Mechanics of Materials 43 (2011) 598–607

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Mechanics of Materials

journal homepage: www.elsevier .com/locate /mechmat

Dynamic mechanical and ultrasonic properties of polyurea

Jing Qiao a,b,⇑, Alireza V. Amirkhizi b, Kristin Schaaf b, Sia Nemat-Nasser b, Gaohui Wu a

a School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, Chinab Mechanical and Aerospace Engineering, Center of Excellence for Advanced Materials, University of California, San Diego, CA 92093, USA

a r t i c l e i n f o

Article history:Received 12 April 2011Received in revised form 9 June 2011Available online 30 June 2011

Keywords:PolyureaMechanical propertiesTime–temperature superposition

0167-6636/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.mechmat.2011.06.012

⇑ Corresponding author at: School of Materials Sing, Harbin Institute of Technology, Harbin 150001,

E-mail address: jingqiao.hit@gmail.com (J. Qiao)

a b s t r a c t

Dynamic mechanical analysis (DMA) and ultrasonic measurements were carried out tostudy the temperature and frequency dependences of viscoelastic properties of polyurea.Master curves of Young’s storage and loss moduli were developed from the DMA data.Relaxation spectra were subsequently calculated by means of two approximate models,and the apparent activation energy of molecular rearrangements was also determinedbased on the temperature dependence of the time–temperature shift factor. Velocity andattenuation of longitudinal and shear ultrasonic waves in polyurea were measured in the0.5–2 MHz frequency range between �60 and 30 �C temperatures. The complex longitudi-nal and shear moduli were computed from these measurements. Combining these resultsprovided an estimate of the complex bulk and Young’s moduli at high frequencies. Theresults of the DMA and temperature and frequency shifted ultrasonic measurements arecompared and similarities and deviations are discussed.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Polyureas are a class of segmented block copolymerderived from the chemical reaction between an isocyanateand an amine. Similar to the segmented polyurethanes,polyurea generally microphase separates into high-Tg

‘‘hard’’ domains and relatively low-Tg ‘‘soft’’ domains(Fragiadakis et al., 2010; Das et al., 2007; Yi et al., 2006;Pathak et al., 2008). The hard segment domains, whichform thread-like, crystalline structures, are typically dis-persed in the continuous soft segment matrix (Das et al.,2007). The hard segments are extensively hydrogen-bonded and function as both reversible physical cross-linksand reinforcing fillers, thus providing good mechanicalproperties (Pathak et al., 2008). By tailoring the underlyinghard and soft domain structure through chemistry, poly-urea offers a wide range of mechanical properties, fromsoft rubber to hard plastic. Together with its rapid poly-merization and fire, abrasion, and corrosion resistance,

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cience and Engineer-China.

.

polyurea has a myriad of applications in the coating indus-try, e.g. on transportation vehicles, pipelines, steel build-ings and marine constructions (Shime and Mohr, 2009).More recently, it has been used either as a protective coat-ing on metallic structures and buildings or an insertedlayer in a blast-tolerant sandwich structure to impart im-proved blast resistance (Tekalur et al., 2008; Mock and Bal-izer, 2005; Amini et al., 2006; Bahei-El-Din and Dvorak,2006; Amini et al., 2010a,b).

As an elastomer, polyurea exhibits viscoelastic behav-ior, which depends strongly on the strain rate (or fre-quency) and temperature, as well as pressure. A numberof studies have been carried out to determine and interpretthe viscoelastic behavior of polyurea, in order to under-stand its performance. Das et al. (2007) researched thedynamic mechanical properties of polyurea at low fre-quencies. Yi et al. (2006) and Sarva et al. (2007) studiedthe stress–strain behavior of polyurea in uniaxial compres-sion over a range of strain rates from 10�3 s�1 to 104 s�1 inquasistatic tests and split Hopkinson pressure bar (SHPB)tests, and found that the flow stress magnitude of polyureaincreases as the strain rate increases. Roland et al. (2007)reported the stress–strain measurements in uniaxial

J. Qiao et al. / Mechanics of Materials 43 (2011) 598–607 599

tension at intermediate strain rates (0.06–573 s�1) using adrop weight test instrument, and subsequently they ex-tended the strain rate to 830 s�1 (Pathak et al., 2008). Morerecently, Shime and Mohr (2009) further evaluated the ratesensitive response of polyurea in a strain rate range of 101

to 103 s�1, using a modified SHPB system.Because of its application as an impact-resistant coat-

ing, the response of polyurea to very high strain rates (highfrequencies) is of some importance. However, the higheststrain rate provided by the SHPB system is limited to104 s�1, according to the published works stated previ-ously. In order to extend the time/frequency scale, Zhaoet al. (2007) determined the compressive relaxation behav-ior of polyurea using quasistatic tests in a servo-hydraulicsystem at temperatures between �49 and 22 �C and thenconstructed a relaxation master curve at 0 �C over areduced time range of 10�10 to 107 s, based on the time–temperature superposition (TTS) principle. Afterwards,they demonstrated its validity with the aid of a simulationmodel and SHPB measurement. For similar work based ontensile relaxation data, see Amirkhizi et al. (2006). On thecontrary, Fragiadakis et al. (2010), Pathak et al. (2008)opposed the applicability of the TTS principle on polyureaand they conducted dielectric spectroscopy measurements(Roland and Casalini, 2007; Bogoslovov et al., 2007), whichcould provide characterization over a wide frequencyrange (10�2–106 Hz) and elevated pressure (1 GPa), toelucidate the segmental dynamics of polyurea.

