If P and S are 2^nd -rank tensors, show that 9^2=81...

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.

Key Concepts

Tensor Rank and Component Counting

A tensor’s rank indicates the number of indices required to label its components. In a three?dimensional space, a second-rank tensor has 3x3=9 components while a fourth-rank tensor has 3^4=81 components. This counting is fundamental when analyzing how tensor components transform and how many independent coefficients are involved in linear relations among tensors.

Linear Transformation of Tensors

When one tensor is expressed as a linear combination of another tensor’s components, the transformation is mediated by a higher-rank tensor. In this process, each component of the output tensor is written in terms of the components of the input tensor through coefficients that are organized in a tensor of rank equal to the sum of the ranks of the indices involved.

Symmetry in Tensors and Reduction of Independent Components

Many physical tensors, such as those in elasticity theory, possess intrinsic symmetry properties which reduce the number of independent components. For example, if a second-rank tensor is symmetric, it has fewer unique components than the full 9. Similarly, the symmetry in the transformation tensor (fourth-rank) leads to a reduction in the number of independent coefficients needed to describe the linear relationship, cutting down redundant terms.

Application of Symmetry Conditions in Tensor Equations

In many applications, especially in the mechanics of materials, symmetry conditions are imposed based on physical principles, like conservation laws. These conditions not only simplify tensor equations by reducing the number of necessary independent coefficients but also ensure the consistency of the mathematical formulation with physical realities, such as symmetry in stress and strain tensors.

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