2.1 Constitutive Theory

Transcription

1 Secon.. Consuve Theory.. Consuve Equaons Governng Equaons The equaons governng he behavour of maerals are (n he spaal form) dρ v & ρ + ρdv v = + ρ = Conservaon of Mass (..a) d x σ j dv dvσ + b = ρ v& + b = ρ Conservaon of Momenum (..b) x d j q du σ : d dvq = ρ u& σ j d j = ρ Conservaon of Energy (..c) x d ogeher wh he symmery of sress (conservaon of angular momenum). Usng he knemac relaon ( ( ) ) v v j gradv + gradv d j = + x j x T d = (..) elmnang he rae of deformaon and reang he velocy as he unknown hese are 5 equaons n he 4 unknowns: densy ρ he hree velocy componens v he 6 ndependen sress componens σ j he hree hea flux componens q and he nernal energy u (he body force b s an exernal supply assumed o be known). Consuve Equaons The remanng equaons necessary o solve a parcular connuum mechancs problem are he consuve equaons. These equaons relae o parcular maerals for example he consuve equaons descrbng he behavour of waer wll be very dfferen o hose descrbng he behavour of a meal. Dfferen consuve equaons are also used for he same maeral; for example he equaons descrbng he flow of molen alumnum are dfferen o hose descrbng he response of alumnum a room emperaure. A physcal body s characersed by he hree basc felds he densy ρ he moon χ (here expressed by he velocy) and he emperaure θ (no nroduced no he equaons as ye). The remanng quanes on he oher hand he sress σ he hea flux q and he nernal energy u depend no only on he behavour of he maeral bu on he knd of maeral ha consues he body. { σ q u} are called consuve quanes and addonal hypoheses mus be nroduced o characerse he response of a parcular maeral he consuve equaons for example of he form Sold Mechancs Par IV 5

2 Secon. ( ρ χ θ ) q = f ( ρ χ θ ) u = f ( ρ θ ) σ = f (..3) 3 χ where f s a ensor funcon f s a vecor funcon and f 3 s a scalar funcon of he feld varables. For nhomogeneous maerals hese funcons can also depend explcly on poson x. Consuve equaons are also known as equaons of sae. The frs of..3 nvolvng emperaure and knec and knemacal varables s known as a hermal equaon of sae whereas he hrd nvolvng he nernal energy s known as a calorc equaon of sae. Unlke he physcal laws..-.. consuve laws mus be valdaed by expermen... Prncples of Consuve Theory When consrucng consuve equaons o model maeral behavour one mus ensure ha he followng are sasfed:. he prncple of maeral objecvy he consuve equaon mus be nvaran under an observer ransformaon (see and 4.. for hyperelasc elasc maeral. maeral symmery he consuve equaon mus be conssen wh any maeral symmeres whch are presen (see 4.5 for hyperelasc maeral 3. he second laws of hermodynamcs he consuve equaon mus no volae he dsspaon nequaly The frs wo of hese wll no be pursued formally here as hey have been dscussed n prevous Chapers. The hrd wll be a man consderaon n much of wha follows n laer secons: a se of felds { ρ χ θ} mus sasfy no only he balance laws and consuve equaons bu mus also sasfy he enropy nequaly Ths s known as he enropy prncple. ρs& + dvq ρs (..4)..3 The Smple Maeral In hs secon a very broad form of consuve relaon called he smple maeral model s derved. In order o derve wo furher gudng prncples are used:. The prncple of deermnsm The prncple of deermnsm saes ha he curren sae a a maeral parcle depends on he complee hermomechancal hsory of he enre maeral body. Sold Mechancs Par IV 5

3 Secon. In order o apply hs prncple frs nroduce he followng noaon: A ( A( (..5) Here s he curren me A ( denoes a funcon A wh ndependen varable s s wh s denong he lengh of me dreced no he pas. For example A () denoes he funcon a he presen me. Le Τ denoe a consuve quany such as sress or energy. Then he prncple saes ha a parcle a he curren me ) = I ( ρ χ θ ( ) Τ (..6) s< Here I s a funconal (a funcon of funcon aken over all parcles of he body and descrbes he maeral response n he broades possble way (broader han he hsory-ndependen funcons of..3). The densy n he curren confguraon ρ ) s relaed o he densy n he reference confguraon o ρ hrough he moon (see Eqn. 3..5) and so..6 can be smplfed ( ) = I ( ρ χ θ ) Τ (..7) s<. The prncple of local acon The feld varables can be expanded n Taylor seres: ρ χ( θ = ρ + ( ) + ( ) ( ) ) = χ( ) ) = θ ) ρ x + θ + ) ) ( ) + ( ) ( ) + ( ) ρ x θ ) ) + L ( ) + L ( ) + L (..8) The prncple of local acon saes ha he curren sae of a maeral parcle depends on he complee hsory of only a small neghbourhood of ha parcle. In ha case..7-8 lead o ( ) = I ρ Gradρ χ ( F θ Gradθ Τ (..9) s< Sold Mechancs Par IV 53

