Prediction of mechanical properties of composites with recycled particles

Mechanics of Composite Materials, Vol. 30, No. 6, 1994 PREDICTION OF MECHANICAL PROPERTIES OF COMPOSITES WITH RECYCLED PARTICLES R. Rikards,* K. Goracy,** A. K. Bledzki,** and A. Chate* INTRODUCTION Every year more than 1 million tons of glass fiber reinforced plastics are produced in Europe. SMC (sheet moulding compound) and BMC (bulk moulding compound) constitute the largest group of reinforced plastics. Typical SMC material consists of 25 % thermosetting resin and 75 % fillers (calcium carbonate and glass fibers). Due to the evident advantages in processing and design engineering, SMC have found many applications, first of all in automotive and electrotechnical industries. The wide application of the SMC raises the problem of recycling. The mechanical, chemical, and thermal stability of SMC, which is an evident advantage in many applications, is linked with the necessity of recycling. Cured thermoset materials due to chemical cross-linking cannot be melted down and remoulded, as can be done with thermoplastics. However, some technical approaches to recycling of thermosets have been proposed. Unfortunately, most of them deal merely with the organic material, which represents only 25 % of the product. The simplest recycling technique is energy recovery, incineration alone or together with another waste or fuel, along with the possibility to recover heat energy from the organic material. The second technique is a feed stock recycling, a kind of plastics recovery technique, with a breakdown of the constituent molecules of the basic raw materials. For the SMC, only the technique of pyrolysis was more or less exactly investigated. Pyrolysis means destructive distillation of the organic resin by heating in vacuum, with the recovery of the organic raw material. The relatively low content of organic material and the high amount of the solid by-product or ash remaining from the pyrolysis or incineration make these techniques inadvantageous for the SMC. Another technique is mechanical recycling. For the SMC, it means mechanically breaking into finer particles of cured material with direct reuse in a new composite of the recycled particles which contain glass fibers, filler, and ground cured resin. This kind of recycling, which uses recycled materials from the old SMC as valuable raw materials, appears to have considerable prospects. This recycling technique, when recycled materials may replace a part of the reinforcing fractions in a new thermosetting moulding material, is called particle recycling. In particle recycling, the recycled material is not just designed to replace the filler; it also has to take over reinforcing functions. So, it is essential that in the process of decomposition the glass fibers are damaged as little as possible. From the technical point of view, there are no problems with preparation of the waste thermoset parts to further reuse. Its seems that the problem of reusing the recycled particles in the new material could be solved. But it should be noted that all these materials with recycled particles used today in industry include only 10-20 % of the weight fraction of the SMC scraps. It is not enough to close the recycling loop for the used parts of automotive and electrotechnical industries. There are two reasons why only 10-20% of the scraps are used in the new materials. The first reason is that there are some technological problems and the second one, that there have not been systematic investigations of new materials with more than 20% of the recycled particles. It is necessary to perform investigations of materials which include a maximum amount of recycled particles. For such materials with a high volume fraction of recycled particles, physical and mechanical properties *Riga Technical University, Institute of Computer Analysis of Structures, Kalku St. 1, Riga, Latvia LV-1658. **Universit~it Kassel, Institut fiir Werkstofftechnik, M6nchebergstr. 3, 34109 Kassel, Germany. Published in Mekhanika Kompozitnykh Materialov, Vol. 30, No. 6, pp. 781-796, November-December, 1994. Original article submitted January 19, 1995. 