ES 240 Solid Mechanics

Transcription

1 ES4 Solid Mhani Fall ES 4 Solid Mhani Joot J Vlaak Shool of Enginring and Applid Sin Harvard Univrity Th not ar largly bad on our not put togthr by Prof Suo whn h taught ES 4 in 6 and on our not dvlopd by Prof Vlaak for ES 46 and ES4 (7, 9) 9/5/ Linar Elatiity-

2 ES4 Solid Mhani Fall Elmnt of Linar Elatiity Rfrn Txt: A lai txtbook i Thory of Elatiity, by SP Timohnko and JN Goodir, MGraw-Hill, Nw York Matrial partil: A olid i mad of atom, ah atom i mad of ltron, proton and nutron, and ah proton or nutron i mad of Thi kind of dription of mattr i too dtaild, and i not ud in th thory of latiity Intad, w dvlop a ontinuum thory, in whih a matrial partil ontain many atom, and rprnt thir avrag bhavior Diplamnt fild Any onfiguration of th matrial partil an b ud a a rfrn onfiguration Th diplamnt i th vtor by whih a matrial partil mov rlativ to it poition in th rfrn onfiguration If all th matrial partil in th body mov by th am diplamnt vtor, th body a a whol mov by a rigid-body tranlation If th matrial partil in th body mov rlativ to on anothr, th body dform For xampl, in a bnding bam, matrial partil on on fa of th bam mov apart from on anothr (tnion), and matrial partil on th othr fa of th bam mov toward on anothr (omprion) In a vibrating rod, th diplamnt of ah matrial partil i a funtion of tim W labl ah matrial partil by it oordinat ( x y, z) tim t, th matrial partil ( x, y, z) ha th diplamnt u ( x y, z, t) v ( x, y, z, t) in th y-dirtion, and w ( x y, z, t), in th rfrn onfiguration At, in th x-dirtion,, in th z-dirtion A funtion of patial oordinat i known a a fild Th diplamnt fild i a tim-dpndnt vtor fild It i omtim onvnint to writ th oordinat of a matrial partil in th rfrn onfiguration a ( x, x, x ), and th diplamnt vtor a u ( x, x, x, t), u ( x, x, x, t), u ( x, x, x, t) If w pla markr on a body, th motion of th markr viualiz th diplamnt fild and it variation with tim Strain fild Givn a diplamnt fild, w an alulat th train fild Conidr two matrial partil in th rfrn onfiguration: partil A at ( x, y, z) and partil B at ( x dx, y, z) + In th rfrn onfiguration, th two partil ar ditan dx apart At a givn tim t, th two partil mov to nw loation Th x-omponnt of th diplamnt of partil A i u ( x y, z, t),, and that of 9/5/ Linar Elatiity-

3 ES4 Solid Mhani Fall partil B i u ( x dx, y, z, t) u ( x dx, y, z, t) u( x, y, z, t) + Conquntly, th ditan btwn th two partil longat by + Th axial train in th x-dirtion i ( x + dx, y, z, t) u( x, y, z, t) u u ε x dx x Thi i a train of matrial partil in th viinity of ( x, y, z) at tim t Th funtion ε ( x, y, z t) i a omponnt of th train fild of th body Th har train i dfind a follow Conidr two lin of matrial partil In th rfrn onfiguration, th two lin ar prpndiular to ah othr Th dformation hang th inludd angl by om amount Thi hang in th angl dfin th har train, γ W now tranlat thi dfinition into a train-diplamnt rlation Conidr thr matrial partil A, B, and C In th rfrn onfiguration, thir oordinat ar A( x y, z) B( x + dx, y, z), and C( x y dy, z) mov by u ( x, y, z, t) and partil C by u ( x y dy, z, t) lin AC about axi z by an angl u ( x, y + dy, z, t) u( x, y, z, t) dy,, x,, + In th dformd onfiguration, in th x-dirtion, partil A u y Similarly, th dformation rotat lin AB about axi z by an angl v ( x + dx, y, z, t) v( x, y, z, t) dx v x ( x, y + dy, z t) u, ( x, y, z t) u,, + Conquntly, th dformation rotat Conquntly, th har train in th xy plan i th nt hang in th inludd angl: u v γ xy + y x For a body in th thr-dimnional pa, th train tat of a matrial partil i dribd by a total of ix omponnt Th train rlat to th diplamnt a 9/5/ Linar Elatiity-

4 ES4 Solid Mhani Fall u ε x, x γ xy v ε y, y u v +, γ y x yz w ε z z v w +, z y γ zx w u + x z Anothr dfinition of th har train rlat th dfinition hr by ε / With thi nw dfinition, w an writ th ix train-diplamnt rlation natly a u u i j ε + ij x j xi xy γ xy Mor gnrally, onidr a body rfrrd to an orthogonal t of ax a hown in th figur Th diplamnt at any point in thi body during dformation ar fully dribd by th diplamnt omponnt (u, u, u ), whih ar aumd to b ontinuou funtion of th oordinat (x, x, x ) Conidr two arbitrary nighboring point P (x, x, x ) and P (x +dx, x +dx, x +dx ) in th untraind body and aum P and P undrgo diplamnt (u, u, u ) and (u +du, u +du, u +du ), rptivly Expanding th diplamnt at P in a Taylor ri about P, w find: du u x dx + u x dx + u x dx, du u x dx + u x dx + u x dx, du u x dx + u x dx + u x dx Th matrix [u i,j ] i th rlativ diplamnt matrix and i in gnral not ymmtri Hr th omma indiat diffrntiation with rpt to th orrponding oordinat Th quation x P P d o d P (x +dx +u +du,x +dx +u +du,x +dx +u +du ) P (x +u,x +u,x +u ) x Diplamnt in dformd body 9/5/ Linar Elatiity-4 x

5 ES4 Solid Mhani Fall an b rwrittn a follow: du u x dx + u + u x x dx + u + u x x dx + u u x x dx + u u x x dx, du u + u x x dx + u dx x + u + u x x dx + u u x x dx + u u x x dx, du u + u x x dx + u + u x x dx + u dx x + u u x x dx + u u x x dx Eah of th trm in th xprion ha a impl phyial maning and rprnt ithr a omponnt of th train matrix [ε ij ] or a omponnt of th rotation matrix [ω ij ], whr: ε ij ( u i, j + u j,i), ω ij ( u i, j u j,i ) Not that th train matrix i ymmtri, whil th rotation matrix i anti-ymmtri It follow thrfor that if w dompo th rlativ diplamnt matrix into ymmtri and antiymmtri part, th ymmtri part rprnt pur dformation, whil th anti-ymmtri part rprnt a rigid body motion 9/5/ Linar Elatiity-5

