Rheological Models. In this section, a number of one-dimensional linear viscoelastic models are discussed.

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1 helgcal Mdels In hs secn, a number f ne-dmensnal lnear vscelasc mdels are dscussed..3. Mechancal (rhelgcal) mdels The wrd vscelasc s derved frm he wrds "vscus" + "elasc"; a vscelasc maeral exhbs bh vscus and elasc behavur a b lke a flud and a b lke a sld. One can buld up a mdel f lnear vscelascy by cnsderng cmbnans f he lnear elasc sprng and he lnear vscus dash-p. These are knwn as rhelgcal mdels r mechancal mdels. The Lnear lasc Sprng The cnsuve equan fr a maeral whch respnds as a lnear elasc sprng f sffness s (see Fg..3.) (.3.) The respnse f hs maeral a creep-recvery es s underg an nsananeus elasc sran upn ladng, manan ha sran s lng as he lad s appled, and hen underg an nsananeus de-sranng upn remval f he lad. Fgure.3.: he lnear elasc sprng The Lnear Vscus Dash-p Imagne nex a maeral whch respnds lke a vscus dash-p; he dash-p s a psncylnder arrangemen, flled wh a vscus flud, Fg..3. a sran s acheved by draggng he psn hrugh he flud. By defnn, he dash-p respnds wh a sranrae prprnal sress: (.3.) where s he vscsy f he maeral. Ths s he ypcal respnse f many fluds; he larger he sress, he faser he sranng (as can be seen by pushng yur hand hrugh waer a dfferen speeds). a nn-lnear hery can be develped by ncludng nn-lnear sprngs and dash ps Sld Mechancs Par I 88

2 Secn.3 Fgure.3.: he lnear dash-p The sran due a suddenly appled lad may be baned by negrang he cnsuve equan.3.. Assumng zer nal sran, ne has (.3.3) The sran s seen ncrease lnearly and whu bund s lng as he sress s appled, Fg Ne ha here s n mvemen f he dash-p a he nse f lad; akes me fr he sran buld up. When he lad s remved, here s n sress mve he psn back hrugh he flud, s ha any sran bul up s permanen. The slpe f he creep-lne s /. sress appled sress remved Fgure.3.3: Creep-ecvery espnse f he Dash-p The lnear relanshp beween he sress and sran durng he creep-es may be expressed n he frm ( ) J ( ), J ( ) (.3.4) J here s called he creep (cmplance) funcn ( J / fr he elasc sprng)..3. The Maxwell Mdel Cnsder nex a sprng and dash-p n seres, Fg Ths s he Maxwell mdel. One can dvde he al sran n ne fr he sprng ( ) and ne fr he dash-p ( ). qulbrum requres ha he sress be he same n bh elemens. One hus has he fllwng hree equans n fur unknwns: Sld Mechancs Par I 89

3 Secn.3,, (.3.5) T elmnae and, dfferenae he frs and hrd equans, and pu he frs and secnd n he hrd: Maxwell Mdel (.3.6) Ths cnsuve equan has been pu n wha s knwn as sandard frm sress n lef, sran n rgh, ncreasng rder f dervaves frm lef rgh, and ceffcen f s. Fgure.3.4: he Maxwell Mdel Creep-ecvery espnse Cnsder nw a creep es. Physcally, when he Maxwell mdel s subjeced a sress, he sprng wll srech mmedaely and he dash-p wll ake me reac. Thus he nal sran s ( ) /. Usng hs as he nal cndn, an negran f.3.6 (wh a zer sress-rae ) leads here s a jump n sress frm zer when he lad s appled, mplyng an nfne sress-rae. One s n really neresed n hs jump here because he crrespndng jump n sran can be predced frm he physcal respnse f he sprng. One s mre neresed n wha happens jus "afer" he lad s appled. In ha sense, when ne speaks f nal srans and sress-raes, ne means her values a, jus afer ; he sress-rae s zer frm n. T be mre precse, ne can deal wh he sudden jump n sress by negrang he cnsuve equan acrss he pn as fllws: ( / ) ( ) d ( ) d ( ) d ( ) ( ) ( ) ( ) ( / ) ( ) d In he lm as, he negral ends zer ( s fne), he values f sress and sran a,.e. n he lm as frm he lef, are zer. All ha remans are he values he rgh, gvng ( ) ( ), as expeced. One can deal wh hs sudden behavur mre easly usng negral frmulans r wh he Laplace Transfrm (see.4,.5) Sld Mechancs Par I 9

