2.1. Relating σ and ̂σ

In thesecond lecture, we had found that traction on an arbitrary plane can be written as: (2) (3)

In the context of stress analysis, traction represents the internal forces acting on a material's surface. When we talk about an arbitrary plane, we refer to any imaginary plane that can be set at any angle. The mathematical representation begins with equations (2) and (3), which define how to calculate this traction based on the stress state of the material. Essentially, we can express the force acting on this arbitrary plane in terms of the stress tensor, which encapsulates all the internal stress states of the material.

Imagine you are pressing your hand against a wall. The force your hand exerts on the wall can be thought of as traction on that specific plane. No matter how you angle your hand, the stresses transmitted through the wall are what we are discussing when we mention traction.

Now, we need to define a relationship between (ɤ₁, ɤ₂, ɤ₃) and (e₁, e₂, e₃). We know that we can always find a rotation tensor R that relates one set of basis vectors to another set of basis vectors: ⇒ɤᵢ = R eᵢ (4)

The relationship between different coordinate systems is crucial in tensor analysis. In equation (4), we introduce the rotation tensor R, which allows us to transform basis vectors from one coordinate system to another. This means if you know where the original set of coordinates (e₁, e₂, e₃) lies, the rotation tensor helps you find their new positions (ɤ₁, ɤ₂, ɤ₃) after rotation. This tensorial relationship enables flexibility in analyzing stresses under different orientations.

Think of it like turning a piece of paper to view different sides. The paper represents a coordinate system, and turning it is like applying the rotation tensor. Even though you are looking at different angles of the same paper, the content on that paper (like stress and forces) doesn’t change.

This means that we have to write each of the vectors and tensors in this coordinate system, i.e. (5) Representation of e in its own coordinate system is trivial. eᵢ will be represented as e₁ and e₂, and e₃ as ... When i = 1, e gets rotated to ɤ₁ as: (6)

To express the vectors and tensors in the new coordinate system, we essentially rewrite them using the same physical quantities but expressed in terms of the new basis vectors. Equation (5) gives us the original representation, while as mentioned, it's simple to note how each eᵢ corresponds to its new orientation ɤᵢ. For instance, when we rotate the first vector, it becomes the first column of the rotation matrix, simplifying calculations involving stress transformation.

Imagine using GPS to navigate. When you input an address, GPS provides directions in a way that's easy to understand (like eᵢ in its original coordinates). If you switch to a different map view or perspective (like rotating coordinates), the physical route remains the same, but your view of it changes based on how it's presented.

Another way to obtain this equation is to realize that the rotation tensor can also be written as R = ∑₃ ɤᵢ ⊗ eᵢ. Further, using the formula derived in lecture 1 to obtain the matrix component of a second-order tensor, i.e., C = (C eᵢ)(eᵢ) we obtain the above equation. Now, our equation for stress matrix component from equation (3) becomes: (9)

This chunk elaborates on an alternate expression of the rotation tensor, enhancing your understanding of how it applies within the realm of stress transformation. By expressing the rotation tensor not just in terms of its physical application but also mathematically as a sum of products between the new and original basis vectors, we can derive the stress matrix's components using previously established formulas. Equation (9) prepares us to relate traction stresses in a new coordinate system.

Consider this like building a model from blueprints. You first visualize or sketch the overall shape (rotation tensor) of the structure (stress components), then break it down into simpler sections or materials (tensors defined in different systems) for easier construction and understanding. Each way of viewing helps in achieving your final structure!

However, by definition, t_k = σ, because t_k represents the mth component of traction t_k and will thus represent the mth entry in the kth column of stress matrix. Hence (10) Finally, writing this equation in matrix form, we get: (11) This is the relation that we use to transform the stress matrix from one coordinate system to another.

To finalize our discussion, equation (10) clarifies that the traction force experienced at any point (t_k) corresponds to stress (σ). This equivalence is vital since it provides a direct means to derive stress values in the new coordinate system using the relationships we've established. The matrix equation in (11) encapsulates these transformations succinctly, allowing us to convert stress matrices efficiently between desired orientations.

Think of it like calculating how much paint you need based on the dimensions of different walls in your house. Based on the area of each wall (representing traction) and your paint type's coverage (stress), you find out how to adjust your order based on whether you're painting one room or multiple rooms at once.

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