0191-5665/94/3006-0563512.50 9 Plenum Publishing Corporation 563 must be determined. Moreover, using recycled particles in the resin transfer moulding (RTM) technology, composites with predicted mechanical properties can be designed. Investigations for using recycled particles as reinforcement have been performed by Bledzki and Goracy [1] and Bledzki et al. [2]. It was demonstrated that in the new composites over 50% of the weight fraction of the recycled particles can be used as a reinforcement. Adding a new resin and some amount of a new glass fiber reinforcement it is possible to ensure sufficiently good mechanical properties of the new composite with recycled particles. To design the new composites with predicted properties, methods of optimization must be applied. Data for the properties of the material, such as stiffness, static and dynamic strength, damping and impact properties, fatigue properties, etc., mostly are obtained from the experiment. Control (design) parameters in the optimum design can be volume fractions of the different particles. Models for the different mechanical properties, as functions of design variables, can be worked out both theoretically and experimentally. There are many investigations in the field of micromechanics of composites. Theoretical models for the effective stiffness of the composites have been much studied. Reviews of these models can be found, for example, in the textbooks of Christensen [3[ and Banichuk et al. [4], in articles by Hashin [5], Ahmed and Jones [6], and McCullough [7]. Mainly two-phase composites were investigated. Some investigations were performed to evaluate the effective stiffness of three-phase composites (see, for example, McCullough [7]). Composites with recycled particles are multi-phase composites. A theory to evaluate effective stiffness of such composites is similar to the well-known theories of the two-phase or three-phase composites. In this case, it is possible to find the effective stiffness of the composite as a function of the control (design) parameters -- voIume fractions of the recycled particles -- from the theory. Damping properties associated with the viscoelastic behavior of the fractions of the composite have also been studied. In most cases, damping models of the composites are based either on the theory of complex moduli (see, for example, Hashin [8]), or the energy approach by Ungar and Kerwin [9]. Gibson [10], Hwang and Gibson [11], Adams et al. [I2], Saravanos and Chamis [13], Gibson [14], and many others modelled the damping of various composites. Using these models the damping properties of the composite also can be obtained as functions of the control parameters -- volume fractions of the particles. However, the reliability of these models is much worse than the reliability of the models for the composite effective elastic stiffness. Strength, damping, fracture, and also stiffness properties of the composite can be modeled using data of experiments. From these data, mathematical models as functions of control (design) parameters can be derived. For this purpose, an approach based on planning of experiments was developed (see Rikards et al. [15] and Rikards [16]). In this method of optimum design, the problem solution is divided into the following stages: choice of control parameters (design variables) and establishment of the domain of search, formulation of plans of experiments for the chosen number of reference points, realization of the experiments in the reference points (physical experiment or theoretical calculation), formulation of mathematical models from the experimental data, design of composite on the basis of the accepted mathematical models, and, as a final verification, experiments at the point of the optimum solution. This approach has been used in the present investigation of mechanical properties of the composite with recycled particles. From the experiments, Young's modulus in the tension and flexure, static strength in tension and flexure, and impact properties of the composite with different volume fractions of the recycled particles are determined. Using both data of experiments and theoretical calculations, a mathematical model, as a function of the volume fraction of recycled particles, is formulated for the elastic stiffness of the composite. An example of minimum weight design of the composite with recycled particles subjected to stiffness constraints is presented. EXPERIMENT In the present investigation, recycled particles from the ERCOM company are used (see Ref. [17]) as the reinforcement and filler. ERCOM produces recycled particles from the old SMC material in six fractions; among them are three powder (marked as P1, P2, P3) and three fibrous (marked as F1, F2, F3) fractions. In Fig. 1, the weight fractions, and in Table 1, the properties of glass, resin, and filler fractions of the ERCOM recycled materials are presented. It can be seen that for all recycled materials the weight fractions of the glass, resin, and filler particles are approximately equal to each other. It should be noted that all recycled materials contain particles of different size (see Table 1, and also the paper by Ehnert [18]). 564 TABLE 1. Properties of the ERCOM Recycled Materials Property I PI I .. P2 P3 1 F1 ! F2 1 F3 Grain size [ram] < 0.2 0.2...0.5 > 0.5 < 0.5 0.5...1.25 > 1.25 Fibre length [mm] < 0.25 0.25...0.3 3...15 0.5...3 3...6 6...20 Density [kg/m a] 1800 1900 1900 1700 1700 1800 Bulk density [kg/m a] 650 300 160 600 450 260 60 [ ~ G l a s s f i b r e Ik~Resin ~ C h a | k 5040 30 Z0 10 0 Glass fibre Resin L Chalk 1 Fig. 1. Weight fractions, %, of glass, resin, and filler in the ERCOM recycled materials. The properties of composite materials with recycled particles depend on the properties and geometry of the fractions used. In the design of composite materials with recycled particles, a compromise between as high as possible recycled material content and reasonable mechanical properties must be obtained. It can be seen in Fig. 2 that three basic forms of recycled particles can be recognized. The first fraction is a powder (see Fig. 2a), the second fraction (see Fig. 2b) is an intermediate one which contains fibers and matrix, and the third is the coarse fraction (see Fig. 2c) containing fibers and also matrix in tile form of a small plates. It has been known for some time that with some combinations of the filler and reinforcement, better properties of the composite can be achieved. It was observed that the major part of the improvement is due to a better packing of filler and reinforcement. For an efficient fiber-reinforced composite Milewski [19] and Katz and Milewski [20], using packing principles and an efficient strain transfer technology, suggested a three-component system: 9component A: 25 vol. % fibers, about 10-15 t~m in diameter with the ratio I/d in the range from 30:1 to 50:1 (here, l is the fiber length and d the fiber diameter); * component B: 10 vol. % spherical filler, R < 1 (R is the ratio of the sphere to the fiber diameters); 9component C: 4 vol.% submicro-fibers, with the ratio 1/d from 10:1 to 20:I and the fiber to sphere diameter 20:1. With such a mixture, the best packing can be ensured. This mean that the largest amount of recycled particles in the new material can be used for a composite with reasonable mechanical properties. Since we do not know the diameters of recycled particles and recycled fibers exactly, the best packing similar to components A, B, C must be determined. For this reason, the method of experiment planning is used. One of the objectives in recycling technology is to design material with predicted mechanical properties. A method of optimum design of composite materials based on experiment planning was outlined by Rikards et al. [15] and Rikards [16]. 565 Fig. 2. Shapes of the recycled particles: fine (powder) fraction (a), intermediate (fibers and matrix) fraction (b), and coarse fraction (c). This method is used in the stage of manufacturing the composite with recycled particles, and later, in the stage of modeling its mechanical properties. In the stage of the material manufacturing, the first step is the choice of the control (design) parameters, and establishment of the limits for these parameters. Let us consider a material manufactured with a combination of the fibrous fractions F1, F3, and the powder fraction P1 (see Table 1). The weight fractions of these materials are preliminary design parameters. In the processing of recycled particle (RP) plates, new polyester resin is added. In addition, the RP plate is covered at its top and bottom with one layer of new glass mat with a random orientation of long continuous fibers. The weight fractions of these newly added materials are measured in the experiment after processing the RP plates. Thus, the weight fractions of these newly added materials are not control (design) parameters. For the processing of the plates with recycled particles (RP plates), the resin transfer moulding (RTM) technology was used. As the matrix material, polyester resin Palatal with high viscosity produced by the BASF company was chosen. On the top and bottom of the plate, one layer of a glass mat (0.300 kg/m 2) was added. For the composite with recycled particles as preliminary design parameters, the amount of the fibrous fraction F3 --, Xl, amount of the powder one P1 --, x 2 , and amount of the fibrous fraction F1 --, x 3, all in grams, were used. All RP plates have a constant thickness. Preliminary design parameters x 1, x 2, x 3 are chosen to be in the following limits: 200 -- x I < 410; 50 < x2 -< 190; 20 < xs -< 90. (1) The next step is the formulation of the experiment plan. The plan of experiment is characterized by the plan matrix Bij, where i is the experiment point and j is variable number. In our case, the number of variables (control parameters) is n = 3 and the number of experiments (number of reference points) is chosen k = 15. Plan matrix Bij for this case is as follows (see Rikards [15, 16]): Br= 566 8 5 11 3 12 10 13 15 6 1 9 2 14 4 7 / 15 2 14 13 1 4 7 12 11 6 5 10 8 3 9 I" 6 10 13 11 7 14 9 8 1 12 2 5 3 4 15 (2) T A B L E 2. Weight Fractions, %, of Different Materials, and Density of the RP Plates Recycled materials New materials Plate No. 1 2 3 