1 introduction to material science

95

Materials and3. Materials and Their Characteristics: Overview

In its most general context, the term mater-ials measurements denotes principles, techniquesand operations to distinguish qualitatively andto determine quantitatively the characteristicsof materials. As materials comprise all natu-ral and synthetic substances and constitute thephysical matter of products and systems, suchas

• machines, devices, commodities• power plants and energy supplies• means of habitation, transport, and commu-nication,

it is clear that materials characterization methodshave a wide scope and impact for science, tech-nology, the economy and society. Whereas in thepreceding chapters the principles of metrology,

3.1 Basic Features of Materials .................... 963.1.1 Nature of Materials....................... 963.1.2 Types of Materials ........................ 973.1.3 Scale of Materials ......................... 993.1.4 Processing of Materials ................. 993.1.5 Properties of Materials .................. 993.1.6 Application of Materials ................ 100

3.2 Classification of MaterialsCharacterization Methods ...................... 101

References .................................................. 102

the science of measurement, are outlined, inthis chapter an overview on the basic features ofmaterials is given as a basis for the classification ofthe various methods used to characterize materialsby analysis, measurement, testing, modelling andsimulation.

Materials measurements are aimed at characterizing thefeatures of materials quantitatively; this is often closelyrelated to the analysis, modelling and simulation, andthe qualitative characterization of materials through test-ing [3.1], see Fig. 3.1.

Measurement Testing

Technical procedureconsisting of thedetermination of attributes,in accordance with aspecified procedure

Result ofanalysis, measurement, testing, modelling, simulation:

characterization of materials byquantities and attributes

Set of operationsfor the purpose ofdetermining the valueof a quantity

Natural or syntheticsubstances;physical matter ofproducts

Materials

Fig. 3.1 Scheme for the characterization of materials

Generally speaking, measurement begins with a def-inition of the measurand, the quantity that is to bemeasured [3.2], and it always involves a comparison ofthe measurand with a known quantity of the same kind.Whereas the general metrological system is based on the

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96 Part A The Materials Measurement System

well-defined SI-Units (see Chapt. 1 of this handbook),for materials there is a broad spectrum of “material mea-surands”. This is due to the variety of materials, theirintrinsic chemical and physical nature, and the many at-tributes, which are related to materials with respect tocomposition, structure, scale, synthesis, properties andapplications. Some of these attributes can be expressed– in a metrological sense – as numbers, like density

or thermal conductivity; some are Boolean, such as theability to be recycled; some, like resistance to corrosion,may be expressed as a ranking (poor, adequate, good,for instance) and some can only be captured with textand images [3.3]. As a background for the materialsmeasurement system and the classification of mater-ials characterization methods, in this chapter the basicfeatures of materials are briefly reviewed.

3.1 Basic Features of Materials

Materials can be of natural origin or syntheticallyprocessed and manufactured. According to their chem-ical nature they are broadly grouped traditionallyinto inorganic and organic materials. Their physicalstructure can be crystalline, or amorphous. Compos-ites are combinations of materials assembled togetherto obtain properties superior to those of their sin-gle constituents. Composites are classified accordingto the nature of their matrix: metal, ceramic orpolymer composites, often designated MMCs, CMCsand PMCs, respectively. Figure 3.2 illustrates withcharacteristic examples the spectrum of materials be-tween the categories natural, synthetic, inorganic, andorganic.

3.1.1 Nature of Materials

From the view of materials science [3.4], the fundamen-tal features of a solid material are described as follows.

Materials Atomic Nature. The atomic elements of theperiodic table which constitute the chemical composi-tion of a material.

Materials Atomic Bonding. The type of cohesive elec-tronic interactions between the atoms (or molecules) ina material, empirically categorised into the followingbasic classes:

• Ionic bonds form between chemical elementswith very different electron-negativity (tendencyto gain electrons), resulting in electron transferand the formation of anions and cations. Bond-ing occurs through electrostatic forces between theions.• Covalent bonds form between elements that havesimilar electron-negativities, the electrons are lo-calised and shared equally between the atoms,leading to spatially directed angular bonds.

• Metallic bonds occur between elements with lowelectron-negativities, so that the electrons are onlyloosely attracted to the ionic nuclei. A metal isthought of as a set of positively charged ions em-bedded in an electron sea.• Van der Waals bonds are due to the different inter-nal electronic polarities between adjacent atoms ormolecules leading to weak (secondary) electrostaticdipole bonding forces.

Materials Spatial Atomic Structure. The amorphous orcrystalline arrangement of atoms (or molecules) in crys-talline structures is characterized by unit cells which arethe fundamental building blocks or modules repeatedmany times in space within a crystal.

Grains. Crystallites made up of identical unit cells re-peated in space, separated by grain boundaries.

Phases. Homogeneous aggregations of matter with re-spect to chemical composition and uniform crystal

Natural

Inor

gani

c

Org

anic

Synthetic

Minerals WoodPaper

CompositesMMC, CMC,

PMC

MetalsCeramics Polymers

Fig. 3.2 Classification of materials

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Materials and Their Characteristics: Overview 3.1 Basic Features of Materials 97

structure: grains composed of the same unit cells arethe same phase.

Lattice Defects. Deviations of an ideal crystal structure:

• Point defects or missing atoms: vacancies• Line defects or rows of missing atoms: dislocations• Area defects: grain boundaries• Volume defects: cavities

Microstructure. The microscopic collection of grains,phases, lattice defects and grain boundaries.

Together with bulk material characteristics, surfaceand interface phenomena have to considered.

3.1.2 Types of Materials

It has been estimated that there are between 40 000 and80 000 materials which are used or can be used in today’stechnology [3.3]. Figure 3.3 lists the main conventionalfamilies of materials together with examples of classes,members, and attributes. For the examples of attributes,sufficient characterization methods are named.

From a technological point of view, the materialscategorized in Fig. 3.3 as families have different charac-teristics relevant for engineering applications [3.5]:

Metallic Materials; Alloys. In metals, the grains as thebuildings blocks are held together by the electron gas.The free valence electrons of the electron gas accountfor the high electrical and thermal conductivity andthe optical gloss of metals. The metallic bonding –seen as an interaction between the sum total of atomicnuclei and the electron gas – is not significantly influ-enced by a displacement of atoms. This is the reason

Composition:Chemical analysisDensity:MeasurementGrain size:Computational modellingWear resistance:3-body-systems-testingReliability:Probabilistic simulation

CuBeCo

CuCd

CuCr

CuPb

Bronze

CuTe

CuZr

Subject Family Class Member Attributes

Natural

Ceramics

Polymers

Metals

Semiconductors

Composites

Biomaterials

Steels

Cast iron

Al-alloys

Cu-alloys

Ni-alloys

Ti-alloys

Zn-alloys

Materials

Fig. 3.3 Materials types with examples of materials attributes and characterization methods (after [3.3])

for the good ductility and formability of metals. Met-als and metallic alloys are the most important groupof the so-called structural materials (see below) whosespecial features for engineering applications are theirmechanical properties, e.g. strength and toughness.

Semiconductors. Semiconductors have an intermedi-ate position between metals and inorganic non-metallicmaterials. Their most important representatives are theelements silicon and germanium, possessing covalentbonding and diamond structure and the similarly struc-tured III–V-compounds, like gallium arsenide (GaAs).Being electric non-conductors at absolute zero, semi-conductors can be made conductive through thermalenergy input or atomic doping which leads to thecreation of free electrons contributing to electricalconductivity. Semiconductors are important functionalmaterials (see below) for electronic components andapplications.

Inorganic Non-Metallic Materials, Ceramics. Atomsin these materials are held together by covalent andionic bonding. As covalent and ionic bonding ener-gies are much higher than metallic bonds, inorganicnon-metallic materials, like ceramics have high hard-ness and high melting temperatures. These materialsare basically brittle and not ductile: In contrast tothe metallic bond model, a displacement of atomisticdimensions theoretically already breaks localised cova-lent bonds or transforms anion–cation attractions intoanion–anion or cation–cation repulsions. Because ofmissing free valence electrons, inorganic non-metallicmaterials are poor conductors for electricity and heat,this qualifies them as good insulators in engineeringapplications.

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% Atomic, molecular % Microelectromechanical % Continuum engineeringsystems systems (MEMS) systems

% Electronic, quantum % Microstructures % Bulk, componentsstructures of materials joined structures

Nanoscale Microscale Macroscale

10–9 10–6 10–3 100 103(m)

Fig. 3.4 Scale of materials: systems and structures

Organic Materials; Polymers, Blends. Organic mater-ials whose technologically most important represen-tatives are the polymers, consist of macromoleculescontaining carbon covalently bonded with itself and withelements of low atomic number (e.g. H, N, O, S). Inti-mate mechanical mixtures of several polymers are calledblends. In thermoplastic materials, the molecular chainshave long linear structures and are held together through(weak) intermolecular (van der Waals) bonds, leading tolow melting temperatures. In thermosetting materials thechains are connected in a network structure and do notmelt. Amorphous polymer structures (e.g. polystyrene)are transparent, whereas the crystalline polymers aretranslucent to opaque. The low density of polymersgives them a good strength-to-weight ratio and makesthem competitive with metals in structural engineeringapplications.

Composites. Generally speaking, composites are hybridcreations made of two or more materials that maintaintheir identities when combined. The materials are cho-

. . .Material type. . .Size rangeShapeToleranceRoughness. . .Batch quantity

Compression

Rotation

Injection

Blow

Subject Family Class Member Attributes

Joining

Shaping

Surfacing

Casting

Deformation

Moulding

Composite

Powder

Rapid prototyping

Process

Fig. 3.5 Hierarchy of processing of materials

sen so that the properties of one constituent enhance thedeficient properties of the other. Usually, a given prop-erty of a composite lies between the values for eachconstituent, but not always. Sometimes, the property ofa composite is clearly superior to those of either of theconstituents. The potential for such a synergy is one rea-son for the interest in composites for high-performanceapplications. However, because manufacturing of com-posites involves many steps and is labour intensive,composites may be too expensive to compete with met-als and polymers, even if their properties are superior. Inhigh-tech applications of advanced composites it shouldalso be borne in mind that they are usually difficult torecycle.

Natural Materials. Natural materials used in engineer-ing applications are classified into natural materials ofmineral origin, e.g. marble, granite, sandstone, mica,sapphire, ruby, diamond, and those of organic origin,e.g. timber, India rubber, natural fibres, like cotton andwool. The properties of natural materials of mineral

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Materials and Their Characteristics: Overview 3.1 Basic Features of Materials 99

Matter Materials

%Structural Materials→ mechanical, thermal tasks

%Functional Materials→ electrical, magnetic, optical tasks

%Smart Materials→ sensor + actuator task

%Solids, Liquids

%Atoms, Molecules

Processing

Manufacturing

Machining

Forming

Nanoscale manipulation

Assembly

Fig. 3.6 Materials and their characteristics result from the processing of matter

origin, such as for example high hardness and goodchemical durability, are determined by strong cova-lent and ionic bonds between their atomic or molecularconstituents and stable crystal structures. Natural mater-ials of organic origin often possess complex structureswith direction-dependent properties. Advantageous ap-plication aspects of natural materials are recycling andsustainability.

Biomaterials. Biomaterials can be broadly definedas the class of materials suitable for biomedical ap-plications. They may be synthetically derived fromnon-biological or even inorganic materials or theymay originate in living tissues. The products thatincorporate biomaterials are extremely varied and in-clude artificial organs; biochemical sensors; disposablematerials and commodities; drug-delivery systems; den-tal, plastic surgery, ear and ophthalmological devices;orthopedic replacements; wound management aids;and packaging materials for biomedical and hygienicuses.

For the application of biomaterials the understand-ing of the interactions between synthetic substrates andbiological tissues are of crucial importance to meet theneeds of clinical requirements. However, medical andclinical aspects of biomaterials are not treated in thisHandbook.

3.1.3 Scale of Materials

The geometric length scale of materials has more thantwelve orders of magnitude. The scale ranges fromthe nanometer dimensions of quantum-well structures –with novel application potentials for advanced commu-nications technologies – to the kilometer-long. structuresof bridges for public transport, pipelines and oil-drillingplatforms for the energy supply of society. Accordingly,

materials measurement methods have to characterizematerials with respect to the

1. Nanoscale, sizes of about 1 to 100 nanometers [3.6],2. Microscale, relevant for micro-devices and micro-

systems having sizes of typically 1 to 1000 micro-meters [3.7],

3. Macroscale materials have the dimensions of all cus-tomary products, devices and plants, ranging fromthe millimeter to the kilometer scale [3.8].

Figure 3.4 gives an overview on materials scales withsome key words.

3.1.4 Processing of Materials

For their use, materials have to be engineered by pro-cessing and manufacture in order to fulfil their purposeas the physical basis of products designed for the needsof the economy and society. There are the followingmain technologies to transform matter into engineeredmaterials [3.9]:

• Machining, i. e. shaping, cutting, drilling, etc. ofsolids,• Net forming of suitable matter, e.g. liquids, moulds,• Nanotechnology assembly of atoms or molecules.

