Minnesota State University, Mankato Cornerstone: A Collection of Scholarly and Creative Works for Minnesota State University, Mankato All Theses, Dissertations, and Other Capstone Projects Theses, Dissertations, and Other Capstone Projects 2017 Size Effect of Clay Filler Particles on Mechanical Properties of Pultruded Polymer Composites Under Shear Loading Jaysimha Reddy Sonti Reddy Minnesota State University, Mankato Follow this and additional works at: https://cornerstone.lib.mnsu.edu/etds Part of the Mechanical Engineering Commons Recommended Citation Sonti Reddy, Jaysimha Reddy, "Size Effect of Clay Filler Particles on Mechanical Properties of Pultruded Polymer Composites Under Shear Loading" (2017). All Theses, Dissertations, and Other Capstone Projects. 755. https://cornerstone.lib.mnsu.edu/etds/755 This Thesis is brought to you for free and open access by the Theses, Dissertations, and Other Capstone Projects at Cornerstone: A Collection of Scholarly and Creative Works for Minnesota State University, Mankato. It has been accepted for inclusion in All Theses, Dissertations, and Other Capstone Projects by an authorized administrator of Cornerstone: A Collection of Scholarly and Creative Works for Minnesota State University, Mankato.
SIZE EFFECT OF CLAY FILLER PARTICLES ON MECHANICAL PROPERTIES OF PULTRUDED POLYMER COMPOSITES UNDER SHEAR LOADING A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering By Jaysimha Reddy Sonti Reddy MINNESOTA STATE UNIVERSITY, MANKATO MANKATO,MINNESOTA December 2017
The thesis paper has been examined and approved on 17/11/2017. Examining Committee: Dr. Jin Y. Park, Professor Date Dr. Shaobiao Cai, P.E. - Assistant Professor Date Dr. Farhad Reza, P.E. - Professor Date Minnesota State University, Mankato i
ABSTRACT Polymer composites are finding a range of applications across different fields of engineering. Hence, there is a need to understand the properties of these polymer composites. Mechanical testing can be done to determine the properties but firstly the composite must be manufactured consuming time and money. However, if the properties gained differ from those expected from composite the obtained data is rejected, and a new composite must be manufactured consuming more time and effort. The micromechanical models play an important role in estimating the properties of the polymer composites. These models estimate the properties of the composite by estimating the properties of the constituents that build up the composite. However, these models fail to explain the effect of particle size on to the mechanical properties of the overall composites. The study reviews different micromechanical theories and focuses on the development of an analytical model that can accurately predict the effect of clay particle size on to the mechanical properties of pultruded polymer composites with clay filler particles and the numerical results are compared to experimental data to validate the analytical model. ii
ACKNOWLEDGMENTS I would like to gratefully thank my advisor Dr. Jin Y. Park, for his support, guidance and encouragement during my graduate studies at Minnesota State University, Mankato. Without him, this thesis work not have been possible. I am also thankful to my other committee members, Dr. Shaobiao Cai and Dr. Farhad Reza for all their time and help along the way. I would like to thank Mr. Kevin Schull for providing a great environment for carrying research in the laboratory. Lastly, I am eternally grateful to my parents and friends for their love and support. iii
Table of Contents 1 Introduction... 1 2 Micromechanical Theories... 6 Theoretical Analysis of Matrix Material... 6 Theoretical Analysis of Roving Layers... 9 Theoretical Analysis of CSM layers... 10 Theoretical Analysis of Properties of Composites... 11 Calculation of Volume Fraction of the Constituents of Composites... 12 3 New Micromechanical Model... 15 4 Process and Approaches... 17 Experimental Analysis... 17 Cross-section of the Sample... 19 Volume fractions and Density of the Composite Specimen... 22 5 Results and Discussion... 25 6 Conclusions... 30 Bibliography... 31 iv
