2.1. Reducing the cuboidal representation of state of stress to a square

There is another simpler way to draw such a state of stress for whom there are no shear components in the third direction: we can just draw a square instead of a cube where the sides of the square denote the faces of the cuboid as shown in Figure 2.

In this chunk, we learn that when there are no shear components in the third direction, we can simplify the representation of stress from a cuboid (a three-dimensional shape) to a square (a two-dimensional shape). This means we can represent the stresses acting on the surfaces of the cuboid using a square diagram, making the visualization simpler. The sides of the square correspond to the faces of the cuboid, which makes it easier to interpret the stress state.

You can think of this simplification like how we sometimes draw simple 2D shapes to represent more complex 3D objects in sketches or diagrams. For instance, when drawing a box, we often use a square to represent the top face of the box rather than trying to illustrate all sides in three dimensions, which can be confusing.

The right edge of the square represents the (+e1) face. Similarly, the top edge represents the (+e2) face and the plane of the square itself represents the e3 face. So, the plane containing the square has just one traction component σ .

Here, we delve into what each part of the square representation means. Each edge of the square corresponds to different faces of the original cuboid. The right edge represents one principal stress direction, the top edge another, and the face of the square corresponds to the third direction. The notation σ signifies that we have a normal stress acting out of the plane where the square is located.

Imagine a piece of paper that is held flat in front of you. You can think of the surface of the paper as representing the stress on a face of a cube. Each edge of the paper could represent the forces acting on that face, where you can easily visualize how pressure is applied to the paper's surface.

We use a dot enclosed by a circle to denote the traction component coming out of the plane as shown in Figure 2. On the edges of the square, we have both the shear and normal components of traction.

In this part, we understand how traction components are represented within the square. A dot inside a circle indicates the normal traction that is acting out from the surface of the square. This is an important notation since it helps in visualizing how these forces interact. Additionally, shear components are represented on the edges of the square, indicating where forces act parallel to the surface, stressing the importance of distinguishing between different types of stress.

Consider a sticker being pressed onto a surface. The pressure pushing directly out from the center of the sticker is like the normal traction, while any sliding or movement of the sticker over the surface represents shear traction.

Keep in mind that such a reduce to square to represent stress matrix is possible only when the third coordinate axis lies along one of the principal directions.

This crucial point emphasizes the condition under which we can apply this simplification. The third coordinate must align with one of the principal stress directions for this representation to be valid. If it is not, the stress state cannot be accurately depicted as a simple square, leading to potential misunderstanding or errors in analysis.

You can think of this as following a rule in sports. For instance, in football, you can only pass the ball in certain ways depending on how players are positioned. If they are not positioned correctly, the pass might go to the wrong player or not work at all. Similarly, if the third coordinate is not aligned correctly, the square representation will not adequately describe the stress state.

--