2. A simpler approach to find the traction force

Let us consider the cylindrical element shown in Figure 2. Earlier, we had done the derivation by also considering the variation of traction at different points on +z and −z planes. If we assume the traction to be constant everywhere on the plane with its value being equal to the value at the center of the plane, we can directly multiply it with the area of the plane to get the total force. We illustrate it now.

In this part, we are introducing a simplified method to calculate the traction force on a cylindrical element. The critical assumption here is that we consider the traction force to be consistent across the entire +z and −z planes. Instead of calculating the variation in traction at different locations across these planes, we treat it as constant and simply take the value at the center of those planes. By multiplying this constant traction value by the area of the plane, we can more easily find the total force acting on it.

Think of a flat sponge being pressed down. If you apply the same pressure everywhere on the sponge's surface, you can easily calculate the total force by just multiplying the pressure (traction) by the sponge's area. However, if you pressed harder in some areas than others, it would complicate things significantly.

As the center of the cylindrical element has coordinates (r, θ, z), the center of the +z plane will be at (r, θ, z + ∆z). Thus, the traction at the center of +z plane will be.

Here, we define the coordinates for our cylindrical element to specify where we are measuring the traction force. The center of the cylindrical element is marked by three coordinates: r (the radial direction), θ (the angular coordinate), and z (the height). When we focus on the +z plane, we note that its center is located slightly above the cylindrical element (up by ∆z). This localization helps in calculating the specific traction value that we will use in our total force calculations.

Imagine standing at the center of a tall cylindrical tower. Your height (z) marks your position in vertical space, while your distance from the tower's central column (r) determines how far you are from it. Your direction facing can be described by the angle (θ), similar to how we identify points on the cylindrical element.

The three basis vectors in the above expression, when evaluated at the center of +z plane will be the same as the one at the center of cylindrical element because the two points have the same θ coordinate.

When we evaluate the basis vectors needed for our calculations, we find that the vectors representing direction at the center of the +z plane remain unchanged from those at the center of the cylindrical element. This consistency occurs because both points share the same angular coordinate (θ). Hence, it allows simplifications in the calculations without any need to adjust for the basis vectors across the two points.

Picture a train that runs on a circular track. No matter how far along the track you go, the basic direction and layout (like the track layout) do not change. So, while you're on different positions (like at +z plane or at the center), the core mechanism of your movement remains the same and allows for straightforward calculations.

Thus, the total force on the two z planes will be obtained after assuming that traction does not vary over different points in a face of the cylindrical element.

By making the assumption that the traction remains constant across the +z and -z planes, we can calculate the total force acting on both planes much more straightforwardly. This simplification helps us bypass complications that would have arisen from varying traction, allowing us to focus solely on the area and the constant value of traction we identified earlier.

Think of a tablecloth being pulled down by a weight. If the pull (traction) is consistent all over, you can easily calculate how much the cloth will feel that weight by multiplying the pull by the area it covers. However, if some corners were being pulled harder than others, you'd have to consider all those differences, complicating your calculations.

Now, we do the Taylor’s expansion of the components of traction about the center of the cylindrical element. As only z coordinate has changed, the Taylor’s expansion will only have terms corresponding to derivatives of z.

In this chunk, we introduce Taylor's expansion to refine our traction components further. This mathematical approach allows us to approximate the behavior of our traction components, focusing solely on how they vary with changes in the z-coordinate. By doing so, we can carefully incorporate the effects of small changes without overcomplicating the analysis.

Consider how you might estimate the height of a tree branch as you climb higher. Initially, you can look at it directly, but as you go up step-by-step, you can use basic math to estimate how much higher you are, without needing to measure everything from the ground each time—this helps keep the process manageable.

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