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(1)CScience Reports of the Yokohama Natiopal University, Sec. I. No. 4, 1955). 't t. Viscoelastic Properties of Polymeric Substances*. By. Keiichi SHIBATA '. Abstract .･'. : ･･-. (1) Measurements on the transient behaviour of polymethyl rnethacrylate reveal that the torsional and bending creeps are linear with logarithmic time. ,. overaconsiderabletimerange. '.･ '' -,.･ t.. /.t ,,sand (2) Comparison between the creep .data and theory ',, of Tobolsky Eyring shows that somewhat ambiguous modifications have to be made for the parameters in their fundamental equation.. (3) By means of Gross' theory, a distribution function of retardation times is deduced from the creep data.. (4) Measurements' of dynamic Young's modulus of po!ymethyl metha! crylate are made with bending vibration frequencies of 500 and 100 cycles and at various temperature ranging from a room temperature up to 1200C. A Young's modulus-temperature diagram constructed from the data of several investigators including the .present ones shows that there is an anomalous. region in the vicinity of 1000C where the Young's modulus is markedly pt. dependent upon temperature and frequency. An attempt is made to correlate distribution function of retardation times obtained from the creep data and the vibrational experiment.. t.. (5) Correlations between the transient and the steady-state behaviours suggested by Dunell and.Tobolsky are discussed on the basis of Gross' theory. If the stress relaxation (or creep) is found tQ be linear with logarithmic time over a considerable time range and the principle of superposition is valid, the simple correlation between the rate of stress relaxation (or creep) and the. elastic loss (or imaginary component of compliance) is derived $o long as the relaxation (or retardation) frequency function is of the rectangUlar form. and the distribution function of relaxation (or retardation) times is of the form of one type of Pearson's distribution function. The experimental verification is made.. * The researches were carried out at Kinoshita Laboratory, Scientific Research lnstitute, Tokyo, from 1948 to 1952..

(2) 20 1･--'･, K.Shibata I. Introduction ' ' In Tecent years, the-great development in the production technique of 'high polymers has brought a renewed interest in the studies of their mechanical behaviours. These substances have properties which place them be･･. 'tween ideal solids and ideal viscous liquids, i.e., they have rigidity as well as. viscosity. Some attempts have been made to develop a general theory which -would describe the mechanical behaviour of bodies which exhibit simultaneous. ･elastic.and viscous behaviours. Although these theories have thrown much useful light on many aspects of general problem, various essential behaviours still remain so far unanalyzed on account of the scarcity of available experi-. mentaldata. ' ' In an experimental study on the mechanical behaviour of viscoelastic. .. 'body there are two ways of attack;they are: (a) The measurement of the -transient phenomena caused by sudden application of a constant load (creep). f. or deformation (stress relaxation). (b) The measurement of the steady:state behaviour under alternating load or deformation.･ Observations are made on the transient behaviours for the duration of about one minute or 'longer; the steady-state behaviour involves measurementS with frequencies. higherthanlc/s. ' '. ' -. The present investigation was undertaken in order'to carry out. (a) measuements of torsional as well as bending creep and dynamic Young's modulus of polymethyl methacrylate, (b) comparison between the experimental results and theories, (c) deduction of a distribution function of retardation times from the }experimental results. g. (d) derivation of a formula which correlates the transient and the -steady-state behaviours.. ,. II. Experimental (Transient Behaviour) Previous experimental researches(i) have laid the groundWork for an 'investigation ofthe transient behaviours of polymeric substances. One of the. main purposes of the present measurement is to acquire information that may conduce'to exact features of the time dependence of rec'overable retarded [deformation (so-called "reversible relaxational strain"). . By use of torsional deformation of a cylindrical rod as strain, it may be '. tt. 1) For instance; Kobeko, Kuvshinsky, Gurevich: Bull. Acad. Sci. U. R.S.S., 3 (1937), 336; S. D. Gehman: J APPL Phys., 19 (1948), 456; A. Tobolsky, H. Eyring: J Chem. Phys.j 11 (1943), 125; H.A. Robinson, R. Ruggy, E. Slantz: 1:'Ampl. Phys., 15 (1944), 343; J. A.. 'Sauer,J.Marin,C.C.Hsiao:XAPPI.Phys.,20(1949),507.''' ''.

(3) Viscoelas'tic Properties of Polymeric Substances 21i possible to make the measurement with high accuracy, shape and dimension of the test piece being unchanged during the 6reep, and to obtain facilities. in making the direct comparison between'the experimental results and the phenomenological theories. As the material to be used' in the pr,esent investigation, polymethyl methacrylate(2) was selected, because its molecular structure. issimpleanditsexperimentaldataarelacking. ' . ･-･･.. Generalarrangementofthe '･' .' ･','--, 7. apparatus is shown in Fig. 1, in which 1 is the cylindrical test piece, its diameter being 8.50mm.. 4. Two shafts arranged in strict. 4. alignment are firmly coupled to it through chucks 2 and 3, and ,. are clamped to a stand with vices 4 and 5, one of the shafts resting through a ball bearing 6. A wheel having a sufficiently small moment. 5. 6. s. 2. 7. 3. 5. e Fig. 1.. of inertia, its diameter being By hanging a weight 8 over the wheel, the test piece is twisted. With the aid of a small mirror 9 aMxed at a desired position on the test piece, the amount of strain is measured by the method 13.6cm, is attached to the shaft.. of telescope and scale.. After ascertaining these was no observable strain due to setting of the test piece by leaving it unloaded fosc a week, a load was applied'and the v. .. measurements were made at -a room temperature. A curve typical of those obtained experlmentally is shown schematically 'm Fig. 2. 0n. .s St3 .. sudden application of a load, there Isqnmstan- S taneous elastic deformation,theamountofwhich. is approximately proportional to. load. This is followed by a. thatofthe ￡ elipt retardeddefor- V. Time mation (creep) which was, inthe present experiFig. 2. ment, still ebservable after 104 mmutes. On sudden removal of the load at any desired time during the creep, an instantaneous partial recovery takesplace, followed by a retarded recovery (creep recovery). There was no observable irreversible deformation in any of the presentexperiments. The' instantaneous deformations due to the application and the removal of the load seem to be of equal amount,(3) and the rigidity calculated from theM gives the values of 1.2--1.1×10iO dyne/cm2. Attention 2) 3). Manufactured by the Fuji Kasei & Co., Ltd., Tokyo. The latter seems to be slightly smaller than the former..

