DEFORMATION
OF SOLIDS

WHEN FORCES ARE APPLIED TO A SOLID BODY,ITS
SIZE OR SHAPE MAY CHANGE.

THE CHANGE OF SHAPE OR SIZE IS CALLED
DEFORMATION.

THE DEFORMATION IS CALLED A TENSILE
DEFORMATION IF AN OBJECT IS STRETCHED OR
COMPRESSIVE DEFORMATION IF AN OBJECT IS
SQUASHED.
IF FORCES ARE APPLIED TO A
MATERIAL IN A WAY AS TO DEFORM
IT ,THE MATERIAL IS SAID TO BE
STRESSED.
AS A RESULT OF THE STRESS, THE
MATERIAL BECOMES STRAINED.
IN THIS CHAPTER WE SHALL BE
CONCERNED ONLY WITH SOLIDS
AND WITH STRESS WHICH RESULTS IN
AN INCREASE IN LENGTH OR A
DECREASE IN LENGTH.
ELASTIC
AND
PLASTIC
BODY

A BODY THAT RETURNS TO ITS
ORIGINAL SHAPE AND SIZE ON THE
REMOVAL OF THE DEFORMING
FORCE( WHEN DEFORMED WITHIN
ELASTIC LIMIT) IS CALLED AN
ELASTIC BODY.

A BODY THAT DOES NOT RETURN
TO ITS ORIGINAL SHAPE AND SIZE
ON THE REMOVAL OF THE
DEFORMING FORCE, HOWEVER
SMALL THE MAGNITUDE OF
DEFORMING FORCE MAY BE IS
CALLED A PLASTIC BODY.
PUTTY,WAX ARE EXAMPLES OF
NEARLY PLASTIC BODIES.
ALL BODIES ARE ELASTIC,THE
DIFFERENCE LIES ONLY IN THE
DEGREE.
NO BODY IS PERFECTLY ELASTIC OR
PERFECTLY PLASTIC.
IN A BODY WHEN ELASTIC BEHAVIOUR
INCREASES, THE PLASTIC BEHAVIOUR
DECREASES AND VICE VERSA.
HOOKES
LAW

HOOKES LAW STATES THAT ,
PROVIDED THE LIMIT OF
PROPORTIONALITY IS NOT
EXCEEDED, THE EXTENSION OF A
BODY IS PROPORTIONAL TO THE
APPLIED LOAD.

FORCE= SPRING CONSTANT X EXTENSION

F=KX

K= SPRING CONSTANT OR FORCE PER UNIT
EXTENSION

SI UNIT - N/m
Spring
constant

Different springs extends by different
amounts for the same load.

Spring constant allows us to compare
the extension of different springs.

Springs with a larger spring constant will
extend less for the same load.

Spring constant depends on the
stiffness of the spring material, thickness,
diameter of turns of the coil ,number of
turns per unit length, overall length of
the spring.

Spring constant varies inversely with the
length of the spring.

Thus more the length of the spring, less
is the spring constant.
Forceextension
graph
THE SECTION OF THE GRAPH FROM THE ORIGIN TO THE
POINT P IS STRAIGHT. IN THIS REGION ,THE EXTENSION OF
THE SPRING IS PROPORTIONAL TO THE LOAD.
POINT B IS REFERRED TO AS LIMIT AS PROPORTIONALITY.
FOR SMALL LOADS,WHEN THE LOAD IS REMOVED,THE
SPRING RETURNS TO ITS ORIGINAL LENGTH. THE SPRING IS
SAID TO HAVE UNDERGONE AN ELASTIC CHANGE.
ELASTIC LIMIT
THE ELASTIC LIMIT ( E) IS THE
MAXIMUM FORCE THAT
CAN BE APPLIED TO A WIRE
SUCH THAT THE WIRE
RETURNS TO ITS ORIGINAL
LENGTH WHEN THE FORCE
IS REMOVED.
BEYOND E, THE SPRING IS
DEFORMED PERMANENTLY
AND THE CHANGE IS SAID
TO BE PLASTIC FOR POINTS
BEYOND E.
INVESTIGATING
COMBINATIONS OF SPRING

WHEN DIFFERENT SPRINGS ARE COMBINED,THEY
WILL HAVE AN EFFECTIVE SPRING CONSTANT.THIS
WILL BE DIFFERENT FROM THE SPRING CONSTANTS
OF THE INDIVIDUAL SPRINGS.
STRESS, STRAIN AND YOUNG
MODULUS OF ELASTICITY

THE VALUE OF THE SPRING CONSTANT FOUND
FROM A FORCE EXTENSION GRAPH MAY VARY
FOR DIFFERENT OBJECTS MADE FROM THE SAME
MATERIAL.

IF WE HAVE TWO SPRINGS BOTH MADE FROM
STEEL,BUT WITH DIFFERENT DIAMETERS OR
LENGTHS, THEN THE SPRING CONSTANT OF
EACH SPRING WILL BE DIFFERENT.

HOWEVER, A DESIGN ENGINEER NEEDS TO BE
ABLE TO COMPARE THE PROPERTIES OF
DIFFERENT MATERIALS IN A WAY THAT DOES NOT
DEPEND ON HOW THE MATERIAL IS SHAPED IN
AN OBJECT.

