Good morning everybody Uhh let us start todayís
topic uhh on elasticity uhh uhh crystal uhh
(struc differ) uhh different crystal structures
So so far we have discussed the different
type of uhh structure of materials for example
we explained in terms of BCC FCC HCP or different
and uhh very basic units of the uhh materials
So now today we will try to discuss this uhh
next topic that is called elasticity part
uhh of different crystals So first I will
try to focus on isotropic elasticity of materials
and then uhh gradually will shift to anisotropic
elasticity of materials and how the orientation
dependence of elastic response is generally
observed in case of cubic and non-cubic crystals
So from the basic part start with the elasticity
So it is known that umm all the objects are
a deformable when it is subjected to some
kind of load but at the same time it is possible
to change their shape or size due to the application
of the external load but how we can change
and (size) size and try to explain in the
mathematical form then we need to define several
parameters to analyse the deformation mechanisms
of solids
So first in terms of elasticity will try to
explain uhh the deformation behaviour so elasticity
is known that it is a reversible process that
means if we apply the load and subsequently
released the load it will come back to its
initial position without any permanent deformation
So most of the materials practically undergoes
very small amount of the elastic deformation
but (el) can go up to large amount of permanent
deformation with the application of the mechanical
load here
So there are 2 possibilities to analyse the
elasticity first if we analyse the load with
respect to the original area or the actual
area at the instant time of load But when
the stress is calculated on the basis of original
area it is called engineering stress or nominal
stress or (256) similarly When it is based
on the original length to calculate the strain
then it can be called as engineering strain
Of course engineering stress and strain diagrams
is specifically useful within the elastic
range but the true stress strain diagram is
specifically significant or important in plastic
range Although there is a (323) for small
deformation case uhh engineering stress strain
and or true stress strain there is a very
small difference is there within the elastic
limit
First we define the stress and strain uhh
over a solid body Stress actually quantify
which is the proportional to the force causing
deformation to that specific solid Now stress
is actually the external force acting of the
object per unit cross-sectional area So we
can measure the stress like that it is force
divided by area that means force per unit
area So unit of stress is uhh in SI it is
Pascal or Newton per meter square which is
same as the unit of pressure
Similarly strain is a measure of degree of
deformation with the application of the load
But for sufficiently small amount of the stress
strain is proportional to the stress (The)
the constant term of the proportionality actually
depends on the materials being deformed and
the nature of deformation We call this proportionality
constants as elastic modulus If we look into
the right-hand side figure we see the L is
the original length and with the application
of the load F it deformed to final length
L plus delta L So specifically increment of
the length with the specific direction that
means parallel to the force vector here is
the delta L
Now here stress is the can be defined the
load over the area so that is a stress and
strain can be defined here what is the increment
of the length as compared to the original
length So of course in this case we can define
stress as a engineering stress or strain as
a engineering strain The dimension of the
strain it is same that having no dimension
of strain
Now if there is a application of the load
in 1 specific direction that is uniaxial direction
then the figure shows the typical stress strain
diagram where the initial state from A to
B the chord actually follow uhh a linear path
but beyond B the diagram becomes non-linear
and upto certain point it breaks that point
is called uhh the rupture or the fracture
happens at (the) at that point
Now strains will disappear if we remove the
load that point B and it will come back to
the initial position A But at the same time
if deformation happens up to the point C and
then afterwards if we remove the load it comes
back to the position D So there exists some
permanent deformation AD That (s) that is
called the material undergoes plastic deformation
so that deformation is not recoverable
So in this case that is (por up) deformation
at point C is called as plastic deformation
but deformation at up to point B can be considered
as a elastic deformation So the largest stress
for which this occurs is called as the elastic
limit but when the strain does not return
to the 0 after the stress is removed the material
is set to behave plastically So there is a
clear distinction between elastic part as
well as plastic deformation
So within the elastic limit the ratio of the
stress and strain is equal to the constant
term that constant term is called as elastic
modulus In the sense that it is a comparison
of what force is applied and how the object
deforms upto certain extent So if we look
into that expression the stress is equal to
elastic modulus and multiplied by strain So
since strain is dimensionless so equation
of stress is equal to the here equation of
uhh unit of modulus of elasticity
So both the modulus of elasticity and stress
having the same unit but when we produce the
stress this stress is whether it is normal
stress or shear stress and the strain whether
it is normal strain or shear strain accordingly
we can define different elastic modulus and
that is limited to within the elastic limit
So 3 different types of elastic modulus is
generally we observe in the solid mechanics
approach when a material is deformed with
the application of some external load So 1
is the Youngís modulus Actually it measures
resistance of a solid to a change in its length