6 ES4 Solid Mhani Fall Th tat of tr at a matrial partil Imagin a thr-dimnional body Th body may not b in quilibrium (g, th body may b vibrating) Th matrial proprty i unpifid (g, th matrial an b olid or fluid) Imagin a matrial partil inid th body What tat of tr do th matrial partil uffr from? To talk about intrnal for, w mut xpo thm by drawing a fr-body diagram Rprnt th matrial partil by a mall ub, with it dg paralll to th oordinat ax Cut th ub out from th body to xpo all th for on it 6 fa Dfin tr a for pr unit ara On ah fa of th ub, thr ar thr tr omponnt, on normal to th fa (normal tr), and th othr two tangntial to th fa (har tr) Now th ub ha ix fa, o thr ar a total of 8 tr omponnt A fw point blow gt u organizd Notation: ij Th firt ubript ignifi th dirtion of th vtor normal to th fa Th ond ubript ignifi th dirtion of th tr omponnt W labl th oordinat a ( x, y, z) It i omtim onvnint to writ th oordinat a (, x x ) x, Sign onvntion On a fa who normal i in th poitiv dirtion of a oordinat axi, th tr omponnt i poitiv whn it point to th poitiv dirtion of th axi On a fa who normal i in th ngativ dirtion of a oordinat axi, th tr omponnt i poitiv whn it point to th ngativ dirtion of th axi 9/5/ Linar Elatiity-6

7 ES4 Solid Mhani Fall Equilibrium of th ub A th iz of th ub hrink, th for that al with th volum (gravity, inrtia fft) ar ngligibl Conquntly, th for ating on th ub fa mut b in tati quilibrium Normal tr omponnt form pair Shar tr omponnt form quadrupl Conquntly, only 6 indpndnt tr omponnt ar ndd to drib th tat of tr of a matrial partil Writ th ix tr omponnt in a by ymmtri matrix: Tration vtor: Imagin a plan inid a matrial Th plan ha th unit normal vtor n, with thr omponnt n, n, n Th for pr ara on th plan i alld th tration Th tration i a vtor, with thr omponnt: t t t t Qution: Thr ar infinit many plan through a point How do w dtrmin th tration vtor on all th plan? Anwr: You an alulat th tration vtor from t t t n n n W now th mrit of writing th omponnt of a tat of tr a a matrix Alo, th ix omponnt ar indd uffiint to haratriz a tat of tr, bau th ix omponnt allow u to alulat th tration vtor on any plan Proof of th tr-tration rlation Thi tration-tr rlation i th onqun of th quilibrium of a ttrahdron formd by th partiular plan and th thr oordinat plan Dnot th ara of th four triangl by A, A x, A y, A z Rall a rlation from gomtry: 9/5/ Linar Elatiity-7

8 ES4 Solid Mhani Fall A An, A An, A An x x y y z z Balan of th for in th x-dirtion rquir that t A A + A + A x xx x xy y xz z Thi giv th dird rlation t x n + n + n xx x xy y xz z Thi rlation an b rwrittn uing th indx notation: t n + n + n It an b furthr rwrittn uing th ummation onvntion: t jn j Similar rlation an b obtaind for th othr omponnt of th tration vtor: t n + n + n t n + n + n Th thr quation for th thr omponnt of th tration vtor an b writtn olltivly in th matrix form, a giv in th bginning of thi tion Altrnativly, thy an b writtn a t n i ij j 9/5/ Linar Elatiity-8

9 ES4 Solid Mhani Fall Hr th ummation i implid for th rpatd indx j Th abov xprion rprnt thr quation Exampl: tr and tration A matrial partil i in a tat of tr with th following omponnt: (a) Comput th tration vtor on a plan intrting th ax x, y and z at, and, rptivly (b) Comput th magnitud of th normal tr on th plan () Comput th magnitud of th har tr on th plan (d) Comput th dirtion of th har tr on th plan Solution W nd to find th unit vtor normal to th plan Thi i a problm in analytial gomtry Th quation of a plan intrting th ax x, y and z at, and i x + y z + Altrnativly, a plan an b dfind by a givn point on th plan, x, and a unit vtor normal to th plan, n For any point x on th plan, x x i a vtor lying in th plan, o that n ( x x ) n x, or ( x x ) + n ( y y ) + n ( z z ) y z A omparion of th two quation of th plan how that th normal vtor i in th dirtion,, Normalizing thi vtor, w obtain th unit vtor normal to th plan: 6/ 7 n / 7 / 7 9/5/ Linar Elatiity-9

10 ES4 Solid Mhani Fall 9/5/ Linar Elatiity- (a) Th tration vtor on th plan i 7 56 / 7 / 7 / 7 / 7 / 7 6 / z y x t t t (b) Normal tr on th plan i th tration vtor projtd on to th normal dirtion of th plan n t n () and (d) Th har tr in th plan i a vtor: 4 7 t n n τ Th dirtion of thi vtor i th dirtion of th har tr on th plan Th magnitud of th har tr i 686

11 ES4 Solid Mhani Fall Str fild and momntum balan Imagin th thr-dimnional body again At tim t, th matrial partil ( x y, z) tat of tr ( x, y, z t) Dnot th ditributd xtrnal for pr unit volum by ( x, y, z, t) ij,, i undr a b An xampl i th gravitational for, b z ρg Th tr and th diplamnt ar timdpndnt fild Eah matrial partil ha th alration vtor u i / t Cut a mall diffrntial lmnt, of dg dx, dy and dz Lt ρ b th dnity Th ma of th diffrntial lmnt i ρ dxdydz Apply Nwton ond law in th x-dirtion, and w obtain that [ ( ) xx ( x, y, z,t )] ( ) yx ( x, y, z,t ) dydz xx x + dx, y, z,t [ ] +dxdz yx x, y + dy, z,t [ ( ) zx ( x, y, z,t )] + b x dxdydz ρdxdydz u +dxdy zx x, y, z + dz,t Divid both id of th abov quation by dxdydz, and w obtain that t xx x + yx y + zx z + b x ρ u t 9/5/ Linar Elatiity-