List of Figures Figure 1: Schematic of Pultrusion Process [11]... 3 Figure 2: Roving Layer and MAT layers of a pultruded composite sample [4]... 4 Figure 3: A sketch showing the matrix containing spherical inhomogeneties [4]... 4 Figure 4: Composite properties structure... 5 Figure 5: Configuration of the tested specimen (Units: mm)... 18 Figure 6: A virgin specimen with a strain rosette application... 18 Figure 7: A tested specimen showing shear failure... 19 Figure 8: Schematic stacking sequence of FRC-1... 20 Figure 9: Schematic stacking sequence of FRC-2... 20 Figure 10: Schematic stacking sequence of FRC-3... 20 Figure 11: Schematic stacking sequence of FRC-4... 21 Figure 12: Cross section of specimen after burning test (Dimensions 25 X 25 X 6.5 mm).. 23 Figure 13: Post burn-out fiber architecture of pultruded composites... 23 Figure 14: S1212 varying with inclusion radius for FRC-1... 26 Figure 15: S1212 varying with inclusion radius for FRC-2... 26 Figure 16: S1212 Varying with inclusion radius FRC-3... 27 Figure 17: S1212 Varying with inclusion radius FRC-4... 27 Figure 18: Variation in shear modulus for FRC-1... 28 v
Figure 19: Variation in shear modulus for FRC-2... 28 Figure 20: Variation in shear modulus for FRC-3... 29 Figure 21: Variation in shear modulus for FRC-4... 29 vi
List of Tables Table 1: Experimental Properties... 19 Table 2: Properties of Constituents... 21 Table 3: Density of Specimens... 22 Table 4: Average Volume Fractions of the Constituents... 24 Table 5: Average Volume fractions of CSM and Roving Layers... 24 Table 6: Comparison of Experimental and Theoretical Shear Modulus... 25 vii
NOMENCLATURE G G r G filler K K r K filler υ filler/m υ void/m υ r/m E m E f E 11/Roving E 22/Roving υ m/roving Shear modulus of matrix Shear modulus of resin Shear modulus of filler particle Bulk modulus of matrix Bulk modulus of resin Bulk modulus of filler particle Volume fraction of filler in matrix Volume fraction of void in matrix Volume fraction of resin in the matrix Young s modulus of matrix Young s modulus of fibers Longitudinal modulus of roving fibers Transverse modulus of roving fibers Volume fraction of matrix in roving υ f/roving Volume fraction of fibers in roving G 12/Roving V 12/Roving E Mat G Mat Shear modulus of roving layer Poisson s ratio of roving layers Young s modulus of CSM mat Shear modulus of CSM mat viii
υ f/mat Volume fraction of fiber in CSM layer υ m/mat L a Volume fraction of matrix in CSM layer Characteristic length of filler particle Radius of the filler particle ix
x
1. Introduction Polymer composites that have complex microstructures are finding important applications in many of the engineering designs and products, ranging from aerospace, automobile to building of structures. These complex microstructures of the composites can be modified and adjusted to obtain desired mechanical properties. Hence, many types of research have been carried out over the past decades to better understand the properties of these microstructures to establish a quantitative relation between the parameters of these microstructures and the mechanical properties of the overall composites. Two such important parameters that needed to be studied are particles and voids that are present in the matrix of polymer matrix composite. Several researchers have proposed analytical models to evaluate the mechanical properties of an isotropic medium consisting of micro-sized inclusions as inhomogeneities, among them Eshelby model, Selfconsistent model and Mori-Tanaka model [1, 2, 3] are widely used to predict the properties. Analytical models were developed to show the effect of filler and void content on the mechanical properties of pultruded polymer composites and recommended to consider the contents of filler and void to estimate the properties of composites [4]. A Platelet-shaped filler model was developed which is the modified version of Mori-Tanaka model to determine the shape effect of the particle on composites [5]. These methods can predict the properties of composite materials without considering the size effect. Some of the experiments conducted on composites that have different sizes of clay particles show variation in the measured mechanical properties of the composites [6, 7]. Also, an analytical model was developed which is a modified version of Secant Moduli method that captures the particle size dependence on the overall plasticity of particulate composites [8]. The study also showed that the size dependence is 1
more observed when the particle size reaches the characteristic length and for large sizes of particles [9]. General description of Polymer micro/nanocomposites can be found in [10]. These composites typically consist of many layers of fiber strands. These fibers are arranged and bonded in a matrix to form a desirable strong polymer composite. The main objectives of this study are to present an Analytical model that can accurately estimate the size dependence of clay filler particles on mechanical properties of the polymer composites under shear loading, and compare the estimated analytical results with the experimental results. 2
Pultruded Polymer Composites Pultruded polymer composites are produced by the process called pultrusion Figure 1. These pultruded composites usually consist of fiber reinforcement, polymer resin, filler particles, small amounts of catalyst and anti-ultraviolet additives and some unwanted voids. The composite can be divided into stacking sequence of two layers. The roving layer which consists of roving fibers and matrix and the second layer is known as CSM layer which is the combination of continuous strand mat in resin matrix these layers are stacked one above the other in sequence as shown in Figure 2. Generally, pultrusion devices pull continuous roving fiber and continuous mats fiber through a resin bath, a performing fixture, a heated cure die, to obtain a final composite product. Figure 1: Schematic of Pultrusion Process [11] 3
Figure 2: Roving Layer and MAT layers of a pultruded composite sample [4] The matrix further consists of resin, filler, voids, a small amount of catalyst, anti-ultraviolet additives and pigments. The filler is added to improve the resin properties such as shrinkage reduction, surface hardness, load transfer and cost reduction. The typical matrix structure is shown in Figure 3. Figure 3: A sketch showing the matrix containing spherical inhomogeneties [4] 4
Figure 4: Composite properties structure The fillers are added to the matrix, it has been shown that dramatic improvements in mechanical properties can be achieved by adding a few percentages of clay particles in the matrix material. Polymer composites containing particles with small aspect ratio have also been studied because of their technological and scientific importance. Many studies have been conducted on the mechanical properties of these particulate filled polymer composites and found that modulus and stiffness can be readily increased by adding micro or nano-sized particle. However, the strength directly depends on the stress transfer between particles and the matrix. So for well-bonded particles, the stress can be effectively transferred to the particles which clearly improves the strength [11-13] also in case of poorly bonded particles the strength may reduce due to the addition of particles [12]. 5
2. Micromechanical Theories There are several methods used to determine the properties of the polymer composites without considering the size effect of the particles, one such method used in this study is a micromechanical analysis of composite materials in which the properties are determined by looking at the properties of individual constituents that make the composite. These methods include the Eshelby model, Mori-Tanaka model and the Self-Consistent model and some of the modified versions of these models which include void and filler contents. Theoretical Analysis of Matrix Material Modified Eshelby Model This model considers resin, filler and voids whereg, G r and G filler are shear moduli and K, K r and K filler are bulk moduli of composite, resin matrix, filler respectively, ν filler/m and ν void/m are volume fraction of filler and void [4]. G = G r {1 υ filler/m(g filler G r ) G r + 2S 1212 (G filler G r ) υ 1 void/m( G r ) (1) G r + 2S 1212 ( G r ) } K = K r {1 1 υ filler/m (K filler K r ) K r + (1/3)S kkll (K filler K r ) υ void/m ( K r ) (2) K r + (1/3)S kkll ( K r ) } S 1212 = ( 3K r + 6G r 15K r + 20G r ) 9K r S kkll = ( ) 3K r + 4G r (3) (4) 6
Modified Mori-Tanaka Model: This model was developed by the researchers T. Mori and T. Tanaka and was based upon Eshelby model. This model is used to relate average stress for inhomogeneities to average stress of matrix where the inhomogeneities are assumed to be randomly oriented and evenly distributed. The modified form of Mori-Tanaka was given in [4] and can be written as fallows. G = G r {1 + υ filler/m (G filler G r ) G r + 2S 1212 ν r/m (G filler G r ) + υ void/m ( G r ) G r + 2S 1212 ν r/m ( G r ) } (5) K = K r {1 + υ filler/m (K filler K r ) K r + (1/3)S kkll υ r/m (K filler K r ) + υ void/m ( K r ) K r + (1/3)S kkll υ r/m ( K r ) } (6) S 1212 = ( 3K + 6G ) 15K + 20G (7) 9K S kkll = ( ) 3K + 4G (8) Where υ r/m is volume fraction of resin in the matrix. Self-Consistent Model The self-consistent model was developed and used for conditions where the volume fraction of the inhomogeneities in the homogenous material increases. This is caused by the addition of more inhomogeneities into the material due to which the interaction between these inhomogeneities 7
