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Stress and Strain is an important concept in physics that causes the deformation of objects. It helps understand how objects change their shape when forces act on them.
- Stress-strain curve describes the relation between stress and strain, which is obtained by applying load to an object.
- Due to external forces, objects can be squashed, twisted, squeezed, sheared, or pulled.
- Stress refers to the force that acts on a unit area of a body.
- Strain refers to the deformity in length of an object when it is subjected to stress.
- Stress and strain are strongly connected because strain takes place due to the formation of stress.
- If you have noticed that material like rubber bands can be stretched easily, it is impossible if you try to stretch an iron rod.
Key Terms: Stress, Strain, Stress Curve, Strain Curve, Hooke’s Law, Equilibrium , Stress and Strain, Stress – Strain Curve, Force, Types of Stress, Types of Strains
What is Stress?
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In mechanics, stress is defined as the force applied per unit area. It is the ratio of internal force F produced when a substance is deformed to the area A where force is applied.
- At equilibrium, the interior force is adequate to the magnitude of the externally applied force.
- The Newton per square meter (Nm2) implies the SI unit for stress.
- The dimensional formula of the compound is ML-1T-2 .
- Dyne-cm2 is the CGS unit in which stress is measured.
Stress σ= F/A
- Where σ is the stress applied
- F is the force applied
- A is the area of force applied.
Types of Stress
Stress applied to a material can be of two types as follows:
- Tensile Stress: The external force per unit area of the material resulting in the stretch of the material is known as tensile stress.
- Compressive Stress: Compressive stress is the force that is responsible for the deformation of the material, such that the volume of the material reduces.
Stress
What is Strain?
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Strain can be referred to as the ratio of the amount of deformation that the body experiences in the direction of force applied to its initial size.
- It's expressed in numbers because it doesn't have any dimensions.
- The relation of deformation in terms of the length of the solid is shown below:
ε=δl/L
where,
- ε = strain due to the stress applied
- δl = modified long
- L = the original length of the material
Types of Strain
Strain experienced by a body can be of two types depending on stress application as follows:
- Tensile Strain: The deformation or elongation of a solid body due to applying a tensile force or stress is known as tensile strain. In other words, tensile strain is produced when a body increases in length as applied forces try to stretch it.
- Compressive Strain: Compressive strain is the deformation of a solid due to the application of compressive stress. In other words, compressive strain is produced when a body decreases in length when equal and opposite forces try to compress it.
Strain
Stress-Strain Curve
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The stress-strain curve delineates the connection between stress and strain for materials. In other words, the curve is a graphical representation that shows the reaction of a material when a load is applied.
- The stress–strain curve helps understand components such as yield strength, Young's modulus, and ultimate tensile strength.
- The form of deformation on an object due to external force can be rotation, compression, stretching and torsion.
- The stress and strain curve can be uniaxial, biaxial, or multiaxial.
Stress and Strain Curve
Explaining Stress-Strain Graph
The different regions of the stress-strain curve are as follows:
Proportional Limit
The area within the stress-strain curve obeys Hooke's law because of the proportional limit. In this limit, the ratio of stress and strain provides the proportionality constant, which is referred to as Young's modulus.
Elastic Limit
The elastic limit is the maximum stress a substance can endure before permanently deforming. When the load is working on the object, it is totally removed, and therefore, the material returns to its original position; that time is understood because of the object's elastic limit.
Yield Point
The point at which the material starts to deform plastically is understood because of its yield point. Once an object's yield point is crossed, plastic deformation occurs. There are two sorts of yield points: (i) upper yield point and (ii) lower yield point.
Ultimate Stress Point
The point at which a material endures maximum stress before failure is understood because of the Ultimate Stress point. Failure occurs beyond this point.
Fracture or verge of collapse
It is the point at the stress-strain curve where the material fails because it is on the verge of collapse.
Stress – Strain Curve
Hooke’s Law
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According to Hooke’s Law, the strain of the material is directly proportional to the stress applied on an object within the elastic limit of that material.
- While studying spring and elasticity, the English scientist Robert Hooke noticed that many materials displayed an identical property when the stress-strain relationship was studied.
- There exists a linear region where the force required to stretch the material is proportional to its extension.
- Mathematically, Hooke’s law is presented as:
F = -k.x
- Where, F = the force
- x = the extension length
- k = spring constant in N/m
Things to Remember
- Stress and strain have a straight proportional relationship up to an elastic limit.
- The relationship is explained by Hooke's law.
- It is a a graphical way to show the reaction of a material when a load is applied by comparing stress and strain.
- Stress ratio is the ratio of minimum stress to maximum stress.
- Strain in a solid is proportional to the applied stress, which must be within the solid's elastic limit.
Previous Year Questions
- A steel wire can withstand a load up to… [BITSAT 2008]
- A load of 1kg weight is a attached to one end of a steel wire of area of… [BITSAT 2008]
- There is some change W length when a 33000N tensile force is applied… [BITSAT 2008]
- The upper end of a wire of diameter… [BITSAT 2010]
- If the ratio of lengths, radii and Young's modulus of steel and… [VITEEE 2009]
- One end of a horizontal thick copper wire of length… [JEE Advanced 2013]
- A copper wire of length 2.2m and a steel wire of length… [KCET 2013]
- A light rod of length 100 cm is suspended from the ceiling horizontally… [AP EAPCET]
- A copper wire and a steel wire of the same diameter and length… [BHU VET]
- When a rod is heated but prevented from expanding the stress developed… [BHU VET]
- Two wires of same length and same material but of radii… [KEAM]
Sample Questions
Ques: What are the kinds of Stress and Strain? (4 marks)
Ans: Types of Stress
- Normal Stress: The restoring force per unit area perpendicular to the body surface is understood because of the normal stress. It is differentiated into two types: tensile and compressive stress.
