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Обверткин И.В., Пасечник К.А., Власов А.Ю. Потенциал использования одностенных и многостенных углеродных нанотру-бок и углеродных нановолокон в качестве модификаторов полимерной матрицы, способных увеличить размеростабильность и технологическую устойчивость композиционного материала // Вестник Пермского национального исследовательского политехнического университета. Механика. - 2021. - № 4. - С. 98-110. DOI: 10.15593/perm.mech/2021.4.10
Obvertkin I.V., Pasechnik K.A., Vlasov A.Y. The potential of using SWCNTs, MWCNTs and CNFs capable of increasing the composite material dimensional and technological stability as modifiers of a polymer matrix. PNRPUMechanics Bulletin, 2021, no. 4, pp. 98-110. DOI: 10.15593/perm.mech/2021.4.10
ВЕСТНИК ПНИПУ. МЕХАНИКА
№ 4,2021 PNRPU MECHANICS BULLETIN
https://ered.pstu.ru/index.php/mechanics/index
DOI: Ш.15593/регт.тесЫ2021.4Л0 УДК 620.1/.2
ПОТЕНЦИАЛ ИСПОЛЬЗОВАНИЯ ОДНОСТЕННЫХ И МНОГОСТЕННЫХ УГЛЕРОДНЫХ НАНОТРУБОК И УГЛЕРОДНЫХ НАНОВОЛОКОН В КАЧЕСТВЕ МОДИФИКАТОРОВ ПОЛИМЕРНОЙ МАТРИЦЫ, СПОСОБНЫХ УВЕЛИЧИТЬ РАЗМЕРОСТАБИЛЬНОСТЬ И ТЕХНОЛОГИЧЕСКУЮ УСТОЙЧИВОСТЬ КОМПОЗИЦИОННОГО МАТЕРИАЛА
И.В. Обверткин, К.А. Пасечник, А.Ю. Власов
Сибирский государственный университет науки и технологий им. академика М.Ф. Решетнева, Красноярск, Россия
О СТАТЬЕ
АННОТАЦИЯ
Получена: 4 мая 2021 г. Принята: 10 декабря 2021 г. Опубликована: 30 декабря 2021 г.
Ключевые слова:
нанокомпозиты, углеродные нанотрубки и нановолокна, термомеханические характеристики, аналитическое моделирование, термический анализ.
В данной работе изучается влияние углеродных наночастиц, таких как углеродные нановолокна (УНВ), одностенные углеродные нанотрубки (ОУНТ) и многостенные углеродные нанотрубки (МУНТ), на величину математического ожидания и дисперсии коробления пластин из волокнистого композиционного материала. Были рассмотрены ламинаты с дезориентацией волокон как асимметричные, и для оценки коробления использовали модель, предложенную Dano и Hyer. Полученные результаты подтверждают, что добавление углеродных наночастиц в качестве модификатора в полимерную матрицу волокнистого композиционного материала позволяет увеличить размеростабильность (математическое ожидание среднеквадратичного отклонения (СКО)) и технологическую устойчивость схемы армирования (дисперсия выборки СКО композиционной пластины). Математическое моделирование коробления пластины с учетом возможной дезориентации волокна показало снижение дисперсии коробления на 12,6 и на 26,6 % при модификации ОУНТ и МУНТ соответственно. Экспериментально определены коэффициент линейного термического расширения (КЛТР) наноструктурированной полимерной матрицы с различными наполнителями. Обнаружено, что углеродные наномодификаторы более эффективны как компенсаторы термического расширения полимерной матрицы в композиционных материалах, армированных углеродными волокнами, чем в полимерной матрице без макроволокон. Добавление 0,05 % УНТ, 1 % МУНТ в эпоксидную смолу позволяет снизить КЛТР на 9,7 и 15,4 % соответственно. В то же время добавление аналогичного количества наночастиц в полимерную матрицу волокнистого композиционного материала снижается КЛТР поперек волокон на 15,56 и 35,8 % соответственно. На основании полученных результатов были построены зависимости КЛТР в трансверсальном направлении полимерного композиционного материала, математическое ожидание СКО и дисперсия выборки СКО точности формы композиционной пластинки от концентрации модификатора. На основании полученных данных можно сделать вывод, что для снижения математического ожидания и дисперсии коробления композиционного материала существует эффективная концентрация, увеличение которой нецелесообразно, несмотря на дальнейшее снижение КЛТР в трансверсальном направлении.
© ПНИПУ
© Обверткин Иван Владимирович - н.с., e-mail: +[email protected], : 0000-0003-4895-5757. Пасечник Кирилл Арнольдович - н.с., e-mail: [email protected], : 0000-0001-6245-1630. Власов Антон Юрьевич - к.ф.-м.н., доц., e-mail: [email protected], : 0000-0002-6360-7382.
Ivan V. Obvertkin - Researcher, e-mail: +[email protected], : 0000-0003-4895-5757. Kirill A. Pasechnik - Researcher, e-mail: [email protected], : 0000-0001-6245-1630. Anton Yu. Vlasov - CSc in Physical and Mathematical Sciences, Associate Professor, e-mail: [email protected], : 0000-0002-6360-7382.
