Property Optimisation of EPDM Rubber Composites Using Mathematical and Statistical Strategies

)is paper describes a study in which EPDM-based rubber composites were investigated aiming at developing formulations subjected to restrictions on cost and the properties of the material. )e contents of components other than calcium carbonate, para&nic oil, and CBS vulcanising accelerator, as well as additives and processing conditions, were kept constant. Fractional factorial design coupled with computational numerical optimisation was used to minimise the number of mixtures. )e results demonstrate that statistical design of experiments and particle swarm optimisation (PSO) algorithms are promising methods to design composition variables. Mixture costs as low as 1.92 US$/kg can be achieved in compositions containing, for example, 107 phr of calcium carbonate, 95 phr of para&nic oil, and 1.13 phr of CBS accelerator. )e corresponding composite propertypredicted values were 66.8 Shore A for hardness, tensile strength of 7.8MPa, 570.8% elongation at break, and 23.0% rebound resilience. )is demonstrates that, in this way, the desired product with speci>ed characteristics can be comfortably manufactured at minimum cost.


Introduction
e modern rubber industry o ers a very wide variety of technological products derived from synthetic elastomers like ethylene-propylene-diene M-class rubber (EPDM rubber).ese products nd applications in di erent elds, namely, in automotive, naval, and mechanical industries.Such rubber compounds (composites) are manufactured from complex mixtures of di erent raw materials (di erent kinds of EPDM elastomers, llers, process oil, vulcanising and protecting agents, etc.), and the production steps involve a variety of processes (e.g., mixing, extruding, cutting, moulding, and vulcanising) [1][2][3][4][5][6].
e performance and manufacture of rubber products has been receiving more and more attention, and the industry has successfully introduced and applied the usage of a series of quality certi cation standards.Also, stringent market and price competition demand shorter product development cycles and reduced costs, which include raw materials and processing, as well as research and development costs.All of that makes it di cult to de ne an adequate new formulation by simple adjustment of older ones, based on rule of thumb or virtue of experience.
e application of statistical design of experiments (DoE) to the industrial formulation of rubber composites can be a convenient and accurate means of obtaining reliable quantitative estimates of properties as the result of any change in contents of raw materials [7,8].e modelling of a given property using the design of mixture experiments is becoming common practice [9][10][11][12][13][14][15][16][17] and was proven, in all cases reported, to lead to greater e ciency and con dence in the results obtained, and to be less demanding in time and both material and human resources.
Many studies on the e ects of raw material changes on the physical properties of rubber composites coupled with DoE can be found in the literature, but few data are available on research carried out using the cost characteristics of rubber compounds.However, although standard requirements for physical and mechanical properties of a rubber compound are mandatory, high costs might preclude the product's competitiveness in the market.us, the rubber compound engineer most often needs to produce an optimised formulation, which ts the requirements of physical and mechanical properties, while subjected to processing and cost constraints [18].In industrially oriented applications of materials like rubbers or ceramics, the technique generally used to optimise equality and inequality property constraints is the graphical overlay of contour plots generated by the regression models of the properties [10,15,[19][20][21].However, as the number of functions and constraints increases, so do the di culties to handle and design the di erent contour plots, and these strategies begin to have a rather limited performance.
Computational optimisation of nonlinear programming problems, which include numerical analysis of continuous and discrete variables, has been an active and important engineering research issue.
e optimisation problem consists in nding out a solution for the objective function and related constraints.e use of particle swarm optimisation (PSO) algorithms for solving nonlinear, multimodal, and nondi erentiable optimisation problems, which are not well tted for conventional optimisation algorithms, has gained increasing attention in recent years [22,23].PSO algorithms fundamentals result from the observation, interpretation, and modelling of the movements of individuals in bird ocks or sh schools, as well as their group behaviour as a swarm.It is a simple algorithm, so only a few lines of the computer program based on simple mathematical operations are needed to deploy the basic tool of PSO. e computer-aided optimisation method provides an e cient way to predict the optimum formulation without using those awkward contour plot graphical overlays.
In this work, a fractional factorial design of experiments was used to study the e ect of ller, process oil, and vulcanising accelerator contents on the mechanical properties (hardness, tensile strength, elongation at break, and rebound resilience) of EPDM rubber composites.Regression models were calculated from the results of the measured properties, under constant processing conditions and contents of other raw materials and additives.e regression models were then used in a PSO algorithm to obtain optimised EPDM rubber formulations subjected to property constraints and cost requirements.

Experiment Design.
A 3 3−1 fractional factorial design was chosen to model the e ect of varying contents of the three factors CC, PO, and VA on the composite properties because it required the minimum number of experiments (nine mixture compositions) for which nonlinear e ects and interactions of all the factors could be investigated [7,8].Given that the contents of all raw materials and additives other than the three factors were kept constant, a new calculation basis was de ned to translate contents of the factors from their usual phr base values, m i , into fractions, X i , as needed in the factorial design, and vice versa.Among the factors limiting phr values, the CC content is the highest of them all (125 phr in the reference product).Hence, this was chosen as the base reference value, M, and contents of all the factors were then expressed as weight fraction relative to that of the CC content, that is, X i m i /M.To design the matrix of mixture experiments, three weight fraction content levels were chosen within the usual ranges for the manufacture of general-purpose products, as shown in Table 2.
STATISTICA statistical software (StatSoft Inc., 2010) was used to determine the geometric and coded notations as well as randomise the treatment combinations, resulting in a standard experiment order.Table 3 shows the mixing ratios of chosen factors for the nine compounds, obtained from the 3 3−1 fractional factorial design.

Mixture Preparation, Moulding, and Property
Evaluation.For each of the nine di erent formulations, in two replications, the selected amounts of raw materials were mixed in a two-roll laboratory mill (Equipabor, Brazil) at 70 °C and 1 : 1.20 speed ratio, as recommended by the ASTM  [25] using a Zwick durometer.e tensile strength (TS) and elongation at break (EB) tests were carried out according to ASTM D 412 Standard [26] using an EMIC DL 2000 testing machine.e rebound resiliency (RS) was determined in accordance with ASTM D2632 using a rebound tester [27].For each mixture, in each replication, the property nal value was taken as the average of the test results obtained for ve di erent test pieces.

Optimisation Strategy.
e experimental results obtained for each property were used to iteratively calculate, with STATISTICA, the coe cients of a regression equation, until a statistically relevant model and response surface was obtained, relating that property value with the weight fractions of calcium carbonate (CC), para nic oil (PO), and vulcanising accelerator (VA) present in the corresponding mixture of raw materials.
A PSO algorithm was developed using the property model equations obtained with STATISTICA and common limiting property values (those of the reference automotive hoses manufactured by NSO Borrachas, Joinville, SC, Brazil), aiming at nding the best composition range (weight fractions) that meets the property limits while minimising costs of the composites.4 presents the values of hardness (HD), tensile strength (TS), elongation at break (EB), and rebound resiliency (RS) obtained for the nine mixtures in two replications.Material costs for the nine mixtures in replication 1 are also shown in Table 4.

Measured Properties and Statistical Analysis. Table
Table 5 shows the results of the variance analysis of the regression equations obtained for HD, TS, EB, and RS, using the nomenclature commonly found in the literature (major statistical properties: p values and coe cient of multiple determination R 2 ) [7,8].It can be seen that, in all cases, the nonlinear models are statistically signi cant at the required level (p value ≤ signi cance level) and present small variability (high coe cients of multiple determination).Although only e ects with p value lower than 0.10 were considered signi cant, p values higher than 0.10 were kept in Table 5 because those e ects should appear in the models.In all cases, the errors could be considered randomly distributed around a zero mean value (i.e., they are uncorrelated), which suggests a common constant variance.On the basis of this analysis, the regression models obtained were accepted to describe the e ect of contents of raw materials (CC, PO, and VA) on HD, TS, EB, and RS, and the nal results are (1).ese equations are all referred to the weight fractions of the components calcium carbonate (X 1 ), para nic oil (X 2 ), and vulcanising accelerator (X 3 ), so that mixing of raw materials can easily be carried out.HD  (1)

Experimental Validation of the Models.
ree extra mixtures, F 1 , F 2 , and F 3 (check-point mixtures), were used to validate the calculated statistical models (the mixtures and their test pieces were prepared following the same procedure as before).Table 6 presents the compositions of those three mixtures and the corresponding measured and predicted values for HD, TS, EB, and RS.It can be seen that the estimates calculated using (1) can be higher or lower than, but are always very close to, the corresponding experimental value (low error), which validates the calculated models.

Cost and Property Optimisation Using PSO Algorithm.
Following the same procedure and reasoning described above for the compounds mechanical properties, a valid and signi cant regression model was also obtained for the cost of mixtures (CT), which can be described by the following equation: CT 2.68 − 0.66X 1 − 0.26X 2 + 0.28X 3 . (2)