4 Secn.3 ( ) C ( ) (.3.7) The creep-respnse can agan be expressed n erms f a creep cmplance funcn: ( ) J ( ) where J ( ) (.3.8) When he lad s remved, he sprng agan reacs mmedaely, bu he dash-p has n endency recver. Hence here s an mmedae elasc recvery /, wh he creep sran due he dash-p remanng. The full creep and recvery respnse s shwn n Fg The Maxwell mdel predcs creep, bu n f he ever-decreasng sran-rae ype. There s n anelasc recvery, bu here s he elasc respnse and permanen sran. sress appled sress remved Fgure.3.5: Creep-ecvery espnse f he Maxwell Mdel Sress elaxan In he sress relaxan es, he maeral s subjeced a cnsan sran a. The Maxwell mdel hen leads { Prblem } ( ) ( ) where ( ) e /, (.3.9) Analgus he creep funcn J fr he creep es, () s called he relaxan mdulus funcn. The parameer s called he relaxan me f he maeral and s a measure f he me aken fr he sress relax; he shrer he relaxan me, he mre rapd he sress relaxan. Sld Mechancs Par I 9

5 Secn The Kelvn (Vg) Mdel Cnsder nex he her w-elemen mdel, he Kelvn (r Vg) mdel, whch cnsss f a sprng and dash-p n parallel, Fg I s assumed here s n bendng n hs ype f parallel arrangemen, s ha he sran experenced by he sprng s he same as ha experenced by he dash-p. Ths me,,, (.3.) where s he sress n he sprng and s he dash-p sress. lmnang, leaves he cnsuve law Kelvn (Vg) Mdel (.3.) Fgure.3.6: he Kelvn (Vg) Mdel Creep-ecvery espnse If a lad s appled suddenly he Kelvn mdel, he sprng wll wan srech, bu s held back by he dash-p, whch cann reac mmedaely. Snce he sprng des n change lengh, he sress s nally aken up by he dash-p. The creep curve hus sars wh an nal slpe /. Sme sran hen ccurs and s sme f he sress s ransferred frm he dash-p he sprng. The slpe f he creep curve s nw /, where s he sress n he dashp, wh ever-decreasng. In he lm when, he sprng akes all he sress and hus he maxmum sran s /. Slvng he frs rder nn-hmgeneus dfferenal equan.3. wh he nal cndn ( ) gves ( ) ( / ) e (.3.) Sld Mechancs Par I 9

6 Secn.3 whch agrees wh he abve physcal reasnng; he creep cmplance funcn s nw J ( ) e, / (.3.3) The parameer, n cnras he relaxan me f he Maxwell mdel, s here called he reardan me f he maeral and s a measure f he me aken fr he creep sran accumulae; he shrer he reardan me, he mre rapd he creep sranng. When he Kelvn mdel s unladed, he sprng wll wan cnrac bu agan he dash p wll hld back. The sprng wll hwever evenually pull he dash-p back s rgnal zer psn gven me and full recvery ccurs. Suppse he maeral s unladed a me. The cnsuve law, wh zer sress, reduces. Slvng leads ( ) ( / ) Ce (.3.4) where C s a cnsan f negran. The here s measured frm he pn where "zer lad" begns. If ne wans measure me frm he nse f lad, mus be replaced ( / wh. The sran a s ) ( ) ( / ) e. Usng hs as he nal cndn, ne fnds ha ( ) e ( / ) e ( / ), (.3.5) The creep and recvery respnse s shwn n Fg There s a ransen-ype creep and anelasc recvery, bu n nsananeus r permanen sran. sress appled sress remved Fgure.3.7: Creep-ecvery espnse f he Kelvn (Vg) Mdel Sress elaxan Cnsder nex a sress-relaxan es. Seng he sran be a cnsan, he cnsuve law.3. reduces. Thus he sress s aken up by he sprng and s cnsan, s here s n fac n sress relaxan ver me. Acually, n rder ha Sld Mechancs Par I 93