plays a more prominent role with larger inhomogeneity volume fraction. The general forms of selfconsistent model are modified [4] and can be written as follows. G = G r + υ filler/m(g filler G r )G G + 2S 1212 (G filler G ) + υ void/m( G r )G G + 2S 1212 ( G ) (9) K = K r + υ filler/m(k filler K r )K K + (1/3)S kkll (K filler K ) + υ void/m( K r )K K + (1/3)S kkll ( K ) (10) S 1212 = ( 3K + 6G ) 15K + 20G (11) 9K S kkll = ( ) 3K + 4G (12) Once we determine the properties of the matrix material we can compute G compite, E composite and V Composite can be found. 8
Theoretical Analysis of Roving Layers The roving layers are made up of unidirectional continuous fibers. These fibers are bonded together in a matrix to form the roving layers. The mechanical properties of the roving layers can be obtained by using one of the micromechanical models. In this study, Chamis models were used to determine the transverse modulus of roving layers. The longitudinal modulus of roving layers can be obtained as follows. E 11/Roving = E f ν f/roving + E m ν m/roving (13) The transverse modulus of the roving layers is determined by using Chamis equation E 22/Roving = E m 1 υ f/roving (1 E m Ef ) (14) The shear modulus of the roving layers can be determined from Chamis model as G 12/Roving = G m 1 υ f/roving (1 G m Gf ) (15) Where E m, G m and E f, G f are Young s and shear modulus of matrix and fiber. The roving layers poisons ratio can also be calculated as V 12/Roving = V f υ f/roving + V m υ m/roving (16) 9
V 21/Roving = V 12/Roving E 22/Roving E 11/Roving (17) Theoretical Analysis of CSM layers The CSM layers are made of randomly oriented continuous fibers. These fibers are bonded into the matrix material to form CSM layers. The fibers in the CSM layers are randomly oriented, hence the properties of this layers can be assumed to be isotropic. The properties of the CSM layers can be determined using Tsai-Pagano model as follows. E Mat = 3 8 E 11 + 5 8 E 22 (18) G Mat = 1 8 E 11 + 1 4 E 22 (19) VMat = E Mat 2G Mat 1 (20) Where E Mat, G Mat and V Mat are Young s modulus, shear modulus and Poisson s ratio of the CSM layers and E 11 and E 22 are longitudinal and transverse moduli of unidirectional fibrous composite containing same volume fraction of identical fibers and matrix as that of a CSM layers under consideration. These are obtained from following equations. E 11/Roving = E f υ f/mat + E m υ m/mat (21) E 22/Roving = E m 1 υ f/roving (1 E m Ef ) (22) 10
Theoretical Analysis of Properties of Composites The combined properties of the composite can be calculated by using classical laminate theory as follows. E 11/composite = η E11 + η Mat (υ 12/Rovingη E22 + υ Mat η Mat ) 2 η E22 + η Mat (23) E 22/composite = η E22 + η Mat (υ 12/Rovingη E22 + υ Mat η Mat ) 2 η E11 + η Mat (24) G 12/Roving = G 12/Roving υ Roving + G Mat υ Mat (25) V 12/Composite = υ 12/Rovingη E22 + υ Mat η Mat η E22 + η Mat (26) Here η E11 = η E22 = E 11/Roving υ Roving 1 V 12/Roving V 21/Roving (27) E 22/Roving υ Roving 1 V 12/Roving V 21/Roving (28) η Mat = E Matυ Mat 2 1 V Mat (29) 11
Calculation of Volume Fraction of the Constituents of Composites The pultruded polymer composites in this study have two different layers, roving and CSM. The volume of the composite can be written in terms of its layers as follows v Composite = v Roving + v Mat (30) v Roving = v f/roving + v m/roving (31) v Mat = v f/mat + v m/mat (32) Here ν Roving, ν Mat, ν f/roving, ν m/roving, ν f/mat and ν m/mat are volumes of roving, mat, roving fiber, roving matrix, mat fiber and mat matrix, respectively. The total volume of the pultruded composites can also be determined as follows v Composite = v f + v m (33) Here v Composite, v f and v m are total volumes of fibers and matrix. The v f and v m can be expressed as show below. v f = v f/roving + v f/mat (34) v m = v m/roving + v m/mat (35) Here v f/roving, v f/mat, v m/roving and v m/mat are volumes of fiber in roving, fiber in mat, volume of matrix in roving and volume of matrix in CSM layer. 12