12 ES4 Solid Mhani Fall Thi i th momntum balan quation in th x-dirtion Similarly, th momntum balan quation in th y- and z-dirtion ar xy x + yy y + zy z + b y ρ v t xz x + yz y + zz z + b z ρ w t Whn th body i in quilibrium, w drop th alration trm from th abov quation Uing th ummation onvntion, w writ th thr quation of momntum balan a x ij j + b j ui ρ t 9/5/ Linar Elatiity-

13 ES4 Solid Mhani Fall Hook law for iotropi matrial For an iotropi, homognou olid, only two indpndnt ontant ar ndd to drib it lati proprty: Young modulu E and Poion ratio ν In addition, a thrmal xpanion offiint α haratriz train du to tmpratur hang Whn tmpratur hang by Δ T, thrmal xpanion au a train α ΔT in all thr dirtion Th ombination of multiaxial tr and a tmpratur hang au train ( ) ε xx E ν + xx yy zz + αδt ( ) ε yy E ν + yy zz xx + αδt ( ) ε zz E ν + zz xx yy + αδt Th rlation for har ar γ xy ( + ν ) ( + ν ) ( + ν ) E, γ xy yz E, γ Rall th notation ε /, and w hav xy γ xy yz zx E zx ε xy + ν E xy, ε yz + ν E yz, ε zx + ν E zx Indx notation and ummation onvntion Th ix tr-train rlation may b writtn a ε ij + ν E ij ν E kkδ ij Th ymbol δ ij tand for whn i j and for whn i j W adopt th onvntion that a rpatd indx impli a ummation ovr, and Thu, + + kk Homognity Whn talking about homognity, you hould think about at lat two lngth al: a larg (maro) lngth al, and a mall (miro) lngth al A matrial i aid to b homognou if th maro-al of intrt i muh largr than th al of mirotrutur A fibr-rinford matrial i rgardd a homognou whn ud a a omponnt of an airplan, but hould b thought of a htrognou whn it fratur mhanim i of intrt Stl i 9/5/ Linar Elatiity-

14 ES4 Solid Mhani Fall uually thought of a a homognou matrial, but rally ontain numrou void, partil and grain Iotropy A matrial i iotropi whn rpon in on dirtion i th am a in any othr dirtion Mtal and rami in polyrytallin form ar iotropi at maro-al, vn though thir ontitunt grain of ingl rytal ar aniotropi Wood, ingl rytal, uniaxial fibr rinford ompoit ar aniotropi matrial Exampl: a rubbr layr prd btwn two tl plat A vry thin lati layr, of Young modulu E and Poion ratio ν, i wll bondd btwn two prftly rigid plat A thin rubbr layr btwn two thik tl plat i a good approximation of th ituation Th thin layr i omprd btwn th plat by a known normal tr z Calulat all th tr and train omponnt in th thin layr Solution Th tr tat at th dg of th lati layr i ompliatd W will nglt thi dg fft, and fou on th fild away from th dg, whr th fild i uniform Thi mphai mak n if w ar intrtd in, for xampl, th diplamnt of on plat rlativ to th othr Of our, thi mphai i miplad if w ar onrnd of dbonding of th layr from th plat, a dbonding may initiat from th dg, whr tr ar high By ymmtry, th fild ha only th normal omponnt and ha no har omponnt Alo by ymmtry, w not that x y Bau th lati layr i bondd to th rigid plat, th two train omponnt vanih: ε ε x y Uing Hook law, w obtain that or ( ν ν ) ε x x y z, E ν x z ν 9/5/ Linar Elatiity-4

15 ES4 Solid Mhani Fall Uing Hook law again, w obtain that ε z E ( ) ( + ν )( ν ν ν ) z y x z ( ν ) E Conquntly, th lati layr i in a tat of uniaxial train, but all thr tr omponnt ar nonzro Whn th lati layr i inompribl, ν 5, it annot b traind in jut on dirtion, and th tr tat will b hydrotati 9/5/ Linar Elatiity-5

16 ES4 Solid Mhani Fall Summary of Elatiity Conpt Playr: Fild Str tnor: tr tat mut b rprntd by 6 omponnt (dirtional proprty) Strain tnor: train tat mut b rprntd by 6 omponnt (dirtional proprty) Str fild: tr tat vari from partil to partil (poitional proprty) Str fild i rprntd by 6 funtion, xx x, y, z;t ( ), xy x, y,z;t ( ) Diplamnt fild i rprntd by funtion, u( x, y, z;t),v( x, y,z;t), w( x, y, z;t) Strain fild i rprntd by 6 funtion, ε xx ( x, y, z;t ),ε xy ( x, y, z;t ), Rul: lmnt of olid mhani Momntum balan Dformation gomtry Matrial law Complt quation of latiity: Partial diffrntial quation Boundary ondition - Prrib diplamnt - Prrib tration Initial ondition: For dynami problm (g, vibration and wav propagation), on alo nd prrib initial diplamnt and vloity fild Solving boundary valu problm: ODE and PDE Idalization, analytial olution: g, SP Timohnko and JN Goodir, Thory of Elatiity, MGraw-Hill, Nw York Handbook olution: RE Ptron, Str Conntration Fator, John Wily, Nw York, 974 nd dition by WD Pilky, 997 Brut for, numrial mthod: finit lmnt mthod, boundary lmnt mthod 9/5/ Linar Elatiity-6

17 ES4 Solid Mhani Fall D Elatiity: Colltd Equation Momntum balan x x + xy y + xz z + b x ρ u t yx x + y y + yz z + b y ρ v t zx x + zy y + z z + b z ρ w t Strain-diplamnt rlation Hook Law ε x u x, ε yz v z + w y ε y v y, ε zx w x + u z ε z w z, ε xy u y + v x ε x E[ x ν ( y + z) ] + αδt, ε zx + ν E zx ε y E [ y ν ( z + x) ] + αδt, ε xy + ν E xy ε z E z ν ( x + y) Str-tration rlation [ ] + αδt, ε yz + ν E yz t t t n n n 9/5/ Linar Elatiity-7

18 ES4 Solid Mhani Fall D Elatiity: Equation in othr oordinat Cylindrial Coordinat (r,, z) Momntum balan u, v, w ar th diplamnt omponnt in th radial, irumfrntial and axial dirtion, rptivly Inrtia and body for trm ar ngltd r r + r r + rz z + r r r r + r + z z + r r rz r + z r + z z + rz r Strain-diplamnt rlation ε r u r, ε z v z + w r ε v r + u, ε zr w r + u z ε z w z, ε r u r + v r v r Sphrial Coordinat (r,, φ) i maurd from th poitiv z-axi to a radiu; φ i maurd round th z-axi in a righthandd n u, v, w ar th diplamnt omponnt in th r,, φ dirtion, rptivly Inrtia trm ar ngltd 9/5/ Linar Elatiity-8

