Strain Transformation

Global 1D Strain

Consider a rod with initial length L which is stretched to a length L'. The strain measure e, a dimensionless ratio, is defined as the ratio of elongation with respect to the original length,

Infinitesimal 1D Strain

The above strain measure is defined in a global sense. The strain at each point may vary dramatically if the bar's elastic modulus or cross-sectional area changes. To track down the strain at each point, further refinement in the definition is needed. Consider an arbitrary point in the bar P, which has a position vectorx, and its infinitesimal neighbor dx. Point P shifts to P', which has a position vector x', after the stretch. In the meantime, the small "step" dx is stretched to dx'. The strain at point p can be defined the same as in the global strain measure, Since the displacement , the strain can hence be rewritten as,

General Definition of 3D Strain

As in the one dimensional strain derivation, suppose that point P in a body shifts to point P after deformation. The infinitesimal strain-displacement relationships can be summarized as, where u is the displacement vector, x is coordinate, and the two indices i and j can range over the three coordinates {1, 2, 3} in three dimensional space. Expanding the above equation for each coordinate direction gives, where u, v, and w are the displacements in the x, y, and z directions respectively (i.e. they are the components of u).

3D Strain Matrix

There are a total of 6 strain measures. These 6 measures can be organized into a matrix (similar in form to the 3D stress matrix), shown here,

Engineering Shear Strain

Focus on the strain exy for a moment. The expression inside the parentheses can be rewritten as, where . Called the engineering shear strain, gxy is a total measure of shear strain in the x-y plane. In constrast, the shear strain exy is the average of the shear strain on the x face along they direction, and on the y face along the x direction. Engineering shear strain is commonly used in engineering reference books. However, please beware of the difference between shear strain and engineering shear strain, so as to avoid errors in mathematical manipulations.

Compatibility Conditions

In the strain-displacement relationships, there are six strain measures but only three independent displacements. That is, there are 6 unknowns for only 3 independent variables. As a result there exist 3 constraint, or compatibility, equations. These compatibility conditions for infinitesimal strain refered to rectangular Cartesian coordinates are, In two dimensional problems (e.g. plane strain), all z terms are set to zero. The compatibility equations reduce to, Note that some references use engineering shear strain () when referencing compatibility equations. Plane State of Strain Some common engineering problems such as a dam subjected to water loading, a tunnel under external pressure, a pipe under internal pressure, and a cylindrical roller bearing compressed by force in a diametral plane, have significant strain only in a plane; that is, the strain in one direction is much less than the strain in the two other orthogonal directions. If small enough, the smallest strain can be ignored and the part is said to experience plane strain. Assume that the negligible strain is oriented in the z-direction. To reduce the 3D strain matrix to the 2D plane stress matrix, remove all components with z subscripts to get, where exy = eyx by definition. The sign convention here is consistent with the sign convention used in plane stress analysis. Coordinate Transformation The transformation of strains with respect to the {x,y,z} coordinates to the strains with respect to {x',y',z'} is performed via the equations, The rotation between the two coordinate sets is shown here, where q is defined positive in the counterclockwise direction. Principal Directions, Principal Strain The normal strains (ex' and ey') and the shear strain (ex'y') vary smoothly with respect to the rotation angle q, in accordance with the transformation equations given above. There exist a couple of particular angles where the strains take on special values. First, there exists an angle qp where the shear strain ex'y' vanishes. That angle is given by, This angle defines the principal directions. The associated principal strains are given by, The transformation to the principal directions with their principal strains can be illustrated as: Maximum Shear Strain Direction Another important angle, qs, is where the maximum shear strain occurs and is given by, The maximum shear strain is found to be one-half the difference between the two principal strains, The transformation to the maximum shear strain direction can be illustrated as: Mohr's Circle Strains at a point in the body can be illustrated by Mohr's Circle. The idea and procedures are exactly the same as for Mohr's Circle for plane stress. The two principal strains are shown in red, and the maximum shear strain is shown in orange. Recall that the normal strains are equal to the principal strains when the element is aligned with the principal directions, and the shear strain is equal to the maximum shear strain when the element is rotated 45° away from the principal directions. As the element is rotated away from the principal (or maximum strain) directions, the normal and shear strain components will always lie on Mohr's Circle. Derivation of Mohr's Circle To establish the Mohr's circle, we first recall the strain transformation formulas for plane strain, Using a basic trigonometric relation (cos22q + sin22q = 1) to combine the above two formulas we have, This equation is an equation for a circle. To make this more apparent, we can rewrite it as, where,       The circle is centered at the average strain value eAvg, and has a radius R equal to the maximum shear strain, as shown in the figure below,