Askeland D.R., Fulay P.P. Essentials of Materials Science & Engineering

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V Virtually any combination of metals, polymers, and ceramics is possible. In many

cases, the rule of mixtures can be used to estimate some of the properties of

a composite.

V Composites have many applications in construction, aerospace, automotive, sports,

microelectronics and other industries.

V Dispersion-strengthened materials contain exceptionally small oxide particles in

a metal matrix. The small stable dispersoids interfere with slip, providing good

mechanical properties at elevated temperatures.

V Particulate composites contain particles that impart combinations of properties to

the composite. Metal-matrix composites contain ceramic or metallic particles that

provide improved strength and wear resistance and assure good electrical con-

ductivity, toughness, or corrosion resistance. Polymer-matrix composites contain

particles that enhance sti¤ness, heat resistance, or electrical conductivity while

maintaining light weight, ease of fabrication, or low cost.

V Fiber-reinforced composites provide improvements in strength, sti¤ness, or high-

temperature performance in metals and polymers and impart toughness to ceramics.

Fibers typically have low densities, giving high speciﬁc strength and speciﬁc mod-

ulus, but they often are very brittle. Fibers can be continuous or discontinuous. Dis-

continuous ﬁbers with a high aspect ratio l=d produce better reinforcement.

V Fibers are introduced into a matrix in a variety of orientations. Random ori-

entations and isotropic behavior are obtained using discontinuous ﬁbers; unidirec-

tionally aligned ﬁbers produce composites with anisotropic behavior, with large

improvements in strength and sti¤ness parallel to the ﬁber direction. Properties can

be tailored to meet the imposed loads by orienting the ﬁbers in multiple directions.

V Laminar composites are built of layers of di¤erent materials. These layers may

be sheets of di¤erent metals, with one metal providing strength, and the other pro-

viding hardness or corrosion resistance. Layers may also include sheets of ﬁber-

reinforced material bonded to metal or polymer sheets or even to ﬁber-reinforced

sheets having di¤erent ﬁber orientations. The laminar composites are always ani-

sotropic.

V Sandwich materials, including honeycombs, are exceptionally lightweight laminar

composites, with solid facings joined to an almost hollow core.

GLOSSARY

Aramid fibers Polymer ﬁbers, such as Kevlar

, formed from polyamides, which contain the

benzene ring in the backbone of the polymer.

Aspect ratio The length of a ﬁber divided by its diameter.

Bimetallic A laminar composite material produced by joining two strips of metal with di¤erent

thermal expansion coe‰cients, making the material sensitive to temperature changes.

Brazing A process in which a liquid ﬁller metal is introduced by capillary action between two

solid-based materials that are to be joined. Solidiﬁcation of the brazing alloy provides the bond.

Carbonizing Driving o¤ the non-carbon atoms from a polymer ﬁber, leaving behind a carbon

ﬁber of high strength. Also known as pyrolizing.

Cemented carbides or cermets Particulate composites containing hard ceramic particles bonded

with a soft metallic matrix. The composite combines high hardness and cutting ability, yet still

has good shock resistance.

C HA P T E R 1 7 Composites: Teamwork and Synergy in Materials580

Chemical vapor deposition (CVD) Method for manufacturing materials by condensing the

material from a vapor onto a solid substrate.

Cladding A laminar composite produced when a corrosion-resistant or high-hardness layer of a

laminar composite is formed onto a less expensive or higher-strength backing.

Delamination Separation of individual plies of a ﬁber-reinforced composite.

Dispersed phase The phase or phases that are dispersed in a continuous matrix of the composite.

Dispersoids Tiny oxide particles formed in a metal matrix that interfere with dislocation move-

ment and provide strengthening, even at elevated temperatures.

Filament winding Process for producing ﬁber-reinforced composites in which continuous ﬁbers

are wrapped around a form or mandrel. The ﬁbers may be prepregged or the ﬁlament-wound

structure may be impregnated to complete the production of the composite.

Honeycomb A lightweight but sti¤ assembly of aluminum strip joined and expanded to form

the core of a sandwich structure.

Hybrid organic-inorganic composites Nanocomposites consisting of structures that are partly

organic and partly inorganic, often made using sol-gel processing.

Matrix phase The continuous phase in a composite. The composites are named after the con-

tinuous phase (e.g., polymer-matrix composites).

Nanocomposite A material in which the dispersed phase is nano-sized and is distributed in the

continuous matrix.

Precursor A starting chemical (e.g., a polymer ﬁber) that is carbonized to produce carbon ﬁbers.

Prepregs Layers of ﬁbers in unpolymerized resins. After the prepregs are stacked to form a

desired structure, polymerization joins the layers together.

Pultrusion A method for producing composites containing mats or continuous ﬁbers.

Rovings Bundles of less than 10,000 ﬁlaments.

Rule of mixtures The statement that the properties of a composite material are a function of the

volume fraction of each material in the composite.

Sandwich A composite material constructed of a lightweight, low-density material surrounded

by dense, solid layers. The sandwich combines overall light weight with excellent sti¤ness.

Sizing Coating glass ﬁbers with an organic material to improve bonding and moisture resistance

in ﬁberglass.

Specific modulus The modulus of elasticity divided by the density.

Specific strength The tensile or yield strength of a material divided by the density.

Staples Fibers chopped into short lengths.

Tapes Single-ﬁlament-thick strips of prepregs, with the ﬁlaments either unidirectional or woven

ﬁbers. Several layers of tapes can be joined to produce composite structures.

Tow A bundle of more than 10,000 ﬁlaments.

Whiskers Very ﬁne ﬁbers grown in a manner that produces single crystals with no mobile dis-

locations, thus giving nearly theoretical strengths.

Yarns Continuous ﬁbers produced from a group of twisted ﬁlaments.

Glossary 581

PROBLEMS

Section 17-1 Dispersion-Strengthened

Composites

17-1 What is a composite?

17-2 What do the properties of composite materials

depend upon?

17-3 Give examples of applications where compo-

sites are used for load bearing applications.

17-4 Give two examples of applications where com-

posites are used for non-structural applica-

tions.

17-5 What is a dispersion-strengthened composite?

How is it di¤erent from a particle-reinforced

composite?

17-6 What is a nanocomposite? How can steels con-

taining ferrite and martensite be described as

composites? Explain.

17-7 Nickel containing 2 wt% thorium is produced

in powder form, consolidated into a part, and

sintered in the presence of oxygen, causing all of

the thorium to produce ThO

spheres 80 nm in

diameter. Calculate the number of spheres per

.(Note: The density of ThO

is 9.86 g/cm

17-8 Spherical aluminum powder (SAP) 0.002 mm in

diameter is treated to create a thin oxide layer

and is then used to produce a SAP dispersion-

strengthened material containing 10 vol%

. Calculate the average thickness of the

oxide ﬁlm prior to compaction and sintering of

the powders into the part.

17-9 Yttria (Y

) particles 750 A in diameter are in-

troduced into tungsten by internal oxidation.

Measurements using an electron microscope

show that there are 5  10

oxide particles per

. Calculate the wt% Y originally in the alloy.

(Note: The density of Y

is 5.01 g/cm

17-10 With no special treatment, aluminum is typi-

cally found to have an Al

layer that is 3 nm

thick. If spherical aluminum powder prepared

with a total diameter of 0.01 mm is used to

produce the SAP dispersion-strengthened mate-

rial, calculate the volume percent Al

in the

material and the number of oxide particles per

. Assume that the oxide breaks into disk-

shaped ﬂakes 3 nm thick and 3  10

4

mm in

diameter. Compare the number of oxide par-

ticles per cm

with the number of solid solution

atoms per cm

when 3 at% of an alloying ele-

ment is added to aluminum.

Section 17-2 Particulate Composites

17-11 What is a particulate composite?

17-12 What is a cermet? What is the role of WC and

Co in a cermet?

17-13 Calculate the density of a cemented carbide, or

cermet, based on a titanium matrix if the com-

posite contains 50 wt% WC, 22 wt% TaC, and

14 wt% TiC. (See Example 17-2 for densities of

the carbides.)

17-14 A typical grinding wheel is 22.5 cm in diameter,

2.5 cm thick, and weighs 2.7 kg. The wheel

contains SiC (density of 3.2 g/cm

) bonded by

silica glass (density of 2.5 g/cm

); 5 vol% of the

wheel is porous. (Note: The SiC is in the form

of 0.04 cm cubes.) Calculate

(a) the volume fraction of SiC particles in the

wheel and

(b) the number of SiC particles lost from the

wheel after it is worn to a diameter of 20 cm.

17-15 An electrical contact material is produced by

inﬁltrating copper into a porous tungsten car-

bide (WC) compact. The density of the ﬁnal

composite is 12.3 g/cm

. Assuming that all of

the pores are ﬁlled with copper, calculate

(a) the volume fraction of copper in the com-

posite,

(b) the volume fraction of pores in the WC

compact prior to inﬁltration, and

before inﬁltration.

17-16 An electrical contact material is produced by

ﬁrst making a porous tungsten compact that

weighs 125 g. Liquid silver is introduced into

the compact; careful measurement indicates that

105 g of silver is inﬁltrated. The ﬁnal density

of the composite is 13.8 g/cm

. Calculate the

volume fraction of the original compact that is

interconnected porosity and the volume fraction

that is closed porosity (no silver inﬁltration).

17-17 How much clay must be added to 10 kg of

polyethylene to produce a low-cost composite

having a modulus of elasticity greater than

827 MPa and a tensile strength greater than

14 MPa? (Note: The density of the clay is 2.4 g/

and that of polyethylene is 0.92 g/cm

17-18 We would like to produce a lightweight epoxy

part to provide thermal insulation. We have

available hollow glass beads for which the out-

side diameter is 0.16 cm and the wall thickness is

C HA P T E R 1 7 Composites: Teamwork and Synergy in Materials582

0.0025 cm. Determine the weight and number of

beads that must be added to the epoxy to pro-

duce a one-kg composite with a density of 0.65

g/cm

.(Note: The density of the glass is 2.5 g/

and that of the epoxy is 1.25 g/cm

Section 17-3 Fiber-Reinforced Composites

Section 17-4 Characteristics of Fiber-Reinforced

Composites

17-19 What is a ﬁber-reinforced composite?

17-20 What ﬁber-reinforcing materials are commonly

used?

17-21 In a ﬁber-reinforced composite, what is the role

of the matrix?

17-22 What do the terms CFRP and GFRP mean?

17-23 Explain brieﬂy how the volume of ﬁber, ﬁber

orientation, and ﬁber strength and modulus

a¤ect the properties of ﬁber-reinforced compo-

sites.

17-24 Five kg of continuous boron ﬁbers are intro-

duced in a unidirectional orientation into 8 kg

of an aluminum matrix. Calculate

(a) the density of the composite,

(b) the modulus of elasticity parallel to the

ﬁbers, and

the ﬁbers.

17-25 We want to produce 4.5 kg of a continuous

unidirectional ﬁber-reinforced composite of HS

carbon in a polyimide matrix that has a mod-

ulus of elasticity of at least 172  10

MPa par-

allel to the ﬁbers. How many kg of ﬁbers are

required? (See Chapter 16 for properties of

polyimide.)

17-26 We produce a continuous unidirectionally re-

inforced composite containing 60 vol% HM

carbon ﬁbers in an epoxy matrix. The epoxy has

a tensile strength of 103 MPa. What fraction of

the applied force is carried by the ﬁbers?

17-27 A polyester matrix with a tensile strength of

90 MPa is reinforced with Al

ﬁbers. What

vol% ﬁbers must be added to insure that the

ﬁbers carry 75% of the applied load?

17-28 An epoxy matrix is reinforced with 40 vol%

E-glass ﬁbers to produce a 2-cm-diameter com-

posite that is to withstand a load of 25,000 N.

Calculate the stress acting on each ﬁber.

17-29 A titanium alloy with a modulus of elasticity of

110  10

MPa is used to make a 454-kg part for

a manned space vehicle. Determine the weight of

a part having the same modulus of elasticity

parallel to the ﬁbers, if the part is made of

(a) aluminum reinforced with boron ﬁbers and

(b) polyester reinforced with high-modulus car-

bon ﬁbers (with a modulus of 4482 MPa).

materials.

17-30 Short, but aligned, Al

ﬁbers with a diameter

of 20 mm are introduced into a 6,6-nylon matrix.

The strength of the bond between the ﬁbers and

the matrix is estimated to be 7 MPa. Calculate

the critical ﬁber length and compare with the

case when 1-mm alumina whiskers are used in-

stead of the coarser ﬁbers. What is the minimum

aspect ratio in each case?

17-31 We prepare several epoxy-matrix composites

using di¤erent lengths of 3-mm-diameter ZrO

ﬁbers and ﬁnd that the strength of the compo-

site increases with increasing ﬁber length up

to 5 mm. For longer ﬁbers, the strength is vir-

tually unchanged. Estimate the strength of the

bond between the ﬁbers and the matrix.

Section 17-5 Manufacturing Fibers and

Composites

17-32 Explain brieﬂy how boron and carbon ﬁbers are

made.

17-33 Explain brieﬂy how continuous-glass ﬁbers are

made.

17-34 What is a coupling agent? What is ‘‘sizing’’ as it

relates to the production of glass ﬁbers?

17-35 What is the di¤erence between a ﬁber and a

whisker?

17-36 In one polymer-matrix composite, as produced,

discontinuous glass ﬁbers are introduced directly

into the matrix; in a second case, the ﬁbers are

ﬁrst ‘‘sized.’’ Discuss the e¤ect this di¤erence

might have on the critical ﬁber length and the

strength of the composite.

17-37 Explain why bonding between carbon ﬁbers and

an epoxy matrix should be excellent, whereas

bonding between silicon nitride ﬁbers and a sili-

con carbide matrix should be poor.

Section 17-6 Fiber-Reinforced Systems and

Applications

17-38 Explain brieﬂy in what sporting equipment

composite materials are used. What is the main

reason why composites are used in these appli-

cations?

17-39 A polyimide matrix is to be reinforced with

70 vol% carbon ﬁbers to give a minimum

Problems 583

modulus of elasticity of 276  10

MPa. Rec-

ommend a process for producing the carbon ﬁ-

bers required. Estimate the tensile strength of

the ﬁbers that are produced.

17-40 What are the advantages of using ceramic-

matrix composites?

Section 17-7 Laminar Composite Materials

Section 17-8 Examples and Applications

of Laminar Composites

Section 17-9 Sandwich Structures

17-41 What is a laminar composite?

17-42 A microlaminate, Arall, is produced using 5

sheets of 0.4-mm-thick aluminum and 4 sheets

of 0.2-mm-thick epoxy reinforced with unidi-

rectionally aligned Kevlar

ﬁbers. The volume

fraction of Kevlar

ﬁbers in these inter-

mediate sheets is 55%. Calculate the modulus of

elasticity of the microlaminate parallel and per-

pendicular to the unidirectionally aligned

Kevlar

ﬁbers. What are the principle advan-

tages of the Arall material compared with those

of unreinforced aluminum?

17-43 A laminate composed of 0.1-mm-thick alumi-

num sandwiched around a 2-cm-thick layer of

polystyrene foam is produced as an insulation

material. Calculate the thermal conductivity of

the laminate parallel and perpendicular to the

layers. The thermal conductivity of aluminum is

0.57

cal

cm  s  K

and that of the foam is

0.000077

cal

cm  s  K

17-44 A 0.01-cm-thick sheet of a polymer with a

modulus of elasticity of 4.8  10

MPa is sand-

wiched between two 4-mm-thick sheets of glass

with a modulus of elasticity of 83  10

MPa.

