Stress Tensor - 2 Video Lecture - Civil Engineering (CE)

so today let's discuss the stress tensor
let's take a block of material and pull
on both ends with a load P now if we
assume this is a solid you might expect
the responsive material to be something
like that you might elongate and get a
little bit longer in this direction a
little bit thinner in that direction
now if we took the same material and
applied the same force but we applied it
to that face the top and the bottom face
we might expect the response to look
something as such so not only does the
magnitude of the force matter because
obviously if I pull on this harder it
might be long gate more but the face
upon which that force acts is important
so this leads us to the idea of the
stress tensor so the stress tensor is
just a generic representation of the
state of stress inside the material so
if we have a cube we have faces in
different directions and the stress
tensor just describes the forces or the
stresses that are acting on each of
those faces
so in an XYZ coordinate system we have
phases which have normal directions
pointing in the x y&z direction on each
of those phases we have XY and Z forces
so the notation the first index X
denotes the direction of the normal
vector so X X X are all point wheel ones
acting on this face why-why-why are all
ones acting on this face and the second
one is the direction the four so xx is
the X force acting on the face with the
normal component in the X direction so
we represent these nine components of
the stress tensor and a matrix
so we represented as such so it has nine
components and this is what we call the
stress tensor so we've seen before we
had scalar quantities things like
temperature pressure density these are
just a single number one number so these
can change with time in space but a
particular point a particular time it's
just a number
vectors were things like velocity and
heat flux these have three numbers right
the XY and z components of the vector
right because velocity has a magnitude
and a direction tenser it's things like
stress like with a vector I you should
put an arrow over it when I'm writing to
denote a tensor all you should put two
lines in it just to remind myself that
is a tensor and it's described by nine
numbers or a three by three matrix so
there's our tensor now if we want to
find the stress vector on any surface
all we have to do is take our stress
tensor and take the dot product with the
normal vector so again the arrows are
vectors the two lines is a tensor and if
I take the dot product of a vector with
a tensor I get back the vector now the
way I usually remember this is just
write things out as sort of a vector
matrix multiply so if we write our
normal vector as a column vector and I
take the vector matrix product of this
vector in dotted with this matrix T then
that gives me a vector which is the
stress vector so as an example of this
let's just consider a face where the
normal vector points out here just to
make sure that this works so are my
normal vector there is 1 0 0 and if I do
the MEC vector matrix multiply I simply
get a vector which has components X Y
and C components which we just extract
the values of this from the stress
tensor off that face so it's kind of a
trivial example but we see we get the
right result and if I use the normal
vector 0 1 0 or 0 0 1 I simply extract
you to the X Y or Z row and again these
rows are all the forces or stresses that
are acting on a face of the normal
vector pointing the X Direction y
direction Z direction now let's consider
a slightly more complicated example
let's look at a surface whose inclined
at 30 degrees so therefore the normal
vector is 1/2 square root of 3 over 2
and 0 so let's just write this out
so we get a vector where the XY and z
components are starting to mix the
values from the stress tensor which are
acting on the X face and the Y face and
you can see that since our normal vector
is closer to the Y normal vector we're
sort of waiting those stresses more by a
factor of square root of three over two
then the ones that are pointing in the X
direction and as this angle changes and
this becomes closer to one we see we
start to pick up more the the components
that are acting on that Y face the ones
that are acting up there on that Y face
so let's see how we might use the stress
tensor in our formulation for fluid
mechanics so if we have our arbitrary
volume a potatoe and if we imagine
everywhere along the surface of this
thing we can go to a point and we can
define a stress vector so this is the
force per unit area that's kind of
acting locally at that point and for
whatever reason it could be different
all over the objects so it just sort of
changes as we go around if we want to
find that total force due to these
surface forces so it would be a vector
all we would have to do is integrate
that stress vector if we somehow by
magic knew it over the surface and I if
I sum up all those little forces all
over the surface I get a net force so
again this is a stress which is a force
per unit area so I integrate over the
area and I get the nest net force but we
know this stress vector is given is
related to the stress tensor so I can
replace the stress vector within the
normal vector right the normal vector
which points everywhere out of our
surface dotted with the stress tensor at
that point now I won't prove it but the
divergence theorem works just as well
for tensors as it does for vectors so
this can be written as the divergence of
the stress tensor integrated over the
volume so this is going to be a key
result that we need to eat or that we're
going to use and if we want to know how
to calculate the divergence of the
stress tensor well if we remember this
gradient operate the del operator right
if we just write it out as a vector
we can do exactly the same thing when we
looked at dotting a tensor with a normal
vector so I'll write out the x y&z
component with my del operator and if I
just simply do my vector matrix
operation I get
obtain the following vector it has x y
and z components and you can see the
form there so when I take the divergence
of a tensor I get a vector and it has
this form where I'm taking the gradient
in the x y&z direction of all these
different nine numbers so taking the
divergence of a tensor gives me a vector
which has this form let's see if we can
prescribe some physical meaning to the
divergence of the stress tensor so let's
go back to our cube so there's our cube
in X Y Z I'm going to place the origin
this cube in the back corner there now
let's just draw the X forces so let's
just consider the X forces each face of
this cube we're going to have some force
but we're going to consider to be sort
of finite in size and then we'll take
the limit as that size goes to zero so
there's Delta X Delta Z Delta Y so each
side is going to have length so each
face of this has each fête all six faces
of this cube have stresses so on this
back face we'll just call that t xx and
on this front face we're going to have a
stress as well pointing in this
direction sort of outward from the
surface and if our stress tensor is
varying in space but over a small
distance we can approximate the stress
here as T xx the value that it was at
zero zero plus a little bit based on its
derivative with respect to x times the
length Delta X so that's just the Taylor
series now we're only doing X forces
only so I won't draw the others just to
this diagram doesn't get cluttered now
let's do the forces acting on the two
faces that have normal directions in the
Y so we'll start at the bottom that one
will be pointing backwards at is T YX
and this one will be pointing that way
and likewise if the stress tensor is not
constant in space
it's going to be given by the value here
at zero zero right plus a little bit
based on its gradient in the direction Y
and then we're going to have likewise
for Z and now it's getting a little more
clutter to draw but I'll draw it anyway
and so there are forces all of them
right so we have two that are acting on
the z face two that are acting on the X
face and two that are acting on the Y
face so let's just sum all those up
let's sum the forces so we'll start with
the blue ones we have t xx plus the
small change from here to there minus
the force that's pointing backwards and
these T X X these are stresses so in
order to get forces I have to multiply
it by Delta Y and Delta Z now let's add
the other ones in
so there is the sum of the forces now we
can see several things cancel here and
what we're left with is that this the
force in the x-direction
is the first component what we calculate
on the last page when we did the
divergence of the stress tensor so that
first component is the net force in the
X Direction times the volume so we can
think of that vergence of the stress
tensor as the net force per unit volume