1.1. Representation of gradient

We need to first express gradients in cylindrical coordinate system. In Cartesian coordinate system, the gradient of a quantity is given by (3) whereas its definition in cylindrical coordinate system is (4)

In this chunk, we focus on how gradients are represented in different coordinate systems. The gradient in a Cartesian coordinate system provides a straightforward calculation based on the x, y, and z coordinates. However, when we transition to cylindrical coordinates, which use the parameters (r, θ, z), the definition of gradient changes slightly. This is important because gradients describe how a quantity changes in space, which is crucial for understanding deformation in materials.

Imagine you are standing on a hill and you want to measure how steep it is at your position. If you were to describe your position on a traditional map (Cartesian coordinates), you would use x and y. However, if you're on a cylindrical hill, you'd describe where you are by how far you are from the center (r), how far around the hill you are (θ), and how high up you are (z). The gradient would give you a measure of the steepness in relation to these cylindrical coordinates.

Notice that the partial derivative with respect to θ is divided by r as θ is non-dimensional and we are taking the gradient in space. We will use the above form to obtain the gradient of the displacement vector u.

In this chunk, we introduce a key detail about cylindrical coordinates: the variable θ (the angular coordinate) is treated differently than r (the radial coordinate). Specifically, the partial derivative with respect to θ is divided by r. This division is because θ is a non-dimensional quantity, meaning it doesn't have physical units like meters or centimeters. Thus, it requires normalization by multiplying or dividing by something with dimensions, which is represented by r, the radius. Understanding this difference is crucial for calculating how vectors change in a three-dimensional space.

Think of walking around a circular track. The distance you cover while walking is represented by r (how far you are from the center), but the actual angle (θ) at which you walk doesn’t change this distance directly. You can't measure how 'far' you walk around in angles without relating it to the distance from the track’s center (r). This is similar to how we adjust our calculations in cylindrical coordinates.

The displacement vector can be written in cylindrical coordinate system by decomposing it along cylindrical basis as follows: u=ue +u e +u e r r θ θ z z

In this chunk, we introduce the concept of the displacement vector in cylindrical coordinates. The displacement vector 'u' is expressed as the sum of its components along the three basis directions: radial (e_r), circumferential (e_θ), and axial (e_z). Each component accounts for how much the displacement in the material occurs along these specific directions. Understanding this decomposition is essential for performing calculations related to strain and stress in materials undergoing deformation.

Imagine you have a balloon that you are inflating. The way it expands can be described in three different ways: how much it grows outward from the center (radially), how it stretches around the circumference (circularly), and how it tallies upwards (axially). These directions help us describe the total change in shape of the balloon—that's similar to how we define our displacement vector in cylindrical coordinates.

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