THEY CAN DO THIS BY USING THE YOUNG
MODULUM OF THE MATERIAL WHICH IS A
MEASURE OF THE STIFFNESS OF THE MATERIAL.

YOUNGS MODULUS IS A CONSTANT FOR A
MATERIAL, AND DOESNT DEPEND ON THE SHAPE
OR SIZE OF THE OBJECT FORMED FROM THE
MATERIAL.
STRESS

TO CALCULATE THE YOUNGS MODULUS OF A
MATERIAL, WE NEED TO BE ABLE TO COMPARE THE
FORCE APPLIED AND THE EXTENSION OBTAINED IN
A WAY THAT DOES NOT DEPEND ON THE
DIMENSIONS OF THE OBJECT.

TO DO THIS WE USE TWO QUANTITIES KNOWN AS
STRESS AND STRAIN WHICH ENABLS US
TOCOMPARE THE DEFORMATION OF DIFFERENT
MATERIALS REGARDLESS OF THE DIMENSIONS OF
THE OBJECT.

THE SAME TENSILE FORCE IS APPLIED TO SAMPLES
OF WIRE OF THE SAME MATERIALS THAT HAVE
DIFFERENT DIAMETERS AND SAME LENGTH.

THEY WILL DEFORM BY DIFFERENT
AMOUNTS,EVENTHOUGH THEY EXPERIENCE THE
SAME FORCE.

A QUANTITY CALLED STRESS TAKES THE DIFFERENCE
IN CROSS SECTIONAL AREA INTO ACCOUNT.
THE STRESS (SIGMA)
THE SRESS IN AN OBJECT IS DEFINED
AS THE FORCE APPLIED PER UNIT
CROSS SECTIONAL AREA.
STRESS= FORCE/CROSS SECTIONAL
AREA NORMA L TO THE FORCE
SI UNIT IS N/m^2 or pascal (Pa)
STRAIN
THE YOUNG MODULUS OF
ELASTICITY

THE VALUES OF STRESS AND STRAIN ALLOW US TO
CALCULATE THE YOUNG MODULUS OF A
MATERIAL.

THIS IS A MEASURE OF THE STIFFNESS OF THE
MATERIAL.IT IS ANALOGOUS TO THE SPRING
CONSTANT IN HOOKES LAW.

YOUNGS MODULUS=STRESS/STRAIN
=FL/Ax


SI UNIT IS Pa
PROVIDED HOOKES LAW IS OBEYED, THE YOUNGS
MODULUS IS CONSTANT.
FOR A GIVEN MATERIAL WE CAN PLOT A GRAPH OF
THE STRESS AGAINST STRAIN.
YOUNGS MODULUS IS CALCULATED USING THE
LINEAR REGION OF THE STRESS- STRAIN GRAPH.
IN THE REGION WHERE CHANGES ARE
PROPORTIONAL, IT CAN BE SEEN THAT STRESS IS
PROPORTIONAL TO STRAIN.
STRESS= E x STRAIN
THE CONSTANT E IS KNOWN AS THE YOUNGS
MODULUS OF THE MATERIAL.

SO WE HAVE A CONSTANT E FOR A PARTICULAR
MATERIAL WHICH WOULD ENABLE US TO FIND
EXTENSIONS KNOWING THE CONSTANT AND THE
DIMENSIONS OF THE SPECIMEN.

THE ADVANTAGE OF USING STRESS AND STRAIN
INSTEAD OF STRESS AND STRAIN IS THAT THE RATIO
OF STRESS AND STRAIN IS THE SAME REGARDLESS
OF THE SIZE AND SHAPE OF THE OBJECT.
THE RATIO E IS A PROPERTY OF THE
MATERIAL.
IT IS A MEASURE OF THE
MATERIALS STIFFNESS-ITS
RESISTANCE TO STRETCHING OR
BENDING.
NOTE THAT SPRING CONSTANT
VARIES FOR A SPRING OF SAME
MATERIAL BUT WITH DIFFERENT
DIMENSIONS.
EXPERIMENT TO FIND THE YOUNG MODULUS OF
A MATERIAL
INTERPRETING STRESS-STRAIN GRAPHS
1.COPPER (DUCTILE)
DUCTILE MATERIAL
(example-copper)
THE LINEAR REGION OF THE GRAPH UPTO THE LIMIT OF PROPORTIONALITY SHOWS
THAT THE MATERIAL OBEYS HOOKES LAW.
THE GRADIENT OF THE LINEAR REGION OF THE STRESS –STRAIN GRAPH GIVES THE
YOUNG MODULUS OF THE MATERIAL.
DUCTILE

UPTO THE ELASTIC LIMIT,THE SAMPLE SHOWS ELASIC
DEFORMATION.

THIS MEANS THAT IF THE LOAD IS REMOVED,THE SAMPLE WILL
RETURN TO ITS ORIGINAL LENGTH.

