Compatibility Condition Video Lecture - Civil Engineering (CE)

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we saw that there are four concepts in
mechanics and four equations that
connects them right that is what we have
been looking at so we have seen all the
four concepts they are seen what forces
we have seen what displacement is and
then we have seen now what strain as'
and we have seen what stresses right so
we have seen the equation that connects
the displacement to the strain say
strain and displacement relation to be
1/2 h plus h transpose and then we have
seen the equation that connects the
stress and the force which is equivalent
equations
which is divergence of Sigma plus Rho be
equal to Rho a and Sigma equal to Sigma
transpose as seen these equations now
what we are going to see is the equation
that connects the strain into the
displacement that is the compatibility
condition
okay this is what we are going to look
at No now what did we say about a
displacement field
we said that this relation field would
be in such a way that for our surface
the surface will move like this but
there won't be opening up of surfaces
are their own be independent raishin of
surfaces right that is a material
particle will be mapped onto a material
particle in the current configuration
dello and be a case or in a material
particle becomes two different material
particles are there won't be a case and
two different material particles will
become fused to a single material
particle okay so that's what we said
about the displacement field if that
were to be a case then what you know is
the order of differentiation of the
displacement field will not matter okay
mathematically what this means is if you
have dou square u X by dou X dou Y this
will be same as dou square u X though I
know X okay that is the order in which
you differentiate a component of the
displacement field will be immaterial
because it is a smooth continuous
displacement field okay so what happens
then is now I have six components of
strain that I can prescribe I ask this
competence of saying that I can
prescribe from which I had to find only
three components or a displacement field
so there's a disparity in the number of
unknowns okay so what happens there'll
be a restriction on how I can prescribe
the component of the strain because this
prescription of the component upstream
should be consistent so that I find a
smooth continuous displacement field
from the prescribed strain components to
give an idea let's look at the plane
strain case let assume the strain is
going by epsilon X 6 function of X comma
Y epsilon XY some function of X comma y
0 epsilon XY some function of X comma Y
epsilon YY some function of X comma Y
zero zero zero zero okay say I prescribe
this epsilon X 6 epsilon XY and epsilon
YY now
this prescription are to be able to find
the company hull displacement D UX and
uy so for this I allow the displacement
field as you to be UX function of X
comma Y ax plus uy function of X comma y
ey so that epsilon X X becomes Rho u X
by dou X epsilon Y Y becomes dou u Y by
dou Y and epsilon XY becomes 1/2 dou X
by dou y plus dou Y by dou X this comes
from the fact that I use this in here to
compute the strain components in terms
of a displacement field ok now let's
take forward these equations I add
epsilon xx as dou X by dou X I add
epsilon YY ass dou u Y by dou Y add
epsilon XY as 1/2 dou u X by dou y plus
dou Y by dou X now let's compute dou
square epsilon XY by dou X dou Y what is
this going to be this is going to be 1/2
Doh cube u X by dou X dou Y square plus
dou Q u Y dou X square dou Y here again
I use the fact that dou X dou Y dou X
are used the cube uy
by dou X dou Y dou X can be written as
the cube view Y by dou X square dou Y
since you offer is smooth and
differentiable
okay it sufficiently smooth and
differentiable because I should have
second derivatives the derivatives of U
of ye possible okay sufficiently smooth
and differentiable okay now what is this
what is two times dou square epsilon XY
by dou X dou Y this is nothing but dou
square epsilon xx by dou Y square plus
this dou square epsilon YY by dou X
square okay so basically what do you
obtain this differential equation
connecting the components of this strain
connecting the components of this chain
right differential equation
connecting the components of the strain
okay so this is called as compatibility
condition
okay if the displacement field is
sufficiently smooth and differentiable
that exist some relationship between the
components of the strain only
prescriptions of strain and satisfy
those relationships are admissible
strain fields okay so that is why the
computed condition tells us this is if I
assume the strangeness plane in terms of
x and y directions similarly I can write
the condition for the strain being plane
in terms of the other two planes okay so
if the plane is plane in the XZ plane
I'll Absalon XZ loaded by dou X dou Z to
be given by dou square epsilon xx by Dow
Z square but I am writing this way
analogy which you can show also will all
true in the real case okay sir on Z Z by
dou X square and similarly I have Rho
square epsilon Y ez by dou Y dou Z to be
equal to dou square epsilon YY by Dow Z
square plus dou square epsilon Z Z by
dou Y square okay so this is the strain
is plane in the X n Z coordinate or the
strain is plane in the Y & Z coordinate
okay apart from this the general set of
comparability conditions in three
dimensions i won't derive this
chené is in general represented as curl
of curl of epsilon being equal to zero
only six of them would be independent
okay
those six are the three that are listed
here is one set of equation the other
equations are what I will give now
those E and a 3 question that we are
before were
so these are the comparability
conditions so these six equations are
the compatibility conditions okay now
since strain is a symmetric tensor you
can write epsilon XY is equal to epsilon
Y X so I aren't bothered about being
precise about writing XY are why is that
r XZ okay so either you can write it
with a symmetric tensor it is for the
same reason that you look at the angle
change a daughter with B between a and B
this will be 2 times epsilon a dotted
with B divided by sine of theta RF from
the definition of transpose this will be
same as 2 times epsilon B dot with a
divided by sine of theta RF so it
doesn't matter with what vector you
prefix the epsilon okay it does not
matter on what vector the linear chain
acts this will be same because epsilon
is equal to epsilon transpose by the way
we are defined epsilon as okay okay so
we have seen what a comparable condition
now is the only equation that is
remaining is going back to this chart we
have to connect the sis and restrain the
equation that connects the stress and
the strain is called as the constitutive
relation
okay
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