So that application of the load here may be
either tensile or compressive but in this
case the uhh load acts normal to the cross-sectional
area
Shear modulus its measures resistance to motion
of the plane of a solid past over (s up) uhh
solid sliding past each other So that will
graphically I will try to explain what is
shear modulus So in this case the load actually
applies which is parallel to the cross sectional
area Third 1 is the bulk modulus this actually
measures the resistance of solids or liquids
to change in their volume
So these 3 elastic modulus we can explain
further that if we look into the figure the
(el) application of the force at cross-sectional
area that ratio actually define the stress
and if tensile loading is acting here so it
can be considered as a tensile stress But
the area of cross-section is normal to the
force direction At the same time there is
a deformation along the direction of the force
that is delta L and that increment of the
deformation with respect to the initial length
can be considered as a tensile strain
So this tensile stress and tensile strain
ratio is considered as a Youngís modulus
Stress can also act in such a way that it
may be the compressive load also So in this
case the Sigma equal to E into Epsilon and
E specifically considered as a Youngís modulus
and this actually indicates the measure of
the stiffness of a solid body
Now the if we look into that diagram that
elastic limit of the substance actually defined
the maximum stress can be applied to the substance
before it permanently deformed that we have
already explained And if we take as a reference
the right-hand side figure here we can see
the stress strain diagram of a specific material
but in this in this case how we can measure
the Youngís modulus and the stress strain
curve Since we are explained that stress is
proportional to the strain to the elastic
limit and that constant of proportionality
we affiliate with the Youngís modulus So
the Youngís modulus is actually represented
by the term of ratio stress by strain but
this (ratio) this relation of stress strain
initial period is normally follow the linear
relation
So up that line AB is a straight line here
and the slope of the (straight line) straight
line is physically represents the Youngís
modulus We can find out in that way that if
we consider the angle theta between the (line)
straight line and the strain axis so tan theta
represents the slope and the tan theta equal
to stress by strain but that definitely within
elastic limit and that ratio actually represents
the Youngís modulus here So higher (E) Youngís
modulus (meet) means there exist higher stiffness
for a specific structure of a solid
Now next is the shear modulus Shear modulus
we can define in such a way that when an object
is subjected to a force tangential to 1 of
its faces while we keep (co) as a constant
in the opposite face and then the stress in
this case is called a shear stress Definitely
in this case application of the load is parallel
to the cross-sectional area here And to a
first approximation and specifically for small
(di) distortion or small deformation in case
no change in volume occurs with this deformation
We define the shear stress as a ratio similar
to normal stress area uhh sorry force divided
by area but this area is the tangential to
the area of A of the force being acting
Now if we look into right-hand side figure
if we try to find out that there is application
of the force on the top of the element and
bottom face is (keeped) kept as fixed Now
with the application of the load A on the
top surface its deformation longer direction
of the load is delta x and that deformation
happens over the reference of height h Now
in this case the shear strain on be defined
as delta x divided by h where delta x is the
horizontal distance that is the shear force
moves and h is the height of the object
Therefore when we try to find out the share
modulus its is basically the ratio of the
shear stress and shear strain So here the
shear stress is defined as the force F divided
by a cross-sectional area A but it is noteworthy
that this cross-sectional area is actually
acting parallel to the application of the
force
So in terms of stress we can find out that
S is F by A that is shear stress and the ratio
of shear strain that is delta x by h So shear
stress is generally represented by Tao that
is equal to G into gamma So here gamma equal
to (t) tan theta which is equal to delta x
by h here and that is the measure of the shear
strain in this case and G is considered as
a shear modulus in this case having the unit
of newton per meter squared or Pascal similar
to pressure
Now the third modulus that is called bulk
modulus which is characterised with the response
of a substance when it is subjected to uniform
squeezing or is there any reduction in the
application of pressure and that actually
acts over the volume of an element So a uniform
distribution of forces occurs (c) specifically
when the object is immersed in a fluid And
the object is subjected to this type of deformation
that is undergoes by the change of volume
without any change of shape
So this volumetric (str) stress which is defined
as the magnitude of the normal force to that
area A And that quantity P F by A here it
is termed as pressure And if the pressure
changes by an amount of delta P which is equal
to the change of force by the area the object
will experience a change of volume uhh delta
V
So that delta V with respect to the original
V that is called volume strain or volumetric
strain and here the bulk modulus is defined
by the ratio of change of pressure and the
volumetric strain So the difference from the
other 2 modulus with respect to the bulk modulus
is that in case of bulk modulus we representing
the stress and strain term acting over the
volume
So B if we generally define the bulk modulus
and that is specific to the solid and that
is the ratio of the volume stress and the
volume strain But this volume stress actually
comes from the change in pressure and volumetric
strains actually change of volume with the
