1.2. Diagonality of matrix for in principal coordinate system

We know that the stress matrix in the coordinate system of principal stress directions becomes diagonal. Similarly, when we represent the strain matrix in the coordinates system of the principal strain directions, it will become a diagonal matrix.

When we talk about matrices in the contexts of stress and strain, particularly in principal directions, we say that they become diagonal. This means all the off-diagonal elements (the elements not on the main diagonal running from the top left to the bottom right) become zero. In simpler terms, the values of stress or strain only depend on the corresponding principal directions and don’t interact with one another. This simplifies calculations because it indicates that in a principal coordinate system, normal stresses or strains occur without shear components.

Imagine a simple set of axes showing the North, South, East, and West directions, where each direction corresponds to forces like wind or pull. Now think about when the wind blows only in the North direction—at that moment, we don't care about the East or West forces because they are effectively 'zeroed out.' Similarly, when we analyze a strain matrix in its principal directions, it simplifies down to just the essential components.

As the off-diagonal components will be zero, this means that if we take the two line elements directed along the principal strain directions, there will not be any change of angle between them.

The significance of having zero off-diagonal elements is that the principal strain or stress directions do not cause any angular change in the body when considered specifically along those lines. This relates to the physical deformation of objects. So when we apply strain along these principal directions, the object changes size but retains its shape, meaning it still 'looks like' a rectangle or cuboid. This lack of angle change is crucial for understanding how materials deform.

Think of a rubber band. If you stretch it evenly along its length (principal direction), it gets longer but maintains its straight shape. There aren't any twists or bends—only stretching. If we stretch it at an angle, it would get distorted. The idea here is that when stress or strain is applied in principal directions, the effect remains smooth and predictable.

To visualize this, we can consider a cuboid whose face normals are along the principal strain directions (given by the eigenvectors of the strain tensor).

Imagine a cuboid, much like a box, aligned such that its faces align with the principal strain directions derived from the strain tensor. When deformation occurs along those directions, the cuboid stretches or compresses but remains a cuboid, simply getting larger or smaller without altering its rectangular shape. This helps solidify the concept of principal strains because it illustrates that while sizes change, the geometric arrangement remains intact.

Consider a loaf of bread. When you press down on the bread from above, it flattens but remains rectangular. If you pinch it from the sides (at an angle), it starts to look more distorted. The principal strain directions are like pressing down directly over the top—affecting size while keeping everything clean and rectangular.

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