Chapter 3, Materials Property Charts Video Solutions, Materials Selection in Mechanical Design

Problem 1

Use Fig. $3.3$ to identify one metal and one polymer with a longitudinal wave speed close to $1000 \mathrm{~m} / \mathrm{s}$.

Mirza Aslam Beig

Numerade Educator

01:01

A component is at present made from a brass, a copper alloy. Use the Young's modulus - Density $(E-\rho)$ chart of Fig. $3.3$ to suggest three other metals that, in the same shape, would be stiffer. 'Stiffer' means a higher value of Young's modulus.

Narayan Hari

Numerade Educator

00:47

Problem 3

Use the Young's modulus - Density $(E-\rho)$ chart of Fig. $3.3$ to identify materials with both a modulus $E>50 \mathrm{GPa}$ and a density $\rho<2000 \mathrm{~kg} / \mathrm{m}^{3}$.

Mirza Aslam Beig

Numerade Educator

01:01

Problem 4

Use the Young's Modulus - Density $(E-\rho)$ chart of Fig. $3.3$ to find (a) metals that are stiffer and less dense than steels and (b) materials (not just metals) that are both stiffer and less dense than steel.

Narayan Hari

Numerade Educator

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Problem 5

Use the $(E-\rho)$ chart of Fig. $3.3$ to identify metals with both $E>100 \mathrm{GPa}$ and $E / \rho>0.02 \mathrm{GPa} /\left(\mathrm{kg} / \mathrm{m}^{3}\right)$.

Lainey Roebuck

Numerade Educator

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Problem 6

Use the $E / \rho$ chart of Fig. $3.3$ to identify materials with both $E>100 \mathrm{GPa}$ and $E^{1 / 3} / \rho>0.003(\mathrm{GPa})^{1 / 3} /\left(\mathrm{kg} / \mathrm{m}^{3}\right)$. Remember that, on taking logs, the index $M=E^{1 / 3} / \rho$ becomes
$$
\log (E)=3 \log (\rho)+3 \log (M)
$$
and that this plots as a line of slope 3 on the chart, passing through the point $E=27$ when $\rho=1000$ in the units on the chart.

Lainey Roebuck

Numerade Educator

15:11

Problem 7

Use the $E-\rho$ chart of Fig. $3.3$ to establish whether woods have a higher specific stiffness $E / \rho$ than epoxies.

Jayashree Behera

Numerade Educator

01:34

Problem 8

Do titanium alloys have a higher or lower specific strength (strength/density, $\sigma_{f} / \rho$ ) than the best steels? This is important when you want strength at low weight (landing gear of aircraft, mountain bikes). Use the $\sigma_{f} / \rho$ chart of Fig. $3.4$ to decide.

Supratim Pal

Numerade Educator

02:47

Problem 9

The design of a cycle safety helmet requires a lining that will crush at a stress of $8-12 \mathrm{MPa}$ and be as light as possible. Use the $\sigma_{f}-\rho$ chart of Fig. $3.4$ to select an appropriate class of material to meet this need.

Chai Santi

Numerade Educator

02:35

Problem 10

Use the modulus-strength $E-\sigma_{f}$ chart of Fig. $3.5$ to find materials that have $E>10 \mathrm{GPa}$ and $\sigma_{f}>1000 \mathrm{MPa}$.

Chai Santi

Numerade Educator

01:23

Problem 11

The undercarriage of a cargo plane requires a metal with a compressive strength above $200 \mathrm{MPa}$ and a high bending strength per unit weight, requiring a high value of the index $\sigma_{f}^{2 / 3} / \rho$. What metals does the chart of Fig. 3.4 suggest?

Narayan Hari

Numerade Educator

05:47

Problem 12

Transport systems require materials that are stiff and strong, but at the same time are light and have enough toughness to survive accidental overloads chart (Fig. 3.6) to identify materials that meet these criteria. Eliminate all ceramics on the grounds that they are brittle. Which material has the best combination of both specific modulus and specific strength? Which metal has the highest specific modulus? Which has the highest specific strength?

Satpal Satpal

Numerade Educator

02:28

Problem 13

Is the fracture toughness, $K_{l c}$, of the common polymers polycarbonate, ABS, or polystyrene larger or smaller than the engineering ceramic alumina, $\mathrm{Al}_{2} \mathrm{O}_{3}$ ? Are their toughness qualities $\mathrm{G}_{1 c}=K_{1 c}^{2} / E$ larger or smaller? The $K_{l c}-E$ chart, Fig. 3.7, will help.

Narayan Hari

Numerade Educator

02:29

Problem 14

The bumpers of cars have to be tough and light. Compare the use of aluminium and of polypropylene for car bumpers. First read the approximate toughness, $G_{1 c}$ for the two materials from the chart of Fig. 3.7. Then divide each of these by the density of the material (read from any of the charts with density as an axis) to get a measure of toughness per unit mass. What conclusions can you draw?

Daniel Matthias

Numerade Educator

02:28

Problem 15

Use the fracture toughness-modulus chart (Fig. 3.7) to find materials that have a fracture toughness $K_{1 c}$ greater than $40 \mathrm{MPa} \cdot \mathrm{m}^{1 / 2}$ and a toughness $\mathrm{G}_{1 c}=K_{1 c}^{2} / E$ (shown as contours on Fig. 3.7) greater than $10 \mathrm{~kJ} / \mathrm{m}^{3}$.

Narayan Hari

Numerade Educator

02:28

Problem 16

The elastic deflection at fracture (the 'resilience') of an elastic-brittle solid is proportional to the failure strain, $\varepsilon_{f r}=\sigma_{f r} / E$, where $\sigma_{f r}$ is the stress that will cause a crack to propagate:
Here $K_{1 c}$ is the fracture toughness and $c$ is the length of the longest crack the materials may contain. Thus
$$
\varepsilon_{f \mathrm{r}}=\frac{1}{\sqrt{\pi c}}\left(\frac{K_{\mathrm{l} c}}{E}\right)
$$
Materials that can deflect elastically without fracturing are therefore those with large values of $K_{1 c} / E_{\text {. Use the }} K_{l c}-E$ chart of Fig. $3.7$ to identify the class of materials with $K_{1 c}>1 \mathrm{MPa} \mathrm{m}^{1 / 2}$ and high values of $K_{1 c} / E_{\text {. Position the }} K_{1 c} / E_{\text {criterion such that all metals are excluded. }}$

Narayan Hari

Numerade Educator

05:48

Problem 17

One criterion for design of a safe pressure vessel is that it should leak before it breaks: the leak can be detected and the pressure released. This is achieved by designing the vessel to tolerate a crack of length equal to the thickness $t$ of the pressure vessel wall, without failing by fast fracture. The safe pressure $p$ is then
$$
p \leq \frac{41}{\pi} \frac{1}{R}\left(\frac{K_{l c}^{2}}{\sigma_{f}}\right)
$$
where $\sigma_{f}$ is the elastic limit, $K_{1 c}$ is the fracture toughness, $R$ is the vessel radius. The pressure is maximized by choosing the material with the greatest value of
$$
M=\frac{K_{1 c}^{2}}{\sigma_{y}}
$$
Use the $K_{1 c}-\sigma_{f}$ chart of Fig. $3.8$ to identify three alloys that have particularly high values of $M$.

Chai Santi

Numerade Educator

06:49

Problem 18

A material is required for the blade of a rotary lawnmower. Cost is a consideration. For safety reasons, the designer specified a minimum fracture toughness for the blade: it is $K_{1 c}>30 \mathrm{MPa} \mathrm{} \mathrm{m}^{1 / 2}$. The other mechanical requirement is for high hardness, $H$, to minimize blade wear. Hardness, in applications like this one, is related to strength:
$$
H \approx 3 \sigma_{y}
$$
where $\sigma_{f}$ is the strength (Chapter 2, Engineering Materials and Their Properties gives a fuller definition). Use the $K_{1 c}-\sigma_{f}$ chart of Fig. $3.8$ to identify three materials that have $K_{1 c}>30 \mathrm{MPa} \mathrm{m}{ }^{1 / 2}$ and the highest possible strength. To do this, position a ' $K_{\mathrm{lc}}$ ' selection line at $30 \mathrm{MPa} \mathrm{} \mathrm{m}^{12}$ and then adjust a 'strength' selection line such that it just admits three candidates. Use the Cost chart of Fig. $3.19$ to rank your selection by material cost, hence making a final selection.