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l^oKHJ^H This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0)
THE POTENTIAL OF USING SWCNTS, MWCNTS AND CNFS CAPABLE OF INCREASING THE COMPOSITE MATERIAL DIMENSIONAL AND TECHNOLOGICAL STABILITY AS MODIFIERS OF A POLYMER MATRIX
I.V. Obvertkin, K.A. Pasechnik, A.Y. Vlasov
Reshetnev Siberian State University of Science & Technology, Krasnoyarsk, Russian Federation
ABSTRACT
In this study, the effect of carbon nanofibers (CNN), single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs) on the warpage expected value and warpage dispersion of a plate made of a fibrous composite material are investigated. Laminates with fiber disorientation were considered as asymmetric and the model proposed by Dano and Hyer was used to evaluate the warpage. The results obtained confirm that the addition of carbon nanoparticles as a modifier to the polymer matrix of a fibrous composite material can increase the dimensional stability (the mathematical expectation of the standard deviation) and the technological stability of the reinforcement scheme (the variance of the standard deviation of the composite plate). Modeling of the warpage of plate, taking into account the possible disorientation of the fiber, showed a decrease in the warpage dispersion by 12.6 and 26.6 % with the modification of the SWNTs and MWCNTs, respectively. The coefficient of thermal extension (CTE) of a nanos-tructured polymer matrix with various fillers were experimentally determined. It was found that carbon nanomodifiers are more effective as compensators for the thermal expansion of the polymer matrix in composite laminates reinforced with carbon fibers than the polymer matrix without macrofibers. The addition of 0.05 % SWCNTs, 1 % MWCNTs to the epoxy resin reduces the CTE by 9.7 and 15.4 %, respectively. At the same time, the addition of a similar amount of nanoparti-cles to the epoxy matrix of the fiber composite reduces the CTE in the transverse direction by 15.56 and 35.8 %, respectively. On the basis of the obtained results, the dependences of the transverse CTE of the polymer composite material, the mathematical expectation of the standard deviation, and the variance of the standard deviation of the composite plate form accuracy on the concentration of the modifier were constructed. According to the obtained data, it can be concluded that in order to reduce the mathematical expectation and the variance of the warping of the composite material, there is an effective concentration, the increase of which is impractical, despite the further decrease in the transversal CTE.
©PNRPU
ARTICLE INFO
Received: 04 May 2021 Accepted: 10 December 2021 Published: 30 December 2021
Keywords:
Nanocomposites, Carbon nanotubes and nanofibers, Thermomechanical characteristics, Analytical modeling, Thermal analysis.
Introduction
Continuous demand for light weight, cost-effectiveness, high strength, dimensional stability and accuracy of products from polymer-matrix composites (PMCs) is the driving force behind the search for new materials. The discovery of carbon fillers made it possible to make a quantum leap in the creation of nanocomposites. Carbon nanoparticles such as nanotubes and nanofibers are materials with a unique structure and an unusual combination of properties, such as high strength and flexibility, high Young's modulus and a negative CTE [1-11]. The unique properties of materials allow them to be highly effective modifiers for polymer matrices directionally changing their mechanical and ther-momechanical characteristics [9, 12, 13].
The paper investigates the effect of carbon modifiers on the warpage of the composite material. Modifiers have a high Young's modulus, a low CTE and high polymer-modifier affinity. The hypothesis of increase in the technological stability of composite material reinforcement schemes when using modified resin is also tested. In addition, it is necessary to pay attention to the change in technological properties of modified polymer matrix, in particular, to the study of their rheology, as well as to the extent of the effect of nanoparticles dispersion and morphology on the
rheology of the polymer material. The dispersion rheology plays an important role in nanocomposites processing due to the large increase in the system viscosity with increasing of the nanomodifier concentration [11-40].
For the first time in history, polymer nanocomposites were studied in [25]. Since then, there have appeared many works devoted to the manufacture and study of the properties of polymer nanocomposites. The authors of the study [41] investigated the modification of epoxy resin with multilayer nanotubes aligned in a magnetic field, and they succeeded in reducing the CTE of the epoxy resin by 11.6 % with random distribution of the modifier and by 12.6 and 10.8 % in the direction of alignment and perpendicular to the direction of alignment in a magnetic field, respectively. The authors of [23] also studied the effect of the volume fraction of MWCNTs on the nanocomposite CTE. They determined that the nanocomposite CTE tends to decrease with increase in the volume fraction of carbon nanotubes. The CTEof composites containing 0 (epoxy resin), 8.5, 10.4, 14.1, 21.2, and 27.0 volume % of MWCNTS is 7.0 10-5; 0.4 10-5; -0.6 10-5; -1.4 10-5; -0.5 10-5 and -0.7 10-5, respectively. It was shown in [42] that the addition of 1 % of carbon nanofibers into epoxy resin increases Young's modulus and flexural strength by 26 and 20 %, respectively, and decreases the nanocomposite CTE by
53 %. The authors of [1] were able to increase Young's modulus by 11 % and decrease the CTE by 32.5 % adding 1 percent of carbon nanofibers for weight (length 30 microns, diameter 20-80 nanometers) into epoxy resin. The authors of [13] were able to increase Young's modulus by 9 % and decrease the CTE of the nanocomposite by 23.7 % adding 1 percent of MWCNTs for weight (length 0.5-2 microns, diameter 8-15 nanometers) into epoxy resin.
Another important factor in the effectiveness of the modifier is the interfacial interaction between the nanofiller and the epoxy matrix. Numerous methods such as function-alization of nanofillers, ultrasonic treatment, and homogeni-zation are used in an attempt to improve bond strength or interfacial adhesion. It was found that the agglomeration of carbon nanoparticles dramatically reduces the mechanical properties of the fabricated nanocomposites. Because of the small size and high aspect ratio of nanoparticles, individual carbon nanoparticles are usually twisted and therefore do not exhibit as high efficiency as possible. Therefore, without effective dispersion and establishment of strong chemical affinity with the polymer matrix, it is impossible to transfer the properties of carbon nanoparticles to the finished product. Although, according to the data of researchers [26], the functionalization of carbon nanotubes reduces their mechanical properties.
