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Strain-Transformation equations are based on the geometry of the deformation of deformable bodies(including some small-angle approximations).

External strain , or normal strain, is defined as a ratio of a total elongation to an original length .


Shear strain is defined as a change in angle between two originally perpendicular line segments that intersect at a point.  When , angle the sheared line makes with its original orientation, equals 90 degrees, the shear strain is infinte.










 
Normal and Shear Strain

The extensional strains and are determined by examining the change in length of short, mutually othogonal line segments and ; and the shear strains and are determined by the changes in right angles that originally exist between these lines.
 
 
General Equations
 






 
 
Principal Strains
 

In-Plane Principal Strains:

 
In-Plane Principal Directions:
 
 
Maximum Shear Strains
 
Maximum In-Plane Shear Strains:
 
In-Plane Shear Directions:
 
 
Mohr's circle for two-dimensional Strain

Like the stress-transformation equations, the strain-transformation equations can be simplified bu introducing the double-angle trigonometric identities. This yields
Therefore
 
Equation of a circle in the plane with center at and radius R, with the angle being a parameter is