Viscoelastic Properties of Polymers a Short Course POLYCHAR 25

Transcription

1 Viscoelasic Properies of Polymers a Shor Course POLYCHAR 5 Michael Hess Deparmen of Polymer Engineering & Science, Chosun Universiy, KwangJu, Souh Korea Deparmen of Maerials Science, Universiy of Norh Texas, Denon, Texas, USA. INTRODUCTION and BASICS The mechanical behaviour of convenional solids is usually described by heir elasic behaviour (limiing case of ideal elasic behaviour). As long as he deformaions are no oo large Hooke's Law applies: ( ) x F E T, () F force, E elasic modulus (Young Modulus) in uniaxial deformaion, T hermodynamic emperaure, angular velociy, x displacemen ( srain) In erms of a shear deformaion, see fig. -4, his reads: σ ( T, )γ G () σ shear sress, G(T, ) shear modulus, γ shear (deformaion). Sress and srain are ensors, see fig. 3. When orhogonal o he plane he sress is called "normal sress". The reciprocal shear modulus is called (shear) compliance J /G. l o l Fig. : he (weigh) force means a sress σ ha causes a deformaion, ha is measured as he srain ε. l 0 is he iniial displacemen and l is he displacemen under load b. l-l o l a Wih many examples provided by Kevin Menard, UNT, Denon, Texas and Perkin Elmer Corp. b The measuremen of hese quaniies is no rivial since in polymers here is relaxaion and creep.

2 0 x 0, ε λ F E F ( ) σ E ε (3a-d) There are differen ypes of polymeric maerials such as hard viscoelasic solids orsof viscoelasic solids and highly viscous liquids (such as pich) ha appear o be a solid on he firs glance bu ha show a very slow flow (creep). Also, he specimen come in a paricular shape ha should probably no be changed. Consequenly, here are differen ypes of sresses and deformaions, more or less complex, all of hem delivering a corresponding modulus. Examples are exension, compression, shear, orsion, bending (3-poin, 4-poin), flexing, ec. These mechanical values, however, are correlaed. For deails see he books of Ferry, or Read and Dean or Menard in "furher reading". Sof maerials he shear modulus is abou Pa allow more degrees of freedom in he choice of sample geomery. The Poisson raio is close o 0.5 so ha all exensional viscoelasic funcions are correlaed wih he shear funcion by he facor 3 (see below). Hard viscoelasic maerials he shear modulus is abou Pa can principally be invesigaed wih he same kind of equipmen, however, some differen feaures may have o be considered when siffness increases by some orders of magniude. The siffness is no only depending on he maerial and emperaure bu also on he shape (plae vs. -bar, for example). The accuracy of a modulus from an experimen can be significanly influenced by he accuracy of he measuremen of he shape of he sample. In paricular a high values of he modulus care has o be aken ha he sample modulus is sill much lower han he modulus of srucural pars of he measuring equipmen, in paricular in dynamic experimens.

3 σ yield sress ε Fig. : schemaic sress-srain curves for differen polymers. The end of he curves marks he yield of he maerial. Deviaions from lineariy documen non-hookean behaviour and are caused by viscose effecs, see ex. There are differen definiions of he srain all of hem become idenical a small deformaions: Cauchy (engineering srain) ε (4a) c 0 3

4 4 Hencky (rue srain) 0 ln h ε (4b) Kineic heory of rubber elasiciy k ε (4c) Kirchoff srain 0 K ε (4d) Murnaghan srain 0 M ε (4e) Fig. 3: simple shear applied o a cube. The deforming force aacks parallel o plane (see fig. 4) and orhogonal o plan. If he deformaion akes place a consan volume he Poisson raio µ 0.5, see ex.

