Materials Selection in Mechanical Design

Michael F. Ashby

Chapter 2

Engineering Materials and Their Properties - all with Video Answers

Educators

Chapter Questions

Problem 1

Sound velocity. The speed of sound in Pyrex (borosilicate) glass is $5610 \mathrm{~m} / \mathrm{s}$. Find the density of this glass from Appendix A and use it to estimate the modulus $E$ of Pyrex.

Mayukh Banik

Mayukh Banik

Numerade Educator

Problem 2

Deflection of beams. A cantilever beam has a length $L=50 \mathrm{~mm}$. It has a rectangular cross-section width $b=5 \mathrm{~mm}$ and a thickness $t=1 \mathrm{~mm}$. It is made of an aluminium alloy. By how much will the end deflect under an end-load of $F=5 \mathrm{~N}$ (roughly that exerted by the weight of 5 apples)? Use data from Appendix A4 for the (mean) value of Young's modulus of aluminium alloys, the equation for the elastic deflection of a cantilever from Appendix B3 and for the second moment of a beam from Appendix B2 to find out.

James Kiss

Numerade Educator

Problem 3

Deflection of beams. The wings of a glider are each $4 \mathrm{~m}$ long. The loadbearing member is the wing spar, a tubular beam running the length of the wing. The wing spar in this glider has a diameter of $140 \mathrm{~mm}$ and a wall thickness of $6 \mathrm{~mm}$. It is made of an aluminium alloy. In flight the wing spar is loaded in bending with (we will assume) a uniformly distributed force per unit length. By how much will the wing tip deflect in calm flight if the loaded glider weighs $1000 \mathrm{~kg}$ ? Use data from Appendix A3 for the (mean) value of Young's modulus of aluminium alloys, the equation for the elastic deflection of a cantilever from Appendix B3 and for the second moment of a beam from Appendix B2 to find out.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 4

Vibration of beams. A cantilever beam with length $L=200 \mathrm{~mm}$, width $b=12 \mathrm{~mm}$ and thickness $t=2 \mathrm{~mm}$ weighs 38 grams and has a natural vibration frequency of $f=42$ hertz. What is the modulus of the material of the beam? You will find equations for natural frequencies of vibration and moments of area in Appendix B, Tables B12 and B2.

James Kiss

Numerade Educator

Problem 5

Forensic materials science. Exercise E2.4 used a natural vibration frequency to measure the modulus of a beam. Use the information in E2.4 to calculate the density of the beam, and then use this and the result of the exercise (the modulus of the beam) as a forensic tool. Scan the modulus values in Table A3 of Appendix A seeking a match between the calculated modulus and those in the table. What subsets of materials match the calculated modulus? Of these, which subset also matches the calculated density?

James Kiss

Numerade Educator

Problem 6

Springs. A spring, wound from stainless steel wire with a wire diameter $d=1 \mathrm{~mm}$, has $n=20$ turns of radius $R=10 \mathrm{~mm}$. How much will it extend when loaded with a mass $M$ of $1 \mathrm{~kg}$ ? Assume the shear modulus $G$ of stainless steel to be $3 / 8 E$ where $E$ is Young's modulus, retrieve this from Appendix A3, and use the expression for the extension of springs from Appendix B6 to find out.

Narayan Hari

Numerade Educator

Problem 7

Torsion of tubes. A thick-walled tube has an inner radius $r_{i}=10 \mathrm{~mm}$ and an outer radius $r_{o}=15 \mathrm{~mm}$. It is made from polycarbonate, $\mathrm{PC}$. What is the maximum torque $T_{f}$ that the tube can carry without the onset of yield? Retrieve the (mean) yield strength $\sigma_{y}$ of PC from Appendix A3, the expression for the torque at onset of yield from Appendix B6 and that for the polar moment of a thick-walled tube from Appendix $B 2$ to find out.