application of this force F without any change
of the shape of the object
Let us look into 1 example to understand the
different modulus or specifically the bulk
modulus uhh in this case A solid was initially
at normal atmospheric pressure that is 1 into
10 to the power 5 Newton per meter squared
and afterwards it was put into the ocean to
a depth but the pressure was 2 into 10 to
the power 7 Newton per meter squared The volume
of the sphere in air is point 5 metre cube
but now we need to estimate the how much does
the volume change once the sphere is submerged
into the water from the atmosphere So bulk
modulus also defined here uhh so it is noteworthy
that all the units of the uhh modulus which
is the units (is) equal to the stress units
of stress
Now bulk modulus if we directly apply is (del)
change of pressure and delta V by V so here
the sign convention is considered as a negative
depending upon the application of the pressure
and whether it is expanding or whether it
is squeezing So change of volume can be represented
as the original volume into change of pressure
divided by the bulk modulus So in this case
change of pressure can also be calculate if
the gauge pressure is known at a specific
depth where the pressure was 2 into 10 to
the power 7 Newton per meter squared
So in this case the difference between the
final and the initial pressure is actually
the 2 into 10 to the power 7 Newton per meter
squared So the change of volume if we put
the numerical values of all the parameters
and we found out change of volume is calculated
as 1 point 6 into 10 to the power minus 4
metre cube Uhh this negative sign actually
indicates there is a decrease in volume with
the application of the pressure
Now we come to that 3-dimensional state of
the stress and of course all these cases the
analysis is limited to within the elastic
limit So so far we have discussed the 1-dimensional
or 1 directional set of the stress now we
try to implement the concept of 1 directional
state of the stress for 3-dimensional case
Now if we look into that figure left-hand
side figure uhh (yeah) material volume if
we consider and on that material volume if
we try to find out what the element of stresses
can be act on this volume of the material
Let us look into that there is an arbitrary
forces basically acting on an element but
this comp1nt of the forces can also be decomposed
into the 3 orthogonal comp1nt and on this
3 orthogonal comp1nt or maybe we can say the
in the Cartesian coordinate system we can
define the general state of the stress Let
us look into that elemental volume here and
what are the different stresses are acting
on this case
First they are the stresses normal stress
is Sigma Y probably it is acting along the
Y direction Sigma X is the uhh acting on X
direction and Sigma Z These are the normal
stresses which acting Now on we see that there
are several shear stress comp1nts as well
and we we if we see there are the shear stress
comp1nts are Tao YX or Tao XY Tao YZ or Tao
ZY Tao XZ or Tao ZX I mean there are 6 shear
stress comp1nts all are (ac) acting over the
surface of this elemental volume But any state
of the force condition can also be represented
within the elemental volume in terms of this
9 comp1nts of the stress that means 6 are
the normal stress comp1nts and sorry 3 are
the normal stress comp1nts and 6 are the shear
stress comp1nts
Uhh here sealed that Sigma X Sigma Y and Sigma
Z are the normal stresses but the rest of
the 6 are the shear stresses It is a convention
is like that Tao XY is the stress on the face
perpendicular to the axis at points in the
positive Y direction So with this Convention
we can define the different shear stress comp1nts
But there are 9 stress comp1nts out of which
only 6 are independent since the shear stress
comp1nt Tao XY is equal to Tao YX Tao YZ is
equal to Tao ZY and Tao ZX equal to Tao XZ
so therefore the stress vectors can be represented
at the 6 comp1nt that is the Sigma X Sigma
Y Sigma Z and 3 shear stress comp1nts Tao
XY Tao YZ and Tao ZX 2728
This is the another way to represents the
3-dimensional state of the stress and sometimes
we define the axis as X1 X2 X3 and the sheer
or normal stress comp1nts can be represented
uhh like that and here is Sigma X which is
equal to Sigma 11 Sigma Y which may be able
to Sigma 22 and Sigma Z which is equal to
Sigma 33 Similarly the shear stress comp1nt
Tao XY can be represented as Sigma 12 Tao
YZ Sigma 23 and Tao ZX (or if) which is equal
to Sigma 31 So throughout our analysis we
can use both of the notation either Sigma
X or either Sigma 1 in that way we can use
to for that analysis of 3 dimensional state
of the stress
Now the Hookeís Law states that stress is
proportional to the strain but how we can
apply in case of 3-dimensional stress state
this Hookeís Law The stress suppose the stress
Sigma X is acting on X direction and that
actually produces 3 strains 1 is the longitudinal
strain or maybe you can say that extension
along the X axis that can be defined as Epsilon
X equal to Sigma X by E But at the same time
it will produce the transverse strains or
maybe contraction along the Y and Z axis which
are related to the Poissonís ratio
Now in case of 3-dimensional stress state
if 1 stress is acting 1 specific direction
(let) and if there is a extension along this
direction so there must be contraction in
other 2 directions to make the volume consistency
over the deformation That is why the Poissonís
ratio actually comes into the picture to consider
the latter contraction in there is a application
of the load in 1 specific direction
So here is (Sigma Y) Epsilon Y and Epsilon
Z can be represented as the negative of the
Poissonís ratio and (mult) multiply with
the strain in X direction so that is is equal
to that Poissonís ratio into Sigma X by E
So the negative sign comes because of the
contraction in other direction as compared
to the extension along in X direction
So here how we can define the Poissonís ratio
in this case The first (3047) indicates the
undeformed state but if we apply the load
then it will try to deform in 1 direction