- Tangential Stress: it's called tangential stress when the elastic restoring force acts parallel to the area .
Types of Strain
- Longitudinal Strain: The strain produced on the body thanks to the deforming force, which results in a change in just the length of the thing is understood as longitudinal or the tensile strain.
- Volumetric Strain: this is often the strain produced on the body thanks to the deforming force, which results in only the change in volume of the thing.
- Shear Strain: thanks to the tangential stress, an angle tilt is caused within the body; this is often referred to as a shear strain.
Ques: What is the connection Between Stress and Strain? (5 marks)
Ans: Hooke's law expresses the connection between stress and strain; it states that the strain in an object is proportionate to the strain applied within the range of the elastic limit of that object.
- The 19th century English scientist Hooke noticed while experimenting with springs and elasticity of the materials, they exhibited an identical property when the stress-strain relationship was studied.
- The linear region where the force required to stretch the fabric was proportionate to the extension of the fabric.
Mathematically, Hooke's law is usually expressed as:
- F = –k.x
Where,
- F is that the force
- x is that the extension length
k is that the constant of proportionality referred to as spring constant in N/m.
- Point A shown within the graph is that the proportional limit, which exhibits the proportional relationship between stress and strain in many of the materials.
- Point B is understood because the elastic limit, which is beyond the proportional limit, where an object returns to its original form when the applied stress is reversed, or the external force is removed. The proportional limit and elastic limit for several of the material is the same or equal.
- Point C within the graph is understood because the yield point, where the strain increases faster than stress, and therefore the material experiences some amount of permanent deformation.
- The stress causes a specified amount of permanent strain at the offset yield strength (point B).
- D denotes the worth of the last word lastingness of the fabric that has been reached. It denotes the utmost stress which will be applied to the fabric before its failure occurs.
Ques: In steel, the Young’s modulus and the strain at the breaking point are 2 x 1011 Nm-2 and 0.15 respectively. The stress at the breaking point for steel is, therefore: (2 marks)
i)2 x 108 Nm-2
ii)3 x 1010 Nm-2
iii)3 x 1012 Nm-2
iv)None of the above
Ans: ii) 3 x 1010 Nm-2
Breaking stress = breaking strain x Young’s modulus
= 0. 15 x 2 x 1011 = 3 x 1010 Nm-2
Ques: The temperature of a wire is doubled. The Young’s modulus of elasticity will be: (5 marks)
i)Also double
ii)Become four times
iii)Remain same
iv)Decrease
Ans: iv) will decrease
Young’s modulus is defined as the ratio of normal stress to longitudinal strain within limit of proportionality.
As Y ∝ 1/ ΔT
Y decreases as when temperature increases, \(\Delta\)T increases.
Ques: The stretching of a coil spring is determined by its shear modulus. Why? (2 marks)
Ans: When a coil spring is stretched, neither its length nor its volume changes, there is only the change in its shape. Therefore, the stretching of the coil spring is determined by the shear modulus.
Ques: Explain Hooke’s Law? (2 marks)
Ans: Hooke’s law states that the extension produced in the wire is directly proportional to the load applied within the elastic limit i.e. Acc to Hooke’s low,
- Stress:
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Strain - Stress: E x Strain
- E: Modulus of elasticity
StrainQues: A steel cable with a radius of 1.5 cm supports a chairlift at a ski area. If the maximum stress is not to exceed 
N 
, what is the maximum load the cable can support? (3 marks)
Ans: Radius of the steel cable, r = 1.5 cm = 0.015 m
Maximum allowable stress = N
Maximum stress = \(\frac{Maximum force}{Area of cross-section}\)
\(\therefore\) Maximum force = Maximum stress 
Area of cross-section
= \(108 \times \pi(0.015)^2\)
= 7.065 x 104 N
Hence, the cable can support the maximum load of 7.065 x 104 N.
Ques: Define shear stress and shear strain? (2 marks)
Ans: Shear stress: Shear stress is the amount of force per unit area perpendicular to the axle of the member.
Shear strain: The shear strain is defined as the length of deformation divided by the perpendicular length in the plane of the force applied.
Ques: Explain failure of thin cylindrical shell subjected to internal pressure? (3 marks)
Ans: Failure of thin cylindrical shell subjected to internal pressure:
When a thin cylindrical shell is subjected to internal pressure its wall is subjected to the following stresses.
- Hoop or circumferential stress.
- Longitudinal stress
When these stresses exceed the permissible limit, the cylinder is likely to fail in the following two ways.
- Split up into two troughs (semicircular halves)
- Split up into two short cylinders
Ques: Elasticity is the property of a body by which it regains its original state on the removal of the deforming force. ‘
a) Why Steel is more elastic than Rubber?
b) The figure given below shows the stress-strain curve for a given material. What are the Youngs modulus and approximate yield strength for this material? (4 Marks)
Ans: a) Youngs moduls of steel is larger than rubber. Hence steel is more elastic than rubber.
b)
Ques: What is the difference between stress and strain? (4 marks)
Ans: The difference between stress and strain are as follows:
| Stress | Strain |
|---|---|
| Stress refers to force applied per unit area of an object, | Strain can be referred to as the ratio of the amount of deformation that the body experiences in the direction of force applied to the initial sizes of the body. |
| The dimension of this quantity is ML-1T-2 | This quantity has no dimensions. |
| The formula for stress is given as: σ = F/A | The formula for strain is given as: ϵ = δl/L |
| It is denoted by σ and its units is Nm-2 | It is denoted by ϵ and has no unit. |
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