At the macrolevel the classical lamination theory (CLT) is most often used to assess the emerging macrostresses in laminated composites [29]. This theory also makes it possible to predict the shape of laminates after curing. It is although important to remember that the classical lamination theory has a number of limitations, in particular, it does not take into account such important factors as the effect of the cooling rate and changes in the matrix properties during the curing process [28].
To assess the stresses arising in a polymer composite, several theories are used, such as elasticity solution [30-32], cylinder theory [33-50], Eshelby theory [39] and energy method [40]. The Mori-Tanaka's model [51, 57, 54] is used in this work to assess the effective elastic properties of the polymer matrix reinforced with nanoparticles [54-71]. CLT is used to assess the effective elastic properties of the composite. The models [58, 71] are used to assess CTE of composite.
Modeling of the warpage of composite material taking into account the fibers disorientation was carried out in [73, 74]. To assess the shape of composite materials the authors of [73] use CLT with the assumption that the structure of the material with disorientated layers will be close to a symmetric one. In [74] the finite element method was used
to model the laminate shape. In this paper we consider disorientated laminates as asymmetric ones and use the model proposed by Dano and Hyer [61].
Previous studies have shown that reducing the warpage of the composite material can be achieved through the use of nanomodifiers. But none of the previous works, as far as the authors know, experimentally studied the effect of nanoaddi-tives on the technological stability of reinforcement schemes.
In this paper, we consider composite laminates with quasi-isotropic reinforcement schemes, which have theoretically near zero warpage, due to the fact that the thermal stresses in these materials are balanced. However, the presence of reinforcing fibers disorientation causes to warpage. Modification of the polymer matrix in order to reduce the difference between the thermomechanical properties of the components allows to reduce the effect of fiber disorientation on the dimensional stability and technological stability. The purpose of this study is to study the effect of carbon nanomodifiers on dimensional stability as the mathematical expectation of the standard deviation of the composite plate form accuracy and technological stability as the variance of the standard deviation of the composite plate form accuracy, taking into account the disorientation of reinforcing fibers. The influence of the selected modifiers on the viscosity of the epoxy matrix, depending on the concentration and func-tionalization, is also studied. The functionalization was performed to add functional oxygen-containing groups to the surface of the modifier.
1. Experimental procedure
1.1. Materials
Manufactured by "INUMIT" CJSC two-component ep-oxy system T67 containing a mixture of epoxy resins based on bisphenol A and epichlorohydrin as well as aromatic and aliphatic diamines was used as the modified resin.
TUBALL ™ SWCNTs from OCSiAl company with an average outer diameter of 1.6 ± 0.4 nm and length of more than 5 ^m [5], multi-walled carbon nanotubes of the "Taunit" series of the Nanotech Center LLC with an outer diameter of 20-50 nm and length of more than 2 ^m [45], as well as carbon nanofibers from 50 to 250 nm with fiber length of up to 0.5 mm obtained at the Institute of Catalysis named after G.K. Boreskov SB RAS, Russia [46] were used as modifiers of the polymer matrix in this work. A comparative overview of various carbon nanomodifiers is given in Table 1.
Table 1
Property MWCNTs SWCNTs CNFs
Diameter (nm) 5-50 [14]; 13-36 [17,181; 12,5 [81 0.6-0.8 [14]; 1,6 ± 0,4 [3,4,5,7] 50-200 [19,20]
Length (^m) 1,8-10,99 [17181 5 [3,4,5] 100 [19,20]
Young's modulus (GPa) 1000 [14]; 270-970 [17, 18]; 800 [8]; 1280 [21] 1500 [14]; 790 [7] 240 [6]
Tensile strength (GPa) 10-60 [15]; 63 [22] 50-500 [15] 2.92 [19,20]
CTE -1,2 10-5 [23]; -1,02 10-5; -0,72 10-5 [10] -2 10-5[24] -0,4110-6 [1]; -110-6[2]
Thermal conductivity (W/mK) 3000-6000 [16] 3000-6000 [16] 1950 [19,20]
Review of the characteristics of carbon nanomodifiers
Table 2
Properties of materials used to calculate the properties of modified epoxy resin and composite materials based on it.
Material Physical and mechanical properties Thermomechanical properties
Epoxy resin T67 Young's modulus, GPa: Em = 3.17 [65] Poisson's ratio: vm = 0.35 [67] CTE: am = 68 10-6
Carbon fibersK500 Young's modulus, GPa: El f = 436 [70]; E2 f = 14 [71]; E3 f = 14 [71] Poisson's ratio: v12 f = 0.2 [67]; v23 f = 0.46 [73]; v13 f = 0.2 [67] Shear modulus, GPa: G12 f = 8.78 [64]; G23 f = 2.1; G13 f = 8.78 [64] CTE: a1 f = -0.910-6; a 2 f = 6.810-6[63, 66]
Carbon fibers IMS65 Young's modulus, GPa: El f = 290 [68]; E2 f = 21.8 [64]; E3 f = 21.8[64]. Poisson's ratio: v12 f = 0.2 [67]; v23 f = 0.46 [69]; v13 f = 0.2[67] Shear modulus, GPa: G12 f = 12.5[64]; G23 f = 8.2; G13 f = 12.5[64]. CTE: a1 f = -0.5610-6 [70]; a 2 f = 1010-6 [66]
MWCNT Young's modulus, GPa: Ef = 1000 [14] Poisson's ratio: vf 12 = 0.2[56] CTE: a f = -1.210-5 [23]
SWCNT Young's modulus, GPa: Ef = 1500 [13] Poisson's ratio: vf 12 = 0.2[54] CTE: a f = -2 10-5 [24]
Some properties of the materials presented in the table 2 were directly measured in the course of the work, some information was provided by the manufacturers of materials, and some data was taken from the research of other authors who studied these materials or materials with similar properties.
