1.1. Formula for volumetric strain

The scalar triple product can also be realized as the determinant of a matrix whose columns are formed by the vectors involved in the scalar triple product. Thus, we can write...

In this chunk, the text introduces the concept of the scalar triple product, which is an important mathematical construct for calculating the volume of a parallelopiped formed by three vectors. This scalar triple product can also be expressed as the determinant of a matrix that consists of these vectors as its columns. The scalar triple product provides a measure of the signed volume of the figure defined by the three vectors. This foundational formula allows us to connect the scalar triple product to the concept of volumetric strain.

Imagine unscrewing a jar lid. The way the three edges of the jar lid move apart can be represented by three vectors. The 'space' these vectors define, like the volume in the jar before you open it, can be calculated using the scalar triple product. The determinant essentially measures how much 'space' you have between those edges.

Substituting the below formula in (6) leads to...

In this part, the text describes how to substitute a specific formula into another one to simplify the expression for volumetric strain. The original formula is manipulated algebraically to arrive at a more convenient form, which is crucial for practical calculations in mechanics. The ongoing substitutions lead to the final formula that expresses volumetric strain in terms of easily measurable quantities.

Think of making a smoothie. You start with the raw ingredients (like fruits and yogurt), which can be considered the original formula. As you blend, you mix and change those ingredients into a smooth drink (the final formula), making it easier to consume and enjoy, just as we simplify equations to make them easier to work with in mechanics.

To obtain its determinant, let us expand it in terms of the first row...

This section explains how to compute the determinant of the deformation gradient matrix by expanding it along the first row. The determinants are key in understanding the geometry of the deformation of materials. Here, the mention of higher-order terms is significant as it highlights that for small displacements, these terms contribute little to the calculations and can often be ignored.

Imagine you are analyzing a small piece of rubber. If you stretch it just a little, the tiny changes (higher-order terms) in its shape are so minimal compared to the overall stretching. In math terms, just like ignoring the negligible adjustments makes our calculations cleaner and simpler, in physical terms, it helps us focus on the main changes happening to the rubber.

Finally, using equation (12), we get the following formula for local volumetric strain...

Here, we arrive at the final expression that defines local volumetric strain. This formula is important because it links the strain to the properties of the material under slight deformations. The emphasis on its independence from the specific lines chosen illustrates that volumetric strain is a unique quantity characteristic of the material state at a point.

Consider filling a balloon with water. No matter how you squeeze the balloon, the amount of water you can fit inside (volumetric strain) depends only on the volume of water, not how you manipulate the balloon. This represents the unique nature of volumetric strain at a point, independent of how you apply pressure.

We can notice that the RHS of this equation is equal to the trace of the displacement gradient matrix...

This chunk identifies that the right-hand side (RHS) of the derived equation corresponds to the trace of the displacement gradient matrix, reinforcing the mathematical underpinnings of volumetric strain. Understanding the trace in this context is valuable because it summarizes the effects of strain and ties together different aspects of the deformation in a concise manner.

Think of the trace as a summary of all the notes you take during a class. Just as the trace boils down complex information into a simple, understandable summary, the trace of the displacement gradient matrix condenses various strain effects into a single, meaningful metric that describes the volumetric strain at a point.

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