The analysis of thermoplastic characteristics of special polymer sulfur composite

Specific chemical environments step out in the industry objects. Portland cement composites (concrete and mortar) were impregnated by using the special polymerized sulfur and technical soot as a filler (polymer sulfur composite). Sulfur and technical soot was applied as the industrial waste. Portland cement composites were made of the same aggregate, cement and water. The process of special polymer sulfur composite applied as the industrial waste is a thermal treatment process in the temperature of about 150–155 ∘C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }\hbox {C}$$\end{document}. The result of such treatment is special polymer sulfur composite in a liquid state. This paper presents the plastic constants and coefficients of thermal expansion of special polymer sulfur composites, with isotropic porous matrix, reinforced by disoriented ellipsoidal inclusions with orthotropic symmetry of the thermoplastic properties. The investigations are based on the stochastic differential equations of solid mechanics. A model and algorithm for calculating the effective characteristics of special polymer sulfur composites are suggested. The effective thermoplastic characteristics of special polymer sulfur composites, with disoriented ellipsoidal inclusions, are calculated in two stages: First, the properties of materials with oriented inclusions are determined, and then effective constants of a composite with disoriented inclusions are determined on the basis of the Voigt or Rice scheme. A brief summary of new products related to special polymer sulfur composites is given as follows: Impregnation, repair, overlays and precast polymer concrete will be presented. Special polymer sulfur as polymer coating impregnation, which has received little attention in recent years, currently has some very interesting applications.

process technology are realized, there is little likelihood that fully impregnated PIC members will ever become economically feasible. This is unfortunate since PIC has such exceptional strength and durability properties [4][5][6][7][8].
Degradation of cement composites often results in initiation and propagation of microcracks and other microdefects. Microcracks develop because of chemical, physical and mechanical interferences. Numerous researches have proved that the presence of microcracks can significantly influence the mechanical properties of cement composites. Since the control and detection of microcracks are not easy, the formation of microcracks is an essential concern for the durability of cement composites. The prediction of material macroscopic properties requires a method capable of quantifying the material microstructural characteristics Let us examine representative volume V of the thermoelastic special polymer sulfur composites as randomly inhomogeneous medium, at each point x (1) for which the Duhamel-Neumann law [41] is given by: the equilibrium equations [41]: and the Cauchy relationship [41]: are valid, where σ i j and ε i j are stress and strain tensors, λi jαβ is the tensor of elastic constants, β i j and α mn are tensors of the coefficients of temperature stresses and the coefficients of linear thermal expansions, respectively, θ is the temperature increase, and u i are the displacements [41]. When the representative volume is acted upon by a uniform load and heating, the stresses and strains that develop within this body form statistically homogeneous random fields satisfying the property of ergodicity. This makes it possible to replace the operation of averaging over an ensemble of realizations. For macroscopic fields, we can write [41]: where λ * i jmn and α * mn are tensors of the effective elastic constants and the temperature expansions coefficients [41].
Consider a special polymer sulfur composites reinforced by randomly distributed and randomly directed ellipsoidal inclusions. This composite may be regarded as a set of subsystems; each subsystem consists of a composite with oriented inclusions, and the symmetry axes of which are directed in a certain manner relative to the initial coordinate system of the composite as a whole. Consider such subsystem and suppose that the inclusions are orthotropic, oriented ellipsoids and the geometric axes of the ellipsoids coincide with the orthotropy axes of the thermoelastic modulus tensor. The orientation of the coordinate axes of the subsystem x 1 , x 2 , x 3 relative to the basic coordinate system of the material x 1 , x 2 , x 3 will be described by directional cosines as follows [41]: The directional cosines may be expressed in terms of Euler angles as follows [41]: a 11 = cos ψ cos ϕ − cos θ sin ψ sin ϕ ; a 21 = − cos ψ sin ϕ − cos θ sin ψ cos ϕ ; a 31 = sin θ sin ψ; a 12 = sin ψ cos ϕ + cos θ cos ψ sin ϕ ; a 13 = sin θ sin ϕ ; a 22 = −sin ψ sin ϕ + cos θ cos ψ cos ϕ ; a 32 = −sin θ cos ψ; a 23 = sin θ cos ϕ ; a 33 = cos θ; (0 ≤ θ < π; 0 ≤ ϕ < 2π; 0 ≤ ψ < 2π).
Then, it follows from the conversion formulas for the components of tensors of four and second ranks on transition from coordinate system x i to x i that [41]: Then, introducing the distribution function that describes the spread of the coordinate axes of the subsystems relative to the Euler angles f (θ, ϕ , ψ), the mean value of the thermoplasticity coefficients may be written in the form of an integral with respect to all three angles [41]: Thus, the approximate determination of the thermoplastic constants of the special polymer sulfur composites may be divided two stages: First, the properties of the special polymer sulfur composites with oriented and randomly distributed inclusions are determined; then the properties of the subsystems oriented in a certain manner, relative to the axes of the basic coordinate system, are calculated, and the effective properties of the whole system are determined by using the specified distribution function. The first stage is based on a model of the special polymer sulfur composites of stochastic structure reinforced by oriented ellipsoids. The effective properties of this special polymer sulfur composites, with orthotropic components, may be determined by using the method of conditional moment functions, and the second stage is based on the Voigt scheme, as outlined above, or on the Rice scheme (the elastic-pliability and the coefficients of linear thermal expansions tensors are averaged) [41].
In this case, Eq. (9) takes the form [41]: where S * i jkl = λ * −1 i jkl , and S * i jkl is determined by analogy with Eq. (8) [41]. The special polymer sulfur composites applied as the industrial waste technology sets new standards and allows you to use sulfur concrete extensively in the road construction industry. Considerable savings owing to longer life and resistance to corrosion plus the thickness of the road base layer are self-evident especially in regions with harsh climate, such as Siberia, Far East and the north of Russia and Canada as discussed elsewhere [19][20][21][22][23][24][25].
The special polymer sulfur composites applied as the industrial waste material technology means not only advantages for the user but also benefits for the environment, including low CO 2 emission and low energy input for the process of waste stabilization and product prefabrication. Virtually waste-free, this technology does not require using water, and every product is 100 % recyclable without any loss or waste as discussed and published elsewhere [30][31][32][33][34][35][40][41][42][43][44][45][46].