Carbon fibers IMS 65 are presented in the form of spread tow fabric. Carbon fiber is spread to a 25 mm width tape. The fabric has arranging the fibers in the woven structure in the straightest orientation possible the fiber properties are used in the most effective way. Due to fabric can presented as two unidirectional orthotropic layers.
1.2. Equipment and research methods
The researched carbon nanoparticles were functioned to increase the polymer-modifier affinity.
To increase the adhesion between carbon nanoparticles, functionalization was performed to add functional oxygen-containing groups to the surface of the modifier. Two methods wers used in this work for the functionalization of nanoparticles.
According to the first method, the classical Hammers oxidation method was used to modify the surface with the addition of functional oxygen-containing groups [43, 44]. 400 ml of concentrated sulfuric acid containing 2.5 g of ammonium nitrate was gradually added into a round bottom flask containing 5 g of carbon nanotubes. After stirring for 20 minutes, 1.5 g of potassium permanganate was added to the ice baths while the temperature did not rise, and the mixture was stirred for an additional 20 minutes. Then the temperature of the mixture was gradually increased to 40 °C and held for 30 minutes. Then 700 ml of deionized water was successively added stirring and kept for 15 minutes. Then the suspension was treated with hydrogen peroxide until the appearance of a yellowish hue and the absence of exothermic reaction. The resulting suspension was then treated with ion exchange resins, filtered and
washed with a total of 7 liters of distilled water. The resulting modified nanotubes were dried on calcium chloride and silica gel.
The second method included holding at 35 °C for 7 days in methylene chloride under reflux. Then carbon nanoparticles were sonicated in a water bath in concentrated sulfuric and nitric acid in a ratio of 3 to 1 for 3 hours after filtration in vacuum and washing with distilled water. Then carbon nanoparticles were collected and dried on calcium chloride and silica gel.
The IR spectra of functionalized and native nanotubes were measured using a NicoletiS10 FT-IR spectrometer (ThermoFisherScientific) in the range of wave numbers 400-4000 cm-1. Before measurements the samples were dried at the temperature of 120 degrees for 12 hours to remove traces of water. The measurement of carbon nanotubes was carried out in mixture with KBr at room temperature.
Images of carbon nanotubes were obtained using a Hitachi HT7700 transmission electron microscope.
The researched nanocomposites modified with various carbon nanoparticles were made using sonication technology. First the epoxy resin was mixed for 30 minutes at 2000 rpm. Then the mixture was treated with ultrasound (UZDN-A, kHz 22 ± 1.65) for 40 minutes, with a pause every five minutes to cool the mixture. During cooling the container with the mixture was immersed in water. After completion of the sonication a hardener was added to the mixture and stirred for 10 minutes. Then air bubbles were removed by degassing the mixture in vacuum for 5 minutes. Then the mixture was cured at the temperature of 80 °C for 3 hours and at the temperature of 120 °C for 6 hours.
The viscosity of the resulting mixture after the degassing process was determined on a Discovery Hybrid Rheometers (DHR - 2) rotary rheometer at the constant mixture temperature of 50 degrees and with increasing a shear rate up to 16.4 s-1.
Samples of a polymer composite material based on modified resin were obtained using the method of vacuum
assisted resin infusion molding (VARIM). Metal tooling was coated with a release agent. Dry carbon fabric was laid out according to the reinforcement scheme. After laying out the required number of patterns, the release fabric was laid out, and the resin supply and discharge lines are installed on the distribution grid. Next, a vacuum bag was formed, connected to a vacuum pump and degassed for 1 hour at the temperature of 50 °C. The epoxy resin was mixed in stoichiometric ratio of 100 / 40.9 resin / hardener. After that, the reinforcing material is impregnated with resin. Then the tooling with impregnated reinforcing materials is cured in vacuum at the temperature of 80 °C for 3 hours and at the temperature of 120 °C for 6 hours.
After the curing process, the CTE of the manufactured samples was determined using a TAInstrumentsQ400 ther-momechanical analyzer. The tests were carried out with constant heating to 100 °C at a constant heating rate of 5 °C/min. At least three samples are used for each fillers concentration.
The volume ratio of the reinforcing element in the composite material was determined using a Discovery TGA55 TA Instruments thermogravimetric analyzer (USA).
The flatness of the sample surface was measured using an Absolute Arm 7520 SE portable coordinate measuring machine with an external CMS 108 scanner with Geomagic software (Romer) with a spatial accuracy of ± 0.033 mm.
2.3. Mathematical modeling of the effect of modification with carbon nanotubes on the physical and thermomechanical characteristics of the composite material
The Mori-Tanaka's method was used to estimate the CTE of the modified epoxy resin [51, 52]. Elements of the Eshelby Tensor for inclusions with an aspect ratio tending to infinity are given in [53]. Expressions for calculating the effective CTE of the modified epoxy resin are given in Appendix 1. Expressions for calculating the effective Young's modulus of the modified polymer matrix are given in Appendix 1. The calculations were based on the characteristics of the materials shown in Table 2.
Mathematical modeling of the effect of modification with carbon nanotubes on the mathematical expectation of the standard deviation of warpage.
Estimation of the mean standard deviation of warpage is performed according to the model for predicting the shape of an asymmetric composite material sample proposed by Dano and Hyer [60, 61], based on polynomial approximation of displacements, extending CLT including geometric nonlinearity and using the Rayleigh-Ritz method to minimize the total potential energy. The description of the model is briefly presented below.
