Identification of Viscoelastic Materials by Use of Wave Propagation Methods

Transcription

1 Digital Comprhnsiv Summaris of Uppsala Dissrtations from th Faculty of Scinc and Tchnology 369 Idntification of Viscolastic Matrials by Us of Wav Propagation Mthods SAED MOUSAVI ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 007 ISSN ISBN urn:nbn:s:uu:diva-834

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3 To my family

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5 List of Paprs This thsis is a summary of th following paprs: I II III IV V VI S. Mousavi, D.F. Nicolas, B. Lundbrg, Idntification of complx moduli and Poisson s ratio from masurd strains on an impactd bar, Journal of Sound and Vibration, Vol. 77, 004, p K. Mahata, S. Mousavi, T. Södrström, On th stimation of complx modulus and Poisson's ratio using longitudinal wav xprimnts, Mchanical Systms and Signal Procssing, Vol. 0, Nov. 006, p S. Mousavi, K. Wlch, U. Valdk, B. Lundbrg, Non-quilibrium split Hopkinson prssur bar procdur for non-paramtric idntification of complx modulus, Intrnational Journal of Impact Enginring, Vol. 31, 005, p K. Wlch, S. Mousavi, B. Lundbrg, M. Strømm, Viscolastic charactrization of compactd pharmacutical xcipint matrials by analysis of frquncy-dpndnt mchanical rlaxation procsss, Europan Physical Journal E, Vol. 18, Spt. 005, p M.-N. Bussac, P. Collt, G. Gary, B. Lundbrg, S. Mousavi, Viscolastic impact btwn a cylindrical strikr and a long cylindrical bar, Intrnational Journal of Impact Enginring, Availabl onlin 4 March 007 S. Mousavi, L. Hillström, B. Lundbrg, Idntification of complx shar modulus from masurd shar strains on a circular disc subjctd to transint torsion at its cntr, submittd for publication in th Journal of Sound and Vibration

6 Th author s main contributions to ths paprs ar as follows: I. Part of thortical work, major part of numrical and xprimntal work, and part of writing. II. III. IV. Part of thortical work, major part of xprimntal work, part of writing. Part of thortical work, numrical work, major part of xprimntal work, and part of writing. Part of xprimntal work. V. Exprimntal work, and part of writing. VI. Thortical work, part of numrical work, xprimntal work, and part of writing.

7 Contnts 1. Introduction Viscolastic Wavs Constitutiv rlations Extnsional and torsional wavs in bars Shar wavs in circular discs Viscolastic Impact Thory Exprimnts Rsults and discussion Idntification Basd on Extnsional and Torsional Wavs in Bars Thory Exprimnts Rsults and discussion Idntification Basd on SHPB Thory Exprimnts Rsults and discussion Idntification Basd on Shar Wavs in a Circular Disc Thory Exprimnts Rsults and discussion Conclusions Summary in Swdish...50 Acknowldgmnts...5 Rfrncs...54

8 Nomnclatur A a b c E f G k M N n P r t V v Z z x W cross-sctional ara lngth of spcimn distanc from th bar/spcimn intrfac, outr boundary wav spd complx xtnsion modulus, lastic constant frquncy complx shar modulus, lastic constant, impuls rspons wav numbr complx modulus normal forc, wav amplitud numbr of quations wav amplitud radial co-ordinat, impdanc ratio tim impact vlocity particl vlocity, displacmnt charactristic impdanc axial co-ordinat axial co-ordinat nrgy Grk damping cofficint wav numbr rlaxation strngth phas angl, Dirac dlta function wav propagation cofficint, shar strain viscosity strain wavlngth Poisson s ratio dnsity

9 strss shar strss, rlaxation tim frquncy Subscripts A strain gaug position B strain gaug position b strain gaug position, bar C strain gaug position cl classical E xtnsional G shar HI high I incidnt LO low lf low frquncy m numbr of sctions N ngativ dirction n numbr of sctions/quations P positiv dirction, provisional R rflctd r Dby paramtr rlaxd T transmittd, shar u Dby paramtr un-rlaxd M masurd 1 first, strikr scond, bar Suprscripts d M dissipativ lastic masurd

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11 1. Introduction Th first systmatic studis of th viscolastic bhaviour of matrials wr carrid out during th 19 th cntury by Wbr, Maxwll and Boltzmann. Matrials with such bhaviour ar of grat importanc whn it coms to supprssing and controlling vibrations in structurs and structural lmnts which may lad to fatigu, nois pollution, instability and othr problms. In ordr to mak us of th uniqu proprtis of ths matrials, on nds xtnsiv knowldg about thir viscolastic proprtis. In rcnt yars, thrfor, a varity of mthods hav bn dvlopd for th charactrization of viscolastic matrials. In nginring applications, th bhaviour of viscolastic matrials can oftn b approximatd as linar. In th linar rgim, an isotropic viscolastic matrial is charactrizd by two indpndnt complx-valud functions of frquncy. Such functions ar,.g., th complx xtnsion modulus E, th complx shar modulus G and th complx Poisson's ratio. Th rlativ magnituds of th complx moduli for a givn matrial hav bn considrd by Thocaris [1], whil th gnral frquncy dpndncis of ths moduli and of th complx Poisson s ratio hav bn xamind by Pritz []. A typical frquncy rang of intrst in nginring applications may b from a fw hundrd Hz to a fw khz. In this frquncy rang, wav propagation mthods ar suitabl for idntification purposs. Svral such mthods for idntification of th complx xtnsion modulus [3-18], th complx shar modulus [19] and th complx Poisson s ratio [0] hav bn dvlopd. Wav propagation mthods basd on masurmnts on bars undr such conditions that th wavs ar not ovrlapping at instrumntd sctions hav bn usd in svral studis [3-9] concrning idntification of complx moduli. It has bn shown that for th cas of xtnsional or torsional wavs, at last two indpndnt masurmnts ar ndd. Sinc ths mthods rquir that th wavs do not ovrlap, long spcimns ar ndd. Mthods that prmit ovrlap of th wavs at masurmnt sctions [10-17] rquir at last thr indpndnt masurmnts. For th cas of flxural wavs in bams such mthods rquir at last fiv indpndnt masurmnts. How- 11