19 ES4 Solid Mhani Fall Momntum balan r r + r r + rφ rin φ + ( r ot r φ r ) r r + r + φ rin φ + r ( φ ) ot + r rφ r + φ r + φ rin φ + ( r + ot rφ φ ) Strain-diplamnt rlation ε r u r ε ϕ w r wot + v in ϕ ε v r + u ε ϕr w w r + u r in ϕ w ε ϕ + uin + vo r in ϕ ε r u r + v r v r 9/5/ Linar Elatiity-9

20 ES4 Solid Mhani Fall Exampl: Lamé Problm A an xampl of a boundary valu problm, onidr a phrial avity in a larg body, ubjtd to rmot hydrotati tnion Th ymmtry of th problm mak th u of a phrial oordinat ytm onvnint a Lit nonzro quantiti th radial diplamnt u, th radial tr r, two qual hoop tr φ, th radial train ε r, two qual hoop train ε ε φ Thy ar all funtion of r b Lit quation U th bai quation ht and implify taking into aount th ymmtry of th problm: d r r Equilibrium quation: + dr r Dformation gomtry: ε r du dr ; ε u r Matrial law: ε r ( E r ν ); ε E ( ν) ν r Rdu to a ingl ODE Th abov ar a t of 5 quation for 5 funtion of r You an follow a numbr of approah to olv thm Wll tak th approah blow W want to obtain a ingl quation in th radial tr, r From th quilibrium quation, w xpr in trm of r : r d r r + dr Thn w u th matrial law to xpr both train in trm of r : 9/5/ Linar Elatiity-

21 ES4 Solid Mhani Fall ε r E ε E ( ν ) d r r νr dr r d dr r ( ν ) + ( ν ) r W an liminat u from th two quation for dformation gomtry Thi rult in an quation in trm of th two train: ( r ) dr ε r d ε / Expr thi quation in trm of th radial tr, and w gt d r 4 d r + dr r dr d Solv th ODE Thi i an quidimnional quation Th olution i of th form r r m Subtitut r r m into th ODE, and w find two root: m and m - Conquntly, th full olution i B r A +, r whr A and B ar ontant to b dtrmind by th boundary ondition Th hoop tr i givn by B A r Th boundary ondition ar Prribd rmot tr: r S a r Tration-fr at th avity urfa: r a r a Upon dtrmining th two ontant A and B, w obtain th tr ditribution a a r S, S + r r 9/5/ Linar Elatiity-

22 ES4 Solid Mhani Fall Str onntration fator Not that th tangntial tr i nonzro nar th avity urfa, whr it tr rah a maximum Th tr onntration fator i th ratio of th maximum tr ovr th applid tr In thi a, th tr onntration fator i / 9/5/ Linar Elatiity-

23 ES4 Solid Mhani Fall A not on ompatibility Th ix indpndnt omponnt of train at ah point ar ompltly dtrmind by th diplamnt fild u (u, v, w) A a rult it i not poibl to hoo th ix train omponnt indpndntly thy hav to atify a numbr of quation that ari whn th diplamnt omponnt ar liminatd from th quation dfining th train omponnt In othr word, on an ak th following qution: If I hoo ix train omponnt (or ix funtion orrponding to th omponnt), do thr xit a orrponding diplamnt fild Th anwr to thi qution i impl: if I hoo th funtion arbitrarily, thr do not gnrally xit a orrponding diplamnt fild Th ondition that nd to b atifid by th train omponnt in ordr to gt a orrponding diplamnt fild an b found by liminating th diplamnt omponnt u, v, w from th dfinition of th train From th dfinition of th variou train omponnt, w gt: ε xx y u x y, from whih w find that ε yy x v y x, γ xy x y u x y + v y x, ε xx y + ε yy x γ xy x y Two mor rlation of th am kind an b found by ylial prmutation of th lttr x, y and z From th drivativ ε xx y z u x y z, γ xz y u y z + w x y, γ xy z u y z + v x z, γ yz x v x z + w x y, it follow furthr that ε xx y z x γ yz x + γ xz y + γ xy z 9/5/ Linar Elatiity-

24 ES4 Solid Mhani Fall Two mor quation of thi kind an b drivd through ylial prmutation of th oordinat Th quation ar known a th ompatibility quation or th ondition of ompatibility bau thy tll you undr what ondition a ral diplamnt fild xit On an how that for imply onntd rgion (i, matrial without hol), th ompatibility quation ar nary and uffiint to aur that th diplamnt xit and ar ingl-valud If th rgion i multiply onntd, additional ondition nd to b impod It hould b mphaizd that th quation ar ompltly uprfluou whn you trat th diplamnt a variabl in th problm; you only nd thm if you want to u th train omponnt intad 9/5/ Linar Elatiity-4

25 ES4 Solid Mhani Fall Prinipal Str Imagin a matrial partil in a tat of tr Th tat of tr i fixd, but w an rprnt th matrial partil in many way by utting ub in diffrnt orintation For any givn tat of tr, it i alway poibl to ut a ub in a uitabl orintation, uh that th tr omponnt on all th ub fa ar normal to th fa, and thr ar no har tr on th ub fa Th ub fa ar alld th prinipal plan, th normal vtor to th fa th prinipal dirtion, and th tr on thm th prinipal tr Exampl Uniaxial tnion Equal biaxial tnion Hydrotati prur Pur har i th am tat of tr a th ombination of pulling and pring in 45 Givn a tr tat, how to find th prinipal tr? Whn a plan i th prinipal plan, th tration on th plan i normal to th plan, namly, th tration vtor t mut b in th am dirtion a th unit normal vtor n Lt th magnitud of th tration b On th prinipal plan, th tration vtor i in th dirtion of th normal vtor, t n Writ in th matrix notion, and w hav t t t n n n Hr both t and n ar vtor, but i a alar rprnting th magnitud of th prinipal tr Rall that th tration vtor i th tr matrix tim th normal vtor, o that n n n n n n 9/5/ Linar Elatiity-5