or maybe extension in 1 direction but other
2 directions probably it can go through the
contraction So the Poissonís ratio is defined
by the lateral strain to the axial strain
and that is negative of Epsilon X by Epsilon
Z or negative of Epsilon Y and Epsilon Z
So it is obvious that the Poissonís ratio
is dimensionless at the lateral (ss) sign
of this Poissonís ratio is that lateral strain
opposite to the longitudinal strain But theoretical
value of Poissonís ratio is point 25 or perfectly
isotropic elastic materials but maximum limit
is half or point 5 but typical values for
most of the metals is observed between point
24 to point 3
So by looking into the Poissonís ratio we
will try to explain the comp1nt of the stress
or strain at different directions So in order
to determine the total strain 1 specific direction
we can apply the principle of superposition
Let us look into that (ss) strain along the
X axis and the (con) it comes from the contribution
with application of the Sigma X Sigma Y and
Sigma Z that means other 2 directions So Sigma
X causes actually Sigma X by E in the X direction
but Sigma Y causes comp1nt of the strain in
X direction and that is multiplied by the
Poissonís ratio into Sigma Y by E
Similarly the strain comp1nt along X direction
due to the application of the stress Sigma
Z is considered as minus Nu Sigma Z by E So
by applying the principle of superposition
along the X axis the total strain comp1nt
is calculated as 1 by E Sigma X minus Nu into
Sigma Y plus Sigma Z In this case there is
a positive deformation (or pos) happens due
to the application of the Sigma X while negative
deformation happens due to the application
of Sigma Y and Sigma Z So effective deformation
is the uhh Epsilon X that is the consist of
both the longitudinal strain and the other
2 lateral strain and that lateral strain actually
represented (in terms) with the effect of
the Poissonís ratio
Now if we look into that tabulated form the
stress is acting Sigma X Sigma Y and Sigma
Z but what are the comp1nt of the strain is
acting X direction what are the comp1nt of
the strain acting in Y direction and comp1nt
of the strain in Z direction So if we see
that comp1nt of the strain in X direction
Epsilon X is equal to Sigma X by E Epsilon
X due to the Sigma Y that is the multiply
by the Poissonís ratio with the (st) strain
to Sigma Y and similar effect can also be
observed into the Sigma Z So these are the
way out to find out the individual strain
comp1nt acting in different direction
So when you apply the principle of superposition
(of) of the effect of individual comp1nt we
can find out the comp1nt of the strain in
XY and Z direction are represented like that
Epsilon X equal to 1 by E Sigma X minus Nu
into Sigma Y plus Sigma Z similarly Epsilon
Y and (s) Epsilon Z But if we look into all
this expression the positive deformation or
positive amount of strain actually comes from
their individual stress comp1nt along that
specific direction So we are talking about
the positive in the sense that Sigma X Sigma
Y or Sigma Z all are positive
But at the same time the negative contribution
actually comes from the other 2 comp1nts of
the stress when we focus on 1 specific axial
direction So E here is the Youngís modulus
but values of the Youngís modulus can also
be obtained from the uniaxial tension test
and that Youngís modulus actually represents
the slope of the uhh very initial curve or
maybe we can say the initial linear part of
the stress strain curve is a measure of the
Youngís modulus
The shear stress also acting on the unit cube
and that is shear stress is proportional to
the shear strain and that proportionality
is represented in terms of the shear modulus
here So similarly all the shear strain comp1nt
(can be) can also be represented individually
with respect to the uhh (s) uhh shear strain
along the specific plane So here G is the
shear modulus and values of the G can also
be obtained from a torsion test as compared
to E in case of tensile test
These are the some basic idea about the uhh
specific values of the elastic constants for
isotropic materials And if we see that modulus
of elasticity shear modulus and Poissonís
ratio are defined for the different materials
and out of these materials tungsten is having
very high modulus of elasticity That means
the stiffness is very high in case of tungsten
other way we can say the slope is very high
on the stress strain diagram in case of tungsten
as compared to the other materials
And similarly if we see roughly observe the
Poissonís ratio is the maximum point 33 and
minimum is point 27 uhh for this materials
And of course the aluminium alloy is having
low amount of the modulus of elasticity (3834)
Yet uhh stiffness is specifically less as
compared to the other materials
Now if we try to estimate the volumetric strain
which is the changing of the volume or unit
volume and that can be calculated as simply
considering 1 rectangular parallelepiped with
edges dx dy and dz Therefore if we consider
the engineering strain here and that is the
engineering strain is a difference between
the final length specifically I am talking
about when we focused on 1 specific direction
so that is what was the final length minus
what was the initial length divided by the
original length or initial length that actually
measured the uhh engineering strain
So that total volume can be calculated as
1 plus Epsilon X into 1 plus Epsilon Y into
1 plus Epsilon Z multiplied by the initial
volume dx dy and dz And now overall volumetric
strain due to the change of the volume can
also be calculate like that that (f) ev equal
to first is the final volume minus initial
volume divided by the initial volume or original
volume and it can be calculated is like that
1 plus Epsilon X into 1 plus Epsilon Y into
1 plus Epsilon Z minus 1
So if we consider that neglect the higher-order
term specifically (ep) multiply Epsilon X
into Epsilon Y and into Epsilon Z So that
quantity is very small (sp) in case of small
strain so we can approximate that volumetric
strain is the linear sum of all the individual