4 5 6 7 8 9 10 I1 12 13 14 15 Polyester resin Glass 37.7 49.7 50.5 39.9 45.8 40.6 57.7 33.6 36.6 45.5 41.5 37.8 31.4 45.1 45.5 8.2 Glass Polyester resin Chalk 24.0 17.4 19.1 22.0 19.6 22.0 14.5 26.2 24.6 20.2 22.2 24.2 26.4 20.5 20.3 19.4 14.0 15.4 18.0 15.5 17.7 11.6 21.0 19.7 16.5 17.5 19.6 20.9 1.6.4 16.5 10.7 10.4 8.9 11.6 10.7 10.9 7.8 11.9 10.4 9.1 10.7 10.1 13.2 9.1 9.0 (mat} 8.5 6.1 8.5 8.4 8.8 8.4 7.3 8.7 8.8 8.2 8.4 8.2 8.8 8.8 Density Sum of recycled ;,, k g / m a materials 54.1 1500 41.8 1460 43.4 1460" 51.6 1490 45.8 1470 50.6 1480 33.9 1460" 59.1 1520" 54.7 1510 45.8 1470 50.4 1510 53.9 1540 60.5 1560 46.0 1510 45.8 1460 *These values of the composite density were estimated approximately. The values of the design variables in the experiment points are calculated by the formula (see Rikards [15, 16]) xt') = x~j~ + (x~ ax - x~ In) (B,~ - 1)/(k - 1). (3) Here, i = 1, 2 . . . . . k and j = 1, 2 . . . . . n (in our case n = 3 and k = 15). For the processing of RP plates, weight fractions of the recycled materials are calculated by formula (3) with the plan matrix (2) and upper and lower limits of the preliminary design parameters (1). Fifteen different RP plates were manufactured. All plates have the same dimensions 400 • 400 • 4 mm. Weight fractions of the different materials of these plates are presented in Table 2. It can be seen that for the plates, the sum of weight fractions of the recycled particles is mainly greater than 50%. The largest amount of glass content is for plates 8 and 13. It is of interest to compare the structure of the material for these different plates. In Fig. 3, the structure of three different materials is presented. Plate 2 (see Fig. 3a) contains only 41.8% of recycled materials and, therefore, the amount of new polyester resin is the greatest. In this case, the packing of the particles is not an optimum one. Plate 15 (see Fig. 3b) contains an intermediate amount of recycled materials with a great share of the filler (chalk) and short fibers. In this case, the packing of the particles is much better but the weight fraction of the new resin is rather high. Plate 1 (see Fig. 3c) contains one of the greatest amounts of recycled particles. In this case, as can be seen from the photography of the structure, the packing of the particles is the best. In the material, there are no regions with a large amount of new resin. Comparison of the structure of composites with different weight fractions of recycled materials gives some information about the composite. More information can be obtained from the data on mechanical properties of the material. Mechanical properties were obtained in experiments with all 15 RP plates. These properties are: tensile strength Yl [MPa], flexural strength Y2 [MPa], tensile Young's modulus Y3 [GPa], flexure Young's modulus Y4 [GPa], and impact resistance Y5 [kJ/m2] 9The results of experiments and the values of the preliminary design parameters x 1, x 2, x 3 (in grams) in the reference points obtained according to formula (3) are presented in Table 3. In the present paper, mainly the stiffness of the composite with recycled particles has been investigated. Other mechanical properties (strength, impact resistance) are given for comparison. So, in Table 3, it can be seen that there is a correlation between the material stiffness and strength. A comprehensive analysis of the strength, impact, and damping properties of the composite with recycled particles will be presented in the next publication. From the experimental results presented in Table 3, it can be seen that all mechanical properties are functions of the preliminary design parameters Yi(Xl, x 2, x3). It should be noted that these design parameters (weight fractions of the recycled materials F3, P1 and F1) are used only in the first stage of the design as preliminary design variables to obtain different materials. Later on, it will be shown that in the stage of modeling and design of elastic properties, it is better to use other design parameters - - volume fractions of the recycled particles. 567 Fig. 3. Structures of different composites with recycled particles: plate 2 (a), plate 15 (b), and plate 1 (c). EFFECTIVE STIFFNESS OF T H E COMPOSITE W I T H RECYCLED PARTICLES There are many investigations for a micromechanical analysis of the effective stiffness of two-phase composites with various reinforcement (see, for example, Ahmed and Jones [6], Hashin [5], Gibson [14]). Also, models for three-phase composites were proposed (see McCullough [7]). All these models were applied to the composites with fractions of the new materials. Similar models can be applied also to composites with recycled particles. To the composite with recycled particles discussed above, the three-phase model is applied. It is assumed that the first phase is all glass