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Chapter 12: Problem 45

A circular specimen of \(\mathrm{MgO}\) is loaded using a three-point bending mode. Compute the minimum possible radius of the specimen without fracture, given that the applied load is \(5560 \mathrm{~N}\left(1250 \mathrm{lb}_{\mathrm{f}}\right)\), the flexural strength is \(105 \mathrm{MPa}(15,000 \mathrm{psi})\), and the separation between load points is \(45 \mathrm{~mm}\) (1.75 in.).

Short Answer

Expert verified
Answer: The minimum possible radius of the MgO specimen under the given conditions is approximately 3.32 mm.

Step by step solution

01

Write down the given information

We are given the applied load \(F = 5560 \mathrm{~N}\), the flexural strength \(\sigma = 105 \mathrm{MPa}\), and the separation between load points \(L = 45 \mathrm{~mm}\). We will also convert the flexural strength to \(\mathrm{N/mm}^2\), so \(\sigma = 10^{3} \times 105 \mathrm{N/mm}^2\).
02

Write down the equation for bending stress in a three-point bending test

To find the minimum possible radius \(R\), we need to find the stress in the specimen. The equation for the bending stress in a three-point bending test is: \(\sigma = \frac{3FL}{2\pi R^3}\)
03

Substitute the given information into the equation

Substitute the given values for applied load, flexural strength, and separation between load points into the equation: \(10^{3} \times 105 = \frac{3 \times 5560 \times 45}{2\pi R^3}\)
04

Solve for the radius R

Now we need to solve the equation for the radius \(R\). First, simplify the equation: \(R^3 = \frac{3 \times 5560 \times 45}{2\pi \times 10^3 \times 105}\) Next, evaluate the right side of the equation: \(R^3 \approx 36.53\) Lastly, find the cube root of both sides: \(R \approx 3.32\) The minimum possible radius of the MgO specimen without fracture under the given conditions is approximately \(3.32 \mathrm{~mm}\).

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Most popular questions from this chapter

Iron titanate, \(\mathrm{FeTiO}_{3}\), forms in the ilmenite crystal structure that consists of an HCP arrangement of \(\mathrm{O}^{2-}\) ions. (a) Which type of interstitial site will the \(\mathrm{Fe}^{2+}\) ions occupy? Why? (b) Which type of interstitial site will the \(\mathrm{Ti}^{4+}\) ions occupy? Why? (c) What fraction of the total tetrahedral sites will be occupied? (d) What fraction of the total octahedral sites will be occupied?

(a) Suppose that CaO is added as an impurity to Li2O. If the Ca2- substitutes for Li, what kind of vacancies would you expect to form? How many of these vacancies are created for every Ca2- added? (b) Suppose that CaO is added as an impurity to CaCl2. If the O2substitutes for Cl, what kind of vacancies would you expect to form? How many of these vacancies are created for every O2 added?

The corundum crystal structure, found for \(\mathrm{Al}_{2} \mathrm{O}_{3}\), consists of an HCP arrangement of \(\mathrm{O}^{2-}\) ions; the \(\mathrm{Al}^{3+}\) ions occupy octahedral positions. (a) What fraction of the available octahedral positions are filled with \(\mathrm{Al}^{3+}\) ions? (b) Sketch two close-packed \(\mathrm{O}^{2-}\) planes stacked in an \(A B\) sequence, and note octahedral positions

In terms of bonding, explain why silicate mate- rials have relatively low densities.

One crystalline form of silica (SiO2) has a cu- bic unit cell, and from x-ray diffraction data it is known that the cell edge length is 0.700 nm. If the measured density is 2.32 g/cm3 , how many Si4 and O2 ions are there per unit cell?
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