Advances in Materials Science and Engineering
In mathematical language, the optimisation problem consists in minimising this objective function (CT) with respect to the design variables X 1 , X 2 , and X 3 , subjected to the nonlinear inequality constraints posed on HD, TS, EB, and RS, as presented in Table 7. ese property ranges (optimisation goals) were chosen having in mind the standard property speci cation range commonly required for the heat-and air-resistant products used as reference.From an industrial competitiveness point of view, property values outside (above) that range are not so interesting, as they certainly imply extra cost.
e result of the PSO algorithm procedure (minimum costs) is returned as a composition range (weight fraction) for each of the raw materials CC, PO, and VA.ese ranges are also presented in Table 7.
An endless number of mixture composition points can be found which meet the requirements of the properties at low cost values.Alternatively a 2D graphical visualisation can be used, by keeping constant one of the factors (feasibility curve).In the present case, the CC content was chosen as the base reference value, and contents of all the factors were expressed as weight fraction relative to that CC content.
us, feasibility curves can be obtained as VA versus PO for various constant CC contents, as shown in Figure 1.
Figure 1 clearly shows that the function describing the in nite number of mixtures with properties within the speci ed ranges is complex and nonlinear.For a constant CC weight fraction equal to 1.00, optimised costs vary from 1.82 to 1.87 US$/kg, and the PO weight fraction can vary between 0.60 and 0.78, whereas the VA weight fraction has a much narrower range, varying from 0.008 to 0.009.Similarly, when the CC level is kept constant at 0.50, the composite costs vary between 2.17 and 2.23 US$/kg.e optimum VA weight fraction varies nearly in the same range, but the optimum PO weight fractions have a different range, in this case from 0.47 to 0.68.Calcium carbonate at lower levels has a more complex e ect on the optimisation and results in higher costs, ranging from 2.53 to 2.58 US$/kg.Although the PO weight fraction range is narrower (0.40 to 0.60), the VA weight fraction has a broader range, from 0.008 to 0.017.
Table 8 presents the raw materials' weight fractions and property-predicted values for three illustrative mixtures within the optimum range and shows the corresponding costs, ranging from 1.92 to 2.37 US$/kg.From those weight fractions' compositions, the formulations of the corresponding optimised EPDM rubber composites can now be calculated back from the reference CC content (125 phr).e full formulations corresponding to the mixtures in Table 8, meeting the

Conclusions
is study showed that fractional factorial design of experiments in two replicates based on three mixture ingredients generally used on EPDM rubber composites and a particle swarm optimisation (PSO) algorithm are promising methods to design composition variables.
For the chosen key raw materials (calcium carbonate, para nic oil, and CBS vulcanising accelerator) and the processing conditions under consideration, the optimisation results readily show that there is an in nite number   In this way, this investigation showed that the speci ed characteristics of the desired product can be subjected to restrictions typical of the manufacture process, and a broad range of compositions can still be selected so that the nal product has minimum cost and can be comfortably manufactured.

Figure 1 :
Figure 1: Feasibility curves relating VA and PO weight fractions for optimised cost of composites with properties within the speci ed ranges, when the CC weight fraction is kept constant at 1.0, 0.5, and 0.0.

Table 1 :
Base composition of industrial EPDM rubber composites.
[24]ances in Materials Science and Engineering D 15 Standard[24].e sheets obtained were conditioned at 25 ± 2 °C for 24 h in a sealed container before the estimation of optimum curing time.Batches were then compressionmoulded to a 90% cure using an electrical resistance heated hydraulic press (model EMIC) at ∼10 MPa and 160 °C during 10 minutes.e hardness (HD) test was carried out according to ASTM D 676 Standard 91.45 − 42.13X 2 + 252.07X 3 + 10.64X 1 X 2 2 ,

Table 3 :
Mixture compositions (weight fractions of chosen factors only) in the 3 3−1 fractional factorial design.

Table 2 :
Factors and levels (weight fraction) adopted for the 3 3−1 fractional factorial design used to de ne EPDM rubber composites.

Table 4 :
Measured values of hardness (HD), tensile strength (TS), elongation at break (EB), rebound resilience (RS), and material costs (CT) for the nine designed mixtures in two replications.
* Value could not be measured.

Table 7 :
Constraint requirements posed on mechanical properties (HD, TS, EB, and RS) and raw materials' (CC, PO, and VA) weight fraction ranges that minimise costs of the composites, as returned by the PSO algorithm.

Table 8 :
Examples of optimised mixture compositions and predicted values of the corresponding properties.

Table 6 :
Composition (weight fraction) of the check-point mixtures F 1 , F 2 , and F 3 and corresponding measured (M) and predicted (P) values of hardness (HD), tensile strength (TS), elongation at break (EB), and rebound resilience (RS).Advances in Materials Science and Engineering of compositions that meet speci ed property values with minimum costs.For example, within the base composition range investigated, mixture costs as low as 1.92 US$/kg can be achieved with the use of 107 phr of calcium carbonate, 95 phr of para nic oil, and 1.13 phr of CBS vulcanising accelerator.e predicted values for the corresponding compound properties are 66.8 Shore A for hardness, tensile strength of 7.8 MPa, 570.8% elongation at break, and 23.0% rebound resilience.

Table 9 :
Examples of optimised EPDM rubber composites with costs between 1.92 and 2.37 US$/kg.
6 Per hundred of rubber, by weight.6Advances in Materials Science and Engineering