strain comp1nts that means ev equal to Epsilon
X plus Epsilon Y plus Epsilon Z So this volumetric
strain is actually varied when there exists
small amount of the strains
Now if we look into very specific cases the
3-dimensional state of the stress 1 specific
case is the plane stress condition In this
case typically exists when Sigma 3 equal to
0 and practically this happens in case of
the things she and when it is loaded in a
plane of the sheet or a thin wall tube which
is loaded by internal pressure there is no
stress on uhh normal to the free surface
So these are the typical conditions of the
plane stress where Sigma 3 equal to 0 that
means third directional stress is 0 here And
if we set either Sigma Z or Sigma 3 equal
to 0 then we can find out the 3 component
of the strain 1 by E into Sigma 1 minus Nu
into Sigma 2 here the Sigma 3 equal to 0 So
we find out that expression of the uhh 3 component
of the strain but there exist 2 component
of the stress in case of plane stress condition
Now it is possible further to rearrange this
equation and it is possible to find out the
value of the 2 stress is here in terms of
the strain component So here we see that Sigma
1 and Sigma 2 represented in terms of the
material properties that means Youngís modulus
and Poissonís ratio as well as strain components
However third strain component can also be
represents by adding the other 2 stresses
So in plane stress condition in summary we
can say that non-0 stresses exist Sigma X
Sigma Y and shear stress is only Tao XY and
non-0 strains Epsilon X Epsilon Y Epsilon
Z and gamma XY So if we see there are 3 component
of the non-0 stresses and there are 4 component
of the non-0 strain in case of plane stress
conditions So looking into that expression
or relation in terms of the elastic modulus
and we rearrange this equations we can find
out in the matrix from like this Sigma equal
to D into Epsilon so stress equal to D into
Epsilon
So D actually related to the properties of
the materials that means D actually is a function
of the material properties like Youngís modulus
and Poissonís ratio here If we see the (4428)
vector Sigma Sigma X Sigma Y and Tao XY stress
component the D matrix and then finally Epsilon
X Epsilon Y and gamma XY So this is the relation
between the stress and strain but individually
the Epsilon Z can also be calculated in terms
of Epsilon X and Epsilon Y So in this case
D matrix for the plane stress case is defined
like this
Similarly another special case uhh that is
called plane strain condition and this prevails
when strain is Epsilon 3 equal to 0 this case
This actually occurs in case of 1-dimension
is much greater than the other 2 dimension
1 example are the long rod or a cylinder with
some restrained ends So if we put the strain
in third direction that is when Epsilon 3
equal to 0 here we can correlate the Sigma
3 that means in terms of Sigma 1 and Sigma
2
Actually it shows there exists a component
of the stress but 1 2 component of the strain
So in general we represents the plane strain
condition like this non-0 stresses are Sigma
X Sigma Y Sigma Z and Tao XY whereas non-0
strain components are Epsilon X Epsilon Y
and gamma XY So for isotropic linear (elasto)
elastic stress strain law of can be represented
Sigma equal to D into Epsilon uhh again D
actually represents that (po) properties of
the materials but in terms of Youngís modulus
and the Poissonís ratio
So in this case the expression of D is different
from that of plane stress conditions but it
actually links between the stress and strain
and here uhh we can link the 3 component of
the stress in terms of 3 component of the
strain but Sigma Z can also be calculated
in terms Sigma X and Sigma Y So looking into
that D matrix different plane stress and the
plane stress condition we can correlate between
the stress and strain when the condition exists
in terms of plane stress or condition exist
in terms of plane strain condition
Now if we look into that 1 specific example
then we will be able to identify different
component of the stress and when it is subjected
to some elastic deformation So a steel specimen
is subjected to elastic stresses represented
by the matrix if we see the different component
of the matrix and here we need to find out
the corresponding strain for example the 3
component of the strain Epsilon X Epsilon
Y and Epsilon Z Since stress are given so
we can find out exactly the stresses from
the given matrix but how to find out the different
stresses from this matrix let us see
The first component of the matrix 11 component
that is is equal to Sigma X uhh the numerical
value is 2 here But if we look into the Tao
XY or if we see the 12 Tao 12 or Sigma 12
that is actually minus 3 If we look into that
there are 2 minus 3 here so that means it
is a produce the symmetric matrix that means
here physically Tao XY equal to Tao YZ is
following here Similarly Tao Y Z equal to
Tao (ZX) ZY or Tao XZ equal to Tao ZX are
following here So basically here this matrix
there are 6 components and if we pick up that
diagonal component (si) as Sigma X Sigma Y
and Sigma Z if we put the numerical values
we can easily estimate the corresponding strain
in this case
I will try to explain the another example
here that consider a plate under uniaxial
tension uhh that is prevented from the contracting
in the transverse direction and need to find
out the effective modulus along the loading
direction under the condition of plane strain
So it is clearly mentioned that the plane
strain condition prevails in this specific
problem Now it is a basically uniaxial tension
testing but 1 direction it is restricted to
deformation so if there is existence of some
restrictions on specific directions that will
try to produce some amount of the stress but
not the strain
So that condition can be represented like
that Epsilon 2 that means second direction
if we consider the loading direction is direction
1 and transverse direction (is) as direction
2 so second direction the deformation is restricted
So in this case there may not be the change
of length so that means strain component will
