10.5 Linear Viscoelasticity and the Laplace Transform
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1 Scn.5.5 Lnar Vclacy and h Lalac ranfrm h Lalac ranfrm vry uful n cnrucng and analyng lnar vclac mdl..5. h Lalac ranfrm h frmula fr h Lalac ranfrm f h drvav f a funcn : L f f L f f f f f c..5. whr h ranfrm varabl h vrbar dn h Lalac ranfrm f h funcn and f h valu f h funcn a m. h Lalac ranfrm dfnd n uch a way ha f rfr ha ju bfr m zr. Sm hr mran Lalac ranfrm ar ummard n abl.5. n whch a cnan. f f n n! n n abl.5.: Lalac ranfrm Anhr uful frmula h m-hfng frmula: L f f Mchancal mdl rvd h Maxwll Mdl h Maxwll mdl gvrnd by h f hr uan..5: h rul acually nly wrk fr funcn wh drvav ar cnnuu alhugh h drvav f h funcn bng ranfrmd may b cw cnnuu. Dcnnu n h funcn r drvav nrduc addnal rm Sld Mchanc Par I 8 Klly
2 Scn.5.5. akng Lalac ranfrm gv.5. and ha bn aumd ha h ran zr a. h hr dffrnal uan hav bn rducd a f hr algbrac uan whch may nw b lvd g ranfrmng back hn gv n...6: Nw xamn h rn a uddn lad. Whn ung h Lalac ranfrm h lad wrn a whr h avd funcn h Andx h rvu cn. hn.5.6 rad.5.7 Ung h Lalac ranfrm gv.5.8 whch h am rul a bfr n Subun unladng a m ay can b dal wh m cnvnnly by urmng anhr lad n h fr. Pung h n h cnuv uan and ung h Lalac ranfrm gv.5.9 ranfrmng back agan ung h m-hfng rul gv.5. Sld Mchanc Par I 9 Klly
3 Scn.5 Sld Mchanc Par I Klly Addng h h ran du h fr lad hn gv h xcd rul.5. h Klvn Mdl akng Lalac ranfrm f h hr uan fr h Klvn mdl n... gv whch yld... h rn a lad.5. h rn anhr lad f magnud.5. h rn bh lad nw gv h cml cr and rcvry rn:.5. analy h rn a uddnly ald ran ubu n h cnuv uan g whch hw ha h rlaxan mdulu f h Klvn mdl.5.5 h Sandard Lnar Mdl Cndr nx h andard lnar mdl whch cn f a rng n r wh a Klvn un Fg..5. Fg...8a. Un ladng n xc h lf-hand rng rch mmdaly. h dah hn ak u h r ranfrrng h lad h cnd rng a lwly n vr m. Un unladng n xc h lf-hand rng cnrac mmdaly and fr h rgh-hand rng lwly cnrac bng hld back by h dah-. h uan fr h mdl ar frm h fgur
4 Scn.5 Sld Mchanc Par I Klly.5.6 Fgur.5.: h andard lnar mdl On can lmna h fur unknwn frm h fv uan ung h Lalac ranfrm gvng.5.7 whch ranfrm back n andard frm.5.8 whch n...6a. h rn a lad J.5.9 and h cr cmlanc
5 Scn.5 Sld Mchanc Par I Klly J.5. N ha a xcd. Fr rcvry n can urm an lad n h fr a m ay:.5. h rn afr m hn.5. h a xcd mly h rcvry rn f h Klvn un. h full rn a hwn n Fg..5.. h m b farly cl nw h rn f a ral maral a dcud n. alhugh d n allw fr a rmann ran. Fgur.5.: Cr-rcvry rn f h andard lnar mdl Nn-cnan Ladng h rn a cmlx ladng hry can b valuad by lvng h dffrnal cnuv uan r h crrndng hrdary ngral. h dffrnal uan can b m aly lvd ung Lalac ranfrm. xaml Cndr h xaml rad arlr ung hrdary ngral a h nd f... Lad f Fg...5 can b hugh f a cnng f h w lad a and b ald a m. Lad cn f a cnan lad ald a m. Fr lad a
6 Scn.5 Sld Mchanc Par I Klly whch gv h rn fr. Fr lad b n ha [n: L ] h rn afr m hn gvn by addng h w rul:.5. Rlanh bwn Cr and Rlaxan akng h Lalac ranfrm f h gnral cnuv uan..9 Q P lad.5. whch can al b wrn n h cnracd frm Q P.5. whr P and Q ar h lynmal n n Q P.5.5 h Lalac ranfrm f h cr cmlanc J J and rlaxan mdulu can b wrn n rm f h lynmal a fllw. Fr h ran du a un lad J. Snc ubun n h abv uan gv
7 Scn.5 P J.5.6 Q Smlarly h r du a un ran and I fllw ha Q.5.7 P J.5.8 hu fr a lnar vclac maral hr a unu and ml rlanh bwn h cr and rlaxan bhavur..5. Prblm. Chck ha h rlan.5.8 J hld fr h Klvn mdl. a Drv h cnuv rlan n andard frm fr h hr-lmn mdl hwn blw ung h Lalac ranfrm h h Sandard Flud II f Fg...8d and h cnuv rlan gvn by n...6d b Drv h cr cmlanc J by cndrng a uddnly ald lad. Sld Mchanc Par I Klly
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