26 ES4 Solid Mhani Fall Thi i an ignvalu problm Whn th tr tat i known, i, th tr matrix i givn, olv th abov ignvalu problm to dtrmin th ignvalu and th ignvtor n Th ignvalu i th prinipal tr, and th ignvtor n i th prinipal dirtion Linar algbra of ignvalu Bau th tr tnor i a by ymmtri matrix, you an alway find thr ral ignvalu, i, prinipal tr,,, W ditinguih thr a: () If th thr prinipal tr ar unqual, th thr prinipal dirtion ar orthogonal (g, pur har tat) () If two prinipal tr ar qual, but th third i diffrnt, th two qual prinipal tr an b in any dirtion in a plan, and th third prinipal dirtion i normal to th plan (g, pur tnil tat) () If all th thr prinipal tr ar qual, any dirtion i a prinipal dirtion Thi tr tat i alld a hydrotati tat Maximum normal tr Why do w ar about th maximum normal tr? Chalk i mad of a brittl matrial: it brak by tnil tr, not by har tr Whn halk i undr bnding, th tnil tr i along th axial dirtion of th halk, o that th halk brak on a plan normal to th axial dirtion Whn halk i undr torion, th maximum tnil tr i 45 from th axial dirtion, o that it brak in a dirtion 45 from th axial dirtion (Th fratur urfa of th halk undr torion i not a plan, bau of om D fft) W ar about th prinipal tr bau brittl matrial fail by tnil tr, and w want to find th maximum tnil tr Lt ordr th thr prinipal tr a a b a b Thi ordring tak into onidration th ign: a ompriv tr (ngativ) i mallr than a tnil tr (poitiv) On an arbitrary plan, th tration vtor may b dompod into two omponnt: on omponnt normal th plan (th normal tr), and th othr omponnt paralll to th plan (th har tr) Obviouly, whn you look at a plan with a diffrnt normal vtor, you find diffrnt normal and har tr You will b dlightd by th following thorm: Of all plan, th prinipal plan orrponding to ha th maximum normal tr Maximum har tr Why do w ar about th maximum har tr? Mot mtal ar dutil matrial: thy fail by plati yilding Whn a matrial i undr a omplx tr tat, it 9/5/ Linar Elatiity-6

27 ES4 Solid Mhani Fall i known mpirially that yilding firt our on a plan with maximum har tr To find th maximum har tr and th partiular plan, you ar hlpd by th following thorm: Th maximum har tr i τ max ( a ) / Th maximum har tr τ max at on a plan with th normal vtor 45 from th prinipal dirtion n a and n A proof of th abov thorm i outlind blow: Conidr a ytm of oordinat that oinid with thr orthogonal dirtion of th prinipal tr,,, Thn onidr an arbitrary plan who unit normal vtor ha omponnt n, a b n, n in thi oordinat ytm Th omponnt of th tr tnor in thi oordinat ytm i a b Thu, on th plan with unit vtor ( n, n, n ), th tration vtor i ( an, bn, n ) Th normal tr on th plan i n + n a + bn n W nd to maximiz undr th ontraint that n + n + n Th har tr on th plan τ i givn by n τ ( a n ) + ( b n ) + ( n ) a n + b n ( + n ) W nd to maximiz τ undr th ontraint that n + n + n 9/5/ Linar Elatiity-7

28 ES4 Solid Mhani Fall Coordinat tranformation and tnor Th dirtion-oin matrix rlating two ba In D-pa, lt, and b an orthonormal bai, namly, δ i j ij Th ba vtor ar ordrd to follow th right-hand rul Lt,, b a nw orthonormal bai, namly, α β δ αβ Lt th angl btwn th two vtor i and b α i α Dnot th dirtion oin of th two vtor by l i α o i α i α W follow th onvntion that th firt indx of l iα rfr to a oordinat in th old bai, and th ond to a oordinat in th nw bai For th two ba, thr ar a total of 9 dirtion oin W an lit l iα a a by matrix By our onvntion, th row rfr to th old bai, and th olumn to th nw bai Not that l iα i th omponnt of th vtor projtd on th vtor α i W an xpr ah nw ba vtor a a linar ombination of th thr old ba vtor: α l + l + l α α α If you ar tird of writing um lik thi, you abbrviat it a l, α iα i with th onvntion that a rpatd indx impli ummation ovr,, Bau th um i th am whatvr th rpatd indx i namd, uh an indx i alld a dummy indx Similarly, w an xpr th old bai a a linar ombination of th nw bai: i l + i + li li Uing th ummation onvntion, w writ mor onily a 9/5/ Linar Elatiity-8

29 ES4 Solid Mhani Fall i l iα α Tranformation of omponnt of a vtor du to hang of bai Lt f b a vtor It i a linar ombination of th ba vtor: f f, i i whr f, f, f ar th omponnt of th vtor, and ar ommonly writtn a a olumn Conidr th vtor pointing from Cambridg to Boton Whn th bai i hangd, th vtor btwn Cambridg and Boton rmain unhangd, but th omponnt of th vtor do hang Lt f, f, f b th omponnt of th vtor f in th nw bai, namly, f f α α Rall th tranformation btwn th two t of bai, i l, w writ that iα α f f i i f l i iα α A omparion btwn th two xprion giv that f f l i α iα In matrix notation, th omponnt olumn in th nw bai i th tranpo of th dirtionoin matrix tim th omponnt olumn in th old bai Similarly, w an how that f i liα f α, or in matrix notation, th omponnt olumn in th old bai i th dirtion-oin matrix tim th omponnt olumn in th old bai Tranformation of tr omponnt du to hang of bai Th tr tnor,, drib th tat of tr uffrd by a matrial partil Rprnt th matrial partil by a ub Th tr omponnt ar th for pr unit ara on 6 fa of th ub Th tr tnor i rprntd by a by ymmtri matrix Th tat of tr of a matrial partil i a phyial objt, and i indpndnt of your hoi of th bai (i, how 9/5/ Linar Elatiity-9