fibers of different length, which contains partly the recycled fractions of glass fibers and partly the new glass mat. The second phase is polyester resin which contains partly the recycled particles of polyester (old) and partly the new polyester added during the processing of RP plates. The third phase is the filler (chalk). This three-phase model is presented in Fig. 4. This model of the matrix-filler-fibers system was outlined by McCullough [7]. The properties of glass fibers (El, pf), filler (Ep, pp), and resin (E r, Pr) are given. Input data are also the volume fractions of fibers c 1 = cf, filler c 3 = cp, and resin c2 = c r. The volume fraction of the voids is denoted by %. First, properties of the modified matrix (see Fig. 4) are calculated using apparent volume fractions c' r and c p of the resin and filler. These apparent volume fractions are related to the actual volume fractions cr and Cp via the relationships (see McCullough [7]) r Cp C'p - - 568 C r -}- Cp Cr ; C: r -- C r -4- Cp ~ T A B L E 3. Preliminary Design Parameters x 1, x 2, x 3, x 4 and Results of Experiments in the Reference Points 1 2 305 260 190 60 45 65 34.3 32.4 101 99 7.12 6.66 7.70 7.03 1518 15.2 3 4 5 6 7 8 9 10 11 12 13 14 15 350 230 365 335 380 410 275 200 320 215 395 245 290 180 170 50 80 110 160 150 100 90 140 120 70 130 80 70 50 85 60 55 20 75 25 ,40 30 35 90 27.8 31.8 39.3 34.7 28.1 31.6 38.2 33.5 37.0 39.5 42.0 37.8 36.3 79 102 103 121 85 107 110 107 111 114 130 120 113 6.05 6.43 7.05 6.37 5.42 5.38 6.93 6.54 6.84 7.04 7.48 6.68 6.92 6.75 6.74 6.87 7.97 6.81 6.98 7.04 7.34 7.64 8.10 8.77 8.03 7.22 17.3 15.2 15.0 15.5 17.0 14.6 14.7 16.1 17.7 14.5 17.1 17.6 15.2 + Le~o1 7o 6 Matrix Filter Modified Matrix = Modified Matrix +N:N Short Fibres Three-phase Composite Fig. 4. The three-phase model of the composite with recycled particles. so that c' r + C'p = 1. The effective stiffness E m of the modified matrix has been calculated by Ishai and Cohen's formula (see Ahmed and Jones [6]), E~ = Er {1 + c'j [m/(m- 1) - (c',)1/31}; here, m = Ep/E r. The resulting properties of an isolated resin/filler system are used to represent the behavior of the modified matrix. Further, the system is assumed to be a two-phase composite of fibers embedded in the modified matrix. It is more complicated to evaluate the effect of the fiber fraction of the composite with recycled particles on the effective moduli. In this composite there are fibers with different length. So, in the recycled material P1 fiber length is/f <_ 0.25 ram, in the recycled material F1 fiber length is 0.5 m m < If < 3 mm, while in the recycled material F3 it is 6 m m _< If ___ 20 mm (see Table 1); and, finally, the RP plate from both sides is covered with one layer of glass fiber mat with long fibers. All fibers are randomly oriented in the RP plate plane. It was investigated by Halpin and Pagano [21] (see also Gibson [14]) that there exists an effect of fiber aspect ratio on Young's modulus of the randomly oriented short fiber two-phase composite. It was observed that the effect is larger for boron/epoxy composites and smaller for glass/epoxy composites. It should be noted that there is no information about volume fractions of the fibers of different fiber aspect ratios obtained during the recycling. Only the weight fraction of the new glass mat and the overall weight fraction of the glass fibers obtained by the recycling (see Table 2) were known. To evaluate this complex distribution of the fibers, the Hirsch model has been used for calculation of Young's modulus (see, for example, Ahmed and Jones [6]): E = Z.Z,. + ( 1 - X) ~:,~. (4) 569 TABLE 4. Material Properties of the Fractions of the Composite with Recycled Particles Material I Young's modulus E, GPa Poisson's ratio 70 4 47.8 0.3 0.3 0.323 E-gl~ss Polyester Filler (chalk) [ Density p, k g / m a 2540 1200 2400 T A B L E 5. Comparison o f Experimental and Theoretical Values of the Young's Modulus in Tension for Composites with Recycled Particles Plate E, GPa No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ~,% Exper. 0.19 0.16 0.15 0.19 0.17 0.19 0.13 0.20 0.20 0.17 0.18 0.20 0.22 0.18 0.17 0.72 0.79 0.80 0.73 0.76 0.73 0.84 0.70 0.70 0.76 0.76 0.73 0.69 0.77 0.75 0.06 0.04 0.05 0.06 0.05 0.06 0.03 0.07 0.07 0.05 0.06 0.06 0.07 0.05 0.05 0.03 0.01 0 0.02 0.02 0.02 0 0.03 0.03 0.02 0 0.01 0.02 0 0.03 7.12 6.66 6.05 6.43 7.05 6.55 5.42 5.38 6.93 6.54 6.84 7.04 7.48 6.68 6.92 (12.1%) (7.4%) (5.9%) (3.7%) (5.1%) (5.4%) (7.9%) (20.1%)* (5.1%) (3.7%) (3.5%) (4.5%) (5.5%) (7.2%) (8.2%) 7.12 6.68 6.68 7.13 6.74 7.12 5.96 7.43 7.43 6.74 6.96 7.30 7.73 6.88 6.74 0 +0.3 +9.4 +9.8 -4.6 +8.0 +9.1 +27.6 +6.7 +3.0 +1.7 +3.6 +3.2 +2.9 -2.7 *The standard deviation of modulus for this material is too large; this point will be excluded from further modeling. Here, E v is