The following report will be on Strain Transformation.
Strain transformation issimilar to stress transformation, so that many of the techniques and derivationsused for stress can be used for strain. We will also discuss methods ofmeasuring strain and material-property relationships. The general state ofstrain at a point can be represented by the three components of normal strain,Ix, Iy, Iz, and three components of shear strain, gxy, gxz, gyz. For thepurpose of this report, we confine our study to plane strain.
That is, we willonly concentrate on strain in the x-y plane so that the normal strain isrepresented by Ix and Iy and the shear strain by gxy . The deformation on anelement caused by each of the elements is shown graphically below. Beforeequations for strain-transformation can be developed, a sign convention must beestablished. As seen below, Ix and Iy are positive if they cause elongation inthe the x and y axes and the shear strain is positive if the interior anglebecomes smaller than 90°. For relative axes, the angle between the x and x'axes, q, will be counterclockwise positive. If the normal strains Ix and Iyand the shear strain gxy are known, we can find the normal strain and shearstrain 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 equationscan be derived for finding the strain at a rotated axes: Ix' = (Ix + Iy)/2 +(Ix - Iy)cos 2q + gxy sin 2q (1) gx'y' = [(Ix - Iy)/2] sin 2q + (gxy /2) cos2q (2) The normal strain in the y' direction by substituting (q + 90°) for q inEq.1. The orientation of an element can be determined such that the element'sdeformation at a point can be represented by normal strain with no shear strain.These normal strain are referred to as the principal strains, I1 and I2 .
Theangle between the x and y axes and the principal axes at which these strainsoccur is represented as qp. The equations for these values can be derived fromEq.1 and are as followed: tan 2qp = gxy /(Ix - Iy) (3) I1,2 = (Ix -Iy)/2 ±{[(Ix -Iy)/2]2+ (gxy/2)2 }1/2 (4) The axes along which maximum in-plane shearstrain occurs are 45° away from those that define the principal strains and isrepresented as qs and can be found using the following equation: tan 2qs = -(Ix- Iy) / 2 (5) When the shear strain is maximum, the normal strains are equal tothe average normal strain. We can also solve strain transformation problem usingMohr's circle. The coordinate system used has the abscissa represent the normalstrain I, with positive to the right and the ordinate represents half of theshear strain g/2 with positive downward.
Determine the center of the circle C,which is on the I axis at a distance of Iavg from the origin. Please note thatit is important to follow the sign convention established previously. Plot areference point A having coordinates (Ix , gxy / 2). The line AC is thereference for q = 0.
Draw a circle with C as the center and the line AC as theradius. The principal strains I1 and I2 are the values where the circleintersects the I axis and are shown as points B and D on the figure below. Theprincipal angles can be determined from the graph by measuring 2qp1 and 2qp2from the reference line AC to the I axis. The element will be elongated in thex' and y' directions as shown below. The average normal strain and the maximumshear strain are shown as points E and F on the figure below.
The element willbe elongated as shown. To measure the normal strain in a tension-test specimen,an electrical-resistance strain gauge can be used. An electrical-resistancestrain gauge works by measuring the change in resistance in a wire or piece offoil and relates that to change in length of the gauge. Since these gauges onlywork in one direction, normal strains at a point are often determined using acluster of gauges arranged in a specific pattern, referred to as a strainrosette. Using the readings on the three gauges, the data can be used todetermine the state of strain, at that point using geometry and trigonometricidentities. It is important to note that the strain rosettes do not measurestrain that is normal to the free surface of the specimen.
Mohr's circle canthen be used to solve for any in plane normal and shear strain of interest. Itis important to mention briefly material-property relation ships. Note that itis assumed that the material is homogeneous, isotropic, and behaves in a linearelastic manner. If the material is subject to a state of triaxial stress, (notcovered in this report) associated normal strains are developed in the material.
Using principals of superposition, Poisson's ratio, and Hooke's law, as itapplies in the uniaxial direction, the normal stress can be related to thenormal strain. Similar relationships can be developed between shear stress andshear strain. This report was a brief summary of strain transformation and therelated topics of strain gauges and material-property relationships. It isimportant to realize that this report was confined to in plane straintransformation and that a more complete study would involve shear strain inthree dimensions, then material-property relationships could be developedfurther. Also, theories of failure were not covered in this report.