However, experimental data are still scarce at frequen-cies higher than 106 Hz. Characterizing rubbery polymersat high frequency is difficult, even at small amplitudes.Typical dynamic mechanical analyzers are limited to fre-quencies below 100 Hz (Roland et al., 2007). When the fre-quency is higher than 1 MHz, we are not aware of anypublished literature on measuring the modulus by apply-ing known values of the stress and measuring the strain(Sinha and Buckley, 2006). Under such conditions, an ultra-sonic technique appears to be a valuable method to mea-sure response in high frequency regions. In addition, theprevious researchers have not reported the relaxationspectrum of polyurea. The relaxation spectrum is aninherent material property and dependent on molecularrelaxation times. Therefore it should be theoretically inde-pendent of the experimental technique (Alvarez et al.,2007). Bulk behavior of polyurea, which is fundamentallydifferent from the shear properties and considerably diffi-cult to measure directly at high frequencies, is also rarelyaddressed.

In the present investigation, we report the results of aseries of tests performed to study the temperature andfrequency dependence of the viscoelastic behavior of poly-urea, including results obtained using dynamic mechanicalanalysis (DMA) and ultrasonic wave measurements. Mas-ter curves are developed from the DMA data and relaxationspectra are then obtained. Ultrasonic wave measurement iscarried out over the frequency range from 0.5 to 2 MHz attemperatures from �60 to 30 �C. Both longitudinal andshear waves are studied. From the calculated complex lon-gitudinal and shear moduli, complex bulk and Young’smoduli at high frequencies are estimated. Finally, theresults of the DMA and temperature and frequency shifted

ultrasonic measurements are compared. The similarities ofthe two sets of measurements are analyzed, and the appli-cability of the theory is discussed.

2. Experimental details

2.1. Material

The polyurea was prepared by the reaction of a polycar-bodiimide-modified diphenylmethane diisocyanate (Ison-ate 2143L, Dow Chemical) and poly(tetramethyleneoxide-di-p-aminobenzoate) (Versalink P-1000, Air Products). Inorder to ensure that the reaction is completed and has pro-duced some cross-linking a stoichiometric ratio of 1.05:1isocyanate to amine was used herein. First, the two compo-nents were degassed separately under 1 torr vacuum untilmost of the entrapped air bubbles were removed. Then,they were mixed for a few minutes while still under vac-uum. Finally, the mixture was cast into a Teflon mold toobtain the test samples. Prior to measurements, the sam-ples were cured at room temperature (25 �C) in an environ-mental chamber maintained at 10% relative humidity fortwo weeks.

2.2. Dynamic mechanical analysis (DMA)

Dynamic mechanical analysis was conducted using a TAInstruments Dynamic Mechanical Analyzer 2980, using thecorresponding software to collect and analyze the experi-mental data. The samples measured approximately 3 mmthick by 10 mm wide and were clamped at a free lengthof 17.5 mm. Both ends were cantilevered, i.e., they wereconstrained from rotation and sliding at both ends byclamping plates and excited into a sinusoidal transversedisplacement at one end with a strain amplitude of15 lm. The experiments were performed over the temper-ature range from �80 to 70 �C, stepping upwards in incre-ments of 3 �C. At each temperature step, five frequencies of1, 2, 5, 10 and 20 Hz were tested sequentially. Thermalsoaking times of 3 min at the beginning of each step min-imized the effects of thermal gradients. Liquid nitrogenwas used to cool the system to sub-ambient temperature.

2.3. Ultrasonic measurement

The ultrasonic system used in the present work wasbuilt upon a personal computer (PC) system. As illustratedin Fig. 1(a), it consists of a Matec� TB-1000 tone-burst sig-nal generating and receiving card, Panametrics� contacttransducers, a 100:1 attenuator, and a Tektronix DPO3014 digital phosphor oscilloscope. The dashed rectanglein Fig. 1(a) represents the temperature control chamber.Toneburst signals of given frequencies are sent from thecard to the transmitting transducer, propagate throughthe sample to the receiving transducer, and eventually sentdirectly to the oscilloscope. The received signals differ inarrival time and amplitude because of both the differentthicknesses of samples sandwiched between the twotransducers and the wave speed in the material. In essence,the small thickness experiment is the calibration needed to

a bFig. 1. Block diagram of ultrasonic experiment setup. The dashed rectangle in (a) represents the temperature control chamber, which contains the sampleand transducers, as shown in (b) where three sample configurations are shown. The first two configurations are used (with one and two sample pieces,respectively, to measure the time of travel for two thicknesses) for longitudinal waves. The bottom configuration is used for shear wave measurements. Thealuminum rod delays the shear wave compared to the longitudinal component. In this configuration, two thin samples of different thicknesses were used.

600 J. Qiao et al. / Mechanics of Materials 43 (2011) 598–607

remove the effect of interface impedance between thetransducer and the sample, as well as the time-delay inthe electronics and the transducers. Therefore comparisonbetween the large and small thickness tests representsonly the effect of extra material on the time-delay (wavespeed) and transmission amplitude (material loss).

2.3.1. Longitudinal wave measurementA pair of V133-RM Panametrics� contact transducers

was used to generate and receive longitudinal ultrasonicsignals. The nominal element size of the transducer is/8:9 � 10:7 mm and the nominal center frequency is2.25 MHz. Braider oil was used to achieve better surfacecoupling between the transducers and the sample. Samplesused in longitudinal wave tests have the dimensions13 � 13 � 6 mm. The temperature of the sample was var-ied stepwise by 10 �C decrements from 30 to �60 �C andat each step was stabilized 10 min to minimize the effectsof thermal gradients. Measurements were made in the fre-quency range from 0.5 to 2 MHz.

-4

-3

-2

-1

0

1

2

3

4

0.0E+00 5.0E-06 1.0E-05 1.5E-05

Am

plit

ude

(V)

Time (s)

Signal 1

Signal 2

t

A

A0

Fig. 2. Received signals of the measurement.

A detailed procedure is provided here. As illustrated inFig. 1(b), first, one sample was inserted between the twotransducers. The received signal was recorded at each tem-perature step. In the second test, two samples (includingthe first one) were inserted between the two transducers.The received signal was also recorded at each temperaturestep. By comparison of the signals from these two tests, thetime shift and amplitude change of the received signalwere obtained. For example, Fig. 2 shows the two receivedsignals at 0 �C and 1.6 MHz. ‘‘Signal 1’’ represents the re-ceived signal when just one sample was used; ‘‘Signal 2’’represents the received signal when two samples wereused. Consequently, the velocity in the sample can be cal-culated according to the relation:

vL ¼ d=t; ð1Þ

where d is the thickness of the second sample (measured atroom temperature), t represents the time shift observedbetween the two tests, as indicated in Fig. 2.

The attenuation coefficient per unit thickness, a (Neper/cm), is given by

a ¼ 1d

lnA0

A; ð2Þ

where A0 and A are the amplitudes of the received signalsfrom the first and second test respectively, as indicated inFig. 2. Average values from at least six pairs of peaks wereused in the following calculation.

2.3.2. Shear wave measurementA pair of V153-RM normal incidence shear wave trans-

ducers was used to generate and receive shear ultrasonicsignals. The nominal element size of the transducer is/17:8 � 16 mm and the nominal frequency is 1 MHz.The transducers were in contact via a thin film of a shearwave couplant on the sample surface. As illustrated in

J. Qiao et al. / Mechanics of Materials 43 (2011) 598–607 601

Fig. 1(b), an aluminum rod was inserted in the wave pathbetween the generating transducer and the sample to de-lay the shear waves compared to the longitudinal waveswhich inevitably accompany the generation of the shearwave by the piezoelectric transducers. This method hasbeen used by some researchers (Cunningham and Ivey,1956; Bezot and Hesse-Bezot, 2002). The dimension ofthe aluminum rod is quite critical (Cunningham and Ivey,1956). Since longitudinal waves travel faster than shearwaves in metal rod, the length of the aluminum rod shouldbe long enough to ensure that the effect of the longitudinalwaves on the receiving transducer is over before the shearwaves arrive. However, if it is too long, the shear signalswill be seriously attenuated, and reflected longitudinalwaves may interfere with the shear waves. The diametermust be such that there are no interference effects due tolongitudinal waves coming from the sides of the rod. Therod used herein is 76.2 mm in diameter and 50.8 mm long.Since shear waves are highly attenuated in elastomers, it isnecessary to use very thin samples. In order to avoid intro-ducing interfaces which are comparable in thickness tothat of the sample, two samples of 0.77 and 1.22 mm thick-nesses were used. Unlike the longitudinal wave measure-ment, only one sample was employed in each test. Theshear wave velocity and attenuation were also calculatedaccording to Eqs. (1) and (2).

Because of the substantial attenuation in polyurea athigher temperatures, shear wave transmission in the fre-quency region from 0.5 to 2 MHz was not reliably mea-sured at temperatures exceeding 0 �C. Consequently,samples were studied from 0 to �50 �C, stepping down-wards in decrements of 10 �C. The hold time at each tem-perature was increased to 20 min due to the utilization ofthe aluminum rod.

3. Theory

Problems involving wave propagation in linear visco-elastic media are generally solved using Laplace or Fouriertransforms; see, for example, Liu and Subhash (2006) orZhao and Gary (1995), respectively. Here we consider thesimple case of the propagation of a single frequency har-monic plane wave through an isotropic homogeneous lin-ear viscoelastic medium. In the frequency domain afterFourier transform the wave equation is

M�o2u=ox2 ¼ �qx2u; ð3Þ

where u is the phasor form of particle displacement fromits equilibrium position (in the x direction for longitudinalwaves, and in a transverse direction for shear waves); q isthe density; M⁄ represents the effective complex modulusfor the considered wave polarization (longitudinal orshear); x is the angular frequency. The wave velocity is gi-ven by the usual relation:

c� ¼ c0 þ ic00 ¼ M�=qð Þ1=2: ð4Þ

The direct solution of Eq. (4) gives:

M0 ¼ q ðc0Þ2 � ðc00Þ2h i

; ð5aÞ

M00 ¼ 2qc0c00: ð5bÞ

where M0 and M00 are the real and imaginary componentsof M⁄, i.e., storage and loss moduli respectively.

