10.7 Temperature-dependent Viscoelastic Materials

Transcription

1 Secin.7.7 Temperaure-dependen Viscelasic Maerials Many maerials, fr example plymeric maerials, have a respnse which is srngly emperaure-dependen. Temperaure effecs can be incrpraed in he hery discussed hus far in a simple way by allwing fr he cefficiens f he differenial cnsiuive equains be funcins f emperaure. Thus, qn..3.9 can be expressed mre generally as p p p q q 2 q2 (.7.) where denes emperaure. quivalenly, ne can allw fr he creep and relaxain funcins be funcins f emperaure in he herediary inegral frmulain. Thus qns read (, ) () J (, ) (, ) () (, ) d ( ) J (, ) d d d ( ) (, ) d d (.7.2).7. xample: The Maxwell Mdel Cnsider a Maxwell maerial whse dash-p viscsiy is a funcin f emperaure. The differenial cnsiuive equain is hen d d (.7.3) d d where is he emperaure-independen spring siffness. This equain is a funcin f bh emperaure and ime. Wih emperaure a funcin f ime,, i is a linear differenial equain wih nn-cnsan cefficiens. Fr cnsan emperaure, i has cnsan cefficiens. Cnsider firs he case f cnsan emperaure. The relaxain mdulus and creep cmpliance funcins can be evaluaed by applying uni srain and uni sress. Frm he previus wrk, ne has (, ) e J, / R, R (.7.4) Thus any given maerial has emperaure-dependen relaxain and creep funcins. Cnsider nw he change f variable Slid Mechanics Par I 337

2 Secin.7 A (.7.5) where A is any cnsan (which can be chsen arbirarily fr cnvenience see laer). This ransfrms qn..7.3 in A d d A (.7.6) d d This is nw an equain wih dependence n nly ne variable,. Frm his equain, ne bains relaxain and creep funcins ( ) e J / R A, R A (.7.7) These equains generae maser curves frm which he differen emperaure-dependen curves.7.4 can be bained. xample Daa Fr example, cnsider a viscsiy which varies linearly ver he range C C accrding he relain A (.7.8) where is a cnsan viscsiy, A. 2 and 2 C (a reference emperaure a. This funcin is pled in Fig..7. belw. which ( ) / Figure.7.: linear dependence f viscsiy n emperaure Slid Mechanics Par I 338

3 Secin.7 Als, le / m. The resuling relaxain and creep funcins f qn..7.4 are pled in Fig..7.2 belw (fr m 5 ). (a) (, ) (b) 5 J (, ) Figure.7.2: emperaure-dependen funcins; (a) relaxain mdulus, (b) creep cmpliance Ne he fllwing, referring Fig..7.2: (i) fr emperaures greaer han he reference emperaure 2 (see qn..7.8), he viscsiy is. This implies ha, fr, he relaxain imes are shrer han fr (see qn..7.4a), Fig..7.2a, and he slpe f he creep curves is greaer han fr (see qn..7.4b)., Fig..7.2b. (ii) fr emperaures smaller han he reference emperaure,. Thus, fr, he relaxain imes are lnger han fr and he slpe f he creep curves is smaller han fr. Slid Mechanics Par I 339

4 Secin.7 Nw chse he cnsan A in qn..7.5 be equal. This ensures ha a he reference emperaure (see.7.8). In her wrds, he maser curves f qn..7.7 and he funcins.7.4 crrespnding cincide (wih he axis and axis cinciden). The maser relaxain and creep curves f qn..7.7 are nw ( ) / J / m. These are pled in Fig..7.3 belw (fr m 5 ). / m e and (a).8 ( ) (b) J ( ) Figure.7.3: maser curves; (a) relaxain mdulus, (b) creep cmpliance All he curves f Fig..7.2 cllapse n he maser curve f Fig..7.3 as fllws: (i) he curves crrespnding he reference emperaure, 2, in Figs..7.2 lie n he maser curves (wih he axis and axis cinciden) (ii) fr a curve wih, if he ime axis f Fig..7.2a,b is sreched (accrding.7.5), he curve will cme lie alng he curve (and hence n he maser Slid Mechanics Par I 34

5 Secin.7 curve); fr a curve wih, if he ime axis f Fig..7.2a is shrunk (accrding.7.5), he curve will cme lie alng he curve (and hence n he maser curve).7.2 Thermrhelgically Simple Maerials The fac ha he relaxain and creep curves f Fig..7.2 cllapsed n he maser curves f Fig..7.3 relied n he change f variable, qn..7.5, reducing he ime and emperaure dependen cnsiuive relain.7.3 an equain in ne variable,, nly, qn This in urn depended criically n he frm f he differenial equain.7.3. Fr example, if he spring siffness in he Maxwell mdel is emperauredependen, he cllapsing f curves is n pssible. Temperaure-dependen viscelasic maerials fr which his cllapsing f curves is pssible are called hermrhelgically simple maerials. In his cnex, he parameer is called he reduced ime. Mre generally, he ransfrmain.7.5 is expressed in he frm (.7.9) a and he funcin a ( ) is called he shif facr funcin. The shif facr is chsen s ha he relaxain and creep curves crrespnding he chsen reference emperaure cincide (as in he Maxwell mdel example abve), i.e. s ha a ( ). The relaxain and creep funcins nw ransfrm as (, ) (, ), J (, ) J (, ) (.7.) Fr emperaures belw he reference emperaure,, a ( ) will be greaer han, and he crrespnding relaxain/creep curves cllapse n he maser curve by shrinking he ime axis, which lks like a shifing f he curve he lef n he curve. On he her hand, fr, a ( ), and he crrespnding curves cllapse by a sreching f he ime axis, which lks like a shifing f he curves he righ n he maser curve. This is summarised in Fig..7.4 belw. The resul f his is ha maerials a high emperaures and high srain raes behave similarly maerials a lw emperaures and lw srain raes. The mehd discussed can als be used when he emperaure is ime-dependen, fr hen he ransfrmain can be expressed as d a (.7.) Slid Mechanics Par I 34

6 Secin.7 s ha d (.7.2) d a leading he same reduced differenial equain. (, ) a a a shrink srech (, ) Figure.7.4: Relaxain mdulus, as a funcin f (a) ime, (b) reduced ime The abve discussin has relaed he differenial cnsiuive equain.7.. The analysis can als be expressed in erms f herediary inegrals f he frm.7.2. Fr example, he equivalen herediary inegral in erms f reduced ime, crrespnding he reduced differenial equain (see qn..7.6 fr he Maxwell mdel equain) is where d ( ) ( ) d (.7.3) d is as befre (see qn..7.7 fr he Maxwell mdel expressin). a Slid Mechanics Par I 342