In addition to these methods, there are also fur-ther technologies, like surfacing and joining, whichare applied to process, shape and assemble mater-ials and products. The design of materials may alsobe supported by computational methods [3.10]. Ithas been estimated that there are at least 1000 dif-ferent ways to produce materials [3.3]. Figure 3.5lists some of the families of processing materialstogether with examples of classes, members, andattributes.

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3.1


Raw materials Engineering materials

%ores%natural substances%coal%chemicals%oil

%metals%ceramics%polymers%structural materials% functional materials

%scrap%waste% refuse

Deposition Performance

Recycling

The earth

Recycling

TechnicalProducts

Fig. 3.7 The materials cycle

3.1.5 Properties of Materials

According to their properties, materials can be broadlyclassified into the following groups [3.11]:

• Structural materials: engineered materials with spe-cific mechanical or thermal properties• Functional materials: engineered materials withspecific electrical, magnetic or optical properties• Smart materials: engineered materials with intrin-sic or embedded “sensors” and “actuators” whichare able to react in response to external loading,aiming at optimising the materials’ behaviour ac-cording to given requirements for the materialsperformance [3.12].

It must be emphasized that the characteristics ofengineered structural, functional, and smart materialsdepend essentially on their processing and manufac-ture, as illustrated in a highly simplified manner inFig. 3.6.

3.1.6 Application of Materials

For the application of materials, their quality, safety andreliability as constituents of products and engineeredcomponents and systems are of special importance. Thisadds performance attributes to the characteristics to bedetermined by materials measurement and testing. Inthis context the materials cycle must be considered.

Figure 3.7 illustrates that all materials (accompa-nied by the necessary flow of energy and information)move in cycles through the techno-economic system:from raw materials to engineering materials and techni-cal products, and finally, after the termination of theirtask and performance, to deposition or recycling. Fromthe materials cycle, which applies to all branches of tech-nology, it is obvious that materials and their properties –to be determined through measurement and testing – areof crucial importance for the performance of technicalproducts. This is illustrated in Table 3.1 for some exam-ples of products and technical systems from the energysector [3.13].

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Materials and Their Characteristics: Overview 3.2 Classification of Materials Characterization Methods 101

Table 3.1 Application examples of materials in energy systems and relevant materials properties [3.13]

Application Materials properties

Mechanical Thermal Electrical Magnetic Optical

Heat engine High-temperature

strength

Electricity generator High-temperature

strength

Nuclear pressure vessel Resistance to

crack growth

Solar energy Heat absorption Photoelectricity Reflectance

Superconductor Ductility; strength High current Magnetic

capacity quenching

Conservation Light weight; Thermal insulation; Semiconductivity Magnetic Low

strength high-temperature efficiency transmission

resistance loss

3.2 Classification of Materials Characterization Methods

From a realization concerning the application of allmaterial, a classification of materials characterizationmethods can be outlined in a simplified manner:

Whenever a material is being created, developed,or produced the properties or phenomena the mater-ial exhibits are of central concern. Experience showsthat the properties and performance associated witha material are intimately related to its compositionand structure at all levels, including which atoms arepresent and how the atoms are arranged in the mater-ial, and that this structure is the result of synthesis,processing and manufacture. The final material mustperform a given task and must do so in an eco-nomical and socially acceptable manner. These mainelements:

• composition and structure,• properties,• performance

and the interrelationship among them define the maincategories of materials characterization methods to beapplied to these elements, see Fig. 3.8.

Figure 3.8 illustrates that the materials charac-terization methods comprise analysis, measurement,testing, modelling, and simulation. These methods

are described in detail in the following parts of thisbook:

• Methods to analyze the composition and structureof materials with respect to chemical composition,nanoscopic architecture and microstructure, surfacesand interfaces are compiled in Part B .• Methods to measure the mechanical, thermal, elec-trical, magnetic and optical material properties aredescribed in Part C .• Methods of testing material performance through thedetermination of mechanisms which are detrimentalto materials integrity, like corrosion, wear, biode-terioration, materials-environment interactions, areoutlined in Part D , which also contains the de-scription of methods for performance control andcondition monitoring.• Methods of modelling and simulation by mathemat-ical and computational approaches – ranging fromMolecular Dynamics Modelling to Monte Carlo sim-ulation – are described in Part E .

Supporting the presentation of the materials characteri-zation methods, in the Appendix relevant InternationalStandards of Materials Measurement Methods are com-piled.

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3.2


AnalysisMeasurement

TestingModellingSimulation

Performance

Materials failure mechanisms:%corrosion% friction and wear%biogenic impact%materials-environment interactions

And performance control by condition monitoring methods:%non-destructive evaluation% lifetime predictions%characterization of safety and reliability

Properties

%Mechanical%Thermal%Electrical%Magnetic%Optical

Composition, Structure

%Chemistry%Microstructure%Surfaces and

interfaces

Fig. 3.8 Categories of materials characterization methods

References

3.1 BIPM: International Vocabulary of Basic and Gen-eral Terms in Metrology (Bureau International PoidsMesures, Paris 1993)

3.2 H. Czichos, W. Daum: Measurement methods andsensors. In: Dubbel Taschenbuch für den Maschi-nenbau, ed. by W. Beitz, K.-H. Grote (Springer,Berlin, Heidelberg 2004) (in German)

3.3 M. F. Ashby, Y. J. M. Brechet, D. Cebon, L. Salvo: Se-lection strategies for materials and processes, Mater.Design 25, 51–67 (2004)

3.4 Encyclopedia of Materials: Science and Technology,ed. by K. H. J. Buschow, R. W. Cahn, M. C. Flem-ings, B. Ilschner, E. J. Kramer, S. Mahajan (Elsevier,Amsterdam 2001)

3.5 H. Czichos (Ed.): Materials. In: HÜTTE Das Inge-nieurwissen (Springer, Berlin, Heidelberg 2004) (inGerman)

3.6 Springer Handbook of Nanotechnology, ed. byB. Bhushan (Springer, Berlin, Heidelberg 2004)

3.7 S. D. Senturia: Microsystem Design (Kluwer, Boston2001)

3.8 Dubbel Taschenbuch für den Maschinenbau, ed. byW. Beitz, K.-H. Grote (Springer, Berlin, Heidelberg2004)

3.9 M. P. Groover: Fundamentals of Modern Manufac-turing (Wiley, New York 2002)

3.10 Computational Materials Design, ed. by T. Saito(Springer, Berlin, Heidelberg 1999)

3.11 N. A. Waterman, M. F. Ashby: The Materials Selector,2nd edn. (Chapman, London 1996)

3.12 M. Schwartz: Encyclopedia of Smart Materials (Wiley,New York 2002)

3.13 Britannica Editors: Materials. In: Encyclopedia Bri-tannica, 2001 edn. (Britannica, Chicago 2001)

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17

Materials Scie2. Materials Science for the Experimental Mechanist

Craig S. Hartley

This chapter presents selected principles of ma-terials science and engineering relevant to theinterpretation of structure–property relationships.Following a brief introduction, the first sectiondescribes the atomic basis for the description ofstructure at various size levels. Types of atomicbonds form a basis for a classification scheme ofmaterials as well as for the distinction betweenamorphous and crystalline materials. Crystal struc-tures of elements and compounds are described.The second section presents the thermodynamicand kinetic basis for the formation of microstruc-tures and describes the use of phase diagramsfor determining the nature and quantity of equi-librium phases present in materials. Principalmethods for the observation and determination ofstructure are described. The structural foundationsfor phenomenological descriptions of equilibrium,dissipative, and transport properties are described.The chapter includes examples of the relation-ships among physical phenomena responsible forvarious mechanical properties and the values of

2.1 Structure of Materials ........................... 172.1.1 Atomic Bonding ........................... 182.1.2 Classification of Materials .............. 212.1.3 Atomic Order ............................... 222.1.4 Equilibrium and Kinetics ............... 282.1.5 Observation and Characterization

of Structure ................................. 31

2.2 Properties of Materials .......................... 332.2.1 The Continuum Approximation ...... 342.2.2 Equilibrium Properties .................. 352.2.3 Dissipative Properties ................... 382.2.4 Transport Properties of Materials .... 432.2.5 Measurement Principles

for Material Properties .................. 46

References .................................................. 47

these properties. In conclusion the chapterpresents several useful principles for experimen-tal mechanists to consider when measuring andapplying values of material properties.

2.1 Structure of Materials

Engineering components consist of materials havingproperties that enable the items to perform the func-tions for which they are designed. Measurements ofthe behavior of engineering components under variousconditions of service are major objectives of experimen-tal mechanics. Validation and verification of analyticalmodels used in design require such measurements.All models employ mathematical relationships that re-quire knowledge of the behavior of materials undera variety of conditions. Assumptions such as isotropy,homogeneity, and uniformity of materials affect bothanalytical calculations and the interpretation of experi-mental results. Regardless of the scale or purpose of themeasurements, properties of materials that comprise the

components affect both the choice of experimental tech-niques and the interpretation of results. In measuringstatic behavior, it is important to know whether relevantproperties of the constituent materials are independentof time. Similarly, measurements of dynamic behav-ior require information on the dynamic and dissipativeproperties of the materials. At best, the fundamental na-ture of materials, which is the ultimate determinant oftheir behavior, forms the basis of these models. Theextent to which such assumptions represent the actualphysical situation limits the accuracy and significanceof results.

The primary axiom of materials science and engi-neering states that the properties and performance of

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2

18 Part A Solid Mechanics Topics

a material depend on its structure at one or more lev-els, which in turn is determined by the compositionand the processing, or thermomechanical history of thematerial. The meaning of structure as employed in ma-terials science and engineering depends on the scale ofreference. Atomic structure refers to the number and ar-rangement of the electrons, protons, and neutrons thatcompose each type of atom in a material. Nanostructurerefers to the arrangement of atoms over distances of theorder of 10−9 m. Analysis of the scattering of electrons,neutrons, or x-rays is the principal tool for measure-ments of structure at this scale. Microstructure refers tothe spatial arrangement of groups of similarly orientedatoms as viewed by an optical or electron microscopeat resolutions in the range 10−6 –10−3 m. Macrostruc-ture refers to arrangements of groups of microstructuralfeatures in the range of 10−3 m or greater, which canbe viewed by the unaided eye or under low-poweroptical magnification. Structure-insensitive properties,such as density and melting point, depend principallyon composition, or the relative number and types ofatoms present in a material. Structure-sensitive proper-ties, such as yield strength, depend on both compositionand structure, principally at the microscale.

This survey will acquaint the experimental mecha-nist with some important concepts of materials scienceand engineering in order to provide a basis for in-formed selections and interpretations of experiments.The chapter consists of a description of the princi-pal factors that determine the structure of materials,including techniques for quantitative measurements ofstructure, followed by a phenomenological descriptionof representative material properties with selected ex-amples of physically based models of the properties.A brief statement of some principles of measurementthat acknowledge the influence of material structure onproperties concludes the chapter. Additional informa-tion on many of the topics covered in the first twosections can be found in several standard introductorytexts on materials science and engineering for engi-neers [2.1–4]. Since this introduction can only brieflysurvey the complex field of structure–property relation-ships, each section includes additional representativereferences on specific topics.

2.1.1 Atomic Bonding

The Periodic TableThe realization that all matter is composed of a fi-nite number of elements, each consisting of atoms witha characteristic arrangement of elementary particles, be-

came widespread among scientists in the 19th and 20thcentury. Atomic theory of matter led to the discoveryof primitive units of matter known as electrons, pro-tons, and neutrons and laws that govern their behavior.Although discoveries through research in high-energyphysics constantly reveal more detail about the struc-ture of the atom, the planetary model proposed in 1915by Niels Bohr, with some modifications due to later dis-coveries of quantum mechanics, suffices to explain mostof the important aspects of engineering materials. In thismodel, atoms consist of a nucleus, containing protons,which have a positive electrical charge, and an approx-imately equal number of electrically neutral neutrons,each of which has nearly the same mass as a proton.Surrounding this nucleus is an assembly of electrons,which are highly mobile regions of concentrated nega-tive charge each having substantially smaller mass thana proton or neutron. The number of electrons is equal tothe number of protons in the nucleus, so each atom iselectrically neutral.

Elements differ from one another primarily throughthe atomic number, or number of protons in the nu-cleus. However, many elements form isotopes, whichare atoms having identical atomic numbers but differentnumbers of neutrons. If the number of neutrons differsexcessively from the number of protons, the isotope isunstable and either decays by the emission of neutronsand electromagnetic radiation to form a more stableisotope or fissions, emitting electromagnetic radiation,neutrons, and assemblies of protons and neutrons thatform nuclei of other elements.

The Periodic Table, shown in Fig. 2.1, classifieselements based on increasing atomic number and a pe-riodic grouping of elements having similar chemicalcharacteristics. The manner in which elements inter-act chemically varies periodically depending on theenergy distribution of electrons in the atom. The ba-sis for this grouping is the manner in which additionalelectrons join the atom as the atomic numbers of the el-ements increase. Quantum-mechanical laws that governthe behavior of electrons require that they reside in thevicinity of the nucleus in discrete spatial regions calledorbitals. Each orbital corresponds to a specific energystate for electrons and is capable of accommodating twoelectrons. Electron orbitals can have a variety of spatialorientations, which gives a characteristic symmetry tothe atom. Four quantum numbers, arising from solutionsto the Schrödinger wave equation, governs the behaviorof the electrons: the principal quantum number n, whichcan have any integer value from 1 to infinity; the az-imuthal quantum number �, which can have any integer

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Materials Science for the Experimental Mechanist 2.1 Structure of Materials 19

value from 0 to (n −1); the magnetic quantum numberm�, which can have any integer value between −� and+�; and the spin quantum number ms which has val-ues ±1/2. The Pauli exclusion principle states that notwo electrons in a system can have the same four quan-tum numbers. As the number of electrons increases withincreasing atomic number, orbitals are filled beginningwith those having the lowest electron energy states andproceeding to the higher energy states.

Elements with electrons in full, stable orbitals arechemically inert gases, which occupy the extreme rightcolumn of the periodic table (group 8). Electronegativeelements, which occupy columns towards the right of

H

1A1

1s1

hydrogen

1.008 2A

H

8A2

1s2

helium

4.003

Li3

[He]2s1

lithium

6.941

Be4

[He]2s2

beryllium

9.012

Na11

[Ne]3s1

sodium

22.99

Mg12

[Ne]3s2

magnesium

24.31

K19

[Ar]4s1

pollassium

39.10

Ca20

[Ar]4s2

calcium

40.08

Rb37

[Kr]5s1

nubidium

85.47

Sr38

[Kr]5s2

strontium

87.62

Cs55

[Xe]6s1

casium132.9

Ba57

[Xe]6s2

barium137.3

Fr87

[Rn] 7s1

francium(223)

Ra88

[Rn]7s2

radium(226)

Ce58

[Xe]6s24f15d1

cerium

140.1

Th90

[Rn]7s26d2

thorium

232.0

3B

Sc21

[Ar]4s23d1

scandium

44.96

Y39

[Kr]5s24d1

yttrium

88.91

La*57

[Xe]6s25d1

lanthanum138.9

Ac ~89

[Rn]7s26d1

actinium(227)

Pr59

[Xe]6s24f3

praseodymium

140.9

Pa91

[Rn]7s25f26d1

protactinium

(231)

4B

Ti22

[Ar]4s23d2

titanium

47.88

Zr40

[Kr]5s24d2

zirconium91.22

Hf72

[Xe]6s24f145d2

hafnium178.5

Rf104

[Rn]7s25f146d2

rutherfordium(257)

Nd60

[Xe]6s24f4

neodymium

144.2

U92

[Rn]7s25f36d1 uranium

(238)

5B

V23

[Ar]4s23d3

vanadium

50.94

Nb41

[Kr]5s14d4

niobium92.91

Ta73

[Xe]6s24f145d3

tantalum180.9

Db105

[Rn]7s25f146d3

dubnium(260)

Pm61

[Xe]6s24f5

promethium

(147)

Np93

[Rn]7s25f46d1

neptunium

(237)

6B

Cr24

[Ar]4s13d5

chromium

52.00

Mo42

[Kr]5s14d5

molybdenum95.94

W74

[Xe]6s24f145d4

tungsten183.9

Sg106

[Rn]7s25f146d4

seaborgium(263)

Sm62

[Xe]6s24f6

samarium

(150.4)

Pu94

[Rn]7s25f6

plutonium

(242)

7B

Mn25

[Ar]4s23d5

manganese

55.94

Tc43

[Kr]5s24d5

technetium(98)

Re75

[Xe]6s24f145d5

rhenium186.2

Bh107

[Rn]7s25f146d5

bohrium(262)

Eu63

[Xe]6s24f7

europium

152.0

Am95

[Rn]7s25f7

americium

(43)

Fe26

[Ar]4s23d6

iron

55.85

Ru44

[Kr]5s14d7

ruthenium101.1

Os76

[Xe]6s24f145d6

osmium190.2

Hs108

[Rn]7s25f146d6

hassium(265)

Gd64

[Xe]6s24f75d1

gadolinium

157.3

Cm96

[Rn]7s25f76d1

curium

(247)

8B

Co27

[Ar]4s23d7

cobalt

58.93

Rh45

[Kr]5s14d8

rhodium102.9

Ir77

[Xe]6s24f145d7

iridium190.2

Mt109

[Rn]7s25f146d7

meitnerium(266)

Tb65

[Xe]6s24f9

terbium

158.9

Bk97

[Rn]7s25f9

berkelium

(247)

Ni28

[Ar]4s23d8

nickel

58.69

Pd46

[Kr]4d10

palladium106.4

Pt78

[Xe]6s14f145d9

platinum195.1

Ds110

[Rn]7s15f146d9

darmstadtium(271)

Dy66

[Xe]6s24f10

dysprosium

162.5

Cf98

[Rn]7s25f10

californium

(249)

11B

Cu29

[Ar]4s13d10

copper

63.55

Ag47

[Kr]5s14d10

silver107.9

Au79

[Xe]6s14f145d10

gold197.0

Uuu111

(272)

Ho67

[Xe]6s24f11

holmium

164.9

Es99

[Xe]6s24f11

einsteinium

(254)

12B

Zn30

[Ar]4s23d10

zinc

65.39

Cd48

[Kr]5s24d10

cadmium112.4

Hg80

[Xe]6s24f145d10

mercury200.5

Uub112

(277)

Er68

[Xe]6s24f12

erbium

167.3

Fm100

[Rn]7s25f12

fermium

(253)

3A

B5

[He]2s22p1

boron

10.81

Al13

[Ne]3s23p1

aluminum

26.98

Ga31

[Ar]4s23d104p1

gallium

69.72

In49

[Kr]5s24d105p1

indium114.8

Tl81

[Xe]6s24f145d106p1

thallium204.4

Tm69

[Xe]6s24f13

thulium

168.9

Md101

[Rn]7s25f13

mendelevium

(256)

4A

C6

[He]2s22p2

carbon

12.01

Si14

[Ne]3s23p2

silicon

28.09

Ge32

[Ar]4s23d104p2

germanium

72.58

Sn50

[Kr]5s24d105p2

tin118.7

Pb82

[Xe]6s24f145d106p2

lead207.2

Yb70

[Xe]6s24f14

ytterbium

173.0

No102

[Rn]7s25f14 nobelium

(254)

5A

N7

[He]2s22p3

nitrogen

14.01

P16

[Ne]3s23p3

phosphorus

30.97

As33

[Ar]4s23d104p3

arsenic

74.92

Sb51

[Kr]5s24d105p3

antimony121.8

Bi83

[Xe]6s24f145d106p3

bismuth208.9

Lu71

[Xe]6s24f145d1

lutetium

175.0

Lr103

[Rn]7s25f146d1

lawrencium

(257)

6A

O8

[He]2s22p4

oxygen

16.00

S16

[Ne]3s23p4

sulfur

32.07

Se34

[Ar]4s23d104p4

selenium

78.96

Te52

[Kr]5s24d105p4

tellurium 127.6

Po84

[Xe]6s24f145d106p4

polonium(209)

7A

F9

[He]2s22p5

fluorine

19.00

Cl17

[Ne]3s23p5

chlorine

35.45

Br35

[Ar]4s23d104p5

bromine

79.90

I53

[Kr]5s24d105p5

iodine126.9

At85

[Xe]6s24f145d106p5

astatine(210)

Ne10

[He]2s22p6

neon

20.18

Ar18

[Ne]3s23p6

argon

39.95

Kr36

[Ar]4s23d104p6

krypton

83.80

Xe52

[Kr]5s24d105p6

xenon131.3

Rn86

[Xe]6s24f145d106p6

radon(222)

Uuq114

(296)

Uuh116

(298)

Uuo118

(?)

Liquids at room temperature Gases at room temperature Solids at room temperature

Lanthanide series*

Actinide series ~

Fig. 2.1 The Periodic Table of the elements. Elements named in blue are liquids at room temperature. Elements named in red aregases at room temperature. Elements named in black are solids at room temperature

the periodic table, have nearly full orbitals and tend tointeract with other atoms by accepting electrons to forma negatively charged entity called an anion. The neg-ative charge arises since electrons join the originallyneutral atom. Electropositive elements occupy columnstowards the left on the periodic table and ionize byyielding electrons from their outer orbitals to form pos-itively charged cations.

Broadly speaking, elements are metals, metalloids,and nonmetals. The classification proceeds from themost electropositive elements on the left of the peri-odic table to the most electronegative elements on theright. A metal is a pure element. A metal that incorpo-

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rates atoms of other elements into its structure withoutchanging its essential metallic character forms an al-loy, which is not a metal since it is not a pure element.The major differences in materials have their origins inthe nature of the bonds formed between atoms, whichare determined by the manner in which electrons in thehighest-energy orbitals interact with one another and bywhether the centers of positive and negative charge ofthe atoms coincide. The work required to remove anion from the substance in which it resides is a measureof the strength of these bonds. At suitable tempera-tures and pressures, all elements can exist in all statesof matter, although in some cases this is very difficultto achieve experimentally. At ambient temperature andpressure, most elements are solids, some are gases, anda few are liquid.

Primary BondsPrimary bonds are the strongest bonds that form amongatoms. The manner in which electrons in the highestenergy levels interact produces differences in the kindsof primary bonds. Valence electrons occupy the highestenergy levels of atoms, called the valence levels. Va-lence electrons exhibit three basic types of behavior:atoms of electropositive elements yield their valenceelectrons relatively easily; atoms of electronegative el-ements readily accept electrons to fill their valencelevels; and elements between these extremes can shareelectrons with neighboring atoms. The valence of an ionis the number of electrons yielded, accepted or sharedby each atom in forming the ion. Valence is positiveor negative according to whether the ion has a positive(cation) or negative (anion) charge.

The behavior of valence electrons gives rise to threetypes of primary bonds: ionic, covalent and metallic.Ionic bonds occur between ions of strongly electropos-itive elements and strongly electronegative elements.Each atom of the electropositive element surrenders oneor more electrons to one or more atoms of the elec-tronegative element to form oppositely charged ions,which attract one another by the Coulomb force be-tween opposite electrical charges. This exchange ofelectrons occurs in such a manner that the overallstructure remains electrically neutral. To a good approx-imation, ions involved in ionic bonds behave as charged,essentially incompressible, spheres, which have nocharacteristic directionality. In contrast, covalent bondsinvolve sharing of valence electrons between neigh-boring atoms. This type of bonding occurs when thevalence energy levels of the atoms are partially full, cor-responding to valences in the vicinity of 4. These bonds

are strongly directional since the orbitals involved aretypically nonspherical. In both ionic and covalent bondsnearest-neighbor ions are most strongly involved andthe valence electrons are highly localized.

Metallic bonds occur in strongly electropositiveelements, which surrender their valence electrons toform a negatively charged electron gas or distributionof highly nonlocalized electrons that moves relativelyfreely throughout the substance. The positively chargedions repel one another but remain relatively stationarybecause the electron gas acting as glue holds them to-gether. Metallic bonds are relatively nondirectional andthe ions are approximately spherical. A major differencebetween the metallic bond and the ionic and covalentbonds is that it does not involve an exchange or sharingof electrons with nearest neighbors.

The bonds in many substances closely approximatethe pure bond types described above. However, mix-tures of these archetypes occur frequently in nature,and a substance can show bonding characteristics thatresemble more than one type. This hybrid bond situa-tion occurs most often in substances that exhibit somecharacteristics of directional covalent bonds along withnondirectional metallic or ionic bonds.

Secondary BondsSome substances are composed of electrically neutralclusters of ions called molecules. Secondary bonds ex-ist between molecules and are weaker than primarybonds. One type of secondary bond, the van der Waalsbond, is due to the weak electrostatic interaction be-tween molecules in which the instantaneous centers ofpositive and negative charge do not coincide. A mol-ecule consisting of a single ion of an electropositiveelement and a single ion of an electronegative element,such as a molecule of HCl gas, is a simple exampleof a diatomic molecule. The center of negative chargeof the system coincides with the nucleus of the chlo-rine ion, but the center of positive charge is displacedfrom the center of the ion because of the presence of thesmaller, positively charged hydrogen ion (a single pro-ton), which resides near the outer orbital of the chlorineion. This results in the formation of an electrical dipole,which has a short-range attraction to similar dipoles,such as other HCl molecules, at distances of the order ofthe molecular dimensions. However, no long-range at-traction exists since the overall charge of the molecule iszero. The example given is a permanent dipole formedby a spatial separation of centers of charge. A tempo-rary dipole can occur when the instantaneous centersof charge separate because of the motion of electrons.

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The resulting attraction forms a weak bond at smalldistances and is typical of van der Waals bonds.

The other secondary bond, the hydrogen bond, in-volves the single valence electron of hydrogen. Inmaterials science and engineering, the most importanttype of hydrogen bond is that which occurs in polymers,which consist of long chains or networks of chemicallyidentical units called mers. When the composition ofa mer includes hydrogen, it is possible for the hydrogenatom to share its valence electron with identical mers inneighboring chains, so that the hydrogen atom is partlyin one chain and partly in another. This sharing of thehydrogen atom creates a hydrogen bond between thechains. The bond is relatively weak but is an importantfactor in the behavior of polymeric materials.

2.1.2 Classification of Materials

It is useful to categorize engineering materials in termseither of their functionality or the dominant type ofatomic bonding present in the material. Since most ma-terials perform several functions in a component, theclassification scheme described in the following sec-tions takes the latter approach. The nature and strengthof atomic bonding influences not only the arrangementof atoms in space but also many physical propertiessuch as electrical conductivity, thermal conductivity,and damping capacity.

CeramicsCeramic materials possess bonding that is primarilyionic with varying amounts of metallic or covalent char-acter. The dominant features on the atomic scale arethe localization of electrons in the vicinity of the ionsand the relative incompressibility of atoms, leadingto structures that are characterized by the packing ofrigid spheres of various sizes. These materials typicallyhave high melting points (> 1500 K), low thermal andelectrical conductivities, high resistance to atmosphericcorrosion, and low damping capacity. Mechanical prop-erties of ceramics include high moduli of elasticity, highyield strength, high notch sensitivity, low ductility, lowimpact resistance, intermediate to low thermal shockresistance, and low fracture toughness. Applicationsthat require resistance to extreme thermal, electrical orchemical environments, with the ability to absorb me-chanical energy without catastrophic failure a secondaryissue, typically employ ceramics.

MetalsMetallic materials include pure metals (elements) andalloys that exhibit primarily metallic bonding. The

free electron gas that permeates the lattice of ionscauses these materials to exhibit high electrical andthermal conductivity. In addition they possess rela-tively high yield strengths, high moduli of elasticity,and melting points ranging from nearly room temper-ature to > 3200 K. Although generally malleable andductile, they can exhibit extreme brittleness, dependingon structure and temperature. One of the most use-ful features of metallic materials is their ability to beformed into complex shapes using a variety of thermo-mechanical processes, including melting and casting,hot working in the solid state, and a combination ofcold working and annealing. All of these processes pro-duce characteristic microstructures that lead to differentcombinations of physical and mechanical properties.Applications that require complex shapes having bothstrength and fracture resistance with moderate resis-tance to environmental degradation employ metallicmaterials.

MetalloidsMetalloids are elements in groups III–V of the peri-odic table and compounds formed from these elements.Covalent bonding dominates both the elements andcompounds in this category. The name arises fromthe fact that they exhibit behavior intermediate be-tween metals and ceramics. Many are semiconductors,that is, they exhibit an electrical conductivity lowerthan metals, but useable, which increases rather thandecreases with temperature like metals. These mater-ials exhibit high elastic moduli, relatively high meltingpoints, low ductility, and poor formability. Commer-cially useful forms of these materials require processingby solidification directly from the molten state followedby solid-state treatments that do not involve signifi-cant deformation. Metalloids are useful in a variety ofapplications where sensitivity and response to electro-magnetic radiation are important.

PolymersPolymeric materials, also generically called plastics, areassemblies of complex molecules consisting of molecu-lar structural units called mers that have a characteristicchemical composition and, often, a variety of spatialconfigurations. The assemblies of molecules generallytake the form of long chains of mers held together byhydrogen bonds or networks of interconnected mers.Most structural polymers are made of mers with anorganic basis, i. e., they contain carbon. They are char-acterized by relatively low strength, low thermal andelectrical conductivity, low melting points, often high

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ductility, and high formability by a variety of tech-niques. These materials are popular as electrical andthermal insulators and for structural applications thatdo not require high strength or exposure to high tem-peratures. Their principal advantages are relatively lowcost, high formability, and resistance to most forms ofatmospheric degradation.

CompositesComposite materials consist of those formed by in-timate combinations of the other classes. Compositescombine the advantages of two or more material classesby forming a hybrid material that exhibits certaindesirable features of the constituents. Generally, onetype of material predominates, forming a matrix con-taining a distribution of one or more other types ona microscale. A familiar example is glass-reinforcedplastic (GRP), known by the commercial name ofFibreglass R©. In this material, the high elastic modulusof the glass fibers (a ceramic) reinforces the tough-ness and formability of the polymeric matrix. Otherclasses of composites have metal matrices with ceramicdispersions (metal matrix composites, MMC), ceramicmatrices with various types of additions (ceramic matrixcomposites, CMC), and polymeric matrices with metal-lic or ceramic additions. The latter, generically knownas organic matrix composites (OMC) or polymer matrixcomposites (PMC), are important structural materialsfor aerospace applications.

2.1.3 Atomic Order

Crystalline and Amorphous MaterialsThe structure of materials at the atomic level canbe highly ordered or nearly random, depending onthe nature of the bonding and the thermomechan-ical history. Pure elements that exist in the solidstate at ambient temperature and pressure always ex-hibit at least one form that is highly ordered in thesense that the surroundings of each atom are identi-cal. Crystalline materials exhibit this locally orderedarrangement over large distances, creating long-rangeorder. The formal definition of a crystal is a sub-stance in which the structure surrounding each basisunit, an atom or molecule, is identical. That is, ifone were able to observe the atomic or moleculararrangement from the vantage point of a single struc-tural unit, the view would not depend on the locationor orientation of the structural unit within the ma-terial. All metals and ceramic compounds and somepolymeric materials have crystalline forms. Some ma-

terials can exhibit more than one crystalline form, calledallotropes, depending on the temperature and pres-sure. It is this property of iron with small amountsof carbon dissolved that is the basis for the heattreatment of steel, which provides a wide range ofproperties.

At the other extreme of atomic arrangement areamorphous materials. These materials can exhibit lo-cal order of structural units, but the arrangement a largenumber of such units is haphazard or random. Thereare two principal categories of amorphous structures:network structures and chain structures. The moleculesof network structures lie at the nodes of an irregularnetwork, like a badly constructed jungle gym. Never-theless, the network has a high degree of connectivityand if the molecules are not particularly mobile, thenetwork can be very stable. This type of structureis characteristic of most glasses. Materials possessingthis structure possess a relatively rigid mechanical re-sponse at low temperatures, but become more fluidand deformable at elevated temperatures. Frequently thetransition between the relatively rigid, low-temperatureform and the more fluid high-temperature form oc-curs over a narrow temperature range. By conventionthe midpoint of this transition range defines the glass-transition temperature.

Linear chain structures are characteristic of poly-meric materials made of long chains of mers. Relativelyweak hydrogen bonds and/or van der Waals bonds holdthese chains together. The chains can move past oneanother with varying degrees of difficulty dependingon the geometry of the molecular arrangement alongthe chain and the temperature. An individual chain canpossess short-range order, but the collections of chainsthat comprise the substance sprawl haphazardly, likea bowl of spaghetti. Under certain conditions of forma-tion, however, the chains can arrange themselves intoa pattern with long-range order, giving rise to crystallineforms of polymeric materials. In addition, some ele-ments, specific to the particular polymer, can bond withadjacent chains, creating a three-dimensional networkstructure. The addition of sulfur to natural rubber in theprocess called Vulcanizing R© is an example.

Materials that can exist in both the crystallineand amorphous states can also have intermediate,metastable structures in which these states coexist.Glass that has devitrified has microscopic crystallineregions dispersed in an amorphous network matrix.Combinations of heat treatment and mechanical defor-mation can alter the relative amounts of these structures,and the overall properties of the material.

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Crystal Structures of Elements and CompoundsBecause ionic bonds require ions of at least two ele-ments, either metallic or covalent bonds join ions ofpure elements in the solid state, although the con-densed forms of highly electronegative elements andthe inert gases exhibit weak short-range bonding typ-ical of van der Waals bonds. Chemical compounds,which can exhibit ionic bonding as well as the othertypes of strong bonds, form when atoms of two ormore elements combine in specific ratios. A stoichio-

Cubic

Tetragonal

Simple

Simple

Body-centered

Body-centered

Monoclinic

Simple End-centered

End-centered

Face-centered

Orthohombic

Simple

Rhombohedral Hexagonal Triclinic

Body-centered Face-centered

Fig. 2.2 Bravais lattices and crystal systems

metric compound contains ions exactly in the ratiosthat produce electrical neutrality of the substance. Inbinary (two-component) compounds, the ratio of thenumber of ions of each kind present is the inverseof the ratio of the absolute values of their valences.For example, Na2O has two sodium atoms for eachoxygen atom. Since the valence of sodium is +1and that of oxygen is −2, the 2 : 1 ratio of sodiumto oxygen ions produces electrical neutrality of thestructure.

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In the solid state, patterns of atoms and moleculesform lattices, which are three-dimensional arrays ofpoints having the property that the surroundings of eachlattice point are identical to those of any other latticepoint. There are only 14 unique lattices, the Bravaislattices, shown in Fig. 2.2. Each lattice possesses threenon-coplanar, non-collinear axes and a characteristic,unique array of lattice points occupied by structuralunits, which can be individual atoms or identical clus-ters of atoms, depending on the nature of the substance.The relative lengths of the repeat distance of latticepoints along each axis and the angles that the axes makewith one another define the seven crystal systems. Fig-ure 2.2 also shows the crystal system for each of theBravais lattices.

Each of the illustrations in Fig. 2.2 represents theunit cell for the lattice, which is the smallest arrange-ment of lattice points that possesses the geometriccharacteristics of the extended structure. Repeating oneof the figures in Fig. 2.2 indefinitely throughout spacewith an appropriately chosen structural unit at each lat-tice point defines a crystal structure. Lattice parametersinclude the angles between coordinate axes, if variable,and the dimensions of the unit cell, which containsone or more lattice points. To determine the number ofpoints associated with a unit cell, count 1/8 for eachcorner point, 1/2 for each point on a face, and 1 foreach point entirely within the cell. A primitive unit cellcontains only one lattice point (one at each corner).The coordination number Z is the number of nearestneighbors to a lattice point.

One of the most important characteristics of crys-tal lattices is symmetry, the property by which certainrigid-body motions bring the lattice into an equivalentconfiguration indistinguishable from the initial config-uration. Symmetry operations occur by rotations aboutan axis, reflections across a plane or a combination ofrotations, and translations along an axis. For example,a plane across which the structure is a mirror imageof that on the opposite side is a mirror plane. An axisabout which a rotation of 2π/n brings the lattice intocoincidence forms an n-fold axis of symmetry. Thischaracteristic of crystals has profound implications oncertain physical properties.

The geometry of the lattice provides a natural co-ordinate system for describing directions and planesusing the axes of the unit cell as coordinate axes andthe lattice parameters as units of measure. Principalcrystallographic axes and directions are those paral-lel to the edges of the unit cell. A vector connectingtwo lattice points defines a lattice direction. A set of

three integers having no common factor that are in thesame ratio as the direction cosines, relative to the co-ordinate axes, of such a vector characterizes the latticedirection. Square brackets, e.g., [100], denote specificcrystallographic directions, while the same three inte-gers enclosed by carats, e.g., 〈100〉, describe families ofdirections. Directions are crystallographically equiva-lent if they possess an identical arrangement of latticepoints. Families of directions in the cubic crystal systemare crystallographically equivalent, but those in noncu-bic crystals may not be because of differences in thelattice parameters.

The Miller indices, another set of three integers de-termined in a different manner, specify crystallographicplanes. The notation arose from the observation by 19thcentury crystallographers on naturally occurring crys-tals that the reciprocals of the intercepts of crystal faceswith the principal crystallographic axes occurred in theratios of small, whole numbers. To determine the Millerindices of a plane, first obtain the intercepts of the planewith each of the principal crystallographic directions.Then take the reciprocals of these intercepts and findthe three smallest integers with no common factor thathave the same ratios to one another as the reciprocalsof the intercepts. Enclosed in parentheses, these are theMiller indices of the plane. For example, the (120) planehas intercepts of 1, 1/2, and ∞, in units of the latticeparameters, along the three principal crystallographicdirections. Families of planes are those having the samethree integers in different permutations, including neg-atives, as their Miller indices. Braces enclose the Millerindices of families, e.g., {120}. Crystallographicallyequivalent planes have the same density and distributionof lattice points. In cubic crystals, families of planes arecrystallographically equivalent.

Although there are examples of all of the crystalstructures in naturally occurring materials, a relativelyfew suffice to describe common engineering materials.All metals are either body-centered cubic (bcc), face-centered cubic (fcc) or hexagonal close-packed (hcp).The latter two structures consist of different stackingsequences of closely packed planes containing identicalspheres or ellipsoids, representing the positive ions inthe metallic lattice. Figure 2.3 shows a plane of spherespacked as closely as possible in a plane.

An identical plane fitted as compactly as possibleon top of or below this plane occupies one of two possi-ble locations, corresponding to the depressions betweenthe spheres. These locations correspond to the uprightand inverted triangular spaces between spheres in thefigure. The same option exists when placing a third

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identical plane on the second plane, but now two dis-tinct situations arise depending on whether the thirdplane is exactly over the first or displaced from it in theother possible stacking location. In the first case, whenthe first and third planes are directly over one another,the stacking sequence is characteristic of hexagonalclose-packed structures and is indicated ABAB. . . Theclose-packed, or basal, planes are normal to an axis ofsixfold symmetry. Figure 2.4 shows the conventionalunit cell for the hcp structure. Based on the hexago-nal cell of the Bravais lattice, this unit cell contains twoatoms.

The c/a ratio is the height of the cell divided bythe length of the side of the regular hexagon form-ing the base. If the ions are perfect spheres, this ratiois 1.6333 = √

(8/3). In this instance, the coordinationnumber of the structure is 12. However, most metalsthat exhibit this structure have c/a ratios different fromthis ideal value, indicating that oblate or prolate ellip-soids are more accurate than spheres as models for theatoms. Consequently, the coordination number is a hy-brid quantity consisting of six atoms in the basal planeand six atoms at nearly the same distances in adjacentbasal planes. Nevertheless, the conventional value forthe coordination number of the hcp structure is 12 re-gardless of the c/a ratio.

When the third plane in a close-packed structureoccurs in an orientation that is not directly above thefirst, the stacking produces a face-centered cubic (fcc)structure. The sequence ABCABC. . . represents thisstacking. The {111} planes are close-packed in thisstructure, the coordination number is 12, and the unitcell contains four atoms, as shown in Fig. 2.5.

The third crystal structure typical of metallic ele-ments and alloys is the body-centered cubic structure,

B-layer C-layer

Fig. 2.3 Plane of close-packed spheres

a

c

Fig. 2.4 Hexagonal close-packed unit cell

which is not a close-packed structure. Figure 2.6 showsthe unit cell of this structure.

The structure has a coordination number of eightand the unit cell contains two atoms.

The density of a crystalline material follows from itscrystal structure and the dimensions of its unit cell. Bydefinition, density is mass per unit volume. For a unitcell this becomes the number of atoms in a unit celln times the mass of the atom, divided by the cell vol-ume Ω:

ρ = n A

ΩN0. (2.1)

The mass of an atom is the atomic weight, A, dividedby Avogadro’s number, N0 = 6.023 × 1023, which is thenumber of atoms or molecules in one gram-atomic orgram-molecular, respectively, weight of a substance.

The (8− N) rule classifies crystal structures of ele-ments that bond principally by covalent bonds, whereN (≥ 4) is the number of the element’s group in thePeriodic Table. The rule states that the element formsa crystal structure characterized by a coordination num-ber of (8− N). Thus, silicon in group 4 forms a crystal

Fig. 2.5 Face-centered cubic unit cell and {111} plane

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Fig. 2.6 Unit cell of bcc structure

that has four nearest neighbors. Crystal structures basedon this rule can be quite complex.

Ionic compounds are composed of cations and an-ions formed from two or more kinds of atoms. Thesecompounds form crystal structures based on two princi-ples: (1) the entire structure must be electrically neutraland (2) the ions can be regarded as relatively rigidspheres of differing sizes forming a three-dimensionalstructure based on the efficient packing of different sizespheres. The radius ratio rule is the principle that de-scribes the crystal structures of many ionic compounds.This rule specifies the coordination number Z whichgives the most efficient packing for spheres whose ra-dius ratios are in a particular range. The radius ratio Remployed for this calculation is the radius of the cationto that of the anion. Table 2.1 illustrates the relationshipof this ratio to the resulting lattice geometry.

Defects in CrystalsThe structures described in the previous sections areidealized descriptions that accurately characterize thearrangement of the vast majority of atomic and mo-lecular arrangements in materials. Within any substancethere will exist irregularities or defects in the struc-ture that profoundly influence many of the propertiesof the material. These defects can exist on the elec-tronic, atomic or molecular scale, depending on thesubstance. A common classification scheme for atomicand molecular defects in materials utilizes their dimen-sionality. Point defects have dimensions comparable toan atom or molecule. Line defects have an appreciable

Table 2.1 Radius ratios and crystal lattice geometry

Z R Exact range Lattice geometry

2 0.000–0.155 0 to 23

√3−1 Linear

3 0.155–0.225 23

√3−1 to 1

2

√6−1 Trigonal planar

4 0.225–0.414 12

√6−1 to

√2−1 Tetrahedral

6 0.414–0.732√

2−1 to√

3−1 Octahedral

8 0.732–1.000√

3−1 to 1 Cubic

extent along a linear path in the material, but essen-tially atomic dimensions in directions normal to thatpath. Surface defects have appreciable extent on a sur-face in two dimensions, but essentially atomic extentnormal to the surface. Surface defects are regions ofhigh local atomic disorder. This characteristic makesthem prone to higher chemical activity than the re-gions that they bound. In addition, the local disorderassociated with the boundaries has different mechan-ical properties than the bulk. These two features ofsurface defects cause them to be extremely importantin affecting the mechanical and often the chemicalproperties of materials. Volume defects occur over vol-umes of several tens to several millions of atoms. Thescale of these defects is generally at the mesoscale andabove.

The simplest kind of point defect occurs when anatom of a pure substance is missing from a lattice site.The vacant lattice site is a vacancy. Atoms of an elementthat leave their lattice sites and attempt to share a latticesite with another atom form interstitialcies. Clearly, in-terstitialcies are closely associated with vacancies. Thiscombination typically occurs under conditions of neu-tron radiation, and the associated damage is a limitingfactor to the use of materials in such environments.

A lattice site occupied by an atom different fromthe host is a substitutional impurity, creating a substi-tutional solid solution with the host as solvent and theimpurity as solute. The concentration of such impuritiesthat a host element can tolerate is dependent in largepart on the relative sizes of the impurity and the hostatoms. Typically, solid solubility is extremely limitedif the atomic sizes of solute and solvent differ by morethan 15%. Interstitial solid solutions occur when smallersolvent atoms occupy some of the spaces (interstices)between solute atoms. Only five elements – C, N, O,H, and B – can be interstitial solutes in metals, princi-pally because of the relative sizes of the atoms involved.The systematic or random insertion into linear poly-mers of mers having different chemical compositionsforms copolymers. Since the substituent mers constitutea disruption in the basic polymer chain, they are substi-

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tutional point defects in the same sense as solute atomsin a solid solution.

The semiconducting properties of metalloids aredue to the thermal excitation of electrons from filled va-lence bands into sparsely populated conduction bands.In a pure substance, this behavior is intrinsic semicon-ductivity. The addition of small amounts of impurities(of the order of one impurity atom per million hostatoms) that have a different chemical valence signif-icantly modifies the electrical conductivity of manysuch substances. These substitutional impurities, whichare electronic defects as well as lattice defects, con-tribute additional electronic energy states that cancontribute to conductivity. This is extrinsic semiconduc-tivity. A donor impurity has more valence electrons thanthe host does, and contributes electron energy states.Acceptor impurities have a lower valence than the host,leaving holes, which can contribute to the transport ofelectrical charge, in the valence energy band. The holesare defects in the electronic structure having equal andopposite charges to the electrons.

Point defects in ionic compounds must maintainelectrical neutrality. One or more defects possessingequal and opposite charges must be present to cancelany local charge caused by the addition or removal ofions from lattice sites. One such defect is the Frenkeldefect, a cation vacancy associated with a nearby cationinterstitial. The excess charge created by the intersti-tial cation balances the charge deficiency due to thevacancy. In principle, such defects are possible with an-ions replacing cations, but the relatively large size ofanions generally precludes the existence of anion in-terstitials. The other important type of point defect inionic materials is the Schottky defect, which is a cationvacancy–anion vacancy pair where the ions have equaland opposite charges. Clearly, the removal of an elec-trically neutral unit maintains local electrical neutrality.Both types of defects can affect electrical and mechani-cal properties of the substance.

Permanent deformation that preserves the numberof lattice sites in crystalline materials generally occursby slip, also called glide, which is the motion of onepart of the crystal relative to the other across a denselypopulated atomic plane, the slip plane, in a denselypopulated atomic direction, the slip direction. For ener-getic reasons this motion does not occur simultaneouslyacross the entire crystal, but commences at a free sur-face or region of high internal stress and propagatesover the atomic plane until it either passes out of thecrystal or encounters a barrier to further motion. At anyinstant during this propagation, a boundary separates

the region of the crystal that has experienced slip fromthat which has not. This boundary forms a linear crys-tal defect called a dislocation. As a dislocation passesover its slip plane, one part of the crystal moves rela-tive to the other by a lattice vector, the Burgers vector,which is generally, but not always, one atomic spacingin the direction of the highest density of atoms. Dis-locations are an important source of internal stress incrystalline materials and their motion is the principalmechanism of permanent deformation. Further infor-mation on the behavior of dislocations can be found instandard references [2.5, 6].

Homogeneous crystalline materials generally con-sist of an aggregate of grains, which are microscopiccrystalline regions having differing orientations. Thatis, the principal crystallographic directions in eachgrain have different orientations relative to an exter-nal coordinate system than corresponding directionsin neighboring grains. Grain boundaries, which aresurfaces across which grain orientations change dis-continuously, separate the grains. Since the grainboundaries contain most of the atomic disorder that ac-commodates the orientation change, they are surfacedefects. Grain boundaries permeate the material, form-ing a three-dimensional network that has a shape andtopology determined by its thermomechanical history.

Close-packed crystal structures, such as face-centered cubic and hexagonal close-packed, exhibitanother form of surface defect called a stacking fault.This occurs when local conditions of deformation orcrystal growth produce a stacking sequence that islocally different from that for the crystal structure.For example, suppose the stacking sequence of close-packed planes in the sequence ABC|ABC. . . is locallydisrupted across the surface indicated by the verticalline so that to the left of the line the crystal is shiftedto cause A planes to assume B stacking, B planes toassume C stacking and C planes to assume A stack-ing. Then the stacking sequence would be ABC|BCA. . .This is topologically equivalent to the removal ofa plane of A stacking, creating a three-layer hcp struc-ture while preserving the face-centered cubic structureon either side of the fault plane. Such a configurationis an intrinsic stacking fault and is associated with theformation and motion of certain kinds of dislocations inface-centered cubic crystals.

Materials can also consist of an aggregate of re-gions having differing chemical composition and crystalstructure from one another. Each region that is chem-ically and physically distinct and separated from the restof the system by a boundary is called a phase. Each

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phase can contain a network of grain boundaries. How-ever, the interphase boundaries that separate the phasesare surface defects that differ from grain boundaries,since the regions they separate differ in not only spa-tial orientation and chemical composition but also, inmost cases, in crystal structure. A multiphase materialhas a system of interphase boundaries whose extent andtopology depend not only on the thermomechanical his-tory of the material but also on the relative amounts ofthe phases present.

Cracks and voids caused by solidification are ex-amples of volume defects. These often arise duringprocessing and can be significant sources of mechanicalweakness and vulnerability to environmental attack ofthe material. Gases formed during fission of nuclear fuelcan also form internal voids during operation, causingswelling and distortion on a macroscopic scale.

2.1.4 Equilibrium and Kinetics

Thermodynamic PrinciplesThe overall composition and processing history de-termine the arrangements of groups of atoms in themicrostructure. The resulting structure of the materialdetermines its response to many conditions of mea-surement. Concepts of thermodynamic equilibrium, therate of approach to equilibrium, and the response ofthe material to its physical and mechanical environmentgovern relationships that connect composition, process-ing, and microstructure.

A thermodynamic system is any collection of itemsseparated from their surroundings by a real or imaginaryinterface, which can be permeable or impermeable tothe exchange of energy. Thermodynamics is the studyof the laws that govern

1. the exchange of energy between the system and itssurroundings

2. the energy content of such a system3. the capacity of the system to do work and4. the direction of heat flow in the system.

The sum of the kinetic and potential energies ofall components of the system is the internal energy U ,which measures the energy content of a system. Thefirst law of thermodynamics relates the rate of changeof internal energy of a system to the rate of heat ex-change with the surroundings and the rate of work doneby external forces

U = Q − P , (2.2)

where Q represents the rate of heat exchanged with thesurroundings, P is the power exerted by external forces,and the superposed dot indicates the derivative with re-spect to time. The second law of thermodynamics statesthat the rate of production of entropy S at an absolutetemperature T , defined as

S = Q

T, (2.3)

must be ≥ 0. For a real system, changes in S referto changes from a standard temperature and pressure.The internal energy less the product of the volume ofthe system V and the pressure on the system, P, is theheat content or enthalpy H . The relationship connectingthese thermodynamic state variables is

H = U + PV . (2.4)

The definition of free energy, which is the capacityof the system to do work, differs when dealing withgaseous and condensed systems of matter. The Gibbsfree energy G employed for condensed systems is re-lated to other thermodynamic variables by

G = H −TS . (2.5)

The following discussion designates free energy by Fwithout reference to the state of the system.

Changes in the free energy of a system occur whenvariables that define the state of the system do work onthe system. Gradients in the free energy of a system withrespect to these state variables are generalized thermo-dynamic forces characterized by adjectives that describethe nature of the variable – chemical, thermal, mechan-ical, electrical, etc. – giving rise to the force. Absoluteequilibrium exists when these forces sum vectorially tozero, i. e., the free energy of the system is an absoluteminimum. A system in equilibrium with one or moretype of force, while not being in equilibrium with oth-ers, is in a state of partial equilibrium. Equilibria can bestable, metastable or unstable. If small changes in thethermodynamic forces tend to alter the state of the sys-tem from its equilibrium condition, the equilibrium isunstable. If such changes tend to restore the system toits equilibrium state, the equilibrium is stable if the orig-inal state represents an absolute minimum in the freeenergy of the system or metastable if the minimum islocal. These concepts are illustrated in Fig. 2.7, whichdepicts various states of mechanical equilibrium.

Although materials are rarely in thermodynamicequilibrium with their surroundings during engineeringapplications, the extent of deviation from its equilibriumstate determines the propensity of a structure to change

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with time. Since changes in structure cause changes inproperties, the suitability of a material for a particularengineering application may also change.

Rate of Approach to EquilibriumIn order for a system that is in metastable equilib-rium in its current state (state 1) to attain the (stable ormetastable) equilibrium state with the next lowest freeenergy (state 2), passage through an unstable equilib-rium state (state 3) whose free energy is higher thanthat of states 1 or 2 is generally required. State 3 isan activated state and constitutes a barrier to the transi-tion of the system from state 1 to the lower free energyof state 2. The free energy difference between states 1and state 3 ΔF(1−3) must be supplied by internal andexternal sources in order to effect the transition fromthe metastable state to the activated state, from whichsubsequent transition to the next lower energy state isassumed to be spontaneous. The nature of the barrierdetermines the free energy difference.

The free energy ΔF(1−3) contains both thermal en-ergy and work done by external fields during activation.The difference between ΔF(1−3) and the work done byexternal sources is called the free energy of activationΔF∗. The Boltzmann factor P is the probability thatthe system will change its state from state 1 to state 3

P = exp(−ΔF∗/kT ) , (2.6)

when ΔF∗ is expressed per atom or mole of the sub-stance. In (2.6) k is Boltzmann’s constant, the universalgas constant per atom or molecule. The Arrhenius equa-tion expresses the rate of change of many chemicalreactions and solid-state processes:

r = ν0 exp(−ΔF∗/kT , (2.7)

in which the rate of change r is the product of the fre-quency of attempts at change ν0 and P is the probabilityof a successful attempt. Figure 2.8 illustrates the pro-cess schematically for the case of a thermally activatedprocess assisted by external stress.

Phase DiagramsThe structure of engineering materials at the microscalegenerally consists of regions that are physically andchemically distinct from one another, separated fromthe rest of the system by interfaces. These regions arecalled phases. The spatial distribution and, to a lim-ited extent, the chemical composition of phases canbe altered by thermomechanical processing prior to theend use of the material. Sometimes service conditions

Force

l

l

y

x

mm

(c)(b)(a)

m

n

ΔF*

ΔF

Fig. 2.7 (a) Metastable, (b) unstable, and (c) stable mechanical equi-librium states for a brick (after Barrett et al. [2.7])

can cause changes in the microstructure of a mater-ial, usually with accompanying changes in properties.Knowledge of the phases present in a material at varioustemperatures, pressures, and compositions is essen-tial to the prediction of its engineering properties andto assessing its propensity for change under serviceconditions. The phase diagram, a map of the thermody-namically stable phases present at various temperatures,pressures, and compositions, presents this informationin a graphical form.

The number of degrees of freedom f (≥ 0) ofa system refers to the number of environmental andcomposition variables that can be changed withoutchanging the number of phases in equilibrium. Thephase rule relates f to the number of distinct chem-ical species or components C and the number of phasespresent Φ:

f = C −Φ +2 , (2.8)

A*A(1-2)

A0

(3)

(2)

(1)

ΔF*ΔF (1-3)

ΔF (1-2)

Fig. 2.8 Free energy (F) versus slipped area (A0) for de-formation processes

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where 2 refers to temperature and pressure. The degreesof freedom in phase diagrams of condensed systemsat constant pressure (isobaric diagrams) is given bya form of equation (2.6) in which 2 is replaced by1, since the pressure is fixed. Either the weight per-cent or atomic percent of the components specifiesthe composition of a multicomponent system. In manyengineering applications, weight percent is preferredas a more practical guide to the relative amounts ofcomponents. In scientific studies of the nature of dia-grams formed from elements having similar chemicalcharacteristics, atomic fraction is preferred. For an n-component system, the weight percent of component ican be calculated from its atomic percent by the relation

(w/o)i = (a/o)i (at.wt.)in∑

j=1(a/o) j (at.wt.) j

× 100% , (2.9)

where (a/o) and (w/o) are the atomic percent and weightpercent, respectively, and at.wt. refers to the atomicweight of the subscripted substance. The summationextends over all components. A similar expression re-lates the atomic percent of a component to the weightpercents and atomic weights of the components.

It is impossible to represent a phase diagram in threedimensions for a material consisting of more than threecomponents. More commonly, diagrams that show thestable phases for a two-component (binary) system atone atmosphere pressure and various temperatures areused as guides to predict the stable phases. This per-mits a two-dimensional map with temperature as theordinate and composition as the abscissa to representthe possible structures. ASM International offers anextensive compilation of binary phase diagrams of met-als [2.8]. A similar compilation is available for ceramicsystems [2.9]. Three-component (ternary) phase dia-grams are more difficult to represent since they requiretwo independent compositional variables in additionto temperature. The most common three-dimensionalrepresentation is a prism in which the component com-positions are plotted on an equilateral triangle base andthe temperature on an axis normal to the base. Ternarydiagrams of many metallic systems have been deter-mined are available as a compilation similar to that forthe binary systems [2.10].

Phase diagrams not only give information on thenumber and composition of phases present in equilib-rium, but also provide the data necessary to calculate therelative amounts of phases present at any temperature.Figure 2.9 illustrates this principle, the inverse lever law,using the silver–copper binary diagram as an example.

From this diagram we note that an alloy of 28.1 w/oCu solidifies at 779.1 ◦C into a solid consisting of twosolid phases: a silver-rich phase α containing 8.8 w/oCu and a copper-rich phase β containing 92.0 w/o Cu.This alloy, which passes directly from the liquid state tothe solid state without going through a region of solidand liquid in equilibrium, is a eutectic. To determinehow much of each phase is present in the eutectic alloyat the solidification temperature, we note that, while theoverall composition of the alloy c0 is 28.1 w/o Cu, it isa mixture of copper- and silver-rich phases, each havinga known composition. Performing a mass balance on theamount of copper in the alloy yields the result:

(%)α =(cβ − c0

)

(cβ − cα

) × 100%

= (92−28.1)

(92−8.8)× 100% = 76.8% (2.10)

for the weight percentage of the silver-rich phase in thealloy. The inverse lever law gets its name from the factthat the numerator in (2.10) is the distance on the com-position axis from the overall alloy composition to thecomposition of the copper-rich phase cβ on the oppo-site side of the diagram from the silver-rich phase whosepercentage is to be determined. The denominator is theentire distance between the compositions of the copper-rich and silver-rich phases. The calculation is analogousto balancing the relative amounts of the phases ona lever whose fulcrum is at the overall alloy composi-tion. While the numerical result shown illustrates thecalculation for the eutectic alloy and temperature, theprinciple applies to any constant temperature as longas we choose the appropriate overall alloy composition

Ag 10 20 30 40 50 60 70 80 90 Cu

961.93°C

1084.87°C

779.1°C

28.1% 92%8.8%

(Cu)

(Ag)

Liquid

Temperature (°C)

wt % copper

1200

1100

1000

900

800

700

600

500

400

300

200

Fig. 2.9 Silver–copper phase diagram at 1 atm pressure

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and the compositions of the phases in equilibrium at thattemperature.

2.1.5 Observation and Characterizationof Structure

In order to implement in the selection and design ofmaterials and processes the axiom that structure atvarious levels determines the properties of materials, itis necessary to have means of examining, measuring,and describing quantitatively the structure at variousscales. The information so obtained then leads to thedevelopment, validation, and verification of models thatdescribe the behavior of engineering structures. The fol-lowing sections describe two types of characterizationtechniques that are useful for this purpose.

Metallographic techniquesMetallographic approaches to studying the structureof materials are either surface techniques or transmis-sion techniques. Surface techniques rely on imagesformed by the reflection of a source of illumination bya prepared surface. The wavelength of the illuminatingradiation and the optical properties of the observationsystem determine the resolution of the technique. Opti-cal metallography employs a reflecting microscope andillumination by visible light to examine the polishedand etched surface of a specimen either cut from bulkmaterial or prepared on the surface of a sample, tak-ing care not to introduce damage during the preparation.Consequently, the examination is generally a destructivetechnique. Magnifications up to approximately 1200 ×are possible with careful specimen preparation. Prop-erly chosen etches reveal the arrangement of grainboundaries, phases and defects intersected by the planeof polish. Contrast arises from the differing reflectivitiesof the constituents of the microstructure. Interpreta-tion of micrographs to relate the observed structureto the properties and behavior of the material requiresexperience and knowledge of the composition and ther-momechanical history of the material.

Scanning electron microscopy (SEM) employsa beam of electrons as an illuminating source [2.11].The de Broglie wavelength of an electron is λ = h/p,where h is Planck’s constant and p is the momentumof the electron. The de Broglie wavelength of elec-trons varies with the accelerating voltage V since foran electron eV = p2/2m, where e and m are the chargeand mass, respectively, of the electron. Electromag-netic lenses use the charge on the electron beam tocondense, focus, and magnify it. Magnifications ob-

tained by this technique range from those characteristicof optical microscopy to nearly 100 times the bestresolution obtained from optical techniques. Contrastamong microconstituents arises because of their dif-fering scattering powers for electrons. Since electronoptics permits a much greater depth of field than is pos-sible with visible light, SEM is the observation tool ofchoice for examining surfaces with a high degree ofspatial irregularity, such as those produced by fracture.While the images are nearly always two-dimensionalsections through three-dimensional structures, stereomicroscopy is widely used in the examination of frac-ture surfaces. Although this method can employ anappreciable range of wavelengths, it is necessary tooperate in a vacuum and to provide a means of dissi-pating any electrical charge on the specimen inducedby the electron beam. In recent years, the developmentof special environmental chambers has permitted theobservation of nonconductive materials at atmosphericpressure.

Transmission metallographic techniques produceimages that are the projection of the content of the ir-radiated volume on the plane of observation. Clearly,such techniques rely on the transparency of the materialto the illumination. This requires the preparation of thinsections of the material, followed by treatments to im-prove the contrast of structural features, then observedby transmission microscopy. Visible light generally suf-fices as the illuminating source for biological materials.Since most engineering materials are not transparent tovisible light, observation by transmitted radiation is lesscommon for these materials. The following section dis-cusses both of the principal exceptions, transmissionelectron microscopy (TEM) and x-ray radiography.

Information obtained from metallographic observa-tions serves a variety of purposes: to determine themicrostructural state of a material at various stagesof processing, to reveal the condition of material ina failed engineering part, and to obtain a semiquan-titative estimate of some types of material behavior.All cases where the images represent plane sectionsrequire analysis by stereological techniques to obtainquantitative information that for relating structure toproperties [2.12]. For a microstructure containing twoor more microconstituents, such as phases, inclusionsor internal cavities, the simplest parametric charac-terization is the volume fraction VV (n) of the n-thmicroconstituent. It is easily shown that this is alsoequal to the area fraction AA(n) of the n-th constituenton an observation plane, which can easily be measuredon a polished metallographic specimen [2.12].

