Effects of geometry and properties of fibre and matrix on overall. composite parameters

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1 Int. Journal of Applied Sciences and ngineering Research, ol. 3, Issue 4, by the authors Licensee IJASR- Under Creative Commons License 3.0 Research article ISSN ffects of geometry and properties of fibre and matrix on overall composite parameters Yi Xiao 1, Ri-Song Qin 2, Manchu Mahalingam 1 1 Research School of ngineering, Australian National University, Canberra, ACT 2601, Australia 2 Fujian Port and Waterway Survey and Research Institute, 283 Yangqiao Road, Fuzhou, China DOI: /ijaser Abstract: To determine the overall material properties, 1, 2 and G 12, of a composite lamina, a finite element (F) simulation approach is presented and its results are compared with those from mechanics of materials. To investigate the effect of fibre s geometry on the overall composite properties, three different fibre s geometries are considered, in conjunction with two matrix materials and four fibre materials. Strand 7, the commercial F analysis software, is employed to conduct F simulation. The F results of the overall material properties of the composite, for different matrix s and fibre s geometries, and material properties, are compared with those from mechanics of materials approach. Key words: Composite lamina. Finite element approach, fiber geometry, fiber matrix. 1. Introduction Composite materials are playing an ever increasing role in supplying materials to satisfy the need for more demanding applications. These materials can be tailored to specific loading conditions which make them ideal to today s climate. Composite materials are superior to conventional materials, such as metals, due to their high strength to weight and stiffness to weight ratios. In classical mechanics, at a macro-level, the material properties are always assumed to be homogeneous on an average basis, whereas at micro-level, i.e. inside the representative volume element (R), the material properties are heterogeneous. That is, at a micro-level, the heterogeneous micro-structure is known and physical laws are known. The task of micromechanics is to find homogeneous material properties at a macro-level based on the information of microstructure. These properties are often called overall material properties or effective material properties, and the process is also known as homogenization. Recently, some investigations have been made to the composite array using methods such as the homogenization method (Qin and Yang 2008). Grassi et al (Grassi et al. 2002) numerically examined the effect of fibre volume fraction of the through thickness Young s modulus. They found that an increase of 2% fibre volume fraction of Z-fibres will induce an increase in the through thickness Young s modulus by a factor of 22-35%. Qin et al (Qin 2004a; Qin 2004b; Yang and Qin 2004; Qin 2005) presented several boundary element micromechanics models for predicting effective properties of materials with defects or inclusions. Antoniou et al (Antoniou et al. 2009) developed a F model to predict mechanical behavior of glass/epoxy tubes under static loading. Xu et al (Xu et al. 2008) conducted an experiment on the plate size in determining the effective modulus. Yang and Qin (Yang and Qin 2001; Yang and Qin 2003) used F method to predict effective elasto-plastic properties of composites. To determine the ranges of effective properties using various micromechanics models, the oigt and Reuss Rule (Gasik 1998) presented a *Corresponding author ( 792 Received on June 17, 2014; Accepted on August 19, 2014; Published on August 29, 2014

2 method to find the upper and lower bounds respectively of the stiffness for a composite material with arbitrary fibre-matrix geometry. Micromechanics models were also used to determine effective properties of piezoelectric materials with cracks (Yu and Qin 1996; Qin and Yu 1997; Qin et al. 1998), microvoids (Qin and Yu 1998), and of human dentine materials (Qin and Swain 2004; Wang and Qin 2007; Wang and Qin 2011). Several other researchers have used representative unit cell models to investigate the dependence of component properties on composite materials (Levy and Papazian 1990; Tvergaard 1990; Bao et al. 1991; Zahl and McMeeking 1991; Li et al. 1995; Feng et al. 2003). In this paper, we examine the effect of geometries and properties of fibres on composites through F unit cell model. In particular, the three types of properties, namely 1, the Young s modulus in the fibre direction; 2, the Young s modulus in the transverse fibre direction; and G 12 the in-plane shear modulus, are examined. By varying the shapes and material properties on the basis of mechanical and physical consideration the effect on the overall material properties can be explored. Moreover, the effect of the length in fibre direction on 1 is investigated and the minimum length required for achieving an acceptable converging result of 1 is obtained. 2. Background formulations In this section, two basic approaches for determining effective properties of fibre composites are briefly described in order to establish notations and provide a common source for reference in later sections. Moreover, material properties used in this work are listed in Section Mechanics of materials approach (MMA) The mechanics of materials method provides an accurate technique to calculate effective material properties of the fibre reinforced composites. These overall material properties can be used to predict the material behavior with various interfaces. The mechanics of materials approach determines the overall material properties due to their respective fibre and matrix volume fractions and constituent material properties. It assumes an average of stresses and strains to examine the global response. The first modulus to be determined is that of the composite material in the fibre direction (denoted as 1 axis), where, ε 1 applies to both the fibres and the matrix according to the basic assumption. 1 = f f + mm (1) q (1) is known as the rule of mixtures for the apparent Young s modulus of the composite material in the direction of the fibres. With the mechanics of materials method, the remain three properties can be determined using 2 m f = + m f f m, G12 GmG f = G + G m f f m ν12 = νmm + νf f (3) The rule appeared in q (2) is known as inverse rule of mixtures. 2.2 Finite element analysis (FA) (2) In order to obtain values for the material properties of fibre reinforced composites using a finite element method, the basic formulations are briefly described in this section. These differ to the mechanics of materials method in that they relate to the basic understanding of stress/strain relations and are not directly 793

