# Mohr's Circle for Plane Strain

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Mechanical Engineering
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Mohr's Circle for Plane Strain

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Mohr's circle analyzes plane strain by plotting points with the abscissa equal to the normal strain and the ordinate equal to half the shearing strain.

The center O of Mohr's circle is defined, with the abscissa of O and the radius equal to the average strain and the radius equation, respectively.

The direction of rotation of the sides associated with the strain components indicates where the corresponding points on Mohr's circle are plotted.

The intersections of Mohr's circle with the horizontal axis correspond to the maximum and minimum principal strains, calculated as the sum and difference of the average strain and the radius, respectively.

During elastic deformation in homogeneous, isotropic materials, the principal strain axes coincide with the stress axes following the application of Hooke's law for shearing stress and strain.

The maximum in-plane shearing strain equals the diameter of Mohr's circle. The components of strain corresponding to a rotation of the coordinate axes through an angle θ are obtained by rotating the diameter of Mohr's circle through an angle 2θ.

## 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.