(4) 22 ' KShibata. should･be paid to the fact, however, that it takes a few seconds at least to. set out the measuement aft6r the application or the removal of a load whereby the instantaneous deformation mentioned above (apparent instantaneous deformation) should involve a portion of the retarded deformations corresponding to the time interval of a few seconds. The exact value of the instantaneous rigidity should be computed from experimental results obtained in supersonic frequency range. (The problem is not treated here.) Some of the observed values are plotted as shown in Figs. 3, 4 and 5 8. ×Io;3. ･-. 7 ct s 5.s3 xl07c g s (3o℃). r.{1>. 6 `. 5. 'X' 4 tig. '3. pt =1.66 ×107c. g. s,(2ooc). 2. /. (2). (b (2tJ. o. e. ･Q･. 10 Time(min)lOO 10VO 10000. /. Fig.. 3. creep recovery curve (lr) corrbsponds to creep curve (1).. (Unloaded after 2280 min-loading.) '. Creep recovery curve (2t) corresponds to creep curve (2).. (UnloadedafterlO080min-loading.) '-. e. taking logarithm of time as abscissas. From the figures, it may be noted that the following equations are quite well. XlOZ!. ･g3 ￡Zl. 2'. ,". LZ/. /. ./.,,.,,.., /. ecS,'l9.8i/m,':'Xl/...d-'gg.l6einlSl'iP/lrgeb";::t:'#.e.?g./],8.e. i. o. Fig･ 4, ' and in the case of the removal of the. ,.

(5) Viscoelastic Properties of Polymeric Substances. Joad7==a'--bao'lnt '.' 't-(2) .. 23･. ziLo--3. nts3s,xJoacgs({;c). t where. 7: strain obssrved,. L. ifo: constant stress,. . t: tlme, a,at, b: constants independent 'oftime.' ･. 3Lgezi￠stI:ft61S}'. It may be expected that ･es=-. Eqs. (1) and (2) would not ; describe the behaviour for ve extremely small as well as. v. 2. at = 1. 66 x107 a g. s. (2(63jC). large time scale. ,. Measurements were made at a room temperature on the. time dependence of the de- 1 pression of middle point of. a Tectangular beam under a. u')tz). constant load applied at that point, both ends of the beam. (Q')･. being supported on knife edges.. '. tt. '. ' 5. Credp redovery curves (lr), <2t) and (3t) correspond respectively to creep curves (1), (2) and (3). (Unloaded after 10 min-loading.). Recoverable creep in the Fig. bending' deformation y of the beam is also described by the equation. t. Ol Timele.mlri')' '{Vq '. y =a+b. 'lnt , (1'). which is similar to Eq. (1) over a considerable time range from a few seconds .. up to3×103 min. The apparentinstantaneous Young's moduli computed from the corresponding deformations provide the values of 3.3×10iO' dyne/cm2 <160C) and'2.9×10iO dyne/cm2 (280C). ･. .. III. Corttparison between Theories and the Experimental Results on Creep. The theories which account for the linear behaviour of creeP with logarithmic time under constant load are clapsified into two groups; they are:. (a) Thedry of'Tobolsky and Eyring(4) in which Newtonian viscosity in the dashpot is replaced by Non-Newtoniap.viscosity (strictly speaking, Eyring's viscosity), and therefore the solqtion of the equation does not satisfy the principle of superposition. (b) Description of the behaviour by use of an 4) A. Tobolsky, H. Eyring: X Chem. Pdys., 11 (1943), 125. ..

(6) ' 24･･'.., K,Shibata tt. infinite number of Voigt elements placed in series which is charact･erized by' a distribution function of retardation times deduced from'experimental data.(5>. The procedure is entirely based on the assumption that the principle of. superpositionisvalid. ... ･. Part 1. Comparison between Theory of Tobolsky and Eyring and the. 'ExperimentalResults. . , .,. ' ' ' Tobolsky and Eyring presented a molecular model to interpret the elastic viscous properties of polymeric substances, i.e., they described creep phe-. nomena in terms of molecular rate and flow processes by using the postulahteiOfnollohwag.ntgheeqtuOat9ilorna!,e Of Strain fOr the i'th type g.f upit process is glve.n by .. ' dd7t = i, dd{i +nixi khT exp(--- dkiT;}-')･2sinh 2gZ)z),･L' (3) ' 'i. '. '. ai:totalstressdistributedovertheunitprocess, ' ni: number of the unit processes in series per unit length along a line perpendicular to a plane over which the stress distributes･. ,. itself, ' .. ' Alk: number of the unit processes per unit area on the plane over .. ). , which the stress distributes itself, - ',.. , Gi:appropriatepartialmodulustobeassocjatedwiththeunit. process, tt-. '. Xi: average distance projected in the direction of stress between･ e. equilibrium positions in the relaxation, process,. dk': free energy of･ activiation for the process,,. '･le:Boltzmann'sconstant, ,.. h. : Planck's constant, -. '. T:absolutetemperature. . . .. '. For small loads, it may be considered that the total stress is distributed over the primary bond in the molecular model (purely elastic) as well as the secondary bond (viscoelastic). In this case the appropriate equations are of. the forms: '. ' d,or,-=3,SOf, ･ (4). ". d. detY,'. '. giddCt2+A2sinhB2o2,. ,,. ' (5). ==p. .. '. '. 5) B. Gross: X AIu?l. Phys., 18 (1947), 212; 19 (1948), 257.. '. '. 1.

(7) Viscoelastic Properties of-?olymeric Substances 25e where '. '. A2=2n2N2khTexp(-dkFT2±) '''' . (6) u. ' ' '. X2 ' . . (7) B2==2A72kT･ .t .. '. ..,.i," ,t,h.e .C.a.S,e,,?,g.a?,P.i',Ca.t,i2.-".,O.f. .a,.C.O.".S.t,a,".t.l9a,d,v,E.q,S･ (4) and (5) are soived. tgh B2(a02-- Gi7) ==tgh B2 (aO 5Gi 70) e-dit (s) '. where A2B2Gl G2. :･(9>. a== Gl+ G2. 7o:thevalueof7att=O. ' '. t. Within the range of. 1)}> e-B2(tro-GiY) E> e-B2(ao-GiYo) (10) one gets an approximate form:. tt. '. tt. or= -Zii' +B,iG,in,{l-+B,iG, lnt (n)(g) On removal of the limitation of e-B2(ao-GiY))}>e-B2(cro-GiYo) g '. ny. 7isexpressedbythefo.11owingequation: . . . or=-X'+B,iG,in-{}-+B,iG,.in(o+t)tt. (i?). t. where - . a-' - ･･･ '. '. ' t tt. o=:-2-e--B2(ao-Giyo) ' ' '. ' '. ･'. It may be expected front Eq. (12) that the straight line representing the relation between 7 and lnt at mediu' m time scale would bend upward against. time axis in short time region. On account' of the absence of this tendency in the present experiment, it seems reasbnable to assume that the value of 0. woUldbemuchsmallerthanafew'. seconds. ' ･. Eq. ?IUIP)P,9glneggtehtast the eXPeriMental formula (1) corresponds to the theoretical. '. 6) Eq. (10) involves the quantities the value of which can not be determined by s. experiment. The range of validity of Eq. (11) is somewhat ambiguous..