The harmonic solution of Eq. (3) is:

u¼ u0 exp ix t� xc�

� �x

n o¼ u0 exp ixt exp

�ixxc0 þ ic00

� �

¼ u0 exp ixt exp�xc00x

c0ð Þ2 þ c00ð Þ2

( )exp

�ixc0x

c0ð Þ2 þ c00ð Þ2

( ): ð6Þ

Comparing Eq. (6) with the attenuated harmonic wavewith phase velocity c, and amplitude attenuation constanta:

u ¼ u0 exp ixt � ixc� ia

� �x

n o; ð7Þ

the real and imaginary parts of the complex velocity are gi-ven by the relations:

c0 ¼ c1þ r2 ; ð8aÞ

c00 ¼ cr1þ r2 ; ð8bÞ

where the dimensionless parameter r is defined by

r ¼ acx: ð9Þ

From Eqs. (5) and (8) we find:

M0 ¼qc2 1� r2

� �1þ r2ð Þ2

; ð10aÞ

M00 ¼ 2qc2r

1þ r2ð Þ2: ð10bÞ

For longitudinal wave transmission, the quantity M⁄ repre-sents complex longitudinal modulus L⁄:

M� ¼ L� ¼ K� þ 43

G�; ð11Þ

where K⁄ is the bulk modulus. In shear tests, the modulusis simply the shear modulus of the material:

M� ¼ G�: ð12Þ

From the two complex moduli K⁄ and G⁄, the complexYoung’s modulus can be calculated from the relation:

E� ¼ 9K�G�

3K� þ G�: ð13Þ

Separating the real and imaginary parts of Eq. (13) will leadto quite complicated expressions for the storage and losscomponents. The calculations in this work were madeusing the complex form of Eq. (13).

4. Results and discussion

4.1. DMA results

Fig. 3 shows the temperature dependence of the storageand loss components of the complex Young’s modulus atfive frequencies obtained from DMA low frequency mea-surements. It can be seen that only one relaxation peak oc-

0

20

40

60

80

100

120

140

160

180

0

500

1000

1500

2000

2500

3000

-100 -80 -60 -40 -20 0 20 40 60 80

You

ng's

Los

s M

odul

us (

MP

a)

You

ng's

Sto

rage

Mod

ulus

(M

Pa)

Temperature (ºC)

1 Hz

2 Hz

5 Hz

10HZ

20Hz

Fig. 3. Young’s storage and loss moduli as functions of temperature atfive frequencies.

0.0

0.5

1.0

1.5

2.0

2.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-10 -5 0 5 10 15 20 25

log

E"*

Tre

f/T

(MP

a)

log

E'*

Tre

f/T

(MP

a)

log ωaT (Hz)

Loss modulus

Storage modulus

Fig. 4. Master curves of Young’s storage and loss moduli at a referencetemperature Tref = 1 �C.

-10

-5

0

5

10

15

20

-100 -80 -60 -40 -20 0 20 40 60 80

log

aT

Temperature (ºC)

Fig. 5. Temperature dependence of the shift factor aT. The filled pointsindicate present results and the unfilled points denote previous resultsreported by Zhao et al. (2007). The solid line is the WLF equation fit to ourpresent results. Tref = 1 �C, C1 = 18.86 and C2 = 161.1.

602 J. Qiao et al. / Mechanics of Materials 43 (2011) 598–607

curs in the temperature region studied, which can beattributed to the glass transition of the soft segment inthe polyurea. The dynamics of the soft domains in thepolyurea is extremely heterogeneous (Fragiadakis et al.,2010). This is due to the fact that relative to their middleportions, at the ends they are covalently bonded to thehard domains which means: (a) they are mechanically con-strained and (b) they have a much higher concentrationand less free volume to move. Therefore, the glass transi-tion is extremely broad, with a DT = Tend � Tonset of about50 �C. As the frequency is increased, the Young’s storagemodulus increases over the temperature covered, whichis more marked in the glass transition region, while theYoung’s loss modulus decreases in amplitude and shiftsto higher temperatures.

The Young’s storage and loss moduli curves are shownin linear scale in Fig. 3. They were then transformed fortime–temperature superposition (TTS) shifting by: (a)changing the horizontal axis to logarithmic frequencyscale, (b) normalizing the modulus values by T/Tref, and(c) plotting the horizontal axis in logarithmic scale. Theywere then shifted independently in the temperature rangefrom �71 to 67 �C. The temperature function aT is the totalamount of shift along the horizontal axis (frequency)needed at each temperature to assemble the entire seriesof experiments into a single master curve based on Tref.Its magnitude represents the ratio of the primary relaxa-tion time (or inverse frequency) at temperature T to thatat temperature Tref. For further fundamental discussionsee Ferry (1980). Master curves and shift factor aT for a ref-erence temperature Tref = 1 �C were determined, as repre-sented in Figs. 4 and 5. The detailed procedure is given inQiao et al. (2011). It is noteworthy that the shifting proce-dure based on the TTS was done with smooth transitionsand therefore its application is qualitatively justified. Threeregions may be identified over the somewhat extendedrange of frequencies in Fig. 4. The storage modulus levelsoff at both low and high reduced frequencies (xaT). Theshifting procedure works almost perfectly in the middle re-gions, but it starts losing accuracy as one gets closer to thetwo ends of the xaT region for the loss modulus curve. Thisis partially due to the fact that the measured loss modulusbecomes flat (as a function of frequency) at lower xaT re-gions and TTS principle becomes unavailable at higher

xaT regions. The loss modulus peak in the frequency regionstudied, indicates the glass transition phenomena.

The temperature dependence of the shift factor, aT, isplotted in Fig. 5. Shift factors reported by Zhao et al.(2007), who constructed a relaxation master curve basedon the quasistatic relaxation of polyurea and subsequentlydemonstrated its validity with the aid of a split Hopkinsonpressure bar test, are also included in Fig. 5. As the compar-ison makes clear, the present shift factors agree well withthe previous results. The present shift factors in the tem-perature range from �71 to 49 �C were fit to the Wil-liams–Landel–Ferry (WLF) equation:

log aT ¼�C1 T � Trefð ÞC2 þ T � Trefð Þ : ð14Þ

Thus, the parameters C1 = 18.86 and C2 = 161.1 wereobtained.

