1. Conditions for applying Mohr's Circle

Mohr’s circle is a graphical way to find normal and shear components of traction on arbitrary planes. The only restriction is that the plane normal has to be perpendicular to one of the principal stress directions.

Mohr's Circle is a valuable graphical method used in mechanics to determine the state of stress on various planes. The main condition for its application is that the normal vector of the plane experiencing stress must be perpendicular to at least one of the principal stress directions. This means that in a certain coordinate system, if you align one of the axes along a principal stress direction, then you can analyze stresses on planes that are normal to that direction.

Imagine you are a carpenter trying to understand the stress on a piece of wood. You can only apply Mohr's Circle when you are analyzing stress on axes aligned perfectly with the wood grain or at a right angle to it. Only in these positions can you reliably evaluate how the wood will react under various forces.

Accordingly, let us consider a coordinate system such that the third coordinate axis is along one of the principal stress directions (the rest two coordinate axes need not be along any principal direction).

When analyzing stress using Mohr's Circle, it's important to set up your coordinate system correctly. You choose one of the coordinate axes to align with one of the principal stress directions. This provides a reference point for understanding how other stresses will affect the material. The other two axes can be oriented in any way; however, their orientation can impact how the stress is calculated.

Think of it like setting up a coordinate grid on a map where north is marked by one axis. No matter how you place the east-west axis, you’ll always know how far you are from due north, which helps in navigation or understanding where you stand on the map.

On such planes, we want to find the normal and shear components of traction. The stress matrix in this coordinate system will be such that its column will be formed by traction on the plane along the third coordinate axis. But that being a principal plane, the third column will not have any shear component.

Once the coordinate system is defined, the next step is to analyze the stress components acting on the planes of interest. The identified planes perpendicular to the principal stress contribute to the formation of the stress matrix. Here, since the plane is aligned with the principal direction, the shear component for that specific direction will be zero, simplifying calculations.

Consider pouring gel in a mold that has one side perfectly aligned with the earth's gravitational pull. The gel will rest evenly on that side (normal component), while the side exposed might allow other forces to act unpredictably, making it challenging to figure out shear forces without proper analysis.

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