23.8: Mohr's Circle for Plane Strain
Mohr's circle is a crucial graphical method used to analyze plane strain by plotting strain on a set of cartesian coordinates, where the abscissa is normal strain ∈ and the ordinate is shear strain γ. Similarly to Mohr’s circle for plane stress, two points X and Y are plotted. Their coordinates are (∈x, -γXY) and (∈Y, γXY), respectively.
Mohr's circle visually represents the strain states under various conditions, which is essential for understanding material behavior. The center of Mohr's circle, labeled O, corresponds to the average normal strain, with the circle's radius derived from the relationship between normal and shearing strains. This helps to visualize how strains transform under different loading conditions by depicting the rotations and shifts in the circle as the coordinate axes rotate.
The points where Mohr's circle intersects the horizontal axis are particularly significant, representing the maximum and minimum principal strains. These principal strains are calculated from the average strain plus and minus the circle's radius, respectively, and indicate the strain limits a material can sustain under a given load. In homogeneous, isotropic materials undergoing elastic deformation, the principal strain axes align with the stress axes, a correlation established by Hooke's law for shearing stress and strain. This alignment aids in predicting material responses under stress.
In addition, the diameter of Mohr's circle represents the maximum in-plane shearing strain. For analysis involving rotated coordinate axes, rotating the diameter XY of Mohr’s circle through an angle 2θ effectively determines the strain components at that orientation for coordinate axes rotated through the angle θ.