Narayan Hari

Numerade Educator

Problem 8

Stress concentrations. A round bar, $20 \mathrm{~mm}$ in diameter, has a shallow circumferential notch with a depth $c=1 \mathrm{~mm}$ with a root radius $r=10$ microns. The bar is made of a low carbon steel with a yield strength of $\sigma_{y}=250 \mathrm{MPa}$. It is loaded axially with a nominal stress, $\sigma_{\text {nom }}$ (the axial load divided by the unnotched area). At what value of $\sigma_{\text {nom will yield first commence at the root of }}$ the notch? Use the stress concentration estimate of Appendix B9 to find out.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

Thermal stress. An acrylic (PMMA) window is clamped in a thick low carbon steel frame at $T=20^{\circ} \mathrm{C}$. The temperature falls to $T=-20^{\circ} \mathrm{C}$, putting the window under tension because the thermal expansion coefficient of PMMA is larger than that of steel. If the window was stress-free at $20^{\circ} \mathrm{C}$, what stress does it carry at $-20^{\circ} \mathrm{C}$ ? Use the result that the bi-axial stress caused by a biaxial strain difference $\Delta \varepsilon$ is

Jacob Paiste

Numerade Educator

Problem 10

Unstable cracks. The PMMA window described in Exercise $2.9$ has a contained crack of length $2 a=0.5 \mathrm{~mm}$. If the maximum tensile stress that is anticipated in the window is $\sigma=20 \mathrm{MPa}$, will the crack propagate? Choose an appropriate equation for crack propagation from Appendix B10 and data for the fracture toughness $K_{1 c}$ of PMMA from Appendix A6 to calculate the length of crack that is just unstable under this tensile stress.

Manik Pulyani

Numerade Educator

Problem 11

Centrifugal stress. A flywheel with a radius $R=200 \mathrm{~mm}$ is designed to spin up to $8000 \mathrm{rpm}$. It is proposed to make it out of ductile (nodular) cast iron, but the casting shop can guarantee only that it will have no crack-like flaws greater than $2 a=2 \mathrm{~mm}$ in length. Use the expression for the maximum stress in a spinning disk in Appendix B7, that for the stress intensity at a small enclosed crack from Appendix B10 and data for cast iron from Appendix A2 and A4 to establish if the flywheel is safe. Take Poisson's ratio $\nu$ for cast iron to be $0.33$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 12

Blunting cracks. Liberty ships, thousands of which were constructed during World War II, were welded rather than riveted. Many were mistakenly made from a steel that became brittle at the temperatures of the Atlantic in winter. They had square hatches with sharp corners from which a crack could
start and propagate across the deck which, being welded, allowed a continuous path not interrupted by riveted plates. It is said that alert seamen, observing such a crack to start, would seize a power drill and drill a hole at the crack tips, effectively blunting it and reducing the stress concentration. If the crack when the seaman first saw it was $1 \mathrm{~m}$ long, and the largest drill he had was $2 r=25 \mathrm{~mm}$ in diameter, what is the stress concentration at the end of the crack once the hole was drilled? You will find the equation for the stress at a distance $r$ from the tip of a crack with a length $2 a$ in Appendix B, Table B10.

Eduard Sanchez

Numerade Educator

Problem 13

Contact forces. A clamp has a hemispherical contact face of radius $R=10 \mathrm{~mm}$ to allow for misorientation. It is used to clamp a heavy lead archaeological treasure for X-ray examination. How much clamping force, $F$, can be applied without damage (meaning plastic deformation) to the treasured object? You will find the both the modulus $E$ and yield strength $\sigma_{\gamma}$ of lead alloys in Appendix A, Table A3 (use mean values), and the equation for force required to trigger plastic deformation beneath a spherical indenter in Appendix B, Table B8.