be 0 but there must be some of a stress component
that means it will try to create the some
amount of stress along direction 2
But if we investigate overall the problem
no stress actually acting normal to the free
surface maybe in the third direction there
does not acting any stress that means the
Sigma 3 equal to 0 Now this condition if we
apply the Hookeís law can find out that restriction
of the strain along direction 2 equal to 0
from that condition we can find out Sigma
2 equal to Nu into Sigma 1 so relation between
the Sigma 2 and Sigma 1
Then (effective) effective strain direction
1 is the represent in terms of Sigma 1 and
Sigma 2 and report here the values of the
relation between Sigma 1 Sigma 2 in the second
part of this equation then we can find out
Epsilon 1 in terms of Sigma 1 and other material
parameters (here I am) that means E and Nu
So therefore and plane strain (condition)
modulus in the direction one can also be represented
by stress by strain in the specific direction
So that means Sigma 1 by Epsilon 1 which can
be represent E by 1 minus Nu square and this
effective value can also be very precisely
evaluated if we know the numerical value of
the Nu if we consider Nu as point 33 then
the plastic strain modulus is calculated as
1 point 12 E So this case the effective modulus
along on specific direction maybe along direction
1 is 12 percent more as compared to the Youngís
modulus which (is) which was measured in case
of uniaxial tensile testing
We look into another example problem like
that a cylinder pressure vessel 10 meter long
has closed ends and all other parameters for
example thickness diameter and the internal
pressure also given Youngís modulus also
given the materials of this vessel is made
of steel and Poissonís ratio are also given
Then if we neglect any effects associated
with the details of the how the ends are attached
we can find out at different amount of the
strain or stresses and what are the change
of the length along different directions
If we look into that what are the stresses
or strain acting in this case if we see the
stress is acting along X direction due to
the internal pressure of this thick cylinder
Also about the circumference there is acting
of the stress Sigma Y but Z direction since
this is a thin sheet so there may not be any
amount of the stress acting on the Z direction
or variation of the stress is negligible due
to the thin sheet pressure vessel assumptions
So of course it is not that the ratio of the
thickness radius to the thickness is very
small here so that it can be considered as
a thin-walled (b) pressure vessel and we can
estimate that stress acting direction X is
pr by 2t where p is the internal pressure
r is the radius and t is the thickness of
the vessel and we can find out Sigma X as
300 mega Pascal Similarly we can estimate
the Sigma Y pr by t where 600 mega Pascal
But as already I mentioned that Sigma Z varies
actually minus p which is the internal pressure
and outside it is 0 so we can assume that
this is sufficiently small that can produce
any significant variation over the thickness
of the wall so Sigma Z can be considered as
0 in this case So when Sigma Z equal to 0
so this can be considered as a plain stress
problem here Now we can found out the different
component of the strain along X Y and Z direction
so here we have mentioned the X Y and Z direction
So if we put all the numerical values here
we can find out the X direction Y direction
and Z direction the strain component but this
strain we can find out that strain in along
X direction that is a change of the length
of the cylinder with respect to the original
length and similarly over the circumference
Sigma Epsilon Y can also be calculated over
the integer of the length and that is the
ratio of the change of the diameter of the
original diameter and similarly Epsilon Z
can also be (calculated) estimated with the
change of the thickness with respect to the
original or initial thickness
So putting all these strain component values
we can find out what is the change of the
length change of the diameter and change of
the thickness here And we just observe that
change of the thickness is very small in this
case as compared to the change of the length
or change of the diameter
We shift to another problem we can find out
that some data which is the measurement from
the strain gauge uhh made on the free surface
of a steel plate indicate that the principal
strain components are point 004 and point
001 What are the principal stresses Here the
assumptions is that no stress normal to the
free surface and the values of the Youngís
modulus and Poissonís ratio for steel are
also given
So we straightforward apply the strain component
along the X direction that is 1 by E uhh in
terms of Sigma Y Sigma (X Y) X and Sigma Z
and (al) also in terms of Nu So since no stress
acting normal to the free surface that means
Sigma Z equal to 0 here and using that we
can find out (and we) we get 2 equations in
terms of the stress and strain (I) I mean
Epsilon X Epsilon 1 stress component or Sigma
X and Sigma 1 these are the stress components
So if we solve it we can find out Sigma Y
and Sigma X in terms of Epsilon X (Eps) Epsilon
Y and Youngís modulus as well as Nu that
means Poissonís ratio
So if we put all the numerical values we can
find out that Sigma X point 965 Giga Pascal
and Sigma Y equal to point 516 Giga Pascal
So this example problems actually give some
practical idea how we can apply uhh 3D elasticity
theory to solve very different kind of problems
so it is a this was a very basic problems
but we I think we can understand how this
applications or expression of the different
strain or stress components is applied to
solve any practical problems
Now we have discussed there exists 3 different
types of (module) modulus as well there are
existence of the Poissonís ratio but using
all this elastic modulus or elastic components
uhh or we can say elastic constants can find
out different correlations among the different
parameters Let us investigate how all these
parameters for example E Youngís modulus
