The main question of this tutorial is "what is stress?" And no, I do not mean the feeling you get when overwhelmed, I mean the internal resistive force that all materials exhibit when a force is applied to it! Whenever we touch, push or pull object, a force is applied to it, and with every object, its material will dictate what happens when that force is applied. If a force is large enough, objects can break, wood can splinter, and metal can bend.
Now, with every force, it will be applied with a given area, for example if one were to push on a table with their fingertip vs pushing on a table with a book, the area distribution of the book would be larger than the area distribution of your fingertip. Now this quantity of force per unit area is often referred to as pressure which is denoted as $ P $ which is equal to a force, $ F $, divided by an area, $ A $, or
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$$ P = \frac{F}{A} $$
However, there is another name for pressure, and that is stress.
There are two types of stress:
In fact, for a 3D space, there are 6 unique components of stress. $ \sigma_{x} $, $ \sigma_{y} $, $ \sigma_{z} $, $ \tau_{xy} $, $ \tau_{yz} $, and $ \tau_{xz} $. $ \sigma_{x} $, $ \sigma_{y} $, and $ \sigma_{z} $ are the normal stresses, and $ \tau_{xy} $, $ \tau_{yz} $, and $ \tau_{xz} $ are the shear stresses.
We can actually describe our stresses in the form of a stress tensor, or matrix. If you are familiar with the moment of inertia matrix, the format is very similar. We can let the matrix $ [\sigma] $ be defined as
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$$ [\sigma] = \begin{bmatrix} \sigma_x & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_y & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_z \end{bmatrix} $$
Now, matrix $ [\sigma] $ is a symmetrical matrix, which means if you draw a line along the diagonal (going from the top left to the top right, the values will be mirrored. In other words, $ \tau_{xy} = \tau_{yx} $, $ \tau_{yz} = \tau_{zy} $, and $ \tau_{xz} = \tau_{zx} $.
I want to talk real quick about the meaning of the subscript for stresses. Notice how $ \tau_{xy} $, for example, has a subscript of $ xy $. Well this isn't randomly chosen because there is an inherent meaning as to how the variables are arranged. The first variable denotes the normal direction to the plane the stress is acting, and the second variable denotes the direction the stress points. Let's take a look at an example of this to see what I mean.
Fig. 1 - Visualization of stresses on an object.
Let's take a look at the green shear stress, $ \tau_{xy} $ on the right side of the square. The shear stress is acting on the right face of the object who's normal vector (perpendicular vector) is pointing in the x direction, hence the first variable of its subscript being x. It is also pointing in the y direction, hence the second variable in the subscript being y.
You may notice for normal stresses though, I only wrote them with one variable as their subscript, $ x $, $ y $, or $ z $. They can be written as $ \sigma_{xx} $, $ \sigma_{yy} $, and $ \sigma_{zz} $, or $ \sigma_{x} $, $ \sigma_{y} $, and $ \sigma_{z} $. They both mean the same thing! This is because a normal stress is pointing along an object face's normal vector. For example, the pink normal stress, $ \sigma_x $ is acting on the right face of the object which has a normal vector pointing in the x direction, and $ \sigma_x $ is pointing in the x direction as well, hence, where both x's come from.
Stresses are very important in this field of study and so we will talk about them in great lengths!