Construction Materials, 4th Ed_Section1

7/29/2019 Construction Materials, 4th Ed_Section1

1/55

ConstructionMaterialsTheir nature andbehaviourFourth edition

Edited byPeter Domone andJohn I Iston

9 ~ r ~ l ~ ~ f rancisLONDON AND NEW YO RK


2/55

h r st published as Concrete Timber and Metals 1979y C ha pm a n a nd Ha llnd editi o n publ ished 1994

by C ha pm an and Ha lled itio n published 200 Iby Spo n Press

h is edition published 2010by Spon PressPa rk Squ a re, Milton Pa rk, Abingdon, Oxon OX 14 4R N

imulta neo usly published in the USA a nd Canadaby Spon Press70 M adi so n Av enu e, Ne w Yo rk , NY 100 16, USAPress is an imprint of the Tay lor 6 Francis Croup , an in forma business

20 10 Spon Pressypese r in Sa bon byra phi craft Li mited , Hong KongPrinted an d bo und in C rea r Brita in by

MPC Boo ks C ro up, UKA ll ri gh ts rese rved. No pa rr of thi s book ma y be reprinted o r reproduced o rutili sed in a ny form or by any elec tro ni c, mecha ni ca l, o r o ther means, nowkn ow n or herea fter in ve nted , in cludi ng photocopyin g an d recording, o r inn y in fo rm a tio n sto rage or ret ri eval sys tem, w ithou t per miss io n in w riti ngfrom the pu b lishers .

his publ ica tion prese nt s materi a l of a br oa d sco pe a nd ap plicahi lit )'.Des pi te stringe nt efforrs by al l con ce rn ed in the pub li shing process , so metypo grap hica l o r editori a l e rro rs may occ ur, a nd rea ders ar e enco ur aged tobr ing th ese ro o ur a ttenti on whe re they reprcse nr err o rs of sub sta nce . Th epu b lisher a nd a uthor discla im a ny lia b ility, in who le o r in pa rr , a ri s ing fro minform at ion co nta ined in this pub lica tion. T he reade r is urged to co nsultwi t h an a ppro priate li censed profess ional pri or to raking any actio n ormak ing a ny in te rpr etat io n th a t is within the rea lm of a licensedpr o fess io nal prac ti ce.Bri tish Library Cata lrJgu ing in Publica tion DataA ca ta logue reco rd fo r thi s boo k is a va ilab le fro m the Br iti sh Libra ryLibrary o f Congress Cata loging-in-P11hlication DataCo nstructio n ma teri a ls : the ir na ture and behav io ur / ledircd hylPeter Domone a nd J. M . !I Iston. - 4th ed.p. em.Includes bibli og raphi ca l refe rences.I . Building ma terial s. I. Domone, P. L. J. II. lllston , J. M.

A403.C636 201062 4 .1 '8-dc22 2009042708ISBN 10: 0-41 5 -4 65 15-X (hbk )ISBN I O: 0-4 15- 4 65 16-8 (pbk )ISBN 10: 0-20 3- 92757-5 (e bk )ISBNl3: 978-0 -41 5-465 15- 1 (hbk )ISBN I3 : 978 -0 -41 5-465 16-8 (pbk )ISBN 13: 978 -0- 20 3-92757- 1 (e bk ) NAZA RBAYE VUNIVERSITY

LIBRARY


3/55

IntroductionWe conventi ona ll y think of a material as being eithera so lid or a fluid. These states of matter are conve nie ntl y based on the response of th e mater ial toan a pplied fo rc e. A so lid will maintain its shapeund er its own weight, and res ist appli ed forces withlittle deform at ion.' An unconfin ed fluid will flow underits own weight or applied force. Fluids can be dividedinto liquids and gases; liquids a re essentia lly incompressibl e and mainta in a fi xed vo lume when pl acedin a co nta iner, wh ereas gases are grea tly compress ibl eand w ill also expa nd to fill the vo lume ava ilabl e.Although th ese di visions of mater ials are often conve nient, we must recogni se th at th ey are not di stinct,and some materials di sp lay mi xe d behav iour, suchas ge ls, which can va ry from near solids to nearliquid s.

In co nstru ct ion we are for th e most par t conce rn ed with so lid s, since we use these to ca rry rh eapplied or self-we ight loads, but we do need tound erstand so me aspects of fluid behaviour, forexa mple when dea lin g wit h fr es h concrete or theflow of water or gas into and through a mater ia l.In term ed ia te viscoelas tic behaviour IS aI oIm portant.

PART 1FUNDAMENTALSRevised and updated by Peter Domone,with acknowledgements to the previousauthors, Bill Biggs, ian McColl andBob Moon

This first part of the book is aim ed at both descr ib ing and exp laining th e behaviour of materials in ge neral, without spec ifica ll y concentrating on any one typeor group of materials. Th at is the purpose of th elater sec tions. This part therefore prov id es the bas isfor the later parts, and if yo u get to grips with th eprinciples then mu ch of what follows will be clearer.In the first chapter we star t with a descript ion ofthe building blocks of a ll mater ials - aroms - andhow they combine in single elements and in compounds to form gases, liquids and solids. We thenintroduce so me of th e pri ncipies of thermodyn ami csan d the processes in vo lved in changes of state, withan emph asis on the change from liquid to so lid. Inthe next two chapters we desc ribe th e behaviour ofsolids w hen subjec ted to load and th en cons id er thestru cture of the various types of solids used in construction, thereby giving an ex planation for and anund erstanding of their behaviour.This is fo llowed in subsequent chapters by consid era tion of rh e process of fracture in more dera il(in cluding a n int roduction to the subjec t of fracturemechani cs), and then by brief di sc uss ions of thebehaviour of liquid s, viscoelas tic materials and ge ls,the nature and behaviour of surfaces and th e elec tricaland th ermal properties of materi als.

1 Bur note rhar th e deformat ion may sr i!! be sign ifica nt on an enginee rin g sca le, as we sha ll see extensively in rhis book.


4/55


5/55

As engineers we are pr ima rily co ncerned with the propert ies o f mate ri a ls a t th e mac ros tru ctura l leve l, bu tin order to understa nd these pro perties (w hich we w illintroduce in C ha pte r 2 ) a nd to modi fy th em to o urad vantage, we need a n understa nding of the structu reof materia ls a t the a tomic level through bonding fo rces,mo lecules a nd mo lecula r a rra nge ment . Some kn o wledge of th e processes in vo lved in c ha nges o f sta te,pa rticul a r ly fro m liquids to so lid s, is a lso va lua ble.

The conce pt o f 'a to mistics ' is no t new. Th e ancientG ree ks - a nd es pec ia lly Democ ritu s (ca. 4 60 Bc) -ha d the idea o f a sing le e lementa ry particle but th eirsc ience did not ex tend to obse rva tio n and ex perim ent.Fo r th at we had to wa it nea rl y 22 ce nturies unrilDa lton, Avogadro a nd Ca nni zzaro fo rmula ted a to micth eo ry as we kn ow it today. Even so , ve ry ma nym yste ries still rema in unreso lved. So in rrea ti ng th esubject in thi s w ay we a re reaching a lo ng w ay bac kin to th e develo pment o f th o ught a bo ut th e uni versea nd th e way in w hich it is put to ge th er. Thi s iscove red in th e fi rs t pa rt of thi s c ha pte r.

Co ncepts o f cha nges o f sta te a re mo re recent .Engineering is mu ch co nc ern ed with cha nge - th echa nge fro m th e unloa ded to th e loa ded sta te, rh eco nsequ ences o f cha nging tempera tu re, environment,etc . Th e first sc ientific studies o f thi s ca n be a ttributedto Ca rno t ( 182 4 ), la te r ex tend ed by such g ia nts asC la us iu s, J o ul e a nd o th ers to pro du ce ideas s uch asth e co nse rva tio n o f ene rgy, mo mentum , etc . Sinceth e ea rl y studies we re ca rri ed o ut o n hea t engin esit beca me kn ow n as the sc ience of th e rm odyna mics, 11 In ma ny enginee rin g co ur ses th er mod yna mics is tr ea rcdas a separa te to p ic, o r no r co nside red a t a ll. Bur, beca useirs a pplica tio ns se t rul es rha r no eng in ee r ca n ig nore, abri ef d iscuss io n is inc luded in thi s chap te r. W ha t ar e th eserul es? Succ inc tl y, they a re: Yo u ca nno t w in , i.e. yo u ca n no t ge t mo re o ur of a

sys tem th a n yo u pu r in . Yo u ca nn o t br ea k even - in a ny cha nge so mething w illbe los t or, to be mo re pr ec ise , it w ill be use less fo r th epurpose yo u have in mind .

Chapter 1Atoms, bonding,energy andequilibrium

but if we ta ke a broa der view it is rea ll y th e a ra nd sc ience o f manag ing, co ntro lling a nd using th et ra nsfer o f energy- w heth e r th e energy o f rhe ato mth e energy of th e rides o r th e energy o f, say, a liftingri g. The second pa rt o f thi s cha pter th ere fore dea lw ith th e co nce pts o f ene rgy as a pplied to changeso f sta te, fro m gases to liquid, bri e fly, a nd fro m liquidro so lid , mo re ex tensive ly, including considera tio no f equilibrium a nd equilibrium di ag ra ms. If th esea t first seem da unting, yo u may skip pas t th esesectio ns o n first rea d ing, bu t co me bac k ro th emas th ey a re impo rta nt .

1.1 Atomic structureAtom s, th e bui lding bl oc k o f e lements, co nsist o f anu c leus sur ro un d ed by a clo ud o f o rbiting elec tr o nsT he nucl eus co nsists o f pos iti ve ly cha rged proto nsa nd neutra l ne ut ro ns, a nd so has a net pos iti vecha rge th a t ho lds th e nega ti ve ly cha rged elec tr o nsw hi ch revo lve a round it , in pos iti o n by a n e lec trosta tic a ttr ac ti o n.2 Th e cha rges o n th e pro to n a ndelec tron a re eq ua l a nd Opp os ite (1. 602 X ] o - JY COUlom bs) a nd th e number of elec tro ns a nd pro to nsare equ a l a nd so th e a to m ove ra ll is e lec trica ll yneut ra l.Pro to ns a nd neutron s have a pprox im a te ly th esa me ma ss, 1 .67 x 1o-n kg, w hereas a n elec tronhas a mass o f 9. 11 X 1o - l l kg , near ly 2000 tim esless. These re la ti ve dens it ies mea n th a t th e size oth e nucleus is very sma ll co mpa red to th e s ize oth e ato m . Alth o ugh th e na tu re o f th e elec t ro n clo udma kes it difficult to defin e th e size o f ato ms prec ise lyheliu m ha s th e sma llest a to m, with a ra dius of a bo u

2 Pa rti c le ph ysicists have di scove red or pos tul a ted a co nsiderab le number of o ther sub -atomic pa rticles, such asqu a rk s, mu o ns, pio ns and neutrinos. It is howeve r suffic ienfo r o ur pur poses in th i s boo k fo r us to co nsider o nl yelectro ns, proto ns a nd neut ro ns.

3


6/55

Fundam enta lsTable 1.1 Avai lable elec tr on states in the fir st fo ur she ll s and sub -s hells of elec trons in the Bohr a tlll ll(a fter Ca lli ste r, 2007)

Principal quantumnumber (11 ) Shell Sub-shell ( /)j K s2 L s

p3 M spd

4 Npdf

0.03 nanometers, while caesium has one of the largest,with a radius of about 0.3 nano metres.An element is characteri sed by: the atomic number, whi ch is th e number of protons in the nucleus, and hence is a lso th e numberof elec trons in orbit; the mass number, wh ich is s um of the number ofprotons and neutrons. For many of the lighter elements these numbers a re similar and so th e massnumber is approximately twice the atomi c numbe r,though this relationship breaks down with increasing atomic number. In some elements the numberof neutrons can vary, lea ding to isotopes; the atomi cweight is the weighted average of the atomi c massesof an element 's na turally occurring isotopes.