(8) K: Shibata. 26. a- 2 + B,iG, ip, '{Y･. bao=. i. (13)'. ' (14). B2 Gl '. 'In order to obtain an equation which is Valid for the creep. recovery, Eqs. <4) and (5) axe solved under the condition that ai+ff2=O. The solution is of. the form:. tghB2G17 2 Ftgh. B2G,v(O) e-at 2. (15). -where. " 'x 7(O) : initial v/alud of the creep recovery. Within the range of. --. 1>e-B2GiY>e-B2GiY(O). '. an approxlmate expression of 7==- B2iGiin -{;: -- B2iG'iint. (16)(7). is obtained. Assuming that Eq. (2) corresponds to Eq. (16), one gets a' == -- B;Gi ln -{}-. bao== 1 B2 Gl'. (17) (18). In the present experiment, the lgad was removed in the stqge where the ･creep was still taking place, ln view of the experimental results that thd slopes of the creep and the'corresponding creep recovery curves plotted against logarithmic time are of the same absolute value, it may be expected -that the straight lines in ty-lnt diagram obtained on the removal of load at. any desired stage of the creep under the same load should be parallel to ･each other. The speculation is verified by the observed values shown in IFig. 6.. The time to when 7 is equal ,to zero in Eq. (2) is given by lnto= a7bao. 'By substituting Eqs. (17) and (18) into this equation, one gets to =2/a.. Remembering that the value of to depends on the time of removal of load, other conditions being the same (of Figs. 5 and 6), one may, conclude that a in the case of creep recovery depends upon the time of unloading, while ct in the case of creep at a constant temperature is characterized by 'the specific properties of the material.. 7) of. t. foot-note (6).. p. '.

(9) Viseoelastie Propenies of'Polymeric Substances. 27. 2 x/oL3. atfL66x/o7cg.s.(26oc). .ses. $/ i. ･x. ' ･ (1). ERIIiiE=t± ot 10 ･IOO Time(mm). o/. Fig. 6. Creep recovery curves (1) and (2) correspond to the same creep curve. (1) : Unloaded after 90 min-loading. (2) : Unloaded after 10 min-loading. 4. The results of the theory of Tobolsky and Eyring, i.e., Eqs. (11) and (16) show that the absolute values of the slopes of creep and the corre`. sponding creep recovery curves plotted against logarithmic time are not only of the same value, but are determined by the characteristic constants of the material. The experimental results support the former, while they deny the latter showing that the slopes are proportional to the amount of ao. Tobolsky and Eyring have already pointed out that B2 is inversely proportional to the amount of ao. Hence, it seems reasonable to corclude that the duality in the. physical meaning of ct mentioned above, originates in that of A2. It has been left unanalyzed, however, which of the parameters in A2 will be responsible for the discrepancy. In short, in order to interpret. the experimental results by the theory of. Tobolsky and Eyring, somewhat ambiguous modifications have to be made for the pararneters A2 and B2 inv61ved in the fundamental equations..

(10) 28" KShibat'a. ' Part 2. Deduction of a Distribution Function of Retardation Times from '. theExperimentalResults. ' ,. Some attempts have been made to find out a set of appropriate retar･dation times in terms of which experimental results on creep would be described.. 'For example, Robinson, Ruggy and Slantzf8) have found three retardation times of polymethyl methacrylate with the aid of graphical analysis of their 'tensile creep curves. However, on account of the uncertainity in drawirig 'tangents to the creep curves, their graphical method gives its ground to the -more elegant one presented recently by Gross,(9) i. e., a mathematical deduction. ･of an appropriate distribution function of retardation times from experimental. vcreep data by considering them as a continuous spectrum. For the determination of the complete form of the distribution function, measurement should 'be made over extremely broad time range extending to short as well as to long time scale, but it is hard to realize. However,'an approximate distri'bution function deduced from a set of creep data for considerable time range may still serve to discuss qualitatively the mechanical behaviours of high. k. ,. polymers. Furthermore, when strain in creep is expressed` by a simple 'function of time, the general method of Gross can be replaced by a simpler ･one described below, so long as the distribution function is concerned.. Assuming that the principle of superposition is valid, the behaviour of .recoverable creep is described by use of an infinite number of Voigt elements .arranged in series, i.e., the creep function "(t) is given by a normalized rdistribution function F(T) of retardation times T as follows, in terms of which the continuous system of Voigt elements is characterized.. ' aJn(t)=B[1-S,eeF(T)e-ti'dT] , , (19) where.Bisanormalizingconstant. L ''' '' ,i The rate of creep {p(t) is given as follows by using the retardation. frequencyfunctionp(s): .,i . . , ,, , '. ' ':' g,(t).{'iL"-:s,Oep(S).-tsd, , ,, (2o) tt'. tt. tt. fs= i/.: retardation ,frequency, '･, . >(BF(T)dT==-p(s)-!gLt. . .. tt. '. ' ' As the deformation was recoverable in'the present measurement, it seems 8) H. A. Robinson, R. Ruggy, E. Slantz: .1: ApPl. Phys., 15 (1944), 343. 9) B. Gross: ,L AIipl. Phys., 18 (1947), 212; 19 (1948), 257.. ". L.

