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.