Another use fu l quantity whe n we co me to con-sid er compounds and chemi cal reac tions is th e mole,which is the amount of a substance that co nta in s6.023 x 1023 atoms of an element or molecul es o fa co mpound (Avogadro's number). This number hasbee n chosen beca use it is the number o f atoms thatis contained in the atomic mass (o r weight ) ex pressedin grams. For example, carbon has an atomic we ightof 12.011 , and so 12.011 grams of ca rbon contain6.023 x 1023 atoms .Th e manner in wh ich the o rbits of the electro nsare di stributed aro und the nucl eus con tr o ls th echaracteristics of the element and th e way in which

atoms bond with other atoms of th e same elementand with atom s from different elements.For our purposes it will be suffi cient to desc ribethe structure of the so-ca lled Bohr atom, whicharose from developments in quantum mechanics in4

Maximum number o f elecrronsNumb er o fenergy stat es (nr1) Per sub -shell Per shell

335I357

2 22 862 186102 32610

14

the earl y pa rt of the 20 th ce ntury. This overcamethe problem of explaining wh y nega tively chargedelectr ons wou ld not co llapse into th e posit ive lycharged nucleus by propos ing th at elec trons revo lvea round the nucl eus in one of a number of di sc reteorbita ls or shells, each with a defin ed or qu a ntisedenergy leve l. Any electr on mov ing betw ee n energylevels or orbita ls would make a qu antum jump witheither emi ssion or abso rpti o n of a discrete amounto r quantum of energy.Each elecrron is cha rac teri sed by four qu a nrumnumbers: the principal quantum number (n = 1, 2, 3, 4 . . . ,whi ch i the quantum shell ro whi ch rh e electronbelongs, also denoted by K, L, M, N . . . , corres ponding to n = 1, 2, 3, 4 . . . ; the secondary qu antum number (I = 0, I , 2 . . .

n - 1), wh ich is the sub-she ll to wh ich the electronbelongs, denoted by s, p, d, f, g, h for I = 1, 2,3, 4, 5, 6, acco rding to its shape; the third qu antum number (m 1) , whi ch is thenumber o f energy sta tes within eac h sub -s he ll ,the total number of which is 2/ + I ; the fourth qu antum number (m ,) which desc ribesthe electron 's direct ion of spin and is either + 1/1or - 11!.Th e number of sub-shell s that occur within each shellth erefore increases wit h an in crease in the principalquantum number (n ), and the number of energy stateswithin eac h subshell (m 1) in creases with an in creasein the secondary quantum number(/ ). Table 1.1 showshow thi s leads to the maximum number elec tronsin eac h shell for th e first four shells.


7/55


8/55

Fundamentals

(a) Between chlorine atomsFig. 1.3 Covalent bonding.

' borrow' or 'share' rwo; sin ce sodium ca n onl y donateone electron, the chemi ca l formula for sodium ox ideIS a20. Magnes ium has two va lence elec trons andso the chemi ca l formula for magnes ium chl oride isMgCI 2 and for magnes ium ox id e MgO. T hu s, th enumber of va lence elec trons determines the rela ti veproportions of elements in compound s.The strengt h of th e ioni c bond is proportional toeAe11 r where eA and are the charges on the ionsand r is the intera tomi c se para ti on. The bond isstrong, as show n by the hi gh melting point of ioni ccompound s, and its strength in creases, as mi ght beexpec ted, where two or more elec trons a re donated.T hu s the melting point of sodium chl oride, aC I,is 80 l C; that of ma gnes ium ox id e, MgO, wheretwo elec t rons are in vo lved, is 2640C; and th at ofzirconium ca rbide, Z rC, where four elec trons a reinvolved, is 3500C. Although ionic bonding inv olvesthe transfer of elec trons betwee n different atoms, th eovera ll neutrality of the materia l is ma inta in ed.The io ni c bond is a lw ays non -direc ti ona l; that is,when a crys ta l is built up of large numbers of ions,th e elec trosta ti c charges a re ar range d sy mmetrica II yaround each ion , with th e result that A ions surroundthemse lves with B ions and vice versa, with a so lidbeing formed. The pattern adopted depends on thecharges on, and th e relative sizes of, the A and Bions, i. e. how many B ions can be comfortablyaccommodated around A ions whilst preserving thecorrect ratio of A ro B ions.1.2.2 COVALENT BONDINGAn obvious limita ti on of the ioni c bond is that it canonl y occur betwee n atoms of different elements, andtherefore it ca nn ot be res ponsible for the bondingof any o f the solid elements. Where both atoms areof the elec tron-acceptor type, i.e. with cl ose to 8ou termost elec trons, octet structures ca n be builtup by the sharing of two o r more va lence elec tronsbetween th e atoms, forming a cova lent bond .

For example, two chl orine atoms, which each haveseven va lence elec trons, can achi eve the octet struc-6

(b) Between oxygen atoms

ture and hence bond together by contributing oneelectron eac h to share with the other (Fig. 1.3a).Oxygen has six va lence elect rons and nee ds tosha re two of these with a neighbour to form a bond(Fig. 1.3b). In both cases a molec ul e with two a ro msis formed (C I2 and 0 2 ), which is rh e normal sta teof these two gaseo us elements and a few others.There are no bonds between th e molec ul es, whi ch iswhy such eleme nts are gases at norma l tempera tureand pressure.Cova lent bonds are ve ry strong and directional; th eycan lead to ve ry strong two- and three-dimensionalstructures in elements where bonds can be fo rmedby sharing electron s with more than one adj acentatom, i. e. which have four, fiv e o r six va lence electrons. Ca rbon and silicon, bo th of whi ch have fourva lenc e elec trons, are two importa nt exa mples. Astructure can be built up with each atom formingbonds with fo ur ad jacent atoms, thu s ach1 ev in g therequired elec tron octet. In practice, the atoms arrangeth emse lves with eq ual angles betwee n a ll the bonds,which produces a tetrahedra l stru ctu re (Fig. 1.4 ).Ca rbon atoms are a rrange d in this way in di amond,which is one of the ha rd est materials kn ow n anda lso has a very hi gh melting point (3500 C).Cova lent bonds are aI o formed between a romsfrom different elements to give compound s. Methane( H4) is a simpl e exa mpl e; eac h hydrogen atomachi eves a stabl e helium elec tron confi gurat io n bysharing o ne of th e four atoms in carbon 's outer shelland the ca rbon atom ac hieves a stable oc tet figurationby shar in g th e elec tron in eac h of th e four hydrogenatoms (Fig. 1.5) . It is a lso poss ibl e for carbon atomsto form long chain s to which other atoms ca n bondalong the length , as show n in Fig. 1.6. This is thebasis of many polymers, whi ch occur extensively inboth natural and manu factured forms.A large number of com pound s have a mi xtureof cova lent and ioni c bonds, e.g. sulphates such asNa 2S0 4 in which th e sulphur and oxygen are cova lently bonded and form sulph ate ions, which forman ioni c bond wi th th e sodium ions. In both the


9/55

0Outer shell ofa single carbonor sil icon atom

tAtoms, bonding, energy and equilibrium

- --+

Fig. 1.4 Covalent bonding in ca rbon or si lica to form a continuous stm cture with (our bonds orientated at equalspacing giving a tetrahedron-based stru cture.

H

Fig. l. S Cova lent bonding in methane, C

t t t-Fig. 1.6 Covalent bonding in carbon chains.

ionic and cova lent bonds th e elec tro ns a re held fairl ystrongly a nd a re no t free to move far, which accountsfor th e low electrical co nducti vity of mater ia ls co n-ta ining such bond s.

1.2.3 METALLIC BONDSMetallic a toms possess few va lence electro ns and thusca nn ot form cova lent bond s betwee n each oth er;instead they obey what is termed th e free-e lectrontheory. In a metallic crysta l th e valence elec tro nsare detached from their atoms and ca n move fre elybe tween th e pos iti ve meta lli c io ns (Fig. 7. 7). T hepositive ion s a re a rranged reg ul ar ly in a crysta lla tti ce, a nd the elec tros ta tic at trac t ion between th epositive ions and the free negative electrons prov idesth e co hes ive str engt h of the meta l. Th e meta lli cbond may thus be regarded as a ve ry spec ia l case

t t

7


10/55

Fundamenta ls

Fig. 1. 7 The free elec tron system in the meta llic bondin a m onovalent metal.

of cova lent bond in g, in which the octet stru cture issa ti sfi ed by a ge nerali sed donation of the va lenceelectrons to form a 'c loud ' that permeates the wholecrys ta l lattice, rath er th an by elec tron shar ing berweenspec ific atoms (true cova lent bonding) or by donationto another atom (ioni c bonding).Since th e elec trostati c a ttracti on between ions andelec trons is non-directional, i.e. the bonding is notloca li sed berween indivi dual pairs or groups of ato ms,metallic crys tals ca n grow eas il y in three dim ensions,and the ions can approac h a ll neighbours eq ua ll yto give max imum stru ctu raI density. T he res ultingstructures are geo metrica ll y simp le by compar isonwith the stru ctures of ioni c compounds, and it isthis simplicity that accounts in part for the ductility(a bility to deform non-reversibl y) of the metallicelements.Meta llic bonding also ex pl a in s the hi gh thermaland elec trical conductivity of metals. Since the va lenceelec trons are not bound to any part icu lar atom, theyca n move thr ough th e lattice und er th e app lica tionof an elect ri c potential, causin g a current fl ow, andcan also, by a se ri es of co llisions with neighbouringelec trons, tra nsmit thermal energy rap idl y throughthe latt ice. Opt ical properties can also be ex pl a in ed.f or example, if a ray of light fall s on a metal, th eelec trons (bein g free) can absorb the energy of th elight beam, thu s preventing it from passing throughthe crys ta l and rendering th e metal opaque. Th eelec trons that have absorbed the energy a re excitedto hi gh energy levels and subsequently fal l back totheir o rigin al va lues with the emi ss ion of the lightenergy . In other words, the light is refl ected backfr om the surface of the metal, whi ch when po lishedis hi ghl y refl ecti ve.Th e ab ility of metals to for m a lloys (of ex tremeimpor tance to enginee rs) is also ex plained by th efree-elec tron th eory. Sin ce the elec trons are not bound,when two metals a re a lloyed there is no question8

of electron exc hange or sharing between atoms inion ic or cova lent bond ing, and hence the o rdinaryva lence laws of combination do not app ly. The princip al limita ti on then becomes one of a tomic size,and prov iding th ere is no great size difference, twometals may be abl e to fo rm a continuous se ries ofa ll oys or so lid so luti ons from I00% A to I00 % B.The rules governing th e compos ition of th ese so luti onsare di sc ussed la ter in th e chap ter.1.2.4 VAN DER WAALS BONDS AND THEHYDROGEN BONDIoni c, cova lent and metal lic bonds a ll occ ur beca useof th e need for atoms ro achi eve a sta bl e electronco nfiguration; they are strong and a re th erefo resometimes know n as primary bonds. However, someform of bondi ng force berween the res ulting mo lec ulesmu st be present since, for exa mple, gases w ill a llli quefy and ultim ately so li dify a t suffi cien t ly lowtempera tures.Such secondary bonds of forc es a re known asWan der Waa ls bonds or Wan der Waa ls forces andare universal to a ll atoms and molec ul es; th ey arehowever sufficiently weak that their effec t is oftenoverwhelm ed when primary bonds are present. Theyar ise as fo llows. Alth ough in Fig. 1.1 we rep rese ntedthe orbiting electrons in di sc rete shell s, the tru epi ct ure is th at of a cloud, the densit y of the cloudat any poi nt bein g related to the probab ility offinding a n elec tron there. Th e elect ron charge is thu s'spread' around the atom, and , ove r a pe ri od oftime, th e charge may be th ought of as sy mmetrica ll ydistributed with in its particul ar cloud.However, the elec troni c charge is mov ing, a nd thismea ns th at on a sca le of nanoseconds the electrosta ticfi e ld aro und the atom is continu ously fluctuating,res ulting in the formatio n of a dynamic elec tric dipole,i.e. the ce ntres of pos iti ve charge and negati ve chargea re no longe r co incident. When ano ther atom isbrought in to prox imity, th e dipoles of the two atomsmay interac t co-operatively with one ano th er (F ig. 1.8)and th e res ult is a weak non -directi onal electrostati cbond.As well as this fluctuating dipole, man y mol ec ul eshave permanent dipoles as a res ult of bo nding betwee ndifferent types of atom. T hese ca n pl ay a considerab le part in the structure of polymers and orga ni ccompound s, where sid e-cha in s an d radical groupsof ions ca n lead ro points of predomi nantly positi veor nega tiv e charges. These w ill exe rt an electrosta ti cattract ion on other op positely charged groups.Th e strongest and most important exa mp le ofdip ole interac ti on occurs in co mpound s betweenhydrogen and nitrogen, oxygen or fluor in e. It occurs


11/55

Momentary dipoles

Attraction

Fig. 1.8 Weak Van der Waa ls linkage between atomse to fluctuating elec trons fields.

@--------8resultingdipole

The water molecule (b) The structure of water. 1.9 Th e hydrogen bond between water mo lecules .

eca use of the sma ll and simple struc ture o f th eydrogen a to m and is known as th e hydrogen bond., for exa mple, hydr ogen links covalently with

n to form water, th e elec tr o n contributed bye hydroge n a to m spend s the grea ter pa rt o f its

me betw een the two atoms. The bond acq ui res adipole with th e hydrogen becoming virtua llypos itively cha rged ion (Fig. J .9a).Since th e hydrogen nucle us is no t screened by any

th er elec t ro n she lls, it ca n attract to itself o th erega ti ve end s o f dipo les, and the result is the hydrogenIt is co nsid e ra bly stronge r (a bout 10 rim es)a n oth e r Van der Wa a ls linkages, bur is mu cher (by 10 to 20 rim es) than a ny of th e primary

onds. Figure 1. 9b shows th e resultant struc tu re ofwhere the hydroge n bo nd forms a second a ry

between th e wa ter mo lec ul es, and ac ts as age betw ee n tw o elec tr o nega tive oxyge n io ns.this re la ti vely insignificant bond is o ne o f the

os t vita l factors in th e evo lutio n a nd su rviva l ofe on Ea rth . It is responsible for th e ab no rm a lly

Atom s, bonding, energy and equilibriumhigh melting a nd bo iling po ints of wa ter a nd for itshigh spec ific hear, which prov id es an essential globa ltempe ra ture co ntro l. In th e abse nce of th e hydrogenbond, wa ter wo uld be gaseous at a mbi ent tempe ra-tures, lik e a mm o ni a a nd hydrogen sulphid e, a nd wewou ld not be here.

Th e hydrogen bo nd is a lso respo nsible fo r th eunique property o f water of ex pansion during freez ingi.e. a density dec rease. In so lid ice, th e co mbina ti onof cova lent and str ongish hydrogen bonds result in athree-dimensional rig id but rela ti ve ly open structu re,but o n melting thi s st ructure is partially des tr oyedand th e water mo lec ules beco me more close ly packed,i.e. th e density increases.

1.3 Energy and entropyThe bonds that we have just described ca n occurbetween atoms in gases, liquids and so lid s and toa large ex tent a re res ponsible for th eir ma ny a ndvar ied p rope rti es. Although we ho pe constructionma ter ia ls do no t cha nge sta te whilst in serv ice, wea re very much co ncerned with such changes durin gth e ir manufacture, e.g. in th e coo ling o f metal s fro mth e mo lten to th e so lid sta te. Some know ledge of th eprocesses and th e rul es governing th em are th erefo reuseful in understan ding th e structure and pr ope rti esof th e mate ri a ls in their 'ready-to use' state.

As engineers, a lth o ugh we co nventi ona lly ex pressour findings in terms of force, defl ection, stress, stra ina nd so o n, these are simpl y a co nventi o n. Fundamen-ta lly, we a re rea lly dea ling with energy. Any cha nge,no marte r how simple, involves an exc hange o f energy.The m ere act o f lifting a bea m invo lv es a cha nge inth e potentia l energy of th e bea m, a change in th estrain energy held in the lifting ca bl es a nd a n inputof mec ha nica l ene rgy from th e lifting dev ice, whichis itse lf tr a n sforming electri ca l or oth er energy intokinetic energy. The ha rness in g and co ntrol of energya re a t th e hea r t o f a ll eng in eer ing .

Thermodynamics teaches us a bo ut energy, a nddraws a ttentio n to th e fact th at every materi a lpossesses a n internal energy assoc ia ted with itsstr u c ture. We begi n this sec ti on by discussing so meof th e th erm odyn a mic principles th at are o f impo rt-a nce to understa nding th e behav io ur patterns.1.3.1 STABLE AND METASTABLE EQUILIBRIUMWe sho uld recogni se that a ll sys tems a re a lwayssee king to minimise the ir energy, i.e. to become morestable. However, although thermod ynamically correct,some changes toward a more stab le co nditio n proceed so s lowl y th a t th e system a pp ea rs to be stab le

9


12/55

Fundamen tal s

>-OlQ;cw P ,Act1 va t1onenergy

________!________---- -- ------r -Free

energy- - - - - - - - - -

Fig. 1. 10 Illustration of activation and free energy.

eve n th o ug h it is no t . Fo r exa mpl e, a sm a ll balls itting in a ho llow at th e top of a hill wi ll remai nth e re until it is lifted o ut and rolled down th e hill.Th e ba ll is in a metastab le sta te a nd requires a sm a llinput of ene rgy to sta rt it o n its way down th e mainslope.

Figure 1.10 shows a ball sitting in a depress ion wi tha potenti a l energy of P 1. It w ill ro ll to a lowe r energysta te P2, but o nl y if it is first li fted to th e top of thehump between th e two ho llows. Some ene rgy hasto be lent to th e ba ll to do thi s, which th e ba ll re turnswhen it ro lls down th e hump to its new pos iti o n.T his borrowed energy is known as th e activationenergy for th e process. Thereafter it possesses freeenergy as it ro ll s down to P2 How eve r, it is losingpo tenti a l ene rgy a ll th e tim e a nd eventu a ll y (say, a tsea leve l) it wi ll ac hi eve a sta bl e equilibrium . However,note t\-vo things . At P1P2, etc. it is a ppa rentl y s tab le,but ac tu a ll y it is metasta ble, as th e re a re o th e r m orestab le sta tes ava ilab le to it , give n th e necessaryact iva ti o n ene rgy. Where does th e ac tivation energycome from ? In materials sc ience it is ex tr ac ted mos tl y(bu t no t ex clusive ly) from heat. As things a re hea tedto higher tempe ra tures th e a to mi c particles reactmo re rap idl y a nd can break o ut of th e ir metastab lesta te in to one where they ca n now lose ene rg y.1.3.2 MIXINGIf whisky and water a re pl aced in th e sa m e co nt a iner,th ey mi x spontaneo usly. The in te rn a l ene rgy o f th eres ulting so luti o n is less th a n th e sum of the twointern a l energies before th ey were mi xed. There is noway th a t we ca n se par a te th em except by distillation,i.e . by hea ting th em up a nd co llec ting th e vapoursa nd se para ting these into alco ho l and wa te r. Wemu st , in fact , put in energy to sepa ra te them. But,s ince energy ca n be neither be c rea ted nor destroyed,th e fac t that we must use energy, a nd quite a lo tof it, to restore the sta tu s quo mu st surely p oseth e question ' Where d oes th e energy come frominitia lly?' Th e a nswer is by no means s im p le but, as10

we sha ll see, eve ry partic le, whether of water orw hi sky, possesses kinetic energies of motion and of1nteracnon.

Wh en a sys tem such as a liq uid is left to it se lf,its intern a l energy remain s constant, but when itin te racts w ith another sys tem it wi ll e ith e r lo se orga in energy. T he tr a nsfe r may in vo lve work or hea to r both and th e first law of t r o d y n a m c ~ , ' th eco nse rva ti o n of energy an d hea t ', requires rh .1r:

dE= dQ- dW ( 1.1 )w here E =in ternal energy, Q = hea t and W = wo rkdon e by th e sys tem o n th e sur ro undings. Wh ;Jt thi ste ll s us is th a t if we ra ise a cupfu l of wate r from20C to 30C it does not ma tter how we do it. Weca n hea r it, stir it w ith paddl es or even pur in aw ho le army of gno mes each equipped w ith a hotwa ter bottle, but the in ter na l energy a t 30C w illa lways be above that a t 20C by exac tl y th e sa mea mo unt. Note that th e first law says nothing .1bourthe sequences of changes th a t a re necessa ry to br in gabout a change in intern a l energy.1.3.3 ENTROPYC lass ica l th e rm odyna mics, as normall y taught toeng in ee rs, regards e ntr opy, S, as a capacity propert yof a sys tem w hich increases in proportion ro th ehea t a bsorbed (d Q) at a give n temperature (T). Hencethe well kn ow n relationship:

dS;? dQJT (1.2)w hich is a perfec tl y good defi niti on but does norgive any sort of pi ctu re of th e mea nin g of entrop yand how it is defin ed. To a mater ia ls sc ienti st entro pyhas a rea l ph ys ica l mea ning, it is a meas ure of th esta te of disorder or chaos in the system. Whisky a ndwater co mbine; thi s simpl y says that, stat istica ll y,th e re a re many ways th a t the ato ms can get mi xedup and on ly o ne possible way in which the whiskyca n stay o n top o f, o r, depending o n ho w yo u pourit , a t th e bo tto m of, th e wa te r. Bo ltzmann showedth a t th e entr opy of a system co uld be representedby:

S = k In N (1.3)where N is th e number of ways 111 which th epartic les ca n be distributed and k IS a co nsta nt(Boltzmann 's co nsta nt k = 1.38 x 10-23 }/K ). Thelogar ithmic relat ionship is impo rt ant; if the mo lecul es of water ca n adopt N 1co nfigurati ons and th oseof whisky N 2 the number of po ss ibl e configur atio nsope n to th e mix ture is not N 1+ N 2 but N 1 x N 2 Itfollows fro m thi s th a t th e entr opy of any c losedsys tem no t in eq uilibrium w ill tend to a ma x imum


13/55

in ce thi s represe nt s th e rn os r pro ba bl e a rr ay o fnfigur ations. Thi s is th e second law o f th ermoyna mics, fo r w hi ch yo u sho uld be very gra teful.yo u rea d th ese word s, yo u are kee pin g a li ve byea thing a ra nd o mly di stributed mi xtu re o f oxygen

nd nitrogen. No w iri s sra ri srica ll y possible th at atme instant a ll th e oxygen a tom s wi ll co ll ect inne co rner o f th e room w hile yo u tr y to ex ist on

ur e nitrogen, bur o nl y sra ri srica ll y poss ible. Th erere so ma ny o th er poss ible di stributi o ns invo lvingmo re ra nd o m a rra ngement o f the tw o gases th a tis mos t li kely th a t you w ill co ntinue to brea thee no rm a l ra nd om mi xtu re..3.4 FREE ENERGY

t m ust be cl ea r tha t th e fund a ment a l tendency fo rntr o py to increase, th a t is, fo r systems to becomeo re ra nd omi sed , mu st sto p so mewh ere a nd so meo w, i.e. th e system mu st reac h equi librium. If no r,e entire uni ve rse wo uld brea k down into chaos .we have seen in the first pa rr of thi s cha pter, th eeaso n for th e ex istence o f liquid s and so lids is th a teir ato ms a nd mo lecul es a re nor to ta lly indifferent

each other a nd, under ce rt a in cond itions andce rta in limita ti ons, wil l associ a te o r bo nd withch o th er in a non-rand om way.As we sta red a bove, from th e first la w o f therm onami cs the cha nge in intern a l energy 1s giveny:

dE= dQ - dWo m the second law o f thermodynami cs the entropyha nge in a reversible pr ocess is:

TdS = d Q (1.4 )ence:

dE= Td S - dW ('I .5 )d iscussin g a system subj ect to change, it is co n

eni ent to use th e co ncept o f free energy. For ir rersible chan ges , th e c hange in free energy is a lwaysega ti ve and is a measure o f th e dri ving fo rce leadin gequilibrium . Sin ce a sponta neo us chan ge mu std to a mo re pro ba bl e sta re (o r else it wo uld nor

appen) it fo llows th at, a t equ ilib ri um , energy ISsed wh ile ent ropy is ma ximised.Th e H elmh o ltz free energy is defin ed as :

H = E - TS (1.6)nd the Gibbs free energy as:

G = pV + E - TS (1.7)nd , ar equilibrium , bo th mu st be a m1n1mum.

Atoms, bond ing, energy and eq ui librium

1.4 Equilibrium and equilibriumdiagramsM os t of the ma te ri a ls that we use a re not pure burco nsist o f a mi x ture o f o ne o r mo re constitu ents.Eac h o f th e th ree mate ri a l sta res o f gases, liquid s a ndso lids may co nsist o f a mi x tu re o f different co mpo nents, e.g . in a lloys of two meta ls. These co mpo nentsa re ca lled phases , w ith each ph ase being ho moge neo us. We need a sc heme th a t a llows us to sum ma ri serhe influences o f tempera ture a nd press ur e o n th ere la tiv e sta bility o f each sta te (a nd , w here necessa ryits co mpo nent ph ases) a nd o n th e tra nsitions th a t canoccur betw ee n th ese . Th e time-hono ured ap proac hro this is with equ ilibrium diagram s. No te th e wo rdequilibrium . T he rm o d yna mi cs te ll s us th a t th i s isth e co ndit io n in w hi ch th e mate ri a l has minimumin te rn a l energy. By defin iti o n, equilibrium di ag ra mste ll us a bout thi s minimum energy sta te th a t a systemis t ry ing to reach, bu t wh en using th ese we sho uldbea r in mind th a t it w ill a lways ta ke a finite timefo r a tra nsiti o n fro m o ne sta te ro a no th er to occuro r fo r a chem ica l reaction ro ta ke place . So metimes ,thi s rim e is va ni shing ly sma ll , as when d yna miteex plodes . At o th er times, it ca n be a few seco nd s,days o r even ce nturi es. G lass mad e in th e MiddleAges is still g lass a nd sho ws no sign o f crys ta ll ising.So, no t eve ry sub stance or mi x ture that we use hasrea ch ed thermody na mic equ ilibrium .We onl y ha ve space here to in t ro duce so me of th ee lements o f th e grea t wea lth o f fund a menta l th eo ryund e rl ying th e fo rm s o f equ i librium dia g ra ms.1.4.1 SINGLE -COMPONENT DIAGRAMSThe temperature-pressure diag ram fo r water (Fig. 1.1 1)is an im po rta nt exa mple of a sing le-co mpo nentdiagra m, a nd we can use thi s to esta b lish so me g roundrules a nd language fo r use la te r.

Th e d iagram is in ' tempera ture-press ure space'a nd a number of lin es are ma rk ed whic h representbo un d a ry co nditio ns betw ee n differing ph ases, i.e.sta tes o f H 20 . The line AD represents combin a t io nso f temperature a nd pressure a t w hich liquid wa te ra nd so lid ice a re in equi li brium , i.e. ca n coex ist . Asma ll hea t input wi ll a lter the propo rtio ns o f icea nd wa ter by melting som e o f th e ice . H o weve r, itis a bso rb ed as a cha nge in in tern a l energy o f th emix ture, th e la tent hea t of melting . The temperatureis no t a lte red , bur if we put in la rge a mo unts o fhea r, so that a ll th e ic e is melted a nd there is so meheat le ft ove r, th e temperature rises an d we endup w ith s lightl y w a rm ed w a te r. Similarly, lin e ABrepresents th e equi li brium betw ee n liquid wa ter a nd

11


14/55

Fundamental s1000

100Ul0 10E I Liquid:_ x: y


15/55

mpos ition line until we a rri ve at B. At thi s temera ture, a tin y number o f sma ll crys ta ls beg in torm . Furth er redu cti on in tempera tur e brings aboutn in crease in th e a mo unt of so lid in equili briumith a diminishin g a moun t of liquid . On a rri ving a ta ll th e liquid has go ne a nd the materia l is tota llylid. Furth er coo ling brings no furth er changes .o te th at there is an importa nt difference betw eens a ll oy and the pure meta ls of which it is com posed.o th C u and N i have we ll defin ed unique meltingr freez in g) tempera tures bur the a lloy so lidifiese r th e tempera tu re range BC; meta llurgists o ftenpea k o f th e ' pas ty ra nge'.We now need to exa mine seve ra l ma tt ers in moreeta il. First, the so lid crys ta ls th at fo rm a re what isown as a 'so lid so lution' . Cu and Ni are chemi ca ll yimila r elements a nd both , when pure, fo rm facentr ed cubi c crysta ls (see Chapter 3) . In thi s case,50 :50 a ll oy is a lso co mp ose d of face -cent red cubi crysta ls but each la tti ce sire has a 50:50 chance ofeing occupi ed by a C u atom o r a N i a tom .If we appl y Gibbs's ph ase rul e a t po int A, C = 2

tw o co mp o nent s, C u & N i) and P = 1 (one ph ase,) a nd so F = 3 (i.e. 3 degrees o f freedom ). Wen therefo re independ entl y a lte r co mpos ition, temerature a nd pressure and the structure rema ins liquid.emember, we have taken pressure to be constantnd so we a re left with 2 prac tica l degrees o f freedom,mp os ition a nd temp era tu re. The same a rgum ent

o lds a t po in t D , but, of co ur se, th e stru ctu re hereth e crys ta lline so lid so lution of C u a nd N i.At a po int between 8 a nd C we have liquid andlid ph ases coexis ting, so P = 2 a nd F = 2. Asefo re, we mu st di scoun t one degree o f fr eedo meca use pressure is ta ken : : con sta nt. This lea vess with F = 1, whi ch mea ns th a t th e sta tu s quo ca ne ma inta in ed onl y by a co upl ed change in both

mpos ition a nd tempera tu re. There fore, it is notnl y th a t th e structu re is tw o ph ase, but also th a te pr opo rti o ns o f liquid a nd so lid ph ases rema ina ltered.We ca n fin d th e propo rti ons o f liquid and so lid

orresponding to a ny po in t in th e tw o ph ase fieldsing th e so-ca lled Lever rule. The fir st step is toaw the constant temperature line through the pointFig. 1.1 2. T hi s in tersects th e phase boundariest Y and Z. The so lid line co nta ining Y represe ntse lowe r limit o f ] 00% liquid , a nd is kn ow n as thequidus . Th e so lid line co nta ining Z is th e upperof 10 0 % so lid and kn ow n as the so lidus.Ne ither th e liquid no r so lid ph ases correspondingpo in t X have a co mp os ition identica l with thatf th e a lloy as a w ho le. The liquid co nta ins mo reu a nd less N i, th e so lid less C u a nd mo re N i. The

Atoms, bonding, energy and equilibriumco mpos itio n o f eac h ph ase is given by th e po ints Ya nd Z, respective ly. Th e pr o po rti ons o f th e ph asesba la nce so th a t th e we ighted ave rage is th e sa me asth e ove ra ll co mpos itio n o f th e a lloy. It is easy toshow th a t:(Weight o f liquid o f co mpos ition Y) x YX =( Weight

o f so lid o f co mpos iti o n Z) x X Z (1.9)T his is s imil a r ro w ha t wo uld be ex pected o f amec ha nica l leve r ba la nced a bo ut X, hence th e na meLever rul e.O ne co nsequ ence o f a ll thi s ca n be seen by reex a mining th e coo lin g o f th e 50:5 0 a ll oy from theliquid phase . Co nsider Fig. 1.13 . At po in t X 1 onth e liquidus, so lid ifica tio n is a bo ut to begin. At atempera tu re infin ites im a ll y below X 1 there w ill besome crys ta ls so lidi fy ing o ut of th e liquid; th eirco mpos ition is given by Z 1 At a tempera tu re a bo utha lfway betw ee n so lidus a nd liquidus {X2), we havea mi x tu re of so lid a nd liquid of co mp os itions Z 2a nd Y2 . In ge nera l, th e pro po rti o n o f liquid to so lidha lfway thro ugh th e freez ing ra nge need no t be""50 :50, but in this case it is. Fina ll y, at a temperatu reinfinites im a lly a bove x3, which is on the so lidus,we have nea rl y 100 % so lid o f comp os ition Z 3roge th e r w ith a va nishin gly sma ll a mount o f liquido f co mpos iti on Y1 When th e temperature fa lls tojust below X 3, th e a lloy is to ta lly so lid and Z 1 hasbecome identica I with X 1.

N o te tw o impo rt a nt e s . First, Z 3 is the sa meas th e ave rage co mpos iti o n we sta rt ed with , X 1Seco nd , so lidifica tio n ta kes place ove r a ra nge o ftempera tures, a nd as it occ urs the co mp os itions o fliquid a nd so lid ph ases cha nge co ntinuo usly. For thisto ac tu a lly ha pp en, sub stantia l amounts o f diffusio n

Solid

Cu CompositionFig. 1.13 Equilibrium phase diagram for Cu-N i(Fig. 1.1 2 redrawn to show com position variationswith temperature) .

Ni

13


16/55

Fundamentals

14001200

Ge..... 1000Ql:sQl0. .EQlf -

800

400

y

, A

........ .. ................... ~ E . l ..

