The Mohr circle is a technique used in engineering and geology to graphically represent a symmetrical tensor (2x2 or 3x3) and calculate moments of inertia, deformations and stresses, adapting them to the characteristics of a circle (radius, center, etc.). Calculation of the absolute maximum shear stress and the absolute maximum strain is also possible.

This method was developed around 1882 by the German civil engineer Christian Otto Mohr (1835-1918).

Two-dimensional case

In two dimensions, the Mohr circle allows the maximum and minimum compression to be determined from two measurements of the normal and tangential stress over two angles that form 45°:.

Using rectangular axes, where the horizontal axis represents the normal stress and the vertical axis represents the shear or tangential stress for each of the previous planes. The circumference values ​​are represented as follows:

The maximum and minimum tensions are given in terms of those magnitudes simply by:.

These values ​​can also be obtained by calculating the own values ​​of the tension tensor, which in this case is given by:.

Spherical tensioners and diverters

Spherical Tensioner.

They give rise to changes in volume but never in shape, that is, their physical meaning is that of forces from different directions converging towards the same point, such as the pressure exerted by water on an object located in the depths, the pressure causes the system to collapse inwards.

Deviator Tensioner They give rise to changes in shape but not volume. In some types of plasticity the yield surface is calculated from the deviatoric tensor; not the complete tensioner.

For plane and quasi-plane solids, the same Mohr circle technique that was used for two-dimensional stresses can be applied. In many cases it is necessary to calculate the moment of inertia around an axis that is inclined; the Mohr circle can be used to obtain this value. It is also possible to obtain the main moments of inertia. In this case, the formulas for calculating the average moment of inertia and the radius of the Mohr circle for moments of inertia are analogous to those for calculating stresses: