Computational and Experimental Determination of the Viscoelastic ...

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ScienceDirect Procedia Engineering 113 (2015) 499 – 505

International Conference on Oil and Gas Engineering, OGE-2015

Computational and experimental determination of the viscoelastic parameters of the dispersed-filled polymeric materials Shil’ko S.V.a,*, Chernous D.A.a, Kropotin O.V.b, Mashkov Yu.K.b a

Metal Polymer Research Intitute of National Academy of Sciences of Belarus, Kirov St. 32а, Gomel 246050, Belarus b Omsk State Technical University, 11, Mira Pr., Omsk 644050, Russian Federation

Abstract The aim of the study is to describe the two-component polymer composites as a viscoelastic heterogeneous medium. The proposed computational and experimental approach to the determination of effective viscoelastic parameters is based on a statistical method of moment functions and Prony model. The validity of the hypothesis of aliasing relaxation times of the matrix and the filler in the absence of the interfacial layer is established, as well as the use of standardized tests in uniaxial tensioncompression at constant temperature in order to determine the parameters of the thermoviscoelasticity equations. To illustrate the proposed approach we determine the viscoelastic parameters of polytetrafluoroethylene, filled with particles of polyethylene terephthalate. By approximating the deformation diagrams in uniaxial tension obtained at different temperatures, we also found the viscoelastic parameters of glass-nylon composite. Temperature independence of the relaxation core dimensionless parameter and the activation energy of the relaxation process is defined © 2015 2015The TheAuthors. Authors. Published by Elsevier © Published by Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Omsk State Technical University. Peer-review under responsibility of the Omsk State Technical University Keywords: polymer composites; relaxation processes; viscoelasticity; strain diagram; temperature

1. Introduction Currently an urgent problem is the problem of creating antifriction polymeric materials for friction units, including a sealing device operating in extreme conditions (aggressive processing medium, high and ultra-low (to the extent of cryogenic) temperatures, lack of lubrication, dynamic loads, etc.). Due to obvious temperature and time dependence of the polymers properties some ways to improve their shape and thermal stability, based on the adding

* Corresponding author. Tel.: +3-752-327-74638; fax: +3-752-3277-5211. E-mail address: [email protected]

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Omsk State Technical University

doi:10.1016/j.proeng.2015.07.342

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S.V. Shil’ko et al. / Procedia Engineering 113 (2015) 499 – 505

of the dispersed filler are widely used. [1] However, the existing mechanical and mathematical models and computational methods to predict the deformation parameters of polymer composites and optimize the content of filler are used insufficiently in materials science and tribo-engineering. In this paper, in order to interpret the mechanical properties of polymer composite materials (PCM) by determining their viscoelastic parameters, and also at elevated temperatures, we provide a description of PCM deformation using the method of moment functions and Prony model. This approach was tested on the example of polytetrafluoroethylene, filled with particles of polyethylene terephthalate, and glass-nylon composite. 2. Application of the method of moment functions to determine the viscoelastic properties of polymer composites One of the most common at the present time mechanical and mathematical models of the stress-strain state of PCM is based on the statistical method of moment functions [2]. In this model, the effective shear modulus Gk and effective bulk modulus Kk of the composite are determined by ratios

Gk

Kk

M G f (1 M )Gm

6M (1 M ) G f Gm

pK 2 pG , ª6 M Gm (1 M )G f 4 pG º pK 2 pG 5 pK pG ¬ ¼

M K f (1 M ) K m

3M (1 M ) K f K m

2

2

3 M K m (1 M ) K f 4 pG

. (1)

Here M is the volume fraction of filler; Gf, Gm, Gk are shear modulus of filler, matrix and composite respectively; Kf, Km, Kk is bulk modulus of filler, matrix and composite respectively. In the expressions (1) for brevity we use the notation

pG

G f Gm

M Gm (1 M )G f

,

pK

K f Km

M K m (1 M ) K f

. (2)

Materials of the matrix and the filler in the composite will be considered as isotropic linearly viscoelastic. For these materials bond component of the stress and the deformation tensors can be described by the following physical equation [3]

V ij t

t

³ 2G t W

deij W dW

0

t

dW G ij ³ K t W

d H W

0

dW

dW ,

(3)

where V ij t is components of the stress tensor; еij(t) is components of strain deviator; G(t), K(t) are functions (cores), describing respectively the shear and bulk stiffness of material. To approximate the functions G(t) and K(t) we use a model representation in the form of Prony series [4] nG

G [ Gf ¦ Gi e i 1

[ OiG

,

K [

nK

K f ¦ Ki e i 1

[ OiK

.