Materials
The initial materials used in technological procedure for special sulfur composite production were: sulfur binder applied as industrial waste (Fig. 1) and technical soot as a filler. Also technical soot applied as industrial waste.

Sulfur
At first, the elementary sulfur was used as a binder in production of special sulfur composite. However, despite excellent mechanical properties after preparation, the samples exhibited low stability, so spalling and failure occurred after a short period [16,[25][26][27][28][29][30]. The development of modified sulfur binder contributed to better endurance of sulfur composite, which led to its primary use for road construction and repairing, and as a building material [19,20].
Sulfur, the basic component for a modified sulfur binder, originates from the technical soot. Because of the importance of preparing the sulfur composite with modified sulfur, both elementary and the obtained modified sulfur were investigated by scanning electron microscope (SEM), type JEOL JSM-5800 with EDX, Fig. 2, and their microstructures were analyzed according to the literature [19][20][21][22][23][24][25].
The results showed that the elementary sulfur was composed of dense orthorhombic crystals of alpha form (S α ), Fig. 2a, while modified sulfur consists of plate monoclinic crystals of beta form (S β ), partially polymerized in zigzag chains, Fig. 2b. Therefore, it was proved that modification of sulfur was achieved.
The composition of the sulfur binder applied as the industrial waste: 97.86 % S 8 fine sulfur, oil: 2.13 %, the ash: 0.01 %, producer "Siarkopol" Tarnobrzeg. The composition of the sulfur binder is presented in Table 1.
The results of the preliminary tests were analyzed, and the special polymerized sulfur in the industries objects having the best properties among the tested composites was selected for further studies. The information about the preparation of special polymerized sulfur is presented in Table 2. The composition of the special polymerized sulfur applied as the industrial waste is presented in Table 3, and its experimentally determined properties is shown in Table 4. The sulfur binder used for investigations, applied as the industrial waste, is shown in Fig. 3. Sulfur came from desulfurization of the fuel of diesel.