(11) Viscoelastic Properties of Polymeric Substances 29, reasonable to assum6 that the amount of strain caused by creep should approach asymptotically to a finite equilibrium value with time. To satisfy thjs condition as well as to be able to represent the behaviour of strain in Sdh cr i:gtitMweorpegalOanthettheerse3P2/i'dM2agasiffoOifoMwusia: (i) may be modified by intro-. uO. 7(t)=a"+b' ln (" +t)-- b' ln (g+t) (21). '. '. '. '. '. '. '. (g,'L7Ggi "' ･'- ' '. '. and " and g are constants with the dimension of time, and are much smaller. than a few seconds and greater than 104 min respectively at a room ,. temperature.. Eq.(21)canberewritteninthefollowingform:' ', '. '･ 7(l)==7,(1+-g)tll..3((gitt]) .-. (22) ''. '. '. '. . ., 7o==ar'+b'in-!ll- ･ .'' '. Hence, the rate of creep g(t) in the experimental formula (22) is ex-. pressed by , '. t. -.. '. '. '. ' (23) q(t'==-il8'{oi.t-glt) ･ '･'. tt By substituting Eq. (23) into Eq. (20) and applying Laplace 4inverse ,. transformation, one gets t. tt･ t ' p(s)==2ililiS:Z]O.O.he'-',(,s,llLt--'gi+t)etsdt=-ll'IJ(e-"s-eTgs)..iV. ' ' Theref6re, ' ' ' '. '. tt. BF(T)d.=. -::17 (e'"fT Iile-gh )d.. where '･ Hence, .. '. . B..-gl:'.l:e-""T7e-ìTcerff'in(g/di. ' ' . (24) , F(T)dT==1.(lg/,y)(e-"il;e-,li'])dT .. ',. '.

(12) 30 ' '' K.Shibat'a. ' In the case of'ordinary creep'where the extremely large value ofttT is nbt taken ìnto account, an' approximate form shown kel･ow may be used, in ''. whichtheterme-gi'inEq.(24)isneg!ected,i.e., ' ' tt. ' ' .' '' F(T)dT=constantx-lie-eiidT･ (25) Eq. (25) is of an unsymmetrical form, its mode being at T--". (One type. ofThePearson's distribution function). '' distribution function (25) has also beeri given by Voglis(iO) as an / after effect on mica and wax. It will be experimental result of dielectric worth making similar measuTements on polymethyl methacrylate. ). IV. Experimental (Steady-State Behaviour) Measurements of dynamie Youpg's modulus of polymethyl methacrylate were made with bending vibration frequencies of 500 and 100 cycles and at various temperatures ranging from a room temFerature up to 1200C. In. ,. accordance with the temperature range where the experiment was carried out,. txmo methods were employed: one was an electromagnetic method which :eOmUipderbaetuarPePiriggioUnP EObo9vOeOCgb.tg9 Other was an ei,ectrostatic one used in the. For excitation of the bending vibtation, an electromagnetic device was used, which consisted of'a small iron plate cemented on the surface of the ' test piece and a magnetic telephone receiver facing to it at an appropriate distance. The oscillatingcurrent was picked-up with another electromagnetic device of the same construction at the free end of the test piece. It was amplified and rectified, and was read on microammeter scale. The mass of iron plate was made negligibly small compared with' that of the test piece. In the present measurements, error due to the mass of iron plate (about O.05g) is estimated to be 1.v2%. Correctionfor rotationalenergy is neglected. WithlOnUt tfiiegngfixCpaenrilmeernrtOsr'. made in the temp6rataure reg'ion aboVe 900C, the. bending vibrations of the beam were excited and picked-up electrostatically by using copper foils and plate electrodesin the places of iron plates and telephone receivers respectively. Measurement of minute capacity chang of. the receivino' condeiLser due to the vibration of the beam was made with. o. Foster-Seeley'scircuit. ･････e ･ ' With either experimental arrangement, a resonance curve was obtained by reading the out-put current at various oscillator frequencies. Dynamic Young's modulus E and the longitudinal viscosity are calculated from the. 10) G.M.Voglis: Zeits.fPhys.,109(1938),52. ･ .. e. ,.

(13) Viscoelastic Properties' Of POIymeric Substances 31, resonance frequency L where the oui-put current exhibits a maximum value, and from the half value breadth of the resonance curve respectively. (The viscosity problem is not treated here.). ' between E and f is shown below: The relation. '. '. E=4"21I, {44.'f2' ' ' (26). where ･. '. '. '. tt. '. l: length of the beam,. ,. ' '. '. 'p: density of the beam, re: radius of gyration of the 'rectangular cross section of the beam, kn: constant fOr each mode of vibrations. The fundamental vibration as well as the first and'the second modes of vibrations were used in the present experiments, the measurements of their frequencies being made with Hay's bridge or Campbell's bridge. In carrying out the measurement, the change of the resonance frequencies ･of the beam of a constant length caUsed by the elevation of temperature was traced. On account of the fact that Young's modulus, accordingly the reso-. nance frequency, decreases with increasing temperature, the tracing of temperature dependence of Young's modulus at a definite frequency is not easy. Hence, the determination of the value of Young's modulus at 500 cycles, for example, was made by interpolating two values obtained at the frequencies of the first and the second modes of vibrations, an appropriate length of the beam keing selected so that the former was lower than and. ' the latter higher than 500 cycles. ･. The observed values obtained in･the present experiments are plotted as. q. shown in Fig. 7 together with those given in literatures. Curves I and II are. the results of the present measurements in which the former was obtained ,. at 500 cycles and the latter at' about 100 cycles. The observed values at 2CO cycles seem almost fall in with those at 500 cycles in the temperature region. lower than 900C. The points 1 and 2 show.the values of the apparent instantaneous Yotmg's moduli. '(of p. 23)-Curves A, B and C represent the temperature dependence of static Young's moduli,' for which little of the experimental conditions was reported. Curve A is the..results of rlieasurenient. by Bartoeìi) using "quick acting testing machine". Curves B apd C show the results obtained by Robinson, Ruggy and Slantz,(i2) with a thin rod a's -the test piece. Curve B shows thevaluecalculated from the tptal elongation under constant load, and curve C shows that from the apparent instantaneous elongation. Hence, it may be presumed' that the points 1' and 2 would lie. tt. tt t '`:･. '. 11) Cited in the paper of Robinson; Ruggy and Slantz. (loc. cit.). 12) loc.cit. .. ' ,. 1. t tt . '.

