Materials Science and Engineering. An Introduction

William D. Callister

Chapter 8

Failure - all with Video Answers

Educators

Chapter Questions

Problem 1

What is the magnitude of the maximum stress that exists at the tip of an internal crack having a radius of curvature of $1.9 \times 10^{-4} \mathrm{~mm}$ ( $7.5 \times 10^{-6}$ in.) and a crack length of $3.8 \times$ $10^{-2} \mathrm{~mm}\left(1.5 \times 10^{-3} \mathrm{in}\right.$.) when a tensile stress of $140 \mathrm{MPa}(20,000 \mathrm{psi})$ is applied?

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

Estimate the theoretical fracture strength of a brittle material if it is known that fracture occurs by the propagation of an elliptically shaped surface crack of length 0.5 mm ( 0.02 in .) and having a tip radius of curvature of $5 \times$ $10^{-3} \mathrm{~mm}\left(2 \times 10^{-4} \mathrm{in}.\right)$, when a stress of $1035 \mathrm{MPa}(150,000 \mathrm{psi})$ is applied.

Narayan Hari

Numerade Educator

Problem 3

If the specific surface energy for aluminum oxide is $0.90 \mathrm{~J} / \mathrm{m}^2$, using data contained in Table 12.5 , compute the critical stress required for the propagation of an internal crack of length 0.40 mm .

Narayan Hari

Numerade Educator

Problem 4

An MgO component must not fail when a tensile stress of $13.5 \mathrm{MPa}(1960 \mathrm{psi})$ is applied. Determine the maximum allowable surface crack length if the surface energy of MgO is $1.0 \mathrm{~J} / \mathrm{m}^2$. Data found in Table 12.5 may prove helpful.

Narayan Hari

Numerade Educator

Problem 5

A specimen of a 4340 steel alloy with a plane strain fracture toughness of $54.8 \mathrm{MPa} \sqrt{\mathrm{m}}$ ( $50 \mathrm{ksi} \sqrt{\mathrm{in}}$.) is exposed to a stress of 1030 MPa $(150,000 \mathrm{psi})$. Will this specimen experience fracture if it is known that the largest surface crack is 0.5 mm ( 0.02 in .) long? Why or why not? Assume that the parameter $Y$ has a value of 1.0 .

Narayan Hari

Numerade Educator

Problem 6

Some aircraft component is fabricated from an aluminum alloy that has a plane strain fracture toughness of $40 \mathrm{MPa} \sqrt{\mathrm{m}}(36.4 \mathrm{ksi} \sqrt{\mathrm{in}})$. It has been determined that fracture results at a stress of $300 \mathrm{MPa}(43,500 \mathrm{psi})$ when the maximum (or critical) internal crack length is $4.0 \mathrm{~mm}(0.16 \mathrm{in}$ ). For this same component and alloy, will fracture occur at a stress level of $260 \mathrm{MPa}(38,000 \mathrm{psi})$ when the maximum internal crack length is $6.0 \mathrm{~mm}(0.24 \mathrm{in}$.$) ? Why$ or why not?

Narayan Hari

Numerade Educator

Problem 7

Suppose that a wing component on an aircraft is fabricated from an aluminum alloy that has a plane strain fracture toughness of $26 \mathrm{MPa} \sqrt{\mathrm{m}}$ $(23.7 \mathrm{ksi} \sqrt{\mathrm{in}}$.). It has been determined that fracture results at a stress of 112 MPa $(16,240 \mathrm{psi})$ when the maximum internal crack length is 8.6 mm ( 0.34 in .). For this same component and alloy, compute the stress level at which fracture will occur for a critical internal crack length of $6.0 \mathrm{~mm}(0.24 \mathrm{in}$.).

Narayan Hari

Numerade Educator

Problem 8

A large plate is fabricated from a steel alloy that has a plane strain fracture toughness of $82.4 \mathrm{MPa} \sqrt{\mathrm{m}}(75.0 \mathrm{ksi} \sqrt{\mathrm{in}}$.). If, during service use, the plate is exposed to a tensile stress of $345 \mathrm{MPa}(50,000 \mathrm{psi})$, determine the minimum length of a surface crack that will lead to fracture. Assume a value of 1.0 for $Y$.

Manik Pulyani

Numerade Educator

Problem 9

Calculate the maximum internal crack length allowable for a Ti-6Al-4V titanium alloy (Table 8.1) component that is loaded to a stress one-half of its yield strength. Assume that the value of $Y$ is 1.50 .

Narayan Hari

Numerade Educator

Problem 10

A structural component in the form of a wide plate is to be fabricated from a steel alloy that has a plane strain fracture toughness of $98.9 \mathrm{MPa} \sqrt{\mathrm{m}}(90 \mathrm{ksi} \sqrt{\mathrm{in}})$ and a yield strength of $860 \mathrm{MPa}(125,000 \mathrm{psi})$. The flaw size reso-. The flaw size resolution limit of the flaw detection apparatus is $3.0 \mathrm{~mm}(0.12 \mathrm{in}$.$) . If the design stress is one-$ half of the yield strength and the value of $Y$ is 1.0 , determine whether or not a critical flaw for this plate is subject to detection.

Manik Pulyani

Numerade Educator

Problem 11

After consultation of other references, write a brief report on one or two nondestructive test techniques that are used to detect and measure internal and/or surface flaws in metal alloys.

Ajay Singhal

Numerade Educator

Problem 12

Following is tabulated data that were gathered from a series of Charpy impact tests on a tempered 4340 steel alloy.
$$
\begin{array}{cc}
\hline \text { Temperature }\left({ }^{\circ} \text { C }\right) & \text { Impact Energy (J) } \\
\hline 0 & 105 \\
-25 & 104 \\
-50 & 103 \\
-75 & 97 \\
-100 & 63 \\
-113 & 40 \\
-125 & 34 \\
-150 & 28 \\
-175 & 25 \\
-200 & 24 \\
\hline
\end{array}
$$
(a) Plot the data as impact energy versus temperature.
(b) Determine a ductile-to-brittle transition temperature as that temperature corresponding to the average of the maximum and minimum impact energies.
(c) Determine a ductile-to-brittle transition temperature as that temperature at which the impact energy is 50 J .

Manik Pulyani

Numerade Educator

Problem 13

Following is tabulated data that were gathered from a series of Charpy impact tests on a commercial low-carbon steel alloy.
$$
\begin{array}{cc}
\hline \text { Temperature }\left({ }^{\circ}\right. \text { C) } & \text { Impact Energy }(\boldsymbol{J}) \\
\hline 50 & 76 \\
40 & 76 \\
30 & 71 \\
20 & 58 \\
10 & 38 \\
0 & 23 \\
-10 & 14 \\
-20 & 9 \\
-30 & 5 \\
-40 & 1.5 \\
\hline
\end{array}
$$
(a) Plot the data as impact energy versus temperature.
(b) Determine a ductile-to-brittle transition temperature as that temperature corresponding to the average of the maximum and minimum impact energies.
(c) Determine a ductile-to-brittle transition temperature as that temperature at which the impact energy is 20 J .

Manik Pulyani

Numerade Educator

Problem 14

A fatigue test was conducted in which the mean stress was $70 \mathrm{MPa}(10,000 \mathrm{psi})$, and the stress amplitude was $210 \mathrm{MPa}(30,000 \mathrm{psi})$.
(a) Compute the maximum and minimum stress levels.
(b) Compute the stress ratio.
(c) Compute the magnitude of the stress range.