Manik Pulyani

Numerade Educator

Problem 14

Estimating thermal properties. You wish to measure, approximately, the thermal conductivity $\lambda$ of polyethylene (PE). To do so you block one end of a PE pipe with a wall thickness of $x=3 \mathrm{~mm}$ and diameter of $30 \mathrm{~mm}$ and fill it with boiling water while clutching the outside with your other hand. You note that the outer surface of the pipe first becomes appreciably hot at a time $t \approx 18$ seconds after filling the inside with water. Use this information, plus data for specific heat $C_{p}$ and density $\rho$ of PE from Appendix A, Tables A2 and A5, to estimate $\lambda$ for PE. How does your result compare with the listed value in Table A7?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 15

Heat exchangers. A power transistor with an area of $A$ of $1 \mathrm{~mm}^{2}$ dissipates 10 watts. It is attached to a heat sink $2 \mathrm{~mm}$ thick, effectively cooled to $30^{\circ} \mathrm{C}$ on the back face by forced airflow. If the heat sink is made of aluminium nitride, how hot will the microchip be at steady state? You will find equations of heat flow in Appendix B, Table B15, and the thermal properties of aluminium nitride in Appendix A, Tables A5 an A6.

Anand Jangid

Numerade Educator

Problem 16

Choosing dielectrics. The capacitance $C$ of a condenser with two plates each of area $A$ separated by a dielectric of thickness $t$ is $$
C=\varepsilon_{r} \varepsilon_{0} \frac{A}{t}
$$
where $\varepsilon_{o}$ is the permittivity of free space and $\varepsilon_{r}$ is the dielectric constant of the material between the plates. Select a dielectric by scanning data in Appendix A, Tables A7 and Table A8, first to maximize $C$ and second to minimize it, for a given $A$ and $t$.

Sarah Mccrumb

Numerade Educator

Problem 17

Piezoelectric actuators. Piezoelectric materials respond to an electric field by change of shape. The strain $\varepsilon$ induced by a field $E$ is
$$
\varepsilon=d_{33} E
$$
where $d_{33}$ is the piezoelectric charge coefficient, listed in Table A8. Its units are $\mathrm{pm} / \mathrm{V}$. A sample of soft PZT (lead zirconium titanate) is exposed to a field of $10 \mathrm{MV} / \mathrm{m}$. How large is the resulting strain? You will find the piezoelectric charge coefficient for soft PZT in Table A8. Use a mean value.

Shubham Verma

Texas A&M University

Problem 18

Pyroelectric temperature sensing. Remote, noncontact, temperature sensing is possible by focussing radiation from the source onto a pyroelectric material and by using the charge that appears on its surface as a measure of temperature. A sensor is required to measure temperature in the range 100 $200^{\circ} \mathrm{C}$. To work in this range the material must have a ferroelectric Curie temperature greater than $200^{\circ} \mathrm{C}$, because otherwise it ceases to be pyroelectric in the sensing range. The signal is maximized by selecting a material with the largest possible pyroelectric coefficient. Use the data for pyroelectrics in Appendix A, Section A8, to make a selection.

Dheeraj Sharma

Numerade Educator

Problem 19

Magnetostrictive actuation. A magnetostrictive actuator has and active core of length $L=20 \mathrm{~mm}$. It is made of Terfenol $\mathrm{D}$, a giant magnetostrictive alloy of terbium, dysprosium and iron. If a field sufficiently large to saturate the core is applied, by how much will the core change in length? Use the mean value of the data for Terfenol D listed in Appendix A, Section A9, to find out.

Jacob Shpiece

Numerade Educator

Problem 20

Material price. It is proposed to replace the cast iron casing of a power tool with one with precisely the same dimension moulded from nylon. Will the material cost of the nylon casing be greater or less than that made of cast iron? Use data from Appendix A3 and A11 to find out.

Mrinal Rana

Numerade Educator

Problem 21

Carbon footprint. Plastics are sometimes portrayed as environmentally poor because most are synthesize from oil or natural gas, implying a larger embodied energy and carbon footprint than other materials. Is this an accurate portrait? Use the data from Appendix A, Table A10 and A2 to compare the carbon footprint of three of the materials most widely used in household products: polypropylene, aluminium and carbon steel. Use mean values of the values in the tables.

James Kiss

Numerade Educator