12 vr, us of boundary conditions can rduc th numbr of masurmnts ndd. Ths mthods allow th us of rlativly short bars. Somtims, th matrial undr invstigation cannot b manufacturd in th siz rquird by wav propagation mthods, or instrumntation of th tst spcimn is difficult or impossibl. Exampls of such matrials ar foams, cmnts usd in biotchnology and pharmacutical xcipints. For such matrials, Split Hopkinson Prssur Bar SHPB tsting offrs a possibility. SHPB tsting is commonly usd whn studying th constitutiv proprtis of matrials at high 10 3 s -1 or vry high >10 4 s -1 strain rats. Th mthod originats from th work by Hopkinson [1] and Davis [], but today s us is mainly du to Kolsky [3] who introducd th placmnt of a short spcimn btwn two long bars, calld prssur bars. Th SHPB tchniqu was originally dvlopd for tsting matrials in comprssion, but it has also bn usd for tsting in torsion and othr cass of load. Th prssur bar matrials commonly usd hav bn high strngth stl or titanium alloys, which hav vry high mchanical impdanc compard with viscolastic matrials such as polymrs. A rviw of problms and dvlopmnts in classical SHPB tsting has bn providd by Gray III [4]. Rcntly, thr has also bn much intrst in tsting soft matrials, which has lad to th us of viscolastic lowimpdanc bars mad of polymric matrials [5-3]. In polymric bars, disprsion and damping ar gnrally significant. For a SHPB tst to b considrd valid, it is rquird that svral conditions concrning th spcimn should b approximatly fulfilld: i Equilibrium should prvail, ii th stats of strss and strain should b uniform, iii th stat of strss should b uni-axial, and iv th ffcts of friction at th bar/spcimn intrfacs should b ngligibl. Ths conditions ar intrdpndnt and somtims in conflict. Thus, i rquirs low whras iii and iv rquir high aspct ratios lngth to diamtr. Th condition ii is in conflict with itslf as axial uniformity rquirs a low and radial uniformity a high aspct ratio. In ordr to achiv th bst possibl rsult with th SHPB tchniqu it is of importanc to hav an incidnt wav that is suitabl for th tst. Rcntly, som ffort has bn mad by Kumar t al. [33] to prscrib th loading and unloading in a SHPB tst. Producing th incidnt puls in a viscolastic SPHB rquirs undrstanding of th procss of impact btwn two viscolastic bodis. Th axial impact btwn flat-ndd cylindrical lastic bars was considrd in dtail by Saint-Vnant in th 1860s. In Papr V th axial impact btwn a cylindrical strikr of finit lngth and a long cylindrical bar, both of linarly viscolastic matrial, is considrd. Thortical rsults 1

13 for this impact procss can srv as a guid for th choic of impact vlocitis, dimnsions and positions of snsors. In Papr III, a modifid SHPB tst procdur for charactrization of viscolastic matrials was dvlopd, and in Papr IV th mchanical proprtis of som viscolastic pharmacutical xcipints wr idntifid by us of this procdur. Th aim of Paprs I and II was to dvlop a procdur, basd on propagation of wavs in a singl tst bar, for th idntification of th complx xtnsion modulus E, th complx shar modulus G and th complx Poisson s ratio. Of principal intrst was th assssmnt of linarity and isotropy. Th procdur was applid to two matrials of importanc in nginring, viz., polymthyl mthacrylat PMMA and polypropyln PP. Idntification of complx shar modulus basd on propagation of torsional wavs in a prismatic bar rquirs a long tst spcimn Papr I. Many matrials usd today ar manufacturd with othr gomtris lik blocks, shts and plats. Thrfor, an idntification mthod has bn dvlopd in Papr VI that maks us of a circular disc spcimn. Such a spcimn gomtry has not bn usd bfor. 13

14 . Viscolastic Wavs.1. Constitutiv rlations Viscolasticity concrns tim-dpndnt dformation bhaviour. In th vicinity of th glass transition tmpratur, long-chain polymrs show strong viscolastic bhaviour such that th strain dpnds not only on th prsnt load condition but also on th ntir load history. Th rlation btwn strss and strain in linarly viscolastic matrials can b xprssd t Y t dt, 1 t 0 whr Y t is th strss rlaxation function and t is tim. Fourir transformation givs ˆ iyˆ ˆ, whr E iyˆ is th complx xtnsion modulus and f is th angular frquncy. Th complx shar modulus G can b dfind similarly. For isotropic linarly viscolastic matrials, E, G and th complx Poisson s ratio ar intrrlatd by E G1. 3 Th ral part of th complx moduli, rfrrd to as storag moduli, rlats componnts of strsss and strains that ar in phas. Convrsly, th imaginary parts, rfrrd to as loss moduli, rlat componnts of strsss and 14

15 strains that ar 90 out of phas. It is common to dfin th loss angls E and G as E tan E, E G tan G, 4 G whr prim and doubl prim indicat ral and imaginary parts, rspctivly, of th complx moduli. Th bhaviour of viscolastic matrials can oftn b dscribd by simpl mchanical modls composd of prfctly lastic springs and prfctly viscous dashpots. Th standard linar solid SLS [34], shown in Fig. 1, is composd of two lastic springs with stiffnss paramtrs E or G, E or d d d G and a viscous dashpot with viscosity. d d Fig. 1. Standard linar solid SLS. E or G and E or G rprsnt th d stiffnss of th springs and th viscosity of th dashpot. Th complx modulus E can b xprssd in trms of th constitutiv d d paramtrs E, E and as d d E E i E. 5 d d E E i This is a paramtric dscription of a viscolastic matrial. For matrials with mor complx bhavior, svral SLS moduls can b combind in sris or in paralll with ach othr. By us of Dby rlaxation paramtrs, Eq. 5 can b rwrittn as Eu Er E Eu, 6 1 i 15

16 d d whr E E rlaxation tim, dfining th tim scal of th rlaxation procss, E u E un-rlaxd modulus at frquncis 1, d d E r E E E E rlaxd modulus at frquncis 1. A matrial may hav svral rlaxation mchanisms. It can b shown [35] that W W E tan, 7 E E whr W and W ar th nrgy dissipatd and th maximum lastic nrgy stord pr cycl of dformation, rspctivly, and E is th phas angl by which th strain lags bhind th strss. Thus, tan E is a masur of th rlativ nrgy dissipation in th matrial. From a paramtrically idntifid complx modulus, on can obtain th nrgy dissipation pak and th rlaxation strngth, quantitis which provid insight into th physical procsss and structurs of th matrials. From Eqs. 6 and 7, th dissipation factor tan can b xprssd in trms of th modl paramtrs. Stting th drivativ of tan with rspct to to zro givs th maximum dissipation frquncy, f d max E E E max, 8 d and th rlaxation strngth, E u Er Er. Th rlaxation strngth can d b writtn in trms of modl paramtrs as E E... Extnsional and torsional wavs in bars Th govrning quation in trms of strain for both xtnsional and torsional wavs in slndr prismatic bars is ˆ z, ˆ z,, 9 z 16

17 whr ˆ z, is th Fourir transform of th rlvant strain componnt ˆ zz z, for xtnsional wavs and ˆ z z, for torsional wavs. Also, is th wav propagation cofficint, 10 M whr M is th complx xtnsion modulus E for xtnsion and th complx shar modulus G for torsion, and is th dnsity of th matrial. Th gnral solution of Eq. 9 is ˆ z, Pˆ z Nˆ z, 11 whr P ˆ and N ˆ ar frquncy-dpndnt complx-valud amplituds of harmonic wavs travlling in opposit dirctions. Th damping cofficint is th ral part of th complx wav propagation cofficint for both xtnsion and torsion, and th wav numbr k is th imaginary part. Th wavlngth is invrsly proportional to th wav numbr k, k..3. Shar wavs in circular discs Axi-symmtric shar wavs in circular discs with constant thicknss ar govrnd by th diffrntial quation vˆ vˆ r r r 1 vˆ 0, 1 r r whr vˆ is th Fourir transform of displacmnt fild v r, t, ct is a complx wav numbr, and c T is th shar wav spd. Th shar strain ˆ is calculatd from th displacmnt fild as ˆ vˆ r vˆ r. Th gnral solution of this quation is a linar combination of th Hankl functions 1 H1 J1 iy1 and H1 J1 iy1, with r. Th gnral solution can b writtn as ˆ 1 vˆ Pˆ H r N H, r 17

