3. Second order tensors and their representation

The tensor product could also be written as follows: (12) Upon contrasting the above form with the expansion of a vector in equation (4), we make a note that, just like a general vector is expressed as a linear combination of three basis vectors, a tensor can be expressed as a linear combination of nine basis tensors. Each of the basis here (e ⊗ e) are themselves tensors.

This chunk discusses how a second order tensor can be represented as a combination of basis tensors. Just like vectors, which can be expressed based on their components along specific basis vectors, tensors are also expressed through their components along basis tensors. Here, the tensor product notation (e ⊗ e) indicates that we're looking at combinations of basis tensors. Each of the nine basis tensors represents a unique aspect of the tensor being analyzed.

Imagine building a complex structure from parts. Just like each part—like beams, rods, etc.—has a specific role in the structure, each basis tensor contributes to forming the complete second order tensor. Together, these parts determine the overall behavior of the structure, just as the coefficients of basis tensors define the characteristics of the second order tensor.

Thus, a general second order tensor can be written as (13) Thenine coefficients C are in general independent of each other. The coefficient C can be thought of as the component of the tensor C along the basis tensor e ⊗ e.

This part highlights that a second order tensor has nine components (C_ij) which can vary independently. It establishes that these coefficients represent specific interactions or properties that make up the tensor. Each C coefficient corresponds to a unique combination of the basis tensors, allowing us to describe the tensor's behavior in different contexts.

Consider the way a fabric can change its properties based on how the fibers are woven together. Each type of weave might impact the fabric’s stretch, strength, and appearance. Similarly, the nine coefficients in a second order tensor represent various properties that can influence the behavior of materials or systems in mechanics, allowing for detailed descriptions of how they respond to forces.

Using (10), we can also say the following for basis tensors, e.g., (14) Thus, just like basis vectors have unique column form, each of the basis tensors has a unique matrix form.

In this chunk, it’s emphasized that basis tensors can be represented in matrix form similar to basis vectors. This provides an intuitive way to visualize and manipulate tensors. The conversion from tensor notation to matrix form allows for easier computations and operations, analogous to how matrices are utilized in linear algebra.

Think of a matrix as a recipe that outlines how to mix different ingredients (basis tensors) to create a dish (the resulting tensor). Each specific combination of ingredients creates a unique flavor, much like how various arrangements and coefficients in the tensor create distinct mechanical properties or responses in materials when subjected to forces.

Using (13) and (14), it is easy to see that the coefficient C in (13) also forms the ith row and jth column of the matrix representation of C in (e ,e ,e ) coordinate system, i.e., . Just like vectors, the matrix form of a tensor changes from one coordinate system to the other but the tensor itself does not change.

This section explains that while the matrix representation of a tensor will differ when viewed in different coordinate systems, the underlying tensor remains unchanged. This demonstrates the property of tensors being independent of the coordinate system. By understanding how tensors transform, we can better apply them across various scenarios in physics, engineering, and beyond.

Consider how you view an object from different angles. While the appearance might change based on your perspective (like the orientation of a matrix), the object itself remains the same. Tensors work similarly; no matter how you look at them, their properties and behaviors recognized within a system stay constant.

As a final remark, tensors can be of any order which are all independent of the coordinate system but their representations change from one coordinate system to the other. Scalars, e.g., are zero-th order tensors and all vectors are first order tensors. We can also have third and fourth (even higher) order tensors.

This closing chunk clarifies the concept of tensor order, emphasizing that tensors can range from scalars (zero-th order) to vectors (first order), and beyond to higher orders. Each increase in order corresponds to a more complex interaction or relationship represented by the tensor. Understanding tensor order helps in categorizing physical phenomena appropriately.

Imagine a family based on different generations. A scalar (like a single person) represents the zero-th order, while a one-child family (vector) is the first order. If you see grandparents (second order) who have multiple children (third order), you realize that as generations continue, the family tree becomes more intricate, similar to how tensors of higher orders reflect more complex relationships and interactions in systems.

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