2.1. Traction contribution

Starting with the plane, an arbitrary point on this face will have coordinates as in the (e₁, e₂, e₃) coordinate system. This is because we need to move by from the centroid to first reach the center of the e face.

In this part, we are introducing the concept of traction on a specific plane of the cuboid. The traction is essentially the force experienced at points along this plane, which can vary depending on the position on the plane. The coordinates (e₁, e₂, e₃) represent a three-dimensional space where these forces are being analyzed. The point of interest on the plane is defined relative to the centroid, which is a central reference point for our analysis.

Imagine you're holding a flat, rectangular piece of cheese. The point in the middle of the cheese is like the centroid. When you press down on different parts of the cheese, the pressure (or traction) changes - some areas might feel firmer, while others might yield more easily. The different responses you observe represent the varying traction on the surfaces.

In general, traction will vary from point to point on this face. For the particular point under consideration, traction will be given by: tn = σn.

Here, we define traction as a function of stress at the specific point of interest on the plane. The notation 'tn = σn' indicates that the traction at point 'n' is directly proportional to the stress at that same point. This shows the relationship between the force acting on the plane and the internal stress distribution within the material.

Think of a sponge. If you squeeze the sponge at one point, that point feels different than the other areas that are not being pressed. The pressure you're applying at that point correlates to the internal stresses in that part of the sponge, just as traction relates to stress in materials.

We want to write this in terms of the stress tensor at the centroid of the cuboid using Taylor’s expansion, i.e., neglecting higher order terms to simplify the analysis.

In this section, we are expanding the stress function around the centroid using a mathematical technique called Taylor’s expansion. This allows us to approximate the value of the stress tensor at different points in the cuboid without diving into complicated calculations. By neglecting the higher order terms, we simplify our model while still capturing the essential behavior of the stresses involved.

Consider a cupcake: if you want to analyze the taste at the edge of the cupcake, you might first taste the center since that's where the flavors are most concentrated. By understanding the center's flavor (the centroid), you can make approximations about the flavor towards the edges without having to taste every single bite.

The total force due to traction on e plane will be obtained by integration of traction over the area of this plane.

The total force acting on the e-plane is calculated by integrating the traction values across the area of the face. This means summing up the tiny forces experienced at every point on the surface, which gives us the overall force exerted on that face of the cuboid. The mathematical symbols and relationships help to define how this integration is set up based on the limits that correspond to the surface dimensions.

Think of calculating the total weight of a pizza by adding up the weight of every slice. Each slice contributes a little weight, and by adding all the slice weights together (just like integrating traction), you can find out how heavy the entire pizza is. This approach applies the same idea to find the total force on a surface.

The first two terms within square brackets in the above equation are independent of ξ and η. So, they can come out of the integration and further integration simply gives the total area of e face.

In this segment, we observe that certain terms in our integrals can be factored out because they do not change with position. This simplification helps in performing the integration step more easily. By taking out these constants, we can focus our calculations on the variables that actually vary based on position, simplifying the area computation considerably.

Consider filling a swimming pool with water. The total surface area of the pool remains constant regardless of how deep the water is. When you're calculating how many liters it would take to fill, you're primarily focused on the surface area rather than the depth at each point. Similarly, here we focus on the area while factoring out variations that do not affect the results.

The total traction force (denoted by Ftra) on all six surfaces will be.

This part summarizes that after conducting similar analyses for each face, we arrive at a total traction force acting on all six surfaces of the cuboid. Each surface interacts with forces in a distinct manner, but through our methodical approach, we can express the contributions to the overall force from each direction.

When lifting a box, all the corners and edges of the box can be seen as surfaces. Each corner might experience different forces as you lift the box due to its weight, but when you determine how much effort it takes to lift, you consider the total weight distributed across all corners. This principle is similar to calculating total force on the surfaces.

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