7 Secn.3 he Kelvn mdel underges an nsananeus sran f, an nfne sress needs be appled, snce he dash-p wll n respnd nsananeusly a fne sress Three lemen Mdels The Maxwell and Kelvn mdels are he smples, w-elemen, mdels. Mre realsc maeral respnses can be mdelled usng mre elemens. The fur pssble hreeelemen mdels are shwn n Fg..3.8 belw. The mdels f Fg..3.8a-b are referred as slds snce hey reac nsananeusly as elasc maerals and recver cmpleely upn unladng. The mdels f Fgs..3.8c-d are referred as fluds snce hey nvlve dashps a he nal ladng phase and d n recver upn unladng. () () (a) () (b) () (c) (d) Fgure.3.8: Three-elemen Mdels: (a) Sandard Sld I, (b) Sandard Sld II, (c) Sandard Flud I, (d) Sandard Flud II 3 he sress requred s ( ) ( ), where () s he Drac dela funcn (hs can be deermned usng he negral represenans f.4 Sld Mechancs Par I 94

8 Secn.3 The dfferenal cnsuve relans fr he Maxwell and Kelvn mdels were n dffcul derve. Hwever, even wh hree elemens, dervng hem can be a dffcul ask. Ths s because ne needs elmnae varables frm a se f equans, ne r mre f whch s a dfferenal equan (fr example, see.3.5). The ask s mre easly accmplshed usng negral frmulans and he Laplace ransfrm, whch are dscussed n Only resuls are gven here: he cnsuve relans fr he fur mdels shwn n Fg..3.8 are (a) (b) (c) (d) (.3.6) The respnse f hese mdels can be deermned by specfyng sress (sran) and slvng he dfferenal equans.3.6 fr sran (sress)..3.5 The Creep Cmplance and he elaxan Mdulus The creep cmplance funcn and he relaxan mdulus have been menned n he cnex f he w-elemen mdels dscussed abve. Mre generally, hey are defned as fllws: he creep cmplance s he sran due un sress: ( ) J ( ), ( ) J ( ) when Creep Cmplance (.3.7) The relaxan mdulus s he sress due un sran: ( ) ( ), ( ) ( ) when elaxan Mdulus (.3.8) Whereas he creep funcn descrbes he respnse f a maeral a creep es, he relaxan mdulus descrbes he respnse a sress-relaxan es..3.6 Generalzed Mdels Mre cmplex mdels can be cnsruced by usng mre and mre elemens. A cmplex vscelasc rhelgcal mdel wll usually be f he frm f he generalzed Maxwell mdel r he generalzed Kelvn chan, shwn n Fg The generalzed Maxwell mdel cnsss f N dfferen Maxwell uns n parallel, each un wh dfferen parameer values. The absence f he slaed sprng wuld ensure flud-ype behavur, Sld Mechancs Par I 95

9 Secn.3 whereas he absence f he slaed dash-p wuld ensure an nsananeus respnse. The generalsed Kelvn chan cnsss f a chan f Kelvn uns and agan he slaed sprng may be med f a flud-ype respnse s requred. In general, he mre elemens ne has, he mre accurae a mdel wll be n descrbng he respnse f real maerals. Tha sad, he mre cmplex he mdel, he mre maeral parameers here are whch need be evaluaed by expermen he deermnan f a large number f maeral parameers mgh be a dffcul, f n an mpssble, ask. Generalzed Maxwell Mdel N N Generalzed Kelvn Chan N N Fgure.3.9: Generalsed Vscelasc Mdels I s evden ha, n general, a lnear vscelasc cnsuve equan wll be f he frm p p p p 3 p 4 ( IV ) ( q q q q q 3 4 IV ) (.3.9) The mre elemens (sprngs/dashps) ne uses, he hgher he rder f he dfferenal equan. qn..3.9 s smemes wren n he shr-hand nan where P and Q are he lnear dfferenal perars P Q (.3.) Sld Mechancs Par I 96

