Relative variations of nonlinear elastic moduli in polystyrene-based nanocomposites
RRelative variations of nonlinear elastic moduli in polystyrene-basednanocomposites
A.V. Belashov a , Y.M. Beltukov a , O.A. Moskalyuk b and I.V. Semenova a , ∗ a Ioffe Institute, 26, Polytekhnicheskaya, St.Petersburg, 194021, Russia b St.Petersburg State University of Industrial Technologies and Design, 18, Bolshaya Morskaya, St. Petersburg, 191186, Russia
A R T I C L E I N F O
Keywords :nonlinear elastic modulipolymer-based nanocompositespolystyreneelastic properties
A B S T R A C T
In this paper we apply the methodology based on the analysis of changes in acoustic wave veloci-ties under static stress for measurements of the third-order elastic moduli in three polystyrene-basednanocomposites with different fillers: SiO particles, halloysite natural tubules, and carbon black par-ticles. The samples were fabricated by the same technology and our data provide information onrelative changes of nonlinear properties of the composites caused by addition of the fillers. The dataobtained for composites are compared with that for commercial grade polystyrene. The substantialvariations of the nonlinear elastic moduli for composites with different types of fillers are demonstratedand analyzed. The results are in a qualitative agreement with theoretical predictions.
1. Introduction
Due to the current progress in engineering functionalnano-, micro- and macro-composites are widely used in var-ious industries, in aerospace, automotive technologies, andpipe line transportation, in particular [1, 2, 3, 4]. Rapidlygrowing applications of composite materials stimulate anal-ysis of various aspects of their mechanical behavior. How-ever, the composite response to intensive dynamic loadingis often hard to predict since it may depend drastically uponmatrix and filler characteristics, filler distribution in the ma-trix and on resulting mechanical characteristics of the com-posite. The reliable prediction of novel composite materi-als behavior under dynamic loading is one of the most de-manded problems of the modern condensed matter physicsfrom the viewpoint of both theoretical description and appli-cations.Nonlinear elastic properties of materials are increasinglyfound to be of high importance in the description of ma-terial response to various loads. A number of researchershave suggested approaches to define nonlinear elasticity us-ing third-order elastic moduli, or nonlinearity parameters,that are applied as measures to evaluate the deviation ofa stress-strain relationship from linear behavior. The onesused most frequently for the description of nonlinear elas-ticity in solids are those introduced by Landau and Lifshitz[5], Murnaghan [6] and Thurston and Brugger [7]. The setsof nonlinear elastic moduli defined in these models are infact closely related and can be calculated one from another.The third-order elastic moduli and their linear combina-tions were already demonstrated to be informative for theprediction of fatigue damage, for description of thermoelas-tic properties of crystalline solids, acoustic radiation stress,radiation-induced static strains, creep, thermal aging, waveprocesses, etc. In general nonlinear parameters were demon-strated to be more sensitive to structural changes in the ma-terial than linear ones. ∗ Corresponding author: [email protected]ffe.ru (I.V. Semen-ova)
