__STRAIN ELLIPSE__

The equation of a circle with unit radius, and its centre at the origin is given by:

x^{2 }+ y^{2 }= 1

If we deform this circle according to the general two dimensional linear transformation, the eq below can be derived:

[ (dx'-by')/(ad-bc)]^{2} + [(-cx'+ay')/(ad-bc)]^{2}
= 1

x'^{2} (d^{2}+c^{2})
2x'y' (bd+ac) y'^{2}
(a^{2}+b^{2})

-------------- - ------------- + --------------- =1

(ad-bc)^{2}
(ad-bc)^{2}
(ad-bc)^{2}

Simplifying the constant terms,

Ax'^{2} -2B
x'y' +Cy'^{2} =1

^{ }^{ Which is
an ellipse, the }^{strain
ellipse}^{ or }^{finite
strain ellipse}^{ with its centre at the origin. It can be
shown that the principal strains are given by the quadratic equation which has
positive roots: }

^{ }^{ }_{(1+e1)}^{2}_{
or}^{ }_{(1+e2)}^{2}_{=[(a}^{2}_{+b}^{2}_{+c}^{2}_{+d}^{2}_{)/2]}^{ }_{ ±}^{
}_{{ [(a}^{2}_{+b}^{2}_{+c}^{2}_{+d}^{2}_{)}^{2}_{
- 4(ad-bc)2]}^{1/2}_{}/2}

^{and that they make angle q
' with x axis where q ' is given by:}

^{ }_{ tan 2 q
'=[2(ac+bd)]/(a}^{2}_{+b}^{2}_{-c}^{2}_{-d}^{2}_{)}

Just as the sides of the original square grid were rotated
during two dimensional homogeneous strain, so is the line which becomes the
major axis and the line that becomes the minor axis of the strain ellipse. These
lines known as the **PRINCIPAL STRAINS** or **PRINCIPAL AXES OF STRAIN **can
be shown to have made an angle before the deformation
with the x axis, where is given by:

tan 2 q = [2(ab+cd)]/[a2-b2+c2-d2]

The ratio of the principal strains is called the **ELLIPTICITY**
of the finite strain ellipse.

The principal axes of strain are therefore rotated through
an angle ( q ' - q
) during the two dimensional homogeneous strain to become the Principal Strains.
This angle designated by w
(omega) is called the **rotational
component** of strain:

w = q ' - q

The two dimensional strain is **IRROTATIONAL** only when b and c of
the constants are equal and the matrix becomes symmetric, since q
' and q are the
equal.

| a b |

| b d |