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Another common measurement of microstructure isthe mean linear intercept Λ(t) or its reciprocal PL(t),the intercept density, of a test line parallel to the unitvector t with the traces of internal boundary surfaces onan observation plane. The intercept density is the num-ber of intersections per unit length of the test line withthe feature of interest. Since microstructures are gen-erally anisotropic PL(t) depends on the orientation ofthe test line. The volume average of all measurementsis an average intercept density, 〈PL〉; its reciprocal isthe mean linear intercept 〈Λ〉. The grain size refers tothe mean linear intercept of test lines with grain bound-aries. When employing this terminology, remember thatthe measurement is actually the mean distance betweengrain boundaries in the plane of measurement and maynot apply to the three-dimensional structure. In addi-tion, the concept of size implies an assumption of shape,which is not included in the measurement. Nevertheless〈Λ〉 remains the most widely used quantitative charac-terization of microstructure in the materials literature.Similar measurements can be made of the intersectionsof test lines with interphase boundaries and internal cav-ities, or pores. The associated mean linear intercepts arephase diameters or pore diameters, respectively. The in-tercept density for a particular class of boundaries alsomeasures the total boundary area per unit volume 〈SV〉through the fundamental equation of stereology:

〈SV〉 = 2 〈PL〉 . (2.11)

This quantity is important when studying propertiesinfluenced by phenomena that occur at internal bound-aries.

Diffraction TechniquesIn an irradiated array of atoms, each atom acts asa scattering center, both absorbing and re-radiating theincident radiation as spherical wavelets having the samewavelength as the incident wave. When the wavelengthof the incident radiation is comparable to the spacingbetween atoms, interference effects can occur amongthe scattered wavelets. A crystalline material acts asa regular array of scattering centers, and producesa pattern of scattered radiation that is characteristic ofthe array and the wavelength of the radiation. Amor-phous materials also produce scattering, but the lack oflong-range atomic order precludes the development ofpatterns of diffracted beams.

Diffracted rays occur by the constructive interfer-ence of radiation reflected from atomic planes. Themodel proposed by W. H. Bragg and W. L. Bragg, il-lustrated in Fig. 2.10, describes the phenomenon.

Radiation of wavelength λ is incident at an angle θ

on parallel atomic planes separated by a distance d.Scattered radiation from adjacent planes interferes con-structively, producing a scattered beam of radiation,when the path difference between rays scattered fromadjacent planes is an integral number n of wavelengths.The relationship describing this phenomenon is Bragg’slaw,

nλ = 2d · sin θ , (2.12)

which is the basis for all measurements of the atomicdimensions of crystals. X-rays, a highly energetic elec-tromagnetic radiation having a wavelength the sameorder of magnitude as the spacing between atoms incrystals, inspired the original derivation of (2.12) [2.13,14]. If the crystal is sufficiently thin to permit diffractedbeams to penetrate it, transmission diffraction patternsoccur on the side of the crystal opposite to the incidentradiation. However, even if the diffracted beams cannotpenetrate the crystal, back-reflection diffraction patternsoccur on the same side as the incident beam. Crys-tal structure analysis employs both types of patterns.One method of identifying unknown materials employsanalysis of diffraction patterns formed by radiation ofknown wavelength to determine the lattice parameterand crystal structure of an unknown substance, whichis then compared with a database of known substancesfor identification [2.15, 16].

Both incident and diffracted X-radiation canpenetrate substantial thicknesses of many materials,rendering this diffraction technique useful for deter-mining the condition of material in the interior of anengineering part. Lattice distortion giving rise to in-ternal stress, averaged over the irradiated volume, canbe determined by comparing the interplanar spacingsof a material containing internal stress with those ofa material free of internal stress [2.17]. This techniqueof x-ray stress analysis is extremely valuable in deter-

d

θθ

θ

Fig. 2.10 Derivation of Bragg’s law

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Materials Science for the Experimental Mechanist 2.2 Properties of Materials 33

mining residual stresses in heat-treated steel parts, forexample. It has the advantage that lattice strain, whichis the source of internal stress, is the direct result ofthe measurement. This is in contrast to the calculationof stress from measurements of strain on the exter-nal surface of a material, which include componentsdue to both lattice strain and the stress-free, perma-nent strain caused by dislocation motion. When stressis calculated from such surface measurements, the as-sumption that dislocation motion is absent or negligibleis implicit [2.18].

Diffraction contrast from scattered x-rays forms thebasis for x-ray topography of crystals, useful in thestudy of the defect structure of highly perfect singlecrystals used in electronic applications [2.19]. Magni-fied diffraction spots from crystals reveal images of thestructure of the material with a resolution comparableto the wavelength of the radiation. Small changes inthe orientation and spacing of atomic planes due to thepresence of internal defects change the diffraction con-ditions locally, creating contrast in the image of thediffracted beam. The availability of high-intensity x-ray sources from synchrotrons has greatly increased theapplicability and usefulness of this technique.

Radiography also forms images by the interactionof material with x-rays, but in this case, the contrastmechanism is by the differential absorption of radiationcaused by varying thicknesses of material throughout anirradiated section as well as by regions having differingdensities, which affects the ability of material to ab-sorb x-rays [2.20]. Radiography can reveal macroscopicdefects and heterogeneous distributions of phases inmaterials. Since contrast is not due to diffraction bythe structure, this process cannot provide informationon its crystal structure or the state of lattice strain in thematerial.

The principles of diffraction apply not only toelectromagnetic radiation but also to the scatteringof highly energetic subatomic particles, such as elec-

trons and neutrons, which have de Broglie wavelengthscomparable to the interatomic spacing of the crys-tals. Electron beams are employed in both transmissionand back-reflection modes to reveal information onthe crystal structure and defect content of materials.The fact that electron beams possess electrical chargemakes it possible to employ electromagnetic lenses tocondense, focus, and magnify them. Transmission elec-tron microscopy (TEM) uses diffracted electron beamstransmitted through material and subsequently magni-fied to reveal features of atomic dimensions [2.21].Since electron beams are highly absorbed by most ma-terials, specimens examined in TEM are only a fewthousand atoms thick. Nevertheless, this technique re-veals much useful information about the nature andbehavior of dislocations, grain boundaries and otheratomic-scale defects. Orientation imaging microscopy(OIM) [2.22] employs back-reflection electron diffrac-tion patterns to form maps called pole figures bydetermining the orientation of individual grains. Thesemaps describe the orientation distribution of grains ina material, which is a major cause of the anisotropy ofmany bulk properties. The greater penetrating power ofneutron beams reveals information from greater thick-nesses of material than x-rays [2.23]. Since these beamsare not electrically charged, their magnification by elec-tromagnetic lenses is not possible. Therefore, theirprincipal use is for measurements of changes in latticespacings due to internal stresses, and not for formingimages.

In addition to revealing the structure of materials,many advanced techniques reveal not only the structurebut also the chemistry of a substance at the microscale.While techniques such as the three-dimensional (3-D)atom probe, Auger electron spectroscopy and imaging,Z-contrast microscopy, and many others complementthose for structural examination and often employ simi-lar physical principles; a complete survey is beyond thescope of this review.

2.2 Properties of Materials

Material properties are attributes that relate appliedfields to response fields induced in the material. Appliedand induced fields can be of any tensorial rank [2.24,25]. Fields are conjugate to one another if the appliedfield does an incremental amount of work on the sys-tem by causing an incremental change in the inducedfield. External fields and the associated field variablesinclude temperature T , stress τ, and electrical (E) and

magnetic (H) fields; their respective conjugate fields areentropy S, strain ε, electrical displacement D, and mag-netic induction B. Cross-effects occur when changes inan applied field cause changes in induced fields that arenot conjugate to the applied field.

Consider a system subjected to a cyclic change ofapplied field variables such that each variable changesarbitrarily while the others are constant, then the pro-

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cess reverses until the variables have their initial values.If such a closed cycle occurs under conditions such thatthe system is continuously in equilibrium with its sur-roundings, the thermodynamic states at the beginningand the end of the process will be the same and no network is done. Properties that relate conjugate fields inthis case are equilibrium properties and depend onlyon the thermodynamic state of a system, not its his-tory. A caloric equation of state relates conjugate fields,which permits the description of the relationship be-tween applied and induced fields in terms of appropriatederivatives of a thermodynamic potential.

If processes that occur during the cycle dissipate en-ergy, they produce entropy and the system is not in thesame state in the initial and final conditions. Althoughthe applied field variables are the same at the beginningand end of the cycle, the conjugate induced variables donot return to their original values. The change of ther-modynamic state occurs because of irreversible changesto the structure of the material caused, at least in part, bychanges in the internal structure of the system that occurduring the cyclic change. Associated material proper-ties are dissipative and exhibit hysteresis. Dissipativeproperties are dependent on time and the history of thesystem as well as its current thermodynamic state. How-ever, in some cases it is possible to relate these proper-ties to appropriate derivatives of a complex dissipativepotential [2.24]. In general, equilibrium properties arestructure insensitive, while dissipative properties arestructure sensitive. The search for additional state vari-ables that depend on the internal structure of thematerial is a subject of much continuing research.

Transport processes are dynamic processes thatcause the movement of matter or energy from onepart of the system to another. These produce a flux ofmatter or energy occurring in response to a conjugatethermodynamic force, defined as the gradient of a ther-modynamic field [2.26]. Attributes that connect fluxesand forces are generally structure-insensitive materialproperties.

2.2.1 The Continuum Approximation

ContinuityThe complex nature of real materials requires approx-imation of their structure by a mathematical conceptthat forms a basis for calculations of the behaviorof engineering components. This concept, the contin-uous medium, replaces discrete arrays of atoms andmolecules with a continuous distribution of matter.Fields and properties defined at every mathematical

point in the medium are continuous, with continuousderivatives, except at a finite number of surfaces sep-arating regions of continuity [2.24].

The process that defines properties at a point inthe medium consists of taking the limit of the volumeaverage of the property over increasingly smaller vol-umes. For sufficiently large volumes the average willbe independent of sample size. However, as the volumesampled decreases below a critical range, the volumeaverage will exhibit a dependence on sample size, whichis a function of the nature of the property being meas-ured and the material. The smallest sample volume thatexhibits no size dependence defines the continuum limitfor the material property. The continuum approxima-tion assigns the value of the size-independent volumeaverage of the property to a point at the centroid ofthe volume defined by the continuum limit. It followsthat, when the size of a sample being tested approachesthe continuum limits of relevant properties, results canbe quite different from those predicted by a continuummodel of the material. This limit depends on the mater-ial, its structural state, and the property being measured,so that all need to be specified when assigning a valueto the continuum limit.

The continuum approximation employs the conceptof the material particle to link the discrete nature of realmaterials to the continuum. By the argument above, thematerial attributes of such a particle must be associatedwith a finite volume, the local value of the contin-uum limit for the attributes. As noted earlier, continuumlimits may differ for different properties; consequentlythe effective size associated with a material particlewill depend on the property associated with the contin-uum. Coordinate systems based on material particles,in contrast to those based on the crystal lattice, mustnecessarily be continuous by the assumption of globalcontinuity of the medium. Despite the inherent differ-ences in concepts of material structure, it is possible todevelop models of many material properties based onatomic or molecular characteristics that interface wellwith the continuum model at the scale of the continuumlimit. These limits can only be determined experimen-tally or by the comparison of properties calculatedfrom physically based models of atomic arrays of var-ious sizes. Such models form essential links betweencontinuum mechanics and the knowledge of structureprovided by materials science.

HomogeneityThe continuous medium used for calculation of thebehavior of engineering structures also possesses the

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property of homogeneity, that is, properties at a ma-terial point do not depend on the location of the pointin the medium. While this assumption is reasonablyaccurate for single-phase materials of uniform com-position, it no longer applies to multiphase materialsat the microscale. In the latter case, properties of thebulk depend on appropriate averages of the propertiesof the constituent phases, which include parameterssuch as the volume fraction of each phase, the aver-age size of the particles of each phase, and in somecases the orientation of the phases relative to one an-other and to externally applied fields. Higher ordermodels of multiphase materials treat each phase asa distinct continuum with properties distinct from otherphases.

Physical fields within the body are also influencedby the nature, orientation, and distribution of the phasespresent. Average values of induced internal fields de-termined from the external dimensions of a body andfields applied to the surfaces may not accurately re-flect the actual conditions at points in the interior ofthe body due to different responses from constituentphases within the material. Material properties deter-mined from the bulk response to these mean fields maydiffer significantly from similar properties measuredon homogeneous specimens of the constituent phases.Measurements of local fields rely on techniques thatsample small, finite volumes to determine local ma-terial response, then inferring the values of the fieldsproducing the response.

Since a quantitative determination of variables thatdetermine the bulk properties of multiphase materialsis often difficult, time-consuming, and expensive, ex-perimental measurements on bulk specimens typicallydetermine these properties. It is important to realize thatapplying the results of such measurements implicitlyassumes that differences in structure of the material atthe microscale have a negligible effect on the resultingproperties. Structural variability due to inhomogeneitycan be taken into account experimentally by determin-ing properties of the same material with a range ofstructures and assigning a quantitative value to the un-certainty in the results.