18 whr P ˆ and N ˆ ar complx-valud amplituds. Bcaus of th asymptotic bhaviour of th Hankl functions, ths complx amplituds can b shown to b associatd with outgoing and ingoing wavs, rspctivly. Th shar strain ˆ can b calculatd using th diffrntiation ruls j j j d / d H1 H 0 1/ H 1 with j 1,. It can b xprssd as ˆ ˆ 1 1 ˆ P H0 r H1 r N H0 r H1 r. 14 r r 18

19 3. Viscolastic Impact 3.1. Thory Th incidnt puls in a SHPB is gnratd by th impact of a strikr, which is prfrably mad of th sam matrial as th bars. For tsting of soft matrials, in particular, th bar and th strikr ar commonly mad of polymric matrials with low charactristic impdanc. Ths matrials oftn show strong viscolastic bhaviour. Thrfor, viscolastic impact btwn a cylindrical strikr and a cylindrical bar is of fundamntal importanc in SHPB tsting of matrials with low charactristic impdanc. By th assumptions that initially plan cross-sctions rmain plan and radial inrtia can b nglctd, th wav motion in th strikr and th bar is govrnd by th two diffrntial quations Nˆ x A i vˆ and vˆ x i Nˆ AE, whr x is an axial co-ordinat with origin at th strikr/bar intrfac as shown in Fig.. Fig.. Axial impact btwn a cylindrical strikr and a long cylindrical bar of a viscolastic matrial. x Th gnral solution of th two govrning quations is Nˆ Nˆ p x N n and ˆ x ˆ 1 ˆ x v Z Np Nn whr N ˆ p and N ˆ n ar amplituds at x 0 associatd with wavs travlling in apposit dirctions and 1 Z A E is th charactristic impdanc. Lt th vlocity rsponss of th impact facs of th strikr and th bar to impulsiv forcs t b G 1 t and G t, rspctivly, with vlocitis and forcs dirctd into th impact facs dfind as positiv. Also, assum provi- 19

20 sionally that th impact facs of th strikr and th bar rmain in contact aftr thir initial contact at tim t 0. As a rsult of impact, th vlocitis of th impact facs thn bcom v1 t V1 G1 t Fp t and v t G t Fp t, rspctivly, whr F p t is a provisional impact forc. As th convolutions G1 t Fp t and G t Fp t ar zro for t 0, and th impact facs hav common vlocity v1 t v t for t 0, th provisional impact forc can b dtrmind from th intgral quation [36, 37] G t G t F t V H 1 p 1 t, 15 whr H t is Havisid s unit stp function. Fourir transformation givs Fˆ p V i 1 Gˆ Gˆ By using th conditions N ˆ L1, 0, Nˆ 0, Fˆ p and v ˆ0, Gˆ ˆ 1 Fp for th strikr and Nˆ 0, Fˆ p, vˆ 0, Gˆ ˆ Fp and N 0 for th bar, on obtains th impuls rsponss ˆ n Gˆ 1 1 L , 1L1 Z 1 1 Gˆ. 17 Z Substitution of ths rlations into Eq. 16 givs Fˆ 1L1 Z1V 1 1 R 1 p, 18 1L1 i 1 R whr R Z Z1 Z Z1 1 r 1 r is th rflction cofficint, rlatd to th normal forc, for wavs in th strikr at th strikr/bar intrfac and r Z 1 Z is th strikr-to-bar charactristic impdanc ratio. Th provisional impact forc F p t is positiv initially. If it rmains nonngativ for all tim, th actual impact forc bcoms F0 t Fp t. If, instad, it changs sign from positiv to ngativ at som finit tim t t0, sparation occurs at this tim and th actual impact forc bcoms 0

21 1 H t t F F0 t 0 p t 19 providd that th contact btwn th strikr and th bar is not rstablishd. With t 0, this xprssion applis also to th cas that F t rmains non-ngativ for all tim. Th conditions N ˆ n 0 and N ˆ 0, F ˆ0 giv th normal forc and th particl vlocity x Nˆ Fˆ, 0 Fˆ 0 x vˆ 0 Z in th bar. Th strain in th bar is givn by N ˆ / A E, i.., Fˆ 0 x ˆ. 1 Z c 3.. Exprimnts Figur. 11 shows th xprimntal st-up. Cylindrical strikrs and a cylindrical bar wr fabricatd from a singl bar of polymthyl mthacryalat PMMA. Th bar was first machind to a diamtr of mm, which mad strikrs and th bar fit into th barrl of an air gun with a tolranc of 0.06 mm. Th nds of th strikrs and th bar wr machind to b flat and prpndicular to th cntr lins. Th lngth of th strikrs usd was 10.0 and mm, and th lngth of th bar was 000 mm. Th bar was supportd by ight Tflon barings mountd on th sam havy aluminium bam as th air gun. Th impact nd of th bar was locatd insid th barrl about 40 mm from th muzzl. Nar th muzzl, axial slots in th barrl allowd th air to scap on both sids of th strikr. In this way no significant air cushion was formd in front of th strikr, and th prssur acting on th rar nd of th strikr during impact was ngligibl. Th slots also mad it possibl to stimat th impact vlocity from th tim of flight of th strikr btwn two axial positions. 1

22 Fig. 3. Exprimntal st-up for viscolastic impact tsts. Th bar was instrumntd with pairs of diamtrically opposit and axially orintd rsistiv strain gaugs A, B and C at distancs xa 10, xb 40 and xc 70 mm, rspctivly, from its impactd nd. Th activ lngth of th strain gaugs was 6 mm, and thy wr connctd to Whatston bridgs followd by amplifirs Masurmnt Group 10B with bandwidth 100 khz so as to mak th output signals proportional to th symmtric componnts of strain. Ths signals wr rcordd with sampling frquncy 1 MHz by a 16-bit data acquisition board. Shunt calibration was usd for ach channl. A sparat impact tst was carrid out in ordr to idntify th complx modulus of th PMMA. In this tst a standard lad projctil was fird with an air rifl against th impactd nd of th bar. Th projctil had a diamtr of 4.5 mm, and lngth 5 mm approximatly. Th wight of th projctil was about 0.4 g. Th complx modulus was idntifid non-paramtrically and d constitutiv paramtrs E, E and d wr obtaind by minimizing th diffrnc btwn th non-paramtric complx modulus and th complx modulus givn by Eq. 5 for th thr-paramtr viscolastic standard modl. This was don in th frquncy intrval 0.5 to 3 khz in th sns of last squars. Th sam numbr of discrt frquncis wr considrd in ach subintrval 0.5-1, 1-, -4,, and 16-3 khz. Th dtails of th xprimnts for viscolastic impact tsts ar can b found in Papr V. Th non-paramtric complx modulus was stimatd from th changs in amplitud and phas of a strain puls travlling through sctions A and C of th bar [7, 11]. Thus, th wav propagation constant is givn by

