Stress Tensor - 1 Video Lecture - Civil Engineering (CE)

in this video about stress tensors I
will talk about how a three-dimensional
stress state is described using a stress
tensor let's start with the basic
definition of stress stress is defined
in general as the force divided by the
area over which the force is acting the
units on stress are Newtons per meter
squared which is also known as Pascal's
now the force could be acting normal to
a face or parallel to a face and this
gives rise to normal stresses or here
stresses in a normal stress the force
acts perpendicular to the area and in a
shear stress the force acts parallel to
the area let's take a look at an example
of each of these so if we consider some
small volume like this then if we have a
force which is acting in this way this
force is perpendicular to this top face
and this bottom face and so we would
call this a normal stress
if instead that force were acting on the
top and bottom faces but in this
direction here then that's what we would
call a shear stress now a component
could be loaded in such a way that there
are both normal stresses and shear
stresses present therefore we need to
consider the three dimensional stress
State so in the three dimensional stress
State we could have normal and shear
stresses acting on any of the faces
we're going to define the components of
stress in the following way with the
notation Sigma IJ where this is defined
in the following way this is the force
acting in the J direction on the I face
so let's just put that in words because
it can be confusing this is the stress
on the I face in the J direction let's
look at an example of a cube where we
have labeled these after defining a
coordinate system
so let's take a look at how these
stresses are defined we first need to
define a coordinate system so we're
going to call this the X 1 Direction
this is the X 2 direction and this the X
3 direction and then let's just remember
that our stresses are defined in the
following way
FJ divided by a I so if we start with
the stresses let's say on the X 3 face
then we can define first Sigma 3 3 this
is the stress acting on the 3 face and
in the 3 direction then we can define
this Sigma 3 2 this is the stress acting
on the 3 face in the 2 direction and
then finally Sigma 3 1 the stress acting
on the 3 face in the one direction so in
the case where the subscripts are mixed
those are our fear stresses now we can
look on the X 2 on the to face so this
would be Sigma 2 2 on the to face in the
2 Direction Sigma 2 3 on the to face and
in the 3 direction and Sigma 2 1 on the
to face and in the one direction
and finally we can consider the stresses
on the face 1 so this normal stress is
Sigma 1 1 this shear stress on the one
face in the 2 direction is Sigma 1 2 and
this other shear stress on the one face
in the 3 Direction is Sigma 1 3 so the
arrows here that are indicating the
shear stresses indicate the direction of
a positive shear stress if the shear
stress were acting in the opposite
direction that would be what we would
call a negative shear stress we can now
take these components of stress and
define what is called the stress tensor
so the stress tensor vus is sometimes
defined with the double underline and
this just collects these stress values
together so along the diagonal are the
normal stresses and then the off
diagonal terms represent the shear
stresses
so this stress tensor has nine
components to it but in order to
maintain equilibrium it's the case that
the shear stresses have to be equal to
one another
so for equilibrium Sigma IJ has to equal
Sigma J I because otherwise this thing
would start spinning so in the end there
are six independent components to the
stress tensor so let's just summarize
about the stress tensor the stress
tensor includes both normal stresses
where the force is perpendicular to the
area it includes shear stresses where
the force is parallel to the area our
stresses are described by Sigma IJ and
we have six independent components of
the stress tensor