Question

The Poisson's ratio of the wall material is 1/3​. When it was subjected to some internal pressure, its nominal perimeter in the cylindrical portion increased by 0.1%, and the corresponding wall thickness became:

a) Decreased by 0.1%
b) Unchanged
c) Increased by 0.1%
d) Increased by 0.3%

Answer 1

When it was subjected to some internal pressure, its nominal perimeter in the cylindrical portion increased by 0.1%, and the corresponding wall thickness became a) decreased by 0.1%

How to find rate?

Poisson's ratio (
`\( \\u \)`) is defined as the ratio of transverse contraction strain to longitudinal extension strain in the direction of stretching force. Mathematically, it is expressed as:

`\[ \\u = -\frac{\varepsilon_{\text{transverse}}}{\varepsilon_{\text{longitudinal}}} \]`

Where:

`\( \varepsilon_{\text{transverse}} \)` = transverse strain,

`\( \varepsilon_{\text{longitudinal}} \)` = longitudinal strain.

The relationship between the change in diameter (
`\( \Delta D \)`), change in length (
`\( \Delta L \)`), and Poisson's ratio is given by:

`\[ (\Delta D)/(D) = -\\u (\Delta L)/(L) \]`

Given that the nominal perimeter (P) is related to the diameter (D) by
`\( P = \pi D \)`, express the change in perimeter as:

`\[ (\Delta P)/(P) = (\Delta D)/(D) \]`

Now, if the nominal perimeter increases by 0.1%, then
`\( (\Delta P)/(P) = 0.001 \)` (0.1%).

Substitute this into the relationship and solve for
`\( (\Delta L)/(L) \)`:

`\[ 0.001 = -\\u (\Delta L)/(L) \]`

Given that
`\( \\u = (1)/(3) \)`, substitute this value:

`\[ 0.001 = -(1)/(3) (\Delta L)/(L) \]`

Now, solve for
`\( (\Delta L)/(L) \)`:

`\[ (\Delta L)/(L) = -3 * 0.001 \]`

`\[ (\Delta L)/(L) = -0.003 \]`

So, the longitudinal strain (
`\( (\Delta L)/(L) \)`) is -0.003 or -0.3%.

Now, the wall thickness is related to the diameter by:

`\( \text{thickness} = \frac{D - \text{inner diameter}}{2} \)`.

If
`\( (\Delta D)/(D) = -0.003 \)`, then:

`\( \frac{\Delta \text{thickness}}{\text{thickness}} = -0.003 \)`.

This means the wall thickness decreases by 0.3%.

Therefore, the correct answer is:

a) Decreased by 0.3%