IF THE STRESS CONTINUES TO INCREASE BEYOND THE ELASTIC
LIMIT, THE SAMPLE SHOWS PLASTIC DEFORMATION AND WILL
NOT RETURN TO ITS ORIGINAL LENGTH ONCE THE LOAD IS
REMOVED.

IN PLASTIC DEFORMATION, INCREASINGLY LARGE EXTENSIONS
ARE OBTAINED FOR EQUAL INCREASE IN APPLIED LOADS.

PLASTIC DEFORMATION OCCURS IN METALS DUE TO THE
INTERNAL ARRANGEMENT OF THE METAL ATOMS CHANGING.

THIS PROCESS HAPPENS MORE EASILY IN STEEL THAN COPPER.

IF THE STRESS INCREASES TOO MUCH, THEN THE SAMPLE WILL
REACH BREAKING POINT AND THE SAMPLE WILL BREAK.
STRESS-STRAIN
GRAPH FOR
GLASS (BRITTLE)
COMPARED
WITH DUCTILE
GLASS IS BRITTILE( NO PLASTIC REGION)
RED LINE ON GRAPH REPESENTS GLASS.
STEEL AND
ALUMINIUM

A MORE RIGID
MATERIAL SUCH AS
STEEL HAS A STEEPER
SLOPE THAN A LESS
RIGID MATERIAL SUCH
AS ALUMINIUM.

STEEL HAS LARGE E.

IF E IS LARGE, THEN
THE MATERIAL
DISTORTS ONLY A
LITTLE UNDER THE
INFLUENCE OF THE
APPLIED STRESS.
STRESS –STRAIN GRAPH OF
A PERMANENTLY
DEFORMED BODY
THE WIRE DOES NOT
RETURN TO ITS ORIGINAL
LENGTH WHEN THE STRESS
IS REMOVED.(THE MASSES
HAVE BEEN REMOVED) THE
MATERIAL IS PERMANENTLY
DEFORMED.
STRESS-STRAIN
GRAPH FOR A
POLYMER(RUBBER)
RUBBER
BAND

RUBBER BAND DOES NOT
OBEY HOOKES LAW .THE
CURVE IS SAID TO BE A
HYSTERESIS LOOP AND THE
AREA BETWEEN TWO
CURVES REPRESENTS LOST
ENERGY.

SOME OF THE WORK DONE
ON THE RUBBER BAND IS
AGAINST INTERNAL
FRICTION ,WHICH
INCREASES THE
TEMPERATURE OF THE
RUBBER BAND.THUS THERE IS
LESS ENERGY AVAILABLE TO
RAISE THE WEIGHTS BACK
UP.
ELASTIC
POTENTIAL
ENERGY

WHENEVER YOU STRETCH A MATERIAL ,YOU ARE
DOING WORK.

THIS IS BECAUSE YOU HAVE TO APPLY A FORCE
AND THE MATERIAL EXTENDS IN THE DIRECTION OF
THE FORCE.

YOU WILL KNOW THIS IF YOU HAVE EVER USED AN
EXERCISE MACHINE WITH SPRINGS INTENDED TO
DEVELOP YOUR MUSCLES.

WE CALL THE ENERGY IN A DEFORMED SOLID THE
ELASTIC POTENTIAL ENERGY OR STRAIN ENERGY.

IF THE MATERIAL HAS BEEN STRAINED ELASTICALLY
,THE ENERGY CAN BE RECOVERED.

IF THE MATERIAL HAS BEEN PLASTICALLY
DEFORMED,SOME OF THE WORK DONE HAS GONE
INTO MOVING ATOMS PAST ONE ANOTHER ,AND
THE ENERGY CANNOT BE RECOVERED.THE
MATERIAL WARMS UP SLIGHTLY.
ELASTIC
POTENTIAL
ENERGY IS EQUAL
TO THE AREA
UNDER THE
FORCE –
EXTENSION
GRAPH.

WORK DONE =FORCE X DISTANCE MOVED IN THE
DIRECTION OF THE FORCE

FIRST,CONSIDER THE LINEAR REGION OF THE
GRAPH WHERE HOOKES LAW IS OBEYED.

THERE ARE TWO METHODS TO CALCULATE THE
WORK DONE.
WE CAN THINK ABOUT THE
AVERAGE FORCE NEEDED TO
PRODUCE AN EXTENSION X.
THE AVERAGE FORCE
=0+F/2=F/2
METHOD 1
ELASTIC PE =WORK DONE= ½ F x
= ½ Kx^2
K is the spring constant
Method 2

The other way to find the elastic potential energy
is to recognize that we can get the same answer
by finding the area under the graph.

The area shaded is a triangle whose area is given
by

Area=1/2 bh

This again gives

Elastic potential energy= ½ Fx = ½ kx^2
WORK DONE IN STRTCHING OR
COMPRESSING A MATERIAL IS
ALWAYS EQUAL TO THE AREA
UNDER THE GRAPH OF FORCE
AGAINST EXTENSION.

IF THE GRAPH IS NOT A STRAIGHT LINE ,WE
CANNOT USE THE Fx RELATIONSHIP, SO WE HAVE
TO RESORT TO COUNTING SQUARES OR SOME
OTHER TECHNIQUE TO FIND THE ANSWER.