23 ln ˆ / x x C A A ˆC and th complx modulus by E. Th impact vlocitis of th 10 and 360 mm PMMA strikrs wr stimatd to b 9.5 and 8.7 m/s. Th vlocitis wr scald with factors 0.98 and 1.03 rspctivly in ordr to facilitat comparison of th strain puls shaps calculatd using paramtrically stimatd complx modulus and masurd strains Rsults and discussion Th non-paramtric and paramtric rsults for th complx modulus E of th PMMA matrial ar shown vrsus frquncy f in Fig. 4, whr also th discrt points usd for paramtric idntification ar indicatd with dots. Th constitutiv paramtrs of th SLS wr stimatd to b E E d 7.3 GPa and d MPas. Fig. 4. Non-paramtric dots and paramtric solid curvs complx modulus E of th PMMA vrsus frquncy f /. Th xprimntal and thortical rsults for th strains A, B and C vrsus tim t, and th corrsponding spctrum A ˆ vrsus frquncy f ar shown in Figs. 5 and 6. It can b sn that thr is a good gnral agrmnt btwn th xprimntal and th thortical rsults. Thus, i thr ar only small dviations in puls shaps, which ar mainly du to oscillations of th masurd strains. Furthrmor, ii th ris and fall of th masurd main pulss ar stp and thir widths corrspond to two transit tims for th dis- 3

24 Fig. 5. Strain vrsus tim t at strain gaug stations A, B and C and strain spctrum ˆ vrsus frquncy f at strain gaug station A of PMMA bar. Comparison of thory thick curvs and xprimnt thin curvs for 10 mm strikr. Masurd impact vlocity scald by factor Fig. 6. Strain vrsus tim t at strain gaug stations A, B and C and strain spctrum ˆ vrsus frquncy f at strain gaug station A of th PMMA bar. Com- parison of thory thick curvs and xprimnt thin curvs for 360 mm strikr. Masurd impact vlocity scald by factor

25 continuous viscolastic wav front through th strikrs, as prdictd by thory. Agrmnt according to i and ii rquirs that th conditions b clos to 1D so that gomtrical disprsion can b nglctd. This mans that th wavlngths of th prdominating wavs must b much largr than th diamtr d of th strikr and th bar, i.. c / f d or f c / d 185 khz. 5

26 4. Idntification Basd on Extnsional and Torsional Wavs in Bars 4.1. Thory An idntification procdur for both xtnsional and torsional wavs can b basd on th us of rdundant masurmnts of rlvant strain componnts. This procdur was usd by Hillström t al. [16] for idntification of complx modulus E. Th procdur usd hr is basd on masurmnt of n surfac strains ˆ1, ˆ,..., ˆ n at sctions z1 z...< z n. Ths masurmnts togthr with Eq. 11 giv a systm of n quations z 1 z z n z 1 z z n ˆ 1 Pˆ ˆ. 3 N ˆ ˆ n Hr, th thr complx valud functions, P ˆ and N ˆ ar to b dtrmind. Thn, by us of Eq. 10, th rlvant complx modulus M can b calculatd. If th numbr of masurmnts corrsponds to th numbr of unknowns, in this cas thr, thr gnrally xists an xact solution for. Du to masurmnt rrors, th rsult for at vry low frquncis may giv larg rrors in E and G. For highr frquncis, th rsults for E ar valid only if th 1D rquirmnt is satisfid. For G, howvr, thr is no such limitation. For an ovr-dtrmind systm, with n 3, thr is no uniqu solution. Howvr, it is possibl to obtain an approximat solution in th sns of last squars by minimizing th rror 6

27 7 Aw w ˆ / ˆ ˆ, ˆ ~, 4 ˆ ˆ ˆ ˆ 1 n, ˆ ˆ ˆ N P w, z n z n z z z z 1 1 A 5 whr ˆ is th column vctor of masurd strains zz for th cas of xtnsion and z for th cas of torsion, ˆ w is that of complx amplituds, and doubl bars dnot th Euclidan norm. Th complx Poisson s ratio can b calculatd using axial and circumfrntial strains masurd at th sam axial position and th rlation, ˆ, ˆ z z zz. 6 Equation 3 can b usd to study th isotropy of th matrial. Du to th linarity of th systm 3 and Eq. 6, th quality of th stimatd Poisson s ratio can b improvd by a joint stimation approach. Thn, th circumfrntial strains do not nd to b obtaind at th sam axial position as th axial strains. Th rquirmnt is a minimum of thr axial strain masurmnts and on in th circumfrntial dirction. With this approach, th systm 3 and Eq. 6 ar rplacd by th systm ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ cm c c n y y y y y y z z z z z z N P m m n n, 7 whr n z ar th axial positions for axial strain masurmnts and m y ar thos for circumfrntial masurmnts. Using th sam last squars proc-

28 dur as abov, on can obtain stimats for E and. In Papr II it has bn shown that th joint stimation approach yilds th bst possibl rsult for compard to a simpl or wightd avraging. 4.. Exprimnts Two diffrnt matrials wr usd in th xprimntal tsts, polymthyl mthacrylat PMMA and polypropyln PP. PMMA is an amorphous polymr with rlativly low loss, and at room tmpratur it is blow th glass transition tmpratur T g. Th dnsity for PMMA usd in th xprimnts was stimatd to b 1183 kg/m 3. PP is a smi-crystallin polymr with rlativly high loss. Its glass transition tmpratur is blow room tmpratur. Th dnsity for th PP bars usd in th xprimnts was 915 kg/m 3. PP bars had bn manufacturd through xtrusion, which ld to som dviation from circular shap of th cross-sctions of th bars. Th matrials usd ar typical for high and low loss matrials and ar commonly usd in nginring applications. Th tsts wr carrid out for bars with dimnsion, 10 mm diamtr and approximatly 1000 mm lngth. Each tst bar was instrumntd at four diffrnt sctions z z to z 5 with strain gaugs for masurmnt of th normal strains z z and th shar strains z. Us was also mad of th circumstanc that at a fr nd z z1 of a bar ths strains ar zro, which providd n 5 strains, two of which wr rdundant, for th idntification of th complx xtnsion modulus E and th complx shar modulus G, rspctivly. In addition, ach bar was instrumntd at a fifth sction z z0 with strain gaugs for masurmnt of th normal strains z z and ndd for th idntification of th complx Poisson s ratio. Fig. 7. Exprimntal stup for idntification of complx modulus E. Th bar was suspndd with thin fishing wirs to minimiz th ffct of supports. It was xcitd by impact. 8

