4.4 Continuum Thermomechanics

Transcription

1 4.4 Contnuum Themomechancs The classcal themodynamcs s now extended to the themomechancs of a contnuum. The state aables ae allowed to ay thoughout a mateal and pocesses ae allowed to be eesble and moe fa fom themal and mechancal eulbum. Some schools of thought would ueston whethe entopy s a state functon at all unde these condtons. Hee, we smply accept the fact that t s. Ths s pat of the atonal themodynamcs appoach and s geneally accepted n the sold mechancs communty The Fst Law The fst law of themodynamcs s, n ate fom, P ext Q * U K (4.4.) whee P ext s the powe of the extenal foces, * Q s the ate at whch heat s suppled (called the themal powe, the non-mechancal powe, o the ate of themal wok), U s the ate of change of the ntenal enegy and K s the ate of change of knetc enegy. The supescpt * s used hee and n what follows to ndcate ates of change of uanttes whch ae not state functons. Recall fom Pat III, En 3.8.2, the mechancal enegy balance, Elmnatng P ext and K fom these euatons leads to Heat supply P P K ext nt (4.4.2) * P Q U nt (4.4.3) It s conenent to wte the total heat supply to a fnte olume of mateal as an ntegal oe the olume. Ths s done by defnng the heat flux to be the ate at whch heat s conducted fom nteo to exteo pe unt aea, Fg The ate of heat enteng s thus nds. Let thee also be a souce of heat supply nsde the mateal, fo s example a adato of heat. Let d be the ate of such heat supply, whee the scala s the heat souce, the ate of heat geneated pe unt olume. Thus, wth the degence theoem, * Q dd d (4.4.4) 42

2 Fgue 4.4.: heat flux ecto and nomal ecto to a suface element Recall also fom Pat III, Ens , the stess powe Pnt σ : dd (4.4.5) Combnng Ens , and expessng the stan enegy ate n the fom of an ntegal (see Pat III, En ) leads to dd dd d Snce ths holds fo all olumes, one has the local fom σ : u d (4.4.6) n ds σ : d d u The Fst Law (4.4.7) The Second Law Entopy The entopy S( x, t) s defned as the scala popety S sd (4.4.8) whee s s the specfc entopy o entopy densty. The change n entopy s due to two uanttes. Fst, ey lke the heat tansfeed nto a body, En , defne the entopy ()* supply S to be the ate of entopy nput, S s nds sd s (4.4.9) whee s s the entopy flux though the element suface and s s entopy supply due to souces wthn the element. Futhe, the entopy flux s taken to be popotonal to the heat flux, and the popotonalty facto s the ecpocal of the non-negate scala absolute tempeatue (and smlaly fo the densty s and the heat supply densty ) so that, usng the degence theoem, 422

3 S nds s d d d d (4.4.) ()* Defne the entopy poducton S to be the dffeence between the ate of change of entopy and the entopy supply: S S S (4.4.) The second law of themodynamcs states that the entopy poducton s a non-negate uantty, The Clausus-Duhem Ineualty Thus one has the Clausus-Duhem neualty: S (4.4.2) S d dt sd d d d (4.4.3) In local fom, the Clausus-Duhem neualty eads as (ntoducng a specfc entopy ()* poducton, s ) s s d (4.4.4) o, eualently { Poblem }, s s d ( ) 2 The Second Law (4.4.5) Ths s the contnuum statement of the Second Law The Dsspaton Ineualty Elmnatng neualty d (and ) fom both the fst and second laws leads to the dsspaton s s u σ : d ( ) Dsspaton Ineualty (4.4.6) 423