AI + Si

Silicon (mass%)

SiLiquid + Si

Fig. 1.14 Equi librium phase diagram for alumin ium-silicon.

must occ ur in both liquid and so lid. Diffusion inso lids is ve ry mu ch slower than th at in liquids andis the source of so me prac ti ca l di ffi cult y. Eith erso lidifica tion mu st occur slow ly enough fo r diffusionin the so lid to keep up or stri ct equilibrium conditionsa re no t met. Th e kin etics of phase tra nsforma ti onsis therefo re of interest, but fo r the moment, we w illcontinue to di scuss ve ry slowl y formed, equilibriumor nea r equilibrium stru ctures .14.3 EUTECTIC SYSTEMSLe r us now exa mine a nother diag ram, that fo ra luminium-s ilico n (A I-S i) a lloys (Fig. 1.14 ). PureAI fo rms face -centred crys tals (see Chapter 3) burSi has the same crys ta l structure as dia mond . Th esea re in compat ibl e and extensive so lid so lutions lik ethose fo r Cu:Ni ca nn ot be fo rmed. Si crys tal s ca ndi sso lve onl y tin y amounts of AI. For our purposes,we ca n igno re this so lubilit y, a lth ough we mi ghtrecog ni se rh ar the semi conducto r industry makesgrea t use o f it , sma ll as it is. AI crys ta ls ca n disso lvea little Si , bur aga in nor ve ry much, a nd we willignore it. Thu s, tw o solid ph ases a re poss ibl e, AIand Si. Wh en liquid, rh e element s di sso lve readil yin the melt in a ny proportions.Co nsid er rh e compos iti on Y. On coo ling to theliquidus linea r A, pure (o r nea rl y pure) crys ta ls ofSi beg in ro fo rm . At B we have so lid Si coex istingwith liquid o f co mpositi on LB in propo rtions give nby the Lever rule. At C we have so lid Si in equilibriumwith liquid of co mpos iti on close to E.

14

Now consid er a lloy X. The sequence is much rh esame except th e first so lid to fo rm is now AI. Wh enthe temperature has fa ll en to a lm ost we haveso lid AI in eq uilibrium with liquid o f co mpos iti onclose to E. ore rh ar borh a lloy X and a ll oy Y,when coo led to TF, conta in substantia l amounts ofliquid of compos iti on E. An infinites im a l drop intemperature below T1 causes thi s liquid to so lidifyinto a mi xture of so lid AI and solid Si. At E wehave 3 phases which can coex ist; liquid , solid AIand so lid Si. Th e sys tem has two components andthu s th e phase rule gives us no degrees of freedomonce we have di scounted pressure. E is a n in va riantpoint; any change in temperature or composition willdi sturb the equilibrium.The point E is kn own as th e eutec tic point and wespea k of the eutec tic co mpos iti on and the eutecti ctemperature, T

1. Thi s is th e lowes t temperature arwhi ch liquid can exist and the eutec ti c a lloy is tharwhich rema in s liquid dow n to T 1 . It so lidifies at aunique temperature, quite unlike Cu- i or AI-S ia lloys of other compos iti ons. All oys close to theeutec tic compos ition (""13 %Si) are wid ely used beca use they can be eas ily cast into complex shapes ,and th e Si di spersed in the AI strengthens it. Eutec tica lloys in o th er systems find simila r uses (cast-ironis of nea r eutectic compos ition) as well as uses asbraz ing a lloys etc.

1.4.4 INTERMEDIATE COMPOUNDSO ften, the bas ic components of a system can fo rmcompounds. In metals we have CuAI2, Fe1C a nd manymore. Some oth er relevant exa mples are: Si0 2 and co rundum, Al20 1, which form mullite,3(AI20 1)2(S i0 2), an important constituenr of firedcl ays, pottery and bricks. Figure 1.1 5 shows theSiOcA I20 1 diagram. It can be th ought of as twodi agrams, one for 'Si0 2-mullite' and th e otherfo r 'mullire-A I20 , ' , jo in ed together. Eac h partdi agram is a simple eutectic system like AI-Si; lime, CaO, and silica , Si0 2, which form thecompounds 2(Ca0 )Si02, 3(Ca0)S i0 2 and o thers,which have grea t tec hnolog ica l significa nce asac ti ve ingredi ents in Portland cement (to be di scussed in deta il in Chapter 13). In a simil a r wayto mullite, th e lime (CaO)-s ilica (Si0 2) di agram(Fig. 1.1 6) can be thought of as a series o f jo in edtogether eutec tic systems.

In many cases we do not have to think abou t thewhole di agram. Figure 1.17 shows the Al-C uAI 2di agram, aga in a simple eutec tic system. A nota bl efea ture is the so-ca lled solvus lin e, AB, whi chrepresents th e so lubility of CuAI2 in so lid c rystals


17/55

2200Liquid

2000Mullite + Liquid

Corundum+ liquid

::l1800 Cry stobalite

"'jj + liquida.E Mull ite + LiquidQ) Corundum-

1600 + mull iteCrystobalite + Mullite

1400 L- - - - - L -- - - - ' - - - - ' - - - ' - - - ' - - - - - - - - ' - - - - -LI>r ' - - - - ' - - - ' - - - - 'Si0 2 20 40 60 80 Al, 03Al 20 3 (mass%)

Mull ite 3AI 20 3:2Si0 2Fig. 1.15 Equilibrium phase diagram for silica (SiO 2) -alumina (AI20 1) .

::J'Q)o._EQ)1-

260022001800 F14001000600Si02

[ C = CaO l CS--v= sio , c2s . _ :Liqcid cs:l":" \ ~ ~ I

20 40 60 80 CaOCaO (mass%)

Fig. 1.16 Equilibrium phase diagram fo r lime (CaO )-silica

Atoms, bonding, energy and equilibrium700

Liquid600

500::J CuAI2coQj solvus._ 400 a+ CuAI2)1-300

B200 AI 10 20 30 40 50Copper (ma ss% )

Fig. 1.17 Equilibrium phase diagram for A I-CuA I2

o f A I. This curves sha rp l y, so th a t ve ry mu ch lessC uAI2 w ill disso lve in AI a t lo w tempe ra tures th a nwi ll a t high tempe ra tures. Thi s is a fortun a te factth a t underlies o ur a bility to a lte r th e mic ros tru c tu reso f so me a lloys by sui ta ble hea t-tr ea tm ents, di sc ussedin m o re d eta il la te r.

We ha ve no t yet cons ide red th e iro n-ca rbo ndi ag ra m , whi c h is pe rh a ps th e mos t impo rt a ntdiag ra m fo r nea rly al l eng inee rs. T his is of pa rti cula rre levan ce in c iv il a nd stru c tu ra l eng inee ring s incestee l in a ll its fo rm s is used ex tensive ly. We wil lleave di sc uss ing thi s until C hapter 11.

ReferencesKinge ry WD , Bowen HK a nd Uhlmann D R (1976 ). Intro-du ction to Ceramics, 2nd ed iri o n, Jo hn W iley and So ns,New York.

Ca llisrer WD (2007). M arer ia ls sc ience and engin eerin g. Aninr rodu crio n 7rh edn ., John Wiley and Sons, New Yo rk .

15


18/55

Chapter 2Mechanical propertiesof solids

We have seen in Chap ter 1 how bonds are formedbetween atoms to for m bulk elements and compound s, and how changes of sta te occur, with anemphasis on the formation of solids fr om moltenliquid s. Th e behaviour of solid s is of particul arinteres t to constru cti on engin ee rs for th e obviousreason that these are use d to produce load- bea ringstructures; in th is chapter we define th e propertiesand rules used to quantify the behaviour of solidswhen loaded. To understand thi s behaviour and th erefore to be able to cha nge it to our advantage weneed to consider some o ther aspects of the st ru ctureand nature of the mater ials beyond those disc ussedin Chapter 1; we wi ll do thi s in Chapter 3.You wil l find it necessa ry to refer to th e defi nitionsetc. given in this chapter when readi ng the subse quent on indi vidual materials. Although we willinclude here some exa mples of the behav iour ofconstruction materials, a ll of the definiti ons andexplana ti ons are app lica bl e to any materia ls be in guse d by engin ee rs of any di sci plin e.

2.1 Stress, strain andstress-strain curvesLoad ing ca us es materi a ls to deform and, if highenough, to brea k clow n and fai l. All load ingon mate ri als can be considered as combinat ions o fthree basic types - tension, compression and shea r.These are no rm ally shovvn di agra mma ti ca lly as inFig. 2.1. -- LI ___e_s_ion__ _,l -_. I Compression I - - -ig. 2.1 Basic types of load.16

Clea rl y the deformation from loadin g on anelement or test specimen will depend on both itssize and th e properties of the material from whichit is made. We can eliminate the effec t of size byconverting: the loa d to stress, cr , defi ned as load, P, divided

by the area, A, to which is applied, i. e.cr = PIA (2.1 )

and th e deform ation to strain , , defin ed as change inlength , t,.l, divided by origina l length , I, 1.e. = :,./11 (2 .2)

These definiti ons are ill ustrated for simpl e tensionin Fig. 2.2a. Compressive stress and stra in a re in

Cross-sectional1 area A

Initial length, I Extension , 6 /Tensile stress, o = PIA Tensile stra in, c = 111

(a) Tensionp Displacement, xf.-1

Loogth] .____ -= - - - 'shear force, P

Loadedarea, A

Shear stress , T = PIA Shear strain , y = xl l(b) Shear

Fig. 2.2 Definitions of tensile and shear stress and strain.


19/55

th e oppos ite direct ions. The equi va lent definiti onsof shea r stress, T, and shear stra in , y, which are notquite so obvious, are shown in Fig. 2.2b.As wi th a ll quan tities, th e dim ensions and unitsmust be co nsidered: stress = load/a rea and therefore its dimensionsare [ForceJ!JLength] 2. Typical units are N/mm2(o r M Pa in rh e Sl system), lb /i n2 and tonf/ft 2 inthe Imperi al sys tem. stra in = change in length/o rigin al length and therefore irs dimensions are [Length] /[LengthJ, i.e. itis dimension less.However, stra in va lu es ca n be very sma ll and it isof ten co nvenient to use either:

percentage st ra in (or % stra in )= stra in x 100or mi cros tra in (Jl s) = stra in x 106

As we ll as th e lin ea r stra in , we ca n also simila rlydefinevo lumetric strain (EJ= change in vo lume(LlV)/o riginal vo lume( V) (2.3)

The relat ionship betwee n stress and strain IS anextremely impor tant character istic of a materia l. Itva ries with the rate of appli cation of srress (o r load);we wi I co nsider four cases:a) steadily increasing - ze ro to fa ilure in a few

minutes, e.g. as in a laborato ry tes tb) permanent or static - constant with time, e.g.th e self we ight of the upper part of a stru ctureacting on th e lowe r partc) impact or dynamic - ve ry fast, las ting a fewmicroseconds, e.g. the impact of a vehic le on acrash barrier, or an expl os iond) cyclic - va riab le wi th loa d reversals, e.g. ea rth quake loa ding - a few cycles in a few minutes,and wave load ing on an o ffsho re structure -many cycles ove r man y yea rs.For the moment, we will confine o urse lves to case(a): steady loa ding to fa ilure in a few minutes. Thisis what is used in th e most common types of laboratory tests th at are used to meas ure or charac teri sea mater ial's behaviour.There a re a wide variety of d ifferent forms ofstress-s tra in behaviour for different materi als;

Fig. 2.3 shows those for some co mmon mater ials.Mo sr have at leas t two distinct regions: An initia l lin ea r or nea r-linear region in which

1000

800

ro- 600Cl.