calculated using Voigt's formula for the two-phase composite (fibers embedded into a modified matrix): E v = c l e f + (1 - el) E,n, and E R is calculated using the Reuss formula for the two-phase composite: EfE~ ER = (1 - c:) E: + cfE~ In formula (4), the empirical parameter X can be evaluated to fit experimental data. To compare the above theory with the experiment data it is necessary to know the properties of the fractions. Material properties of the fractions are presented in Table 4. Using data presented in Tables 2 and 4, volume fractions c i of the constituent materials can be calculated (see Gibson [141): c, = m J 6 o / p , ) . Here, m i and Pi are the weight fraction and density of the i-th constituent, respectively. It should be noted that the sum of c i must be equal to unit. If it is less it means that there are voids in the composite. The void volume fraction c4 is c4 = 1 - 570 (cl + c2 + c3)- TABLE 6. Theoretically Calculated Values of Young's Modulus E of the Composite with Recycled Particles in the Points of Experiments. Variables x 1 and x: (5) are Taken within the Limits (7) Volume fractions of E, GPa fibre powder particles, x 1 particles, 7.12 5.82 8.05 5.88 7.55 5.81 6.35 6.32 7.66 6.95 7.02 6.43 7.51 6.33 6.93 0.17 0.11 0.24 0.10 0.22 0.12 0.14 0.16 0.21 0.20 0.18 0.13 0.23 0.15 0.19 X2 0.09 0.05 0.07 0.075 0.055 0.03 0.06 0.02 0.08 0.025 0.065 0.085 0.035 0.04 0.045 Using the three-phase model outlined above, the effective stiffness of all 15 RP plates has been calculated. Results of this analysis are presented in Table 5. Experimental results for Young's modulus in tension were presented above (see Table 3, material property Y3). It is assumed that in formula (4), the parameter X = 0.15. In Table 5, for the experimental Young's modulus the percentage of the standard deviation also is presented (in brackets). In the last column, the difference in percent between theory and experiment, 5 =--E-E'*P E . 100% is shown. It is seen from the data of Table 5 that experimental and theoretical results are in good agreement. The difference does not exceed 10% except for plate 8. It should be noted that plate 8 contained some processing imperfections. A set of experimental data for the composites with other recycled materials shows that the empirical parameter X is in the limits 0.15 <_ X < 0.18. Hence, a simple three-phase model with simple formulas for calculation of the modified matrix and composite stiffness can be used for theoretical estimation of Young's modulus of the composites with recycled particles. M O D E L I N G AND D E S I G N OF STIFFNESS OF T H E C O M P O S I T E W I T H R E C Y C L E D P A R T I C L E S One of the objectives in recycling technology is to design material with predicted mechanical properties. A method of optimum design of composites based on experiment planning was outlined in [15, 16]. In the first stage of design, weight parts of the recycled materials F3, P1, and F1 in the limits (1) were used as preliminary design parameters. These parameters were selected because using such parameters it was possible to fabricate 15 RP plates. However, for the final design these parameters are not suitable. By this means after verification of experiment and theory, the glass fiber volume fraction c t and the powder (filler) volume fraction c 3 were taken as final parameters, i.e., as the design variables Xl, x2: xl = c l ; x ~ = c3. (5) These parameters can be used as test (or optimization) parameters in the procedure of the composite elastic stiffness design. Plan of experiment (2) with three variables (weight parts of the recycled materials F3, P1, F1) and 15 points was used for the other parameters (weight fractions), not for the volume fractions of the particles. In this case, points of experiment (for the weight fractions of the recycled materials) in the domain of the final design variables x 1 and x 3 are not distributed as regularly as possible. Moreover, points of experiment in some regions of the domain of final design variables are very close 571 to each other. So, it is impossible to determine a mathematical model, for example, of the Young's modulus, as a function of volume fractions of the particles using only data of physical experiment. This is because this experiment was carried out in the reference points of the plan for the weight fractions of the recycled materials. Additional points of experiment should be used. In these additional points, experimental values of the stiffness of the material can be obtained not from physical experiment but can be calculated using the theory outlined above and verified for the 15 reference points. In the present investigation, we are following the principle that data partly are obtained from physical experiment and partly from theoretical modeling. These additional