Mohr's circle

Mohr's circle is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor.

His work inspired fellow German engineer Christian Otto Mohr (the circle's namesake), who extended it to both two- and three-dimensional stresses and developed a failure criterion based on the stress circle.

, i.e., the stresses acting on a plane with a different orientation passing through that point of interest —forming an angle with the coordinate system

From the balance of angular momentum, the symmetry of the Cauchy stress tensor can be demonstrated.

The previous derivation for the equation of the Mohr Circle using Figure 4 follows the engineering mechanics sign convention.

is the shear stress acting on the face with normal vector in the positive direction of the

In the physical-space sign convention, positive normal stresses are outward to the plane of action (tension), and negative normal stresses are inward to the plane of action (compression) (Figure 5).

In the Mohr-circle-space sign convention, positive shear stresses rotate the material element in the counterclockwise direction, and negative shear stresses rotate the material in the clockwise direction.

Two options exist for drawing the Mohr-circle space, which produce a mathematically correct Mohr circle: Plotting positive shear stresses upward makes the angle

on the Mohr circle have a positive rotation clockwise, which is opposite to the physical space convention.

That is why some authors[3] prefer plotting positive shear stresses downward, which makes the angle

on the Mohr circle have a positive rotation counterclockwise, similar to the physical space convention for shear stresses.

To overcome the "issue" of having the shear stress axis downward in the Mohr-circle space, there is an alternative sign convention where positive shear stresses are assumed to rotate the material element in the clockwise direction and negative shear stresses are assumed to rotate the material element in the counterclockwise direction (Figure 5, option 3).

This way, positive shear stresses are plotted upward in the Mohr-circle space and the angle

in the object under study, as shown in Figure 4, the following are the steps to construct the Mohr circle for the state of stresses at

As expected, the ordinates of these two points are zero, corresponding to the magnitude of the shear stress components on the principal planes.

Thus, the magnitude of the maximum and minimum shear stresses are equal to the value of the circle's radius

(Figure 4) is half the angle between two lines joining their corresponding stress points

This double angle relation comes from the fact that the parametric equations for the Mohr circle are a function of

The second approach involves the determination of a point on the Mohr circle called the pole or the origin of planes.

Once the pole has been determined, to find the state of stress on a plane making an angle

The normal and shear stresses on that plane are then the coordinates of the point of intersection between the line and the Mohr circle.

The orientation of the planes where the maximum and minimum principal stresses act, also known as principal planes, can be determined by measuring in the Mohr circle the angles ∠BOC and ∠BOE, respectively, and taking half of each of those angles.

Solution: Following the engineering mechanics sign convention for the physical space (Figure 5), the stress components for the material element in this example are: Following the steps for drawing the Mohr circle for this particular state of stress, we first draw a Cartesian coordinate system

These points follow the engineering mechanics sign convention for the Mohr-circle space (Figure 5), which assumes positive normals stresses outward from the material element, and positive shear stresses on each plane rotating the material element clockwise.

Knowing both the location of the centre and length of the diameter, we are able to plot the Mohr circle for this particular state of stress.

-axis are the magnitudes of the minimum and maximum normal stresses, respectively; the ordinates of both points E and C are the magnitudes of the shear stresses acting on both the minor and major principal planes, respectively, which is zero for principal planes.

To confirm the location of the Pole, we could draw a line through point B on the Mohr circle parallel to the plane B where

For instance, the line from the Pole to point C in the circle has the same inclination as the plane in the physical space where

In the same way, lines are traced from the Pole to points E, D, F, G and H to find the stress components on planes with the same orientation.

Figure 1. Mohr's circles for a three-dimensional state of stress
Figure 2. Stress in a loaded deformable material body assumed as a continuum.
Figure 3. Stress transformation at a point in a continuum under plane stress conditions.
Figure 4. Stress components at a plane passing through a point in a continuum under plane stress conditions.
Figure 5. Engineering mechanics sign convention for drawing the Mohr circle. This article follows sign-convention # 3, as shown.
Figure 6. Mohr's circle for plane stress and plane strain conditions (double angle approach) . After a stress analysis, the stress components , , and at a material point are known. These stress components act on two perpendicular planes and passing through . The coordinates of point and on the Mohr circle are the stress components acting on the planes and of the material element, respectively. The Mohr circle is then used to find the stress components and , i.e., coordinates of any stress point on the circle, acting on any other plane passing through . The angle between the lines and is double the angle between the normal vectors of planes and passing through .
Stress components on a 2D rotating element . Click to see animation . Example of how stress components vary on the faces (edges) of a rectangular element as the angle of its orientation is varied. Principal stresses occur when the shear stresses simultaneously disappear from all faces. The orientation at which this occurs gives the principal directions . In this example, when the rectangle is horizontal, the stresses are given by The corresponding Mohr's circle representation is shown at the bottom.
Figure 7. Mohr's circle for plane stress and plane strain conditions (Pole approach). Any straight line drawn from the pole will intersect the Mohr circle at a point that represents the state of stress on a plane inclined at the same orientation (parallel) in space as that line.
Figure 8
Figure 9
Figure 10. Mohr's circle for a three-dimensional state of stress