30 ES4 Solid Mhani Fall you ut a ub to rprnt th partil) Howvr, th omponnt of th tr tnor do dpnd on your hoi of th bai How do w tranform th tr omponnt whn th bai i hangd? Conidr th tr tat of a matrial partil, and th tration vtor on a givn plan In th old bai, and, dnot th omponnt of th tr tat by, th omponnt of th unit vtor normal to th plan by n j, and th omponnt of th tration vtor on th plan by t i Uing th ummation onvntion, w writ th tration-tr quation a ij t i n ij j Rall that w obtaind thi rlation by th balan of for on a ttrahdron In th languag of linar algbra, w all th tr a a linar oprator that map th unit normal vtor of a plan to th tration vtor ating on th plan Similarly, in th nw bai,,, dnot th omponnt of th tr tat by αβ, th omponnt of th unit vtor normal to th plan by n β, and th omponnt of th tration vtor on th plan by t α For balan rquir that t (a) α αβ n β W now xamin th rlation btwn th omponnt in th old bai and tho in th nw bai Th tration i a vtor, o that it omponnt tranform a t α l i αt Inrt i ti ijn j into th abov, and w obtain that t l i n Th unit normal vtor tranform a n j l n jβ β t Conquntly, w obtain that α liα ijl jβn β α α ij (b) j Equation (a) and (b) ar valid for any hoi of th plan Conquntly, w mut rquir that l i l αβ α ij jβ Thu, th tr-omponnt matrix in th nw bai i th produt of thr matri: th tranpo of th dirtion-oin matrix, th tr-omponnt matrix in th old bai, and th dirtion-oin matrix 9/5/ Linar Elatiity-

31 ES4 Solid Mhani Fall 9/5/ Linar Elatiity- A pial a: th nw bai and th old bai diffr by an angl around th axi Th ign onvntion for follow th right-hand rul Th dirtion oin ar,,,, o, in,, in, o Conquntly, th matrix of th dirtion oin i [ ] o in in o iα l Th omponnt of a vtor tranform a o in in o f f f f f f Thu, o in in o f f f f f f f f + + Th omponnt of a tr tat tranform a o in in o o in in o Thu,

32 ES4 Solid Mhani Fall + + o + in + o in in + o o + in in + Stat of plan tr o An vn mor pial a i that th tr omponnt out of th plan ar abnt, namly, + + o + + o in + o in in In Br Stion 74, th quation ar rprntd by graphially (i, Mohr irl) In thi a, i on prinipal dirtion, and th prinipal tr in thi dirtion i zro To find th othr two prinipal dirtion, w t, o that th two prinipal dirtion ar at th angl p from th - and -dirtion Thi angl i givn by tan p Th two prinipal tr ar givn by + ± + W nd to ordr th two prinipal tr and th zro prinipal tr in th dirtion Th maximum har tr i th half of th diffrn btwn th maximum prinipal tr and th minimum prinipal tr 9/5/ Linar Elatiity-

33 ES4 Solid Mhani Fall 4 Salar, vtor, and tnor Whn th bai i hangd, a alar (g, tmpratur, nrgy, and ma) do not hang, th omponnt of a vtor tranform a f f i l, α iα and th omponnt of a tnor tranform a l l αβ ij iα jβ Thi tranformation dfin th ond-rank tnor By analogy, a vtor i a firt-rank tnor, and a alar i a zroth-rank tnor W an alo imilarly dfin tnor of highr rank Multilinar algbra Th rul of th abov tranformation i bad on only on fat: th tr i a linar map from on vtor to anothr vtor You an gnrat a nw tnor from a linar map from on tnor to anothr tnor You an alo gnrat a tnor by a bilinar form, g, a bilinar map from two vtor to a alar Of our, a multi-linar map of vral tnor to a tnor i yt anothr tnor Conquntly, all tnor follow a imilar rul undr a hang of bai W writ thi rul again for a third-rank tnor: g g l l l αβγ ijk iα jβ kγ Invariant of a tnor A alar i invariant undr any hang of bai Whn th bai hang, th omponnt of a vtor hang, but th lngth of th vtor i invariant Lt f b a vtor, and f i b th omponnt of th vtor for a givn bai Th lngth of th vtor i th quar root of f i f i Th indx i i dummy Thu, thi ombination of th omponnt of a vtor i a alar, whih i invariant undr any hang of bai For a vtor, thr i only on indpndnt invariant Any othr invariant of th vtor i a funtion of th lngth of th vtor Thi obrvation an b xtndd to high-ordr tnor By dfinition, an invariant of a tnor i a alar formd by a ombination of th omponnt of th tnor For xampl, for a ymmtri ond-rank tnor ij, w an form thr indpndnt invariant: 9/5/ Linar Elatiity-

34 ES4 Solid Mhani Fall,, ii ij ij ij jk ki In ah a, all indi ar dummy, rulting in a alar Any othr invariant of th tnor i a funtion of th abov thr invariant Exri For a nonymmtri ond-rank tnor, giv all th indpndnt invariant Writ ah invariant uing th ummation onvntion, and thn xpliitly in all it trm Exri Giv all th indpndnt invariant of a third-rank tnor Th tat of train of a matrial partil i a ond-rank tnor In th old bai, th oordinat of a matrial partil ar ( x, x, x ) diplamnt fild ar u i ( x, x, x, t) partil ar ( x, x, x ), and th omponnt of th diplamnt fild ar u ( x, x, x, t) Rall that α l i αui, and th omponnt of th In th nw bai, th oordinat of th am matrial u and x l x j jβ β, o that α u x α β l iα ui x j x j x β l l iα jβ ui x j Thu, th gradint of th diplamnt fild, u / x, form th omponnt of a ond-rank tnor Conquntly, th tat of train, bing th ymmtri part of th diplamnt gradint, i a ond-rank tnor Whn th bai i hangd, th omponnt of th train tat tranform a ε ε l l αβ ij iα jβ i j 9/5/ Linar Elatiity-4

35 ES4 Solid Mhani Fall Hlmholtz fr nrgy of a rod undr uniaxial tnion Conidr a rod, initial lngth L and ro-tional ara A Whn a loading mahin appli a for f to th rod, th lngth of th rod bom L W rord th funtion f ( L) xprimntally, whih nd not b linar Whn th lngth hang from L to L + dl, th loading mahin do work fdl to th rod W modl an lati olid with a Hlmholtz fr nrgy, ( L T) dl, and th tmpratur inra by dt, th fr nrgy inra by F, Whn th lngth inra by df fdl SdT Hr S i th ntropy, whih an alo b maurd xprimntally by mauring th hat aborbd by th rod in a low loading pro, ds δq/t Dfin th tr and train a f, ε L L L A Dfin th fr nrgy dnity, w, a th fr nrgy pr unit volum, namly F w A L W an imilarly dfin th ntropy dnity S A L Hr w hav ud th initial ara and initial lngth to dfin th tr, th train, and th fr nrgy dnity With th dfinition, w an rwrit dw dε dt df fdl SdT a Th fr nrgy dnity i a funtion of th train and th tmpratur: ( ε T) w w, 9/5/ Linar Elatiity-5

