4.3. Multiplying two second order tensors

There are various ways in which two second order tensors can be operated together. We will consider the usual multiplication of two tensors which yields another second order tensor:

This chunk introduces the concept of multiplying two second order tensors to produce another tensor. It emphasizes that there are multiple methods for tensor operations, but here we will focus on the standard multiplication method. The result of this operation is not just a single value or vector, but another entire tensor, preserving the tensor nature.

Think of two recipes for making compound dishes. Each recipe (tensor) involves mixing various ingredients (components). When you combine the recipes by following a standard procedure (multiplying tensors), you'll end up with a new dish (a new tensor) that has its own unique taste and properties.

Using Kronecker delta property, we remove the summation over j and replace j with k everywhere:

In this step, the Kronecker delta function helps simplify the expression during the multiplication of tensors. It essentially 'filters' terms so that only specific ones are taken into account, thus streamlining the multiplication process and making it easier to compute the resulting tensor.

Imagine you're sorting through a box of toys and only want to keep the red ones. You can think of the Kronecker delta as a filter that helps you quickly identify and retain only the relevant toys, just like how it helps to keep track of specific index terms while multiplying tensors.

As this is the expression for C and any general second order tensor C can be written as Σi Σl C_ij e_i ⊗ e_l (using equation(13)). Noting this in the above expression, we see that the expression within the bracket is nothing but C_ij, i.e.

This chunk reveals how a second order tensor can be expressed in terms of its components and basis tensors. It shows that the result of the multiplication can be understood as a reorganization of the original tensor components into a structured expression, reinforcing the relationship between tensor multiplication and individual tensor components.

Imagine you have a set of building blocks (tensor components) arranged in various shapes (basis tensors). When you bring two sets together (multiply the tensors), you're essentially rearranging the blocks to form new shapes (the new tensor), but the types and amounts of blocks remain the same.

Writing this for all components together, we would get... Thus, we see that when we multiply two tensors, their matrix forms multiply in the usual way. The matrix form of the resultant tensor is simply the multiplication of the matrix forms of individual tensors.

Here, the text concludes the multiplication process by showing that the resulting tensor's matrix form is derived from the conventional matrix multiplication of the individual tensor matrices. This links back to the linear algebra principles that most students are already familiar with, indicating a sense of continuity and stability in tensor operations.

Consider combining two sets of colored marbles (matrix forms of tensors). When you multiply them, you're merging all colors according to specific rules, resulting in new blends (the resultant tensor). The physical act of sorting and counting reflects the logical approach to tensor multiplication.

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