20/55

Fundamentals

rJ)rJ) cr ,iJj

Tangent modulus atcr , = tan e

StrainSecant mod ulus between

cr , and cr2 = tan eFig. 2.4 Definitions of tange nt and seca nt 111oduli ofelasticit y.

defined as follows, are used ro calcul ate deflectionsand movement under loa d.2.2.1 THE ELASTIC MODULIFor linea r elastic materia ls, stress is proportiona lto stra in (Hooke's law) and for uniax ial tension o rcompression we can defin e:Young 's m odulus (E)= slope of stress-straingrap h = srress/s tra in = a ! (2.4 )[E is a lso known as the modul es of elas ticity,the E-modu lus or simpl y th e stiffn ess.ISince strain is dimension less , th e dimensions of E areth e sa me as those of stress i.e. I Force]/!Length 12 . Convenient SI units to avo id la rge numbers a re kN /mm 2or CPa.For materia ls that have non -lin ear elas tic behaviour (q uire a few, particu larly non-metals) a modulusva lue is still use ful and there are some alterna ti vedefiniti ons, illustrated in Fig. 2.4: The tangent modulus is th e slope of the tangentto the curve at any stress (which should bequ oted) . A spec ial case is the tangent modulus atth e origin i.e. at ze ro stress. Th e secant m odulus is the slope of the straightlin e joining two points on th e curve. No te that

stress levels co rres ponding to the two points mu stbe given. If only one stress is given, then it isreasonabl e to assume that the oth er is ze ro .E-va lu es fo r construction materials range from0.007 CPa for rubber to 200 CPa for stee l (di amondis stiffer still at 800 CPa, bur this is har dl y a con-struction materiaI). Va Iues therefore va ry ve rywide ly, by more than 4 orders of magnitude fromrubber to stee l. 1

1We di sc uss va lu es for th e majo r groups of materi a ls inthe relevant pa rts of the book, and then make compa riso ns of this and o th er key prope rties in Chapter 6 1.

18

For shea r load1ng and defo rm a tion, th e equivalentto E is th eshear modu lus (G)=s hea r stress( r )/shea r stra in (y) (2.5 )

G, which is sometimes ca ll ed the modulu s of rigidity, is ano th er elastic constant for the mate rial, andit has a different num erical va lu e to E.The bulk modulus is used when es timating thechange in volume of a material under load. In thecase of unifo rm st ress on a material in a ll direc ti onsi. e. a press ure ({J) as might be found by submerge nceof the spec im en to some depth in a liquid:

Th e vo lumetric strain (_.) = chan ge(reduction ) in vo lume/o rigin al volume (2.6 )and

th e bulk modulus (K) = pi, (2. 7)2.2.2 POISSON'S RATIOWhen a materi al is loa ded or stressed in one di rection, it will deform (or st ra in ) in the direction ofth e load, i.e. longitudina ll y, and perpendicularto the load i.e. latera ll y. Th e Poisson 's ra tio is therat io of the strain in th e direction ro. Thus inFig. 2.5 :

, = x/L (ex tension) ,. = - y/a and = -z lh

(Th e - ve sign indicates contract ion .)Th e y and z direct ions are both perpendicu lar toth e dir ec tion of load ing, x,

Poisson's ra tio (u ) = - /, = -) , (2.8 )The Poisson 's rati o is ano th er elas ti c co nstan t fo rth e materia l. The minus sign ensures that it is apositi ve number. Va lu es for co mmon material s varyfrom 0. 15 to 0.49 (see Tab le 6 1.1 in Chapter 6 1 .

)-Xz- tio ( Q;;,-:=====L=====f r;Longitudinal strain ,,= x/L Lateral strain Ey = - yla = r, = - z/b

Poisson's ratio , u = -/, = - ,/f,Fig. 2.5 Definition of Poisson .s rat io.


21/55

W'e should note that the above definiti ons o fE G and u assume that the materia l has simila rp;operri es in a ll dir ec tions (i.e. it is isotropic) andrh erdo re there is a sin gle va lu e of eac h elas ti cco n t.ln t fo r any direc ti on of loa ding. Anisotropicmate ri als, i.e . those which have different propertiesin d ii fferent direc ti ons, e.g. timber, will have different v.1lu es of , C and u in eac h di rect ion, andclea r-lv the di rec tion :111d we ll as the va lu e itse lf mustthen be stared.2.2.3 RELATIONSHIPS BETWEEN THE

ELASTIC CONSTANTSTh e fo ur elas tic constants that we have now defin ed,E, C, u and K, mi ght at first glance seem to describedifferent aspects of behaviour. It is poss ibl e to proveth at rhey are nor independent and that they arerelated by rh e simple ex pressio ns:and = 2C( I + u )K = /3( 1 - 2u) (2.9)(2. 10)Th e p roo f of these express ions is nor unduly difficult(see fo r exa mple Case, Chil ve r and Ross, 1999) butwh at is more imp ortant is the consequ ence that ifyou know, or have meas ured, any two o f the co nstants then yo u can calculate the va lue o f the o thers.Many materi als have a Po isson's ra ti o betwee n 0.25and 0.3 5, and so the shea r modulus (C ) is o ftenabout 40 % of the elas tic modu lu s () .

Equ a tion 2.10 tells us so mething about the limitsto rh e va lu e of Poisson 's ra ti o. We have defin edthe hul k modulus, K, by consid ering the case o fthe c hange in volume o f a spec im en und er pressure(equation 2.7) . This change must a lways be a reduc-ti on, as it would be in conceiva bl e for a material toexpand under pressure - i. e. in the same direc tionas th e pressure. Th erefore K must always be pos iti veand since E is a lso positi ve (by definiti on) then(1 - 2u ) mu st be positive, and sou 0.5 ALWAYS! (2.11 )

A material with u > 0.5 cannot ex1st; if you havecarried out some tes ts or done some ca lcula ti onsth at give such a va lu e, then you must have made ami stake. It also fo ll ows that if u = 0.5 then K isze ro and the materi aI is inco mpressibl e.2.2.4 WORK DONE IN DEFORMATIONTh e wo rk done by a load when defo rmin g a material, although not an elastic constant, is another use fulvalue. Work is force x di stance, and so

W = J: d e (2.12 )

Mechanical properties of solidswh ere W =wo rk done by the loa d Pin ca usin g anex tension e.

Th e wo rk done on unit volume of the materialof length I and cross-section A is:

W = Pcl e/A I = - - = ad J,. j" P de J[o o A I o (2. 13)whi ch is th e a rea und er rh e stress-s tra in curve .T hi s wo rk must go somewhere, and it is sto redas in te rn a l strain energy within rh e mate ri a l. Withelas tic deforma tion, it is ava ilabl e to return themate ri a l to its zero state on unloa din g; in pl as ticdefo rm a ti o n, it permanentl y defo rms the mater ia la nd , eve ntua ll y, it is sufficient to ca use frac ture. Wew ill ex pl o re the relationship between this energyand frac tu re in more deta il in Cha pter 4.

2.3 Plastic deformationAs we have sa id, defo rmati on is plas tic if it res ultsin permanent deformation after loa d removal.In ve ry broad terms, materials ca n be di vided intoth ose th a t are:

Ductile - whi ch have la rge pl as ti c deforma ti onbefo re fa ilure (say stra ins> 1% )a nd th ose tha t a re:

Brittle - with little o r no pl as tic deforma tionbefo re fa ilure (say stra in s < 0.1 %)So me exa mples of stress- str a in cur ves of eachtype of materia l have bee n shown in Fig. 2.3. Itfo llows fr om equ a ti on (2.1 3) tha t duct ile materialsrequire mu ch grea ter amount s o f work and have

mu ch grea ter amounts o f inte rn al stra in energybefo re fa ilure. Th ere are clea rl y some intermedi atemateri a ls, but eng in ee rs genera lly prefer to useductil e materi als that g ive warning o f di stress befo refa ilure in th e event of overload. Brittle materials fa ilsuddenl y w ithout warning - o ften catastrophica ll y.Significant pl as tic deform a tion obviously occurso nl y in du c til e materi als. We ca n use the idea lise dstress-str a in curve for mild steel show n in Fig. 2.6to illu strate co mmon features of this behaviour. There is a sharp and di stinct end to the linea relas tic behaviour (point A), ca lled the limit ofelastic ity or the yield point . Th ere is a region of in creasin g st ra in with littleo r no increase in stre ss (AB ), often very short . Un loading in the plastic region (say from a

po in t x ) produces behaviour pa ra llel to the initi al(linear ) elas tic behavi our. Reloa din g produces19


22/55

Fundamenta ls

(f)(f)

Ui

0

D

Failure

Y':; yl + - + 1 + ~ - - - - - - - - - - - + 1 1 Stra ina t Plas tic

Fig. 2.6 Stress-s train wrue for mild steel.

simil ar elas ti c behaviour up to th e unloa d point,and the deformation then continues as if the un load/reloa d had nor occurred, i.e. rh e materi al' remembers' where it was . Another feature of pl as tic behaviour, not appare ntfr om Fig. 2.6, is that the deforma ti on rakes pl aceat consta nt vo lume, i.e. the Po isson 's ra ti o is 0.5for deformation beyond th e yield point.

Two important implications for engineers are:l. The stress at the yield po in t A, ca lled rh e yieldstress (crJ , is an important property fo r designpurposes. Working stresses are kept safe ly belowthis.2. If, before use, rh e materi al is loaded or strainedto say a point x beyo nd rh e constant stressreg ion, i.e. beyond B, and then unl oa ded, itend s up at point y. If it is th en use d in this stare,th e yield stress (i.e. at x ) is grea ter than theo riginal va lu e (at A) i. e. rh e mater ial is 's tronge r'.T hi s is know n as work hardening or strainhardening (or sometimes co ld working) to di s-tingui sh it from other methods of strengtheningth at in vo lve heat treatment (which we w ill di s-cuss in Chapter 8). The working stresses ca nth erefore be in creased. Th e draw bac k is that thefailure stra in of the work- ha rd ened materia l(fr om y to fa ilure) is less tha n that of the origin al mate rial (from 0 to failure) and so th ereforeis the tota l work ro fr acture. Th e wo rk -hardenedmaterial is th erefore more britt le.If there is no dist in ct end to the elasti c behaviouri.e. the graph gra du a ll y becomes non-lin ea r, thenan a lternative to th e yield stress ca lled the proofstress is use d in stead. This is defin ed and ob ta in edas show n in Fig. 2.7:1. A tange nt is draw n to the stress- stra in cur ve atth e o rigin .20

(f)(f)

Ui0.1% proof

stress

0.1%(2)

Fig. 2 . 7 Determination of proof stress .Strain

2. A low va lu e of stra in is se lec ted - norma ll y either0. 1% (as in the fig ure) or 0.2 % .3. A lin e is drawn through this point para llel to thetangent at the or igin .4. The stress va lu e a t the point where this intersectsthe stress-s tra in cur ve is the 0. 7% proof stress.(If a stra in va lue of 0.2 % is chosen, th en theres ult is th e 0.2 % proo f stress.)

2.4 Failure in tensionT he form of fa ilure in uniax ial tension depend s onwhether a materi al is brittle o r ductile. As we havealready said, brittle materials fa il wit h little o r nopl as tic defo rm a ti on; failure occurs sudd enl y w ithoutwarning, and th e fracture surface is perpendicu larto the direc ti on of loading (F ig. 2.8).Ductile mater ials nor on ly und ergo la rge str ain sbefore fai lure, but often have an apparent reducti on of stress before failure (i. e. beyond point D inFig. 2.6) . Up to the max imum stress (D ), the elongation is unifo rm , but after this, as th e load starts todecrease, a loca lised narrowing or necking can beseen somewhere along the length (Fig. 2.9a).As the stress continues to fall (bur still at increasin g stra in ) the di ameter at the neck also dec reases,un ti l, with ve ry ductile materials, it reac hes a lm ostze ro before fa ilure, whi ch rak es the form of a sharp

- .._I - - - - ~ ~ 1 ..._- - - - ~ 1 -Fig. 2.8 Brillle fai lure in tension.