4 Secon. The densy and densy graden n he reference confguraon are smply consan funcons of and he dependence on hese can be ncorporaed no he explc dependence on so..9 can be expressed as ( ) = I χ ( F θ Gradθ Τ (..) s< Fnally a consuve equaon canno depend on he moon.e. on he curren poson x snce oherwse objecvy would no be sasfed so one has ( ) = I F θ Gradθ Τ (..) s< Ths s he consuve equaon for a smple maeral. I s also called a maeral of grade. One can also have consuve equaons wh dependence on hgher order gradens as appear n Eqn...8. Such maerals are called maerals of grade n n = 3K. The expansons n..8 are wh respec o he coordnaes; local effecs spaally bu long-erm effecs emporally. One can also expand he feld varables emporally (wh respec o ); n hs case one arrves a maeral models wh shor-erm memory of dfferen grades whch are approprae o dfferen classes of vscous flud. The explc dependence on n.. denong a non-homogeneous maeral can be omed for clary gvng ( ) ) = I F θ Gradθ Τ Smple Maeral (..) s< The hermoelasc maeral s a smple maeral for whch he funconal I reduces o a funcon a he curren me wh no hsory dependence: ) = f ( F θ Gradθ ) Τ (..3) A specal case s he Cauchy elasc maeral whose consuve relaon s emperaure- σ = σ F. ndependen ( ) A vscoelasc maeral s a smple maeral. For example one parcular form of he funconal I for a vscoelasc maeral would be ( ) dτ ( ) = f F τ ) θ τ )Gradθ τ ) Τ (..4) A maeral undergong plasc deformaon s no a smple maeral. For a smple maeral knowledge of he deformaon a all mes s enough o deermne he maeral s response. However n plascy knowledge of he deformaon alone s no suffcen snce one mus also know he proporon of elasc and permanen deformaon n any complee deformaon. Sold Mechancs Par IV 54

5 Secon...4 Inernal Varables The funconal form of he vscoelasc maeral n..4 s explc. One can also defne he funconal relaons mplcly n he form of dfferenal equaons. In ha case one mus supplemen he varables ( F θ Gradθ ) wh furher varables whch ncorporae he hsory-dependence of he maeral. These new varables are called nernal varables. The deformaon graden (and/or knemacal varable and emperaure are measurable and conrollable quanes. They are accessble o drec observaon and for hs reason hey are called exernal varables. The nernal varables on he oher hand are no conrollable or observable; hey are hdden. These addonal sae varables whch can be scalars vecors or ensors descrbe n some way aspecs of he nernal (mcro-) srucure of maerals assocaed wh rreversble/dsspave (non-elasc) effecs. The evoluon of he nernal varables over me as descrbed by dfferenal equaons replcaes ndrecly he hsory of he deformaon. For example consder a connuum elemen of a fbrous-ype maeral compressed o a sran and hen held here. The sran s he exernal knemacal varable. However hs does no gve all he nformaon for even hough he sran s now held fxed he fbres of he maeral mgh end o slde over each oher n an aemp o fnd more favourable posons so ha he force requred o hold he elemen a he fxed sran decreases.e. sress relaxaon occurs. There wll be frcon beween he fbres so he fbre slppage s accompaned by dsspaon. The amoun of slppage whch has occurred up o he curren me can be represened by an nernal varable. The response of he maeral can now be expressed n he form () = f ( F θ Grad θ α α K) Τ (..5) where he α are he nernal varables. Eqn...5 s hen supplemened by evoluon equaons for he nernal varables dfferenal equaons descrbng how hey change over me. The nernal varable approach s preferred o he explc funconal approach snce allows for a wder number of processes o be modelled; only n he case of lnear vscoelascy can he dfferenal equaons of he nernal varable approach be convered no explc funconals such as..4. The Prncple of Equpresence When usng nernal varables s helpful o follow he gudelne offered by he prncple of equpresence whch saes ha all consuve equaons for a maeral should nclude he same ls of ndependen varables unless he ncluson of one of more of hese varables s shown o be unnecessary or volaes some physcal law. For example one would have he same ls for boh he sress and nernal energy: Sold Mechancs Par IV 55

6 Secon. σ = f u = g ( F θgradθ α α ) ( F θgradθ α α ) (..6) Sold Mechancs Par IV 56