29 Signals from th strain gaugs wr amplifid and filtrd prior to digitisation using a digital oscilloscop. Th xprimntal st-up for axial impact tsts is shown schmatically in Fig. 7 and that for torsional impact tsts in Fig. 8. In th lattr tsts, a torsional wav was gnratd by torsional impact at on nd of th bar. Th impactd nd of th bar was machind and scurd in a mtallic support to prvnt latral movmnt during th impact. Data from th xprimnts wr convrtd to strains using shunt calibration. Th xprimnts wr carrid out with two diffrnt lvls of xcitation, high and low, in both th axial and th torsional and impact tsts. Th nonimpactd nd of th bars was always kpt fr. In all tsts, th bars wr xcitd using an air rifl. Th xprimnts ar dscribd in dtail in Papr I. Fig. 8. Exprimntal st-up for idntification of complx shar modulusg. Th bar was suspndd with thin fishing wirs to minimiz th ffct of supports. Th impactd nd was scurd for latral movmnt using a mtallic support. For th joint stimation, a bar with lngth 000 mm and diamtr 10 mm was instrumntd with both axial and circumfrntial strain gaugs at four diffrnt sctions, 10, 470, and mm from th impactd nd. Th ight strain signals obtaind from ths gaugs wr rcordd with a sampling intrval of 0. s. Th sam rcording quipmnt and xprimntal stup as in Fig. 7 was usd Rsults and discussion Som of th rsults for th 10 mm bars ar prsntd hr. Complt rsults for 10 and 0 mm bars ar prsntd in Papr I. Th rsults obtaind for th complx moduli E, G and th Poisson's ratio vrsus frquncy for th 9

30 10 mm bars ar shown in Fig. 9. For PMMA, th rsults ar prsntd up to 40 khz, and for PP thy ar prsntd up to 15 khz. For both matrials, th complx Poisson's ratio is approximatly constant. Th ratios of th complx xtnsion moduli E E and th complx shar HI LO Fig. 9. Complx Extnsion modulus E, complx shar modulus G and complx Poisson s ratio for PMMA and PP vrsus frquncy for 10 mm tst bars. moduli G HI GLO at high indx HI and low indx LO lvls of impact xcitation ar shown vrsus frquncy in Fig. 10 for PMMA and PP. For both matrials, ths ratios ar approximatly ral and qual to unity, which indicats that th rsponss ar vry clos to linar undr th conditions of th tsts. Th ratio G 1 E is shown vrsus frquncy in Fig. 11 for PMMA and PP. This ratio is xpctd to b ral and qual to unity for an isotropic matrial. For th 10 mm PMMA bar, this ratio is vry clos to bing ral and qual to unity. For th 10 mm PP bar, it is approximatly ral but somwhat smallr than unity This dviation from unity indicats a crtain dgr of anisotropy. It is blivd that som anisotropy may b du to th xtrusion procss usd for fabrication of th PP tst bars. 30

31 Fig. 10. Ratios of complx xtnsion moduli EHI ELO and complx shar moduli G HI G LO of PMMA and PP at diffrnt lvls of impact xcitation vrsus frquncy for 10 mm tst bars. Fig. 11. Ratio bars. G 1 E for PMMA and PP vrsus frquncy for 10 mm tst Th rsults for xtnsion modulus ar valid for frquncis that fulfil th 1D rquirmnt, whil th rsults for complx shar modulus is valid for all frquncis. For PMMA at 40 khz, th wavlngth of quasi-longitudinal wavs is approximatly 55 mm or 5.5 diamtrs for th 10 mm tst bar. For PP at 15 khz, th wavlngth of such wavs is approximatly 10 mm or 1 diamtrs for th 10 mm tst bar. For th joint stimation of complx modulus E and Poisson s ratio, th rsults obtaind for PMMA from Eq. 7 ar shown in Fig. 1. In th frquncy rang 5-15 khz, th quality of th rsult for Poisson s ratio is 31

32 considrably improvd compard with th rsults shown in Fig 9. It has bn shown in Papr II that th joint stimation improvs th quality of th stimatd Poisson s ratio and is spcially hlpful for cass whr th xcitation is poor or th masurmnt of axial and circumfrntial strain componnts at th sam position is difficult or impossibl. Fig. 1. Rsults for PMMA from joint stimation of complx Extnsion modulus and Poisson s ratio as functions of frquncy. Th rsults ar basd on an avrag of fiv xprimnts with ight masurd signals, four in th axial and four in th circumfrntial dirction. 3

33 5. Idntification Basd on SHPB 5.1. Thory Considr th SHPB shown in Fig. 13. A cylindrical spcimn with lngth a and cross-sctional ara A is placd btwn two cylindrical bars with cross-sctional ara A b. Th bars and th spcimn ar co-axial, and th contacts at th bar/spcimn intrfacs ar assumd to b prfct. Th matrial of th spcimn is linarly viscolastic with complx modulus E, whr is th angular frquncy, and dnsity. Th matrial of th bars is linarly viscolastic with complx modulus E b and dnsity b. A wav Fig. 13. Bars and spcimn. is gnratd in th lft bar and at th tst spcimn it is partially transmittd into th right bar and partially rflctd back into th lft bar. Th strain wav tim historis for th incidnt wav IM t, th rflctd wav RM t and th transmittd wav TM t ar rcordd at a distanc b from th spcimn/bar intrfacs. It is considrd that th wavlngths and b of wavs in th spcimn and th bars, rspctivly, ar much longr than th diamtrs such that on- 33

34 dimnsional conditions prvail. Lt th ˆIM, ˆRM and ˆTM b th Fourir transforms of th masurd strain historis and lt b, Z b b th wav propagation cofficint and charactristic impdanc of th bars, rspctivly. Thn, th rlations hold for particl vlocitis and normal forcs in th bars indics 1 and rfr to lft and right bar, rspctivly, Nˆ 1 Nˆ I x b 1 Nˆ R x b 1, 1 vˆ 1 Z b Nˆ I x b 1 Nˆ R x b 1, 8 Nˆ ˆ ˆ 1 Z x x T T b ˆ b b N, v N, 9 whr x 1 and x ar axial co-ordinats with origins at th bar/spcimn intrfacs, N ˆ I and N ˆ R ar th amplituds of th incidnt and rflctd wavs at th first bar/spcimn intrfac, and N ˆ T is th amplitud of th transmittd wav at th scond bar/spcimn intrfac. From ths rlations and thos btwn normal forcs, normal strsss and normal strains in th bars, th amplituds of th incidnt, rflctd and transmittd wavs at th bar/spcimn intrfacs can b xprssd in trms of th masurd strains as Nˆ I A E b b b ˆ b IM, Nˆ R A E b b b b ˆ RM, Nˆ T A E b b b b ˆ TM. 30 Th normal forc N ˆ x, and th particl vlocity v ˆ x, in th spcimn ar givn by Nˆ Nˆ P x Nˆ N x, 1 vˆ Nˆ P Z x Nˆ N x, 31 whr x is an axial co-ordinat with origin at th cntr of th spcimn and and Z ar th wav propagation cofficint and charactristic impdanc of th tst spcimn. Using th continuity of normal forcs and particl vlocitis at th bar/spcimn intrfacs and Eqs. 8 to 31, on can xprss th rlation btwn th masurd strains ˆRM and ˆTM as 34