Filler
The filler used in this production was technical soot "Seva Carb" (granulation: 0.330-0.990 μm). Technical soot with maximum grain size of 0.990 μm was used as an filler. Chemical analysis indicated that the filler mainly consisted of carbon (99.90 %).

Cement
Ordinary portland cement CEM I 32,5 R was used in this study.
The aggregate was first charged into the mixer and mixed with some tap water. Then, the cement and the rest of the mixing water were added and the homogenization continued. The overall mixing time was about 6 min. The concrete mixture was poured into molds and compacted by a vibration table. The specimens were demolded 24 h after casting and then cured in a moist room at a temperature of 20 ± 2 • C with 95-98 % relative humidity for 27 additional days, before being subjected to the tests. The prism-shaped samples with dimensions 40 × 40 × 160 (mm), were prepared.
Each reported value is the average of three readings obtained from three different samples, to ensure the reliability of the test results.
Mechanical strength (compressive and flexural) of the concrete samples was conducted using the "Tecnotest" Modena-Italy press with maximum load of 2000 kN, and method for testing the strength of concrete according to the standard (Fig. 4).

Impregnation technique
It was prepared by melting special polymerized sulfur (polymer coating) at the temperature 150-155 • C and then by cooling to the ambient temperature. The samples and elements were immersed for 5-10 min and remaining specimens for 0-5 hours in molten special polymer sulfur coating. The specimens were then removed from the steel vessel, and excess liquid sulfur on the surface was wiped off. The samples and elements were cooled in water for 20 min in order to crystallize the sulfur in the surface pores and prevent loss of sulfur by evaporation and were then left at room temperature to cool in the air. The specimens were weighed before and after impregnation and sulfur loading calculated. The impregnated specimens look shiny greenish to dark gray depending upon the original color of the specimens (Fig. 2). However, the rough texture is not very much affected. The total process time is 00:20-00:30 hours (20-30 min.) for samples and elements. Details were passed in the literature Książek [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].
The samples concrete and cement mortar was impregnated with special polymerized sulfur (polymer coating) applied as the industrial waste view of Fig. 5.
The impregnation caused saturation of the pores of cement composites and the strain hardening of surface.

Thermoplastic properties
The thermoplastic properties of special polymerized sulfur were examined. Sulfuric binder used in investigations as the industrial waste is shown in Fig. 3. The thermoplastic properties of special polymerized sulfur as a function of temperature is shown in Fig. 6 [19][20][21][22][23][24][25]. Special polymerized sulfur possesses the lemon yellow sintered microcrystals. Chemical element of the special polymerized sulfur and the spectral lines of special polymerized sulfur, applied as the industrial waste is shown in Fig. 7 [19-30].  [19][20][21][22][23][24][25][26][27][28][29][30] In 2008-2015, author reported on research with special polymerized sulfur (polymer coating)-infiltrated concrete and building mortar and noted that waste sulfur has economical advantages over organic polymer. When the impregnation was vacuum assisted, an exceedingly strong and durable concretes and building mortars could be produced by precasting. Author did not recommend his particular formulation for cast-in-place use. In effect, the waste sulfur was used as a substitute for some of the portland cement, because the concrete and building mortar, being infiltrated, had a water-cement (w/c) ratio of 0.70, about 4.0 bag/yd. 3. In 2008-2015, it was concluded that at least in Poland this infiltrated concrete and building mortar would probably be less costly than portland cement concrete having a w/c of 0.40. The sulfur infiltrated concrete had a compressive strength of over 0.000 lb./in. 2 (700 kg/cm 2 ), withstood 200 cycles of freezing and thawing without damage and was exceedingly resistant to chemical attack. Apparently, the special polymerized sulfur fills the capillaries and prevents absorption of water or chemicals, and this prevents the critical saturation that causes freeze-thaw distress. A personal communication from author indicated that there had been no change in the properties of this laboratory concrete in five years [4][5][6][7][8].
The impregnation caused saturation of the pores of cement composites and the strain hardening of surface.

