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Chapter 7: Problem 928

A uniform cube is subjected to volume compression. If each side is decreased by \(1 \%\) Then what is bulk strain ? (A) \(0.01\) (B) \(0.06\) (C) \(0.02\) (D) \(0.03\)

Short Answer

Expert verified
The bulk strain is approximately -0.03, so the correct answer is (D) \(0.03\).

Step by step solution

01

Calculate the initial volume

Let's denote the initial side length of the cube as s. The initial volume of the cube V₁ = s³
02

Find the new side length

As each side is subjected to a volume compression of 1%, the new side length, s', will be 99% of its original length: s' = 0.99s
03

Calculate the new volume

The new volume of the cube, V₂, can be found using the new side length: V₂ = (s')³ = (0.99s)³
04

Determine the change in volume

Now, we can find the change in volume, ΔV, by subtracting the initial volume from the new volume: ΔV = V₂ - V₁ = (0.99s)³ - s³
05

Calculate the bulk strain

The bulk strain is defined as the ratio of the change in volume to the initial volume: Bulk Strain = \( \frac{ΔV}{V₁} \) Substituting the expressions for ΔV and V₁ we found earlier, we get: Bulk Strain = \( \frac{(0.99s)^3 - s^3}{s^3} \)
06

Simplify the expression

Now, let's simplify the expression for the Bulk Strain: Bulk Strain = \( \frac{(0.99^3)s^3 - s^3}{s^3} \) Since the denominator and the numerators all have the same s³, we can cancel them out: Bulk Strain = \( 0.99^3 - 1 \)
07

Calculate the numerical value

Now, we will compute the numerical value of the Bulk Strain: Bulk Strain = \( 0.99^3 - 1 ≈ -0.0297 \)
08

Select the correct answer

Since the calculated Bulk Strain ≈ -0.0297 ≈ -0.03, the correct answer is: (D) \(0.03\)

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