The mechanical behavior of polyurea is dictated byproper combination of time (frequency) and temperature.The results in Figs. 4 and 5 represent the proper way ofcombining these two variables based on the method of re-duced variables (Ferry, 1980).

0

50

100

150

200

250

300

350

-80 -60 -40 -20 0 20 40 60

Eα

(KJ

/mo

l)

Temperature (ºC)

WLF Equation

Numerical derivative

Fig. 7. Temperature dependence of the apparent activation energy for thea-relaxation processes in polyurea.

J. Qiao et al. / Mechanics of Materials 43 (2011) 598–607 603

The relaxation spectrum U(s), represents the contribu-tion to moduli by relaxation mechanisms whose time con-stants are around s (Ferry, 1980):

E0 xð Þ ¼ Ee þZ 1

�1Ux2s2= 1þx2s2� ��

d lns; ð15aÞ

E00 xð Þ ¼Z 1

�1Uxs= 1þx2s2� ��

d lns; ð15bÞ

The relaxation spectrum U can be calculated from the fre-quency dependence of E0 and E00, by the first approximationformula suggested by Ferry (1980):

UðsÞ ¼ E0 d log E0=d log x� �

1=x¼s; ð16aÞ

U sð Þ ¼ E00 1� d log E00=d log x� �

1=x¼s: ð16bÞ

where s is the relaxation time and d representsdifferentiation.

A modified second approximation, given by Williamsand Ferry (1953), provides the following formulas fordetermining U:

UðsÞ ¼ AE0 1� d log E0

d log x� 1

� �1=x¼s

; ð17aÞ

UðsÞ ¼ BE00 1� d log E00

d log x

� �1=x¼s

: ð17bÞ

0

0.5

1

1.5

2

2.5

log

(MP

a)

logτ (s)

First approximation-E'

First approximation-E"

(a)

0

0.5

1

1.5

2

2.5

-20 -15 -10 -5 0 5 10

-20 -15 -10 -5 0 5 10

log

(MP

a)

logτ (s)

Second approximation-E'

Second approximation-E"

(b)

ΦΦ

Fig. 6. Relaxation spectrum (U) obtained from reduced mechanicalspectra of E0 and E00 by first (a) and second (b) approximation formula.

The factors A and B are given by

A ¼ ð1þ jm� 1jÞ=2C 2�m2

� �C 1þm

2

� �� 2 6 m 6 2;

ð18aÞ

B ¼ ð1þ jmjÞ=2C32� m

2

� �C

32þ m

2

� �� 1 6 m 6 3;

ð18bÞ

where m is the slope of a plot of log U (from the firstapproximation calculation) against log s. Calculations ofU derived from the data of Fig. 4 based on the first approx-imation and the modified second approximation formulasare depicted in Fig. 6. There is good agreement betweenthe values from the storage (E0) and loss moduli (E00). Thisagreement is a confirmation of the internal consistencyof the experimental data (Williams and Ferry, 1954), sincethe storage and loss moduli are based on two independentphysical measurements; nevertheless they result in thesame relaxation spectrum.

An apparent activation energy for the a-relaxation pro-cesses, Ea, can be determined from the temperature depen-dence of E0 and E00 through the shift function aT (Wang andPloehn, 1996):

Ea ¼ Rd½ln aTð Þ�d 1=Tð Þ ¼ 2:303R

d½log aTð Þ�d 1=Tð Þ ; ð19Þ

where R = 8.314 JK�1 mol�1 is the gas constant. Ea for poly-urea was obtained by calculating the numerical derivativeof log(aT) and was plotted as a function of temperature inFig. 7. In the case of log(aT) described by WLF equation,Eq. (19) is converted into the form:

Ea ¼ 2:303RC1C2T2

C2 þ T � T0ð Þ2; ð20Þ

where C1 = 18.86, C2 = 161.1 and T0 = 1 �C. The calculatedEa from Eq. (20) is also shown in Fig. 7. The two sets of dataare in good agreement, except at temperatures close to thetwo ends of the temperature region. This is due to the low-er accuracy of the shifting procedure as stated previously.It can be seen the calculated Ea increases with decreasingtemperature as the glass transition is approached. Thisbehavior primarily reflects changes in a segmental ormonomeric friction coefficient, which involves volume

2800

(m

/s)

-60C

604 J. Qiao et al. / Mechanics of Materials 43 (2011) 598–607

and energy requirements for local co-operative motions(Bueche, 1953; Fox and Flory, 1950).

1600

1800

2000

2200

2400

2600

0 0.5 1 1.5 2 2.5

Lon

gitu

dina

l Wav

e V

eloc

ity

Frequency (MHz)

-50C

-40C

-30C

-20C

-10C

0C

10C

20C

30C

Fig. 9. Longitudinal wave velocity as a function of frequency.

