4.1. Mathematical form

Let us add equations (11) and (12):

(18)

Thus, the sum of radial and hoop stresses turns out to be a constant through the thickness of the tube, however they individually vary through the tube’s thickness.

In this section, we derive the relationship between two stress components: radial stress (σ_rr) and hoop stress (σ_θθ). By adding the equations that describe these stress components, we find that while their individual values change across the thickness of a hollow cylinder, their sum remains constant. This constant behavior highlights the uniformity in the internal response of the cylinder under pressure.

Imagine blowing up a balloon. As the balloon inflates, the pressure inside creates stress on the walls of the balloon. While the pressure might feel the same at different points on the surface, the material can stretch differently in various areas. This analogy illustrates how the sum of the stress on the balloon's surface is consistent, even as each area responds differently.

Let us now solve the equilibrium equation (1) directly in terms of stress components as follows:

(using (18)). (19)

From equation (11), we know that σ_rr is a function of r alone.

Here, we transition to using our derived constant from the previous chunk to express the equilibrium of forces in terms of stress components. Specifically, since we have established that σ_rr depends solely on the radial distance (r), we can simplify our calculations by treating σ_rr as a function of r only. This makes the equilibrium calculations more straightforward by reducing the number of variables we need to manage.

Think of this like measuring how the weight of water in a tub affects the pressure at different points on the tub's bottom. If the tub is uniformly filled, we can predict the pressure using just the depth (distance from the surface) without worrying about other aspects like the shape of the tub, because the pressure increases uniformly as you go deeper.

Thus, we get that is also a constant, say C. Further, from the definition of strain components, we know that and . Thus, we get:

(16)

Integrating (15) twice, we finally get:

(17)

In this part, we find that the quantity we derived behaves like a constant under the defined conditions. By exploring the strain components related to these stress components through integration, we can establish a mathematical expression that describes the behavior of the hollow cylinder when subjected to internal pressures. This leads to a more comprehensive understanding of how the material deforms under loads.

It's similar to stretching a rubber band: if you know how much you stretched (strain), you can figure out how much force you applied (stress). Here, we analyze the relationship mathematically, representing how consistent stretching over a certain area leads to predictable material behavior.

Whenever we solve a differential equation, we get unknown integrating constants. To obtain those constants, one has to apply boundary conditions. Similarly, we need to identify the boundary conditions for our deformation problem.

Boundary conditions are essential constraints applied to the differential equations we generate from our models. They define the behavior of our system under specific conditions, such as forces or constraints applied at the edges of the structure. In this hollow cylinder example, the internal pressure and lack of external pressure at the outer surface guide how we calculate the unknown constants that originate from integrating our equations.

Consider a race track where cars can only drive in from certain points (boundary conditions). The way cars behave as they enter and move around the track can be predicted based on where the track is laid out, just as knowing the pressures on our cylinder helps us predict how it behaves under stress.

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