2.2. Body force contribution

We also need the body force contribution. 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.

Here, the body force is simply the force acting on a unit volume of material. When calculating the total force acting on an object, you need to consider how much volume that object occupies and integrate the body force across that volume.

Imagine a sponge submerged in water where the water exerts equal pressure on all parts of the sponge. This pressure can be seen as a body force acting throughout the volume of the sponge.

The body force, denoted by b, can also vary within the volume of the cuboid in general. So, it will be a function of the position vector of the point of interest in the cuboid.

Body forces can change depending on where in the material you are looking. For instance, in a gravitational field, the force acting on an object may change slightly if it is at a different height. Thus, the body force is dependent on the location within the object.

Think of a hilly landscape where the gravitational pull on a rock differs at the top of a hill compared to the base. The force acting on it varies based on its position.

A general point in the cuboid will have position vector (x + γe1 + ξe2 + ηe3). Thus, the total force would be:

This position vector describes a point's location in three-dimensional space relative to a chosen origin. Using this, we can calculate the force acting at that specific point in the body, accounting for different position-based variations.

Imagine you're measuring the force felt by a ball at various depths in a swimming pool. The farther down you go, the greater the water pressure at that specific point, which symbolizes how pressure can be influenced by position.

The higher-order terms in Taylor's expansion when integrated will give terms of order less than the volume of the cuboid. In mathematics, 'o' is used to denote a smaller-order term.

When using Taylor’s expansion, we simplify our calculations by ignoring terms that become insignificant as the volume shrinks. The notation 'o(∆V)' indicates that these terms diminish quicker than the size of the volume we’re analyzing.

It's like calculating how much cake you need for a party. If you're planning for a small crowd, the icing on each slice might be negligible compared to the overall size of the cake, allowing you to focus on the cake itself more than the minor details.

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