36 ES4 Solid Mhani Fall For a givn olid, th funtion w ( ε,t) i dtrmind xprimntally On w know thi funtion, w an obtain th tr-train rlation by taking th diffrntiation: w ( ε,t ) ε T Similarly, w an alo alulat th ntropy dnity by w ( ε,t ) T ε W nxt rtrit ourlv to mall train, o that w an xpand th funtion w ( ε,t) into a Taylor ri in th train: w ( ε, T) w + wε + wε Th offiint w, w, w ar funtion of th tmpratur W will only go up to th quadrati trm in train Th tr i obtaind by taking partial diffrntiation: w + ε w Thu, w idntify w a Young modulu E W alo idntify w a th ridual tr o Ralling th xprimntal obrvation of thrmal xpanion, w may wih to writ thi ridual tr a w Eα ( T T rf ), whr α i th offiint of thrmal xpanion, and T rf i a rfrn tmpratur at whih th tr vanih In mot irumtan, E and α only dpnd on th tmpratur wakly, and ar rgardd a matrial ontant Th nrgy dnity thn bom w( ε,t ) w + o ε + Eε Th ntropy dnity i thn givn by 9/5/ Linar Elatiity-6

37 ES4 Solid Mhani Fall w o T dw o dt + d o dt ε + de dt ε Exprimntally, th ntropy dnity may b dtrmind by mauring hat apaity, namly, th hat aborbd by a unit volum of th olid pr unit hang in tmpratur, whn th train i hld ontant Thu, d dt T Uing th xprimntally maurd valu of, w an dtrmin w, w, w Th abov analyi an b gnralizd in many dirtion to aount for divr xprimntal obrvation For th tim bing, w will fou on th tr train rlation of a linarly lati olid, and mak on important gnralization: w will onidr an aniotropi olid undr a multiaxial tr tat W will drop th tmpratur dpndn, and throw in thrmal xpanion train at th nd of th analyi Whn w ar not partiularly intrtd in th tmpratur dpndn of th fr nrgy, following a larg body of litratur, w will all th Hlmholtz fr nrgy th lati nrgy Exampl: Elati nrgy dnity of a blok undr a impl har Conidr a blok of hight H and ro-tional ara A Subjt to a har for F, th blok har by an angl Whn th har angl hang from to + d, th loading mahin do th work FH d to th blok For an lati olid th work i tord a th lati nrgy in th blok Dfin th har tr and th (nginring) har train a τ F A, γ Th nrgy pr unit volum i a funtion of th har train, w( γ ) From th abov, w hav dw τdγ Th har tr i th diffrntial offiint of th nrgy dnity funtion Whn th blok i mad of a linarly lati olid, undr har load, th tr-train rlation i τ Gγ Conquntly, th nrgy dnity funtion i 9/5/ Linar Elatiity-7

38 ES4 Solid Mhani Fall w γ ( γ ) G Thi rult hold only for linar lati olid in pur har ondition Multi-axial tr tat Th advantag of th nrgy dnity funtion bom lar undr th multi-axial tr tat Th nrgy dnity i alar, and th train i a tnor Th nrgy dnity i a funtion of all omponnt of th tain tnor: (,, ) w w ε ε (For popl familiar with thrmodynami, w dal nrgy at a ontant tmpratur, o that th nrgy i th Hlmholz fr nrgy) Th omponnt of tr tnor ar diffrntial offiint: dw ε pq d pq That i pq w ( ε, ε, ) ε pq In linar latiity, w aum that tr i linar in train Thu, th nrgy dnity i a quadrati form of th tain tnor, writtn a w C ijkl εijε kl Hr th C ijkl offiint ar th omponnt of a fourth-rank tnor alld th tiffn tnor Without loing any gnrality, w an aum th following ymmtri: C C C C ijkl jikl ijlk klij If w ount arfully, w hould hav indpndnt omponnt for a gnrally aniotropi lati olid Th omponnt of th tr tnor ar linar in th omponnt of th train tnor: 9/5/ Linar Elatiity-8

39 ES4 Solid Mhani Fall pq C pqijεij W an alo invrt thi rlation to xpr th train in trm of th tr: ε pq S pqij ij Hr S pqij ar th omponnt of a fourth-rank tnor alld th omplian tnor Thy hav th am ymmtry proprti Str-train rlation in a matrix form W an alo writ th abov quation in anothr form Th tat of train i pifid by th ix omponnt: ε, ε, ε, γ, γ, γ x y z yz zx xy In thi ordr w will labl thm a ε, ε, ε, ε 4, ε 5, ε 6 Th ix train omponnt an vary indpndntly Th lati nrgy pr unit volum i a funtion of all th ix train omponnt, ( ) Thi i th nrgy dnity funtion Whn ah train omponnt hang w ε,ε,ε,ε 4,ε 5,ε 6 by a mall amount, dε i, th nrgy dnity hang by dw dε + dε + dε + 4 dε dε dε 6 Hr w u th nginring train for th har, rathr than th tnor omponnt W do o to avoid th fator in th abov xprion offiint of th nrgy dnity funtion: Eah tr omponnt i th diffrntial i w ε i If th funtion w( ε,ε,ε,ε 4,ε 5,ε 6 ) i known, w an dtrmin th ix tr-train rlation by th diffrntiation Conquntly, by introduing th nrgy dnity funtion, w only nd to pify on funtion, rathr than ix funtion, to dtrmin th tr-train rlation Linar lati olid: Th abov onidration apply to olid with linar or nonlinar trtrain rlation W now xamin linar lati olid For th tr omponnt to b linar in th train omponnt, th nrgy dnity funtion mut b a quadrati form of th train omponnt: 9/5/ Linar Elatiity-9

40 ES4 Solid Mhani Fall w ij ε i ε j ( ε ε + ε ε + ε ε ) i, j Hr ij ar 6 ontant Th ro trm om in pair, g, ( + )ε ε Only th ombination + will ntr into th tr-train rlation, not and individually W an all + by a diffrnt nam A onvnint way to ay that thr i only on indpndnt ontant i to jut lt W an do th am for othr pair, namly, ij ji Th matrix ij i ymmtri, with indpndnt lmnt Conquntly, ontant ar ndd to pify th latiity of a linar aniotropi lati olid Bau th lati nrgy i poitiv for any nonzro train tat, th matrix ij i poitiv-dfinit Rall that ah tr omponnt i th diffrntial offiint of th nrgy dnity funtion, i w/ ε i Th tr rlation bom i ij ε j j In th matrix notation, w writ ε ε ε ε 4 ε 5 ε 6 Th phyial ignifian of th ontant ij i now vidnt For xampl, whn th olid i undr a uniaxial train tat, ε, ε ε ε 4 ε 5 ε 6, th ix tr omponnt on th olid ar ε, ε, Th matrix ij i known a th tiffn matrix Invrting th matrix, w xpr th train omponnt in trm of th tr omponnt: 9/5/ Linar Elatiity-4