In Murnaghan’s approximation the nonlinear elastic be-havior of an isotropic solid material is described by threenonlinear, third-order moduli ( 𝑙, 𝑚, 𝑛 ) and two linear, 2ndorder, Lamé constants ( 𝜆, 𝜇 ). First measurements of theseconstants were performed by Hughes and Kelly, based onthe simplified Murnaghan’s theory, see [8] for details. Inbrief, the methodology is based on the acousto-elastic ef-fect and applies the analysis of the dependence of velocity oflongitudinal and shear ultrasound waves in the sample uponthe applied static stress, providing data for determination ofnonlinear elastic moduli of the material. This technique wasthen widely applied the measurements of nonlinear elasticmoduli of various materials, see e.g. [9, 10, 11] and is stillone of the most frequently used approaches.Other approaches developed to assess nonlinear elas-tic properties of materials utilize dynamic acousto-elasticity[12, 13], second harmonic generation [14], Brillouin scatter-ing [15], coda wave interferometry [16], strain solitary [17],Lamb [18] and Rayleigh [19] waves. However, no standardmeasurement procedure has been approved by now.There are many theoretical and numerical studies on thelinear elastic moduli of nanocomposites [20, 21, 22, 23].However, theoretical models of the nonlinear elastic proper-ties of composite materials are still in an active development[24, 25, 26, 27, 28, 29]. A theory, which takes into accountnonlinear elastic properties of both the matrix and the fillerwas developed recently for the case of spherical inclusions[30]. The experimental validation of theoretical predictionsis highly desirable, however by now measurements of non-linear elastic properties of composites are very rare. Andto the best of our knowledge no data on that for polymericcomposites has been published as yet.In this paper we apply the methodology based on theanalysis of changes in acoustic wave velocities under staticstress for measurements of the third-order elastic moduliin three polystyrene-based nanocomposites with differentfillers. The samples were fabricated by the same technologyand our data provide information on relative changes of non-
A.V. Belashov et al.:
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Page 1 of 7 a r X i v : . [ c ond - m a t . d i s - nn ] M a y elative variations of nonlinear elastic moduli in polystyrene-based nanocomposites linear properties of the composites provided by addition ofthe fillers. The results obtained for composites are comparedwith those for commercial grade pure polystyrene. The cor-relation of experimental data with theoretical predictions isanalyzed.
2. Methodology for evaluation of thethird-order elastic moduli
