3.2. Extracting the axis and angle of local rotation

Let w denote the axial vector of W. Then, upon comparing I + W with (25), we can conclude the following: (28)

In this part of the section, the axial vector 'w' is introduced, which is related to the anti-symmetric part of the displacement gradient tensor, denoted as 'W'. By comparing this with the earlier defined rotation tensor, 'I + W' helps us understand how the axial vector corresponds to local rotation. Essentially, the axial vector captures the rotation's direction and magnitude, similar to how the angle of a turn is defined.

Think of it like using a wheel. When you spin a wheel, it rotates around an axis at its center. Here, the axial vector is like the center of the wheel; it tells you where the rotation is happening, whereas the angle of rotation tells you how much the wheel has turned.

Let us now look at the matrix form of the skew symmetric tensor W in (e1,e2,e3) coordinate system: (29)

This chunk discusses how the skew symmetric tensor 'W' can be represented in a 3D coordinate system. The matrix representation gives us insights into the properties of the tensor, specifically how it behaves under different transformations. Skew symmetric tensors have characteristics that allow them to represent rotational effects mathematically, which is crucial in understanding local rotation.

Imagine trying to describe how a spinning top behaves. Each point on the top is rotating but in a different way based on its distance from the center. The matrix representation is similar to plotting the positions of points on the spinning top to see how they move differently while still being part of the same rotation.

The column form of the axial vector of W will then be (30). The magnitude of this vector will then be the angle of rotation and the unit vector in its direction will be the axis of local rotation.

Here, we delve into the practical implications of the axial vector. Not only does it give us the direction of the rotation, but its magnitude directly corresponds to the angle through which the rotation occurs. Understanding the magnitude and direction is essential for applying these concepts in real-world scenarios, such as in robotics or mechanical systems.

Consider a door hinge. The door rotates around the hinge (the axis), and the angle through which it swings is the magnitude of that rotation. If you push the door lightly, it swings a little (small angle); if you push harder, it swings wider (larger angle). This analogy helps to visualize how local rotation works.

Let us consider an example where our coordinate system is (e1,e2,e3) and the displacement components are given as follows: u1 = u1(X1,X2), u2 = u2(X1,X2), u3 = 0. (31)

This section introduces a practical example to illustrate the concepts learned. We analyze a scenario where the displacement components are defined in a two-dimensional plane, effectively simplifying the problem by assuming no displacement in the third direction. This allows for a clearer calculation of the axial vector of the local rotation, tying the theoretical concepts to a concrete example.

Think of this example as a sheet of paper (e1 and e2 directions), where you can only move the paper up or down (u1 and u2). The thickness of the paper (e3 direction) doesn’t change. This simplification makes it easier to see how the paper could twist (rotate) without changing its thickness, similar to how we analyze local rotation in mechanics.

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