2.1. Force on +z and −z Plane

On +z plane, the traction vector (t+z) is given by

$t_{+z} = \sigma_{zz} e_z + au_{rz} e_r + au_{ heta z} e_{ heta}$ (1)

In this section, we start by discussing the traction vector acting on the +z plane of a cylindrical element. The notation used represents the different components of the traction vector: σ is the normal stress in the z-direction, τ represents shear stresses, and e indicates the unit basis vectors in their respective directions. Basically, this equation describes how forces act on this plane, which is essential for understanding how stress affects the material.

Imagine pressing down on a sponge with your hand. The pressure you apply creates a force (the traction vector) in the vertical direction (upwards or downwards). In our case, the sponge represents the cylindrical element, where your hand applies force in different directions, similar to how stress vectors interact within the element.

The total force due to this traction can be obtained by integrating it over +z plane. The difference between the outer and inner radius of this face is $\Delta r$ and the total angle subtended by the face at the center is $\Delta \theta$. The coordinates of a general point on the top surface is denoted by $(r + \xi, \theta + \eta, z)$ (where r and θ coordinates vary while the z coordinate remains fixed). Thus, the area of the infinitesimal element is:

$dA = d\xi \times (r + \xi) d\eta = (r + \xi) d\xi d\eta$ (2)

Next, we calculate the total force acting on the +z plane by integrating the traction vector over the area of the plane. To do this, we need to specify how the coordinates change across the top surface of the cylinder and define an infinitesimal area (dA). This area is calculated based on the radial and angular changes, represented by dξ and dη, giving us a clear geometric representation of how we consider the area for our force calculations.

Think about a pizza cutter slicing through a pizza. The area where the cutter applies pressure (traction) can be thought of as dA. As you move the cutter (analogous to integrating over the area), you compute the total force exerted on the pizza (total force on +z plane). As the cutter moves, it exposes more area for the pressure to act upon—this is similar to how we compute total force over the cylindrical element.

For −z plane, all the traction components act along negative basis directions but an arbitrary point on −z plane will have coordinates $(r + \xi, \theta + \eta, z - \Delta z)$. Thus, the total force on −z plane will be...

On the −z plane, the situation is similar, but the traction components act in the opposite direction. The coordinates of a point on this plane are adjusted to reflect the change in the z component, indicating we're effectively looking at the forces acting downwards, as the negative basis directions come into play. Although the tensions and coordinates are transformed, they are analyzed similarly to those on the +z plane.

Returning to our sponge analogy, if you now pull the sponge in an upward motion, the force applied from below (like your fingers) creates a different effect than if you apply downward pressure. The angle and direction of those forces change significantly, similar to how we analyze different planes in cylindrical coordinates.

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