The following report will be on Strain
Transformation. Strain transformation is

similar to stress transformation, so
that many of the techniques and derivations

used for stress can be used for
strain. We will also discuss methods of

measuring strain and
material-property relationships. The general state of

strain at a point can
be represented by the three components of normal strain,

Îx, Îy, Îz, and
three components of shear strain, gxy, gxz, gyz. For the

purpose of this
report, we confine our study to plane strain. That is, we will

only
concentrate on strain in the x-y plane so that the normal strain
is

represented by Îx and Îy and the shear strain by gxy . The deformation on
an

element caused by each of the elements is shown graphically below.
Before

equations for strain-transformation can be developed, a sign
convention must be

established. As seen below, Îx and Îy are positive if they
cause elongation in

the the x and y axes and the shear strain is positive if
the interior angle

becomes smaller than 90°. For relative axes, the angle
between the x and x'

axes, q, will be counterclockwise positive. If the
normal strains Îx and Îy

and the shear strain gxy are known, we can find the
normal strain and shear

strain at any rotated axes x' and y' where the angle
between the x axis and x'

axis is q. Using geometry and trigonometric
identities the following equations

can be derived for finding the strain at a
rotated axes: Îx' = (Îx + Îy)/2 +

(Îx - Îy)cos 2q + gxy sin 2q (1) gx'y' =
[(Îx - Îy)/2] sin 2q + (gxy /2) cos

2q (2) The normal strain in the y'
direction by substituting (q + 90°) for q in

Eq.1. The orientation of an
element can be determined such that the element's

deformation at a point can
be represented by normal strain with no shear strain.

These normal strain
are referred to as the principal strains, Î1 and Î2 . The

angle between the x
and y axes and the principal axes at which these strains

occur is represented
as qp. The equations for these values can be derived from

Eq.1 and are as
followed: tan 2qp = gxy /(Îx - Îy) (3) Î1,2 = (Îx -Îy)/2 ±

{[(Îx -Îy)/2]2+
(gxy/2)2 }1/2 (4) The axes along which maximum in-plane shear

strain occurs
are 45° away from those that define the principal strains and is

represented
as qs and can be found using the following equation: tan 2qs = -(Îx

- Îy) / 2
(5) When the shear strain is maximum, the normal strains are equal to

the
average normal strain. We can also solve strain transformation problem
using

Mohr's circle. The coordinate system used has the abscissa
represent the normal

strain Î, with positive to the right and the ordinate
represents half of the

shear strain g/2 with positive downward. Determine the
center of the circle C,

which is on the Î axis at a distance of Îavg from the
origin. Please note that

it is important to follow the sign convention
established previously. Plot a

reference point A having coordinates (Îx , gxy
/ 2). The line AC is the

reference for q = 0. Draw a circle with C as the
center and the line AC as the

radius. The principal strains Î1 and Î2 are the
values where the circle

intersects the Î axis and are shown as points B and D
on the figure below. The

principal angles can be determined from the graph by
measuring 2qp1 and 2qp2

from the reference line AC to the Î axis. The element
will be elongated in the

x' and y' directions as shown below. The average
normal strain and the maximum

shear strain are shown as points E and F on the
figure below. The element will

be elongated as shown. To measure the normal
strain in a tension-test specimen,

an electrical-resistance strain gauge can
be used. An electrical-resistance

strain gauge works by measuring the change
in resistance in a wire or piece of

foil and relates that to change in length
of the gauge. Since these gauges only

work in one direction, normal strains
at a point are often determined using a

cluster of gauges arranged in a
specific pattern, referred to as a strain

rosette. Using the readings on the
three gauges, the data can be used to

determine the state of strain, at that
point using geometry and trigonometric

identities. It is important to note
that the strain rosettes do not measure

strain that is normal to the free
surface of the specimen. Mohr's circle can

then be used to solve for any in
plane normal and shear strain of interest. It

is important to mention briefly
material-property relation ships. Note that it

is assumed that the material
is homogeneous, isotropic, and behaves in a linear

elastic manner. If the
material is subject to a state of triaxial stress, (not

covered in this
report) associated normal strains are developed in the material.

Using
principals of superposition, Poisson's ratio, and Hooke's law, as it

applies
in the uniaxial direction, the normal stress can be related to the

normal
strain. Similar relationships can be developed between shear stress and

shear
strain. This report was a brief summary of strain transformation and
the

related topics of strain gauges and material-property relationships. It
is

important to realize that this report was confined to in plane
strain

transformation and that a more complete study would involve shear
strain in

three dimensions, then material-property relationships could be
developed

further. Also, theories of failure were not covered in this
report.