According to Hughes and Kelly [8] in the case when noshear deformation in the sample takes place and strain tensoris diagonal, velocities of longitudinal and shear ultrasonicwaves propagating in the sample along the 𝑥 direction canbe expressed in the form: 𝜌 𝑉 𝑥 = 𝜆 + 2 𝜇 + (2 𝑙 + 𝜆 ) Tr 𝜀 + (4 𝑚 + 4 𝜆 + 10 𝜇 ) 𝜀 𝑥𝑥 , (1) 𝜌 𝑉 𝑦 = 𝜇 + ( 𝜆 + 𝑚 ) Tr 𝜀 + 4 𝜇𝜀 𝑥𝑥 + 2 𝜇𝜀 𝑦𝑦 − 𝑛 𝜀 𝑧𝑧 , (2) 𝜌 𝑉 𝑧 = 𝜇 + ( 𝜆 + 𝑚 ) Tr 𝜀 + 4 𝜇𝜀 𝑥𝑥 + 2 𝜇𝜀 𝑧𝑧 − 𝑛 𝜀 𝑦𝑦 , (3)where 𝜌 is the undeformed sample density, 𝑉 𝑥 is longitudi-nal wave velocity, and 𝑉 𝑦 , 𝑉 𝑧 are velocities of shear wavespolarized parallel and perpendicular to the uniaxial stressand traveling perpendicular to the stress axis. Assume thestatic uniaxial stress 𝑇 is applied along the 𝑦 axis, thenthree non-zero components of the strain tensor can be writ-ten as: 𝜀 𝑦𝑦 = − 𝑇 ∕ 𝐸 , 𝜀 𝑥𝑥 = 𝜀 𝑧𝑧 = 𝜈𝑇 ∕ 𝐸 , where Young’smodulus 𝐸 = 𝜇 (3 𝜆 + 2 𝜇 )∕( 𝜆 + 𝜇 ) and Poisson coefficient 𝜈 = 𝜆 ∕(2 𝜆 + 2 𝜇 ) . Introduce the effective longitudinal andshear moduli: 𝑀 𝑥 = 𝑉 𝑥 𝜌 , 𝐺 𝑦 = 𝑉 𝑦 𝜌 , 𝐺 𝑧 = 𝑉 𝑧 𝜌 andwrite the set of equations (1)–(3) in the form: 𝑀 𝑥 = 𝜆 + 2 𝜇 + 𝛼 𝑥 𝑇 , 𝛼 𝑥 = − 2 𝑙 − 𝜆𝜇 (2 𝑚 + 𝜆 + 2 𝜇 )3 𝜆 + 2 𝜇 , (4) 𝐺 𝑦 = 𝜇 + 𝛼 𝑦 𝑇 , 𝛼 𝑦 = − 𝑚 + 𝜆 𝜇 𝑛 + 2 𝜆 + 2 𝜇 𝜆 + 2 𝜇 , (5) 𝐺 𝑧 = 𝜇 + 𝛼 𝑧 𝑇 , 𝛼 𝑧 = − 𝑚 − 𝜆 + 𝜇 𝜇 𝑛 − 𝜆 𝜆 + 2 𝜇 , (6)where 𝛼 𝑥 , 𝛼 𝑦 and 𝛼 𝑧 are dimensionless slope coefficients ofthe dependencies of corresponding moduli as functions ofapplied transverse stress. The linear elastic moduli 𝜆 , 𝜇 canbe found from the longitudinal and shear velocities at zerostrain, while nonlinear ones 𝑙 , 𝑚 , 𝑛 can be calculated fromthe slope coefficients 𝛼 𝑥 , 𝛼 𝑦 and 𝛼 𝑧 . Thus, the whole set ofmeasurements provides data on both the linear and nonlinearelastic moduli of the specimen material. In the general casewe have: 𝑙 = − 3 𝜆 + 2 𝜇 𝛼 𝑥 − 𝜆 ( 𝜆 + 𝜇 ) 𝜇 (1 + 2 𝛼 𝑦 ) + 𝜆 𝜇 (1 − 2 𝛼 𝑧 ) , (7) 𝑚 = −2( 𝜆 + 𝜇 ) ( 𝛼 𝑦 ) + 𝜆 ( 𝛼 𝑧 ) , (8) 𝑛 = −4 𝜇 ( 𝛼 𝑦 − 𝛼 𝑧 ) . (9)As a simplified alternative to the analysis of the set ofthird-order moduli nonlinear elastic properties of materialscan be evaluated using the nonlinearity parameter 𝛽 com-prising a combination of Murnaghan’s and Lame’s elasticmoduli (see e.g. [31, 16]): 𝛽 = 3∕2 + ( 𝑙 + 2 𝑚 )∕( 𝜆 + 2 𝜇 ) . (10)Note that the 𝑛 modulus is omitted in the parameter 𝛽 . Con-versely, it can be easily shown from equation (4) that if thePoisson coefficient 𝜈 equals to 1/3 (it holds for most of glassypolymers), then a slightly different parameter 𝛾 can be intro-duced and obtained just from measurements of longitudinalwave velocity as a function of the applied uniaxial stress: 𝛾 = ( 𝑙 − 2 𝑚 )∕( 𝜆 + 2 𝜇 ) = 1 − 𝛼 𝑥 . (11)As it was recently shown [30] in nanocomposites on thebase of glassy polymers, such as PS, PMMA or polycar-bonate 𝑙 modulus demonstrates the most profound changes,while 𝑚 and 𝑛 moduli show the least variations. That is whyin many practical cases the estimation of 𝛾 value obtainedfrom measurements of just longitudinal wave velocity as afunction of applied stress can provide data for rough tenta-tive estimation of potential changes in nonlinear propertiesof a nanocomposite.
