Dependence of mechanical characteristics from composition and structure and optimization of mechanical fracture energy of polymer composite material based on high-molecular rubbers

By means of numerical experiment the authors investigate dependence of conventional rupturing stress and mechanical fracture energy at uniaxial tension from fractional composition of dispersed filler, plasticizer volume fraction in polymer binder, effective density of transverse bonds, applied to development of covering for different purposes and with advanced service life in temperature range from 223 to 323 K. They compare mechanical characteristics of polymer composite materials (PCMs) based on highand low-molecular rubbers. It was shown that rupturing stress of high-molecular rubber-based PCM is of a higher magnitude than the stress of low-molecular rubber-based one at almost invariable rupturing deformation. Numerical simulation by variation of composition parameters and molecular structure enables evaluation of its maximum fracture energy which is 1000 times higher than mechanical fracture energy of similar composites based on low-molecular rubbers.


INTRODUCTION
ractional composition of dispersed filler is very essential both for formation of rheological properties of suspensions based on viscous-liquid polymer binders and mechanical characteristics of 3D-linked filled polymer composite materials (PCMs).At that basic formulation parameter is effective extent of volume filling -  / m , where  is volume fraction of filler solids,  m -maximum volume filling depending on particle shape, fractional composition and physicochemical interaction on filler -binder interface.

F
Structural and mechanical dependence of conventional stress  ( )from relative elongation degree  ( ) in case of no lamination between filler particles and polymer binder, for example in road asphalt covering, was proved in previous work [1].Nowadays it is important to develop polymer composite material [2] as a covering for roofs of residential and industrial buildings, sports grounds, and road asphalt with advanced service life in comparison with commonly used PCM.To accelerate development of mentioned polymer composites, it is necessary first of all to investigate relation between molecular structure of polymer binder, effective extent of volume filling, mechanical characteristics and mechanical fracture energy of PCM [3].Previously we have shown that effective mole concentration of transverse chemical and intermolecular bonds is key structural parameter of 3D-linked elastomers filled with solids and based on low-molecular rubbers with terminal functional groups [4].Unfortunately mechanical characteristics of 3D-linked rubbers are usually evaluated without construction of envelopes of sample rupture points according to T. L. Smith at uniaxial tension and different temperatures and deformation rates [3,5].The latter enable to predict service life of advanced PCM in different coverings.This work is intended to investigate numerically the dependence of conventional rupturing stress and mechanical fracture energy at uniaxial tension from fractional composition of dispersed filler, plasticizer volume fraction in polymer binder, effective density of transverse bonds, applied to development of covering for different purposes and with advanced service life in temperature range from 223 to 323 K.

THEORETICAL PART
omputer prediction (calculation) for diagram of uniaxial tension of filled elastomer sample is based on description of its structural and mechanical behavior [1].At that dependence of conventional (related to initial cross-section of the sample) stress (σ) from relative elongation (α), which is related to deformation (ε) by the expression: α = 1+ε/100%, comprises following basic structural parameters and deformation conditions: , where  is polymer density, M c -average internodal molecular weight of the polymer binder; φr =1-φsw -polymer volume fraction in the composite binder,  sw -plasticizer volume fraction in the binder; R -universal gas constant; T ∞ -equilibrium temperature, at which concentration of "physical" (intermolecular) bonds (ν ph ) is negligible; Tg -glass transition temperature of the polymer binder; T -sample test temperature (of numerical experiment); accepted for science and industry; φ -filler volume fraction; φ m -maximum filler volume fraction, depending on fractional composition, particle shape and physicochemical interaction on the filler -binder interface.Basic equation is as follows: where accumulation kinetics of internal composite damage as lamination between the polymer binder and filler particles is described in Gaussian function; n -quantity of volume fractions φi of delaminated dispersed filler fractions (types);

