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A short class with Sarah and her students. The other ways are under More.
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Today, we're going to explore the concept of traction vectors on the +z plane. Can anyone tell me what a traction vector represents?
Isn't it the force per unit area acting at a point on a surface?
Exactly! And in a cylindrical coordinate system, the traction vector on the +z plane can be expressed as: t+z = σ e + τ e + τ e. Does anyone remember what these components represent?
I think σ is normal stress, while τ represents shear stress components.
Good memory! We’ll check these components as we go deeper into the calculations.
Next, let’s look at how we calculate the total force on the +z plane by integrating the traction vector over the area. Can anyone describe how we approach the integration?
Do we start with an infinitesimal area element and then integrate it over the total area?
Correct! The area element on the +z plane is dA = (r+ξ) dξ dη. Now, can anyone explain why we use this expression for the area element?
Because dA needs to account for the radial distance multiplied by the small angle sweep in the direction.
Great explanation! Now we’ll use this area element to find the total force on the +z plane.
Now let's move to the -z plane. Who can share what changes when calculating the force on this plane compared to the +z plane?
The traction components would act in the negative direction, right?
Exactly! If the coordinates for a point on -z are (r+ξ, θ+η, z - ∆z), we express the total force considering both the negative components and the integration over the area again. Always remember: the direction of the vectors matters!
So, when we do this for both planes, we can combine them to examine the overall forces, right?
Yes! That's an essential step when we derive the linear momentum balance.
Traction vector's the force we collect, on surfaces it's what we detect.
Imagine a cylindrical tin. On the top, the force acts strong. On the bottom, it’s just as wrong. Forces accumulate, as we integrate, making movement, we illustrate.
To remember the steps: T-I-F — Traction, Integrate, Force!
T-FAC: Traction For Area Calculation for radial and angular contributions.
Term
What is a traction vector?
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Term
How do we calculate the total force on the +z plane?
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A three-dimensional coordinate system defined by the distance from a fixed axis and the angle around that axis.
A vector that represents the force per unit area acting on a surface.
A differential element of area used in calculus to perform integration over a surface.
Components of stress, including normal stress and shear stress, that describe the internal forces acting within a material.
This section discusses the calculation of forces acting on the +z and -z planes in a cylindrical coordinate system and introduces the necessary equations for deriving linear momentum balance.
In this section, we derive and analyze the forces acting on the upper (+z) and lower (-z) planes of a cylindrical element. It includes the definitions of traction vectors and their integration over these planes to formulate the total forces acting on them, crucial for understanding linear momentum balance in cylindrical coordinates.
In this section, we focus on the forces acting on the +z and -z planes in a cylindrical coordinate system. The traction vector on the +z plane, represented as t+z, is defined using the stress components acting on this plane. The total force acting on the +z plane is calculated by integrating the traction vector over its defined area.
We start with the traction vector:
t_{+z} = au_{zz} e_z + au_{rz} e_r + au_{ heta z} e_{ heta}$$ where `σ`, `τ` represent various stress components. To find the total force on the +z plane, we consider an infinitesimal area element `dA`, factor in both the radial and angular dimensions, and subsequently perform the integration to yield the resultant force. We repeat a similar process for the -z plane, noting that the traction components will be directed negatively. Finally, by combining forces from both planes, we set up for further derivation of the linear momentum balance. This section sets the groundwork for understanding the dynamics of cylindrical structures in applied mechanics.