5 Fig. 4: componens of he sress ensor which represen he forces acing from differen direcion on differen faces of a cubical elemen. Simple shear is a homogeneous deformaion, such ha a mass poin of he solid wih co-ordinaes X, X, X3 in he undeformed sae moves o a poin wih coordinae x, x, x3 in he deformed sae, wih x X + gx x X x3 X3 where g is a consan. For he definiions of he non-ulimae mechanical properies of polymers see A. Kaye, R. F. T. Sepo, W. J. Work, J. V. Alemán, A. Ya. Malkin. The (simple) shear γ ha resuls afer applicaion of he sress σ is given by he quoien x/h, see fig. 3 and fig. 4, and for small deformaion angles γ here is: γ anγ γ (5) The individual sress (respecively deformaion) componens combine o he oal sress σ ij (srain γ ij ) and can be expressed by he marix: σ σ σ 3 σ ij σ σ σ 3 (6) σ 3 σ 3 σ 33 In fig. plane /3 slides in direcion (as indicaed in fig. ) and sress and srain are: σ ij p σ 0 σ p 0 0 0, p γ ij 0 γ 0 γ (7a, b) since only a displacemen in srain componens he planes and occurs. -p is an isoropic compressive pressure ha occurs on applicaion of he shear sress, and γ γ. Dimensional changes caused by longiudinal deformaion usually come wih changes of he cross secion. This is described by he Poisson raio µ. The Poisson raio correlaes he Young modulus wih he shear modulus, respecively he bulk modulus B: ( + µ ) 3 ( µ ) E G B (8) 5

6 so ha for elasomers: µ 0.5 E 3G (9) The volume change on deformaion is for mos elasomers negligible so ha µ0.5 (isoropic, incompressible maerials). In a sample under small uniaxial deformaion, he negaive quoien of he laeral srain (εla) and he longiudinal srain (εlong) in he direcion of he uniaxial force. Laeral srain εla is he srain normal o he uniaxial deformaion. ε ε long la µ (0) E/GPa sof rubber 0.00 polysyrene 3 copper 0 diamond 050 Tab. : Young modulus of differen maerials a ambien emperaure µ 0.5 no volume change during srech 0 no laeral conracion ypical for elasomers ypical for plasics Tab. : ypical values of he Poisson raio The bulk modulus B is derived from he coefficien of isohermal compressibiliy: so ha wih eq. 8: V β V p T p and B V β V E β E 3B 3 T () µ () In elasomers he modulus is relaed o he number N el of elasically acive chains by: k b E Nel kb T J Bolzmann consan K (3) 6

7 The mechanical behaviour of convenional fluids is described by Newon's Law (limiing case of ideal viscous behaviour): x σ η( T, ) γ η( T, ) (4) h η viscosiy, γ shear rae. Eq. 0 describes a linear velociy gradien v h x h in he fluid as shown in fig. 4: Fig. 5: linear velociy gradien in a Newonian fluid. Polymers ypically show boh, viscous and elasic properies. Viscous behaviour can be represened by a dashpo and elasic behaviour by a spring so ha a visco-elasic maerial can be modelled by appropriae combinaion of dashpo(s) and springs. There are wo basic combinaions: he Maxwell-elemen and he Voig-Kelvinelemen, see fig. 6. 7

8 Fig. 6: a Maxwell- and a Voig-Kelvin-elemen wih he corresponding creep behaviour (a consan sress). γ is synonymous wih ε. For a dashpo one obains for he deformaion rae from eq. 4: For a spring here is from eq. 3d: dε σ d η (5) dε dσ (6) d E d so ha a Maxwell-elemen wih spring and dashpo in series is described by: dε d E dσ σ + d η (7) c Wih he definiion of he relaxaion ime : η E (8) c The relaxaion ime is he ime afer which he sress has reached /e of he iniial sress. 8

9 dε d dσ E d shor- ime changes in srain + σ E long-ime changes in srain (9) A Voig-Kelvin-elemen wih a dashpo parallel o a spring is described by: dε σ η + E ε d (0) d Wih he definiion of he reardaion ime : η E (8) dε d Sress a consan srain, 0, can show relaxaion, and srain a consan dσ d sress 0 can show reardaion. Wih hese condiions eq. 9 and 0 are inegraed: σ ε E e 0 σ 0 e () σ ε e E () Combinaion of Maxwell-and Voig-Kelvin-elemens are suied o describe he behaviour of visco-elasic maerials, e. g. by he following 4-elemen model, see fig. 7: d The reardaion ime is he ime required for he o deform o (-/e) of he oal creep. 9

10 Fig. 7: 4-elemen model consising of a Maxwell and a Voig-Kelvin-elemen in series. Afer he sress has relaxed afer he ime here is only a parial recovery ha is conrolled by he reardaion and he corresponding creep. The creep-funcion of he 4-elemen model in fig. 6 is hen given by: ε ε + ε + ε 3 σ E elasic conribuion σ σ + e + E η 3 viscoelasic erm viscos conribuion (3) The dynamic behaviour of Maxwell-and Voig-Kelvin-Elemens can be summarised as follows wih he periodic deformaion given in erms of he angular frequency πν, where ν is he frequency in s -. For explanaion of he ' - and ''-erms see laer. Maxwell-elemen (4 a-g) 0