IsotropyThe usual model of a homogeneous continuous mediumalso assumes that properties of the material areisotropic, i. e., they are independent of the directionof measurement. This assumption generally resultsin the minimum number of independent parametersthat describe the property and simplifies the math-

ematics involved in analysis. However, isotropy isinfrequently observed in real materials, and neglect ofthe anisotropic nature of material properties can lead toserious errors in estimates of the behavior of engineer-ing structures.

Properties of materials possess symmetry elementsrelated to the structure of the material. The follow-ing Gedanken experiment illustrates the meaning ofsymmetry of a property [2.25]. Measure the propertyrelative to some fixed set of coordinate axes. Then oper-ate on the material with a symmetry operation. If thevalue of the property is unchanged, then it possessesthe symmetry of the operation. Neumann’s principlestates that the symmetry elements of any physical prop-erty of a crystal must include the symmetry elements ofthe point group of the crystal. Notice that the principledoes not require that the crystal and the property havethe same symmetry elements, just that the symmetryof the property contains the symmetry elements of thecrystal. Since an isotropic property has the same valueregardless of the direction of measurement, it containsall possible point groups.

As mentioned above, bulk properties of poly-crystalline and multiphase materials are appropriateaverages of the properties of their microconstituents,taking account of the distribution of composition, size,and orientation. A common source of bulk anisotropyin a material is the microscopic variation in orienta-tion of its grains or microconstituents. An orientationdistribution function (ODF) describes the orientationdistribution of microscopic features relative to an exter-nal coordinate system. Depending on the feature beingmeasured, quantitative stereology or x-ray diffractionanalyses provide the information required to determinevarious types of ODFs. Used in conjunction with theproperties of individual crystals of constituent phases,ODFs can describe the anisotropy of many bulk proper-ties, such as the elastic anisotropy of metal sheet formedby rolling.

2.2.2 Equilibrium Properties

Although the definition of equilibrium properties is phe-nomenological, without reference to the structure of thematerial, values of the properties depend on the struc-ture of materials though models described in treatises oncondensed-matter physics and materials science [2.27].Properties described in the following sections are allfirst order in the sense that they arise from an as-sumed linear relationship between applied and inducedfields. However, most properties have measurable de-

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pendencies on other applied fields. Such field-inducedchanges cause second-order effects, such as opticalbirefringence and the electro-optical effect, where first-order properties depend on strain and electrical field,respectively. For compactness, this review will notconsider these effects, although many of them can pro-vide the basis for useful measurements in experimentalmechanics.

The usual description of equilibrium properties em-ploys conjugate effects. In principle, any change in anapplied field produces changes in all induced fields,although some of these effects may be quite small. Fig-ure 2.11, adapted from Nye [2.25] and Cady [2.28],illustrates the interrelationship among first-order effectscaused by applied and induced fields. Applied field vari-ables appear at the vertices of the outer tetrahedron,while the corresponding vertices of the inner tetrahe-dron represent the conjugate induced field variables.Lines connecting the conjugate variables represent prin-cipal, or direct, effects.

Specific HeatFor a reversible change at constant volume, an incre-mental change in temperature dT produces a change ofentropy dS:

dS =(

Cε

T

)

dT , (2.13)

where Cε is the heat capacity at constant strain (vol-ume) and T is the absolute temperature. To maintainconstant volume during a change in temperature, it isnecessary to vary the external tractions on the surfaceof the body (or the pressure, in the case of a gas). Whenmeasurements are made at constant stress (or pressure),the volume varies. The heat capacity under these con-ditions is Cτ . Specific heat is an extensive, isotropicproperty.

E

H

T

D S

ε

τ

B

Fig. 2.11 Relationships among applied and induced fields

Electrical PermittivityA change in electrical field dE produces a change inelectrical displacement dD:

dDi = κij dE j , (2.14)

where the suffixes, which range from 1 to 3, representthe components of the vectors D and E, represented bybold symbols, along reference coordinate axes and sum-mation of repeated suffixes over the range is impliedhere and subsequently. The second-rank tensor κ is theelectrical permittivity, which is in general anisotropic.

Magnetic PermeabilityA change in magnetic field dH produces a correspond-ing change in magnetic induction or flux density dB:

dBi = μij dHj , (2.15)

where μ is the second-rank magnetic permeability ten-sor, which is in general anisotropic. Equations (2.14)and (2.15) apply only to the components of D and B thatare due to the applied electrical or magnetic field, re-spectively. Some substances possess residual electricalor magnetic moments at zero fields. The integral formsof (2.14) and (2.15) must account for these effects.

Hookean ElasticityAn incremental change in stress dτ produces a corre-sponding change in strain dε according to:

dεij = sijk� dτk� , (2.16)

where s is the elastic compliance tensor. In this case, theapplied and induced fields are second-rank tensors, sothe compliances form a fourth-rank tensor that exhibitsthe symmetry of the material. The elastic constant, orelastic stiffness, tensor c, is the inverse of s in the sensethat cijmnsmnkλ = 1/2(δikδ jλ + δiλδ jk). Equation (2.16)is Cauchy’s generalization of Hooke’s law for a generalstate of stress. A material that obeys (2.16) is an idealHookean elastic material.

Most engineering calculations involving elastic be-havior assume not only Hooke’s law but also isotropy ofthe elastic constants. This reduces the number of inde-pendent elastic constants to two, i. e. cijmn = λδijδmn +μ(δimδ jn + δinδ jm), where λ and μ are the Lamé con-stants. Various combinations of these constants are alsoemployed, such as Young’s modulus, the shear modu-lus, the bulk modulus and Poisson’s ratio, but only twoare independent in an isotropic material. The Chapter oncontinuum mechanics in this work lists additional rela-tionships among the isotropic elastic constants [2.29].

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While isotropy is a convenient approximation formathematical calculations, all real materials exhibitelastic anisotropy. Single crystals of crystalline mater-ials display symmetry of the elastic coefficients thatreflects the symmetry of the crystal lattice at the mi-croscale [2.25]. The single-crystal elastic constants areoften expressed in a contracted notation in which thetwo suffixes of stress and strain components becomea single suffix ranging from 1 to 6. The independentterms of the corresponding elastic constant tensor be-come a 6 × 6 array, which is not a tensor. This notationis convenient for performing engineering calculations,but when the elastic constants are not isotropic, changesin the components of the elastic constant matrix dueto changes in the coordinate system of the problemmust be calculated from the full, fourth-rank tensor de-scription of the constants [2.25]. Anisotropic crystallinematerials can display a lower elastic symmetry at themesoscale than at the single-crystal scale due to pre-ferred orientation of the crystallites introduced duringprocessing.

Cross-EffectsAll conjugate fields can be included in the descriptionof the thermodynamic state. Define a function Φ suchthat

Φ = U − τijεij − E� D� − Bk Hk −TS . (2.17)

Then using the definition of work done on the systemby reversible changes in the conjugate variables and thefirst and second laws of thermodynamics, we have

dΦ = −εij dτij − D� dE� − Bk dHk − S dT . (2.18)

Since Φ = Φ(τ, E, H, T),

dΦ =(

∂Φ

∂τij

)

E,H,T

dτij +(

∂Φ

∂E�

)

τ,H,TdE�

+(

∂Φ

∂Hk

)

E,τ,TdHk +

(∂Φ

∂T

)

E,H,τ

dT

(2.19)

it follows that(

∂Φ

∂τij

)

E,H,T

= −εij ,

(∂Φ

∂E�

)

τ,H,T= −D� ,

(∂Φ

∂Hk

)

E,τ,T= −Bk ,

(∂Φ

∂T

)

E,H,τ

= −S .

(2.20)

Further differentiation of (2.19) and (2.20) with respectto the independent variables gives relationships among

the coefficients of the form

−(

∂2Φ

∂τij∂T

)

E,H=

(∂εij

∂T

)

σ,E,H

=(

∂S

∂τij

)

T,E,H= β

E,Hij (2.21)

for the thermal expansion coefficient tensor β at con-stant electrical and magnetic field.

Thermoelasticity illustrates cross-effects that oc-cur among equilibrium properties. Consider the sectionof the tetrahedron containing only the applied fieldvariables stress and temperature and the associated in-duced field variables, strain and entropy, illustratedin Fig. 2.12.

In this illustration, the line connecting stress andstrain represents the elastic relationships and the lineconnecting entropy and temperature denotes the heatcapacity. The indirect, or cross-effects are the heat ofdeformation, the piezocaloric effect, thermal expansion,and thermal pressure, as shown in the figure.

Ten independent variables specify the thermody-namic substate of the material. These can be the ninestress components and temperature, nine strain com-ponents and temperature, or either of the precedingwith entropy replacing temperature. Specification often independent variables determines the ten dependentvariables. Nye [2.25] shows that the relationships con-necting the effects become:

εij = sTijk�τk� +βijΔT , (2.22)

ΔS = βijτij + (Cτ /T )ΔT , (2.23)

where sT is the isothermal compliance tensor. Extensionof this argument shows that the full matrix of mater-

a)

b)

StrainHeat of deformation

Thermal pressure

Piezocaloric effect

Elasticity Heat capacity

Thermal expansion

Entropy

Tempe-rature

Stress

ε S

Tσ

c T/CC/T

α α

f

–f

s

Fig. 2.12a,b Equilibrium properties relevant to thermoelastic be-havior: (a) quantity and (b) symbol

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dicat materiali

Line


ial coefficients defined by relationships similar to (2.21)and written in the form of (2.22) is symmetric.

The atomic basis for the linear relationships in(2.22) arises from the nature of the forces betweenatoms. Although the internal energy of a material isin reality a complex, many-body problem that is be-yond the capabilities of even modern computers tosolve, summation over all atoms of the bond energiesof nearest-neighbor pairs offers a physically reasonableand computationally manageable alternative. The en-ergy of an atom pair exhibits a long-range attraction anda short-range repulsion, resulting in a minimum energyfor the pair at a distance equal to or somewhat greaterthan the interatomic spacing in the material. In the sim-plest case, a diatomic molecule of oppositely chargedions, the Coulomb force between unlike charges pro-vides the attractive force and the mutual repulsion thatarises when outer shells of electrons begin to overlapis the repulsive force. The minimum energy of the pairis called the dissociation energy, and is the energy re-quired to separate the two ions to an infinite spacingat absolute zero. For the example cited the energy isradially symmetric, so that the energy–distance rela-tionship is independent of the relative orientation of theions, and the energy minimum occurs at the equilibriuminterionic spacing. Figure 2.13 illustrates this behaviorschematically.

It follows from (2.16)–(2.20) that

sijk� = − ∂2Φ

∂τij∂τk�, (2.24)

which can be expressed in terms of the strain relativeto the equilibrium spacing by (2.16). Thus the elasticcompliances are proportional to the curvature of theenergy-distance curve of ionic pairs, measured at theequilibrium interionic distance. Extension of this modelto condensed systems of atoms having more complexbonds leads to modification of the details of the re-lationships. The elastic constants of single crystals ofsingle-phase cubic alloys can be estimated by an exten-sion of this concept [2.30].

The energy–bond length relationship shown inFig. 2.13 applies only at absolute zero. At finite tem-peratures, the energy of the pair is increased above theminimum by an amount proportional to kT . This im-plies that for any finite temperature T there exist twovalues of the interionic spacing with the same energy.The observed spacing at T is the average of these two.Most materials exhibit a curve that is asymmetric aboutr0, the equilibrium spacing at T = 0 causing the meanspacing to change with temperature. Generally the na-

0 0.1 0.2 r0 0.4

Attractive energy

Repulsive energy

Minimum energy

0.5 0.6 0.7 0.8

Lattice energy (eV)

Interionic spacing (nm)

15

12

9

6

3

0

–3

–6

–9

–12

–15

Fig. 2.13 Energy–bond length relationship for an ionicsolid

ture of this asymmetry is such that the mean spacingincreases with temperature, as illustrated in Fig. 2.13,leading to a positive coefficient of thermal expansion.

Different values of the elastic compliances re-sult from measurements under isothermal (ΔT = 0) oradiabatic (ΔS = 0) conditions. The adiabatic compli-ance tensor sS is determined by setting ΔS = 0 andsolving for ΔT in the second of (2.22), eliminatingΔT from the first equation and defining sS as the re-sulting coefficients of σ. The relationships among theisothermal and adiabatic quantities are

sSijk� − sT

ijk� = −βijβk�

(Cτ

T

)

. (2.25)

Since the coefficients of the thermal expansion tensorare positive for most materials, adiabatic compliancesare typically smaller than isothermal compliances.Nye [2.25] gives expressions such as (2.25) relating dif-ferences between various properties with different setsof applied field variables being constant. This reason-ing emphasizes the fact that the definitions of theseproperties include specification of the measurementconditions.

2.2.3 Dissipative Properties

Equilibrium processes are reversible and conservative,whereas nonequilibrium processes are irreversible andnonconservative. In the latter cases, work done on thesystem dissipates in the form of heat, generating en-tropy. This introduces history and rate dependence inthe response of the system to external fields and influ-

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1 introduction to material science