Total potential energy of laminate W:
W = J rad (vol).
Where ra is strain energy density.
ra = \CnkiSSki -jAT.
(1)
(2)
Where CjJU are elastic constants, ztj and skl are strains,
a^ is the CTE, AT is the difference between strees-free
and current temperatures as AT = 95 °C. Stress-free temperature is maximum curing temperature equal to 120 °C. Current temperature is 25 °C.
CLT implies a plane stress strain state, that is, normal stresses in the z direction and shear stresses not in the plane of the laminate are assumed to be zero, resulting in the following expression for the total potential energy:
Lx Ly H
HI, j
W = i i i 26llSX + Q'2SxSy + Q16SxySx + 22 +
- Lx - Ly H 2 2
2 2 2
+&£xySy + 2Q66S2y -(ax + 612ay + 016axy)sxAT- (3)
-(Q12ax + Q22ay + Q26axy )Sy AT -(Q16ax + Q26ay + Q66axy )Sxy ATdzdydx.
Where Qv are components of the stiffness matrix of each layer, Lx and Ly are sample width and length, H is total laminate thickness. The strain components sx, s and s^ are determined by the expressions:
s x = s0 + zkx,
e y = e°, + zky,
Sxy = exy + zkxy ■
(4)
(5)
(6)
The quantities s° , s° , sxy h kx, ky, kxy are the deformation of the middle surface and the curvature, respectively, determined by expressions that take into account geometric nonlinearity in accordance with the theory of von Karman:
du0 11 dw
-+-I
dx 2( dx
0 1 [ dw0
S y = dy + 2 [ dy
< =
du dv dw dw
dy dx dx dy
dV kx =--T- = ~a,
x dx2
^2 0 i d w
ky ™ = "b'
d2 w0
kxy ="2 = "C
dxdy
(7)
(8)
(9)
(10)
(11)
(12)
According to the work of Dano and Hyer, the components of the middle surface deformation can be approximated by polynomials of the third order, consisting of the terms the sum of the powers x and y of which is an even number:
+ a„x +
s y = +<
= 2d9 +
3 xy + d4 y ,
x2 + d7 xy + d8 y2, c 2
---h 2d. + 2d6
xy +
1 f ac 21 2
1 f bc.
2 ^ 2
(13)
(14)
(15)
y
Substituting expressions (13), (14), (15) into equation (3), we obtain the equation of the total potential energy in the form of a function of 14 unknowns:
W = W(a, è,c,d1,<
;,d7,d8,d9,d10,d11 ) (16)
To ensure a state corresponding to the minimum potential energy, it is necessary:
dw = Y—ox = 0.
1=1 dxt '
(17)
Where xt - a, b,c, (17), it is required:
To ensure condition
ox
0,
(18)
where i = 1, 2, ... 14. This leads to a system of 14 nonlinear equations with 14 unknown constants xt, the solution of which makes it possible to find the sample shape of an asymmetric composite material. By sequential execution of simple algebraic transformations the system of nonlinear equations is reduced to a system of 3 nonlinear equations with three unknown constants a, b, c. The Newton-Raphson method is used to solve the system. The method requires specifying an initial approximation. The found solution will depend on the choice of this approximation, especially if the sample under consideration has more than one stable shape, for example, bistable samples can have two stable shapes and unstable one [62]. The resulting solution is stable in the case of a positive definite Jacobi matrix:
_a(, ¿„f)
j=■
Э(а, b,.. ,dn )
(19)
The displacement of the analyzed sample points in the z direction is assumed to be:
w (
;(x, y ) = 0,5 (ax2 + by2 + cxy The standard deviation of warpage:
s =
U -1
w - w
(20)
(21)
where l = 1... N is the number of control points on the sample; x and y are coordinates of control points; wt is warpage at control point l ; w ' is the arithmetic mean for a sample of N points.
The mathematical expectation of the standard deviation of warpage is:
E [s] = Ë 5i • P,
where is the standard deviation of warpage with the angles of layers disorientation corresponding to the i design case; P is the probability value corresponding to the i design case. The number n corresponds to the number of placements Akm _ mk , where k is the number of layers in the sample, m is the number of discrete steps A0' in the range from -0' to +0' of the expected maximum disorientation angle. The probability Pi is equal to the product of the values of the probability Pjj (j _ 1...k) of the disorientation occurrence
within the boundaries of the discrete step A0' of each layer.
The value of the probability Pu is determined using the equation for the normal distribution law:
P
rb 1
J aT~l
(23)
Where a and b are the boundaries of the discrete step A0'; g0' is standard deviation of the distribution; ^ is the mathematical expectation of the distribution. In view of the previously accepted 0' as the range of possible disorientation, we will write 3g0' «0'; ^ is taken equal to 0.
For each possible combination of disorientation angles we determine the curvature parameters kx, ky and kxy using the Dano
and Hyer model. Next, we determine the probability density function p (a) of the deviation of warpage, the true functions of
which are continuous. Due to the finite number of calculated values of a, we set a discrete range Act and summarize the values of the probabilities included in the ranges of calculated cases.
The numerical solution was performed in the Python software environment according to the presented equations.
2. Results and Discussion
FTIR spectra of native and functionalized carbon nanoparticles shown in Figure 1 were obtained. The spectra contain intense bands at 3431 cm-1 (stretching vibrations of the -OH group), which can be attributed to hydroxyl groups
3000 2000
Wavenumber, cm"1 Fig. 1. FTIR spectra of native and functionalized CNTs
i=1
and to a trace of water in the potassium bromine tablet used for analysis. The bands at 1724 and 1736 cm-1 probably corresponding to C = O were attributed to stretching vibrations of carboxylic acid groups (-COOH). The bands in the 1250-950 cm-1 range are probably related to the vibration of the C-O bond in a different chemical environment. The relative increase and partial separation of the bands in this wave region indicate increase in the number of surface oxygen-containing functional groups [57], which provide electrostatic and chemical interaction of carbon nanoparticles with a polymer matrix, allowing more uniform distribution in the volume.