35 ˆ ˆ RM TM 1 Z Zb sinh a Zb Z. 3 At low frquncis and/or for a short spcimn, sinh a a, and th complx modulus E of th spcimn can b approximatd by E E a lf E 1 b Ab ba A ba ˆ A A ˆ bb RM TM 1, 33 If th charactristic impdanc of th spcimn is also low, on obtains from Eq. 3 E E a cl 1, E b Z Z Ab ba ˆ A ˆ b 1. TM RM, 34 This approximation is th on usd in th classical cl SHPB procdur, which is basd on th assumption that th spcimn is in a stat of quilibrium. Th dviations of E lf and E cl from E at low frquncis can b obtaind 3 by substituting th strain ratio 3 with sinh a a a / 3! into Eqs. 33 and 34. This givs 1 Z E E lf 1 1 a, a 1, 35 6 Zb Z 1 cl 1 1 a Zb 6 E E, a

36 5.. Exprimnts Two sris of SHPB tsts wr carrid out, on with PP spcimns Tabl 1 and two diffrnt bar matrials, PMMA and aluminium AL, and th othr with four matrials rprsnting a broad rang of xcipints usd in dirct comprssion of pharmacutical tablts. Microcrystallin cllulos Avicl PH10, FMC, MCC is a frquntly usd bindr/fillr du to its high comprssibility and good strngth charactristics [38]. From th starch catgory, maiz starch Starch 1500, Colorcon was slctd. Starchs ar also widly usd bindrs/fillrs in th pharmacutical industry du to thir availability and low cost. Sorbitol Roqutt was slctd as a commonly usd pharmacutical sugar alcohol. Th fourth matrial usd was th cllulos drivativ thyl cllulos Fluka Chmika. This matrial is usd as both a tablt coating and a matrix formr in tablts dsignd to hav a controlld drug rlas profil. It dos not disintgrat or dissolv whn administrd orally as do th othr thr tablt typs. Tabl 1. Tst Conditions for PP spcimn. Tst Spcimn siz Strain amplituds Bar Diamtr Lngth matrial RM 6 [mm] [mm] [ 10 TM 6 ] [ 10 ] 1A PMMA B PMMA C PMMA D PMMA A AL B AL C AL D AL A strikr was fird by mans of an air gun to gnrat th incidnt wav. Th matrial of th strikr was th sam as that of th bars usd in ach tst. For th PP tsts, th diamtr of th bars, th tst spcimn and th strikrs wr approximatly 0 mm. Th dnsitis of AL and PMMA bars wr 700 and 1183 kg/m 3 rspctivly, whil th PP tst spcimn had dnsity 915 kg/m 3. Th nominal spcimn lngths in th PP tsts wr 10, 0, 50 and 100 mm. Th bars usd in th xprimnts had a lngth of approximatly 000 mm, and thy wr supportd on th tst bnch by 13 Tflon barings, Fig. 14. Th mass of th powdr usd to mak ach pharmacutical tablt was varid and rsultd in tablt thicknsss that rangd from 3.5 mm t o 6.76 mm. Th xprimnts ar dscribd in dtail in Paprs III and IV. 36

37 Fig. 14. Exprimntal st-up for SHPB tsts. Lngths in mm. In ordr to achiv a good contact at th bar/spcimn intrfacs, a thin layr of silicon-basd lubricant was applid on th contact surfacs prior to th tsts. Two pairs of strain gaugs whr mountd on ach bar at distancs of 600 and 1000 mm from th intrfacs. Th strain historis masurd at 600 mm wr usd throughout th xprimnts. For th tst with AL bars, th lastic modulus E was usd to calculat th wav propagation cofficint b. Th lastic modulus of th AL bars was st to 71 GPa for all tsts. For th PMMA bars, th two pairs of strain gaugs mountd on th first bar wr usd to dtrmin b for ach tst by us of Eq.. Sinc th contact condition btwn th bars and th tst spcimn was important for achiving good rsults, th bars and spcimn wr pr-loadd to 0.3 MPa approximatly. Th Fourir transform of th isolatd rflctd and transmittd wavs wr calculatd using a fast Fourir transform algorithm. Th st-up usd for th PP spcimn and AL bars was also usd for th pharmacutic-tablt tsts. Howvr, AL bars with diamtr 11.3 mm wr usd to agr with th standard siz of th tablts. Th strikr usd was a 40 mm long AL cylindr. Th strain history of incidnt, rflctd and transmittd wavs wr rcordd with 1 MHz sampling rat. All tsts wr carrid out in ambint tmpratur of 0.5 C. 37

38 5.3. Rsults and discussion Th rsults prsntd in this sction concrns PP spcimn with lngth 10 and 50 mm and th four xcipints matrials MCC, Sorbitol, thyl cllulos and Starch. Mor dtaild rsults ar givn in Papr III and Papr IV. Th strain historis RM t and TM t rprsnting th isolatd rflctd and transmittd wavs ar shown in Fig. 15 for PMMA bars and in Fig. 16 for AL bars. For th AL bars, th rflctd signal from th tst spcimn is strong du to th charactristic impdanc mismatch btwn th AL bars and th PP spcimn. For th PMMA bars, th rflctd wav is smallr as rsult of bttr impdanc match. Th lngth of th prssur bars imposs a limitation on th possibl lngth of th rcordd signals. Th unavoidabl truncation of th tail of th transmittd and rflctd signals giv ris to a truncation rror obsrvd on th stimatd complx moduli E, Elf and E cl. Th truncation rror is mor significant for long spcimns and manifsts itslf as oscillations in th stimatd rsults. S Fig. 17 Tst C. In Papr III this topic is covrd xtnsivly. Fig. 15. Strain historis RM t and TM t rprsnting rflctd and transmittd wavs in tsts with PMMA bars and PP spcimns of lngths 10 and 50 mm Tsts 1A and 1C. Fig. 16. Strain historis RM t and TM t rprsnting rflctd and transmittd wavs in tsts with AL bars and PP spcimns of lngths 10 and 50 mm Tsts A and C. 38

