3. Mathematical form of u

As u and u are functions of z and r respectively, we can rewrite equation (5) as z r (9)

This chunk introduces the idea that the variables u (displacement) are dependent on the coordinates z (longitudinal direction) and r (radial direction). The equation provided (equation 9) describes how to express these displacements in the context of the cylinder's equilibrium. Essentially, it sets the stage for analyzing the distribution of strain within the hollow cylinder based on its geometry.

Imagine stretching a balloon; the amount it stretches depends on how far you are from the center (the radius) and along the length of the balloon. Similarly, in our hollow cylinder, the displacement changes based on the radial and longitudinal positions, which we capture mathematically.

From equation (2), we can infer that σ does not depend on z. Also, σ does not have any term that has zz zz dependence on θ. Thus, it is a function of r only, i.e., (10)

This chunk emphasizes that the stress σ in the context of the hollow cylinder is independent of the longitudinal position (z) and the angular position (θ). It states that σ is solely a function of the radial position (r). This simplification is critical because it allows us to focus on a one-dimensional analysis of stress as it varies only with the distance from the center of the cylinder.

Think of a thick rubber band stretched from both ends. Regardless of where you pinch it, the stretch in the rubber band at any point solely depends on how far you are from the center and not along the length (z) or rotational angle (θ) of the band.

Accordingly, the term dependent on z in the LHS must be a constant or u' must be a constant. As u' denotes longitudinal strain in axial direction, we will denote it by ϵ, an unknown but constant parameter.

This chunk discusses that since stress does not depend on z, the derivative of u with respect to z must be constant. This derivative, which represents axial strain (denoted as ϵ), is introduced as a key concept. It underlines that the strain in the axial direction is uniform throughout the length of the cylinder, which simplifies our calculations and analysis in further sections.

Consider a rubber band being stretched lengthwise. After a certain point, no matter how much more you pull, the amount of stretch remains the same throughout its length. Similarly, the cylinder experiences constant axial strain, making it easier to analyze.

The expressions of σ and σ given in (3) and (4) now become rr θθ (11) (12) Subtracting them, we get (13)

Here, we transform the equations of stress given earlier (equations 3 and 4) into a more usable form to capture the radial and hoop stresses. By subtracting one expression from the other (equation 12 from equation 11), we derive a new relationship that helps in solving equilibrium equations. This step is critical for subsequently understanding how these stress distributions affect the overall behavior of the hollow cylinder under pressure.

Imagine two different rubber bands of various thicknesses being stretched. By examining the differences in their behavior as they are pulled, we can understand how stress varies within each band. This subtraction of stress expressions allows us to isolate how different factors contribute to overall stress.

Integrating (15) twice, we finally get (17) where C and D are the unknown integrating constants.

This chunk discusses the integration of the previously derived equation from the differential equations showing stress. By integrating the relationship, we can find the displacement responses in the radial direction, capturing how the hollow cylinder deforms. The constants C and D are essential as they will help specify particular solutions based on boundary conditions in subsequent discussions.

Think of a sponge being squeezed; the way it deforms continuously as pressure is applied can be likened to the integration of stress across the cylinder. The constants (C and D) will correspond to the sponge's individual characteristics like size and material type affecting how it expands or contracts under pressure.

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