However, a high degree of functionalization causes the appearance of a large number of defects in graphene layers, especially in single-walled nanotubes. Figure 2 shows images of functionalized and native carbon nanotubes, which demonstrate the absence of the structure inherent in nano-tubes after their modification.
The viscosity of epoxy systems was investigated at various concentrations of modifiers. The epoxy resin has distinct pseudoplastic behavior with decreasing viscosity whenthe shear rate increases. When carbon nanoparticles are added, the viscosity of the mixture increases. This effect is most noticeable at low shear rates, and is partially offset by increasing shear rates. These changes can be explained by the formation of a randomly distributed percolationnetwork. Organized structures are destroyed at high shear rates, which makes it possible to reduce the viscosity due to the alignment of modifiers in the direction of flow [47]. Different tendency of modifiers to form a percolation network explains their different effect on the dispersion viscosity. Native single-walled carbon nanotubes are characterized by the greatest influence on the viscosity of the modified resin from the studied materials, functionalized multi-walled nanotubes are characterized by the least one.
Figure 3 shows the dependence of the modified epoxy resin viscosity on the concentration of carbon nanotubes or nanofibers measured at a constant mixture temperature of 50 degrees. As it can be seen from image 3, even at high concentration of functionalized MWCNTs the dispersion
viscosity is low and suitable for processing by such methods as vacuum infusion and injection molding. Sharp increase in viscosity with increase in the concentration of native CNTs is explained by the fact that at a given concentration a percolation network is formed. The network prevents the mobility of the polymer chain, which leads to increase in viscosity [48-50]. As Figure 3 shows, functionalized SWCNTs have less effect on the rheology of the modified resin, which may be due to their greater tendency to agglomeration and destruction of the SWCNT structure upon modification. Agglomerating into large formations, nanomodifiers reduce their effective amount in the polymer matrix. To introduce functionalized SWCNTs into the resin, we used the "Solution processing" method [47], since after the drying stage the CNTs are not dispersed into the polymer matrix during the functionalization of their surfaces by the Hammers method. The addition of CNFs into the polymer matrix does not significantly affect its rheological properties, regardless of the CNFs functionalization. At CNFs concentration of 10 %, the viscosity of the mixture reaches 0.45 Paxs.
0 0J 0.4 0.6 0.8 1 1,2 1.4 1.6 1,S
The concentration, %
+ native MWCNT modified MWCNT + native SWCNT
* modified SWCNT + native CNF
Fig. 3. Influence of the carbon nanoparticles concentration on the viscosity of the modified resin
We made samples of the modified resin with various functionalized and native fillers. The effect of modifica-
tion with various carbon nanoparticles on the CTE of the binder is shown in Table 3. Multi-walled nanotubes modified by the second method turned out to be the most effective. At the concentration of 1 %, they reduce the CTE of the polymer matrix by 15.4 %. Native SWCNTs reduce the CTE of epoxy resin by 9.7 % at the concentration of 0.05 %.
Native SWCNTs and functionalized MWCNTs were used as the most effective modifiers for further research. Table 4 shows the measured and calculated values for the selected modifiers.
Samples of fibrous polymer composite material based on unidirectional carbon fibers and modified and unmodified epoxy resin were made. The fiber volume fraction in the unidirectional composite material was determined using thermogravimetric analysis. The CTE of the material across and along the carbon fibers was measured. The results are presented in Table 5.
The calculated data obtained from the research using the models [59, 72] showed high convergence with the measured data. To assess the convergence of the calculated and measured CTE values in the transverse direction, we used the values measured out-of-plane of the sample in order to neutralize the possible effect of fiber disorientation during the sample manufacture.
Taking into account the possible disorientation of fibers in the plane, mathematical modeling of the warpage of plate was carried out on the basis of the properties presented in the table 6. The CLT was used to estimate the effective elastic properties of the composite. The models described in [59, 72] were used to estimate the effective CTE of the composite. The samples were made of spread tow fabric based on carbon fiber IMS 65 and unmodified or modified epoxy resin and have a reinforcement scheme [0/60/-60]. The samples have thickness 0.24 mm and the size of plate in the XY plane: 200x200 mm.