Calculate the modulus of elasticity of the com-

posite parallel and perpendicular to the sheets.

17-45 A U.S. quarter is 2.3 cm in diameter and is

about 0:16 cm thick. Assuming copper costs

about $2.43 per kg and nickel costs about $9

per kg. Compare the material cost in a compo-

site quarter versus a quarter made entirely of

nickel.

17-46 Calculate the density of a honeycomb structure

composed of the following elements: The two

2-mm-thick cover sheets are produced using

an epoxy matrix prepreg containing 55 vol%

E-glass ﬁbers. The aluminum honeycomb is

2 cm thick; the cells are in the shape of 0.5 cm

squares and the walls of the cells are 0.1 mm

thick. Estimate the density of the structure.

Compare the weight of a 1 m  2 m panel of the

honeycomb compared with a solid aluminum

panel of the same dimensions.

Design Problems

17-47 Design the materials and processing required

to produce a discontinuous, but aligned, ﬁber-

reinforced ﬁberglass composite that will form

the hood of a sports car. The composite should

provide a density of less than 1.6 g/cm

and a

strength of 138 MPa. Be sure to list all of the

assumptions you make in creating your design.

17-48 Design an electrical-contact material and a

method for producing the material that will

result in a density of no more than 6 g/cm

, yet

at least 50 vol% of the material will be con-

ductive.

17-49 What factors will have to be considered in

designing a bicycle frame using an aluminum

frame and a frame made using C—C compo-

site?

C HA P T E R 1 7 Composites: Teamwork and Synergy in Materials584

Appendix A: Selected Physical Properties of Some Elements

Metal

Atomic

Number

Crystal

Structure

Lattice

Parameters

(A)

Atomic

Mass

g/mol

Density

(g/cm

)

Melting

Temperature

(

Aluminum Al 13 FCC 4.04958 26.981 2.699 660.4

Antimony Sb 51 hex a ¼ 4.307

c ¼ 11.273

121.75 6.697 630.7

Arsenic As 33 hex a ¼ 3.760

c ¼ 10.548

74.9216 5.778 816

Barium Ba 56 BCC 5.025 137.3 3.5 729

Beryllium Be 4 hex a ¼ 2.2858

c ¼ 3.5842

9.01 1.848 1290

Bismuth Bi 83 hex a ¼ 4.546

c ¼ 11.86

208.98 9.808 271.4

Boron B 5 rhomb a ¼ 10.12

a ¼ 65.5



10.81 2.3 2300

Cadmium Cd 48 HCP a ¼ 2.9793

c ¼ 5.6181

112.4 8.642 321.1

Calcium Ca 20 FCC 5.588 40.08 1.55 839

Cerium Ce 58 HCP a ¼ 3.681

c ¼ 11.857

140.12 6.6893 798

Cesium Cs 55 BCC 6.13 132.91 1.892 28.6

Chromium Cr 24 BCC 2.8844 51.996 7.19 1875

Cobalt Co 27 HCP a ¼ 2.5071

c ¼ 4.0686

58.93 8.832 1495

Copper Cu 29 FCC 3.6151 63.54 8.93 1084.9

Gadolinium Gd 64 HCP a ¼ 3.6336

c ¼ 5.7810

157.25 7.901 1313

Gallium Ga 31 ortho a ¼ 4.5258

b ¼ 4.5186

c ¼ 7.6570

69.72 5.904 29.8

Germanium Ge 32 diamond cubic 5.6575 72.59 5.324 937.4

Gold Au 79 FCC 4.0786 196.97 19.302 1064.4

Hafnium Hf 72 HCP a ¼ 3.1883

c ¼ 5.0422

178.49 13.31 2227

Indium In 49 tetra a ¼ 3.2517

c ¼ 4.9459

114.82 7.286 156.6

Iridium Ir 77 FCC 3.84 192.9 22.65 2447

Iron Fe 26 BCC

FCC (>912



BCC (>1394



2.866

3.589

55.847 7.87 1538

Lanthanum La 57 HCP a ¼ 3.774

c ¼ 12.17

138.91 6.146 918

Lead Pb 82 FCC 4.9489 207.19 11.36 327.4

Lithium Li 3 BCC 3.5089 6.94 0.534 180.7

(continued)

585

Metal

Atomic

Number

Crystal Struc-

ture

Lattice

Parameters

(A)

Atomic

Mass

g/mol

Density

(g/cm

)

Melting

Temperature

(

Magnesium Mg 12 HCP a ¼ 3.2087

c ¼ 5.209

24.312 1.738 650

Manganese Mn 25 cubic 8.931 54.938 7.47 1244

Mercury Hg 80 rhomb 200.59 13.546 38.9

Molybdenum Mo 42 BCC 3.1468 95.94 10.22 2610

Nickel Ni 28 FCC 3.5167 58.71 8.902 1453

Niobium Nb 41 BCC 3.294 92.91 8.57 2468

Osmium Os 76 HCP a ¼ 2.7341

c ¼ 4.3197

190.2 22.57 2700

Palladium Pd 46 FCC 3.8902 106.4 12.02 1552

Platinum Pt 78 FCC 3.9231 195.09 21.45 1769

Potassium K 19 BCC 5.344 39.09 0.855 63.2

Rhenium Re 75 HCP a ¼ 2.760

c ¼ 4.458

186.21 21.04 3180

Rhodium Rh 45 FCC 3.796 102.99 12.41 1963

Rubidium Rb 37 BCC 5.7 85.467 1.532 38.9

Ruthenium Ru 44 HCP a ¼ 2.6987

c ¼ 4.2728

101.07 12.37 2310

Selenium Se 34 hex a ¼ 4.3640

c ¼ 4.9594

78.96 4.809 217

Silicon Si 14 diamond cubic 5.4307 28.08 2.33 1410

Silver Ag 47 FCC 4.0862 107.868 10.49 961.9

Sodium Na 11 BCC 4.2906 22.99 0.967 97.8

Strontium Sr 38 FCC

BCC (>557



6.0849

4.84

87.62 2.6 768

Tantalum Ta 73 BCC 3.3026 180.95 16.6 2996

Technetium Tc 43 HCP a ¼ 2.735

c ¼ 4.388

98.9062 11.5 2200

Tellurium Te 52 hex a ¼ 4.4565

c ¼ 5.9268

127.6 6.24 449.5

Thorium Th 90 FCC 5.086 232 11.72 1775

Tin Sn 50 FCC 6.4912 118.69 5.765 231.9

Titanium Ti 22 HCP a ¼ 2.9503

c ¼ 4.6831

47.9 4.507 1668

BCC (>882



C) 3.32

Tungsten W 74 BCC 3.1652 183.85 19.254 3410

Uranium U 92 ortho a ¼ 2.854

b ¼ 5.869

c ¼ 4.955

238.03 19.05 1133

Vanadium V 23 BCC 3.0278 50.941 6.1 1900

Yttrium Y 39 HCP a ¼ 3.648

c ¼ 5.732

88.91 4.469 1522

Zinc Zn 30 HCP a ¼ 2.6648

c ¼ 4.9470

65.38 7.133 420

Zirconium Zr 40 HCP a ¼ 3.2312

c ¼ 5.1477

91.22 6.505 1852

BCC (>862



C) 3.6090

Appendix A586

Appendix B: The Atomic and Ionic Radii of Selected Elements

Element

Atomic Radius

(A) Valence

Ionic Radius

(A)