(4)

Here Gf, Kf are long shear modulus and long bulk modulus, respectively; OiG, OiK are parameters with the meaning of the time for the shear and bulk relaxations respectively.

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To determine the effective viscoelastic properties of the composite it is convenient to use dynamic integrated modules, which describe the stress state of the material under harmonic deformation with frequency Z. Using the Prony series in the form (4) for the dynamic shear modulus G* and dynamic bulk modulus K* the following relations can be obtained

G* (Z ) G / (Z ) iG // (Z ), K * (Z )

K / (Z ) iK // (Z ),

§ · n n G iG G iG OiG ¸ , G // (Z ) GZ G / (Z ) G ¨1 ¦ , ¦ 2 G 2 ¨ i 1 1 Z2 OG 2 ¸ i 1 1 Z O i i © ¹ K / (Z )

§ · n G iK ¸ , K // (Z ) K ¨1 ¦ 2 ¨ i 1 1 Z2 O K ¸ i ¹ ©

n

KZ ¦ i 1

G iK OiK

1 Z 2 OiK

2

(5)

.

In the last equality we use the notation G

n

Gf ¦ Gi , G iG i 1

Gi , G

K

n

Kf ¦ Ki , G iK i 1

Ki K .

(6) G

K

Here G, K are the instant shear modulus and instant bulk modulus of the material respectively; Gi , Gi are the dimensionless parameters of relaxation under shear and volume deformation. The substitution of dynamic functions (6) for the matrix and filler materials in equation (1), (2) instead of the corresponding elastic moduli allows us to determine the calculated dependences of complex dynamic modules Gk* and Kk* of the considered composite on the frequency Z. The explicit form of these functions is not presented here due to bulkiness. To perform the calculation it is reasonable to reduce the dependences Gk*(Z) and Kk*(Z) to the same form (5), as for the composite component. The proposed specification does not consider the interphase layer resulting from the complex physical and chemical processes at "filler – polymer matrix" interface. However, this simplifying assumption suggests that the spectrum of composite’s relaxation times is formed by the superposition of the spectra of the respective starting components. Consequently, the series (5) for the considered composite will take the form

Gk* (Z )

Gk/ (Z ) iGk// (Z ), K k* (Z )

K k/ (Z ) iK k// (Z ),

nk nk § G ikG · G GO Gk ¨1 ¦ , Gk// (Z ) Gk Z ¦ ik 2ik 2 , 2 2 ¸ i 1 1 Z Oik © i 1 1 Z Oik ¹ K nk nk § G ik · G KO K k/ (Z ) K k ¨1 ¦ , K k// (Z ) K k Z ¦ ik 2ik 2 . 2 2 ¸ i 1 1 Z Oik © i 1 1 Z Oik ¹

Gk/ (Z )

(7)

In the last equalities the following notations are used

nk

2 nm n f , Oik

OimG , i d nm ; ° K ° °O(i nm ) m , nm i d 2nm ; ® G °O(i 2 nm ) f , 2nm i d 2nm n f ; ° K O , 2nm n f i d nk . ° ¯ (i 2 nm n f ) f

(8)

Here nk, nm, nf are the number of terms in Prony series for the composite, the matrix and the filler respectively; Oik, Oim, Oif are relaxation times for the composite, the matrix and the filler respectively. The instant modules Gk, Kk in (7) are defined by the equations (1), (2) when we replace the complex functions of frequency with the corresponding to the instant elasticity modulus of the component. At the known instant modules