23/55

(a) Necking in du ctile materials in the reducing stressregion of the stress-s train curve

(b) Chisel-point failure in ve ry ductile materia ls

cone cup(c) 'Cup and cone' failure in medium -s trength metals

Fig. 2.9 Necking and fa ilure in ductile materia ls intension.

po int (Fig. 2.9b) . This fo rm of failure is ex treme ,a nd occur s onl y in very ductile mate ri a ls such aspure meta ls, e.g. lea d an d go ld, o r chewing gum(t ry it for your self). Th ese materia ls tend to be wea k,and mos t ductile structural mate ri a ls fail a t a stressa nd str a in some way do wn th e fa lling pa rt o f thecurve but with th e stress we ll a bove zero . N eckin gstill occur s after th e max imum stress and fa ilureoccur s a t th e na rrowest sec ti o n in th e fo rm o f a'cup and co ne ' (Fig. 2. 9c). Th e inn er part of thefa ilure sur face is pe rpendicul a r to th e a pplied loa d ,as in a britt le fai lure, a nd th e crac ks first form here.The ou ter rim , at a bo ut 45 to thi s, is th e fin a l ca useof th e fa ilure .

2.5 True stress and strainThe behav io ur show n in Fig. 2.6 shows th e fai lureoccurring a t a lowe r stress th an th e max imum , i.e.th e ma te ri a l see m to be ge tting wea ker as it appr oaches fa ilure. It fact, the op posite is occurring,and th e reason why th e str ess appea rs to fa ll isbeca use of th e way we ha ve ca lcula ted it. We ha vedefin ed stress as loa d/ar ea an d Fig. 2.6 has beeno bta in ed by dividin g th e loa d (P) by th e o ri gi nala rea before loa ding (A 0 ) . The stress th a t we haveo btained sho uld stri ctl y be ca lled th e nom inal stress( rnonJ, i.e.

(2. 14)In fac t , th e c ross-sectional a rea (A) is reducingth ro ugho ut th e loa ding i.e. A < A0 . At any loa d,

Mechanical properties of solids

Truefailurestress

U5Nominal

failure st ress

True ---.__ ,. /stress : .... ....

Nominalstress

/.

StrainFig. 2.10 True and nominal stress/strain behaviour.

the true stress (cr".,,.) wi ll th ere fo re be hi ghe r th anthe nom ina l stress, i.e.(2. 15)

In the elas tic and plas t ic reg io ns th e redu c ti o n isuniform alo ng th e length (th e Po isson's ratio effect )but th e mag nitude o f th e str a ins in vo lve d are suchth a t th e difference betw ee n th e no mina l and th eac tu a l a rea is very sma ll. However, o nce neckin gsta rts th e a rea of th e neck reduces at a rate such th a tth e true stress co ntinues to increase up to fai lure,as shown in Fig. 2. 10.In th e case of s tra in , th e rela ti o n betw ee n th eincrement of cha nge in length (de), th e incrementof stra in (d) a nd th e length (/) is, by definiti o n

d = dell (2 .1 6)a nd so th e tru e strain () is

- J I n 1, I /0 (2.17)where /0 is th e initi a l lengthTrue st ra in is not difficult to calcu late, bu t meas urement of th e cross-sec tional a rea th ro ughout th eloa din g, and hence ca lculatio n of th e true str ess, ismore difficult, and th erefor e true s tress-stra in gra phsare ra re ly o btai ned, except perh a ps fo r researc hpurposes. Howeve r, measurement of the size o f th eneck a fte r frac tu re is easy, w hich ena bles th e truefrac tu re str ess to be rea dil y o bta ined.

2.6 Behaviour in compressionThe elastic behav iour an d co nstants di scussed 111sec ti o n 2.2 appl y equ a lly to tensile and compressiveloa ding. Th ere are however differences in the observedbehav io ur during plas tic deforma ti o n and failure.

21


24/55

Fundamentals

Test machineplatensFrictionalrestraintforces

Fig. 2.11 No n-uni(or111 plas tic de(onnation ina CO IIIfJress ion tes t.

2.6.1 PLASTIC DEFORMATION OFDUCTI LE MATERIALSVa lu es of yield stresses for ducti le materi als aresimilar to those in tension, but the subse quentbe haviour in a labo ratory tes t is influenc ed by theloa ding sys tem. Te st machines app ly the loadthrough large blocks of stee l ca ll ed platens, whi chbear on the specimen. These are stiffer than thespec im en and therefo re the latera l expan sion ofthe spec im en is opposed by friction at the platen/spec im en interface. This causes a confining forceor res traint at either end of the specimen. Th e effectof this fo rce reduces with distance from the plateni.e. towa rd s the centre of the spec im en, with theres ult rhar a cylindrica l specimen of a duct il e material of say, mild stee l w ill plastically deform in to abarre l shape, and the s id es w ill not stay straight , asin Fig. 2. 11 .Continued load in g of ductil e mater ia ls to hi gherand hi gher stress w ill simpl y resu lt in a flatter andfl atter disc i.e. more and more pl as tic deforma tion,bur no failure in rh e sense of cracking or breakdownof structure. In fact the area is increasin g, and there-fore very hi gh loads are required to keep the tru estress (see section 2.5) increasing. Tests can thereforeeas ily reac h the capac ity of the tes t machin e.2.6.2 FA ILUREOF BRID LE MATER IALSFa ilure stresses of brittle materia ls in compressionare much higher than those in tension- up to twentytimes higher for some materials, e.g. concrete. Thisresults from a ve ry different crack in g and failuremec hani sm. Cracking is a pu lling apar t of two surfaces, and therefore occurs by the ac ti on of a tensilestra in . In uni-axial compressive loading, the stra in s22

l

t

Compressivestress and strain

Lateral tensi lec::=:> stra in

Stable crackdevelopment andgrowth

Fig. 2.12 Multiple crack pallem in a bril! le n z a t e i < ~ l in compress ion leading to higher strength than intension.

in the d irection of lo adin g are obv iously compressive and it is the la tera l strain s that are tensile(Fig. 2. 12 ). T he cracks are form ed perpendicular torh ese stra in s, i.e. parallel to the load direction. Asin gle small crack will not immediately grow rocause fai lure, :111d a whole network of cracks nee dsto be formed, grow and intersect befo re comp letemater ia l breakdown occurs. Thi s requires a muchhi gher stress than that necessa ry to ca use the si nglefa ilure crack under tensile loa ding.T here is a further effect resu lting from rh e fric tion res tra int of the platens discussed above thatcauses the fa ilure stress (i.e. the apparent comp res sive strength ) to be dependent on the spec im engeo metry, pec ifica ll y the height/w idth ra ti o. In thep:n t of the spec im en nea r the platen, thi s restraintopposes and reduces rh e latera l tensile st rain. Thi sin creases the lo ad required for co mpl ete brea kclow n, i.e. failure (in effect, this part of the spec im enis und er a tri-axial compressive st ress system).Th e effect of the restraint reduces with di stancefrom the plate n (fig. 2.13). Short fat specim enswi ll have most of their volume exper ienc in ghi gh restraint, whereas the cen tral part of longer,thinner spec im ens will be nearer to a uni -ax ial stresssystem, and will therefore fa il at a lower averageapp lied st ress.The typical effect of the height/width ratio isshown in Fig. 2. 14 from tes ts on concre te; thestrength (i.e . the failure st ress) expressed relative tothat at a height/width ra ti o of 2. We will di sc ussmeasurement of the compressive strength of concretein more detail in Chap ter 2 1.


25/55

l

t

} . .eg1on of h1ghrestraint fromplaten friction}

Reg ion of lower::l restraint from platenfriction

Fig. 2.13 Variation of restraint from platen friction inco mpression test lead ing to the size effect oncomp ress ive fai lure stress.

1.8

1.60:0c 1.42:'u; 1.2;o Width;:: 1

0.80 1 2 3 4

Height:width rat ioFig. 2.14 Th e effect of height/width ralio on !heompress ive strength of bri tt le materials.

2. 7 Behaviour under constantload - creepCo nstant loa d o r str ess is a very co mm o n occ urrence, e.g. th e stress du e to th e se lf-w eight of astru cture. Materia ls respond to th is stress by ani m mediate strain defor mation, no rm a II y e lastic,fo llowed by a n increase in strain with time, ca lledcreep. Typica l behaviour is ill ustra ted in Fig. 2.15.A str ess app l ied at time t 1 and ma inta ined a t ao nsta nt leve l unti l remova l a t time t2 res ults in : an initi a l elas tic strain o n stress app lica tion (related

to the str ess by th e modu lus of elastici ty) an increase in thi s stra in du e to creep during th e

pe ri o d of co nsta nt stress- fairl y rap id at first butth e n at a dec reas ing rate

Mechanical properties of solids

Elastic recovery___._....;C_reep recovery

)-o


26/55

amentals

PrimarycreepSecondarycreep

Failu re, creeprupture""" Tertiarycreep

Time2.16 Sub-div isions o f creep curves.

tertiary creep:ex ponent, wh ich usua lly liesbetw een 3 and 8at hi gh stress leve ls, afte r aperi od o f tim e (whi ch can bevery length y) there may be anin creas in g rate with tim e leading to fa ilu re, a process knownas creep rupture. Thi s onl yocc urs if the stress is hi gh,typ ica lly more than 70-80%of th e failure str ess meas uredin a sho rt -term test.

so me situ a ti ons, th e stra in is co nstant e.g. able str etched betwee n two fi xed supports or aensioned bo lt cl amp ing two metal pla tes toge ther.e stress reduces with time, as show n in Fig. 2. J 7,process known as stress relaxa tion . In ex tremeth e stress redu ces to ze ro, i.e. th e ca bl e oro lt beco me slac k.During creep and the stress relaxa t ion th e matera l is, 111 effect, fl ow in g, a lbeit a t a very slow ra te.

c t IU5 t, Tim eenen Stress relaxationU5

t, Time. 2.1 7 Schematic o f stress relaxation at constanttra in .

It therefore ap pea rs to be behav ing somewhat li kea liquid . Such mi xe d so lid /liquid behav iour is ca ll edviscoelas ticity; we w ill be discuss ing thi s more detailin Chap ter 5.2.8 Behaviour under cyclicloading - fatigue2.8. 1 FAT IGUE LIFE AND S/N CURVESCyc li c loa ding is ve ry common, e.g. wind and wav eload in g, ve hi cle loa ding on roads and bridge s. Wecan define the charac teristics of th e loading as show nin Fig. 2. 18, in which:p = period of loadingfrequency = lip (e.g. in cyc les/sec or Hertz)

cr lll.X = max imum applied stress


27/55

Fro m resring, it has bee n found th at: N is ind ependent o f frequ ency (except a r veryhi gh frequ encies, above 1 kH z) N is dependent o n the stress ra nge (5) ra th er than

th e individual va lu es o f 0 111 .\X o r 0 11, 11 , pr ov id ed 0 111_ ,o r 0 11 , ; 11 does nor approac h the yie ld strength . Forexa mp le, the fa tigue li fe und er a st ress cyc lingbetw ee n -5 0 a nd +5 0 MPa is th e sa me as tha tund er stress cycling betw een +25 and + 125 MPa a hi gher stress range res ults in a sho rter fa t igueli fe , i.e. N increases with d ecreas ing S. T he rela tions hip is of ten o f th e form

S.N ' = C (2.21 )wh ere a and C are constants (a is betw een 0 .12a nd 0.0 7 fo r mo st materi a ls).