39 Th stimatd complx Extnsion moduli E, E lf and E cl for PP basd on Eqs and us of AL bars ar shown in Fig. 17. Th corrsponding rsults for PP with PMMA bars ar shown in Fig. 18. Th modulus E stimatd in a Papr I is also shown for comparison. In ordr rf Fig. 17. Estimations E, E lf and E cl of th complx modulus of PP vrsus frquncy f obtaind with AL bars and spcimns of lngths 10 and 50 mm Tsts A-C. Estimation E shown for comparison. rf Fig. 18. Estimations E, E lf and E cl of th complx modulus of PP vrsus frquncy f obtaind with PMMA bars and spcimns of lngths 10 and 50 mm Tsts 1A-1C. Estimation E shown for comparison. rf to avoid significant 3D ffcts and nsur th validity of th rsults, th frquncy intrval 0-10 khz was chosn for th rsults shown in Figs. 17 and 18. Th wavlngths for AL, PMMA and PP at 10 khz wr approximatly 6, 11 and 9 diamtrs, rspctivly. Th stimatd complx modulus E obtaind with AL and PMMA bars ar in good agrmnt with th rfrnc data from Papr I. In Fig. 17 Tst C th rsult for th 50 mm PP spcimn shows a wak oscillation in both ral and imaginary part, which is du to ffct of truncation of th rflctd and transmittd wav signals. Equation 35 shows that th low-frquncy stimation E lf is accurat if i a 6. In contrast, Eq. 36 shows that th classical stimation E cl is 39

40 accurat only if also ii Z / Z b 1. Th first condition xprsss closnss to quilibrium in th spcimn, but th scond dos not. Thus, th classical stimation, which is basd on th assumption of quilibrium, is not ncssarily accurat vn if th condition of quilibrium is approximatly fulfilld. Nar impdanc matching, E cl is highly inaccurat no mattr how wll quilibrium is satisfid, whil, in contrast, E lf is highly accurat almost rgardlss of quilibrium. In th tsts carrid out, th quilibrium condition is wll satisfid at low frquncis and/or for short spcimns whil th impdanc condition is not. Thus, for th PMMA bars Z / Z b is always gratr than 0.4, which givs ris to larg rrors in E cl at all frquncis. For th AL bars this quantity is always gratr than 0.07, which givs ris to a much smallr but still finit rror at all frquncis. Ths obsrvations ar consistnt with th rsults obtaind for th complx modulus Figs. 17 and 18. Typical rcordd strain historis for transmittd and rflctd wavs from th tsts with pharmacutical tablts ar shown in Fig. 19. Fig. 19. Typical rcordd strain wavs from a tst with a MCC tablt. RM t and TM t ar tim-domain data for th rflctd and th transmittd wav, rspctivly. Figur 0 displays th ral and imaginary parts of th paramtric and nonparamtric complx modulus as functions of frquncy for th invstigatd tablt matrials. Th figur shows that a rlaxation procss, locatd at ~ 5-10 khz, is prsnt in all matrials. A clos study of th location in frquncy, th strngth and vn th shap of th rlaxation can yild information about th matrial and its rlaxing units. For xampl, th location in frquncy of th rlaxation is dtrmind by factors such as th siz of th rlaxing units and th strngth of th intraction forcs associatd with ths units. Th rlaxation strngth is dtrmind in part by th numbr of rlaxing units pr unit matrial volum [39]. 40

41 Fig. 0. Fittd paramtric curvs solid lins for th complx modulus suprimposd ovr masurd data dashd [ E ] and dottd [ E ] lins of a MCC, b sorbitol, c thyl cllulos and d Starch Th dnsity, modl paramtrs, frquncy of maximum dissipation f max and rlaxation strngth for th pharmacutical tablt matrials ar givn in Tabl. Tabl. Data for pharmacutical matrials givn as man valus of thr dtrminations with standard dviations in parnthsis Matrial MCC PH10 sorbitol thyl cllulos Starch 1500 Dnsity d E d E kg/m 3 MPas GPa GPa khz f max

42 6. Idntification Basd on Shar Wavs in a Circular Disc 6.1. Thory Th shar strain in a disc is dscribd by Eq. 14, whr P ˆ, N ˆ and ar considrd unknowns. It is assumd that th disc is loadd by a transint shar strss ˆ a, ˆ a at its innr boundary r a and that its outr boundary is ithr fr or locatd at infinity. M It is now assumd that th shar strains ˆ r, ˆ 1 1 and ˆ r, ˆM ar masurd at th radii r r1 and r r, rspctivly, with a r1 r b. If th complt shar strain history of th wav gnratd at th innr radius r a can b masurd at th radii r r1 and r r bfor a rflctd wav from th outr radius r b has rachd th radius r r, thn on can considr th disc infinit and lt N ˆ 0. Thn, by Eq. 14, on M M can xprss th ratio of th masurd strains as ˆ ˆ 1 / 1 /, whr th functions 1 and ar indpndnt of th xcitation ˆ. This quation rwrittn as a M M ˆ 0 37 ˆ 1 1 can b solvd for and thn, by using th rlations ct G, on can obtain th non-paramtric rsult c and T G 38 for th complx shar modulus providd that numrical difficultis do not occur. Th complx shar modulus G can b idntifid paramtrically using th thr-paramtr SLS 4

43 d d G i G, p G 39 d d G G i d d with paramtr vctor p [ G, G, ]. This modl is illustratd in Fig. 1. Th optimum paramtr vctor p opt is dtrmind by minimizing th rror T T 1 p p p p p p 40 n with rspct to p, whr i M M p ˆ [, p] ˆ [, ] 41 1 i i i 1 i p is th complx-valud rsidual at i, n is th numbr of frquncy componnts and 1/ i, p i 4 G i, p is th complx valud wav numbr. 6.. Exprimnts Exprimntal idntification tsts wr carrid out with th st-up shown in Fig. 1. A disc mad of PP with 6 mm thicknss and outr diamtr 1000 mm was attachd to a hub at its cntr. Th hub itslf was wldd to a prismatic stl shaft with lngth 6470 mm and diamtr 0 mm. At its opposit nd, th shaft was prloadd in torsion btwn two clamps. Whn th clamp closst to th disc was suddnly rlasd, by fracturing a pr-notchd bolt, a torsional wav was gnratd in th shaft. Th wav, with lngth approximatly twic th intr-clamp distanc, propagatd towards th disc, at which it was partially rflctd. As a rsult, a transint torqu that producd an outgoing shar wav loadd th disc. Th shar strains wr masurd at two diffrnt locations r 1 60 and r 96 mm using strain gaugs. Th gaugs wr connctd to a bridg amplifir and th signals from th amplifir wr 43

44 fd to a data acquisition board. Thr xprimntal idntification tsts wr carrid out at room tmpratur of approximatly C. Fig. 1. Exprimntal st-up. Lngths in mm. Th amplitud of th load was controlld by th prload of th shaft high, low, and th duration of th load by th distanc btwn th clamps long distanc 165 mm, and short distanc 110 mm. In Tsts a, b and c, th prload of th shaft and th distanc btwn th clamps wr takn as highlong, low-long and low-short, rspctivly. Both paramtric and nonparamtric idntifications wr carrid out in th intrval 3-0 khz Rsults and discussion Th rsults of th xprimntal idntification tsts ar shown in Figs. -4. M M Figs. a-c show th masurd shar strains 1 and from Tsts a- c, rspctivly. Th idntification was basd on th masurd shar strains btwn th two full vrtical lins so that rflctd wavs from th rim of th disc wr xcludd. Non-paramtric and paramtric rsults for th complx shar modulus G from Tsts a-c ar shown in Figs. 3 a-c, rspctivly, whr th uppr curvs in ach diagram ar th ral parts and th lowr curvs ar th imaginary parts. Irrgularitis can b obsrvd nar th frquncy zro and at crtain non-zro frquncis. In th thr cass, th paramtric rsults appar to b mor rprsntativ than th non-paramtric ons. 44

