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A short class with Sarah and her students. The other ways are under More.
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Today, let's start with finding the unknown constants in our expression for the longitudinal displacement, represented as u.
Why do we need to find these constants?
Good question! Finding these constants is essential because they help us relate the displacement to the physical properties of the material under certain boundary conditions.
What are those boundary conditions?
We have two key conditions: the internal pressure acting on the inner surface and zero pressure on the outer surface. These will help us solve for the constants.
So, does that mean the constants we find are specific to the pressure applied?
Exactly! Each scenario can lead to different values for C and D based on the applied stresses and material properties.
Can we summarize the steps for solving this?
Sure! Identify the boundary conditions, apply them to our expressions, and solve for constants C and D through integration. Remember, this helps us understand how the hollow cylinder behaves under pressure.
Now that we have a grasp of constants, let's look into applying the boundary conditions. What do we find at the inner surface?
We find that the stress is equal to the internal pressure.
Exactly! So we can set the radial stress equal to -P and move ahead with our equations.
And at the outer surface?
Right, there is no traction, so the stress is zero there. This gives us the second boundary condition to work from.
So these conditions lead to equations we can solve for our constants?
Precisely! It’s all about substituting these values into our derived equations and solving for the constants.
Can we write this in the equation format?
Of course! Let’s summarize these boundary conditions mathematically: σ (r1) = -P and σ (r2) = 0.
Now, let’s derive the final expression for radial displacement, u. What have we gathered so far?
We have our constants and the boundary conditions merged with our displacement equations.
Right! Plug those values into the derived equation for u. What do you observe?
It incorporates Young's modulus and Poisson's ratio, showing how they affect displacement.
Great insight! Do you think the equation changes if there is no pressure?
Yes, when P = 0, it simplifies, showing the effect of Poisson’s ratio even without external pressure.
Can we visualize how this displacement looks physically?
Absolutely! We can understand radial deformation as a result of internal forces, demonstrated visually through deformation under pressure.
Young's stays stiff, follows the lift, as boundary conditions play a key shift.
Imagine a hollow cylinder being pushed from the inside. It stretches in the middle, while holding its pressure at the ends, revealing how boundary conditions define its fate.
Remember 'C' for Constants, 'P' for Pressure, 'D' for Displacement. Each plays a role in the mechanics puzzle.
BYP - Boundary Conditions, Yield, and Pressure.
Term
What is the significance of Young's Modulus?
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Define Poisson's Ratio.
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What do boundary conditions provide in physics?
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The axial deformation experienced by a material or structure when subjected to loads.
Conditions at the boundary of a system which are necessary to solve differential equations.
A measure of the stiffness of a solid material, defined as the ratio of stress to strain.
The ratio of transverse strain to axial strain in a material subjected to axial loading.
Stress that occurs in the radial direction within a material, particularly in cylindrical structures.
The section deals with deriving the final expression for the longitudinal deformation in a hollow cylinder, considering various constants and boundary conditions.
In this section, we focus on finding the constants in the expression for the longitudinal displacement in a hollow cylinder. The analysis involves applying boundary conditions, deriving expressions for axial and circumferential stresses, and ultimately results in the final expression for axial deformation, integrating parameters including Young’s modulus and Poisson’s ratio.
This section delves into deriving the final form of the longitudinal displacement (2) in a hollow cylinder under internal pressure. We start by recalling the expression derived previously and focus on determining the unknown constants (C, D) to finalize the solution.
The finalized equation for u comprehensively incorporates the effects of eigenstates like Young’s modulus and Poisson’s ratio, culminating in an expression that signifies the radial displacement effect even in the absence of applied pressure, a crucial insight into material behavior under torsional and axial loads.
This development is critical as it not only provides the last piece of the puzzle for analyzing hollow cylinder deformation but also serves as a basis for understanding material properties in engineering applications, especially in extension, torsion, and inflation scenarios.