(14) K. 'Shibata.. 32. ¥,. 'L-'･K･ fx D.2. IC5boy. x aimoy ' lO". A. .A. E. R･ 2. b. .. N.t. ca. = pt. 1oe ,. = rs o g･. c. pt 2E). g. ･> l re 2no jooot. tIFt 't ･1llJ kN X. 1oe. B. INN llx l. x xNN N. xNxN. XNN NXN Ns)-N:. p. NN ... h..... 30 '.60, 90 120 .150･.180. Temperaturc oa ' ' Fig. 7,. -. '. on the extension of curve C. The points D and K are the experimental data given by Weber and Goeder,(i3) the former being calculated ftom the apparent instantaneous elongation as well as compression amount bf a rod due to loading, and the latter being computed from the compressional vibration ekcit'ed by.Kundt's method or a similar method mentioned in this section. tt is seen that the data obtained by Weber and Goeder show good agreement with the present experimental results' . Point L is the result of measurement. ' penettating by Schneider and Burton,(i4) made at 5 Mc/s by observing the '. 13) Weber, Goeder: Phys. Rev., 61 (1942), 94. No description on frequencies. 14) Schneider, Burton: .1: Ampl. Phys., 20 (1948), 48. There is no description on temperature. Presumably, the measurements were made at a room temperature..

(15) Viscoelastic Properties of Polymeric Substances 33 power of supersonic wave through a plate at various incident angles. Dotted curves represent･ the recalculated values from data which have been presented.. by Alexandrov and Lazurkin<i5) in terms of deformation amount of a test piece with definite dime4sion due to the oscillating exteTnal force with constant amplitude. Figures 1,10, 100 and 1000 on the dotted curves show the frequencies per minute. The-calculations were made by assuming thattheamount of 4%' of the strain cprresponds to E==1.2×107 dyne/cm2.(i6) However,. whatever the assumption may be, there was an inevitable discrepancy between. the results given by Alexandrov and Lazurkin and those given -by Robinson,, Ruggy and Slantz, the cause of which still 'remains unsolved.. Though Fig. 7 seems to be a random colleotion of the fragmental results: given in literatures, it may still serve to obtain some information on tempera--. `. ture and frequency dependences of Young's moduli. An inspection of Fig. 7 reveals the facts that: (1) There is a certain temperature range where a marked 'dependence･ of Yoopg's moduli on temperature as well as on frequency is observed. For. example, at 1000C the dynamic Young's modulus (at 100 cycles) shows the value of about one hundred times greater than that of the apparent instantaneous Young's modulus. These behaviours may be attributed to.the elastic. dispersion. .' ,. e. ,. ,(2) Outside of the said temperature range, temperature and' frequency dependence of Young's moduli seem to be rather slight. The steady value of: Young's modulus in the lower temperature region is about one thousand. times greater than that in the higher. These mechanical behaviours of: polymethyl methacrylate may be expected naturally by treating them as relaxational phenomena. (3) In the simplest model (mentioned in Part 1 of Section III) representing the main features of the mechanical behaviour of high polymer, thetotal stress which is in equilibrium with the exteimal force, is divipted intotwo partial stress, one fi (purely elastic) being distributed on the primhry. bond and the other fe (relaxational) on the secondary bond. Let･ Ei and Eb･ be Young's modulus of the primary and the secondary bond respectively.(i7>' The total elongation under constant load is characterized by El, while the. instantaneous elongation by Et+&. The reasonable value of El estimated from Fig. 7 is of the order of 107 dyne/cm2.(i8) At present, the knowledge-. 15) Alexandrov, Lazurkin: Acta Physicochimica, 12 (1940), 547. . 16) In order to bring the dot.ted curve 1 to the group of curves B and C as close as･. possible,thesaidcorrespondenceismadetentatively. ' ･. 17) It maY be considered that the temperature dependence of those is father slight. 18) lt may be remembered that the value is of the same order of that of raw rubber'. ttt ataroomtemperature. . ' .,'.･. '.

(16) 34 KShibata. concerning the value of Ela(i÷'EL+Eb) is limited to the fact that it should be. greater than the value of 6×10iO dyne/cm2, and its exact value is left undetermined on account of the-lack of measurements at ･considerably high frequencies and at low temperatures. In the simplest model mentioned above, it is considered that the value of the partial stress attributed to the secondary. bond would approach asymptotically to zero with time. The propriety of this speculation also still remains undetermined owing to the lack of facilities. in making measurements under extremely long-time load application. To determine the propriety by measurements of smaller time scale, the measurements should be made at higher temperatures whereby distinct irreversible flow would be observed and dithculties of its separation from the total deformation would not be overcome.(i9). '. V. Correlation between Transient and Steady-State Behaviours. , (. Part 1.. In Part 1 the correlations between the transient and the steady-state behaviours are treated by a method which is suggested by that of Gross.(20) Mathematically, the latter problem is completely analogous to the steady-. stateresponse'ofanelectricalnetworktoanalternatipgvoltagesource. ' Assume that the total deformation y(t) under Constant Ioad Pb is expressed. by the f611owing equation: ' y(t)=,EF,[i+th,(t)]A'', . (27) '. where. '. '. g. Ets: instantaneous Young's modulus,. . F: formfactor, e iPf･(t):creepfunction. '. The deformation y(t) due to the application of var'iable load P(t) is given by Gross with the aid of the principle of superposition as follows:. '. "(t)=E'[P(t)+St.-..I'(')'q(t--T)d'] ' (28). where q(t) is the rate of creep. When the material is subjected to the sinusoidal stress, i.e., 19) In practice, it is dithcult to-distinguish flow from delayed elasticity. It is not reasonable to regard the deformation as flow because of its appa'rent linear behaviour with time. 20) loc. cit. .. ,.

(17) Viscoelastic Properdes of Polymeric Substances. 35. P( t)=:.Pb eiwt, . ' '. the steady-state admittance G(itu)' is expressed by G(ito)== ]i;l [1+A(tu)--2n3 (tu)]. (29). .･･1. '. '. t, , -1-. y (t)= G (i tu)･P (t). '. '. and '. ' '. ' A (a))==SoeOcos ((DT)･{p(T) dT,. (30) B (a))=S,OOsin (Q)T) ･ {p (T) dT . ,. Similarly, the steady-state solution of stress for alternating deformation ;So gisiVeexnpraeSssfeOalQbWyS' i'e', aSSUMing that the stress p(t) at fixed. '' p(t)=. .t '. ' tt ' iill[i-v}(t)]yo '. where ', ' ' '. ' '. ' ',. deformation ' ' ' (27,). L. '. V(t): relaxation function,. '. oneobtainswiththeaidoftheprincipleofsuperposition . ". -. P(`'= tilii [`"(t)-!t.`.Yf"'ip(`")dr]. where '. (28,). '. ip (t)=!Zdi'FP"- : rate of relaxation.. Putting y(t)Fyo･eiblt and P(t)=Z(ito)･y(t), the steady-state. impedance. Z(ito)isgivenby ･ z(itu)= lill [1-A(to)+iB"(to)]. ' (29,). where A-(a))==S,OOcos (a) T)･ip (T) dT, B(,,),. S,ee,in (. .).ip (.) dr.. (3ot).

