4.1. Verification of Stress Matrix

Let us verify the new stress matrix by directly finding the traction on one of the planes. Consider traction on the plane with normal as $\hat{n}_2$:

In this chunk, we are introducing a method to verify the new stress matrix. The verification occurs by calculating the traction on a specific plane where the normal vector is denoted as $\hat{n}_2$. This is an important step because it helps ensure that the calculated stress matrix behaves correctly under different operations performed on it.

Think of this as checking a math solution. After solving an equation, you would want to substitute back the solution to see if it satisfies the initial equation. Similarly, verifying the stress matrix ensures that it is consistent with the physical behavior of stress acting on a plane.

This is again a vector equation. Let’s write this down in the (e₁, e₂, e₃) coordinate system. To get the traction vectors, we get back to our stress matrix given in the (e₁, e₂, e₃) coordinate system and take its columns.

Here, we are translating the concept of traction into a vector equation and expressing it in terms of our chosen coordinate system ($e₁, e₂, e₃$). The column of the stress matrix corresponds to the traction vector that we want to analyze. This step emphasizes the necessity of recalling our initial stress matrix definition to extract useful vectors for computation.

Imagine you are reading a recipe in different units (like cups and grams). Even though the units change, the ingredients (tractions) are still the same. This chunk illustrates how we can transform the stress matrix while keeping the underlying physical meanings intact.

To obtain the column form of $\hat{n}_2$, we look at the second column of our rotation matrix. To get $\hat{n}_2 · e$, we take the ith component in this column.

In this piece, we focus on identifying how we can derive the normal vector from the rotation matrix. The rotation matrix contains the adjustments made to our original coordinate system, and by taking the appropriate column, we can extract the specific components that help us understand the new configuration of our normal vector after rotation.

Think of how a compass works. The compass needle always points north, but depending on where you tilt it (like rotating a coordinate system), you need to check the new direction it points to. In this analogy, the components in the rotation matrix represent how the direction of your normal vector changes due to rotation.

Alternatively, we can also find this dot product by looking at the orientation of the two coordinate systems shown in figure 1 and using the basic dot product formula: a·b=||a|| ||b|| cos(angle between them).

This chunk introduces an alternative approach for verification. By employing the dot product formula, we can describe the angle between vectors and verify the interaction between them in a mathematical manner. The dot product method reinforces the connection between geometry (the angle) and algebra (the formula), providing another layer of understanding.

Consider the concept of shadows. If you think of one light source (a vector) and a tall building (another vector), the way the shadow falls (dot product) can be influenced by the angle of the sun. Just like in physics, we use angles and lengths to understand how two objects interact in space.

This verifies what we had gotten using transformation formula. Remark: This zero column represents in (e₁, e₂, e₃) coordinate system. But being a ‘zero vector’ means that its representation in any coordinate system will be a zero column. Therefore, the 2nd column of $\hat{n}$ is a zero column!

This final chunk wraps up the verification process, confirming that the results obtained from the transformation are accurate. It discusses the significance of finding a zero column in the stress matrix, explaining that a zero vector remains unchanged across all coordinate systems. This consistency is crucial for the reliability of our analysis.

You can think of a perfectly leveled table. No matter how you look at it from different angles, if one side is truly flat (zero height), it will always remain flat regardless of your perspective. The 'zero column' analogy helps to understand that some physical properties will always be true, irrespective of how we look at them.

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