Question

If \(P\) and \(S\) are \(2^{\text {nd }}\) -rank tensors, show that \(9^{2}=81\) coefficients are needed to write each component of \(\mathbf{P}\) as a linear combination of the components of \(\mathbf{S} .\) Show that \(81=3^{4}\) is the number of components in a \(4^{\text {th }}\) -rank tensor. If the components of the \(4^{\text {th }}\) -rank tensor are \(C_{i j k m},\) then equation (7.5) gives the components of \(P\) in terms of the components of \(S\). If \(P\) and \(S\) are both symmetric, show that we need only 36 different non-zero components in \(C_{i j k m} .\) Hint: Consider the number of different components in \(P\) and \(S\) when they are symmetric. Comment: The stress and strain tensors can both be shown to be symmetric. Further symmetry reduces the 36 components of \(\mathbf{C}\) in (7.5) to 21 or less.

Short answer

For symmetric \(\mathbf{P}\) and \(\mathbf{S}\), we need 36 different non-zero components in \(C_{ijkm}\).

Step-by-step solution

Understanding rank and components

A tensor of rank 2, like \(\mathbf{P}\) and \(\mathbf{S}\), has components that can be represented in a 3x3 matrix. Each tensor will have \(9\) components since \(3^2 = 9\).

Linear combination of components

Since each component of \(\mathbf{P}\) is a linear combination of the components of \(\mathbf{S}\), and \(\mathbf{S}\) itself has \(9\) components, we need \(9^2 = 81\) coefficients to describe this linear relationship.

Components of a 4th-rank tensor

A 4th-rank tensor in 3-dimensional space has \(3^4 = 81\) components because for each of the 4 indices, there are 3 possible values.

Symmetry consideration

When \(\mathbf{P}\) and \(\mathbf{S}\) are symmetric, it means \(P_{ij} = P_{ji}\) and \(S_{ij} = S_{ji}\). Therefore, instead of 9 independent components, each matrix has 6 independent components because the repeated indices do not contribute additional independent components.

Symmetric 4th-rank tensor components

For a 4th-rank tensor \(C_{ijkm}\) with symmetry, we analyze the distinct components. Since each pair of indices (\(ij\) and \(km\)) can be independently chosen from the 6 distinct components of a 2nd-rank symmetric tensor, we need \(6 \times 6 = 36\) components.

Key concepts

Tensor Rank

The rank of a tensor describes the number of indices needed to identify each component. Tensors can be thought of as generalizations of scalars (rank 0), vectors (rank 1), and matrices (rank 2). For example, consider the tensors \( \mathbf{P} \) and \( \mathbf{S} \), which are both second-rank tensors. Here, each component is accessed using two indices, hence they are rank 2.
The number of components in a tensor depends on its rank and the dimensions of the space it is defined in. For a second-rank tensor in a three-dimensional space, the components can be arranged in a 3x3 matrix, resulting in \( 3^2 = 9 \) components.
Understanding tensor rank is crucial because it tells us about the tensor's complexity and the amount of data it holds. A higher rank tensor in the same space will have exponentially more components.

Components of a Tensor

Each tensor is made up of components that can be represented in various forms such as arrays or matrices. For instance, a second-rank tensor, like our \( \mathbf{P} \) or \( \mathbf{S} \), can be visualized as having 9 components in a 3x3 matrix.
When we increase the rank of a tensor, such as moving to a fourth-rank tensor, the number of components increases drastically. In a three-dimensional space, a fourth-rank tensor has \( 3^4 = 81 \) components. These components can conceptually be pictured as filling a 3-dimensional array with dimensions 3x3x3x3.
These components are essential building blocks of tensors, determining their properties and how they interact with each other in mathematical operations.

Linear Combination in Tensors

A linear combination in tensors is similar to linear combinations in vectors and matrices. It involves expressing each component of one tensor as a sum of other tensor components, scaled by specific coefficients.
In our exercise, each of the 9 components of the tensor \( \mathbf{P} \) is a linear combination of the 9 components of \( \mathbf{S} \). To define this relationship, we need \( 9^2 = 81 \) coefficients because each component of \( \mathbf{P} \) depends on every component of \( \mathbf{S} \). This is akin to needing 81 equations to fully describe the transformation between \( \mathbf{P} \) and \( \mathbf{S} \).
Understanding how to form and interpret these linear combinations is fundamental for many applications in physics and engineering, especially in fields dealing with stress and strain tensors.

Symmetry in Tensors

Symmetry in tensors simplifies their representation and reduces the number of independent components. A symmetric tensor won't change when indices are swapped. For example, for second-rank tensors \( \mathbf{P} \) and \( \mathbf{S} \, P_{ij} = P_{ji} \) and \( S_{ij} = S_{ji} \), reducing their independent components from 9 to 6.
For a fourth-rank tensor \( C_{ijkm} \) with symmetry, the analysis changes as we consider distinct components grouped into pairs. Since each pair can be independently selected from 6 components, we need \( 6 \times 6 = 36 \) components.
Additionally, further symmetries (like stress and strain tensors being doubly symmetric) can reduce these components even more, sometimes down to 21 or even fewer. Recognizing these symmetries helps in simplifying complex tensor equations and computations.