3. Materials and experimental setup
The grained 585 polystyrene (Nizhnekamskneftekhim,Russia) was used as a polymer matrix for composite sam-ples. The following materials were applied as nanofillers:silicon dioxide (SiO ) particles Aerosil R812 modified bysilazane (Evonic Industries, Germany); Carbon Black (CB)particles P-805E (Ivanovskiy tekuglerod and rubber, Russia)and Halloysite Natural Tubules (HNT) (NaturalNano Inc.,USA). The typical sizes of filler particles were as follows:SiO ∼ ∼
80 nm in diameter, HNT ∼
100 nm in diameter, 0.5 – 1.2 µm long. Spherical parti-cles, SiO and CB, were introduced to the PS-matrix at theconcentrations of up to 20% weight and halloysite tubulesof up to 10% weight. The chosen filler concentrations wereshown to provide the most essential changes of linear elasticproperties without significant agglomeration [32].Composite samples were fabricated by melt technologyin the form of plates mm , see [33, 34] for fab-rication details. Similar plates were fabricated by the sametechnology from pure polystyrene and used for reference.As known one of the critical requirements for good qual-ity polymer nanocomposite is a well dispersion of filler par-ticles in the polymer matrix and lack of agglomeration [35].The particle distribution in the PS matrix was controlled byobservation of micrographs of cryo-cleaved surfaces of thecomposite samples using a Carl Zeiss Supra-55 scanning A.V. Belashov et al.:
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Figure 1:
Microphotographs of cryo-cleaved surfaces of sam-ples.
Table 1
Data from static tensile tests of PS-based composites.
Strength Tensile StrainSample at break, modulus, at break, 𝜎 𝑏 , MPa 𝐸 , GPa 𝜖 𝑏 , % PS pure 5 6 ± ± ± ± ± ± ± ± ± ± ± ± electron microscope. Figure 1 presents representative mi-crophotographs of the composite samples.The influence of fillers on the static mechanical proper-ties of nanocomposites in tension was studied using Instron5940 universal testing system (USA). Basing on the tensiletest data, the strength at break 𝜎 𝑏 , strain at break 𝜖 𝑏 , andstatic tensile elastic modulus 𝐸 have been determined, seeTable 1. As can be seen from these data the introductionof more rigid particles leads to a more profound increasein the elastic modulus. The elastic modulus rose from 1.6GPa for pure PS to 3.0 GPa for composites with 20% CB.Also the CB particles provided high strength and strain atbreak of composites, with strength at the break being evenhigher than that of pure PS. Thus the chosen concentrationsof nanofillers on one hand provided a noticeable change ofmaterial elasticity but on the other hand did not cause itssubstantial agglomeration. The analysis of nonlinear elastic properties of fabricatednanocomposite samples was performed through measure-ments of changes in velocities of longitudinal and shear ul-trasonic waves depending upon the applied static stress. Theschematic of the experimental setup is shown in Fig. 2. Eachspecimen used in measurements was formed from the threefabricated plates of a nanocomposite, adhesively bonded bythe ethylcyanoacrylate adhesive Superglue. The specimenshad a bar shape with the length of about 5 cm. A spec-imen and a stress gauge were clamped between the jaws
Figure 2:
Experimental setup for measurements of ultrasoundvelocity as a function of applied stress.
Table 2
Measured longitudinal and shear ultrasonic wave velocities.
Specimen Longitudinal, Shear,m/s m/s
PS commercial 2316 ± ± ± ± ± ± ± ± ± ± of the stress unit. Special metallic vice grips indicated bygray color in Fig. 2 provided uniform stress on a certainarea of the specimen of the length 𝑑 . Two piezoelectrictransducers were applied to the end faces of the specimenand were used for generation and detection of longitudinal(piezoelectric transducers P121 by Amati Acoustics, Russia)and shear (piezoelectric transducers V154-RB by Olympus,USA) waves. The input signals were provided by the AM300Dual Arbitrary Generator (Rohde & Schwarz), output sig-nals were recorded by RTB2002 digital oscilloscope (Rohde& Schwarz). In the first set of measurements longitudinal and shearwave velocities were obtained for each of the specimens atzero applied stress. The pulsed longitudinal/shear waveswere generated at the input end face of the specimen anddetected at the output one. Ultrasonic wave velocity wasestimated as a ratio of the specimen length and time delaybetween the fronts of the input and output pulses. The ob-tained values of longitudinal and shear wave velocities in thespecimens are summarized in Table 2.A different approach was utilized for measurements ofvelocity variations as a function of the applied static stress.In this set of experiments, sine ultrasonic waves with thefrequency of 2.25 MHz were applied to the input end faceof the specimen and detected at the output end face. Shiftsof the sine wave at stepwise increase of the applied staticstress were recorded providing data on changes of wave ve-