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Gk, Kk the dimensionless parameters GikG, GikK are determined by the least squares method, based on the condition of the most closely matching the Prony series (7) to the calculated dependences Gk*(Z) and Kk*(Z). 3. Determination of the viscoelastic parameters of PCM based on PTFE and PET As an example of using the proposed approach for predicting the effective viscoelastic characteristics let us consider a polymer, a polymer composite material, in which the matrix phase is PTFE (polytetrafluoroethylene), and as filler material is used polyethylene terephthalate (PET). For these components the volumetric creep can be neglected compared to the shear creep [3]. Wherein GimK = GifK = 0. In accordance with the results of [5], the viscoelastic parameters of the considered composite matrix at 200С take the following values: G = 31.5 MPa; K = 4,7 GPa; G1 = 0.154; G2 = 0.165; O1 = 0.078 s; O2 = 3.448 s. According to [6] at the same temperature the relevant parameters for PET G = 1.376 GPa; K = 6.05 GPa; G1 = 0.34; G2 = 0.39; O1 = 0.01 s; O2 = 3.48 s. Fig. 1 shows the calculated dependences of mechanical loss tangent on the frequency of shear deformation for the considered composite and its components. In calculating the volume fraction of the filler is taken to be 60%. In the considered frequency range of deformation an average relative error of approximation of the calculated frequency dependence of the effective dynamic complex shear modulus G*(Z) by the series (7) was 4.4%. In this example G1k = 0.115; G2k = -5.108; G3k = 0.086; G4k = 5.375. The performed calculations showed that at low volumetric creep of matrix and filler the validity of this assumption is retained for the composite (at GimK = GifK = 0 condition is satisfied GikK | 0). Thus, the considered example confirms the ability to use the hypothesis of aliasing relaxation times in the calculation of effective viscoelastic parameters. 0,16 tgE

3

0,14

0,12

0,1

1

0,08

4

0,06

0,04

2 0,02

0 0

1

2

3

4

5

6

7

Z,8 Hz Гц

Fig.1. Frequency dependence of the mechanical loss tangent under shear deformation for the composite (curve 1), the matrix (curve 2) and the filler (curve 3). Curve 4 – result of the approximation of the frequency dependences of the composite’s effective parameters by Prony series (7).

4. Determination of the viscoelastic parameters of PCM based on polyamide For the experimental determination of viscoelastic characteristics of PCM a dynamic mechanical analysis (DMA) [4, 5] can be used as well as a standardized mechanical sampling testing for uniaxial tension-compression. In particular, Figure 2 shows the experimental stress-strain diagram for glass-nylon composite PA6VS at different temperatures. Each diagram is the average result of three experimental curves. The average statistical error was 5.5% for the 200С, 7.2% for 600С and 8.5% for the 1000С. Standard samples were in the form of blades with

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working part length of 5 cm and a cross section of 15 mm2. Stretching at a rate of 25 mm / min (speed of axial strain H z = 0,0083 s-1) was performed on the test machine INSTRON 5567 equipped with a heat chamber. Vz, MPa МПа 60

20

50

60

40

100 30

20

10

0 0

0,5

1

1,5

2

2,5

Hz, 3%

Fig. 2. Stress-strain diagrams for glass-nylon composite The numbers on the curves correspond to the sample temperature in degrees Celsius. Solid curves - experimental plots; dash - the calculation byformula (11).

In mathematical description of the uniaxial stress state of the sample instead of the functions G(t), K(t) in the physical equations (3) it is advisable to use the function E(t), which is an viscoelastic analogue of the Young’s modulus. This function is set by Prony series (4), it defines the relationship of the axial stress Vz and axial strain Hz t

³ E (t x)

V z (t )

0

E (t )

n

d H z ( x) dx, dx

Ef ¦ Ei e i 1

(9)

t

OiE

.

(10)

Without reducing the generality of further calculations, at the description of the deformation of glass-nylon composite we restrict with the only relaxation time (n = 1) in a series (10). Then under uniaxial stress state and the constant strain rate H z of the sample the equation (9) takes the form

V z (H z )

ª § § H · ·º E «1 G H z GOH z ¨¨1 exp ¨ z ¸ ¸¸» . © H z O ¹ ¹¼» © ¬«

Here

E

n

Ef ¦ Ei i 1

Ef E1 , G

E1 . E

(11)

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Included in (11) the parameters E, O, G are determined by matching comparing the calculated and experimental diagrams of stretching the sample at a constant temperature in the range of reversible deformation. For this it is necessary to minimize the functional form

)( E , G , O )

Ht

³ V

exp z

V z (H z ) d H z , 2

(12)