11 ( ) ( ) ( ) ( ) ( ) ( ) an δ η η J J J J G G G G e G G J J Voig-Kelvin-elemen (5 a-g) ( ) ( ) ( ) ( ) ( ) ( ) an δ η η η + + J J J J G G G G e J J

12 ( ) γ σ G ( T, T, ) γ σ η( ηt ) ( ) x h (, T, γ) γη T, η( T, ) x h Fig. 8: modulus and viscosiy In an ideal elasic body sress and deformaion are in phase, sress and srain are consan over he ime. This is no he case in viscoelasic maerials which show boh properies simulaneously o a smaller or greaer exend, fig. 9. Fig. 9: example for a viscoelasic maerial exposed o a dynamic sress experimen where here is a phase delay beween applied sress and srain response. This delay can be described by a phase angle δ. This behaviour is in paricular imporan in dynamic deformaions, see laer.

13 A sufficienly low emperaures when chain-and chain segmen mobiliy are frozen in, ha is below he glass-ransiion emperaure (see laer), polymers behave like common elasic maerials. The (elasic) deformaions in ha sae are characerised by changes of bond lengh and bond angles. The only in macromolecular subsances observed rubber elasiciy is no caused by an energeic disorion of bond lengh or bond angles bu by enropic effecs: perurbaion of a random coil leads o a sae of lower enropy since he number of accessible quanum saes (conformaions) is resriced by e. g. an exension. Rubber elasiciy can be observed a emperaures higher han he glass ransiion emperaure if he polymer chains are long enough and if cross-links of any kind are presen. The cross-links can be permanen or emporary, chemical or physical of naure. They cause phenomena like relaxaion and creep (reardaion). A sress a consan srain relaxes, a srain a consan sress reards and he maerial creeps. The ypical mechanical response of maerials are shown in fig. 0: 3

14 Fig. 0: Typical response of differen ypes of maerial on an applied sress (op). γ is synonymous o ε. The broken lines refer o uncrosslinked maerial, he solid lines o crosslinked maerial: Normal (ideal) energy-elasic behaviour, he srain follows he sress wihou delay (ideal spring), case a). Normal (ideal) viscous behaviour, no elasic behaviour (ideal dashpo), case d). Case c) shows ypical rubber elasiciy wih a high deformaion and a fracion of irreversible flow. Case b) resembles case c, however, here is a delayed response and afer removal of he applied sress here is a significan relaxaion of he sample sress over a quie long ime long ime, again wih some irreversible flow in he crosslinked sample. This behaviour can be characerised as parially blocked rubber elasic, is ermed "leaher-like" and is observed around he glass ransiion emperaure, see fig.. An overhead foil, fresh from he copymachine, sill warm, is in his leaher-like sae. In principle, any polymer can depending on he emperaure exis in any of hese saes as long as he hermal sabiliy allows his. A polymer sample esed for he emperaure-dependence of is mechanical modulus a a consan frequency will in principle go hrough mos of hese saes depending on he chain lengh (disribuion), degree of crosslinking, degree of crysalliniy and hermal sabiliy. A frequency-scan a fixed emperaure will principally deliver he same informaion (emperaure-frequency-equivalence principle). A consan frequency he emperaure is scanned and observed when he resonance modes corresponding o he measuring frequency are called. A consan emperaure here is jus a frequency sweep and he resonance cases are moniored. Fig. : emperaure of hermal ransiions (measuring frequency Hz) and he corresponding molecular moions. G' is he real par (sorage modulus) of he 4