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Figure 3: (a) Typical recorded sinusoidal signals obtained forthe specimen of commercial grade PS at zero stress and at T= 10 MPa. (b) Convolution between the two signals shownin (a) allows for estimation of temporal shift between the twosignals as Δ 𝑡 𝑃 = 0 . µs. locity. Relatively small changes of the applied stress (up to15 MPa) did not cause significant variations of longitudi-nal/shear wave velocity and sine wave shifts did not exceedits period. The advantage of sinusoidal shape of the ultra-sonic wave in this case allowed us to choose a single fre-quency and to observe a signal of a constant shape whichdid not vary with the applied stress.Calculation of the sine wave shift induced by the appliedstress can be performed by convolution of the signal ob-tained at zero stress 𝑆 ( 𝑡 ) and that recorded at higher stress 𝑆 𝑃 ( 𝑡 ) : 𝑅 𝑃 ( 𝑡 ) = 𝑆 𝑃 ( 𝑡 ) ∗ 𝑆 ( 𝑡 ) . If two sinusoidal signals areshifted from each other at time Δ 𝑡 𝑃 (Fig. 3(a)), global max-ima of their convolution 𝐶 𝑃 ( 𝑡 ) will be shifted from 𝑡 = 0 ata distance Δ 𝑡 𝑃 as shown in Fig. 3(b). Since the time delay Δ 𝑡 𝑃 = 𝑑𝑉 𝑃 − 𝑑𝑉 is a result of wave velocity change from theinitial value 𝑉 to the final one 𝑉 𝑃 and occurs on a specimenlength 𝑑 where stress had been applied, the ultrasonic wavevelocity at stress 𝑃 can be calculated as: 𝑉 𝑃 = 𝑑 Δ 𝑡 𝑃 𝑉 𝑃 + 𝑑 𝑉 . (12)
4. Results and discussion
The described methodology was used for measurementsof wave velocity as a function of applied static stress 𝑇 forthree types of waves: 1) longitudinal waves 𝑉 𝑥 ( 𝑇 ) , 2) shearwaves orthogonal to the direction of applied stress 𝑉 𝑧 ( 𝑇 ) and 3) shear waves parallel to the direction of applied stress 𝑉 𝑦 ( 𝑇 ) . Taking into account the PS density ( 𝜌 = 1060 kg/m )or the corresponding composite density, effective longitu-dinal and shear moduli at several stress values were calcu-lated for each specimen. An example of the complete setof these dependencies obtained for the sandwich specimenmade of fabricated plates of pure polystyrene (PS pure) isshown in Fig. 4. The obtained sets of data were used for cal-culations of the second- and third-order moduli of specimensmade of pure polystyrene and three PS-based nanocompos-ites: PS + 20% SiO , PS + 10% HNT and PS + 20% CB.For comparison measurements were also performed in abulk non-layered specimen made of a commercial grade pure Figure 4:
Example of measured dependencies of effectivelongitudinal (a) and shear (b,c) moduli on applied stress T forthe PS pure specimen. Figure (c) also demonstrates the repro-ducibility of measurements for a single investigated specimen,with colored symbols indicating different sets of measurements. polystyrene. The results obtained are summarized in Ta-ble 3. The data obtained by Hughes and Kelly [8] for purepolystyrene are shown for comparison.The following conclusions can be made from the analy-sis of data in Table 3. First of all the elastic characteristicsof the modern commercial grade polystyrene differ notice-ably from those of the one examined in 1950-s by Hughesand Kelly [8]. That concerns both the linear and nonlin-ear parameters. Parameters of the sandwich made of adhe-sively bonded plates of pure polystyrene fabricated by melttechnology slightly differ from those of commercial gradebulk specimen, that can be due to several factors, amongwhich are the difference in polymer structure and fabrica-tion process and the influence of adhesive layers. Sheets ofcommercial grade PS were fabricated by the extrusion tech-nology followed by cold rolling. While laboratory sampleswere fabricated by melt injection into the heated mold un-der pressure followed by self-cooling. These technologiescause differences in the formed polymer structure that resultin variations of the nonlinear elastic moduli, which are moresensitive to structural changes than the linear ones. Notably,the linear elastic moduli of the commercial and laboratorypolystyrene samples demonstrate almost no difference. Wehave to mention also that differences in both linear and non-linear elastic parameters