5.1. Relation between traction on planes with opposite normals

By definition, t−i is the traction on −e plane whereas ti is the traction on plane. So, these two are on the same plane but with normals pointing in the opposite direction.

This chunk introduces the relationship between traction acting on two different planes that share a common intersection but have normals pointing in opposite directions. The traction on the plane with normal −e is denoted as t−i, while the traction on the plane maintaining the normal direction e is denoted as ti. This means the forces acting on that same internal section of the body are counterbalancing each other.

Imagine two friends pushing against each other in a game of tug-of-war. Each friend's force represents traction on their respective side. Just as both apply forces opposite to one another, the traction in this context works the same way, showcasing an action-reaction pair.

The tractions on these planes will be equal and opposite due to Newton’s third law of motion (because they form an action-reaction pair): t−i = −ti.

According to Newton's third law, for every action, there is an equal and opposite reaction. Therefore, the traction exerted on one plane (t−i) will create an equal but opposite traction (−ti) on the other plane. This principle ensures that the forces are balanced, leading to stable internal interactions within the material.

Think of a balloon filled with air. When you squeeze one side of the balloon, it reacts by pushing back with an equal force on the opposite side. This illustrates Newton’s third law, mirroring the idea that the actions of t−i and −ti balance each other.

Using equation (13) in equation (12), our formula finally becomes: [Final Formula]. Note: The body force and the acceleration terms dropped out! Thus, the above formula holds even if the body force is present or the body is accelerating!

Here, the content explains how the previously introduced equation relates to the overall formula. When we substitute the relationship (13), we simplify our calculations, leading to a formula that still holds true even in the presence of forces (like weight) or acceleration (like when something is speeding up). This reinforces the validity of the traction relationship under varying conditions.

Consider a car accelerating down the road. Even though it is subject to gravity and friction (body forces), the rules of how it interacts with the ground and air still apply in a predictable way. Just like with traction, the underlying principles govern the interactions despite the car's movement.

--