Narayan Hari

Numerade Educator

Problem 15

A cylindrical 1045 steel bar (Figure 8.34) is subjected to repeated compression-tension stress cycling along its axis. If the load amplitude is $66,700 \mathrm{~N}\left(15,000 \mathrm{lb}_{\mathrm{f}}\right)$, compute the minimum allowable bar diameter to ensure that fatigue failure will not occur. Assume a safety factor of 2.0 .
Figure 8.34 Stress magnitude $S$ versus the logarithm of the number $N$ of cycles to fatigue failure for red brass, an aluminum alloy, and a plain carbon steel. (Adapted from H. W. Hayden, W. G. Moffatt, and J. Wulff, The Structure and Properties of Materials, Vol. III, Mechanical Behavior, p. 15. Copyright (01965 by John Wiley \& Sons, New York. Reprinted by permission of John Wiley \& Sons, Inc. Also adapted from ASM Handbook, Vol. 2, Properties and Selection: Nonferrous Alloys and Special-Purpose Materials, 1990. Reprinted by permission of ASM International.)

Ameer Said

Numerade Educator

Problem 16

A $6.4 \mathrm{~mm}(0.25 \mathrm{in}$.$) diameter cylindrical rod$ fabricated from a 2014-T6 aluminum alloy (Figure 8.34) is subjected to reversed tensioncompression load cycling along its axis. If the maximum tensile and compressive loads are $+5340 \mathrm{~N}\left(+1200 \mathrm{lb}_{\mathrm{f}}\right)$ and $-5340 \mathrm{~N}\left(-1200 \mathrm{lb}_{\mathrm{f}}\right)$, respectively, determine its fatigue life. Assume that the stress plotted in Figure 8.34 is stress amplitude.

Narayan Hari

Numerade Educator

Problem 17

A $15.2 \mathrm{~mm}(0.60 \mathrm{in}$.) diameter cylindrical rod fabricated from a 2014-T6 aluminum alloy (Figure 8.34) is subjected to a repeated tensioncompression load cycling along its axis. Compute the maximum and minimum loads that will be applied to yield a fatigue life of $1.0 \times 10^8$ cycles. Assume that the stress plotted on the vertical axis is stress amplitude, and data were taken for a mean stress of 35 MPa ( 5000 psi ).

Narayan Hari

Numerade Educator

Problem 18

The fatigue data for a brass alloy are given as follows:
(FIGURE CAN'T COPY)
(a) Make an $S-N$ plot (stress amplitude versus logarithm cycles to failure) using these data.
(b) Determine the fatigue strength at $4 \times 10^6$ cycles.
(c) Determine the fatigue life for 120 MPa .

Ameer Said

Numerade Educator

Problem 19

Suppose that the fatigue data for the brass alloy in Problem 8.18 were taken from bending-rotating tests, and that a rod of this alloy is to be used for an automobile axle that rotates at an average rotational velocity of 1800 revolutions per minute. Give the maximum torsional stress amplitude possible for each of the following lifetimes of the rod: (a) 1 year, (b) 1 month, (c) 1 day, and (d) 1 hour.

Ameer Said

Numerade Educator

Problem 20

The fatigue data for a steel alloy are given as follows:
$$
\begin{array}{cc}
\hline \begin{array}{c}
\text { Stress Amplitude } \\
\text { [MPa }(\text { ksi }) \text { ] }
\end{array} & \begin{array}{c}
\text { Cycles to } \\
\text { Failure }
\end{array} \\
\hline 470(68.0) & 10^4 \\
440(63.4) & 3 \times 10^4 \\
390(56.2) & 10^5 \\
350(51.0) & 3 \times 10^5 \\
310(45.3) & 10^6 \\
290(42.2) & 3 \times 10^6 \\
290(42.2) & 10^7 \\
290(42.2) & 10^8 \\
\hline
\end{array}
$$
(a) Make an $S-N$ plot (stress amplitude versus logarithm cycles to failure) using these data.
(b) What is the fatigue limit for this alloy?
(c) Determine fatigue lifetimes at stress amplitudes of $415 \mathrm{MPa}(60,000 \mathrm{psi})$ and 275 MPa $(40,000 \mathrm{psi})$.
(d) Estimate fatigue strengths at $2 \times 10^4$ and $6 \times 10^5$ cycles.