4 ()* The tem s s the specfc dsspaton (o ntenal dsspaton) and s denoted by the symbol. The Clausus-Duhem neualty can smply be wtten as s (4.4.7) Multplyng En acoss by the densty leads to s (4.4.8) s u σ : d ( ) Each tem hee has unts of powe pe unt (cuent) olume. The tem nsde the fst backet s called the mechancal dsspaton (pe unt olume). The tem nsde the second backet s the dsspaton due to tempeatue gadents,.e. heat flow, and s called the themal dsspaton (pe unt olume). Note that the themal dsspaton wll always be poste f and ae of opposte sgn. Integatng oe a olume leads to s d σ : d u sd d (4.4.9) dsspaton mechancal dsspaton themal dsspaton ()* The tem s n En s often denoted by the symbol and also temed the dsspaton. Ths s a dsspaton pe unt olume. When the defomatons ae small, the olume changes ae neglgble. When the defomatons ae appecable, howee, the olume and densty change, and t s bette to wok wth specfc uanttes such as. The dsspaton neualty s n tems of the ntenal enegy. In tems of the specfc fee enegy u s, one has s s σ : d ( ) (4.4.2) The Clausus-Plank Ineualty In many applcatons the themal dsspaton s ey much smalle than the mechancal dsspaton (n fact t s zeo n many mpotant applcatons see secton below). If ths s the case then the themal dsspaton ate can be gnoed, and one has the stonge fom of the second law, n tems of ntenal enegy and fee enegy, 424

5 s u σ : d s σ : d Clausus-Plank neualty (4.4.2) whch s known as the Clausus-Plank neualty. Eualently, one can ague that the pocesses of mechancal dsspaton and heat flow ae ndependent, so that each ae sepaately eued to be non-negate, agan leadng to En Ths ssue wll be exploed moe fully n Pat IV, whee t wll ndeed be shown that the themal dsspaton s ey often decoupled fom the mechancal dsspaton, wth both beng eued to be sepaately poste. Usng the fst law, En can be ewtten n the altenate fom s d (4.4.22) whch s an eoluton euaton fo s (showng how t eoles oe tme) Specal Themodynamc Pocesses Some mpotant lmtng pocesses ae consdeed next. Reesble Pocesses In a eesble pocess, s. The Clausus-Plank neualty now becomes u s σ : d o σ : d s (4.4.23) Fom the Fst Law, d s (4.4.24) Ths s the entopy supply (wth zeo tempeatue gadents) and coesponds to the ( ) classcal themodynamc (fo whch, also, ) expesson ds Q /. Isentopc Condtons Fo an sentopc pocess, the entopy s constant and emans constant, so s. In ths case, the dsspaton s 425

6 u σ : d ( ) (4.4.25) Isothemal Condtons In an sothemal pocess, the absolute tempeatue emans constant,. Ths can be acheed, fo example, by keepng the mateal s suoundngs at constant tempeatue, and loadng the mateal ey slowly, so that any tempeatue dffeences whch ase between the mateal and suoundngs ae allowed to dsappea. The Clausus-Plank neualty becomes Eulbum Condtons (4.4.26) s u σ : d o σ : d As mentoned n 4.2.3, a mateal whch s unaffected by extenal condtons has no wok done to t o heat suppled and the fst law then states that the ntenal enegy s constant. In that case, when the entopy has eached a maxmum and the dsspaton s zeo, thee s no moe change n any of the state aables, and eulbum has been eached. Adabatc Condtons In an adabatc pocess, o. Ths can be acheed, fo example, by ey apd loadng, so that thee s no tme fo heat exchange wth the suoundngs. Unde these condtons (and takng also ), the fst law eads σ : d u (ecall that the ntenal enegy change s eual to the wok done n an adabatc pocess). The dsspaton neualty educes to o s s (4.4.27) s (4.4.28) If the pocess s both adabatc and sentopc, then s. An adabatc eesble pocess s eualent to an sentopc eesble pocess Summay A summay of the themomechancal theoy, showng the aous laws and elatons whch ae noled, and how they ae nteconnected, s gen n Fg below. 426

7 Balance of Momentum F ma tds bd d s dσ b Wok-Enegy Pncple W W K ext ext nt P P K nt t ds b d σ: dd d s Fst Law W Q U K * Pext Q U K tds bd nds d s s ud d Second Law () ( ) S S S ()* S S S ()* s d sd ds d n ()* s s d 2 s σ : dd dd d u d σ : d d u Clausus-Duhem (Dsspaton) Ineualty s suσ : d ()* s s σ: d ()* Clausus-Planck Ineualty ()* s su σ : d ()* s s σ : d Fgue 4.4.2: Themomechancs 427