We analyze the volume fraction of the matrix. The volume of matrix is expressed in terms of constituent s volumes as follows v m = v r + v filler + v void (36) Here v r, v filler and v void are volumes of resin, filler and voids contained in the composite respectively. In order to obtain the volume fractions, divide both sides of the equation with v Composite then the following relations are obtained. υ m = υ r + υ filler + υ void (37) υ r = υ filler = υ void = ν r ν Composite (38) ν filler v Composite (39) ν void ν Composite (40) Where υ r, υ filler and υ void are volume fraction of resin, filler, and void within a composite, respectively. Also, the volume fraction of resin, filler and void with respect to matrix material is labeled as υ r/m,υ filler/m and υ void/m υ r/m = υ r υ r + υ filler + υ void (41) 13
υ filler/m = υ void/m = These volume fractions should add up to unity as shown below. υ filler υ r + υ filler + υ void (42) υ void υ r + υ filler + υ void (43) υ r/m + υ filler/m + υ void/m = 1 (44) 14
3 New Micromechanical Model As seen in the literature most micromechanical models uses volume fractions and properties of constituents to determine the composite properties. The classical micromechanical models can be modified to achieve the size effect inclusion into these models. The classical micromechanical model can be seen in Eq. (1) where S1212 is Eshelby tensor for spherical inclusion given by Eq. (3). The Eshelby tensors for spherical inclusion were derived as in [9]. These Eshelby tensors were derived by following the theory proposed by Cheng [13]. S 1212 = ( 3K r + 6G r 15K r + 20G r ) 3h(a + h)k a 5a 3 (k + 2μ) e h [a cosh a h h sinh a h ] (45) h = γ(2μ + k)/4μk (46) Where K, μ are micropolar moduli and a is the radius of spherical filler particle. The Eshelby tensor component listed by [9] includes classical Eshelby tensor along with a new term that can capture the size effect which is a function of the size of the filler particle. The new model proposed is a combination of the classical micromechanical model which considers both fillers and voids as shown in Eq. (47) and also consists of non-classical Eshelby tensor. G = G r {1 ν filler/m(g filler G r ) G r + 2S 1212 (G filler G r ) ν 1 void/m( G r ) (47) G r + 2S 1212 ( G r ) } 15
Here S 1212 = ( 3K r + 6G r 15K r + 20G r ) 3h(a + h)k a 5a 3 (k + 2μ) e h [a cosh a h h sinh a h ] (48) h = γ(2μ + k)/4μk (49) The new combination of these micromechanical models captures the effects of both voids that are present in the matrix while also considering the effect of size of the filler particle on to the overall mechanical properties. Unlike the classical models, these models also predict the mechanical properties with different sizes of the filler particles. 16
4 Process and Approaches The research is achieved by two different ways first experimental analysis is made in order to find the mechanical properties of the polymer composites and to evaluate the volume fractions of individual constituents that build up the composites, secondly, the mechanical properties of the composite are evaluated using analytical equations by varying the size of particles. Experimental Analysis The composite specimens are tested to evaluate the shear properties, an asymmetric four-point bending shear fixture and test method [14] were used to obtain these shear properties of materials. Six samples of the polymer composites specimen were tested. The dimensions and shape of the specimen tested are shown in Figure 5. An MTS 810 material testing system was used for applying compression on to the material. In order to determine the shear properties in the pure shear strain region a two-element stacked rosette (CEA- 06-062WT-350) manufactured by micro measurement group is placed on the specimen as shown in fig no the strain gauges are mounted and cured on to the specimen Figure 6 by using M-bond 200 manufactured by micro measurement by following the step by step procedure provided by the manufacturers. The strain gauges are connected to the MTS conditional amplifier in order to record the shear strain in the composite specimen. The compression load was applied at a rate of 0.15 mm/min and the data was scanned at an interval of every one second. The principal strains are along a ±45º direction to the longitudinal axis. A linear regression method was used to determine the shear modulus from the slope of the stress-strain curve between 0.1% to 0.6% strains with the number of data points always greater than 50. The broken sample is shown in Figure 7. 17