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Att

enua

tion

per

wav

elen

gth

(Nep

er)

Frequency (MHz)

-60C

-50C

-40C

-30C

-20C

-10C

0C

10C

20C

30C

0.0

0.5

1.0

1.5

2.0

2.5

0.6 1 1.4 1.8 2.2

0.6 1 1.4 1.8 2.2

Att

enua

tion

(N

eper

/cm

)

Frequency (MHz)

-50C

-20C

30C

(a)

(b)

4.2. Ultrasonic measurement

The longitudinal wave velocity and attenuation coeffi-cient per unit thickness as a function of temperature at1.6 MHz are depicted in Fig. 8. It can be seen that the lon-gitudinal wave velocity increases monotonically withdecreasing temperature, while the attenuation coefficientper unit thickness peaks around the glassy transition at avalue of about 1.8 Neper/cm. Similar behavior has been ob-served by Nolle and Mowry (1948) and Ivey et al. (1949)for some rubbers. These curves have relatively the sameform at all measured frequencies.

Fig. 9 shows the longitudinal wave velocity as a func-tion of frequency at all temperatures. It can be seen thatthe longitudinal wave velocity is relatively insensitive tofrequency in the region studied. The frequency dependenceof attenuation per wavelength and attenuation coefficientper unit thickness is illustrated in Fig. 10. Only data pointsfor which the sample thickness is more than twice thewavelength are shown. The measured attenuation coeffi-cient per unit thickness is a nearly linear function of fre-quency with the slope proportional to the wave speed(Fig. 10(b)). Therefore, attenuation per unit wavelength isalmost constant as a function of frequency; see Fig. 10(a).Figs. 9 and 10(b) suggest that material properties are fre-quency-insensitive under these conditions.

From the primary data on longitudinal wave velocityand attenuation coefficient per unit thickness, the longitu-dinal storage modulus L0 and longitudinal loss modulus L00

are calculated from Eq. (10), as shown in Fig. 11.The shear wave measurement was conducted at 5 tem-

peratures (�50, �40, �20, �10, 0 �C) using 8 separate volt-age pulses with apparent excitation frequencies of 0.6–2.0 MHz (0.2 MHz step size). However, the transmittedshear stress wave frequency content was not substantiallydifferent from one excitation to others. The apparent fre-quency of the transmitted stress waves seems to be closeto 1 MHz, the design frequency of the transducers. We be-lieve that this effect is due to the significant impedancemismatch between the transducers and samples, whichmakes transmission more difficult at other frequency val-

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1500

1700

1900

2100

2300

2500

2700

-80 -60 -40 -20 0 20 40

Att

enua

tion

(N

eper

/cm

)

Lon

gitu

dina

l Wav

e V

eloc

ity

(m/s

)

Temperature (ºC)

Velocity

Attenuation per thickness

Fig. 8. Longitudinal wave velocity and attenuation coefficient per unitthickness in polyurea as a function of temperature at 1.6 MHz.

Fig. 10. Attenuation per wavelength (a) and attenuation coefficient perunit thickness (b) of longitudinal wave as a function of frequency.

ues. At each temperature, we calculated the average valuesof velocities and attenuations per wavelength at allapparent excitation frequencies and found the apparentfrequency of the stress wave. We averaged the measuredattenuation and wave velocity for those cases that wereclose to 1 MHz. The rest of the data is not reported andused here. Figs. 12 and 13 show the velocity and attenua-tion per wavelength of shear wave as a function oftemperature respectively. Similar to the longitudinal wavevelocity, shear wave velocity increases rapidly withdecreasing temperature. However, the attenuation perwavelength of shear wave decreases as temperaturedecreases.

2.5

3.5

4.5

5.5

6.5

7.5

8.5

L' (

GP

a)

Frequency (MHz)

-60C

-50C

-40C

-30C

-20C

-10C

0C

10C

20C

30C

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2 2.5

0.6 1 1.4 1.8 2.2

L"

(GP

a)

Frequency (MHz)

-60C

-50C

-40C

-30C

-20C

-10C

0C

10C

20C

30C

(a)

(b)

Fig. 11. Longitudinal storage (a) and loss moduli (b) as a function offrequency at several temperatures.

0

200

400

600

800

1000

1200

1400

-60 -40 -20 0 20 40

Shea

r W

ave

Vel

ocit

y (m

/s)

Temperature (ºC)

Fig. 12. Temperature dependence of shear wave velocity at 1.0 MHz. Thesolid line is power law fit (c = 1.594 � 1012 T�3.913; T, temperature inKelvin) used only to estimate the velocities at higher temperatures(10, 20, 30 �C).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

-60 -40 -20 0 20 40

Att

enua

tion

per

wav

elen

gth

(Nep

er)

Temperature (ºC)

Fig. 13. Temperature dependence of attenuation per wavelength of shearwave at 1.0 MHz. The solid curve is a second-order fit (a = 9.32 �10�5�T2 + 1.55 � 10�2�T + 0.967; T, temperature in centigrade) only usedto estimate the attenuation per wavelength at other temperatures(�30, 10, 20, 30 �C).

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-60 -40 -20 0 20 40

G"

(GP

a)

G' (

GP

a)

Temperature (ºC)

G'

G"

Fig. 14. Shear storage and loss moduli as functions of temperature at1.0 MHz. The hollow points are based on the curve-fitting done in Figs. 11and 12.

J. Qiao et al. / Mechanics of Materials 43 (2011) 598–607 605

The shear storage and loss moduli were also calculatedfrom the velocity and attenuation using Eq. (10) and theirtemperature dependence is shown in Fig. 14. The points at�30 �C and temperatures higher than 0 �C are based on theinterpolated and extrapolated portions of the velocity andattenuation curves in Figs. 12 and 13. The shear storagemodulus increases sharply as the temperature decreases.The calculated shear loss modulus at �50 �C is much high-er than that at �40 �C, which might be due to experimentalerror.