(18) 'K.] Shibata. :36. Expressing the complex elastic modulus by. Eee (- in)=E,+2LEI,, (31) 'where. Et: dynamic modulus, Ett: elastic loss, one gets the following relations between modulus and impedance:. , E,=E;(1-A), ,E,,=:EtB. (32) The relation between impedance and admittance functions is given by. G(itu)･Z(ito)=1 (33). '. or by algebraic form: "-. A-=A(1+A)+B2B,.. B (1+A)2+B2' (1+A)2+B2 , ,. (34) '. '. '. tt. These correlations between the transient and the steady-state behaviours obtained by Gross leadto the deduction of the latter from the measurements ･on the former, i.e., when q(t) is obtained from the experimental results on -creep, A and B are calculated from Eq. (30) whereby A and B are computed from Eq. (34). The discussion below refers to the frequency dependence of. Young's modulus of a substance, the creep of which is described by the. .following equation, a modified form of Eq. (1,), '. y(t)=a+b･ln (0+t). (35). ,g. Eq. (35) can be rewritten in the form:. y(t)Fy(O)[1+y(bo)ln"o+t]' '' '. , (36). '. where y(O) = a+b･ln 0.. By comparing' Eq. (36) with Eq. (27), one obtains '' aP'(t)= b ln 0 +t. y(o). (37). 0. Therefore,. q(t)=y(bo)(loge,,) tto !-t+Co (3s) ' j. i.

(19) Viscoelastic Properties of Polymeric Substances 37 '. '. The value of b is obtained as the slope of the straight line showing the relation between y and Int, while the value of y(O) is left unknown so long as the measurements ofcreep is concerned. However, suppose that the value. of the instantaneous Yo'ung's modulus Et(i72!i+Ele) assumes the smallest value (6×10iO dyne/cm2),<2i) the estimated value of y(O) is calculated from the following formulas:. y(O) == F. Po, F=. l3. (39). - Et 48I .. where Z: moment of cross section of the beam. -. I: effective length 6f Jthe beam. For example, from the exFerimental data that the creep curve of a beam g. with l=25cm and Z==(O.477)3×1.50/12 exhibits a slope of b==2.6×10'2 under Lconstant "load P =500×980 dyne, one obtains. ' --iZl･4i ×lol,O,` ×soo×gso ,,.o.2o N(O)F-E .pb=. '. '. Hence, C = O.434 ×. b ==O.050 y(o). Putting q(t)= C/(t+0), A(tu) and B(tu) are calculated from Eq. (30) as. '. q. follows: '' ''. ' ' , '.1 A(oj).,,cS,"ec3s+tottdt=lc(-6.stuo'･ci(ip")--gintus･si(too)),'. .... . (40),. B (.).,,cSoe'e s"m+ait. ,dl=c(--cos ."･si (. o) + sin too･ci (to o)) .. ' ' ' The results of numerical calculations of A/C and B/C are plotted in 'Fig. 8. Combining Eq. (40) and Eq. (34) A- is obtained whereby the value. t. t. of E7E? (ratio of the dynamic Young's modulus to the instantaneous one) is calculated from Eq.(32). The. calculated values are plotted in Fig. 8, putting C:r-1 and C=･ O.1.. ". Fig. 8 gives a suggestion to determine, (1) the propriety of the speculation that, in order to represent the cTeep behaviour in shorter time region, 'the factox lnt can be replaced by ln(e+t), (2) the reasonable value of ", by. comparing the experimental results obtained from the measurements of. t. 21)ofp.34. '. '. ･.･ ,... .t. '.

(20) K. Shibata. 38. s. rty,[i. t5. tlll. {.o. C :oA. tA ". "4N. ,". a5. x. t- A. .. tr.. oo .(ISis oo `= va co =% Fig. 8. ･. Young's moduli over a wide frequency range with the curve of 1--A(=Ei/Ets). However, there is so far no experimental result to refer except those given in the present paper (4.63×10iO dyne/cm2 at 500 cycles, 4.48×10iO dyne/cm2 at 200 cycles) and the value of Schneider and Burton (5.64×10iO dyne/cm2 at 5 Mc/s). Putting Ele==6･･-7×10iO dyne/cm2, a tentative attempt on the estimation of B from the above experiMental data gave the values of, 10-7--. 10-8 sec, but in those cases the deviation from the curve was never less. than 10%.. e. . Part 2. According to Dunell and Tobolsky,(22) Kuhn(23) had shown how, in case of a material for which the creep curve is linear with logarithmic ti.me, one might predict the,dynamic losses in free vibrations. They have pointed out 22) Dunell, Tobolsky: J Chem. Phys., 17(1949), 1001, letter. No description on the deduction in''their letter.. 23) Kuhn, Kttnzle, Preissmann: Hizlv. Chim. Acta, 30 (1947), 307;464;839. Up to the present (1950) no further information is obtained except brief abstract on Chem. Abst., 41 (1947), 3652; 4945; 5743.. ". ,.

(21) Viscoelastic Properties of Polymeric Substances 39 that the rectangular distribution function' of relaxation times, in terms of. which Kuhn described an approximate mechanical system, had previously been extensively investigated by Becker(24) foer ferromagnetic hysteric behaviour, and also that if st･ress relaxation is linear when plotted against logarithmic time, by using the same function, the following simpler correlation between stress relaxation and dynamic properties is obtained: ' '. ndyn.w=2. T × (slope of relaxation curve plotted vs. Iogio time) (41)' Eo 10g,lo. where 1/b K tu K 1/a, opdNn: dynamic viscosity coefficient,. eo: fixed strain at which the stress relaxation measurement iQ carried out, tu: angular frequency, a, b: constants which are, respectively, smaller than and greater. `. ' thanthesmallestandthelargestrecordedtimes. According to Weiss, the ferromagneticsubstance containsa large number of small domains having intrincic value of magnetic intensity. In the absence of an external field, the resultant magnetization of the entire crystal is zero. owing to the random orientation of domains. The magnetization of the crystal due to the external magnetic field is attributed to the instantaneous. and retarded orientations of these domains, and it is considered that the '. 1. principle of superposition is valid for retarded behaviour characterized by the t distribution function of retardation times. Moreover, experiments on magnetic. behaviours have revealed that the magneti,c induction curve B(t) is linear with logarithmic time overabroadtime range and the loss factor is independent of frequency over a broad frequency range. Studies on the mechanical properties of polymeric substances have also revealed that the transient behaviours are linear with logarithmic time and that the elastic loss is relatively independent of frequency over a considerable frequency range.(as) Thus, the magnetic behaviours of ferromagnetic substances seem to be completely analogous to the mechanical behaviours of high polymers, and yet little attention has been paid to the analogy. Owing to the lack of facilities in obtaining information concerning details. of the theories of Kuhn as well as Dunell and Tobblsky, an atteMpt･on the derivation of the said correlation is made on the basis of Gross' theory, by 24) Becker, D6ring: F2rromagnetismus (J Storingeag Berlin), (1938), 103.. 25) A Gemant, W. Jackson: Phil. Mizg., 23 (1947), 950: Dillon, PrettYman, Hall: .1: A2PL Phys., 15 (194tl), 309: of letter of Dunell and Tobolsky. (loc. cil.).

