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Chapter 6: Problem 14

Using the solution to Problem \(6.13\), rank the magnitudes of the moduli of elasticity for the following hypothetical X, Y, and Z materials from the greatest to the least. The appropriate \(A, B\), and \(n\) parameters (Equation \(6.25)\), for these three materials are shown in the following table; they yield \(E_{N}\) in units of electron volts and \(r\) in nanometers: $$ \begin{array}{cccc} \hline \text { Material } & \boldsymbol{A} & \boldsymbol{B} & \boldsymbol{n} \\\ \hline \mathrm{X} & 2.5 & 2.0 \times 10^{-5} & 8 \\ \mathrm{Y} & 2.3 & 8.0 \times 10^{-6} & 10.5 \\ \mathrm{Z} & 3.0 & 1.5 \times 10^{-5} & 9 \\ \hline \end{array} $$

Short Answer

Expert verified
Question: Rank the magnitudes of the moduli of elasticity for hypothetical materials X, Y, and Z from greatest to least. Answer: Z > X > Y

Step by step solution

01

Recall the equation

Recall the equation for the modulus of elasticity (E) in terms of A, B, and n from Eq. 6.25: $$E_{N} = A - \frac{B}{r^n}$$ where \(E_{N}\) is the modulus of elasticity, A and B are constants, r is the atomic separation, and n is the exponent.
02

Plug in the values

Plug in the given A, B, and n values for materials X, Y, and Z: Material X: A = 2.5, B = 2.0 × 10\(^{-5}\), n = 8 Material Y: A = 2.3, B = 8.0 × 10\(^{-6}\), n = 10.5 Material Z: A = 3.0, B = 1.5 × 10\(^{-5}\), n = 9
03

Calculate the moduli of elasticity

For each material, calculate accordingly: For Material X: $$E_{N_{x}} = 2.5 - \frac{2.0 \times 10^{-5}}{r^{8}}$$ For Material Y: $$E_{N_{y}} = 2.3 - \frac{8.0 \times 10^{-6}}{r^{10.5}}$$ For Material Z: $$E_{N_{z}} = 3.0 - \frac{1.5 \times 10^{-5}}{r^{9}}$$ Since we don't have a value for r (atomic separation), we cannot directly calculate the modulus of elasticity. However, we can still compare the contributions of the terms in the equations to each other.
04

Rank the magnitudes

We will rank the magnitudes of the moduli of elasticity based on the impact of the terms in the equation: For large r (atomic separation), the second term of the equation becomes small, and the modulus of elasticity approximates the A value. Based on the A values, the ranking would be: Z > X > Y For small r, the second term becomes more significant. However, since Material Z has a larger A value and Material Y has a smaller B value with a large exponent, this ranking will not change. Therefore, the ranking order remains: Z > X > Y

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Most popular questions from this chapter

As noted in Section 3.15, for single crystals of some substances, the physical properties are anisotropic; that is, they are dependent on crystallographic direction. One such property is the modulus of elasticity. For cubic single crystals, the modulus of elasticity in a general \([u v w]\) direction, \(E_{u v w}\), is described by the relationship $$ \begin{gathered} \frac{1}{E_{u v w}}=\frac{1}{E_{(100}}-3\left(\frac{1}{E_{(100}}-\frac{1}{E_{(111)}}\right) \\\ \left(\alpha^{2} \beta^{2}+\beta^{2} \gamma^{2}+\gamma^{2} \alpha^{2}\right) \end{gathered} $$ where \(E_{100}\) and \(E_{(111)}\) are the moduli of elasticity in \([100]\) and [111] directions, respectively; \(\alpha, \beta\), and \(\gamma\) are the cosines of the angles between \([u v w]\) and the respective \([100],[010]\), and [001] directions. Verify that the \(E_{\langle 110\rangle}\) values for aluminum, copper, and iron in Table \(3.3\) are correct.

A cylindrical specimen of a titanium alloy having an elastic modulus of \(107 \mathrm{GPa}\) (15.5 \(\times\) \(10^{6} \mathrm{psi}\) ) and an original diameter of \(3.8 \mathrm{~mm}\) (0.15 in.) will experience only elastic deformation when a tensile load of \(2000 \mathrm{~N}\) \(\left(450 \mathrm{lb}_{\mathrm{f}}\right)\) is applied. Compute the maximum length of the specimen before deformation if the maximum allowable elongation is \(0.42\) \(\mathrm{mm}(0.0165\) in.).

A bar of a steel alloy that exhibits the stress-strain behavior shown in Figure \(6.21\) is subjected to a tensile load; the specimen is 300 mm (12 in.) long and has a square cross section \(4.5 \mathrm{~mm}(0.175 \mathrm{in}\).) on a side. (a) Compute the magnitude of the load necessary to produce an elongation of \(0.45 \mathrm{~mm}\) \((0.018\) in.). (b) What will be the deformation after the load has been released?

A cylindrical rod \(100 \mathrm{~mm}\) long and having a diameter of \(10.0 \mathrm{~mm}\) is to be deformed using a tensile load of \(27,500 \mathrm{~N}\). It must not experience either plastic deformation or a diameter reduction of more than \(7.5 \times 10^{-3} \mathrm{~mm}\). Of the materials listed as follows, which are possible candidates? Justify your choice(s). $$ \begin{array}{lccc} \hline \begin{array}{c} \text { Material } \end{array} & \begin{array}{c} \text { Modulus } \\ \text { of Elasticity } \\ \text { (GPa) } \end{array} & \begin{array}{c} \text { Yield } \\ \text { Strength } \\ \text { (MPa) } \end{array} & \begin{array}{c} \text { Poisson's } \\ \text { Ratio } \end{array} \\ \hline \text { Aluminum alloy } & 70 & 200 & 0.33 \\ \text { Brass alloy } & 101 & 300 & 0.34 \\ \text { Steel alloy } & 207 & 400 & 0.30 \\ \text { Titanium alloy } & 107 & 650 & 0.34 \\ \hline \end{array} $$

A cylindrical specimen of an alloy \(8 \mathrm{~mm}\) (0.31 in.) in diameter is stressed elastically in tension. A force of \(15,700 \mathrm{~N}\) (3530 lb_{f } \()\) produces a reduction in specimen diameter of \(5 \times 10^{-3} \mathrm{~mm}\left(2 \times 10^{-4}\right.\) in.). Compute Poisson's ratio for this material if its modulus of elasticity is \(140 \mathrm{GPa}\left(20.3 \times 10^{6} \mathrm{psi}\right)\).
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