2.2. Taylor’s Series Expansion

As the stress components are varying in the domain of integration, we can use Taylor’s expansion about the center of the cylindrical element, i.e., at (r,θ,z).

In this chunk, we are discussing the context in which Taylor's series is applied. When evaluating the stress components in a cylindrical element, these stresses vary across the element. To simplify the calculations, we can employ Taylor's series, which allows us to approximate complex functions using polynomials. Specifically, we expand the stress components around a certain point—here, the center of the cylindrical element—characterized by coordinates (r, θ, z). The expansion helps in managing the variations in stress across the element, providing a way to analyze these variations mathematically.

Think of a curved surface, like a hill. If you want to analyze changes in elevation at various points on the hill, you can start at a specific point (the center) and use a simple mathematical model (like a straight line) to represent the slope around that point. Similarly, the Taylor series expansion allows us to create a simple model that represents the changes in stress at the center of the cylindrical element, making calculations more manageable.

In equation (3), e.g., we need to Taylor expand both the stress components as well as basis vectors. Ignoring higher order terms, we get...

This chunk carries forward the idea of applying the Taylor series. Here, we not only expand the stress components but also the basis vectors associated with the cylindrical coordinates. The Taylor expansion involves disregarding the higher-order terms (those that are very small compared to the first few terms) to simplify our calculations. This means we will focus on the primary terms that will have the most significant effect on the results, which allows us to capture the essential behavior of the system without getting bogged down in complicated calculations.

Imagine you are trying to predict the temperature changes in a room throughout the day. Instead of looking at every slight fluctuation, you might take an average and only focus on the most significant changes—like the times when the sun shines through the window or when the air conditioning kicks in. By simplifying the situation like this, you gain insights without overwhelming details. The same principle applies in engineering when we simplify stress calculations.

As e and e change only with θ, their partial derivatives with respect to r and z will be zero...

In this chunk, we delve into the nature of how the basis vectors e_r and e_θ change. The statement highlights that their variation is only dependent on the angular coordinate θ, which allows us to state that their partial derivatives with respect to the other coordinates (r and z) are zero. This simplification is critical as it greatly reduces the complexity when we want to differentiate these vectors in our equations, focusing solely on their dependency on the angle.

Consider a spinning top, where the position of the top changes along a circular path based on its angle. However, the distance from the center (like the radius) doesn’t affect its spin speed—the spin depends entirely on the angle. Just like that, here the basis vectors' adjustment is reliant only on the angle θ, simplifying our understanding and analysis in the calculations.

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