s modulus (E) in GPa

Shear modulus (G) in GPa

Bulk modulus (K) in GPa

Glass

55

23

37

Steel

200

84

160

Iron

91

70

100

Lead

16

5.6

7.7

Aluminium

70

24

70

Read More,

Solved Examples on Stress and Strain

Example 1: A body is under tensile stress, its original length was L m, after applying tensile stress its length becomes 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

Longitudinal strain = (3L/4)/L

Longitudinal strain = 0.75

Example 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

Strain = △L/L = 0.012 %

Strain  = 0.012/100

△L = (0.012/100) x 2.5

△L = 0.3 m

Example 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

Stress = F/A

Stress = 150/10

Stress = 15 N/m2

FAQs on Stress and Strain

What is Stress and Strain?

Stress is the restoring force acting per unit area and strain is the ratio of change in dimension due to stress to the original dimension

What is unit of Stress and Strain?

The unit of stress is N/m2 and Pascal while strain has no units

What is Stress and Strain Formula?

The formula of stress is Restoring Force/ Area while the formula of strain is Change in Dimension/Original Dimension

What is the relation between stress and strain?

The relation between stress and strain is that they are directly proportional to each other up to an elastic limit. Hooke’s law explains the relationship between stress and strain. According to Hooke’s law, the strain in a solid is proportional to the applied stress and this should be within the elastic limit of that solid.

What does the stress-strain curve show?

A stress-strain curve is a graphical way to show the reaction of a material when a load is applied. It shows a comparison between stress and strain.

What is the use of the stress-strain diagram?

The stress-strain diagram provides a graphical measurement of the strength and elasticity of the material. Also, the behaviour of the materials can be studied with the help of the stress-strain diagram, which makes it easy to understand the application of these materials.

What is the yield point in the stress-strain graph?

The yield point is defined as the point at which the material starts to deform plastically. After the yield point is passed, permanent plastic deformation occurs.

How to make a stress-strain curve?

The stress-strain curve is obtained by gradually applying load to a test coupon and measuring the deformation, from which the stress and strain can be determined.

Why are bridges declared unsafe after a long time of use?

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.



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