15 complex shear modulus, Λ is he logarihmic decremen. For explanaion see ex and fig There is a direc relaion of he viscoelasic properies of a polymer and molecular moions, in paricular cooperaive moions. This is caused by he fac ha each deformaion of a polymer chain changes is equilibrium conformaion, hence giving rise o an enropy-driven endency o resore he iniial sae. There are always four pars in he emperaure-modulus curve of an amorphous polymer: he measable glassy solid (frozen liquid) a low emperaures followed by he glassrubber (or brile-ough-) ransiion, he more or less pronounced rubber-elasic plaeau, and finally he erminal flow range. The firs ransiion in fig. 8 coming down from high emperaures is ermed α ransiion. In semi-crysalline polymers his is he crysallisaion/meling process. In amorphous polymers such as in fig.8 he glass ransiion emperaure is he sronges ransiion (α-relaxaion). These ransiions are also called relaxaions since coming from low emperaures hey describe he onse of he molecular moion as indicaed in fig. 8. In paricular he glass ransiion indicaes he onse of cooperaive chain-segmen moions (abou 5 chain segmens) and is a coninuous ransiion leading from a solid-like sae o a liquid-like sae (or vice versa). The glass ransiion is no an equilibrium ransiion, see below. As a maer of fac here is no "he" glass ransiion emperaure since here is an infinie number of glass saes (hence glas ransiion emperaures) depending on he hermal hisory. Annealing changes he physical properies of a glass. The relaxaion behaviour can be moniored a a fixed emperaure wih a frequency sweep or i can be moniored a a fixed frequency bu wih a emperaure sweep. In he firs case resonance is observed when he applied frequency maches a corresponding molecular moion a his emperaure, in he second case resonance is observed when he energy provided by he applied emperaure fis in wih a molecular moion ha maches wih he chosen frequency. This reflecs a imeemperaure relaion Bolzmann's ime-emperaure superposiion principle (TTS), see fig. XXX his, however, is no generally valid, only if all relaxaion processes are affeced by he emperaure in he same way. Only in hese cases ime and emperaure are equivalen. There are numerous examples where here are deviaions from TTS, see fig. 3. The emperaure-dependence of he relaxaion processes menioned above can be described by he Williams-Landel-Ferry equaion (WLF) as long as he resricion menioned does no apply. The (semi-empirical) WLF equaion can be derived using he free volume heory, and a quaniaive descripion is frequenly possible in he mel in a emperaure range from Tg o Tg+00 K. The derivaion goes back o he early work of Doolile 3 on he viscosiy of non-associaed pure liquids. The imporance of a relaion like he WLF equaion becomes clear recalling he fac ha he experimenal echniques usually only cover a raher narrow ime slo, e. g. 0 0 s 0 5 s ( corresponding o a frequency range). The ime-emperaure superposiion principle allows an esimae of he relaxaion behaviour and relaed properies of polymers such as he mel viscosiy over a wide emperaure range (e.g. 0-4 hrs 0 hrs) wih he WLF-equaion and he shif facor. 5

16 Fig. : TTS: superposiion of he individual relaxaion curves a differen emperaures as indicaed on he lef o one maser curve a 5 C on he righ. The inser shows he emperaure-dependence of he shif parameer ha is required o make all curves fi ino one maser curve. Considered a cerain generalised ransiion emperaure T 0 (frequenly he glass ransiion emperaure), A T is called he reduced variables shif facor, where 0 is he ime required for he ransiion and η 0 he corresponding viscosiy. The oher values are hen valid for a differen sae. η ln ln η0 A T ln A T is no only relaed wih he viscosiy bu wih many oher ime-dpenden quaniies a he ransiion emperaure respecively anoher emperaure, see below. 0 (6) 6

17 η lg lg ηg respecively : η lg lg ηs s g lg A lg A C T C ( T Tg) Tg ( T Ts) T Ts (7a, b) Ts [K] (Tg + 50) The index s indicaes he siuaion a an arbirary emperaure up o 50 K above Tg. The numerical consans are empirical and valid for a number of linear amorphous polymers more or less independen of heir chemical naure. The consans C and C depend on he polymer. The "universal" consans are C 7.44 and C 5.6 and give good resuls for many polymers. Some examples were lised by Aklonis and McKnigh 4 : C C Tg/K polyisbuylene Naural rubber Polyurehane (elasomer) polysyrene Poly(ehyl mehacrylae) Tab. 3: WLF-consans and Tg from Aklonis and McKnigh In his way, he WLF-equaion enables a deerminaion of he frequency dependence of a deerminaion of a physical eniy of polymers ha depends on he free volume, such as he glass ransiion emperaure, he deerminaion of which is frequency dependen. For example an increase of he measuring frequency by a facor 0 (or a decrease of he ime frame by a facor of 0) near Tg he glassransiion emperaure is found abou 3 K higher: From eq. 7a one obains: 7

18 T lim Tg ( lg A ) T T Tg ( T Tg) lg lg AT g ν g lg ν T ν g lg ν (8a-c) wih he differen measuring frequencies ν g and ν, e. g. Hz and 0 Hz, respecively. The shif facor is a funcion of he emperaure and ofen obains values beween

19 Failure of TTS Log J* Fig. 3: in conras o fig. he individual relaxaion curves do no fi o form one single maser curve bu hey "branch-off" for longer imes indicaing ha no all relaxaion processes show he same emperaure-dependence. 9