shear modulus Poissonís ratio K or D it is
a bulk modulus can also be correlated
So it is a very first thing if we look into
that expression of Epsilon X Epsilon Y and
Epsilon Z if we add it step forward and find
out this relations But for isotropic elasticity
the following expression also (6011) that
we already discussed these things but here
point to be noted that shear strain for example
shear component YZ is the shear stress Y shear
modulus which is equivalent to the 2 times
of the (s s s) uhh normal strain component
that means Epsilon Y Z So for an isotropic
material it can be says that G shear modulus
is not independent of Youngís modulus and
Nu it should depends on that
Now let us look into the relation between
G E and Nu that means between the shear modulus
Youngís modulus and Nu this 3 constant term
First we will try to focus on that the state
of a pure shear condition So state of pure
shear condition there exist only one shear
stress that is Tao XY of course with respect
to the other components of the other 5 3 normal
stress and 2 shear components as 0
So with the state of the pure shear and if
we analyse the (6126) also we can find out
the principal stress components in this case
Sigma 1 is equal to Tao XY Second principal
stress component is the minus Tao XY and Sigma
3 is definitely 0 So this is the state of
the 3 principal stresses Now when this principal
stresses for pure shear condition is known
to us and we can find out the Epsilon 1 that
means shear strain along direction one can
be represented in terms of the stress components
and if we straightforward put the stress component
we can find out the Epsilon 1 equal to 1 plus
Nu by E into Tao XY
But we know that Epsilon 1 at the (safe) same
time the normal strain component is the half
of the shear strain component and shear stress
component is related to the shear stress and
the shear modulus So from this relation we
can find out this half of the equal to half
is equal to 1 plus Nu by E into G So from
here we can find out the relation between
G in terms of Youngís modulus and Nu So this
is the straightforward relation between the
shear modulus Youngís modulus and Poissonís
ratio
Similarly we can find out for in case of isotropic
elasticity the bulk modulus and with respect
to the Youngís modulus or Poissonís ratio
So uhh it is already discussed that bulk modulus
can be defined minus of dp that means change
of pressure over the volumetric strain If
the negative sign actually indicates depending
upon the direction of the stress as well as
is there is a contraction of the volume with
the application of this type of loading condition
or this type of pressure for this type of
stress condition
So in solid mechanics that Delta V by V is
the volumetric strain is equal to 1 by B into
Sigma M Sigma M can be considered as a mean
stress in this case or Delta V by V is the
volumetric strain but this mean stress actually
equal to the 1 third of summation of the stresses
along X Y and Z direction So for an isotropic
material it can be says that he is not independent
of E and Nu so how we can represents the bulk
modulus in terms of E and Nu from this basic
concept of the uhh relation between the mean
stress and the volumetric strain
Now suppose volumetric strain actually produced
by the hydrostatic stress and in the hydrostatic
stress component the mean stress component
is actually called the Sigma X Sigma Y and
Sigma Z So when there is application of the
hydrostatic state of the stress that means
(if we) if we consider is as a in terms of
pressure so irrespective of the X Y and Z
this 3 Cartesian coordinate system all are
equal in this case
So but if we try to estimate the volumetric
strain here if we see that lxo lyo lzo was
initial length dimension of an object and
final dimension was lx ly and lz So therefore
V by V0 that means final length by initial
length can also be (re) uhh represented in
terms of the logarithm and individual component
But if we know this volumetric (s) strain
actually represented in terms of the Epsilon
X Epsilon Y and Epsilon Z but here all the
terms are (log) uhh true strain
So it is obvious that in case of the true
strain and if the deformation is very small
the value of the volumetric strain or uhh
individual strain component the true strain
is actually equal to the engineering strain
component So here we find out the right-hand
side expression that we can find out the logarithm
V by V0 is actually the volumetric strain
and that is approximation is Delta V by V0
that is actually considered as a uhh engineering
in the perspective of engineering strain
Now Epsilon X can also be represented in terms
of the Sigma X Sigma Y Sigma Z and Nu and
if we find out the (represent) replace Sigma
X Sigma Y and Sigma Z as against stress value
we can find out the Epsilon X in terms of
mean stress and in terms of Poissonís ratio
and Youngís modulus
So from this relation we can find out that
the definition of the bulk modulus Delta V
by (v) V0 which is equal to 1 by V into Sigma
M and right-hand side also there is a Sigma
M So here we can find out that bulk modulus
is a ratio of the E divided by the 3 into
1 minus 2Nu So these are the expression of
the bulk modulus in terms of Youngís modulus
and Poissonís ratio Now if B greater than
0 that means 1 minus 2Nu should be greater
than 0 which uhh indicates that Nu should
be less than half
So for isotropic if bulk modulus is B is positive
then we can say Nu should be less point 5
But if B is negative then that actually indicates
that (at) an increase in pressure should increase
in volume But we have already mentioned that
practical values of the bulk modulus is (most)
for most of the materials (is) uhh sorry (Nu)
Nu that means uhh Nu here the Nu Mu is the
Poissonís ratio (here) that is actually limit
is point 5uhh in uhh in case of solid materials
but practical for all the materials it lies
between around point 24 to point 3 in between
So we can derive so many relations between
all these 4 parameters but here is the summary
E in terms of shear modulus and uhh Nu that