Figure 5: Configuration of the tested specimen (Units: mm) Figure 6: A virgin specimen with a strain rosette application 18
Figure 7: A tested specimen showing shear failure The experimentally determined shear moduli are listed in below Table 1. Where the standard deviation of FRC-1 specimen is less 6%. Table 1: Experimental Properties Specimen Groups (6 Specimens each) Note: FRC-2, FRC-3, FRC-4 data have been taken from [4] Shear Modulus (GPa) FRC-1 4.40 FRC-2 4.60 FRC-3 4.50 FRC-4 4.50 Cross-section of the Sample The cross-section of the pultruded polymer composite and the reinforcement scheme is seen in the Figure 8-11 and it contains a combination of E-glass roving, CSM layers, the matrix part consists of vinyl ester resin and kaolin clay filler particles. The mechanical properties of these constituents are listed in the below Table 2. 19
Figure 8: Schematic stacking sequence of FRC-1 Figure 9: Schematic stacking sequence of FRC-2 Figure 10: Schematic stacking sequence of FRC-3 20
Figure 11: Schematic stacking sequence of FRC-4 Young's Modulus {GPa) Table 2: Properties of Constituents Shear Modulus (GPa) Bulk Modulus (GPa) Density Poisson's Ratio ( g/cm 3 ) E glass fiber 72 29 0.25 2.56 48 Vinylester resin 3.85 1.40 0.37 1.26 4.79 Kaolin clay filler 20 7.69 0.30 2.60 16.67 21
Volume fractions and Density of the Composite Specimen The density of the composite specimen is obtained by performing the experiment ASTM D 792 [15]. Small coupons of samples were cut from the specimens used for shear testing as shown in fig no. The dimensions of the specimen are shown the figure. The average densities of the tested coupons are listed in Table 3. Table 3: Density of Specimens Number of samples Average Density (g/cm 3 ) STD (g/cm 3 ) FRC-1 6 1.83 0.013 FRC-2 6 1.76 0.019 FRC-3 6 1.79 0.017 FRC-4 6 1.83 0.017 The mass fractions and volume fractions are calculated for these specimens following the procedure described by [16]. In this method, the specimen is contained in a crucible Figure 12 and is ignited until only ash and carbon remain. Once the resin is completely removed the analysis of the remaining laminate is performed. The volume and volume fractions are calculated by using the measured weights and densities of the constituents of the composites as described in [16].Table no shows the average volume fractions of the constituents of the composites. 22
Figure 12: Cross section of specimen after burning test (Dimensions 25 X 25 X 6.5 mm) Figure 13: Post burn-out fiber architecture of pultruded composites 23
Table 4: Average Volume Fractions of the Constituents Constituents Symbol FRC-1 FRC-2 FRC-3 FRC-4 Roving E-glass fibers υ f/roving 0.150 0.120 0.183 0.177 Mat E-glass fibers υ f/mat 0.190 0.170 0.152 0.154 Vinylester resin υ r 0.591 0.587 0.578 0.588 Kaolin clay fillers υ filler 0.150 0.110 0.080 0.075 Total voids υ void 0.004 0.009 0.007 0.007 Note: FRC-2, FRC-3, FRC-4 data have been taken from [4] Table 5: Average Volume fractions of CSM and Roving Layers Layer Symbol FRC-1 FRC-2 FRC-3 FRC-4 Roving υ Roving 0.48 0.42 0.55 0.53 CSM υ CSM 0.52 0.58 0.45 0.47 24
5. Results and Discussion In this section, the micromechanical method will be validated by some numerical examples. Initially, the properties of the matrix containing resin, filler and void are determined by using modified Eshelby model [4], and the properties of roving layers and CSM layers are determined by using Chamis micromechanical model and the overall composites are evaluated by using laminate theory. The volume fractions are evaluated as described in [16] and are shown in table [6]. Table 6: Comparison of Experimental and Theoretical Shear Modulus Classical micromechanical model Ratio Size effect included Ratio Sample exp G 12 G 12 G 12 FRC-1 4.38 3.80 1.13 4.40 0.99 FRC-2 4.60 3.91 1.17 4.48 1.02 FRC-3 4.50 3.94 1.14 4.49 1.00 FRC-4 4.50 3.94 1.14 4.76 0.94 The results in Table 6 shows the shear modulus of the composite specimens. It can be seen that the results are compared between experimental and analytical models. A significant difference can be seen between the classical micromechanical models and the experimentally determined shear modulus for different composites specimens. The modulus obtained from the new model gives more close results when compared with results obtained from experiments. In the next case, the size of the filler particle is varied by assuming a constant characteristic length of 17µm and varying the radius between 1to100 times the characteristic length the trend can be observed as the model captures the size effect of the particle as the ratio of a/l increases the Eshelby tensor component also increase. The Figures 14-17, shows the trends of Eshelby tensor and a/l. It can be seen from the figures that as the size of the particle (a) increases the value of 25
