1.6. An alternate method to derive AMB

There is also a simpler but approximate way to come to this final outcome which is given in many textbooks. Consider the same cuboid again. The coordinate system (e1, e2, e3) is shown on the right side of Figure 2. We have drawn stress matrix components on some of the planes as shown in Figure 2.

This chunk introduces a simpler method to derive the Angular Momentum Balance (AMB) related to the stress matrix. The author suggests that this method is often found in textbooks and utilizes a cuboid model with a designated coordinate system. By visualizing the cuboid's stress matrix components on different planes, students can get a clearer picture of where forces apply.

Imagine a box filled with water. We can visualize how the water applies pressure on the sides of the box just like how the stress matrix applies forces on the planes of the cuboid.

On the right plane, we have σ11, τ21, and τ31. On the top plane, we have σ22, τ12, and τ32. On the bottom face, we have σ33, τ32, and τ21. But, these three act in the −e2, −e1, and −e3 directions respectively. This is because this is the −e2 plane.

Here, it identifies the various stress components acting on different planes of the cuboid. The notation σ and τ refers to normal and shear stresses respectively. It clarifies how these stresses interact with the coordinate system chosen. Each stress has a specific direction, indicating how it applies on the face of the cuboid.

Think of squeezing a sponge. The water inside pushes against the walls of the sponge in different directions, just like the stresses act on different planes of our cuboid.

Let us now find the moment due to these tractions about e3 axis passing through the centroid of the cuboid. The normal components σ11, σ22, and σ33 pass through the centroid and thus do not contribute to this moment.

This discusses how to calculate the moment from traction forces about the e3 axis (the vertical axis). It explains that normal stress components do not contribute to the moment about this axis because they act directly through the centroid. Instead, shear components must be considered for calculating moments.

Consider a seesaw at a playground. If you push down in the center (normal stress), the seesaw will not rotate. However, if you push down on one side (shear stress), it creates a moment that causes it to tilt.

The moment due to τ21 will be traction times the area of the face (∆x ∆x) on which it acts times the distance from the center (z). The component τ12 has two contributions from e1 and −e1 planes both in the same direction (+e3). The moment due to two τ components acts in −e3 direction.

This section focuses on how to calculate the moment derived from shear stress components acting in different directions. It explains the multiplication of forces by area and distance to calculate the resultant moments that contribute to angular motion around the centroid.

If you imagine applying pressure at the side of a door with your hand, the area you push against multiplied by how far your hand is from the hinges (center) helps to turn the door. That's how moments work with these shear forces.

Finally, moment due to body force will be a smaller order term. The rate of change of angular momentum about e3 axis is also a smaller order term. When we shrink the cuboid, their contributions will vanish quickly than the contribution to moment due to traction components.

This chunk explains that although body forces contribute to the moment, their effect reduces significantly compared to the traction components as the cuboid shrinks in size. Also, the changes in angular momentum become less significant at smaller scales.

Think about how a small gust of wind (body force) has less effect on a heavy swinging door (angled momentum from traction) than the force you apply by pushing it. As the door gets smaller and lighter (like a smaller cuboid), the gust's influence diminishes even more.

The total moment about e3 axis (passing through the center of the cuboid) will then be derived, leading to an angular momentum equation, which is a simpler derivation but it involves a strong assumption of traction components not varying on cuboid’s faces.

In this final chunk, the total moment around the e3 axis is summarized, resulting in the derived equation for angular momentum balance (AMB). It's noted that this derivation relies on the assumption that traction components remain constant on the surfaces of the cuboid.

This is akin to a simplified method of understanding how forces balance and cause rotation in a system—like estimating the turning of a spinning top by recognizing only the largest forces acting on it, without considering smaller inconsistencies that could throw it off balance.

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