2. LMB Formulation

For the Linear Momentum Balance, we need to sum the total force on the cylindrical element due to tractions and body force and equate it to its rate of change of linear momentum.

This introduction highlights the essential aspects of Linear Momentum Balance (LMB). In classical mechanics, LMB states that the forces acting on an object equal the change in momentum of that object over time. Here, we specifically look at a cylindrical element, which is useful when dealing with bodies shaped like cylinders. We begin by considering all forces acting on the element (due to surface tractions and internal body forces) and relate this to how the momentum of the system changes.

Imagine a cylindrical water bottle. If you push down on it (applying force), the water inside will respond by moving. This is similar to how we analyze forces on the cylindrical element and their effect on momentum.

On +z plane, the traction vector (t+z) is given by t+z = σ e + τ e + τ e. The total force due to this traction can be obtained by integrating it over +z plane...

We start by considering the forces acting on the top surface of the cylindrical element (the +z plane). The traction vector describes how forces are distributed across this surface. The force must be calculated by integrating the traction vector over the surface area of this plane, accounting for components of stress in different directions. This analysis allows us to quantify the total force acting upwards on the cylindrical element.

Think of pushing down on a flat surface like a table with varying pressure across your palm. Each part of your hand exerts a different amount of force. Integrating the pressure over your palm gives the total force applied and how it is distributed, similar to the analysis of forces on the cylindrical surface.

As the stress components are varying in the domain of integration, we can use Taylor’s expansion about the center of the cylindrical element...

Since the stress components change within the cylindrical element, we use Taylor series expansion to express these components at points close to the center of the element. This mathematical technique helps us approximate the values of these components at nearby points, facilitating simpler integrations and calculations for LMB.

Imagine you're observing a moving car from different vantage points along a street. The car's speed seems different depending on your perspective. By using the Taylor series, it's like approximating the car's speed at various points, helping you understand how it moves without needing to know its exact speed at every point.

We show how to obtain the partial derivatives of basis vectors. The θ coordinate of two points differs by ∆θ...

In cylindrical coordinates, as you change the angular coordinate (θ), the direction of basis vectors also changes. By calculating the partial derivatives of these basis vectors with respect to θ, we understand how the basis vectors adjust when moving around the circular path of the cylindrical element. This becomes critical when integrating forces and releases the necessity to deal with constant basis vectors.

Consider how your finger moves around a circular pizza. The angle of your finger changes as you curve around the pizza's edge. Similarly, the basis vector direction changes as you vary the angle θ in cylindrical coordinates, and calculating these shifts helps in understanding related calculations.

We can now plug in the derivatives of basis vectors into equation (6) to get various expressions for the total force...

After obtaining the derivatives of basis vectors, these expressions are substituted back into the equations governing the total force acting on the cylindrical element. This leads to simplified expressions that can be handled mathematically and illustrate how different stress components interact with the coordinates of the cylindrical system, providing clear insights into LMB.

Think of this process like assembling pieces of a puzzle. As you fit each piece together (in this case, the mathematical expressions), you begin to see the complete picture of how forces and stresses distribute over the cylindrical element.

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