41 ES4 Solid Mhani Fall ε ε ε ε 4 ε 5 ε Th matrix ij i known a th omplian matrix Th omplian matrix i alo ymmtri and poitiv dfinit Th omponnt of th tiffn tnor rlat to th orrponding omponnt of th tiffn matrix a, C C 4 C, 44 Howvr, th orrponding rlation for omplian ar, S S 4 S, 44 4 Aniotropi latiity An iotropi, linar lati olid i haratrizd by two ontant (g, Young modulu and Poion ratio) to fully pify th tr-train rlation Som olid ar aniotropi, g, fibr rinford ompoit, ingl rytal Eah tr omponnt i a funtion of all ix train omponnt Conquntly, ontant ar ndd to pify th latiity of a linar aniotropi lati olid For a rytal of ubi ymmtry, uh a ilion and grmanium, whn th oordinat ar along th ub dg, th tr-train rlation ar ε ε ε ε 4 ε 5 44 ε 6 Th thr ontant, and 44 ar indpndnt for a ubi rytal Iotropi olid i a pial a, in whih th thr ontant ar rlatd, ( )/ 44 9/5/ Linar Elatiity-4

42 ES4 Solid Mhani Fall For a fibr rinford ompoit, with fibr in th x dirtion, th matrial i iotropi in th x - and x -dirtion Th matrial i aid to b tranvrly iotropi Fiv indpndnt lati ontant ar ndd Th tr-train rlation ar ε ε ε ε 4 ε ( )/ ε 5 6 Invariant and iotropi matrial If a matrial i iotropi, th lati nrgy dnity w hould b a funtion of th invariant of th train tnor For a linar matrial, w i quadrati in train Th train tnor an form only two invariant quadrati in it omponnt: ε ijεij and Conquntly, for an iotropi, linarly lati olid, th lati nrgy dnity tak th ( ε ) kk form w µε ( ) ijεij + λ ε kk, whr µ and λ ar known a th Lamé ontant Taking diffrntiation, w obtain th trtrain rlation: µε + λε δ ij ij kk ij A omparion with th tr-train rlation in th uniaxial tr tat how that Exri E νe µ, λ ( + ν ) ( + ν )( ν ) It i raonabl to rquir that th lati nrgy dnity b a poitiv-dfinit quadrati form of th train tnor That i, w > for any tat of train, xpt that w whn th train tnor vanih For an iotropi, linarly lati olid, onfirm that thi poitivdfinit rquirmnt i quivalnt to rquir that E > and < ν < / Exampl: tr in an pitaxial film Both ilion (Si) and grmanium (G) ar rytal of ubi unit ll Th dg lngth of th unit ll of Si i a Si 548Å, and that of G i 9/5/ Linar Elatiity-4

43 ES4 Solid Mhani Fall a G 5658Å A G film nm thik i grown pitaxially (i with mathing atomi poition) on th [] urfa of a µ m thik Si ubtrat Calulat th tr and train omponnt in th G film Th rptiv lati ontant ar (in GPa) Si: 658, 69, G: 85, 48, Solution Bau th Si ubtrat i muh thikr than th G film, th train in th ubtrat ar muh mallr than tho in th film W will nglt th mall train, and aum that th ubtrat i undformd Lt axi b normal to th film urfa, and ax and b in th plan of th film, paralll to th ub dg of th rytal ll To rgitr on atom on anothr, G mut b omprd in dirtion and to onform to th undformd atomi unit ll iz of Si Th two in-plan train in th G film ar asi ag ε ε 4% a G Thr will b an longation normal to th film, ε All har train vanih Aording to > th gnralizd Hook law, th tr normal to th film urfa rlat to th train a ε + ε + ε Phyially it i vidnt that thr i no tr normal to th urfa of th film, Inrting into th abov xprion, w obtain that ε ε + % Th two in-plan tr omponnt ar qual, givn by ε + ε + ε, or + ε 9/5/ Linar Elatiity-4

44 ES4 Solid Mhani Fall Inrting th numrial valu, w obtain that may gnrat diloation in th film 56GPa Thi i a hug tr that Frrolati pha tranition Thi part go byond linar latiity Suppo w hav th following xprimntal obrvation A rytal ha a rtangular ymmtry at a high tmpratur Whn th tmpratur drop blow a ritial valu, T, th rytal undrgo a pha tranition Th rytal at a low tmpratur aquir a pontanou train in har Bau of th ymmtry, th har train an go both dirtion (Skth th gomtry) W modl thi rytal with a fr nrgy dnity 4 w( γ, T) A( T T ) γ + Bγ, 4 whr A and B ar poitiv ontant Du to ymmtry, th rytal i qually likly to har in two dirtion, o that w kp th vn powr in th train γ (Skth w a a funtion of γ at two tmpratur, on abov T and th othr blow T ) Whn T > T, th offiint of th lati olid, with th har modulu ( T ) bhavior of th rytal γ trm i poitiv, o that th rytal bhav lik uual A Th T 4 γ trm i unnary to drib th Whn T < T, th offiint of th γ trm i ngativ, and th nrgy i no longr minimal at γ Intad, th nrgy i minimal at two nonzro train, known a th pontanou train, ± γ In thi a, th nough Th tr-train rlation i ( γ, T) w τ A( T T ) γ + Bγ γ 4 γ trm will nur that nrgy go up again whn th train i larg (Skth thi funtion Mark th pontanou train and th hytri loop) Stting τ, w find th pontanou train: ( T T) B γ ± A / Bau th matrial i nonlinar, th har modulu i no longr a ontant, and i givn by 9/5/ Linar Elatiity-44

45 ES4 Solid Mhani Fall ( γ, T) τ µ A( T T ) + Bγ γ At th pontanou train, th har modulu i givn by ( T T) µ A Skth th har modulu a a funtion of th tmpratur, both blow and abov th ritial tmpratur 9/5/ Linar Elatiity-45