Solid Mechanics: Strain 
Principal Strains from Mohr's Circle 
A chief benefit of Mohr's circle is that the principal strains e_{1} and e_{2} and the maximum shear strain e_{xyMax} are obtained immediately after drawing the circle,
where,

Principal Directions from Mohr's Circle 
Mohr's Circle can be used to find the directions
of the principal axes. To show this, first suppose that the normal
and shear strains, e_{x},
e_{y}, and e_{xy}, are obtained at a given
point O in the body. They are expressed relative to the
coordinates XY, as shown in the strain element at right
below.
The Mohr's Circle for this general strain state is shown at left above. Note that it's centered at e_{Avg} and has a radius R, and that the two points (e_{x}, e_{xy}) and (e_{y}, e_{xy}) lie on opposites sides of the circle. The line connecting e_{x} and e_{y} will be defined as L_{xy}. The angle between the current axes (X and Y) and the principal axes is defined as q_{p}, and is equal to one half the angle between the line L_{xy} and the eaxis as shown in the schematic below,
A set of six Mohr's Circles representing most strain state possibilities are presented on the examples page. 
Rotation Angle on Mohr's Circle 
Note that the coordinate rotation angle q_{p} is defined positive when
starting at the XY coordinates and proceeding to the
X_{p}Y_{p} coordinates. In contrast, on the
Mohr's Circle q_{p} is
defined positive starting on the principal strain line (i.e. the
eaxis) and proceeding to the XY
strain line (i.e. line L_{xy}). The angle q_{p} has the opposite sense
between the two figures, because on one it starts on the XY
coordinates, and on the other it starts on the principal
coordinates.
Some books avoid the sign difference between q_{p} on Mohr's Circle and q_{p} on the stress element by locating (e_{x}, e_{xy}) instead of (e_{x}, e_{xy}) on Mohr's Circle. This will switch the polarity of q_{p} on the circle. Whatever method you choose, the bottom line is that an opposite sign is needed either in the interpretation or in the plotting to make Mohr's Circle physically meaningful. 
Strain Transform by Mohr's Circle  
Mohr's Circle can be used to transform strains
from one coordinate set to another, similar that that described on
the plane
strain page.
Suppose that the normal and shear strains, e_{x}, e_{y}, and e_{xy}, are obtained at a point O in the body, expressed with respect to the coordinates XY. We wish to find the strains expressed in the new coordinate set X'Y', rotated an angle q from XY, as shown below:
 
To do this we proceed as follows:
 
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