points are calculated using a plan of experiment with two variables and 15 points (n = 2, k = 15). The plan matrix in this case is as follows (see Rikards [15, 16]): B~ 8 = 2 15 1 15 7 11 12 13 3 5 7 12 11 9 8 10 14 4 3 9 1 13 2 4 14 6 10 t 5 6 " (6) The domain of search for the design variables x 1 = cl; x 2 = c 3 lies within the following limits: 0.10 <_ xl -< 0.24; 0.02 < x2 --< 0.09. (7) The values of the design variables in the points of experiment are calculated using formula (3) with n = 2 and k = 15, In Table 6, results of calculations using a three-phase model are presented. In the present analysis, the same properties of the composite particles (see Table 4) as in the previous section were used. In theoretical calculations, it is assumed that the composite is without voids (c4 = 0) and that the volume fraction of the polyester resin is c~= 1 - ( c 1 + c 8 ) . Using these data (Table 6) and the program RESINT (see Rickards [15, 16]) the mathematical model for the Young's modulus of the composite with recycled particles was obtained (coefficient of correlation c = 96.7%, standard deviation a = 0.024 GPa): E = 3.62 + 15.58xI + 9.46x2. (8) It is understandable that from the theoretical model, with an experiment plan (6) using the program RESINT, a very simple formula for Young's modulus of the composite was obtained. The correlation coefficient in this case is also rather high. The second mathematical model was obtained using both experimental (see Table 5) and theoretical (see Table 6) data. It should be noted that from the experiment (see Table 5) only 14 points are used, to formulate the mathematical model of the composite. Plate 8 was excluded. This means that in this case, 14 points from the physical experiment and 15 points from the theoretical calculations were used for the modeling. Using these ~lata and the program RESINT, the mathematical model for the Young's modulus of the composite given by (c = 60.4%, a = 0.243) theformula E = 3.68 + 14.94xl + 8.55x2. (9) From a comparison of formulas (8) and (9), it can be seen that there are only small differences in the models for the Young's modulus of the composite. In the design of composite with recycled particles, formula (9) can be used, because this formula represents both theoretical and experimental data. The reliability of such a model is higher than that of a model obtained by theoretical modeling only. Let us consider an example of minimum weight design of the composite with recycled particles subjected to the stiffness constraint. Design variables are fiber and filler volume fractions x = (x 1, x2). As the objective function F(x) for the minimum weight design, the density of the composite can be used. Constraints are the limits of modeling (7), and the stiffness constraint. The minimum weight design problem is as follows: minimize F(x) ~ p f x l + ppx2 + Pr [1 - (xi + x~)] 572 (10) ..p,kgtm3 ~ x~ 0.3 1600 lSO0 ,,ool 9 - - - - 0.2 I 1300I 5 ir I, I : G '~---- I~-- o, ' 7 E)GPa 8 Fig. 5. Minimum weight design: density of the composite as a function of the predicted Young's modulus. subjected to the constraints 0.10 < xl -< 0.24; 0.02 _< x2 -< 0.09; E(x) = 3.68 + 14.94x~ + 8.55x2 --- E.. (11) Here, E, is the predicted Young's modulus of the composite. As a model of the composite stiffness expression, (9) is used. In the design, material properties of the fractions presented in Table 5 are taken as initial data. Optimization problem (10), (1 l) was solved using the program SUPEX (see Rikards et al. [15, 16]). For the predicted Young's modulus E, = 7 GPa, solution of the optimum design problem is: Xl* = 0.21; x2* = 0.02; F(x ~) = P = 1506 kg/m3; E(x*) = 7 GPa. Results of optimization with different Young's moduli are presented in Fig. 5. It can be seen that for the composite with higher predicted stiffness, the density increases. If there are restrictions on the composite weight, stiffness is also limited. For example, if the composite density limit o = ~500 kg/m 3 is not exceeded (see Fig. 5), it is impossible to design a composite with the stiffness larger than E, = 7 GPa. DISCUSSION AND CONCLUSION Experimental and theoretical modeling of a composite with recycled particles has been presented. For the design of mechanical properties of the composite, an approach based on planning of experiments was used. The objective of this method is to obtain simple models for the material behavior as functions of design variables. Selection of design variables is the most important stage of the design, In the case of a composite with recycled particles, weight parts of recycled materials were used as preliminary design variables. Experiments were carried out according to the plan of experiment. Static stiffness, strength, and impact properties of the composite have been determined. Young's modulus of