Deduction of correlation for particular cases
Several particular cases of a special polymer sulfur composites with disoriented inclusions are considered. Consider uniformly disoriented inclusions and the mean constants λ * i j and β * i j corresponding to Voigt averaging are calculated. Setting f (θ, ϕ, ψ) = 1.0 in Eq. (9), it is evident that the problem reduces to averaging the sum of the products of the directional cosines [41].
After substituting Eq. (8) into Eq. (9) and taking into account Eq. (7), the result obtained is [41]: Thus, it is evident that the special polymer sulfur composites consist of an isotropic medium in a macrovolume. The mean elastic pliability matrix S * i j and the coefficients of linear thermal expansions α * mn , corresponding to Rice averaging, may be calculated analogously [41].
Thus, it is evident from the relations obtained that such a special polymer sulfur composites are transversely isotropic. Note that, when θ * = π/2, the case in which the inclusions are uniformly disoriented is obtained [41].
Then, taking account of Eqs. (28), (26) reduces to Eq. (21). Note also that the mean plastic-pliability matrix S * i j and the coefficients of linear thermal expansions α * mn corresponding to Rice averaging may be calculated analogously [41].

Numerical investigations: results and analysis
As a numerical example, consider an arbolite based on straw particles and pores in special polymer sulfur composites with the plastic constants, respectively [41]: Here k 2 and k 3 -the ratio of the semi-axes of ellipsoids (these parameters characterize the dimension of inclusions) and c 0 -concentration of the pores in special polymer sulfur composites [41].
The results in Fig. 8 are calculated by the Voigt scheme. The dependencies of Young's modulus E * 3 , Poisson's ratio ν * 12 , and shear modulus G * 12 from concentration of inclusions c 1 and for certain values of k 2 , k 3 are shown in Fig. 9. The continuous curves correspond to uniform disorientation of the inclusions (UD) and disorientation of ellipsoids in the plane x 1 x 2 (DP), and the dashed curves correspond to the case when the b c a Fig. 8 A view the dependencies of Young's modulus E * 3 , Poisson's ratio ν * 12 , and shear modulus G * 12 from concentration of inclusions c 1 and for certain values of k 2 , k 3 are calculated by the Voigt scheme [41] inclusions are uniformly and continuously distributed within the interval 0 Analogous results given by the Rice scheme are shown in Fig. 9. It is evident from the graphs that the numerical values of the macroscopic constants E * 3 , G * 12 given by Voigt averaging are higher than those given by Rice averaging for all concentrations of the reinforcing inclusions [41].
The dependence of the constant ν * 12 is more complicated in character. Note also that, the greatest discrepancy corresponds to concentrations of inclusions in the range 0.3-0.7. In some cases of considerable discrepancy between the Voigt and Rice results, the Hill approximation may be used, i.e., the arithmetic mean of the Voigt and Rice values [41].
Research was also successful in developing processes for partial depth impregnation that could be used in the field for impregnating bridge decks and other cement composites surface. The process required drying the concrete to remove moisture from the surface, applying the special polymer sulfur composites on a thin sand layer that held the special polymerized sulfur during the time the monomer was being imbibed into the concrete, a b c Fig. 9 A view the dependencies of Young's modulus E * 3 , Poisson's ratio ν * 12 , and shear modulus G * 12 from concentration of inclusions c 1 and for certain values of k 2 , k 3 are calculated by the Rice scheme [41] and then polymerizing the monomer using steam heat. The process was capable of producing impregnated depths of 10-50 mm. The concrete surfaces were much more resistant to water absorption, had much higher abrasion resistant, and in generally were much more durable. The downside was that the process normally took about one day and subjected the concrete to high temperatures (thermal process) during drying that resulted in microcracking, and was rather cumbersome. Many bridge decks in the Poland were successfully impregnated during the 1980-1985, but the development of polymer concrete overlays provided a simpler, faster and less costly method for impregnated concrete surfaces [4][5][6][7][8]16,[19][20][21][22][23][24][25].
More recently, however, there has been a successful commercial application of partial depth impregnation. The special polymer sulfur composites vacuum impregnation system uses a thermoplastic membrane that is applied over the surface of a slab, statue, column or wall and vacuum to evacuate the air from the pores.