Table 3
Influence of carbon nanoparticles on the epoxy resin CTE
Modifier The most effective concentration, % CTE of modified resin (reduction relative to pure resin)
Native CNFs 4 64,09 10-6 (7,1 %) ± 1,7110-6
Functionalized CNFs 4 63,1010-6 (8,6 %) ± 1,5310-6
Native SWCNTs 0,05 62,3-10-6 (9,7 %) ± 1,64 10-6
Functionalized SWCNTs 0,05 64,6110-6 (6,3 %) ± 0,95 10-6
Native MWCNTs 1 67,98 10-6 (1,4 %) ± 1,42-10-6
Functionalized MWCNTs 1 58,37 10-6 (15,4 %) ± 0,78 10-6
Table 4
Characteristics of CNT-modified epoxy resin
Test sample Measured CTE, С-1 Predictive CTE, С-1 Predictive Young's modulus, Em GPa
Т67+0,01 % non-functionalized SWCNTs 66,99 10-6 ± 1,59-Ю-6 66,83-Ю-6 3,281
Т67+0,05 % non-functionalized SWCNTs 62,3-10-6 ± 1,64 ■ 10-6 62,96-Ю-6 3,72
Т67+0,2 % functionalized MWCNTs 65,02 10-6 ± 1,7-10-6 58Д9-10"6 3,879
Т67+0,4 % functionalized MWCNTs 63,83 10-6 ± 1,57-Ю-6 53,58-Ю-6 4,539
Т67+1 % functionalized MWCNTs 58,37-Ю-6 ± 0,78 10-6 47,84-Ю-6 6,414
Т67+2 % functionalized MWCNTs 61,5310-6 ± 0,56-Ю-6 44,43-Ю-6 9,478
Table 5
Properties of unidirectional composite material
Indicators Unidirectional layer of composite based on carbon fiber k500 (Fiber volume fraction 57 %)
Т67 Т67+0,05 % unmodified SWCNTs Т67+1 % modified MWCNTs
Calculation Measurement Calculation Measurement Calculation Measurement
Longitudinal CTE, a,j C-1 -0,52-10-6 -1Д9210-6 ± 0Д6 10-6 -0,492 10-6 -1Д9710-6 ± 0Д110-6 -0,365 10-6 -1Д0-10-6 ± 0,28 10-6
Change in the indicator relative to composite with unmodified resin, % - -0,38180 7,85
Transverse CTE, a2 C-1 - 33,575-Ю-6 ± 1,61-Ю-6 - 26,572-Ю-6 ± 1Д610-6 - 21,43-10-6 ± 0,1910-6
Change in the indicator relative to composite with unmodified resin, % - 20,85 36,17
Out-of-plane, a3 C-1 38,8110-6 42/735-10-6 ± 1Д5-10-6 35,82-10-6 36,083-Ю-6 ± 1,79-Ю-6 27,39-Ю-6 27,435-10-6 ± 1,1410-6
Change in the indicator relative to composite with unmodified resin, % - 15,56 35,8
Table 6
Properties of an elementary layer of composite material
Indicators Layer of composite based on carbon fiber IMS65 (Carbon fiber content 47 %)
T67 T67+0,05 % unmodified CNFs T67+1 % modified MWCNTs
Longitudinal modulus, E1 ,GPa 138.55 138.84 140.27
Transverse modulus, E2 ,GPa 11.96 12.25 13.68
In-plane shear modulus, G12 ,GPa 4.44 4.88 6.24
Major Poisson's ratio, v12 0.279 0.273 0.251
Longitudinal CTE, a1, C-1 0.268 10-6 0.339 10-6 0.608 10-6
Transverse CTE, a 2, C-1 48.57 10-6 44.83 10-6 34.1910-6
0,18
0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 Standard deviation
-9- 0 % - a -1 % MWCNT 0,05 % SWCNT
Fig. 4. Graph of the probability density function
60
0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 The concentration, % - —b— /'--»- Aa2 Avar (Standard deviation)
Fig. 5. Graph of the dependence of the variation in the indices of the standard deviation change sample on the concentration of multi-wall nanotubes
The warpage were obtained for each combination of the reinforcing fibers orientations on the basis of the data presented. The mathematical expectation of the standard deviation values of warpage is 0.1418 for the unmodified resin, 0.1372 for the samples based on the resin modified with single-walled nanotubes, 0.119 for the samples based on the resin modified with multi-walled nanotubes. It corresponds to decrease in standard deviation by 3.24 % and 15.95 % when modified with single-walled and multi-walled nanotubes, respectively. The sample variance also decreased by 12.6 % in the case of single-walled nanotubes, and by 26.6 % in the case of multi-walled ones. The
size of each sample is equal to A^ = 2744, with a maximum disorientation angle of 2 degrees. The maximum disorientation angle was determined in terms of the dependence of the efficiency of modification of the composite material polymer matrix with carbon nanotubes, expressed by decrease in the mathematical expectation of the standard deviation compared to the same indicator for the composite material based on unmodified resin, on the maximum disorientation angle. The efficiency of modification has nonlinear dependence and decreases with increasing disorientation angle. For single-walled nanotubes, at the concentration of 0.05 %, the reduction in the modification efficiency was from 11 % to 5 % with increase in the disorientation angle from 0 to 2 degrees, and 37 % and 20 % for multi-walled nanotubes with similar increase in the disorientation angle.
Figure 4 shows the graphs of the probability density function for the standard deviation of the simulated surface of a composite plate based on unmodified and modified epoxy resins, taking into account the fibers disorientation.
Further, on the basis of the models described above we constructed the relations of the accuracy of the composite plate surface, expressed in terms of the standard deviation change, to the concentration of the modifier, taking into account the possible disorientation of the reinforcing fibers. When constructing the relations, we relied on the calculated thermoelastic characteristics of the modified epoxy resins and did not take into account the decrease in the efficiency of the modifier with the increase in the concentration in relation to the epoxy resin CTE. Despite this, the effectiveness of the modifier in improving the accuracy of the surface decreases with increasing its concentration.
Figure 5 shows the relations f of the standard deviation change of the composite plate to the concentration of the modifier, the derivative of f with respect to the modifier concentration, the relation of the change in the variance of the standard deviation sample (AVar (Standard deviation)) to the modifier concentration and the relation of the change in the calculated transverse CTE of an elementary layer of the composite material (Aa2) to the modifier concentration. On the basis of data provided it can be concluded that it is inappropriate to use modifier concentrations above 1 % in order to reduce the warpage and the sample variance.
