# Essay, Research Paper: Strain Transformation

## Engineering

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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.

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.

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