Christine Anacker

Numerade Educator

01:26

Problem 19

Bells ring because they have a low loss (or damping) coefficient, $\eta$; a high damping gives a dead sound. Use the Loss coefficient-Modulus $(\eta-E)$ chart of Fig. $3.9$ to identify material that should make good bells.

James Kiss

Numerade Educator

05:56

Problem 20

A foundation is needed on which to mount vibration-sensitive measuring equipment. It is suggested that the foundation be made from concrete because of its ability to damp vibration. Is this assumption justified? Use Fig. $3.9$ to find out.

Susan Hallstrom

Numerade Educator

05:26

Problem 21

The track of a tracked vehicle is made up of hardened steel segments bonded to rubber connectors. When the vehicle is in motion the rubber is subjected to a tension-compression cycle as the track passes over the front and rear drive sprockets. If the vehicle travels at $20 \mathrm{kph}$, each unit of the rubber is stretched and compressed roughly 1 per second. The thermal conductivity of rubber is extremely low. If the maximum strain $\varepsilon_{\max }$ in the rubber is $\pm 0.4$ and no heat is lost from it, how hot will the rubber get after 30 minutes? Take the volume specific heat $C_{p} \rho$ of the rubber used for the track to be $2.5 \times 10^{6} \mathrm{~J} / \mathrm{m}^{3} \mathrm{~K}$, its modulus $E$ is $2 \times 10^{-3} \mathrm{GPa}$ and its loss coefficient $\eta$ is $0.5$.

Michael Cao

Numerade Educator

01:49

Problem 22

Use the Loss coefficient-Modulus ( $\eta-E)$ chart (Fig. 3.9) to find metals with the highest possible damping.

Jane Diner

Numerade Educator

02:43

Problem 23

Use the Thermal conductivity-Electrical resistivity $\left(\lambda-\rho_{e}\right)$ chart (Fig.
3.10) to find three materials other than diamond with high thermal conductivity, $\lambda$, and high electrical resistivity, $\rho_{e}$.

Mohammad Mehran

Numerade Educator

01:05

Problem 24

It is easier to measure the electrical resistivity of a metal than to measure its thermal conductivity. A new alloy has an electrical resistivity $\rho_{e}$ of $55 \mu \Omega \mathrm{cm}$. Use the Wiedemann-Franz relationship, plotted on Fig. 3.10, to estimate its thermal conductivity, $\lambda$.

Suzanne W.

Numerade Educator

02:27

Problem 25

A novel strontium-silica glass has a thermal conductivity $\lambda$ of $2.1 \mathrm{~W} / \mathrm{m} \mathrm{K}$ (exceptionally high for a glass). What, approximately, would you expect its thermal diffusivity $a$ to be? Use the empirical relationship between $\lambda$ and $a$ to find out, then examine where it would lie on Fig. 3.11.

Akshaya Rs

Numerade Educator

06:54

Problem 26

A new design of frying pan is made from glass ceramic 9606, a material with excellent thermal shock resistance originally developed for missile radomes. The pan bottom is $4.2 \mathrm{~mm}$ thick. When placed on a gas cooker with a pat of butter in it, it is observed that it takes 7 seconds before the butter starts to melt. Use the rules of thumb of Chapter 2, Engineering Materials and Their Properties and this chapter 3 to estimate, first, the thermal diffusivity of glass ceramic 9606 , and from this, its thermal conductivity.

Morgan Cheatham

Numerade Educator

04:23

Problem 27

The window through which the beam emerges from a high-powered laser must obviously be transparent to light. Even then, some of the energy of the beam is absorbed in the window and can cause it to heat and crack. This problem is minimized by choosing a window material with a high thermal
conductivity $\lambda$ (to conduct the heat away) and a low expansion coefficient $\alpha$ (to reduce thermal strains), that is, by seeking a window material with a high value of
$$
M=\lambda / \alpha
$$
Use the $\alpha-\lambda$ chart of Fig. $3.12$ to identify the best material for an ultra-high powered laser window.