The samples were made of fabric based on fiber IMS 65 and unmodified or modified epoxy resin and have a reinforcement scheme [0/60/-60]. The samples have thickness 0.24 mm and the size of plate in the XY plane: 200x200 mm. The standard deviation of warpage was 0.1636 for the unmodified resin and 0.1329 for the resin modified with singlewalled nanotubes, which corresponds to decrease in the standard deviation of warpage by 18.77 %.
Conclusions
We have measured the rheological properties of dispersion and thermomechanical properties of the obtained nanocompo-sites. The most effective modifiers and their concentrations were determined. For native nanotubes, the effective concentration is 0.05 % which provides decrease in the CTE of the polymer matrix by 9.7 % with increase in viscosity to 0.83 Pa*s. Further increase in concentration is impossible due to the achieved high viscosity. For modified MWCNTs, concentration of 1 % is effective, providing decrease in CTE by 15.4 % and increase in viscosity up to 0.23 Pa*s. The thermomechanical properties of nanostructured laminated composite materials were investigated. It was found that carbon nanomodifiers are more effective as compensators for the thermal expansion of the polymer matrix in the fiber/epoxy laminated composites. The transverse CTE of unidirectional composite material decreased by 15.56 % in the cases of using native caibon nanotubes and by 35.8 % in the cases of using functionalized MWCNTs. Mathematical modeling of the warpage of plate, taking into account the possible disorientation of the fiber, showed decrease in standard deviation by 3.24 % and 15.95 % and decrease in sample variance by 12.6 % and 26.6 % when modified with single-walled and multi-walled nanotubes, respectively. On the basis of the obtained results, the dependences of the transverse CTE of an elementary layer of the composite material, the mathematical expectation of the standard deviation, and the variance of the standard deviation of the composite plate on the concentration of the modifier were constructed. According to the obtained data, it can be concluded that in order to reduce the mathematical expectation and the variance of the warping of the plate, there is an effective concentration, the increase of which is impractical, despite the further decrease in the transversal CTE. The results obtained confirm that the addition of carbon nanoparticles as a modifier to the polymer matrix can increase the accuracy and technological stability of the fiber composite quasi-isotropic reinforcement scheme.
Declaration of interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Appendix 1
The Mori-Tanaka's equation for the effective CTE of a filler-reinforced polymer matrix [55]:
a = am + Vf{Cm + (Cf - Cm )•[( - Vf )•S + Vr I ]p
Cf •(af - am )■
Where am and af are the CTE of the resin and modifier, respectively, Vf is the volumetric content of the filler, Cm is the stiffness matrix of the resin, Cf is the stiffness matrix of the filler, S is the Eshelby tensor, I is the unit matrix.
Eshelby tensor S for the case of reinforcement with infinitely long elliptical cylinder in simplified form is [53]
S =
С1111 С 1122 С 1133 0 0 0
С 2211 С 2222 С 2233 0 0 0
С 3311 С 3322 С 3333 0 0 0
0 0 0 С 2323 0 0
0 0 0 0 С 1313 0
0 0 0 0 0 С 1212
С = С = С = 0
"1111 1122 "1133 "
С -
2222
1
2 (1 "»m) 1
С -
3333 \
2 (1 )
С —
2233
—
3322
1
2 (1 "4m )
b + 2ab
(a + b)2 ' v m'a +b
a1 + 2ab (a + b )2
' b2
(a + b) v m'a + b
+ (1" 2»„ )
+ (1" 2»„)
"(1" 2»m)
a + b b
2 (1 "»m )
С —
"3311
С —
2211
""(I " 2»m )
С -
2323
2 (1 "»m )
(a + b) m a + b
1 2»ma
2(1 "»m) a + b
1 2»mb
2(1 "»m) a + b
a2 + b2 1" 2» - +--
2 (a + b )2
V =
S1313 2 (a + b ) S b
S1212 2 (a +b),
um is the Poisson's ratio of the matrix, а and b are the
main and auxiliary semi axes, respectively.
The effective CTE of a polymer matrix modified with three-dimensionally randomly distributed filler can be determined from the equation:
c 1 2
a = — -a.. +---a...
3 11 3 22
Where a11 and a22 are the longitudinal and transverse CTEs determined from the Mori-Tanaka's equation.
The Mori-Tanaka's equation for the effective Young's modulus of a polymer matrix reinforced with filler [55]:
C = [(l-)• Cm + • Cf • A]•[(-)•1 + • A]"
A = [I + S • Cm-1 •(Cf - Cm )
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Благодарности. Работа выполнена коллективом научной лаборатории «Интеллектуальные материалы и конструкции» в рамках государственного задания Министерства науки и высшего образования Российской Федерации на реализацию проекта «Разработка многофункциональных интеллектуальных материалов и конструкций на основе модифицированных полимерных композиционных материалов, способных функционировать в экстремальных условиях» (проект № ФЭФЭ-2020-0015). Изображения углеродных нанотрубок были получены с помощью просвечивающего электронного микроскопа Hitachi HT7700 в Красноярском краевом центре исследовательского оборудования Федерального исследовательского центра «Красноярский научный центр СО РАН».
Конфликт интересов. Авторы заявляют об отсутствии конфликта интересов.
Acknowledgments. This work was carried out by the team of the scientific laboratory "Smart Materials and Structures" within the state assignment of the Ministry of Science and Higher Education of the Russian Federation for the implementation of the project "Development of multifunctional smart materials and structures based on modified polymer composite materials capable to function in extreme conditions" (Project No. FEFE-2020-0015). Images of carbon nanotubes were obtained using a Hitachi HT7700 transmission electron microscope in the Krasnoyarsk Regional Center of Research Equipment of Federal Research Center "Krasnoyarsk Science Center SB RAS".
Conflict of interest. The authors declare no conflict of interest.
