California State University, Northridge

College of Computer Science and Engineering

Department of Civil and Manufacturing Engineering

STRAIN TRANSFORMATION

Submitted to: Nazaret Dermendjian

Submitted for: Applied Mechanics 340

Date requested: April 6, 1999

Date due: April 27, 1999

Date submitted: April 27, 1999

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, ?x, ?y, ?z, 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 ?x and ?y 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, ?x and ?y 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 ?x and ?y 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:

?x’ = (?x + ?y)/2 + (?x – ?y)cos 2q + gxy sin 2q (1)

gx’y’ = [(?x – ?y)/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, ?1 and ?2 . 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 /(?x – ?y) (3)

?1,2 = (?x -?y)/2 ? {[(?x -?y)/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 = -(?x – ?y) / 2 (5)

When the shear strain is maximum, the normal strains are equal to the average normal strain. These values are determined from the following equations:

gmax / 2 = {[(?x – ?y) / 2]2 + (gxy / 2)2}1/2 (6)

?avg = (?x + ?y) / 2 (7)

We can also solve strain transformation problem using Mohr’s circle. The coordinate system used has the abscissa represent the normal strain ?, 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 ? axis at a distance of ?avg from the origin. Please note that it is important to follow the sign convention established previously. Plot a reference point A having coordinates (?x , 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 ?1 and ?2 are the values where the circle intersects the ? 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 ? 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,?x, ?y, gxy, 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, sx, sy, sz,(not covered in this report) associated normal strains ?x, ?y, ?z, are developed in the material. Using principals of superposition, Poisson’s ratio, ?lat = -n?long , and Hooke’s law, as it applies in the uniaxial direction ? = s/E , 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.