Aluminum 1.432 þ3 0.51

Antimony 1.45 þ5 0.62

Arsenic 1.15 þ5 2.22

Barium 2.176 þ2 1.34

Beryllium 1.143 þ2 0.35

Bismuth 1.60 þ5 0.74

Boron 0.46 þ3 0.23

Bromine 1.19 1 1.96

Cadmium 1.49 þ2 0.97

Calcium 1.976 þ2 0.99

Carbon 0.77 þ4 0.16

Cerium 1.84 þ3 1.034

Cesium 2.65 þ1 1.67

Chlorine 0.905 1 1.81

Chromium 1.249 þ3 0.63

Cobalt 1.253 þ2 0.72

Copper 1.278 þ1 0.96

Fluorine 0.6 1 1.33

Gallium 1.218 þ3 0.62

Germanium 1.225 þ4 0.53

Gold 1.442 þ1 1.37

Hafnium 1.55 þ 4 0.78

Hydrogen 0.46 þ1 1.54

Indium 1.570 þ 3 0.81

Iodine 1.35 1 2.20

Iron 1.241 (BCC) þ2 0.74

1.269 (FCC) þ3 0.64

Lanthanum 1.887 þ3 1.016

Lead 1.75 þ4 0.84

Lithium 1.519 þ1 0.68

Magnesium 1.604 þ2 0.66

Manganese 1.12 þ2 0.80

þ3 0.66

Mercury 1.55 þ2 1.10

Molybdenum 1.363 þ4 0.70

Nickel 1.243 þ2 0.69

(continued)

587

Element

Atomic Radius

(A) Valence

Ionic Radius

(A)

Niobium 1.426 þ4 0.74

Nitrogen 0.71 þ5 0.15

Oxygen 0.60 2 1.32

Palladium 1.375 þ4 0.65

Phosphorus 1.10 þ5 0.35

Platinum 1.387 þ2 0.80

Potassium 2.314 þ1 1.33

Rubidium 2.468 þ1 0.70

Selenium 1.15 2 1.91

Silicon 1.176 þ4 0.42

Silver 1.445 þ1 1.26

Sodium 1.858 þ1 0.97

Strontium 2.151 þ2 1.12

Sulfur 1.06 2 1.84

Tantalum 1.43 þ5 0.68

Tellurium 1.40 2 2.11

Thorium 1.798 þ4 1.02

Tin 1.405 þ4 0.71

Titanium 1.475 þ4 0.68

Tungsten 1.371 þ4 0.70

Uranium 1.38 þ4 0.97

Vanadium 1.311 þ3 0.74

Yttrium 1.824 þ3 0.89

Zinc 1.332 þ2 0.74

Zirconium 1.616 þ4 0.79

Note that 1 A ¼10

8

cm ¼ 0.1 nanometer (nm)

Appendix B588

Answers to Selected Problems

CHAPTER 2

2-6 (a) 1:037  10

atoms.

2-7 (a) 1:078  10

atoms/ton.

2-8 (a) 5:70  10

atoms. (b) 0.948 mol.

2-10 3.

2-16 0.086.

2-27 Al

, with strong ionic bonding, has lower coe‰cient.

2-29 Weak secondary bonds between chains.

CHAPTER 3

3-3 (a) 1:426  10

8

cm. (b) 1:4447  10

8

cm.

3-5 (a) 5.3355 A. (b) 2.3103 A.

3-7 FCC.

3-9 BCT.

3-10 (a) 8 atoms/cell. (b) 0.387.

3-15 0.6% contraction.

3-19 A: ½00

1. B: ½120. C: ½111. D: ½211.

3-21 A: ð1

11Þ. B: ð 030Þ. C: ð102Þ.

3-25 ½

110, ½110, ½101, ½101, ½011, ½011.

3-27 Tetragonal—4; orthorhombic—2; cubic—12.

3-29 (a) (111). (c) ð0

12Þ.

3-31 [100]: 0.35089 nm, 2.85 nm

1

, 0.866. [110]: 0.496 nm,

2.015 nm

1

, 0.612. [111]: 0.3039 nm, 3.291 nm

1

,1.