    
  / 0.5 t i i s i -function parameter, α 0.5i -sample elongation corresponding to the half volume of delaminated particles with i -type filler fraction (type),  si -mean-square dispersion for i -type filler fraction (type).Value α 0.5i determines delimitation initiation,  si characterizes composite integrity violation rate.As long as there is no the interfacial Delamination in the elastomeric composite, limiting (rupturing) mechanical characteristics σ b , α b can be evaluated by considering of deformation rate and size of average polymer binder interlayer between filler solids: where indexes «f» and «o» stand for filled and free state elastomeric, respectively.In terms of rupturing deformation the following relation results: which was obtained earlier by summarizing of multiple experimental data [1].Effective concentration of transverse bonds is equal to the total concentrations of chemical (determinant for 3Dlinking of initial linear polymers) and physical (intermolecular, depending on polymer structure and experiment temperature) bonds: For example, if νeff =1mol/m 3 and T= 293K rupturing deformation of 3D-linked elastomeric binder  0 b is almost 1000%.Stated curve is part of the computer program as a calibrating one [8].If sample integrity has been damaged up to the rupture, limiting mechanical characteristics are maximum point values for hump-shaped tension diagrams (σmax, εmax) or values of constrained maximum point assuring serviceability of partly disrupted composite, despite of possible sample tension.To evaluate dependence of material service time from internal damage level of filled polymer composite material by tension, we use envelopes of sample fracture points if no laminations (σb, εb) or maximum points mentioned above (σmax, ε max ).These envelopes were introduced by Smith as following dependences "logσ b,max -logε b, max " [9,10].Problem of fractional composition optimization for dispersed components of polymer material (for given average particle sizes) considering fulfillment of optimization conditions for other performance characteristics could be defined in the following nonlinear programming statement: q d are vectors of volume fractions, porosities and particle sizes of dispersed components in the polymer material, respectively,  opt j is optimal volume fraction of the filler in the composition, φ jv -volume fraction of v-th fraction with j-type filler in the composition, mj -fraction number of j-type dispersed component,   min max , jv j v are upper and lower limits for solids volume fractions in the composition, respectively, opt x j is optimum mass concentration of solid dispersed ingredients in the polymer composition for corresponding characteristics set, for example mechanical, γ j are densities of dispersed components, In is variety of indexes for filler types being part of polymer material formulation.Because of the difficulty, given problem which includes limitation of equation types is converted into the nonlinear programming problem with limitation of in equation types.At that quantity of optimizable independent variables is n = (mj) -m, where m is quantity of polymer material solids types.Normalizing ratio: is automatically fulfilled in case of problem solution.Then we determine the vector of optimum volume fractions for filler fraction in the composition: where  opt jv is optimum volume fraction of v-th fraction for j-type filler.Transformation to optimum mass concentrations of the corresponding solids is carried out by formula: where is the sum of mass concentrations (ratios) of polymer material solids.Mechanical fracture energy (W) of PCM depending on rupturing elongation degree (α b ) is calculated by formula [11]: Mathematical problem statement for maximum fracture energy search when limitations of other characteristics are fulfilled can be described as following nonlinear programming problem: where:

