2. Relating externally applied distributed load on body's surface to stress tensor

Suppose we have an arbitrary body which is clamped at some part of the boundary and a load is applied on some part of the boundary by an external agent. As the load is usually distributed over an area of the boundary of the body, it has the unit of traction which we write as t₀ as shown in Figure 3.

In this section, we start by considering a body that has been fixed or clamped at certain parts, meaning it cannot move. An external load, represented as 't₀', is applied on some parts of its surface. This load does not act at a single point but is spread out over an area, creating what we call 'distributed load'. The significance of understanding this load is that it directly influences the stress that develops in the body.

Imagine a heavy book resting on a table. The weight of the book exerts a downward force on the table surface, which is similar to how an external load acts on the body. Just as the table supports the book's weight, the body we are discussing supports the distributed load applied to it.

Our aim is to relate the state of stress (at the point where external load is acting) to the applied load itself. If we solve the stress equilibrium equations, we can find stress everywhere in the body. But at points on the boundary where the external load acts, we can find the state of stress partially without solving the stress equilibrium equations.

The goal here is to establish a connection between the external load applied to the body and the internal stresses that develop within it. Normally, to find stress throughout the entire body, one would solve the stress equilibrium equations. However, at the surfaces where external loads are applied, we can directly relate the applied load to the stress present without having to go through the entire equilibrium analysis.

Consider a sponge being pressed down by your hand. Even as you press, you can feel how much stronger the sponge feels at the point of contact. This sensation can be likened to stress; the pressure your hand applies is the external load that directly influences the internal stress of the sponge.

Let us set up our coordinate system such that we have e₃ axis along the surface normal as shown in Figure 3. We have drawn the part of the body near the surface. On the top part of this surface, external load t₀ acts. We are looking at an infinitesimal part of the body, so we can safely assume that t₀ is constant on the top surface.

To analyze the stresses correctly, we position our coordinate system with the e₃ axis aligned with the normal to the surface where the load is applied. By doing this, we simplify our analysis. Since the load t₀ is applied to an infinitely small area of the surface, we assume that this load remains consistent across that area when making calculations.

Think of it like stacking books on a table. If you place a single book evenly on the table, the weight from that book is distributed consistently across the surface it touches. Setting up a coordinate system in this way helps us analyze how the weight (or stress) travels through the table (or body) in relation to where the force is applied.

The total force on this small part of the body will be the vector sum of the traction forces applied from the remaining part of the body, externally applied load, and the body force.

Next, we apply Newton’s second law to this small section or 'pillbox' we are analyzing. According to Newton's second law, the total force acting on an object is equal to its mass times its acceleration. In our case, we sum up all the forces acting on our infinitesimal section, from external loads to internal traction forces.

Imagine holding a small box with weights inside. The total force you feel is a combination of your grip strength and the weight of the contents. Similarly, the total force on our infinitesimal part involves contributions from both outside forces and the material forces acting within it.

We now divide both sides of the above equation by Δ∆ and then take the limit as Δ₃ approaches zero.

To derive a meaningful relationship between the applied load and the stress, we continue by simplifying our forces and then conducting a limit process. By dividing by the area elements and shrinking our analysis down to a single point, we enable ourselves to extract information about the stress directly from the external load applied.

Imagine you're squishing a balloon. When you press down on one part, the air inside redistributes. By focusing on smaller areas as you press harder (eventually narrowing down to a single point), you can see how that pressure influences the entire balloon's structure at that point. This limit process helps us connect the narrow area of focus to the broader effects of the external load.

As mentioned earlier, this equation is also used as boundary condition for solving the stress equilibrium equation.

The relationship formed through our analysis shows that the internal forces (traction) that develop in response to an external load are equivalent to the stress at the surface. This is crucial as it serves as a boundary condition when solving complex stress equilibrium equations, providing a concrete foundation to ensure our stress analysis is accurate.

Think of a sports team. The performance of each player impacts the outcome of the game. Here, the external load (like an opponent) directly influences how the internal dynamics (team strategy) evolve. The relationship we found ensures that our stress calculations are grounded in real external conditions, akin to assessing how each player needs to adjust based on the team strategy during a game.

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