20 Fig. 4: he emperaure-dependence of he relaxaion frequency of he α e -(glass), respecively β-ransiion of polysyrene. The slope gives access o he energy of acivaion of he process. The energy of acivaion can give an idea of he origin of he ransiion, see ex. While second-order ransiions as defined by Ehrenfes 5, 6 The apparen energy of acivaion is calculaed from he slope of an Arrhenius plo according o eq. 9: Ea ln + cons. R T The apparen acivaion energy for he ransiions in polysyrene shown in fig. 3 is 35.7 kj/mol for he α-process and 46.5 kj/mol for he β-process. The acivaion energy of relaxaion processes near he glass-ransiion emperaure is usually higher compared wih oher relaxaion processes in a glass. Kovacs 7 has derived eq. (30) for he apparen acivaion energy of molecular relaxaions due o he onse of cooperaive moions of main-chain segmens in amorphous polymers (such as aacic polysyrene): E a RT d ln A dt T (9) (30) e The ransiion a he highes emperaure is frequenly ermed α-ransiion 0

21 Acivaion Energy ells us abou he molecule For example, are hese T g s or a T g and a T β? El as omer Sa mple Because we can calculae he E ac for he peaks, we can deermine boh are glass ransiions. Fig. 5: Discriminaion beween a glass ransiion and a secondary relaxaion. The apparen acivaion energy of glass ransiions is higher compared wih secondary relaxaion processes which are correlaed wih smaller molecular moions (such as side group roaions) ha are usually no cooperaive.

22 The meling ransiion a firs order ransiion is no frequency depending. In cases where i is difficul o measure a sample beyond is meling ransiion because he sample shape disinegraes because of he mel flow, orsional braid-analysis or a comparable echnique migh be used o deermine he ransiion emperaure. This echnique analyses he mechanical properies of he polymer suppored by an iner maerial. This can be a exile maerial soaked wih he sample, a hin meal foil meal (ransiions in lacquer layers or polymer surface layers of a few micromeer hickness can be analysed) or braids of glass hreads or fabric can be used. However, care has o be aken because ineracions of he subsrae wih he suppor can influence he ransiion (emperaure, srengh, ec.). The glass-ransiion emperaure is said o be he emperaure a which he moion of groups of segmens (such as a few repeiion unis) freeze in a cooperaive way, where he viscosiy diverges ec.. There are hree very differen heories approaching he phenomenon of he glass ransiion. These are summarised in ab. 4 wih heir advanages and disadvanages. The crieria for a second order ransiion according o Ehrenfes are usually no fulfilled and here is a srong evidence for is kineic characer. advanages disadvanages hermodynamic heory 8 Variaion of Tg wih molecular mass, plasicizer and cross-link densiy are prediced wih some accuracy for measuremens kineic heory, 3 Frequency-dependence of Tg are well prediced Hea capaciies can be deermined Free-volume heory 9,0, Time-emperaure superposiion principle Expansiviy (below and above) can be relaed wih Tg A rue second order ransiion is prediced bu poorly defined Infinie ime scale required No Tg prediced for infinie ime scale The acual molecular moions are poorly defined Fox and Flory 0 have shown ha he (number-average) molar mass of a polymer significanly influences he glass ransiion emperaure, so ha polymerizaion and cross-linking processe (gelaion) are refleced by Tg, e. g. during he curing process of a hermose. T g K Tg (3) M n

23 Tg is he glass ransiion emperaure a infiniely high molar mass, K is a consan individual for any paricular polymer. According o eq. 3 he glass ransiion emperaure rises and i can happen ha he polymerisaion reacion sops because of he frozen molecular mobiliy. The imeemperaure-ransformaion diagram, fig. 6 developed by Gillham, describes he processes in a curing hermose in deail. Fig. 6: curing behaviour of a hermose displayed as a ime-emperaureransformaion-reacion diagram as an example for he long-ime behaviour of a (crosslinked) amorphous polymeric maerial afer Gillham 3. There are numerous mehods o measure ransiions in polymers and, as poined ou above, he measuring frequency plays an imporan role. The smaller he frequency (or he heaing rae) he closer is he deermined value o he equilibrium value of he propery under consideraion. Some examples are given in fig. 7. 3

