Stress transformation rules
Let us consider an arbitrary plane inside an infinitesimal element. Let this plane be inclined at an angle 
to the vertical face of the element. A free body diagram of the region to the left of this plane is shown in the figure below.
Stress transformation
We get
Now let us look at a section that is perpendicular to the one we have looked at. This situation is shown in the figure below.
Or,
From equations (2) and (4) we see that the shear stresses are equal. However the normal stresses on the two planes are different as you can see from equations (1) and (3).
Maximum normal stresses
What is the orientation of the infinitesimal element that produces the largest normal stress and the largest shear stress? This information can be useful in predicting where failure will occur.
To find angle at which we get the maximum/minimum normal stress we can take the derivatives of 
and 
with respect to 
and set them to zero. So we have
The angle at which 
is a maximum or a minimum is called a principal angle or 
.
Now, from the identities (or we can think in terms of a right angled triangle with a rise of 
and a run of 
Clearly is a maximum while is a minimum value.
Principal stresses
The normal stresses corresponding to the principal angle 
are called the principal stresses.
We have
These principal stresses are often written as 
and 
or 
and 
where 
The value of the shear stress 
for an angle of 
is
Hence there are no shear stresses in the orientations where the stresses are maximum or minimum.
Maximum shear stresses
Similarly, we can find the value of 
which makes the shear stress a maximum or minimum. Thus
In that case
We can show that this is the maximum value of 
.
Note that, at the value of 
where 
is maximum, the normal stresses are not zero.
Mohr's circle
Mohr's idea was to express these algebraic relations in geometric form so that a physical interpretation of the idea became easier. The idea was based on the recognition that for an orientation equal to the principal angle, the stresses could be represented as the sides of a right-angled triangle.
Recall that
We can represent this in graphical form as shown in the figure below. In general, the locus of all points representing stresses at various orientations lie on a circle which is called Mohr's circle.
Notice that we can directly find the largest normal stress and the small normal stress as well as the maximum shear stress directly from the circle. In three-dimensions there are two more Mohr's circles.
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