NUMERICAL EXPERIMENTATION
ab. 1 presents optimized parameter values of silica fraction mixtures in the PCM which are calculated using computer program being developed by the authors [8].Numerical experiments are carried out for: ν ch = 3 mol/m 3 , which is chosen according to predesigned as most effective one.Plasticizer volume fractions in the polymer binder are chosen in form of three values 0.05; 0.1; 0.3 to provide development of the PCM with low viscosity.The last one is required for good component mixing.Tab. 2 presents data required for numerical experimentation.According to the Tab.3, PCM based on 3D-linked high-molecular rubbers, filled with four silica fraction mixture at plasticizer volume fraction 0,05 in the binder, fully complies with given requirements.Previously the authors have proposed polymer composite material for road covering based on low-molecular rubbers of butadiene + dieneepoxyetherurethane [12].For comparison of mechanical behavior of low-molecular and high-molecular rubber-based PCMs, Fig. 2 presents envelopes of rupture points for polymer composite materials according to T. L. Smith.M the temperature 223 K, and rupturing deformation increases from 12% to 35%; at 273 K -from 1100 to 900 kJ; rupturing deformation -from 16% to 46%; at 323 K -from 400 to 350 kJ, rupturing deformation -from 25% to 55%.Fig. 3 presents for the first time by the authors constructed envelopes mechanical destruction energy values for the four fractions of silicon dioxide at a volume fraction of the plasticizer -φsw = 0.05 and temperatures of numerical experiment 223 K, 273 K, 323 K. Envelopes of rupture points of polymer composite material according to T. L. Smith qualitatively characterize material service time.In contrast to that, envelopes of mechanical fracture energies quantitatively evaluate service Life of PCM.Envelopes of mechanical fracture energies for PCM based on low-molecular and high-molecular rubber mixtures are constructed for comparison (Fig. 4).It was shown, that mechanical fracture energy of PCM based on high-molecular rubber mixture is 1000 times higher than of the low-molecular rubber-based one.8.It is stated, that mechanical fracture energy of polymer composite material based on high-molecular rubbers is 1000 times higher than of the low-molecular rubber-based one

;
r m -vectors of cross-linking mole concentration, polymer volume fraction in the binder and effective extent of volume filling, respectively; φ sw -plasticizer volume fraction, coherent with polymer volume fraction (φ m ) as ratio   I nvariety of indexes for composition; n -quantity of composition calculation types.

Fig. 1
presents envelopes of rupture points for the PCM according to T. L. Smith for different filler fraction values at 0.05 plasticizer volume fraction in the binder and cross-binding concentration νch = 3 mol/m 3 at different temperatures of numerical experiment.It is obvious, that PCM filled with optimum mixture of four silica fractions meets the requirements to different coverings, especially to road asphalt.

Figure 2 :
Figure 2: Envelopes of rupture points for polymer composite materials according to T. L. Smith: 1 -based on low-molecular rubbers BR-KTR + PDI-3B; 2 -based on high-molecular rubbers IR + BR-It is obvious that rupturing stress of PCM based on high-molecular rubber mixture is of a higher magnitude than of the low-molecular rubber-based one (12 against 1.2 MPa), at almost the same rupturing deformation.

Table 1 :
Optimum parameter values of silica fraction mixtures.

Table 2 :
Data required for numerical experimentation.Analysis of numerical experimentation results has shown, that increase of plasticizer volume fraction from 0.05 to 0.3 (experiment temperature is 223 K; four -fraction silica) causes decrease of rupturing stress (from 12 to 9 MPa) at slight deformation increase (from 26 to 27%,).Experiment temperature increase does not change this tendency (Tab.3).Increase of silica fraction quantity in the PCM from two to four causes decrease of conventional rupturing stress from 28 to 11 MPa at rupturing deformation increase from 12 to 26% (numerical experiment temperature is 223 K, 0.1 vol. Tab. 3 presents numerical experimentation results about dependence of rupturing stress from rupturing deformation of the PCM depending on plasticizer volume fractions in the binder from 0.05 to 0.3 vol.fraction, on filler fraction T fraction plasticizer).If Т = 273 К, 0.1 vol.fraction plasticizer, conventional rupturing stress decreases from 12 to 4 MPa at deformation increase from 17 to 36%; if Т = 323 К -from 3 МPа (22%) to 1 МPа (45%).Basic requirements to different coverings, especially to road asphalt, are the followings: rupturing stress -ap.8 MPa and deformation -at least 12 -15% at the temperature 223K.

Table 3 :
Numerical experimentation results about dependence of rupturing stress from rupturing deformation of the PCM.

Table 4 :
Numerical experimentation results by determination of maximum mechanical fracture energies for PCM.Analysis of simulated results has shown, describes dependence between mechanical fracture energies and PCM fraction composition.It is obvious, that increase of fraction quantity causes decrease of fracture energy from 1800 to 1650 kJ at Plasticizer volume fraction φ sw = 0,05 vol.fraction