of these two materials are essen-tially lower than those of the modern commercial grade PSand the one used in [8]. This observation also supports ourearlier experimental evidence that layers of ethylcyanoacry-late Superglue adhesive do not affect markedly the resultingmaterial nonlinearity, that was demonstrated in terms of pa-rameters of bulk nonlinear strain solitary waves propagatingin layered polymeric waveguides [36, 37].With this result in mind let us compare elastic parame-ters of nanocomposites with those of pure polystyrene sand-wich fabricated by the same technology (PS pure in Table3). As can be readily seen from the Table all the threenanofillers cause noticeable rise of the second order Lamemoduli 𝜆 and 𝜇 except for the latter for PS with SiO par-ticles. The Young’s modulus 𝐸 calculated from these datacomprises 3.85 GPa for PS pure and 4.23, 4.29 and 4.58 GPafor nanocomposites with SiO , HNT and CB particles re-spectively. Thus in terms of linear elastic properties the addi- A.V. Belashov et al.:
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Page 4 of 7elative variations of nonlinear elastic moduli in polystyrene-based nanocomposites tion of 10% HNT and 20% CB to the polystyrene matrix pro-vided efficient reinforcement of the material. The Young’smodulus 𝐸 of composites obtained by ultrasonic measure-ments exceeded the static one (Table 1) by approximately 2GPa. It corresponds to a typical frequency dependence ofelastic moduli of polymer materials [38].Changes in the nonlinear, third-order moduli 𝑙 , 𝑚 and 𝑛 , demonstrate more complex behavior. In general for allthe composite samples changes in the nonlinear moduli weremore profound than changes in the linear, 2-nd order mod-uli 𝜆 and 𝜇 . Among the nonlinear moduli the 𝑙 modu-lus demonstrated prominent variations in all the compositeswith the maximal change of about 84 % observed in the CB-containing one. The 𝑛 modulus showed high relative varia-tion, of about 91 % only in PS+HNT samples, while varia-tions of the 𝑚 modulus were not so high and reached 46% inPS+HNT and PS+CB nanocomposites.The values of nonlinearity parameter 𝛽 calculated usingequation (10) show comparatively small difference betweenthe laboratory and commercial grade polystyrene samplesand, surprisingly, a very small difference between the purelaboratory PS and PS+HNT composite as compared withhigher difference of those and PS+SiO composite. The pa-rameter 𝛾 (Eq. (11)) demonstrates more straightforward be-havior in these composites. The change of 𝛾 values in HNT-containing composite is higher than that in SiO -containingone. Meanwhile, both 𝛽 and 𝛾 absolute values show pro-found rise in the CB-containing composite as compared withpure PS.The obtained linear and nonlinear moduli of PS + 20%SiO and PS + 20% CB can be compared with the developedtheory of nonlinear elastic properties of composites withspherical inclusions [30]. This theory predicts the effectivevalues of elastic moduli 𝜆, 𝜇, 𝑙, 𝑚, 𝑛 using the known valuesof these moduli for the nanoparticles and the surroundingmatrix. The values for the matrix were taken from our ex-periment for pure PS (see Table 3). To estimate the elas-tic properties of SiO nanoparticles, we used the propertiesof bulk amorphous silica: Young’s modulus 𝐸 = 72 GPaand shear modulus 𝜇 = 31 GPa [39]. For CB nanoparticleswe took the Young’s modulus 𝐸 = 80 GPa and Poisson ra-tio 𝜈 = 0 . [40]. In both cases, the nanoparticles are muchstiffer than the polymer matrix. Thus, the influence of non-linear moduli of nanoparticles to effective nonlinear moduliof nanocomposite is negligible. Indeed, for any macroscopicdeformation of the composite, the deformation of nanopar-ticles is much smaller than that of the surrounding matrix.To estimate the volume fraction of nanoparticles we usedthe density 𝜌 = 2200 kg/m and 𝜌 = 1900 kg/m for SiO and CB respectively. The resulting values are presented inTable 3. Most of the theoretical predictions give values rel-atively close to the experimental ones. However, the experi-ment demonstrates more profound influence of nanoparticlesonto the modulus 𝑙 and smaller influence on moduli 𝑚 and 𝑛 . This deviation can be explained by the approximation ofhomogeneous distribution, which is used in the theory andbecomes rough at the nanometer scale for polymer materials. The observed slight discrepancy of theoretical and experi-mental values can also be due to unknown real parametersof filler particles. The values taken from [39, 40] can beconsidered in our case as estimates only.