Summary and conclusion
The process of special polymer sulfur composites applied as the industrial waste production is a thermal treatment process in the temperature of about 150-155 • C. The result of such treatment is special polymer sulfur composites in a liquid state (thermoplastic properties).
This paper presents methods and results for determining the effective thermoplastic constants of special polymer sulfur composites with isotropic porous matrix reinforced by disoriented ellipsoidal orthotropic inclusions. The method is based on the use of conditional statistical averaging procedure and permits determining effective properties of essentially inhomogeneous special polymer sulfur composites possessing geometrical and physical anisotropy. The solution is constructed in two stages. First, the problem of the effective properties of special polymer sulfur composites with randomly distributed and oriented inclusions is solved. Then the problem of the effective properties of this composites is solved on the basis Voigt or Rice scheme. The effect of shape and concentration of inclusions, concentration of pores in matrix and the type of inclusion orientation has been studied [41].
The presented state of the art in the area of the PIC (special polymer sulfur composites-impregnated cement composites) shows the material complexity of the subject and the gap between polymer concretes and portland cement concretes regarding standardization, formal guidelines for production and recommendation for use. This determines the needs for the further works [4][5][6][7][8].
The result of such treatment is special polymer sulfur composites (polymer coatings) in a liquid state. Special polymer sulfur composites (polymer coatings) a liquid state are mixed with previously heated extender (thermal and plastics processes).
Recent research has led to the development of durable special polymer sulfur composites building mortars, concretes and coatings. All of the methods of using special polymer sulfur composites as a binder for rigid concrete rely on the reaction of one or more modifiers to stabilize, in the hardened state, at least a portion of the special polymer sulfur composites in its less brittle, less dense form. The durability of the concrete produced appears to depend on the modifying system used. In all cases, the special polymer sulfur composites must be heated to a liquid state to react with the modifier and to mix with and coat the aggregate and filler.
Special polymer sulfur composites (polymer coatings)-impregnated cement composites can develop high strength, attain strength in a 0-5 hours, require no special curing, resist acids and organic liquids, has no known undesirable reactions with aggregates, and requires no limitation on the ambient temperature at the time of placement. When its use becomes economically feasible, concrete impregnated special polymer sulfur composites (polymer coatings) will be an excellent material for use in pavement repairs and bridge deck overlays.
Solidified impregnant in the small pores of solids is normally in metastable equilibrium. Spatial restrictions prohibit the formation of regular crystal habit. Because of the formation of noncrystalline modifications, the free energy is increased. In addition, owing to the large surface-to-volume ratio, the surface-free energy is also large. In the absence of concave menisci or strong interaction with the matrix that would reduce the energy, the tendency for spontaneous exudation from the pores is increased.
Special polymer sulfur composites (polymer coatings) do not interact strongly with siliceous surfaces, so that water or other liquids can penetrate the porous network and adsorb on the substrate surface, weakening further the interaction between it and sulfur. Water adsorbs also on sulfur, creating a lubricating effect and high stresses due to surface energy decrease. Both effects facilitate extrusion, and the latter causes destruction.
Because of the relatively slow penetration of water, non-isotropic expansion creates strains beyond the plastic limits. Impregnated porous solids of small size can be successfully utilized for predicting the behavior of large sized systems.
Cement composites impregnated with special polymer sulfur composites (polymer coatings) are widely and successfully used in the repairs of the reinforced concrete structures as well as in their surface protection. Polymer mortars and mortars impregnated with special polymer sulfur composites (coatings) are used for making industrial floors. The main advantages are here excellent adhesion to the various materials, tightness and frost resistance, and in the case of the resin concretes also short time to exploitation readiness as discussed elsewhere [4][5][6][7][8].
The limitation can be relatively high setting shrinkage, as well as some differences between the properties of the repaired concrete and repair material-particularly high thermal expansion, creep, sometimes limited thermal resistance and ageing resistance.

Compliance with ethical standards
Funding This study was not funded by any grant.

Conflict of interest
The author declares and assures that no conflict exists.
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