You can know about the theory of Mohr's circles from any text books of Mechanics of Materials. The following two are good references, for examples.

1. Ferdinand P. Beer and E. Russell Johnson, Jr, "Mechanics of Materials",
Second Edition, McGraw-Hill, Inc, 1992.

2 . James M. Gere and Stephen P. Timoshenko, "Mechanics of Materials",
Third Edition, PWS-KENT Publishing Company, Boston, 1990.

The 2-D stresses, so
called plane stress problem, are usually given by the three stress components
s_{x
}, s_{y
}, and t_{xy}
, which consist in a two-by-two
symmetric matrix (stress tensor):

(1)

What people usually are
interested in more are the two prinicipal stresses sThen, we have the following relationship:

(2)

and
(3)

Equivalently, the above
two equations can be rewritten as follows:
(4)

and
(5)

The expression for the normal stress s(6)

From the relations for s(7)

(8)

This circle is with radius (9)

which corresponds to the
point at which q = 0 and
the point
(10)

which corresponds to the
point at which q = q(11)

and the point
(12)

which corresponds to the
point at which q = 90(13)

which corresponds to the
point at which q = q(14)

Besides, finally one can also read the maxium shear stress
as
(15)

which corresponds to the
apex of the Mohr's circle at which q = q(16)

What people usually are
interested in more are the three prinicipal stresses sImagine that there
is a plane cut through the cube in Fig. 3 , and the unit normal vector
**n**
of the cut plane has the direction cosines v_{x
}, v_{y }, and v_{z
}, that is

(17)

then the normal stress
on this plane can be represented by
(18)

There exist three sets
of direction cosines, (19)

The three eigenvalues
of Eqn (19) are the roots of the following characteristic polynomial
equation:
(20)

where
(21)

(22)

(23)

In fact, the coefficients
A, B, and C in Eqn (20) are invariants as long as the stress state is prescribed(see
e.g. Ref. 2) . Therefore, if the three roots
of Eqn (20) are (24)

(25)

(26)

Numerically, one can always find one of the three roots of
Eqn (20) , e.g. (27)

(28)

(29)

Similar to Fig. 3, one can imagine a cube with their
faces normal to **n**_{1},
**n**_{2},
or **n**_{3
}. For example, one can do so in Fig. 3 by replacing
the axes **X**,**Y**, and **Z** with
**n**_{1},
**n**_{2},
and **n**_{3
}, respectively, replacing the normal stresses
s_{x}
, s_{y}
, and s_{z}
with the principal stresses s_{1
}, s_{2
}, and s_{3
}, respectively,
and removing the shear stresses t_{xy}
, t_{yz}
, and t_{zx}
.

Now, pay attention the new cube with axes **n**_{1},
**n**_{2},
and **n**_{3
}. Let the cube be rotated about the axis **n**_{3
}, then the corresponding transformation of stress may
be analyzed by means of Mohr's circle as if it were a transformation of
plane stress. Indeed, the shear stresses excerted on the faces normal to
the **n**_{3
}axis remain equal to zero, and the normal stress s_{3
}is perpendicular to the plane spanned by **n**_{1
}and **n**_{2
}in which the transformation takes place and thus, does
not affect this transformation. One may
therefore use the circle of diameter *AB* to determine the normal
and shear stresses exerted on the faces of the cube as it is rotated about
the **n**_{3
}axis (see Fig. 4). Similarly, the circles of diameter
*BC* and *CA *may be used to determine the stresses on the cube
as it is rotated about the **n**_{1
}and **n**_{2 }
axes, respectively.

What if the rotations are about the axes rather than principal axes? It can be shown that any other transformation of axes would lead to stresses represented in Fig. 4 by a point located within the area which is bounded by the bigest circle with the other two circles removed!

Therefore, one can obtain the maxium/minimum normal and shear stresses from Mohr's circles for 3-D stress as shown in Fig. 4!

Note the notations above (which may be different from other references), one obtains that

(30)

(31)

(32)

Note that in Fig. 4,