(22) 40 ' KShibata using a frequency function of the rectangular form and also the function (24).･. ' t tt. '. Case 1. Stress relaxation at fixed strain.. In Gross' theory in which the creep is treated by a continuous system of Voigt's model while the relaxation by that of Maxwell's model, the mathe-. matical expressions describing the relations between the creep and the dynamical properties are quite similar to those between the stress relaxation and'the dynamic loss. Hencei assuming that the principle of superposition is valid, the relaxation function V(t) is expressed in terms of the distribution function F(T) of relaxation times T as follows: ili (t)==R[1--- j,OOI-;' (T) e-ti' dT]. (19,). ･-. '. where B: ratio of partial elastic stress to the relaxation. Therefore, the rate of relaxation is expressed in. final. }. value of stress. terms of relaxation. fr.equency function P(s) as follows:. !'. ' '' ip(t)=tdr-=s,oepL-(,),-ts,ds. (2or). tt t/.tt t. where. s==1/T: relaxation frequency, i. B l7<T) dT==-P(S). From. Eq. (20,) and-Eq. (30'), B(to) ･is expressed as follows-: B(to) -s. i. g. ds rg;･.. 6co-p'(s) .9+dS,,.. t. (42). Suppose that the distribution function of relaxation times is of the form:. 11. Y -.p 17(T)==iln(T2/7'1) ,7･' KO ' ,e. (Tl <T < T2) (43) (Tl > T, T > rr2). where 1/ln(T2/Ti) is a nomalization factor. The corresponding relaxation frequency function P(s) is rectangular form :･. given by a.

(23) Viscoelastic Properti,es of ?olymeric Substances 41 ' - <s< --;') 'in(-￡-2/Ti)'(t7i･2' 7i.i. B-(s)=:, ' (44) o ,(i>s,s>km) Combining Eq. (44) with Eq. (42), within the range of ., K' -IL K },. ￠ the function B(tu) is expressed by. ' B(tu)=in(.TB-2/T,)Sllli,("S;E3'k2'fitiin(TB-,/T,}'{}' (45). '. rr. '. From Eqs. (45) and (32), the elastic loss E" is obtained as follows:. E'l== Et BC>tEle i. (.'`,9/.,) '{l' .<`6). ,. ' Eq. (46) shows that when the relaxation frequenecy function is of a rectangular form, the elastic loss is independent of frequency within the range. of7i<KJ-<KT2. ' ' ' tu. Next, combining Eq. (19') with Eq. (43), within the range of Tl <K l <g( T2. tt relaxation ' one gets the following relaxation function for the rectangular frequency function: '. ' KIF(t)g>{B[i+i.(.i,/.,){O･577･'･･+in-Glli-}] ''(47) '. t'. whereO.577......isEuler'sconstant. ' .･' "･,' ' ., Substituting Eq. (47) into Eq. (27'), the total stress P(t) under constant. deformation is expressed by ' ' ' p(t)c)yyoEle[!-B-I1+ 1.0'i.7,7/.''i) -- ln ln.,'/2.,) )]. ' ' '-YoEleBi.(i/.,)'lnt･'･. '. ･<4s) '. If the stress relaxation ls found to be linear with logarithmic time, i. e.,. P(t) == a' --- B' lnl (49) ' one may make the correspondence between Eq. (48) and Eq. (49) in which the constants a' and B' in Eq. (49) are expressed as ･follows:.

(24) K.-Shibata. 42. ct'==iyoEle[1----i(!}(1+ 1.0i5.727/i'i) -- lnl(n.2'/2.,) )] (50). ,(9'=yo El B. ･ (51). L- 1. ln (T2/Tl). Eq. (51) can be rewritten in the form of. ln(T2/Tl) yo. Et P ..AL' (s2). From Eq. (52) and Eq. (46), the relation between the elastic loss E" and the slope of relaxation curve B' plotted against lnt is obtained as follows:. Yo 2. E,r >,,f B' .Z (s3). When the relaxation curve is plotted against logt, the following equation is obtained.. Ett cbe 2 yo logZO. i. T.B' (s3,). Case 2. Creep under constant stress.. If the measurements on creep behaviour are shown by Eq. (1'), similar mathematical tTeatment gives the following relation between the imaginary component of compliance B/Ele and the slope b of the creep curve plotted against logt:. ]li} B(tu)== 2pbZog2o (s4)`26'. t. Thus, when the transient behaviours are linear with logarithmic time over a considerable time range and the principle of superposition is valid, the simple correlations between'the transient and the steady-state behaviours are derived so long as the frequency functions are of rectangular form. If the form of frequency function is flat over a considerable time range, it may be replaced by an appropriate rectangular function without significant error. It seems therefore most reasonable to conclude that the correlation mentioned above still remains to be approximately valid over the corresponding range, In part 2 of Section III, a distribution function of retardation times is given, in terms of which the linear relation between the creep under constant load and logarithmic time can be described within the ordinary time range. If. the contribution of g is neglected, one gets from Eq. (40) 26) No literature concerning the frequency independence of imaginary component of. compliance has been given.' '. '. 4.