The storage and loss components of the complex bulkmodulus at 1.0 MHz are calculated from Eq. (12) and areplotted as a function of temperature, as shown in Fig. 15.Similar to the longitudinal and shear storage moduli, thebulk storage modulus is demonstrated to increase withdecreasing temperature. The magnitudes of shear storagemoduli are approximately 3% of the bulk storage modulusat higher temperatures, as expected for a nearly incom-pressible material. Near the glass transition, this percent-age increases as the temperature decreases. The bulkviscous loss is non-zero at all temperatures.

From the complex bulk and shear moduli, the Young’sstorage and loss moduli are evaluated and master curvesat a reference temperature Tref = 1 �C are constructed byreducing these calculated data based on the TTS principle.The shift factor obtained from the DMA data is used here.Correction to the moduli was made by multiplying the dataat temperature T by a factor (Tref/T). Comparison betweenmaster curves of the complex Young’s modulus fromDMA data and ultrasonic measurement data is shown inFig. 16. The two master curves of Young’s storage modulusare in agreement with each other. In contrast, the agree-ment of the two master curves is poor in the case of

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

2.5

3.0

3.5

4.0

4.5

5.0

5.5

-60 -40 -20 0 20 40

K"

(GP

a)

K' (

GP

a)

Temperature (ºC)

K'

K"

Fig. 15. Bulk storage and loss moduli as functions of temperature at1.0 MHz. The hollow points are based on the curve-fitting done in Figs. 11and 12.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-10 -5 0 5 10 15 20 25

log

E'(

E")

*Tre

f/T

(M

Pa)

log ωaT (Hz)

E'-DMA

E"-DMA

E'-Ultrasonic measurement

E"-Ultrasonic measurement

Fig. 16. Master curves of Young’s storage and loss moduli at a referencetemperature Tref = 1 �C. The hollow points are from the interpolated andextrapolated portions of the ultrasonic measurements.

606 J. Qiao et al. / Mechanics of Materials 43 (2011) 598–607

Young’s loss modulus. The Young’s loss modulus from theultrasonic measurement is greater than predicted fromthe DMA measurement and TTS analysis.

5. Summary and conclusions

In this work, the viscoelastic properties of polyureawere examined by dynamic mechanical analysis and ultra-sonic measurements. The former was carried out over thetemperature range from �80 to 70 �C at five frequencies.Master curves were constructed for the complex Young’smodulus. Relaxation spectra were subsequently calculatedby means of two successive approximation methods. Tem-perature dependence of the apparent activation energywas also determined, which shows an increase approach-ing the glass transition. The apparent activation energyrepresents the relaxation mechanism controlled by theavailability of free volume. The profile agrees with earlierWLF expectations and is distinguished from a constant va-lue as postulated in a rate kinetic model (Williams et al.,1955). Velocity and attenuation of both longitudinal andshear ultrasonic waves were measured in polyurea in a fre-quency range 0.5–2 MHz and the temperature range �60–30 �C. The complex longitudinal, shear, bulk and Young’smoduli were computed from the ultrasonic measurement

data with the aid of relationships between mechanicalproperties. Master curves of Young’s storage and loss mod-uli were deduced from these values by using the shiftfactors determined from the DMA measurements and com-pared with those derived from the DMA data.

It is noteworthy that by applying the time–temperaturesuperposition principle, one can predict the shear stiffnessat ultrasonic frequencies using only DMA data. However,the predicted loss clearly is below the experimental mea-surements at high frequencies. We believe that at higherfrequencies, the hard domains of polyurea become resona-tors and therefore contribute to higher loss throughout thematerial. It is expected that by control of size, properties,and distribution of hard domains in block copolymer andintroduction of other nano-scale or micro-scale inclusions,the loss spectrum can be enhanced and tailored towardsspecific applications.

Acknowledgments

The authors wish to thank Mr. Jon Isaacs for his helpsetting up the ultrasonics experiments. This work has beenconducted at the Center of Excellence for Advanced Mate-rials (CEAM) at the University of California, San Diego. Thiswork has been supported in part through ONR under GrantNo. N00014-09-1-1126 to the University of California, SanDiego.

References

Alvarez, G.A., Quintero, M.W., Eckert-Kastner, S., Alshuth, T., Reinecke, H.,2007. Measurement of high frequency linear viscoelastic functions ofa nitrile-butadiene rubber compound by ultrasound spectroscopy. J.Polym. Sci. B. Polym. Phys. 45, 91–102.

Amini, M.R., Isaacs, J.B., Nemat-Nasser, S., 2006. Effect of polyurea on thedynamic response of steel plates. In: Proceedings of the 2006 SEMAnnual Conference and Exposition on Experimental and AppliedMechanics, St Louis, MO.

Amini, M.R., Isaacs, J., Nemat-Nasser, S., 2010a. Investigation of effect ofpolyurea on response of steel plates to impulsive loads in directpressure-pulse experiments. Mech. Mater. 42, 628–639.

Amini, M.R., Isaacs, J., Nemat-Nasser, S., 2010b. Experimentalinvestigation of response of monolithic and bilayer plates toimpulsive loads. Int. J. Impact Eng. 37, 82–89.

Amirkhizi, A.V., Isaacs, J., McGee, J., Nemat-Nasser, S., 2006. Anexperimentally-based viscoelastic constitutive model for polyurea,including pressure and temperature effects. Philos. Mag. 86, 5847–5866.