the composite with recycled particles was investigated in more detail. From comparison of experimental and theoretical results, it was found that a three-phase model gives good results for the estimation of the composite stiffness. Volume ratios of the fractions of the composite were used as the final design variables. For these design variables, a new plan of experiment was outlined. In the reference points of this new experiment plan, the composite stiffness using a threephase model was calculated additionally. Using data of the physical experiment and theoretical calculations, a simple mathematical model for the composite stiffness was determined. Using this simple model, the minimum weight design of a composite with predicted Young's modulus was performed to demonstrate the possibilities of the method of experiment planning. Using the method of experiment planning, materials with more complex behavior can be modeled and designed, for example, composite fatigue behavior, delamination crack growth in the sandwich and laminated composites, etc. can be taken into account. Finally, it should be noted that simple models obtained by modeling based on experiment planning can be used only within the limits of the design variables established at the initial stage of modeling. 573 Acknowledgments. This ~vork was supported by the Commission of European Union through Contract ERB CIPA CT 930110 "Recycling Technologies of Fiber-Reinforced Thermoset Plastics." REFERENCES . 2. 3. 4. 5. . 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 574 A. K. Bledzki and K. Goracy, "The use of recycled fiber composites as reinforcement for thermosets," Mech. Composite Materials, 29, No. 4, 473477 (1993). A. K. Bledzki, K. Kurek, and C. Barth, "Development of a thermoset part with SMC reclaim," Proc. 50th Annual Techn. Conf. ANTEC, Society of Plastics Engineers, Detroit, 3-7, May, 1991 (1992), pp. 1558-1559. R. M. Christensen, Mechanics of Composite Materials, John Wiley and Sons, New York (1979). N. Banichuk, V. Kobelev, and R. Rikards, Optimization of Structures of Composite Materials [in Russian], Mashinostroenie Publishing House, Moscow (1988). Z. Hashin, "Analysis of composite materials," J. Appl. Mech., 50, September, 481-505 (1993). S. Ahmed and F. F. R. Jones, "A review of particulate reinforcement theories for polymer composites, J. Material Sci., 25, 49334942 (1990). R. L. McCullough, "Micro-models for composite materials- particulate and discontinuous fiber composites," in: Delaware Composites Design Encyclopaedia, Vol. 2, Technomic Publishing Co., Lancaster-Basel (1990), pp. 93-142. Z. Hashin, "Complex moduli of viscoelastic composite. II. Fiber reinforced materials," Intern. J. Solids Struct., 6, No. 6, 797-807 (1970). E. E. Ungar and E. M. Kerwin, Jr., "Loss factors of viscoelastic systems in terms of energy concepts," J. Acoust. Soc. Am., 34, No. 7, 954-958 (1962). R. F. Gibson, "Damping characteristics of composite materials and structures," Proc. 6th Annual ASM/ESD Advanced Composites Conf., Detroit, Michigan, October 8-11 (1990), Vol. 11, pp. 441-449. S. J. Hwang and R. F. Gibson, "Micromechanical modeling of damping in discontinuous fiber composites using a strain energy, finite element approach," J. Eng. Materials Technol., 109, No. 1, 47-52 (1987). R. D. Adams, M. A. O. Fox, R. J. L. Flood, R. J. Friend, and R. L. Hewitt, "The dynamic properties of unidirectional carbon and glass fiber-reinforced plastics in torsion and flexure," J. Composite Materials, No. 3,594603 (1969). D. A. Saravanos and C. C. Chamis, "Unified micromechanics of damping for unidirectional and off-axis fiber composites," J. Composite Technol. Res., 12, No. 1, 3140 (1990). R. F. Gibson, Principles of Composite Material Mechanics, McGraw-Hill, Singapore (1994). R. Rikards, A. K. Bledzki, V. Eglais, A. Chate, and K. Kurek, "Elaboration of optimal design models for composite materials from data of experiments," Mech. Composite Materials, 28, No. 3, 435-445 (1992). R. Rikards, "Elaboration of optimal design models for objects from dat:~ of experiments," in: Optimal Design with Advanced Materials, P. Pedersen (ed.), Elsevier Science Publishers, Amsterdam (1993). "ERCOM closes the recycling loop on SMCs," World Plastics and Rubber Technology, Cornhill Publications Ltd., Hong Kong (1994). G. P. Ehnert, Das Recycling von SMC -- heute und morgen, Proc. 25 AVK-Tagung. Berlin, November, 1993 (1993), P7-3-P7-8. J. W. Milewski, "Whiskers and Short Fiber Technology," Polymer Composites, 13, No. 3, 223-236 (1992). H. S. Katz and J. W. Milewski, Handbook of Fillers and Reinforcement for Plastics, J. W. Milewski (ed.), Van Nostrand Reinhold, New York (1978). J. C. Halpin and N. J. Pagano, "The laminate approximation for randomly oriented fibrous composites," J. Composite Materials, 3, 720-724 (1969).