Manik Pulyani

Numerade Educator

Problem 21

Suppose that the fatigue data for the steel alloy in Problem 8.20 were taken for bendingrotating tests, and that a rod of this alloy is to be used for an automobile axle that rotates at an average rotational velocity of 600 revolutions per minute. Give the maximum lifetimes of continuous driving that are allowable for the following stress levels: (a) 450 MPa (65,000 psi), (b) $380 \mathrm{MPa}(55,000 \mathrm{psi})$, (c) $310 \mathrm{MPa}(45,000 \mathrm{psi})$, and (d) 275 MPa ( $40,000 \mathrm{psi}$ ).

Ameer Said

Numerade Educator

Problem 22

Three identical fatigue specimens (denoted A , B , and C ) are fabricated from a nonferrous alloy. Each is subjected to one of the maximumminimum stress cycles listed below; the frequency is the same for all three tests.
$$
\begin{array}{ccc}
\hline \text { Specimen } & \boldsymbol{\sigma}_{\max }(\text { MPa }) & \boldsymbol{\sigma}_{\min }(\boldsymbol{M P a}) \\
\hline \text { A } & +450 & -150 \\
\text { B } & +300 & -300 \\
\text { C } & +500 & -200 \\
\hline
\end{array}
$$
(a) Rank the fatigue lifetimes of these three specimens from the longest to the shortest.
(b) Now justify this ranking using a schematic $S-N$ plot.

Manik Pulyani

Numerade Educator

Problem 23

Cite five factors that may lead to scatter in fatigue life data.

Narayan Hari

Numerade Educator

Problem 24

Briefly explain the difference between fatigue striations and beachmarks both in terms of (a) size and (b) origin.

Ameer Said

Numerade Educator

Problem 25

List four measures that may be taken to increase the resistance to fatigue of a metal alloy.

Narayan Hari

Numerade Educator

Problem 26

Give the approximate temperature at which creep deformation becomes an important consideration for each of the following metals: tin, molybdenum, iron, gold, zinc, and chromium.

Ameer Said

Numerade Educator

Problem 27

The following creep data were taken on an aluminum alloy at $480^{\circ} \mathrm{C}\left(900^{\circ} \mathrm{F}\right)$ and a constant stress of $2.75 \mathrm{MPa}(400 \mathrm{psi})$. Plot the data as strain versus time, then determine the steady-state or minimum creep rate. Note: The initial and instantaneous strain is not included.
$$
\begin{array}{cccc}
\hline \begin{array}{c}
\text { Time } \\
(\boldsymbol{m i n})
\end{array} & \text { Strain } & \begin{array}{c}
\text { Time } \\
(\boldsymbol{m i n})
\end{array} & \text { Strain } \\
\hline 0 & 0.00 & 18 & 0.82 \\
2 & 0.22 & 20 & 0.88 \\
4 & 0.34 & 22 & 0.95 \\
6 & 0.41 & 24 & 1.03 \\
8 & 0.48 & 26 & 1.12 \\
10 & 0.55 & 28 & 1.22 \\
12 & 0.62 & 30 & 1.36 \\
14 & 0.68 & 32 & 1.53 \\
16 & 0.75 & 34 & 1.77 \\
\hline
\end{array}
$$

Manik Pulyani

Numerade Educator

Problem 28

A specimen $1015 \mathrm{~mm}(40 \mathrm{in}$.) long of a low carbon-nickel alloy (Figure 8.31) is to be exposed to a tensile stress of 70 MPa $(10,000 \mathrm{psi})$ at $427^{\circ} \mathrm{C}\left(800^{\circ} \mathrm{F}\right)$. Determine its elongation after $10,000 \mathrm{~h}$. Assume that the total of both instantaneous and primary creep elongations is 1.3 mm ( 0.05 in .).

