1. Cylindrical Coordinate System

Cylindrical coordinate system turns out to be very handy in studying deformation of bodies having cylindrical shapes as we will see later. It is therefore of importance to derive balance laws in cylindrical coordinate system. Figure 1a shows a generalized cylinder whose radius may change over the axis of the cylinder. The axis of the cylinder coincides with z axis. Any general point of the cylinder will have coordinates (r,θ,z). The position vector of this point is re + ze.

The cylindrical coordinate system is a three-dimensional coordinate system that is particularly useful for analyzing problems involving cylindrical objects. In this system, any point in space is defined by three coordinates: r (the radial distance from the z-axis), θ (the angular position around the z-axis), and z (the height along the z-axis). This is especially advantageous for bodies that have a cylindrical shape, such as pipes or tubes, as it simplifies the mathematics involved. For example, the position vector helps in visualizing where any point is located within that cylindrical space.

Imagine a soda can. The height of the can corresponds to the z-coordinate, the distance from the center of the can corresponds to the r-coordinate, and the angle around the can where you place a dot corresponds to the θ-coordinate. This analogy can help visualize how we can locate any point in the cylindrical shape of the soda can.

The basis vectors for a cylindrical coordinate system are (e_r, e_θ, e_z) whereas in the Cartesian coordinate system, the basis vectors are (e_1, e_2, e_3). Although both sets of basis vectors are orthonormal, there is a big difference in their properties: the basis vectors of Cartesian coordinate system are fixed in direction and do not change from one point to the other but in cylindrical coordinate system, two of the basis vectors (e_r, e_θ) change when θ coordinate of a point changes.

In cylindrical coordinates, the basis vectors represent the directions for the coordinates. The vector e_r points radially outward from the center axis, e_θ points tangentially around the axis (increasing θ), and e_z points up and down along the height of the cylinder. Unlike the Cartesian system, where the direction of basis vectors remains the same at any point, in cylindrical coordinates, the radial and angular vectors change their direction as you move around the z-axis. This flexibility helps accommodate the curvature of cylindrical shapes.

Think of a clock face. The hour hand (like e_r) points outwards from the center to the hour, changing direction as the time changes. The minute hand (like e_θ) moves around the center, altering its angle. The vertical position (like e_z) remains fixed as time passes, just like the height of the clock remains constant.

In cylindrical coordinate systems similarly, we need to analyze cylindrical elements with their face normals parallel to e_r, e_θ, and e_z. A cylindrical element for the generalized cylinder is shown in Figure 3.

To effectively analyze forces and other physical quantities in a cylindrical coordinate system, we use cylindrical elements. These are small cylindrical sections whose surface normals align with the basis vectors. Each face of the cylindrical element corresponds to one of the three basis directions (radial, angular, and vertical). By breaking down the forces across these faces, it becomes easier to compute the resulting effects such as stress and strain on the body.

Consider a water bottle. If you imagine slicing a small section from the bottle lengthwise, you create a cylindrical element. Each side of this slice represents sections where forces might act differently due to internal pressure or external forces, allowing you to study the forces within that miniature section of the bottle.

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