means Poissonís ratio and E can be represented
in terms of bulk modulus and Poissonís ratio
and uhh uhh similarly E can be represent in
terms of shear modulus and bulk modulus So
so many correlations can also be possible
among all these 4 parameters in case of isotropic
elasticity
So there is a few comments on this isotropic
elasticity that for crystalline material E
is generally regarded as being relatively
insensitive to change of the microstructure
but for BCC and FCC Youngís modulus in case
of iron differ by a relatively very small
amount Heat treatment (at) practically having
the large effect on the hardness and the yield
strength but very little effect on the elastic
properties
The elastic behaviour of the polymer is actually
very different from that of the metals So
this isotropic elasticity or all the parameters
so far we have discussed it is a basically
applicable for the solid materials and most
of the engineering materials we can used all
this relations and main thing is that we need
to uhh remember that all this uhh correlation
we have derived assuming there is a exist
some isotropic elasticity
Now further extension of the isotropic elasticity
uhh when we consider there is a thermal expansion
that actually produces some amount of the
thermal strength So thermal so actual if there
is a application of the temperature (7023)
difference within the body itself then increment
of the length or maybe linear strength can
also be represented by the difference or in
terms of thermal expansion coefficient alpha
and what is the temperature difference
So practically in solid materials if that
existence of the thermal load thermal load
in the sense of this (difference) difference
of the temperature it actually produce some
amount of the strain that is called the thermal
strain And when you (super) uhh this thermal
strength can be directly added to the mechanical
strain component uhh we can find out or we
can modify the linear strain component along
X direction which is the first part indicates
in the due to the mechanical load and second
part indicates due to the thermal load
So this generalisation actually useful for
finding the stresses that actually uhh observed
in case of constrained bodies when they are
heated or (they are) they are cooled 1 typical
examples of the bimetallic strip used for
sensing temperature depends on the difference
of the thermal expansion of the 2 materials
Let us look into 1 example to explain the
effect of the thermal expansion So first a
brass rod is restrained but stressfree at
the room temperature which is 20 degree Youngís
modulus is given the coefficient of thermal
expansion also mentioned (or) at what temperature
does the stress reach to minus 172 mega Pascal
So here in this case in absence of any mechanical
load only the strain will be produced due
to the thermal load So that thermal strain
can be defined like that (ex) thermal expansion
coefficient multiplied by the difference in
temperature
So first figure actually indicates that original
length which is having (at) the temperature
room temperature T0 at 20 degrees centigrade
Now if there is a difference of temperature
then it will try to expand (if) freely if
there is no constraint or no obstacle on this
deformation umm So second figure actually
indicates the length or (s) of the material
when it is in final temperature but in this
case no constant is applied to the material
If we look into the third figure if it is
constant if it is resist to move or if it
is resist to deformation due to the application
of the temperature change then definitely
it will create some amount of the stress here
So third figure actually indicates some amount
of the compressive stress will be generated
if we try to restrict the elongation or deformation
(mo on) on specific direction So that thermal
strain is useful to estimate the amount of
the stress here
So Sigma can be considered as the Youngís
modulus into the thermal strain here and that
is the Youngís modulus into coefficient of
the thermal and the temperature difference
and that thermal strain can also be calculate
from the available data and from there we
can find out what is the final temperature
if we will try to produce there is a compressive
stress of 172 mega Pascal here So it is observed
that final temperature here to produce the
172 mega Pascal compressive stress the temperature
should raise to 106 degrees centigrade
So in summary we can say that for single crystal
the properties actually vary with the direction
and we know that atomic arrangement of the
single crystal is different in different directions
so there is a difference of the Youngís modulus
at different direction if we look into that
picture So at the diagonal direction for BCC
iron it is a the Youngís modulus is 273 Giga
Pascal but along the edge the Youngís modulus
is 125 Giga Pascal
So if we here we find out the difference the
Youngís modulus at 2 different directions
That means that (si) perfect single crystal
actually follow some anisotropic behaviour
And second thing is that polycrystal that
properties may or may not vary with the direction
If grains are randomly oriented that we can
consider as a isotropic properties But if
grains are textured then 1 properties are
maybe mechanical properties may be strong
1 specific direction as compared to the other
direction
So in summary the depending upon the application
whether it is need to use elastic (properties)
uhh elastic properties uhh in case of anisotropic
or plastic properties in case of isotropic
that actually depends on the different structure
So in summary we can say that single (crys)
crystal structure follow in different direction
the different properties that means anisotropic
properties hold good in this case
But in case of polycrystal in general if there
is no textured structure then we can follow
the isotropic properties and we can apply
the theory of the isotropic elasticity in
this case to evaluate or to analyse the different
properties or to correlate the amount of the
stress strain using the material properties
So thank you very much for your kind attention
Thank you