3-32 (100): 1:1617  10

/cm

, packing fraction 0.7854.

(110): 1:144  10

/cm

, packing fraction 0.555.

(111): 1:867  10

/cm

, 0.907.

3-35 (a) 0.2797 A. (b) 0.629 A.

3-44 Fluorite. (a) 5.2885 A. (b) 12.13 g/cm

. (c) 0.624.

3-46 Cesium chloride. (a) 4.1916 A. (b) 4.8 g/cm

. (c) 0.693.

CHAPTER 4

4-1 4:97  10

vacancies/cm

4-3 (a) 0.002375. (b) 1:61  10

vacancies/cm

4-5 (a) 1:157  10

vacancies/cm

. (b) 0.532 g/cm

4-7 0.345.

4-9 8.265 g/cm

4-11 (a) 0.0081. (b) one H atom per 123.5 unit cells.

4-13 (a) 0.0534 defects/unit cell. (b) 2:52  10

defects/cm

4-14 (a) ½0

11, ½011, ½110, (b) ½110, ½101, ½101.

4-16 ð1

10Þ, ð110Þ, ð011Þ, ð011Þ, ð101Þ, ð101Þ.

4-18 Expected: b ¼ 2.863 A, d ¼ 2:338 A. (110)[111]: b ¼

7.014 A, d ¼ 2:863 A. Ratio ¼ 0:44.

4-26 (a) K ¼ 0:13 MPa/

ﬃﬃﬃﬃ

, s

¼ 415:7 MPa. (b) 705.7 MPa.

4-28 (a) 128 grains/in

. (b) 1,280,000 grains/in

4-30 3.6.

CHAPTER 5

5-4 1:08  10

jumps/s.

5-6 (a) 59,230 cal/mol. (b) 0.055 cm

/s.

5-12 (a) 0.02495 at% Sb/cm. (b) 1:246  10

Sb/cm

 cm.

5-14 (a) 1:969  10

H atoms/cm

 cm. (b) 3:3  10

atoms/cm

 s.

5-16 1.245  10

3

g/h.

5-18 198



5-28 0.01 cm: 0.87% C. 0.05 cm: 0.43% C. 0.10 cm: 0.18% C.

5-30 907



5-32 0.53% C.

5-34 2.9 min.

5-36 12.8 min.

5-39 667



5-41 51,286 cal/mol; yes.

CHAPTER 6

6-3 (a) Deforms. (b) Does not neck.

6-5 (b) 4740 N.

6-7 82,750 N.

6-9 (a) 5 cm.

6-10 15.2 m.

6-14 (a) 80 MPa. (b) 90.1 MPa. (c) 4267.7 MPa. (d) 4.6%.

(e) 3.4%. (f) 80 MPa. (g) 83.2 MPa. (h) 0.53 MPa.

6-16 (a) 274 MPa. (b) 417 MPa. (c) 172 GPa. (d) 18.55%.

(e) 15.8%. (f) 397.9 MPa. (g) 473 MPa. (h) 0.17 MPa.

6-18 (a) e

¼ 30:0077 cm. (b) d

¼ 0:9999 cm.

6-20 (a) 546.6 MPa. (b) 159; 430 MPa.

6-22 (a) 41 mm; will not fracture.

6-25 29.8.

6-30 No transition temperature.

6-33 Not notch-sensitive; poor toughness.

CHAPTER 7

7-4 0.99 MPa

ﬃﬃﬃﬃ

7-5 No; test will not be sensitive enough.

7-11 68 N.

7-13 d ¼ 4 cm.

7-17 22 MPa; max ¼þ22 MPa, min ¼22 MPa, mean ¼

0 MPa; reduce fatigue strength due to heating.

589

Askeland D.R., Fulay P.P. Essentials of Materials Science &amp; Engineering

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Askeland D.R., Fulay P.P. Essentials of Materials Science & Engineering