Stress Distribution in Compound Cylinders

In summary, the problem discusses a long, hollow, thick elastic cylinder surrounded by a thin elastic band. An internal pressure is applied to the inside radius of the thick cylinder and the stress in the cylinder is given by a complex equation involving various parameters. The solution to this problem requires using plane strain equations instead of plane stress equations.
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Homework Statement


A long, hollow, thick elastic cylinder (E , v) is surrounded by a thin elastic band of thickness t made of a different material ([itex]E_2[/itex] , [itex]v_2[/itex]). The fit is ideal, such that neither a gap nor a pressure exists. An internal pressure p is then applied at the inside radius (r=a) of the thick cylinder. Assuming that the radial displacements is continuous at the interface (r=b), prove that the stress in the thick cylinder are given by
[itex]\sigma_{rr} = -p\frac{\beta(1+(b/r)^2)+(1-(1-((b/r)^2)}{\beta(1+(b/a)^2)+(1-((b/a)^2)}[/itex]

where
[itex]\beta= \frac{(1-v^2)}{(1+v) v-(bE)(1-v_b^2)/(E_b h)}[/itex]

Homework Equations


Lame's Equations. If a long cylinder is subject to internal pressure [itex]p_i[/itex] at [itex]r_i[/itex] and external pressure [itex]p_o[/itex] at [itex]r_o[/itex] then the radial stress distribution is

[itex]\sigma_{rr}=\frac{p_i r_i^2-p_o r_o^2}{r_o^2-r_i^2}+\frac{r_i^2 r_0^2(p_o-p_i)}{r^2(r_o^2-r_i^2)}[/itex]

and the tangential stress distribution
[itex]\sigma_{tt}=\frac{p_i r_i^2-p_o r_o^2}{r_o^2-r_i^2}-\frac{r_i^2 r_0^2(p_o-p_i)}{r^2(r_o^2-r_i^2)}[/itex].

and finally the displacement is

[itex]u_r=\frac{1-v}{E}\frac{(p_i r_i^2 - p_o r_o^2)}{r_o^2-r_i^2} r + \frac{1+v}{E}\frac{r_i^2 r_o^2 (p_i-p_o)}{r_o^2-r_i^2}\frac{1}{r}[/itex].

The Attempt at a Solution



I attempted to match the displacements at the interface and solve for the interface pressure [itex]P[/itex]. However, I can not get the correct solution with this approach.
The displacement of the inner cylinder at the interface is given by
[itex]u_r(r=b)=\frac{1-v}{E}\frac{(p_i a^2 - P b^2)}{b^2-a^2} b + \frac{1+v}{E}\frac{a^2 b^2 (p_i-P)}{b^2-a^2}\frac{1}{b}[/itex].

Canceling terms yields
[itex]u_r(r=b)=\frac{1}{E(b^2-a^2)}(2p_i a^2 b - (1-v)P b^3 - (1+v)P a^2 b)[/itex]
The displacement at the interface of the outer cylinder is given by:

[itex]\sigma_\theta ≈ E \epsilon_\theta→ u(r=b) = \frac{Pb^2}{E_2 t}[/itex]

because the outer elastic band is thin. Equating the displacements I get the interface pressure to

[itex]P = \frac{2 p_i a^2}{\frac{Eb(b^2-a^2)}{E_2 t}+(1-v)b^2+(1+b)a^2}[/itex].

I know this is the wrong interface pressure by check the solution given in the problem statement and through FEA analysis.
 
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Answer

There is a typo in my post above because [itex](1+b)[/itex] should be [itex](1+v)[/itex] in the denominator of the expression for [itex]P[/itex]. The mistake I made was using the lame's equations for plain stress, while the problem wanted to plane strain solution. The expression for [itex]P[/itex] matches the given result by making the plane stress to plane strain substitutions [itex]E\rightarrow E/(1-v^2)[/itex] and [itex]v\rightarrow v/(1-v)[/itex].
 
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1. What is the concept of stress distribution in compound cylinders?

The concept of stress distribution in compound cylinders is the study of how forces are distributed and transmitted through a cylindrical object that is composed of multiple layers or materials. This is important in understanding the structural integrity and performance of compound cylinders in various applications.

2. How does the geometry of a compound cylinder affect stress distribution?

The geometry of a compound cylinder, such as the thickness and arrangement of its layers, can greatly impact stress distribution. A thicker outer layer, for example, can bear more stress and distribute it to the inner layers. Similarly, the shape and orientation of the layers can affect how stress is distributed.

3. What factors can influence stress distribution in compound cylinders?

Aside from geometry, other factors that can influence stress distribution in compound cylinders include the material properties of each layer, the magnitude and direction of applied forces, and the boundary conditions at the ends of the cylinder. These factors all play a role in determining how stress is distributed throughout the cylinder.

4. How is stress distribution in compound cylinders analyzed?

The analysis of stress distribution in compound cylinders involves mathematical and computational methods, such as finite element analysis, to model the behavior of the cylinder under different loading conditions. Experimental techniques, such as strain gauges, can also be used to measure stress distribution in real-world scenarios.

5. What are the practical applications of understanding stress distribution in compound cylinders?

Understanding stress distribution in compound cylinders is crucial in many engineering and scientific fields, such as in the design and construction of pressure vessels, pipes, and aerospace structures. It also has applications in the medical field, such as in the development of prosthetic implants, where the distribution of stress can affect the longevity and performance of the implant.

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