5. Storing information of traction on infinite planes at a point

Theorem: If we know traction on three independent planes we can then find the value of traction on any other plane.

This theorem states that knowing the traction on just three independent planes at a point allows us to determine the traction on any other plane passing through that point. Essentially, we don't need to measure traction on every single plane; instead, we can use the information from three specific planes to infer the rest.

Think of how knowing the coordinates of three points in space lets you determine any point in that space. Similarly, knowing the traction on three planes provides enough information to understand the traction throughout the area around that point.

Consider a point x on our body. Suppose we know the value of traction on three different planes at the point x and then we want to know what is the value of traction on any other plane. We draw a tetrahedron whose vertex is at x and three edges at this point are perpendicular and along the coordinate axes.

Here, we visualize a tetrahedron that has its vertex at point x where traction is being measured. This tetrahedron has three edges aligned along the coordinate axes, corresponding to the three planes from which we know the traction values. The tetrahedron’s geometry helps us to understand how traction relates between these planes and the planes we want to analyze.

Imagine a pyramid with the tip (apex) representing the point where we measure forces. The triangular base represents the three planes where we know traction. From this pyramid, we can infer properties of other angles and bases, similar to how we infer traction on other planes.

We have to always look at the outward normal of a plane with respect to the body. Now, if we know the tractions on the three planes −e1, −e2, and −e3 and we want to find the traction on the tilted plane (having normal as n), first we’ll apply Newton’s 2nd law of motion to the tetrahedron.

This chunk emphasizes the importance of understanding forces acting on the tetrahedron. By using Newton's second law, we analyze the momentum and forces acting through the faces of the tetrahedron. This analysis allows us to calculate the resulting traction on a tilted plane using the known tractions from the three basic planes.

Consider pushing an object; the force you apply affects its movement. Similarly, the forces acting on the faces of the tetrahedron impact the traction we need to find. It's like finding how much push is needed to keep an object moving in a straight line when being pushed from various angles.

The total external force will consist of both contact forces and non-contact forces. Let the area of face OBC be A1, traction on it be t−1, area of face OAC be A2 and traction on it be t−2, area of face OAB be A3 and traction on it be t−3 and area of face ABC be An and traction on it be tn.

In this chunk, we break down the forces acting on the tetrahedron's faces. Each face will experience traction forces based on the area of that face and the known traction values. This helps us to set up the equations needed to derive the traction on the tilted plane.

Imagine a building with different walls exerting forces due to wind pressure and other factors. Each wall will respond differently based on its area and position. Similarly, each face of our tetrahedron reacts differently to forces, which we must analyze to understand total traction.

We can also relate the areas A1, A2, A3 in terms of An. If we project the area A along the direction e1, that projected area is the same as the area of OBC which is A1. We can prove that: A_i = A_n (n·e_i).

This section discusses how to relate different areas based on projections. By projecting the areas along the direction vectors of the planes, we can compute the relationships among these areas, which helps us understand how traction on the tilted plane is influenced by the traction on the known planes.

Consider shadows on a wall; how a shape’s shadow changes depending on the light's angle provides a way to think about projecting areas. By understanding how areas relate through projection, we can better grasp how to infer unknown traction values.

When we apply the limit to our earlier equations, the terms proportional to h will vanish. Thus, in the h → 0 limit, the terms corresponding to body force and acceleration drop out.

This finalizes our understanding of how to derive the traction on the tilted plane from the known values. The limit process helps simplify calculations and eliminates variables that complicate the model, leading us to a practical formula that can be used in various applications in mechanics.

Think of zooming in on a curve; as you zoom in closely, the curve begins to appear straight, simplifying your calculations. Similarly, as we minimize the size of our tetrahedron, we find a cleaner relation for traction on the tilted plane.

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