Narayan Hari

Numerade Educator

Problem 29

For a cylindrical low carbon-nickel alloy specimen (Figure 8.31 ) originally $19 \mathrm{~mm}(0.75 \mathrm{in}$.) in diameter and 635 mm ( 25 in .) long, what tensile load is necessary to produce a total elongation of $6.44 \mathrm{~mm}(0.25 \mathrm{in}$.) after 5000 h at $538^{\circ} \mathrm{C}\left(1000^{\circ} \mathrm{F}\right)$ ? Assume that the sum of instantaneous and primary creep elongations is $1.8 \mathrm{~mm}(0.07 \mathrm{in}$.$) .$

Narayan Hari

Numerade Educator

Problem 30

If a component fabricated from a low carbon-nickel alloy (Figure 8.30) is to be exposed to a tensile stress of 31 MPa ( 4500 $\mathrm{psi})$ at $649^{\circ} \mathrm{C}\left(1200^{\circ} \mathrm{F}\right)$, estimate its rupture lifetime.

Narayan Hari

Numerade Educator

Problem 31

A cylindrical component constructed from a low carbon-nickel alloy (Figure 8.30) has a diameter of $19.1 \mathrm{~mm}(0.75 \mathrm{in}$.). Determine the maximum load that may be applied for it to survive $10,000 \mathrm{~h}$ at $538^{\circ} \mathrm{C}\left(1000^{\circ} \mathrm{F}\right)$.

Narayan Hari

Numerade Educator

Problem 32

From Equation 8.19, if the logarithm of $\dot{\epsilon}_s$ is plotted versus the logarithm of $\sigma$, then a straight line should result, the slope of which is the stress exponent $n$. Using Figure 8.31 , determine the value of $n$ for the low carbon-nickel alloy at each of the three temperatures.

Narayan Hari

Numerade Educator

Problem 33

(a) Estimate the activation energy for creep (i.e., $Q_c$ in Equation 8.20) for the low carbonnickel alloy having the steady-state creep behavior shown in Figure 8.31. Use data taken at a stress level of $55 \mathrm{MPa}(8000 \mathrm{psi})$ and temperatures of $427^{\circ} \mathrm{C}$ and $538^{\circ} \mathrm{C}$. Assume that the stress exponent $n$ is independent of temperature. (b) Estimate $\dot{\epsilon}_s$ at $649^{\circ} \mathrm{C}(922 \mathrm{~K})$.

Ameer Said

Numerade Educator

Problem 34

Steady-state creep rate data are given here for some alloy taken at $200^{\circ} \mathrm{C}(473 \mathrm{~K})$ :
$$
\begin{array}{lc}
\hline \dot{\boldsymbol{\epsilon}}_s\left(\boldsymbol{h}^{-1}\right) & \boldsymbol{\sigma}[\boldsymbol{M P a}(p s i)] \\
\hline 2.5 \times 10^{-3} & 55(8000) \\
2.4 \times 10^{-2} & 69(10,000) \\
\hline
\end{array}
$$
If it is known that the activation energy for creep is $140,000 \mathrm{~J} / \mathrm{mol}$, compute the steady-state creep rate at a temperature of $250^{\circ} \mathrm{C}$ $(523 \mathrm{~K})$ and a stress level of $48 \mathrm{MPa}(7000 \mathrm{psi})$.

Ameer Said

Numerade Educator

Problem 35

Steady-state creep data taken for an iron at a stress level of $140 \mathrm{MPa}(20,000 \mathrm{psi})$ are given here:
$$
\begin{array}{lc}
\hline \dot{\boldsymbol{\epsilon}}_{\mathbf{s}}\left(\boldsymbol{h}^{-1}\right) & \boldsymbol{T}(\boldsymbol{K}) \\
\hline 6.6 \times 10^{-4} & 1090 \\
8.8 \times 10^{-2} & 1200 \\
\hline
\end{array}
$$
If it is known that the value of the stress exponent $n$ for this alloy is 8.5 , compute the steady-state creep rate at 1300 K and a stress level of $83 \mathrm{MPa}(12,000 \mathrm{psi})$.

Ameer Said

Numerade Educator

Problem 36

Cite three metallurgical/processing techniques that are employed to enhance the creep resistance of metal alloys.

Narayan Hari

Numerade Educator