Bahei-El-Din, Y.A., Dvorak, G.J., 2006. A blast-tolerant sandwich platedesign with a polyurea interlayer. Int. J. Solids Struct. 43, 7644–7658.

Bezot, P., Hesse-Bezot, C., 2002. Nice shear modulus of rubber: anultrasonic approach. KGK, Kautsch. Gummi Kunstst.⁄⁄⁄ 55, 114–117.

Bogoslovov, R.B., Roland, C.M., Gamache, R.M., 2007. Impact-induced glasstransition in elastomeric coatings. Appl. Phys. Lett. 90, 221910.

Bueche, F., 1953. Segmental mobility of polymers near their glasstemperature. J. Chem. Phys. 21, 1850–1855.

Cunningham, J.R., Ivey, D.G., 1956. Dynamic properties of various rubbersat high frequencies. J. Appl. Phys. 27, 967–974.

Das, S., Yilgor, I., Yilgor, E., Inci, B., Tezgel, O., Beyer, F.L., Wilkes, G.L., 2007.Structure-property relationships and melt rheology of segmented,non-chain extended polyureas: effect of soft segment molecularweight. Polymer 48, 290–301.

Ferry, J.D., 1980. Viscoelastic Properties of Polymers. Wiley.Fox, T.G., Flory, P.J., 1950. Second-order transition temperatures and

related properties of polystyrene. I. Influence of molecular weight. J.Appl. Phys. 21, 581–590.

Fragiadakis, D., Gamache, R., Bogoslovov, R.B., Roland, C.M., 2010.Segmental dynamics of polyurea: effect of stoichiometry. Polymer51, 178–184.

J. Qiao et al. / Mechanics of Materials 43 (2011) 598–607 607

Ivey, D.G., Mrowca, B.A., Guth, E., 1949. Propagation of ultrasonic bulkwaves in high polymers. J. Appl. Phys. 20, 486–492.

Liu, Q., Subhash, G., 2006. Characterization of viscoelastic properties ofpolymer bar using iterative deconvolution in the time domain. Mech.Mater. 38, 1105–1117.

Mock, W., Balizer, E., 2005. Penetration protection of steel plates withpolyurea layer. In: Presentation at the Workshop on PolyureaProperties and Enhancement of Structures under Dynamic Loads,Airlie, VA.

Nolle, A.W., Mowry, S.C., 1948. Measurement of ultrasonic bulk-wavepropagation in high polymers. J. Acoust. Soc. Am. 20, 432–439.

Pathak, J.A., Twigg, J.N., Nugent, K.E., Ho, D.L., Lin, E.K., Mott, P.H.,Robertson, C.G., Vukmir, M.K., Epps III, T.H., Roland, C.M., 2008.Structure evolution in a polyurea segmented block copolymerbecause of mechanical deformation. Macromolecules 41, 7543–7548.

Qiao, J., Amirkhizi, A.V., Schaaf, K., Nemat-Nasser, S., 2011. Dynamicmechanical analysis of fly ash filled polyurea elastomer. J. Eng. Mater.Technol. 133, 011016-1–011016-7.

Roland, C.M., Casalini, R., 2007. Effect of hydrostatic pressure on theviscoelastic response of polyurea. Polymer 48, 5747–5752.

Roland, C.M., Twigg, J.N., Vu, Y., Mott, P.H., 2007. High strain ratemechanical behavior of polyurea. Polymer 48, 574–578.

Sarva, S.S., Deschanel, S., Boyce, M.C., Chen, W., 2007. Stress-strainbehavior of a polyurea and a polyurethane from low to high strainrates. Polymers 48, 2208–2213.

Shime, J., Mohr, D., 2009. Using split hopkinson pressure bars to performlarge strain compression tests on polyurea at low, intermediate andhigh strain rates. Int. J. Impact Eng. 36, 1116–1127.

Sinha, M., Buckley, D.J., 2006. Physical Properties of Polymers Handbook.Springer.

Tekalur, S.A., Shukla, A., Shivakumar, K., 2008. Blast resistance of polyureabased layered composite materials. Compos. Struct. 84, 271–281.

Wang, J., Ploehn, H.J., 1996. Dynamic mechanical analysis of the effect ofwater on glass bead-epoxy composites. J. Appl. Polym. Sci. 59, 345–357.

Williams, M.L., Ferry, J.D., 1953. Second approximation calculations ofmechanical and electrical relaxation and retardation distributions. J.Polym. Sci. 11 (2), 169–175.

Williams, M.L., Ferry, J.D., 1954. Dynamic mechanical properties ofpolyvinyl acetate. J. Colloid Sci. 9 (5), 479–492.

Williams, M.L., Landel, R.F., Ferry, J.D., 1955. The temperature dependenceof relaxation mechanisms in amorphous polymers and other glass-forming liquids. J. Am. Chem. Soc. 77, 3701–3707.

Yi, J., Boyce, M.C., Lee, G.F., Balizer, E., 2006. Large deformation rate-dependent stress–strain behavior of polyurea and polyurethanes.Polymer 47, 319–329.

Zhao, H., Gary, G., 1995. A three dimensional analytical solution of thelongitudinal wave propagation in an infinite linear viscoelasticcylindrical bar. Application to experimental techniques. J. Mech.Phys. Solids 43, 1335–1348.

Zhao, J., Knauss, W.G., Ravichandran, G., 2007. Applicability of the time–temperature superposition principle in modeling dynamic responseof a polyurea. Mech. Time Depend. Mater. 11, 289–308.