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Assignment No.02
Assignment #
(Strength of Materials)
(Session 2007-2011)
Submitted To:
Sir Dr. ABDUL GHAFOOR BHATI
Submitted By:
HASSAN ALI
(Roll # 12942)
(BS Mechanical engineering and Technology)
(4th Semester)
Subject:
Strength of Materials
Department of Mechanical Engineering and Technology
Government College University Faisalabad
Assignment On: Hooke's Law
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Table of Contents
1. Introduction
2. Hooke's Law
3. Hooke's Law in Shear
4. Elastic Modules
5. Limit of proportionality
6. Limitation of Hooke's Law
7. Uses of Hooke's Law
8. More Real-World Scenarios
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Hooke’s Law
Introduction:
Hooke's Law is a principle of physics that states that the force needed to extend or compress a spring by some
distance is proportional to that distance. The law is named after 17th century British physicist Robert Hooke,
who sought to demonstrate the relationship between the forces applied to a spring and its elasticity. He first
stated the law in 1660 as a Latin anagram, and then published the solution in 1678 as ut tensio, sic vis – which
translated, means "as the extension, so the force" or "the extension is proportional to the force")
This can be expressed mathematically as
F= -kX,
where F is the force applied to the spring (either in the form of strain or stress); X is the displacement of the
spring, with a negative value demonstrating that the displacement of the spring once it is stretched; and k is the
spring constant and details just how stiff it is.
Hooke's Law:
Below the proportionality limit of the stress-strain curve, the relationship between stress and strain is linear. The
slope of this linear portion of the stress-strain curve is the elastic modulus, E, also referred to as the Young's
modulus and the modulus of elasticity. Hooke's law expresses the relationship between the elastic modulus, the
stress, and the strain in a material within the linear region:
σ=Eε
Where σ is the value of stress and ε is the value of strain.
Below the proportionality limit of the stress-strain curve, the relationship between stress and strain is linear. The
slope of this linear portion of the stress-strain curve is the elastic modulus, E, also referred to as the Young's
modulus and the modulus of elasticity. Hooke's law expresses the relationship between the elastic modulus, the
stress, and the strain in a material within the linear region:
σ=Eε
Where σ is the value of stress and ε is the value of strain.
Explanation:
The statement regarding strain & stress being proportional to each other in the elastic region is known as
Hooke's law. "Within Elastic limit stress and strain are directly proportional to each other".
i.e., Stress a Strain within elastic limit.
The constant ratio of stress to strain within elastic limit is called elastic modules.
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Note : Unit of Elastic modules is same as that of stress (as strain has no limit) viz. n/m2.
The behavior of an elastic object, with increasing stress is represented in the Stress - Strain Curve shown below.
OP = Linear Elastic region
PQ = Rising Fatigue region
QR = Total Yield, with breaking into pieces
P = Elastic limit
Q = Yield Point
Linear Elastic region
In this region if stress is increased from O to A & is decreased back from A to O then, the strain increases in
direct proportion from, O to C and decreases from C to O in the same proportion. In this region the object retain
its elasticity, strain is the directly proportional to strain. The limit point 'P' of this region is called elastic limit.
Rising Fatigue region
In this strain region increases with the same amount as, the increase in stress; eventually the substance looses its
elasticity partially, and therefore becomes partially plastic. This deformation remains permanent, even if the
stress is reduced back from B to A and then from A to O. The end point of this region 'Q' is called the Yield
Point.
The total yield region
The region QR, represents infinitely large increase in strain produced when stress is increased even by small
amount; the substance has completely lost its elasticity & has become completely plastic; hence it yields
completely to deforming forces. The substance in this region can be broken into pieces.
Hooke's Law in Shear:
Hooke's law also has a form relating shear stresses and strains:
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τ=Gγ
Where τ is the value of shear stress, γ is the value of shear strain, and G is the shear modulus of elasticity. The
elastic modulus and the shear modulus are related by:
Where ν is Poisson's ratio.
Hooke's law is analogous to the spring force equation, F = k δ. Essentially, everything can be treated as a spring.