24 Fig. 7: comparison of some mehods o deermine hermal ransiions in amorphous, crysalline and semi-crysalline polymers. All of hem can be carried ou a differen frequencies. In differenial scanning calorimery (DSC), for example, he frequency is given by he heaing rae, in dynamic-mechanical (or dielecric measuremens) he frequency of he mechanical sress (or dielecric polarisaion) is a direc parameer of he experimen besides he emperaure or he pressure). Calorimeric mehods are covered in anoher lecion of his course as are volumeric mehods, see also Hess 4. One way among ohers (see "Furher Reading") o deermine dynamicmechanical properies is he free decay of a orsional oscillaion performed in a pendulum such as shown in fig. 8. 4

25 G G + i G Fig. 8: a orsional pendulum as example for equipmen o deermine dynamic mechanical properies. The srip-shaped sample specimen ( 7cmxcmx0.05cm) is wised by abou 5 and hen allowed for a free (damped) oscillaion. The ampliude of subsequen maxima of he oscillaion makes i possible o deermine he logarihmic decremen Λ and he sorage modulus G'(T), he loss modulus G''(T) and he damping D (loss angen, an δ, where δ is he phase angle of he delay of he deformaion behind he sress). In fac he dynamic modulus is a complex physical eniy: G * + * G i G (3) and he loss angen is given by: All imporan equaions are summarised in figs G an δ (33) G 5

26 Fig. 8: free-damping experimen he logarihmic decremen Λ is calculaed from wo subsequen exremes of he oscillaion. 6

27 Fig. 9: calculaion of he sorage modulus G'(T) and he loss modulus G''(T) from a free-damping oscillaion. Θ is he momenum of ineria. 7

28 Fig. 0: definiions of G' and G'' from he differenial equaion of free oscillaions. 8

29 Sorage modulus G' (or E') and loss modulus G'' (or G'') can be explained by fig. :,, E E Fig. : visualisaion of he meaning off he soragemodulus E' (T)(here he Young modulus as example) and he loss modulus E''(T). The loss-energy is dissipaed as hea and can be measured as a emperaure increase of a bouncing rubber ball. Figure by couresy of K. Menard. A. Kaye, R. F. T. Sepo, W. J. Work, J. V. Alemán, A. Ya. Malkin, 998 IUPAC Recommendaion, Pure and Applied Chemisry (998) 70, M. L. Williams, R. F. Landel, J. D. Ferry, J. Am. Chem. Soc. (955) 77, A. K. Doolile, J. Appl. Phys. (95), 47 4 J. J. Aklonis, W. J. McKnigh, Inroducion o Polymer Viscoelasiciy, Wiley-Inerscience (983) New York 5 P. Ehrenfes, Proc. Kon.Akad. Weensch. Amserdam (933) 36, 53 6 P. Ehrenfes, Leiden Comm. Suppl. (933) A. J. Kovacs, J. Polym. Sci. (958) 30, 3 8 E. A. DiMarzio, J. H. Gibbs, J. Polym. Sci. (963) A, 47 9 H. Eyring, J. Chem. Phys (936) 4, 83 0 T.G. Fox, P. J. Flory, J. Polym. Sci. (954) 4, 35 R. Simha, R. F. Boyer, J. Chem. Phys. (96) 37, 003 J. K. Gillham, Encyclop. Polym. Sci. Technol. (986) nd ed. 3 J. K. Gillham, Polym. Eng. Sci. (979) 9, M. Hess, Macromol. Symp. (004) 4, 36 9

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31 FURTHER READING Kevin P. Menard, Dynamic-Mechanical Analysis, CRC-Press (999) Boca Raon W. Brosow, Performance of Plasics, Carl Hanser Verlag (000) Munich I. M. Ward, Mechanical Properies of Solid Polymers, Wiley (983) New York J. J. Aklonis, W. J. McKnigh, Inroducion o Polymer Viscoelasiciy, Wiley- Inerscience (983) New York N. W. Tschloegl, The Theory of Viscoelasic Behaviour, Acad. Press (98) New York D. Ferry, Viscoelasic Properies of Polymers, Wiley (980) New York B. E. Read, G. D. Dean, he Deerminaion of Dynamic Properies of Polymers and Composies, Hilger (978) Brisol L. E. Nielsen, Polymer Rheology, Dekker (977) New York L. E. Nielsen, Mechanical Properies of Polymers, Dekker (974) New York L. E. Nielsen, Mechanical Properies of Polymers and Composies Vol. I & II, Dekker (974) New York A. V. Tobolsky, Properies and Srucure of Polymers, Wiley (960) New York 3