2. Linear Momentum Balance

Think of a small cuboid with its centroid at an arbitrary point x as shown in Figure 1. Let us apply Newton’s second law to this cuboid.

In this first chunk, we start with a simple cuboid, which is a three-dimensional shape. The centroid of this cuboid is considered as the point where the mass is evenly distributed in all directions. When we apply Newton's second law, we look to calculate the forces acting on this shape in relation to its mass. Specifically, Newton's second law relates the forces on an object to its acceleration, indicating that the sum of these forces equals the mass of the object multiplied by its acceleration.

Imagine holding a small box. When you push it, the box accelerates in the direction of the push, demonstrating Newton's second law: more force will cause a stronger acceleration. The concept of the cuboid helps us analyze forces acting on it just like we analyze the forces acting when we apply a push on the box.

There will be traction on the six faces of this cuboid that will be applied by the other part of the body... The total force due to traction on e plane will be obtained by integration of traction over the area of this plane.

In this chunk, we discuss traction, which refers to the forces applied at the surfaces of the cuboid. Each face experiences different amounts of traction depending on the surrounding context and forces acting on it. By integrating the traction over the surface area of a face, we can calculate the total force acting on that particular face. This is important because it allows us to account for variations in traction due to different loading conditions and helps transport this information through the model.

Think of a sponge in water. The sponge's surface is being pushed by the water from all directions, creating a force at each face of the sponge. If you press harder at certain points, those points feel different amounts of pressure, much like the traction along the faces of our cuboid.

As body force is defined as force per unit volume, we need to integrate over the volume of the cuboid to get the total force due to body force which we denote by Fb.

The body force refers to forces acting throughout the volume of the cuboid, rather than just at its surfaces. These forces can vary from point to point within the body due to changes in density or gravitational effects. To find the total body force acting on the cuboid, we must integrate the body force density across the volume of the cuboid. This gives us a comprehensive view of the internal forces acting on all points in the cuboid.

Imagine a balloon filled with air. The air inside applies pressure equally across the entire inner surface of the balloon. The total internal force from all the air molecules acting on the surfaces of the balloon is similar to the body force acting throughout the entire volume of our cuboid.

The total linear momentum will be obtained by the volume integration of the linear momentum of a small volumetric point of the cuboid: ...We get: ⃗p = ρv(x)∆V+o(∆V).

In this part, we focus on linear momentum within the cuboid. Momentum is the product of mass and velocity. We perform a volume integration to compute the overall momentum considering that both density (mass per unit volume) and velocity might vary inside the shape. Thus, we arrive at a formulation that expresses the total linear momentum in terms of the density and velocity at the centroid of the volume.

Consider a train moving along its tracks. The total momentum of the train depends on the mass of the individual cars and their speed. If some cars are loaded while others are empty, the distribution of mass and the speeds would change, affecting the overall momentum just like how we compute it in our cuboid.

Now, substituting equations in Newton’s second law... This is Newton’s second law applied to a cuboidal volume.

In this segment, we consolidate our findings by substituting expressions for total forces and linear momentum into Newton's second law. This leads us to create a balance equation that emphasizes how the change in momentum is influenced by net external forces. When we shrink the volume of the cuboid to an infinitesimal level, we hone in on a core principle within continuum mechanics: understanding how materials would behave under stress and forces acting over them.

Think of it like fine-tuning a balance scale. As you adjust the weights on one side (net forces), you must also account for how much the scale tips (change in momentum). In the case of our cuboid, we're looking to find this balance at a very small scale, just like ensuring we can measure even the tiniest adjustments on the scale.

Let us write the LMB equation in (e1,e2,e3) coordinate system... Equation (19) is the tensor form of the Linear Momentum Balance.

Lastly, we translate our linear momentum balance equation into a specific coordinate system, which allows us to express it in a matrix form. This matrix representation helps in organizing the forces and momentum components with respect to each axis of the cuboid. Each equation derived from this matrix can be understood as balance conditions for force acting in each direction, giving us a detailed view of how physical laws apply across the methods of solution in mechanics.

Imagine organizing a team project based on tasks assigned to each team member (coordinate system). Each member is responsible for part of the project (like forces in different directions). When we gather their updates (momentum balance), we can see how the project is progressing as a whole and identify if adjustments are needed on individual tasks, similar to how we balance forces acting in multiple directions.

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