2. Formula for transformation of a stress matrix

The transformation of a stress matrix refers to finding out a relationship between the stress matrices in two different coordinate systems. The stress matrix in (e₁, e₂, e₃) coordinate system is: (1) Here, tₗ represents the traction on the j-th plane. So, the first column is the representation of t₁ in (e₁, e₂, e₃) coordinate system. Also, notice that i-th row corresponds to component of that particular traction along i-th basis vector.

The stress matrix is basically a way to represent the stresses on material in different coordinate systems. When we talk about transformation, we mean that we can express these stresses from one coordinate perspective to another. In this section, equation (1) shows how the traction tₗ (which is just a force per unit area acting on the j-th plane) can be arranged in a matrix format in one coordinate system (e₁, e₂, e₃). Each column in this stress matrix corresponds to the stresses acting on different planes defined by these coordinate axes. Overall, we can use this matrix to convert stress representations from one coordinate system to another effectively.

Imagine a sports team playing on different fields. Each field may have different markings, but the same players play the same positions on each field. The stress matrix serves as a 'field marking' for the stresses in a material, helping us visualize how those stresses would appear regardless of which 'field' (or coordinate system) we are using.

Now, we need to define a relationship between (e₁, e₂, e₃) and (ē₁, ē₂, ē₃). We know that we can always find a rotation tensor R that relates one set of basis vectors to another set of basis vectors: ēᵢ = Rₑᵢ (4) This is a tensor equation and thus coordinate-free. It does not correspond to any one coordinate system and can be used with respect to any coordinate system.

To switch from the original basis vectors (e₁, e₂, e₃) to a new set of basis vectors (ē₁, ē₂, ē₃), we utilize a mathematical concept known as the rotation tensor (R). This transformation allows us to express physical quantities in a new coordinate system without being tied to a specific orientation. The equation ēᵢ = Rₑᵢ shows how each new basis vector (ēᵢ) can be derived from the original basis vectors (eᵢ) by applying this rotation tensor. The beauty of this equation is that it's general – it applies regardless of how we orient our coordinate systems.

Think of the rotation tensor like a dance instructor guiding dancers to change formations. Regardless of the position they start in, the instructor helps them transition smoothly to a new position. In the same way, the rotation tensor helps translate our understanding of stress from one orientation to another.

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. In fact, the above relation holds for transformation of matrix form of any second-order tensor.

The relation described in the equation (11) offers a concrete way to apply the transformation of stress matrices. By representing the transformation in matrix form, we can systematically manipulate the matrices associated with the stresses when switching from one coordinate system to another. This method can be extended to not just stress matrices, but to any second-order tensor that represents physical quantities in engineering and physics. Essentially, matrix mechanics allows us to simplify and standardize how we transform and analyze these quantities across different coordinate systems.

Consider how global companies operate branches in various regions. Each branch might use its metrics, but they all report standardized data back to the headquarters for simplification. The same approach is taken with stress matrices – although the values may change for different coordinate systems, they're all transformed back into a standardized format that makes analysis and comparisons easier.

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