45 M M Fig.. Masurd shar strains 1 and vrsus tim t in xprimntal idntification tsts mploying load with a high amplitud and long duration, b low amplitud and long duration, and c low amplitud and short duration. Th idntification was basd on th masurd shar strains btwn th two vrtical lins. Tabl 3. Standard linar solid paramtrs in xprimntal tsts. Paramtr Start valu Estimatd valu Tst a Tst b Tst c G GPa d G GPa d MPas Considring th boundary condition at th rim of th disc, on can idntify th complx shar modulus without bing rstrictd to outgoing wavs. This procdur is a two-dimnsional analogu of a procdur usd for bar spcimns,.g., [5, 6, 7], and has th advantag of not rquiring a minimum siz of th disc spcimn. Howvr, it is computationally complx and may lad to rducd accuracy. 45

46 Fig. 3. Complx shar modulus G vrsus frquncy f from xprimntal idntification tsts mploying load with a high amplitud and long duration, b low amplitud and long duration, and c low amplitud and short duration. Uppr curvs in ach diagram show ral parts and lowr curvs imaginary parts. Non-paramtric thin curvs with dot marks and paramtric thick curvs rsults. Th paramtric rsults can b compard in Fig. 4. Th rsults of paramtric idntification ar shown also in Tabl 3. Fig. 4. Paramtric complx shar modulus G vrsus frquncy f from xprimntal idntification tsts mploying load with high amplitud and long duration dottd curvs, low amplitud and long duration dashd curvs, and low amplitud and short duration solid curvs. Uppr curvs show ral parts and lowr curvs imaginary parts. 46

47 Th disturbing inaccuracis at crtain frquncis in Fig. 3 dpnd on th duration of th loading puls Fig.. Ths frquncis, at which th xcitation of th disc is wak or nonxistnt, can b movd out of th frquncy rang of intrst by using an xcitation puls that is sufficintly narrow. If problmatic frquncis ar prsnt, th accuracy of th paramtrically stimatd complx shar modulus can b improvd by xcluding masurmnt data at and around ths frquncis. 47

48 7. Conclusions In Papr I, th complx xtnsion modulus, th complx shar modulus and th complx Poisson s ratio at room tmpratur hav bn idntifid for polymthyl mthacrylat PMMA and polypropyln PP in th approximat frquncy rang 1 to 40 khz for PMMA and 1 to 15 khz for PP. Th rsponss of both matrials wr found to b vry clos to linar and narly isotropic undr th conditions of th tsts. Th dviation from isotropy was largr for PP than that for PMMA. Th obsrvd dviation from isotropy is blivd to b partly du to th xtrusion and cooling procsss usd for fabrication of th PP tst bars. In Papr II it has bn shown that th joint stimation approach will improv th ovrall quality of th stimatd complx modulus E and Poisson s ratio. Whn th xcitation lvl is high, th last-squars stimation of complx modulus and simpl or wightd avraging of complx Poisson s ratio giv rsults of similar quality. Whn th xcitation lvl is low or th optimization problm is poorly conditiond numrically, howvr, th diffrnc btwn th rsults obtaind with th diffrnt mthods can b significant. For th non-quilibrium SHPB procdur studid in Papr III, th following can b concludd. i Th non-quilibrium SHPB procdur dvlopd, and its simplifid low-frquncy vrsion, can b usd to stimat th complx modulus in trms of masurd strains associatd with th rflctd and transmittd wavs. ii Th quality of th rsults is snsitiv to truncation and imprfct contact conditions at th bar-spcimn intrfacs. iii With PMMA bars, good rsults wr obtaind up to 10 khz for spcimns with aspct ratios 0.5, 1.0 and.5, and fair rsults wr obtaind for spcimns with aspct ratio 5.0. iv With AL bars, vry good rsults wr obtaind up to 10 khz for spcimns with aspct ratios 0.5 and 1.0, and fair rsults wr obtaind for th spcimn with aspct ratio.5. iv Th classical SHPB procdur, basd on quilibrium in th spcimn, ovrstimats th magnitud of th complx modulus at all frquncis. Th non-quilibrium SHPB procdur was usd in Papr IV to idntify th complx xtnsion modulus for thin pharmacutical tablts for frquncis up to 0 khz. In addition to providing a masur of th tablt stiffnss 48

49 as a function of frquncy, a rlaxation was obsrvd in th matrials that could b rlatd to th intr-particl binding mchanisms. With hlp of paramtric data, th intrprtations of ths rlaxations show th potntial of this mthod in achiving a bttr undrstanding of th binding mchanisms and physical proprtis of compactd matrials. Conclusions from th study of viscolastic impact in Papr V can b summarisd as follows: i Th impact forc consists of a main puls that is ithr isolatd or followd by a tail with finit or infinit lngth. Th ris and fall of th main puls ar discontinuous, and th width of this puls corrsponds to two transit tims through th strikr for th viscolastic wav front. ii Th tail of th impact forc and th phnomnon of multipl impacts can b avoidd by choosing strikr-to-bar charactristic impdanc ratio r sufficintly small. This is of particular intrst in SHPB tsting. iii Thr is good agrmnt btwn xprimntal and thortical rsults for strains in a PMMA bar impactd by strikrs of th sam matrial and with th sam cross-sctional ara. In Papr VI, it has bn shown thortically and xprimntally that: i Th complx shar modulus of an isotropic and linarly viscolastic matrial can b idntifid on th basis of th volution of an outgoing shar wav btwn two radial positions on a disc at which shar strains associatd with th wav ar masurd. ii Th two-dimnsional wav solutions usd ar xact in th sns of thr-dimnsional thory, and thrfor thr is, in principl, no frquncy byond which th thortical basis is not valid. iii Th mthod rquirs a minimum disc siz which is rlatd to th duration of th load. iv Th non-paramtric rsults bcom inaccurat at frquncis nar zro and at frquncis whr th xcitation is wak or non-xistnt. v Th formr frquncis can b movd towards zro and th lattr can b movd outsid th frquncy rang of intrst by sufficintly dcrasing th duration of th loading puls. Altrnativly, th rquird siz of th disc can b rducd. vi If thr ar problmatic frquncis within th rang of intrst, th rsults of paramtric idntification ar mor accurat and rprsntativ than thos of non-paramtric idntification. vii Paramtric rsults from xprimntal tsts with loads having diffrnt amplituds and durations agr wll with ach othr in accord with th assumd linarity of th tstd Polypropyln matrial. viii Th complx shar modulus of an isotropic and linarly viscolastic matrial can b idntifid similarly also without rstriction to outgoing wavs. This procdur has th advantag of not rquiring a minimum siz of th spcimn but is computationally complx and may lad to rducd accuracy. 49