Hooke's Law can be rearranged to give the deformation (elongation) in the material:
Axial Elongation
(from normal stress)
Angle of Twist
(from shear/torsional
stress)
Poisson's Ratio
As load is applied to a material, the material elongates and the cross-sectional area is reduced. This reduction in
cross-sectional area is called lateral strain, and it is related to the axial strain by Poisson's ratio, ν. For a
circular specimen this reduction in area is realized as a reduction in diameter, and the Poisson's ratio is
calculated as:
So, pulling on it in the x-direction causes it to shrink in the y & z directions. This property of a material is
known as Poisson's ratio, and it is denoted by the Greek letter nu, and is defined as:
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Poisson's ratio only applies within the elastic region of the stress-strain curve, and it is typically about 0.3 for
most metals. The theoretical maximum limit of Poisson's ratio is 0.5.
Elastic Modules:
For the three types of deformation viz. Tensile, Shear and Bulk the elastic modules can also be defined using
Hooke's law.
Young's Modulus: Y
The constant ratio of Tensile stress to Tensile Strain, within the Elastic limit, is called the Young's modulus.
Modules of Rigidity:
The constant ratio of shearing stress to shearing strain, within the elastic limit, is called the modulus of Rigidity.
Bulk Modulus: K
The constant ratio of Bulk stress to Bulk Strain, within, the elastic limit, is called the Bulk modulus :
Limit of proportionality:
Spring constant is a measure of the stiffness of a spring up to its limit of proportionality or elastic limit. The
limit of proportionality refers to the point beyond which Hooke's law is no longer true when stretching a
material. The elastic limit of a material is the furthest point it can be stretched or deformed while being able to
return to its previous shape. Once a material has gone past its elastic limit, its deformation is said to be inelastic.
The higher the spring constant, the stiffer the spring. The spring constant is different for different elastic objects.
For a given spring and other elastic objects, the extension is directly proportional to the force applied. For
example, if the force is doubled, the extension doubles. This works until the limit of proportionality is exceeded.
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When an elastic object is stretched beyond its limit of proportionality, the object does not return to its original
length when the force is removed. In this instance, the relationship between force and extension changes from
being linear, or directly proportional, to being non-linear.
Non-linear extension occurs more in some materials than others. Materials like clay or putty usually show non-
linear extension.
Force-extension graphs
Linear extension and elastic deformation can be seen below the limit of proportionality. Non-linear extension
and inelastic deformation can be seen above the limit of proportionality. The limit of proportionality is also
described as the 'elastic limit'. The gradient of a force-extension graph before the limit of proportionality is
equal to the spring constant.
Stress-strain curves are useful to understand the tensile strength of a given material.
LIMITATIONS OF HOOKE’S LAW:
The law isn’t a universal principle and only applies to the materials as long as they aren’t stretched way past
their capacity.
It ceases to apply past the elastic limit of a material. If the material is stretched past its elastic limit, it causes
permanent deformation.
The law is only accurate for most solid bodies only and if the forces and deformations are small.
USES OF HOOKE’S LAW:
Used in all branches of science and engineering
Foundation for seismology, acoustics and molecular mechanics.
Fundamental principle behind manometer, spring scale, balance wheel of the clock.
More Real-World Scenarios:
While masses on springs are commonly found in physics classes – and serve as a typical scenario for
investigating Hooke's law – they are hardly the only instances of this relationship between deforming objects
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and force in the real world. Here are several more examples where Hooke's law applies that can be found
outside the classroom:
Heavy loads causing a vehicle to settle, when the suspension system compresses and lowers the
vehicle towards the ground.
A flagpole buffeting back and forth in the wind away from its fully upright equilibrium position.
Stepping onto the bathroom scale, which records the compression of a spring inside to calculate how
much additional force your body added.
The recoil in a spring-loaded toy gun.
A door slamming into a wall-mounted doorstop.
Slow-motion video of a baseball hitting a bat (or a football, soccer ball, tennis ball, etc., on impact
during a game).
A retractable pen that uses a spring to open or close.
Inflating a balloon.