5. Conclusions
The observed response of nonlinear elastic parametersto addition of different filler particles to polystyrene matrixdemonstrates substantial influence of nanofillers onto non-linear properties of polymer-matrix composites. As shownthe linear elastic moduli are affected by such structuralchanges of the material to considerably lesser extent thannonlinear ones. The variation of some nonlinear moduli canreach almost 100% due to the presence of nanofillers in stud-ied composites.It was also demonstrated that the introduced parameter 𝛾 ,besides of being measured in a relatively simple way, seemsto be more sensitive to changes of material nonlinearity thanthe known parameter 𝛽 , especially for polymers and polymercomposites in which all the nonlinear elastic moduli usuallyhave the same sign.For nanocomposites with spherical inclusions (PS+SiO and PS+CB), the nonlinear modulus 𝑙 and the parameter 𝛾 demonstrated the most prominent change, especially forPS+CB (more than 80%). The absolute values of the non-linear modulus 𝑚 and the parameter 𝛽 also increased innanocomposites. At the same time, the variation of the mod-ulus 𝑛 was less significant.The nonlinear moduli obtained for nanocomposites withspherical inclusions are in a qualitative agreement with thetheoretical predictions calculated using the nonlinear elastic-ity theory. However, the experiment revealed that the largestnonlinear modulus 𝑙 demonstrates higher sensitivity to thepresence of nanoinclusions than that predicted by the the-ory.For nanocomposites with one-dimensional inclusions(PS+HNT), the variations of nonlinear moduli was quali-tatively different. The deviation of the modulus 𝑙 was oppo-site and not so significant as for composites with sphericalnanoparticles. The parameter 𝛾 also had an opposite changeand became close to zero. At the same time, the modulus 𝑛 increased by almost 100%. Acknowledgments
The financial support from Russian Science Foundationunder the grant
Data Availability
The raw/processed data required to reproduce these find-ings cannot be shared at this time as the data also forms partof an ongoing study.
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Table 3
Second- and third-order moduli of PS-based nanocomposites obtained from ultrasonicmeasurements and theoretical modeling.
Material 𝜆 , 𝜇 , 𝑙 , 𝑚 , 𝑛 , 𝛽 𝛾 GPa GPa GPa GPa GPaPS [8] .
89 ± 0 .
01 1 .
38 ± 0 .
01 −18 . . . . . . . . . PS commercial .
80 ± 0 .
02 1 .
44 ± 0 .
01 −46 . . . . . . . . . . PS pure .
76 ± 0 .
02 1 .
45 ± 0 .
01 −44 . . . . . . . . . . PS+20%SiO .
35 ± 0 .
02 1 .
58 ± 0 .
01 −64 . . . . . . . . . . PS+10%HNT .
02 ± 0 .
02 1 .
62 ± 0 .
01 −40 . . . . . . . . . . PS+20%CB .
58 ± 0 .
02 1 .
71 ± 0 .
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