Zhaojie Xu

Numerade Educator

02:25

Problem 28

An external panel of a satellite is made of the Grade 9 titanium alloy Ti$3 \mathrm{Al}-2 \mathrm{~V}$, with a yield strength $\sigma_{\gamma}$ of $550 \mathrm{MPa}$ and Poisson's ratio $\nu$ of $0.36$. The panel heats up when the face of the satellite is directly exposed to the sun and cools when the sun is obscured by Earth. The panel is rigidly constrained around its edges by attachment to the frame of the satellite, which is maintained at a constant temperature in order to stabilize the electrics it contains. If the temperature of the panel can vary from $-80^{\circ} \mathrm{C}$ to $+120^{\circ} \mathrm{C}$, is there any risk that the resulting thermal stress might cause the panel to yield? Read the value of the thermal stress parameter $\alpha E$ for titanium alloys from Fig. $3.13$ and use it to find out. (Here $\alpha$ is the thermal expansion coefficient and $E$ is Young's modulus.)

Rajesh Singh

Numerade Educator

05:49

Problem 29

Use the Maximum service temperature $\left(T_{\max }\right)$ chart (Fig. 3.14) to find polymers that can be used above $200^{\circ} \mathrm{C}$.

Nicholas Sacco

Numerade Educator

00:46

Problem 30

Which class of magnets has the highest energy product? Which has the lowest? Use the chart of Remanent induction and Coercive force, Fig. 3.18, to find out.

Anna Zeng

Numerade Educator

01:15

Problem 31

If magnetostrictive strain is a linear function of the field $H$, what strain would appear in a Galfenol rod if it was subjected to a field of $100 \mathrm{~A} / \mathrm{m}$ ? Use data read from the chart of Fig. $3.19$ to find out.
$2.5 \times 10^{-4}(0.025 \%)$. A field of $100 \mathrm{~A} / \mathrm{m}$ might produce a strain of $1 / 8$ of this, or about $0.003 \%$.

Shahab Ullah

Numerade Educator

09:32

Problem 32

a. Use the Young's modulus-Relative cost $\left(E-C_{\eta, R}\right)$ chart (Fig. 3.26) to find the cheapest materials with a modulus, $E$, greater than $100 \mathrm{GPa}$.
b. Use the Strength-Relative cost $\left(\sigma_{f}-C_{R}\right)$ chart (Fig. 3.27) to find the cheapest materials with a strength, $\sigma_{f}$, above $100 \mathrm{MPa}$.
'A set of the charts can be downloaded from www.grantadesign.com. All the charts shown in this chapter were created using Granta Design's CES EduPack Materials Selection software. With it you can make charts with any pair (or combination) of properties as axes.
${ }^{2}$ Most material properties are best viewed on log scales because the ranges are so large. Instead of incrementing by steps of $1, \log$ scales increment by steps of a factor of 10 .
${ }^{3}$ Very low density foams and gels (which can be thought of as molecular-scale, fluid-filled, foams) can have lower moduli than this. As an example, gelatin (as in Jello) has a modulus of about $10^{-5} \mathrm{GPa}$. Their strengths and fracture toughness, too, can be below the lower limit of the charts.
${ }^{4}$ This can be understood by noting that a solid containing $N$ atoms has $3 N$ vibrational modes. Each (in the classical approximation) absorbs thermal energy $k T$ at the absolute temperature $T$, and the vibrational specific heat is $C_{\mathrm{p}} \approx C_{\mathrm{v}}=3 N k(\mathrm{~J} / \mathrm{K})$ where $k$ is Boltzmann's constant $\left(1.34 \times 10^{-23} \mathrm{~J} / \mathrm{K}\right)$. The volume per atom, $\Omega$, for almost all solids lies within a factor of two of $1.4 \times 10^{-29} \mathrm{~m}^{3}$; thus the volume of $N$ atoms is $\left(N C_{\mathrm{p}}\right)$ $\mathrm{m}^{3}$. The volume specific heat is then (as the chart shows):

Eduard Sanchez

Numerade Educator