Transformation of Strains

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

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