Problem 13 - A First Course in Continuum Mechanics

Question

Let \(\boldsymbol{A}\) denote the change of basis tensor from a frame \(\left\\{e_{i}\right\\}\) to a frame \(\left\\{e_{i}^{\prime}\right\\}\) with representation \([\boldsymbol{A}]\) in \(\left\\{\boldsymbol{e}_{i}\right\\} .\) Let \(\boldsymbol{S}\) be a secondorder tensor with representation \([S]\) and \([S]^{\prime}\) in \(\left\\{e_{i}\right\\}\) and \(\left\\{e_{i}^{\prime}\right\\}\), respectively. Show that $$ [\boldsymbol{S}]^{\prime}=[\boldsymbol{A}]^{T}[\boldsymbol{S}][\boldsymbol{A}] $$

Step-by-Step Solution

Answer

\([\boldsymbol{S}]' = [\boldsymbol{A}]^T [\boldsymbol{S}] [\boldsymbol{A}]\) follows from the tensor transformation rule for changing basis.

Step 1: Understand the Change of Basis

To change the representation of a tensor with respect to a new basis, you use the transformation matrix which is the change of basis tensor. In this case, tensor \textbf{S}\'s representation changes from \([S]\) to \([S]'\) when the basis changes from \(\{e_i\}\) to \(\{e_i'\}\), using the change of basis tensor \(\textbf{A}\).

Step 2: Understand the Relationship Between Basis and Components

The components of a tensor in a new basis can be found by pre-multiplying by the transpose of the change of basis matrix and post-multiplying by the change of basis matrix itself. This is because the components transform contravariantly, and the basis transforms covariantly.

Step 3: Apply the Transformation Rule for Second-Order Tensors

For second-order tensors, the transformation rule can be expressed in matrix form, which in this case dictates that \([S]' = [A]^T[S][A]\). Here, you multiply the transpose of the change of basis matrix \(\textbf{A}\), denoted as \([A]^T\), by the matrix representation of \(\textbf{S}\), denoted as \([S]\), and then by the change of basis matrix \([A]\) itself to get the representation of \(\textbf{S}\) in the new basis.

Key Concepts

Tensor Transformation

When you're introduced to the idea of tensor transformation, the essence lies in understanding how tensors behave under a change of basis. Imagine you're in a room, and you decide to describe the location of furniture from where you are standing (this is your original basis). If you move to another corner (this is your new basis), you'll need a new description for the furniture placement, even though the furniture hasn't moved at all. This analogy applies to tensors, which are mathematical objects that, like the furniture in a room, have components that depend on the 'viewpoint' or basis chosen.

In the exercise provided, the change of basis tensor, denoted as \( \boldsymbol{A} \), is the mathematical tool that allows us to describe a tensor in a new basis. This process involves using \( \boldsymbol{A} \) to shift the tensor's description from one frame \( \lbrace e_i \rbrace \) to another \( \lbrace e_i' \rbrace \). The transformed tensor retains its intrinsic properties, but its components in the matrix form have changed to fit the viewpoint of the new basis.

Second-Order Tensors

To understand second-order tensors, think of them as numeric grids that carry information about certain quantities in two dimensions, like a sheet of checkerboard that holds numbers at each square. Mathematically, a second-order tensor can be visualized as a 2D array or matrix, which provides more complexity than vectors (first-order tensors) as they relate to quantities that have not just one direction but an orientation in space as well.

They can represent various physical phenomena, such as stress and strain in materials, but what's critical to grasp is that these tensors have components that change when the basis changes. In the given exercise, the tensor \( \boldsymbol{S} \) is represented in two different bases, and the aim is to show how its components in one basis can be used to find its components in the other basis using specific transformation rules.

Matrix Representation of Tensors

Matrix representation is a convenient way to handle tensors, especially when it comes to computations. Imagine playing the piano, where each key has a specific note; similarly, in the matrix representation of a tensor, each element corresponds to a specific component of the tensor. This orderly arrangement allows for operations on tensors to be expressed through familiar matrix operations.

In our exercise example, the tensor \( \boldsymbol{S} \) is represented by the matrices \( [S] \) and \( [S]' \) in two different bases. The representation \( [S] \) is like the familiar tune in the original key, while \( [S]' \) is akin to the same tune transposed into a whole new key. Linking both representations is the change of basis tensor \( \boldsymbol{A} \) that acts as the transposition sheet for our tensor-matrix melody.

Covariant and Contravariant Components

In the realm of tensor mathematics, we meet two unique types of components: covariant and contravariant. These are like the two sides of a coin, each side defining the tensor components from a different perspective. Covariant components (the 'heads') adapt with the basis change, shrinking and stretching as the basis does, while contravariant components (the 'tails') flex against the basis change to maintain the 'length' of a vector.

What is remarkable in our exercise scenario is how these components interact during a basis change. The matrix \( [\boldsymbol{A}] \) helps us transition from one side of the coin to the other, translating contravariant components into covariant components by pre-multiplication with its transpose \( [\boldsymbol{A}]^T \) and post-multiplication with \( [\boldsymbol{A}] \) itself. This pas de deux between contravariant and covariant parts is what creates the new, transformed representation of the tensor under the new basis.