0

where Vzexp – the experimental values of axial stress; Ht – strain corresponding to the yield point. The minimization of the functional (12) was carried out in the MathCad software. The value of Ht for the glassnylon composite was 2% at 200С, 2.2% at 600С and 3% at 1000С. The results of model identification (11) are specified in the table. The table also shows the values of average relative deviation ' of calculated axial stress values from the experimental data in the range of deformation Hz < Ht, allowing to speak about the temperature independence of the parameter G. The decrease in relaxation time with the temperature increase can be described by an exponential function [5] O (T )

§ U · ¸. © RT ¹

O0 exp ¨

(13)

Here O0 – kinetic parameter (testing period); U – activation energy of the relaxation process; R – universal gas constant; T – thermodynamic temperature [Kelvin]. Table 1. The viscoelastic parameters of glass-nylon composite at various temperatures.

T,0С

E, GPа

G

O, s

', %

20

5.02

0.44

1.27

4.32

60

2.88

0.41

0.35

6.03

100

1.04

0.43

0.16

8.11

In the expression (13) an assumption about the independence of activation energy on temperature has been made. In accordance with Table 1. we have O0 = 60.03 micro s; U = 24.25 kJ/mol. The reduce of instant Young’s modulus for glass-nylon composite in this temperature range can be described as a linear function of the form

E(T ) 19,6 0,05T (ÃÏ à) .

(14)

For a more precise description of the temperature dependence of Young’s modulus and the relaxation time it is required to conduct additional experiments at other temperatures. Consideration of only uniaxial stress state of the sample does not allow determining the full range of viscoelastic characteristics of the material; in addition it is required to obtain similar diagrams of shift or bulk loading. In some cases. the amount of required experimental work can be reduced. Thus. for the near-incompressible material (Q o 0.5) the lateral deformation coefficient (the analogue of Poisson’s ratio) is a constant. Then the parameters E. O. G. defined under uniaxial stress state. are associated with the appropriate parameters G. OG. GG characterizing the shear deformation. as follows

G

E , OG 2(1 Q )

O, G G

G.

(15)

If for some material the volumetric creep can be neglected compared to the shear creep [3]. then instead of relations (15) the following equalities will be performed

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E

9 KG , O 3K G

OG

3K G

3K G 1 G

G

, G

1

O 1 G G . OG

(16)

Here K is the bulk modulus of the material. 5. Conclusion The computational-experimental approach to determine of effective viscoelastic characteristics of twocomponent PCM. based on the statistical method of moment functions and Prony model was developed. The validity of the hypothesis of aliasing relaxation times of composite’s individual components under the assumption of the absence of an interfacial layer in the filled polymer is established. However. due to the interfacial layer’s significant role in processes of composites deformation the in-depth study of viscoelastic properties taking into account these factors is the subject of interest. The possibility of using the standardized tests of samples in uniaxial tension-compression at constant temperature in order to determine the thermoviscoelasticity parameters of polymeric materials in physical equations is discussed. In particular for glass-nylon composite the temperature independence of dimensionless parameter of the relaxation core is shown and the validity of assumptions about the independence of the activation energy relaxation process on the temperature is defined. Acknowledgements The research is supported by Belarusian republican foundation for fundamental research (project Т14Р-209) and Russian foundation for fundamental research (project 14-08-90022). References [1] J. Manson. L. Sperling. J. Polymer Blends and Composites. Plenum press. New York and London. 1976. [2] A.N. Guz’. L.P. Khoroshun. G.A. Vanin. Mechanics of Composite Materials and Elements of Constructions. In 3 Volumes. Vol. 1. Mechanics of materials. Naukova Dumka. Kiev. 1982. [3] E.I. Starovoitov. Fundamentals of the Theory of Elasticity. Plasticity and Viscoelasticity. BelSUT. Gomel. 2001. [4] R. Christensen. Mechanics of Composite Materials. Wiley. New York. 1979. [5] D.A. Chernous. S.V. Shil’ko. The Modified Takayanagi Model of Deformation for Dispersed-Filled Composites. Part 3. Determination of Thermal and Elastic Parameters of Polymer Matrix. Mechanics of Composite Materials and Structures 20(2014) 124–131. [6] S.V. Shil’ko. D.A. Chernous. S.V. Panin. Method of Determining the Thermoviscoelastic Parameters of Polymers and Elastomers. Journal of Engineering Physics and Thermophysics 87 (2014) 984-987.