(25) Viscoelastic Properties of Polymeric Substances 43 .･ //gi,B(tu)=c'f-lli-･ Therefore, within the range of oK 5 Kg, 'the' imaginary 6omp6nent .'of. '. compliance is expressed by. ` , ]￡iB(to'=ii}C{;-==±y(o)biogLo"`liL '. tt. Tb .･ '.'. (55) =2P(o)log!o'.. ' In view of the fact that Eq. (55)is of the same form as Eq. (54), it t. ,. seems most reasonable that the two constants " and g which chracterize the function (24), correspond respectively to retardation times Ti and T2 in the rectangular function. When the contribution of g is taken into account, the function B(tu) is of the form:. B(tu)=C{---costu0･si(tu8)+sinto0･Ci(tu") +cos tug･si (tug)-sin tug･ ci(tug)}. Li･s.:s,:.:x.g-:.a-z;-･. lim sinx･Ci(x) =O, b X-O /i.m.. Ci(x)==o.. ". Assuming 0K 8 K4-, an approximate formof B(tu)c>tdC･ g is' obtained, ,. ' valid. ' and Eq. (55) still remains to be approximately By a quite similar treatment on the`case of linear stress relaxi tion (Eq.. (49)), the following results are obtained within the range of 0MK ts Kg-, 0 and g being replaced by the corrsponding constants B and g, respectively.. Eii.EbB(tu)-EtC-lg.,. l(g)'(iogf,).{iL ' ･"(56) '. -. rB' 2 y(O) logio. where. e. c-= pB('o) iogf,.

(26) 44･ '-' i K.Shibata Eq. (56) i's identical with Eq. (53').'. Comparison between the losS factors, predicted from Eq. (53') (stress relaxation) and from observed values, was made on rubber-like substances and textile fibers by Dunell and Tobolsky. In most of the cases, the losses predicted ficom stress relaxation data were less than the observed dynamic. losses. ,. In order to make the similar comparison between the imaginary coin-. ponents of compliance predicted from Eq.(54) (creep) and from the observed value, it is required that the complex elastic modulus Ele(1--A)+iEtB obtained. by dynamic experiment is tranformed into B/Ele by the following transforr mation formula which is valid for any type of distribution function, i.e., '. B .･[EleB]. -E't == [Ele(1-A)]2 + [EbB]2･ In ordinary case, the second term in denominator can by neglected with-. -. -out significant error.. Attention should be paid to the fact, for example, that when the transformation is made on the rectangular function :p-(s), the obtained function p(s) is generally of less fiat form. Hence, it may by possible that B(tu) is ,independent of frequency while B(.tu) is less independent of frequency, and that the time range, where the relaxation is linear with logarithmic time, is fairly broader than that where the creep behaves linearly. (Transformation problem is not treated here.) The verificatioh of reliability of Eq. (54) by the use of data in literatures. on polystyrene, polyvinylchloride, polymethyl methacrylate and ebonite has ngt proved successful. Since the data in literatures are those obtained for different purposes, it is tihought advisable to carry out fitting measurements to answer the present purpose. By using the experimental data obtained on one and the same specimen, the comparison is made as shown in Table 1. From Table 1, it may be said that, except in case of polystyrene, there are still some discrepancies between the observed and predicted values. These discrepancies would arise from the difference of time scales in the measurements. ie., in'the case of creep, it took about one minute at least between loading and the first reading, while the measurements of the steadystate behaviour were made at frequencies of one half cyc!e or higher. In general, particularly in case ofrubber, the predicted values are a little. ･smaller than the observed, the results being similar to those of Dunell and. Tobolsky mentioned above. The elastic loss G't of rubber increases with frequency even in lower frequency range,(27) while the rigidity Gr maintains its value whereby the value of G'7(G'2+G"2) increases with frequency (G'>. '. 27) Dillon, Prettyman, Hall: f APPI. Phys., 15 (1944), 309.. '. th. ,.

(27) Viscoelastic Properties of Polymeric Substances. 45.. Table 1. Comparison between the imaginary components of compliance predicted from Eq. (54) and from observed values.. l .. Material. Deformation. Imaginary part of compliance. Observed. Predicted. 28. 480 170. 1. 1 × 10- 12 1. 2 × 10-12. b. 1.5×10-12 a. Torsion. 20. 1/2. 8.9×10-12 c. 4.7×10-12 a. Polystyrene. Torsion. 25. 1/2. 1.7×10-i2 c. 1 5×10"12 a. Rubber (fi1:er). Torsion. 30. 1/2. 7.-9×10-9 c. 2.4×10-9 a. (Commercial product). Elongation. 20. 1. 96. 8.3×10-9 d. 2.7×10'9 e. Phenolite. Torsion. 27. Y2. 1.0×lo-11 c. 1.4×10nt11 a. Rubber string. .. ature O' C. Frequency. Bending. Polymethyl methacrylate. L. Temper-. a: cf. Section rl. b: cf. Section IV.. c: Measurement was made by means of multiple pendulum oscillator; S. Iwayanagi, T. Hideshima (cooperators in the present investigation): Presented before the 47-th Scientific Lecture Meeting of Scientific Research Insititute, Tokyo. (1951).. d: Measurement was made by means of inverted pendulum; unpublished. (1952). e:unpublished.<1952) . ' ' '. Attention should be paid to the fact that in general, particularly in case of rubber, the deformation in the cage of creep is apt to be greater than that of vibration. The. measurements on rubber were made under dead deformation. e. `. Notes on materials: Rubber: Manufactured by the Research Section of Sumitomo Denko & Co., Ltd.; fi11er 15%; hardness 44; test piece was made in a cylindrical rod, both ends being cemented on brass disks. Phenolite: Supplied by Kawada, Tsuji Laboratory, Scientific Research Institute, Tokyo; condensed by ,ammonia catalyzer and suMciently hardened,. G"). Hence, a better agreement between the predicted and the observed values may be expected when the calculation is made by using the value of G't corresponding to lower frequencies. In case of polymethylate, the predicted value is greater than the observed in bending, while the former is smaller than the latter in torsion. In view. of the fact that polymethyl methacrylate exhibits the secondary anomalous dispersion at a room temperature,(as) it seems reasonable to attribute this inverse behaviour to the making of steady-state measurements at dfferent frequencies. 28) S. Iwayanagi, T. Hideshima: Presented before the Division of Polymer Physics at the Meeting of the Physical Society of Japan, Tokyo. April 1951..

(28) 46 K. Shibata In case of phenolite, the linear behaviour of creep with logarithmic time was somewhat irregu!ar.. Acknowledgment. .. In conclusion, the author expresses his sincere appreciation to Dr. M. Kinoshita, the chief of the laboratory, for his encouragement and also to Mr.. S. Iwayanagi and Mr. T. Hideshima for their cooperation throughout the study. Thanks are also due to Mr. H. Nakane and Mr. K. Sato for their. assistanceincarryingoutthemeasurements. ' . This study was partially financed by the Scientific Research Fund of the Ministry of Education.. k. -. sc. n. '.

(29)