10 Secn.3 n n P p, Q q (.3.) A vscelasc mdel can be creaed by smply enerng values fr he ceffcens p, q, n.3.9, whu referrng any parcular rhelgcal sprng dashp arrangemen. In ha sense, sprngs and dashps are n necessary fr a mdel, all ne needs s a dfferenal equan f he frm.3.9. Hwever, he use f sprngs and dashps s helpful as gves ne a physcal feel fr he way a maeral mgh respnd, raher han smply usng an absrac mahemacal expressn such as eardan and elaxan Specra Generalsed mdels can cnan many parameers and wll exhb a whle array f relaxan and reardan mes. Fr example, cnsder w Kelvn uns n seres, as n he generalsed Kelvn chan; he frs un has prperes, and he secnd un has prperes,. Usng he mehds dscussed n.4-.5, can be shwn ha he cnsuve equan s (.3.) Cnsder he case f specfed sress, s ha hs s a secnd rder dfferenal equan n (). The hmgeneus slun s { Prblem 3} / / h ( ) Ae Be (.3.3) where /, / are he egenvalues f.3.. Fr a cnsan lad, he full slun s { Prblem 3} / / e e ( ) (.3.4) Thus, whereas he sngle Kelvn un has a sngle reardan me, qn..3.3, hs mdel has w reardan mes, whch are he egenvalues f he dfferenal cnsuve equan. The erm nsde he square brackes s evdenly he creep cmplance f he mdel. Ne ha, fr cnsan sran, he mdel predcs a sac respnse wh n sress relaxan (as n he sngle Kelvn mdel). In a smlar way, fr N uns, can be shwn ha he respnse f he generalsed Kelvn chan a cnsan lad s, neglecng he effec f he free sprng/dashp, f he frm Sld Mechancs Par I 97

11 Secn.3 ( ), N / e (.3.5) where, are he sprng sffness and dashp vscsy f Kelvn elemen, N, Fg The respnse f real maerals can be mdelled by allwng fr a number f dfferen reardan mes f dfferen rders f magnude, e.g.,,,,,. If ne cnsders many elemens, qn..3.5 can be expressed as ( ) / e, N (.3.6) In he lm as f negran frm N, leng d d / d d d, and leng d / d ne has, changng he dummy varable, / e d () (.3.7) The represenan.3.7 allws fr a cnnuus reardan me, n cnras he dscree mes f he mdel.3.5. The funcn s called he reardan specrum f he mdel. Dfferen respnses can be mdelled by smply chsng dfferen frms fr he reardan specrum. An alernave frm f qn..3.7 s fen used, usng he fac ha d / dln where. / : () e d ln (.3.8) A smlar analyss can be carred u fr he Generalse Maxwell mdel. Fr w Maxwell elemens n parallel, he cnsuve equan can be shwn be (.3.9) Cnsder he case f specfed sran, s ha hs s a secnd rder dfferenal equan n (). The hmgeneus slun s, analgus.3.3, { Prblem 4} / / h ( ) Ae Be (.3.3) Sld Mechancs Par I 98

12 Secn.3 where agan /, /, and are he egenvalues f.3.9. Fr a cnsan sran, he full slun s { Prblem 4} / / ( ) e e (.3.3) Thus, whereas he sngle Maxwell un has a sngle relaxan me, qn..3.9, hs mdel has w relaxan mes, whch are he egenvalues f he dfferenal cnsuve equan. The erm nsde he square brackes s evdenly he relaxan mdulus f he mdel. By cnsderng a mdel wh an ndefne number f Maxwell uns n parallel, each wh vanshngly small elasc mdul, ne has he expressn analgus.3.7, / e d and s called he relaxan specrum f he mdel. ( ) (.3.3) T cmplee hs secn, ne ha, fr he w Maxwell uns n parallel, a cnsan sress leads he creep sran { Prblem 5} / / / / () e e, (.3.33) Sld Mechancs Par I 99