The fatigue perfo rmance of mate ri a ls is no rma llygiven as SIN curves i.e. g raph s o f str ess range (5)vs. fat ig ue li fe (N). Typica l SIN curves for mi ld steelan d a cop per a lloy a re show n in Fig. 2.19. Fatiguelives a re o ften very long (e .g. th o usa nd s, tens o ftho usa nd s, o r hundreds of th o usa nd s of cyc les) soa log sca le is norm a lly used for N.

The indi vidua l da ta points show n for mi ld steelillustr a te the co nsiderab le sca tt er that is obtainedfrom res t prog ra mm es . Apart fro m the o bv ious supe rior perfo rm ance of stee l, the best-fit line th roug hthe data shows a d isco nti nui ty ar about 240 MPal107cycles w here it becomes parallel to th e x-ax is forhig her fa tigue lives. T his mea ns th a t at va lues of Sbe lo w 240 MPa the fa tigue life is infinite, wh ich isvery use ful for des ign purposes . T his str ess range isca lled th e fa tigue endurance limit, a nd is a typ ica lcha rac te ri stic of ferrous meta ls. No n-ferro us meta ls

500Mild steel

-ro 4000.. VJ 300Ql _ ! i L I _ ~ e _ ~ ~ ~ ~ ~ ~ ~ _ e ! i ~ i __ _________ _Ol 200enen


28/55

entalsa ilure w ill occur w hen thi s sum reac hes I. T hisill be ac hieved w ith con t inued cyc lic load ing,w hi c h o ne o r a ll of n 1, n 2 o r n 1 may increase,epend ing o n the na tu re o f th e load ing . T he ge nera lpressiOn of M iner's ru le is th a t fo r fa ilure:

(2.22)

.9 Impact loadingtr uctu res a nd components of structures can beub jec ted to ve ry rap id rates of app licat ion oftress a nd s tr a in in a nu m be r of circ um sta nces,uch as exp los ions, miss ile o r vehicle impac t, a nds la m . Ma te ria ls ca n respo nd to such im pac ta ding by:a n ap parent inc rease in e las ti c mod ulus, bu r th i sis a third or fou rt h o rd er effec t only - a 10 4 rimesincrease in loadi ng ra te g ives o nl y a 10 % inc reasein e las ti c mod ulusa n inc rease in br ittl e be hav io ur, lead ing to fas tbrittle frac ture in norma ll y du cti le ma ter ia ls . Thi sca n be very da nge ro us - we thin k we a re usinga d uct ile mater ia l th a t has a high wo rk to frac turea nd g ives wa rning of fai lure, but thi s reac t s toim pac t load ing like a brittle mate ria l. T he effec tis enh a nce d if th e mate ri a l co nta ins a pre-ex istingdefect such as a crac k .

he lat te r effect ca nn o t be p redicted by ex tr a po la tg th e resu lts of la bo ra to ry tens ile or co m pr essiones ts such a those desc r ibed ea rlier, a nd im pac t tes toced ur es have bee n deve lo ped to assess the be

av io u r of spec ime ns co nta ining a mac hin ed no tch,h ic h acts a loca l stress raiser. T he C ha rpy test fo reta ls is a goo d examp le of such a tes t . In thi s, a

eavy pendu lum is released a nd strikes the sta nd a rdpec im en a t th e botto m o f i ts sw ing (Fig. 2.2 1). T hepec imen brea ks a nd the ene rgy needed fo r th e fracure is de te rmin ed fro m the difference between th etar ting a nd fo llow- th ro ugh pos itio ns o f th e pendu m . The energy a bsor bed in th e frac ture is ca llede C ha rp y im pac t va lue. As we have di sc ussedrlier in th e chap te r, br ittle ma ter ia ls req ui re lessnergy fo r fa ilur e th a n du ctil e ma te ria ls, a nd a npac t va lue of 15J is no rm a lly used as a somew hat

rbitrary di visio n between th e two, i.e. br itt le materls have a va lue below th is, a nd du cti le mater ia lsbove.An exa m ple o f the use o f th e rest is in deter min inghe effec t o f tempera tu re on du ct ile/brittle behav io ur.a ny ma ter ia ls th at a re du cti le a t no rm a l tempe rares have a tend ency to brit tleness a t red ucing

6

Fig. 2.2 1 Charpy impact test spec im en ((rumdimensions specified in BS eN ISO 148-3 :2008).

5>-OlQjcQ)t5-a.(U

.r:::0

150 Transit iontemperatureBrittle

4100

50

15 - - - - - - - - -I

+ - - - ~ - - ~ ~ T ~ ~ - - ~ - - - - - - 75 - 50 - 25 0 25 50 75Temperature (deg C)

Fig . 2.22 Varia tion of the Char{J)' impact energ y o fa steel with temperature (afier Ro llas on , / 9(, 1 .

tempera tu res. T his effect fo r a part ic ul a r stee l isshow n in Fig. 2.22 . Th e decrease in du ctility w ithfa llin g tempera tu re is ra pi d, with the 15J d ivisionocc urring at abo ut -20C, whi ch is ca lled rh e transi-tion temperature. It wo uld , fo r exam pl e, mean th a tth is steel sho uld not be used in such st ructu res aso il pr odu ction insta lla tions in Arcti c co nditi o ns.Impac t behav io ur and fas t frac tu re a re a n im po rt ant pa rt o f th e subj ect ca lled fractu re mechan ics,whi ch see ks to desc ribe a nd predict how a nd whycrac kin g and frac tu re occ ur. We will co nsider thisin more dera il in Cha pter 4.


29/55

2.10 Variability, characteristicstrength and the Weibulldistribution

Engin eers are continua ll y faced with un certaint y.T hi s may be in the es timati on of the loa ding on astructure (e .g. what is the des ign loa d due to a hurrican e that has a sma ll but finite chance of occurringso metime in the next 100 yea rs?), analys is (e .g. whatass umptions have bee n made in the computer modelling and are they val id ?) or with the constructionmaterials themsel ves . When dea ling with un certaintyin materials, with natura l materials such as timberwe have to cope with nature's own va riations, whichca n be la rge . With manufactured materia ls, no matterhow we ll and care full y the production process iscont ro lled, they a ll ha ve some inherent var iabilityand are th erefore not uni fo rm. Furthermore, whenca rr ying out tes ts on a se t o f samples to assess thisva riability there wi ll also be some una vo id abl eva ri ati on in the tes ting procedure itse lf, no matterhow carefull y the tes t is carried out o r how skilfulthe opera ti ve . Clea rl y there must be procedures todea l with thi s un ce rta inty and to ensure a satisfacto ry ba lance betwee n sa fety and economy. Stru cturalfa ilure ca n lead to loss of li fe, but th e constru cti oncos ts must be acceptable.In this sec tion, after some ba sic sta ti stica l consid erations for describing variabi lity, we will di sc usstwo approac hes to coping with varia tions of strength- characteristic strength and the Weibull di stribution . We will take strength as be ing the ultimate orfa ilu re stress of a mater ia l as meas ured in , say, atension, compress ion or bending tes t (a lthough theargum ent s a pply equ a lly to other pro perties suchas the yield or proo f stress o f a materi al).2. 10.1 DESCRIPTIONS OF VARIABILITYA se ries of tests on nominall y id entica l specim ensfrom either the same or successive batches of materia l usuall y gives va lu es of strength th at are equa llyspread about the mean value wi th a norma l orGa uss ian distr ibution, as show n in Fig. 2.23.T he mean va Iue, 0 "" is clefi ned as th e a ri th meticave rage of a ll the res ults, i.e.:


30/55

damen tals

- 3s

2.1 %

- $

0 .1%

s 2sFailure stress (cr)

g. 2.25 Proportion o f resu lts in th e reg ions o f th el distribution curve.

impo rt a nt properties of thi s eq ua tio n a re:The c urve encloses th e whole population of da ta,a nd therefore no t surpri singly, in teg rat ing th ea bove eq ua tion betwee n th e limits of -oo a nd+co gives a n answer o f 1, o r 100 if th e pro babi li rydensity is ex pressed as a p erce ntage.50 % of th e res ul ts fa ll below th e mea n a nd 50 %above, but a lso , as show n in Fig. 2.25: 68. 1% of results lie within o ne sta nd ar ddev ia tio n of the mea n 95 .5% of results lie within tw o sta nd ard

dev ia ti o ns o f th e mea n 99 .8% of res ult s lie within three sta nd arddev ia ti ons of th e mea n.CHARACTERISTIC STRENGTHgua ra nteed minimum va lue of stress below wh ich

sa mpl e w ill ever fai l is imposs ibl e to define- theture of the no rm a l di stributi o n curv e mea ns th a tere wi ll a lwa ys be a chance, a lbeit very small, o ffa ilure below a ny str ess va lue. A va lu e o f str esslied th e characteristic strength is th erefo re used ,

is defined as th e str ess be low which a n acly sma ll numbe r of resul ts w ill fa ll. Enginee r-g judgement is used to define 'acceptabl y sm a ll '.th i s is ve ry sma ll , th en th ere is a ve ry low ri sk ofilure, but th e low str ess wil l lea d to increaseda rea a nd hence grea ter cos t . If it isgher, th en th e structure ma y be c hea per but th erean increased ri sk of fa ilure.Clea rl y a bala nce is th erefo re required betwee na nd eco nom y. For ma ny ma ter ial s a s tr esslow which 1 in 20 of th e res ul ts occurs is co ned acceptable, i.e. th ere is a 5 % fa ilu re ra te.of the normal di stributi o n curve showsa t th is str ess is 1.64 sta ndard devia tions be low

5% ofresults

MarginOchar Om

Failure stress (cr)

Fig. 2.26 Definition of characteristic strength (a ,.1._, )and margin for a 1 in 20 (5%) failure rate criterion.

th e mea n. This distance is ca lled th e marg in and so,as show n in Fig. 2.2 6:cha rac teri stic st rength = mea n strength - ma rg in

(2.27)where k, the standard devia tio n multiplica tion factor,is 1.64 in this case.

The value o f k varies acco rdin g to th e c hosenfailure ra te (Tab le 2.1), and , as we sa id above, judgement and consensus are used to ar r ive at a n acceptab lefailure rate. In practice, thi s is no t a lways the sa mein a ll circumstances; for exa mple, .5 % is typi ca lfor concrete (i.e. k = 1.64 ), an d 2% for timbe r (i.e.k = 1.96 ).There is a further step in dete rmining a n a llow

ab le stress fo r design purposes. The str ength dataused to determine th e mea n and sta nd a rd dev ia tionfor th e above ana lysis w ill norm a ll y ha ve beenob ta in ed from labo ra to ry tests on sma ll specimens,which genera ll y w ill have no apparent defects o rdam age . Th ey there fo re represent th e best th at canbe ex pected fro m th e mate ri a l in idea l o r near id ea lcircumstances. In prac tice, st ructu ra I eIemen ts andmembers co ntai n a la rge vo lume o f mate ri a l, which

Table 2.1 Va lu es of k, th e stand ar d deviatio nmultiplication factor, fo r various fa ilure ratesFai lure rate (%)50161052

k

011.281.641.962.33


31/55

has a grea ter chance of containing man ufac turingand hand ling defects. Thi s size effec t is taken intoaccoun t by reducing th e characteristic strength bya partial materials' safety fac tor, YmIt i nor ma l prac ti ce for Ym o be give n as a va lu egreater th an one, so the characteristic strength hasto be divided by y"' to give th e a llowab le stress.Hence:

a llowab le design stress= characteristic strength/y"'= (mean strength - marg in )/y"' (2.28)As with th e failure rate, the va lu e of Ym is based onkn ow ledge and experience of the performance of

Construction Materials, 4th Ed_Section1

Documents

Transcript of Construction Materials, 4th Ed_Section1