foreign
so we talked about how the total strain
in a body due to deformation
can be decomposed into deviatoric and
volumetric components
now let's study the deformation of a
solid body and see how this
decomposition
is done mathematically
let's consider a small block of solid
whose dimensions are lx ly
and lz in the three principal directions
now let's subject it to a very small
deformation
since it's oriented in the principal
directions
there are no shear deformations
as a result the solid changes its
dimensions
by dlx dly and dlz
in three principal directions the
resultant change in volume
can be calculated using this equation
in this equation v is nothing but the
volume of the block before undergoing
any deformation
and it's a product of the three initial
principal dimensions
l x l y l z let's plug this relation in
the earlier equation
and divide it by v on both sides and it
reduces to this one
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notice that in this new form these three
terms are nothing but
the normal strains in the three
principal directions
and the term on the left hand side is
the ratio of change in
infinitesimal volume to the original
volume
which is nothing but the volumetric
strain
so from this equation the volumetric
strain
is nothing but the summation of all the
three principal strains
and finally another point that is worth
noting
is that the shear strains do not change
the volume of the body
only the normal strains result in a
change in the volume of the body
so we talked about the deviatoric and
volumetric strains now
let's talk about the stresses
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usually there are two stress components
corresponding to each of these strains
they are the deviatoric stresses and
hydrostatic pressure
the stresses developed in the body due
to derivative stains
are called as deviatoric stresses and
the ability of the material
to resist deviatoric strain is nothing
but
elastic modulus
similarly the stresses developed in the
body due to
any change in volume is called as
volumetric stress
or the hydrostatic pressure and the
ability of the material
to resist volumetric strain is nothing
but bulk modulus
mathematically it's nothing but the
average
of the three principal deviatoric
stresses
to understand the physical meaning of
these quantities
let's consider a simple scenario let's
take this cube
and apply two types of loads and
boundary conditions
in the first case we constrain the block
in only one direction
and keep it free to deform in all the
other directions
in this state much of the deformation is
deviatoric
so majority of the stress is deviatoric
in the second case we constrain the
block in all the directions
and apply the same force in one
direction
in this case the material is not allowed
to deform
in any other direction so the material
can deform
only when there is a change in volume so
all the stresses developed in the body
is volumetric in nature
this is the physical interpretation of
deviatoric
and volumetric stresses
you