3.1. Traction contribution

Let us begin with torque due to traction on the plane with normal along e (denoted by T1). For an arbitrary point y on this face, the arm of the force will be y − x and force on a small area around y would be t1 dA. Hence

$T_1 = ext{t}_1 (y - x) dA$ (25)

This chunk introduces the concept of torque due to traction on a specific face of a cuboid. Here, traction refers to the stress acting at a point on that plane. The equation relates the torque (T1) to the traction (t1) and the distance from the point of consideration to the centroid (y - x). The key concept here is that the torque experienced by the plane is dependent on both the force acting on it and the leverage, which is the distance from the centroid to the point y.

Imagine a door that acts as the face of the cuboid. When you push on the door handle (which is at point y), the amount of force you exert (t1) multiplied by the distance from the hinges (x) determines how effectively you can push the door open (torque). A push closer to the hinges requires more effort, while pushing at the handle is much easier. In this analogy, the door's handle is like the arbitrary point on the face where the traction acts.

We have again used Taylor’s expansion here about the center of the cuboid. In the integration above, the first term within the small bracket crossed with first two terms within the square brackets will be constant and can come out of the integration. So, their integration just gives surface area of ê face for these terms. Other terms will either cancel or will be of lower order than ΔV. In fact, the second term in the final expression is also of lower order than ΔV as it is multiplied by Δx along with ΔV. So this term with all terms after it can be clubbed together as o(ΔV), i.e., (26)

This portion explains how we handle the complexity of calculating the total torque. Using Taylor's expansion allows us to approximate the behavior of the system around the centroid. The key point is that when integrating, some terms can be simplified as they do not depend on the variable being integrated. The concept of lower-order terms is essential here; these terms become negligible as the volume approaches zero, allowing us to focus on the essential contributions to torque.

Think of mixing ingredients in a large bowl. If some ingredients are in large clumps (higher-order terms), they won't mix well compared to the finer particles (lower-order terms) that dissolve easily. When you blend the ingredients, those clumps will eventually break down or become insignificant as you keep mixing, just like how terms in our formula become less important when we consider very small changes in volume.

Adding torques on the +e and -e faces, we get

$T_1 + T_{-1} = e imes ext{σ}_e \, ΔV + o(ΔV)$ (28)

This chunk discusses how we can combine the torque contributions from both the positive and negative sides of the plane (e and -e). The equation shows that the total torque results from the cross product of a unit normal vector (e) and the stress (σ) multiplied by the volume element (ΔV). The term o(ΔV) includes negligible contributions not affecting the overall torque value, reinforcing the importance of significant factors in our calculations.

Consider using a seesaw. When you and your friend sit on opposite ends, your weights create torques around the pivot point in the center. If your friend shifts slightly, that small change in position (like the o(ΔV) terms) hardly affects the balance, but as both of you (torque from both faces) push down on your sides, the seesaw changes its angle significantly, just as σ influences the torque created.

Doing a similar analysis for the remaining four faces, we obtain

$T = ext{...}$ (29)

The last chunk indicates that a similar method is applied to analyze the torque contributions from all four remaining faces of the cuboid. While the specific expression is not given, it implies a comprehensive assessment of how each face contributes to the total torque experienced by the body is necessary for a complete understanding of the mechanics at play.

Picture a multi-sided dice. Each face contributes equally to how the dice rolls. If you want to know how it will land, you need to understand the influence from all sides, not just one. Similarly, when analyzing the torque on the cuboid, the contributions are summed up to reflect the balance and effect of forces acting across all planes.

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