S1212 also increases, unlike the classical part which is constant and is independent of size. When the radius of the particle gets smaller with constant characteristic length S1212 gets much smaller compared to the classical, however, as the size of the particle is increased the value of S1212 increases from below and tries to approach classical S1212. 0.150 0.1468 0.145 0.140 0.1395 S 1212 0.135 0.1325 0.130 0.1267 0.1240 0.125 0.1242 0.120 0 20 40 60 80 100 a/l Figure 14: S1212 varying with inclusion radius for FRC-1 0.15 0.1476 0.145 0.14 0.1401 S 1212 0.135 0.13 0.125 0.12 0.1328 0.1267 0.124 0.1241 0 20 40 60 80 100 a/l Figure 15: S1212 varying with inclusion radius for FRC-2 26
0.15 0.1467 0.145 0.14 0.1395 S 1212 0.135 0.1324 0.13 0.1266 0.124 0.125 0.1241 0.12 0 20 40 60 80 100 a/l Figure 16: S1212 Varying with inclusion radius FRC-3 0.15 0.1467 0.145 0.14 0.1395 S 1212 0.135 0.1324 0.13 0.1266 0.124 0.125 0.1241 0.12 0 20 40 60 80 100 a/l Figure 17: S1212 Varying with inclusion radius FRC-4 The graphs of FRC-3 and FRC-4 are almost similar as they have almost same volume fractions of the individual constituents. The shear modulus of the composites increases with the increase in the volume fraction but as the size increases the modulus starts decreasing. This effect is caused due to weak bonding for particles with a larger radius. 27
G 12 (GPa) 5 4.8 4.6 a/l=1 a/l=50 a/l=100 4.4 4.2 4 3.8 3.6 3.4 3.2 3 0. 0 5 0. 1 0. 1 5 0. 2 0. 2 5 0. 3 V f (Filler Volume Fraction) Figure 18: Variation in shear modulus for FRC-1 G 12 (GPa) 5 4.8 4.6 4.4 4.2 4 3.8 3.6 a/l=1 a/l=50 a/l=100 3.4 3.2 3 0. 0 5 0. 1 0. 1 5 0. 2 0. 2 5 0. 3 V f (Filler Volume Fraction) Figure 19: Variation in shear modulus for FRC-2 28
G 12 (GPa) G 12 ( GPa) 5 4.8 4.6 a/l=1 a/l=50 a/l=100 4.4 4.2 4 3.8 3.6 3.4 3.2 3 0. 0 5 0. 1 0. 1 5 0. 2 0. 2 5 0. 3 V f (Filler Volume Fraction) Figure 20: Variation in shear modulus for FRC-3 The Same trend can be observed in Figures 18-22. The different composite specimens show the same effect with the change in the size of particle and volume fraction the modulus varies, this is because the smaller particle sized filler has a higher total surface area for a given loading. This shows that strength increases with increase in surface area of the filled particles through a more efficient stress transfer mechanism. 5 4.8 4.6 a/l=1 a/l=50 a/l=100 4.4 4.2 4 3.8 3.6 3.4 3.2 3 0. 0 5 0. 1 0. 1 5 0. 2 0. 2 5 0. 3 V f (Filler Volume Fraction) Figure 21: Variation in shear modulus for FRC-4 29
6 Conclusions and Further works This research reveals the results obtained from an analytical and experimental investigation in determining the properties of pultruded polymer composites. It presents the procedure of analyzing, evaluating and validating the micromechanical models. The new model equations can predict the mechanical properties of matrix considering size effect, filler and void in the matrix, the properties of the matrix are evaluated using classical micromechanical model and newly proposed analytical equations. The results of this research suggest considering the size of the particle while estimating the properties of the composites to get a more accurate prediction of the properties. The analytical and experimental results were examined and the results show that the proposed model captures the size effect and also predicts the properties more closely to the experimentally determined results than the classical micromechanical theories. The components of the averaged Eshelby tensor are found to be decreasing as the inclusion radius decreases with constant characteristic length and also this component of Eshelby tensor values are found to be increasing as the inclusion size is increased and approaches the classical tensor value when the inclusion size is large enough. The size effects of the voids have not been considered in this study hence future works can be dedicated to create a new model that can predict the effect of void sizes on the properties of the composites. 30
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