Mohr's Circle for Plane Strain

Points on Mohr's circle are plotted using normal strain as the abscissa and half the shearing strain as the ordinate. Two points, X and Y, are plotted with coordinates (∈x, -γXY) and (∈Y, γXY), respectively. The direction of rotation of the sides associated with strain components determines where corresponding points appear on the circle.

The center O of Mohr's circle corresponds to the average normal strain, while the radius is derived from the relationship between normal and shearing strains. These geometric properties allow visualization of how strains transform under different loading conditions and help predict material responses under stress.

Principal strains are the maximum and minimum normal strains a material experiences. They correspond to the intersections of Mohr's circle with the horizontal axis and are calculated as the average strain plus and minus the circle's radius, respectively. These values indicate the strain limits a material can sustain under a given load.

The maximum in-plane shearing strain equals the diameter of Mohr's circle. This relationship provides a direct graphical method to determine the maximum shear strain a material experiences during plane strain deformation, which is essential for assessing material behavior under complex loading conditions.

To find strain components for rotated coordinate axes, rotate the diameter XY of Mohr's circle through an angle 2θ, where θ is the rotation angle of the coordinate axes. This graphical rotation directly yields the transformed strain components at the new orientation.

In homogeneous, isotropic materials undergoing elastic deformation, principal strain axes coincide with stress axes following Hooke's law for shearing stress and strain. This alignment is fundamental to understanding material behavior and helps engineers predict how materials respond to applied loads.

Both methods use graphical representations on Cartesian coordinates to analyze material behavior. Like Mohr's circle for plane stress, the plane strain version plots two points and uses geometric properties to determine principal values and transformed components. Understanding plane stress concepts helps students grasp plane strain analysis.