8 4.4.7 The use of Themomechancs n Deelopng Mateal Models Mateal models and the conseuences of the laws of themomechancs wll be dscussed n depth n Pat IV. Hee, as an ntoducton, the case of small-stan elastc/themoelastc mateals wll be touched upon befly. The classcal themodynamc expesson fo the wok done to a system s W pdv. Ths was genealsed to the contnuum statement fo the powe exeted on an nfntesmal element, σ : d. In the same way, the knematc aable of the classcal themodynamc system, V, s genealsed to the case of a contnuum by consdeng the state to be a functon of the stans. When the stans ae small, the ate of defomaton s eualent to the tme ate of change of the small stan tenso: The dsspaton neualtes ae then d ε, d (4.4.29) s u,, s u σ : ε o s,, s σ : ε (4.4.3) Reesble Pocess In the case of eesble pocesses: su,, s,, s u σ: ε o s σ: ε (4.4.3) Consde now a mateal whose state s completely defned by the set of state aables (, ), so the fee enegy state functon s (, ). Ths defnes the themoelastc mateal. The ntenal enegy s expessed n canoncal fom as u u( s, ) (see secton 4.3). One then has u u u s and s (4.4.32) 428

9 Thus u u s, s u u s : s σ ε ε o s, s : σ ε ε (4.4.33) Consde now the fee enegy euaton hee (the agument whch follows apples also to the ntenal enegy euaton). The state aables ae (, ) and the othe popetes ae state functons of these aables; these nclude all the tems nsde the backets, that s, s,, and also the patal deates. Fo any patcula state, these popetes wll hae cetan alues. On the othe hand, no matte the state, the aables,,, can (n theoy) be assgned abtay alues: negate, zeo o poste. The tems whch pemultply, ae completely ndependent of these aables ndeed, ths fact s bult nto the model unde consdeaton: the state s a functon of (, ) only and s not, fo example, dependent on the alues of,,,. On the othe hand, s not a state functon and n fact may be a functon of the tempeatue gadents. The only way that En can be satsfed n geneal, then, s fo (4.4.34) and the followng elatons must hold (compae these wth and 4.3.8): u u, and s s, (4.4.35) Consttute Relatons fo Small-Stan Reesble Pocesses Note that the densty (and olume) changes fo small stans may be neglected, so that the densty n can be taken to be the cuent densty o the densty n the undefomed confguaton,. 429

10 4.4.8 Themomechancs n the Mateal Fom The dsspaton neualty s deed hee fo the case of the mateal descpton. The Fst Law In ode to ewte the enegy balance euatons n mateal fom, fst ntoduce the scalas (what follows s analogous to the defntons of tacton and stess wth espect to the cuent and efeence confguatons) Q ( n) ( N) n Q N (4.4.36) Hee s the Cauchy heat flux of En , defned pe unt cuent suface aea ds wth outwad nomal n, and Q the Pola-Kchhoff heat flux, defned pe unt efeence suface aea ds and outwad nomal N. The ate of heat tansfe nto the mateal can now be wtten as ethe of s nds Q NdS (4.4.37) T Usng Nanson s fomula, Pat III, En , nds JF NdS, the Cauchy and Pola- Kchhoff heat flux ectos ae elated though Q J F. The combnaton of the mechancal enegy balance wth the fst law,.e. En , then eads (see also Pat III, En ) FdV DQdV RdV V V V S P : u dv (4.4.38) V whee RdV d, o, n local fom, V P : F DQ R u (4.4.39) o S : E DQ R u (4.4.4) Note that, compang the spatal and mateal foms, V DQdV dd, DQ Jd, DQ d (4.4.4) 43

11 The Second Law Analogous to En , the second law can be expessed n mateal fom as d dt V Q R sdv D dv dv V V (4.4.42) o, analogous to 4.4.6, one has the dsspaton neualty Isothemal Condtons In an sothemal pocess, The second of these can be wtten as P : F ( Q Gad ) s s u (4.4.43) P : F s u o P : F (4.4.44) PF : wth (4.4.45) whee the ate of fee enegy and the dsspaton ae pe unt efeence olume Objectty By defnton, the scalas heat Q, ntenal enegy U, entopy S and tempeatue ae objecte, that s they eman unchanged unde an obsee tansfomaton It follows that the heat flux ecto s also objecte, tansfomng accodng to By defnton, the ecto entopy flux s s objecte, that s t tansfoms accodng to Poblems. Show that d d ( ) 2 2. Show that the elaton Q J F s consstent wth the elaton 4.4.4, DQ Jd. 43