3.2.1. Representation of stress tensor in Cartesian coordinate system

Cartesian coordinate system means that our coordinate axes are e1, e2, and e3 (perpendicular axes).

In a Cartesian coordinate system, we define the axes, typically denoted as e1, e2, and e3, to be mutually perpendicular. This is important because it allows us to easily visualize and analyze how forces and stresses act in three-dimensional space.

Think of a box-shaped room where each wall faces one of the three axes: one wall is aligned with the e1 axis (let's say the front wall), one with the e2 axis (the side wall), and the last with the e3 axis (the ceiling). This setup helps in understanding how an object interacts with its environment.

We want to know the stress tensor at a given point x in our body and represent the stress matrix in the Cartesian coordinate system.

The stress tensor is a mathematical description of the internal forces within a material at a specific point, denoted as point x. It captures how these forces act on the material. In the Cartesian coordinate system, we express this tensor as a matrix that helps us visualize and compute the stress in different directions around that point.

Imagine holding a sponge at a point with your fingers pressing in from different sides. The way the sponge deforms at that point due to your fingers is analogous to the stress tensor—different forces are acting in various directions, and the stress tensor helps describe this.

Think of a cuboid around the point x as shown in Figure 3 on the left. It is centered at x. Its six faces are along e1, e2, e3, -e1, -e2, -e3 respectively.

To better understand the stress tensor, we visualize a small cuboid centered at point x. This cuboid has six surfaces (or faces), each of which corresponds to a side of the Cartesian axes. This visualization helps us analyze the forces acting on each face and how they relate to the internal stress at point x.

Consider a dice placed on a table. Each face of the dice corresponds to the cuboid's faces, and the weight of the dice applies pressure on the table. The way the table interacts with the weight can be likened to how stresses interact with material at point x.

The traction that acts on e1 plane is t1. It has three components as shown in Figure 3. The components along e1 are σ11, τ12 and τ13.

The stress tensor consists of different components characterized by normal and shear stresses. For the plane aligned with the e1 axis, σ11 represents the normal stress (acting perpendicular to the face), while τ12 and τ13 represent shear stresses (acting parallel to the face) on that plane. This differentiation is crucial in understanding how materials behave under various loading conditions.

Imagine slicing a chocolate cake. If you push down with a knife (normal stress), it will compress the cake where the knife contacts it (like σ11). But if you slide the knife to the side without pressing down (shear stress), it will cause the cake layers to slide against each other (like τ12 and τ13), demonstrating how shear and normal stresses work in real life.

We can conclude that σ11 is normal to the plane whereas τ12 and τ13 are in the plane. Thus, σ11 is called the normal component of traction and τ12 and τ13 are called the shear components of traction.

Normal components of stress act perpendicular to surfaces, causing compression or tension, while shear components act parallel to the surfaces, causing distortion. This distinction is critical for predicting how materials will deform or fail under load, enabling engineers to design safer and more effective structures.

Think about pushing two books stacked on top of each other. If you push straight down (normal force), you compress the books (similar to σ11). If instead, you try to slide the top book to the side (shear force), you are mimicking the effect of τ12 and τ13. Understanding these concepts helps engineers prevent structural failures in buildings and bridges.

The shear components of the traction are in the plane, and they try to shear the body.

Shear forces play a crucial role in understanding material behavior. They cause layers within a material to slide past one another, leading to deformation. Recognizing and calculating shear stress is essential for ensuring that materials can withstand applied loads without failing.

Consider how gel-like products sometimes slide out when pressed. If you apply a force to the side of the gel (like cutting through it), it slides smoothly along its surface due to shear forces. Recognizing how shear stress influences the material's integrity is essential for engineers designing with such materials.

Remark: Positive σ11 implies it is tensile in nature and negative σ11 means it is compressive in nature.

Understanding whether stress is positive or negative is vital for analyzing material behavior. A positive stress indicates that the material is being pulled apart (tensile), while negative stress indicates compression, which can lead to buckling or failure. Recognizing these stress conditions helps engineers assess how much force materials can safely withstand during use.

Imagine stretching a rubber band (positive tensile stress) versus squeezing a balloon (negative compressive stress). In both scenarios, you can see how the material reacts differently depending on whether you are stretching it or compressing it.

We need to find t−2. We have already seen in previous lectures that: t−2 = −t2.

Finding the traction on the opposite face of a plane involves understanding that forces balance out by acting in opposite directions. For example, if a force acts on one side of a cuboidal volume, the equal and opposite reaction force acts on the opposite side. This fundamental principle helps maintain equilibrium in structures.

Consider balloon inflation. As air fills a balloon, the pressure on one side pushes outward; simultaneously, an equal force presses in from the inner side, ensuring the balloon remains balanced. In structural engineering, recognizing these reactive forces is critical for safe design and load management.

We should keep in mind that we wanted to find out the stress matrix at the point x and these six planes that we have here are not passing through the point x.

To accurately determine the stress matrix at point x, the planes through which traction is analyzed must intersect with point x. If they do not, the traction values do not accurately represent the stresses present at point x, making it crucial to focus on the correct geometry during analysis.

Think about taking a photo of a person with a camera: the lens must focus directly on the person's face for the image to be clear. Similarly, in mechanics, the traction must be calculated accurately at the specific location to understand its behavior and ensure safety.

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