3 related to the volume fractions as stated in Section ffective Young s modulus in fibre direction 1 and major Poisson ratio ν 12 1 and ν 12 can be determined by considering the loading case shown in Figure 1, where a stress σ 1 is applied in the fibre direction of the composite. The ffective Young s modulus and major Poisson ratio are, then, evaluated by σ σ d / σ d ε d / ε d ε = = =, ν 12 = = = ε1 1 ε1 / ε ε 1 ε1 / ε1 d d d d (4) Figure 1: Composite loaded in fibre direction If the left end is fixed and be defined as L represents the average displacement at the right end, the average strain can ε = L / L (5) Therefore, the major task for F calculation is to determine L and the average stress σ 1. This has been implemented into our F program ffective Transverse Young s modulus 2 Considering the loading case shown in Figure 2, the ffective Transverse Young s modulus 2 is defined as σ d / σ2 2 = = = ε2 ε d / σ d 2 2 ε d 2 2 where ε 2 = W / W. the major task for F calculation is to determine W and the average stress σ 2. (6) 794

4 Figure 2: Composite loaded in transverse direction ffective shear modulus G 12 The in-plane shear modulus of a fibre reinforced composite can be determined by considering the loading case shown in Figure 3, a shear stress is applied over the boundary of the composite. G 12 τd / τ = = = γ γd / τd γd (7) Figure 3: Composite loaded with shear stress Considering the shear strain can be defined as u/y, where u is displacement in fibre direction and y stands for the vertical coordinate originated at the bottom of the composite, the task of F calculation, in this case, is to evaluate shear stress and the displacement u. From these equations, we can determine 1, 2 and G 12, noting that all other stress and strains are taken as zero except for the stress and strain along the direction for the material property being determined. 2.3 Material properties A list of common composite fibre and matrix materials were selected to model the geometries above. Fibres: 1) Carbon Fibre; 2) Kevlar; and 3) -glass, and 4) S-glass) Matrix: 1) poxy Resin; and 2) Polyester Resin Tables 1 and 2 listed the mechanical properties of materials used in FA: 795

5 Table 1: Fibre Properties Fibre Material Carbon Fibre Kevlar 49 -glass S-glass Young s Modulus 294GPa 131GPa 78GPa 89GPa Poisson s ratio Table 2: Matrix Properties Matrix Material (Resins) poxy Polyester Young s Modulus (tensile) 2.415GPa 2.467GPa Poisson s ratio Results and discussion To study effects of properties of each component and fibre s geometry on the overall properties of the composite, following 8 cases are considered: 1) carbon fibre with poxy Resin; 2) carbon fibre with Polyester Resin; 3) Kevlar49 with poxy Resin; 4) Kevlar49 with Polyester Resin; 5) -glass with poxy Resin; 6) -glass with Polyester Resin; 7) S-glass with poxy Resin; and 8) S-glass with Polyester Resin. For each case mentioned above, 6 combinations of geometry are involved: a) Rectangular matrix with circular fibres (RMCF); b) Rectangular matrix with hexagonal fibres (RMHF); c) Rectangular matrix with triangular fibres (RMTF); d) Square matrix with circular fibres (SMCF); e) Square matrix with hexagonal fibres (SMHF); and f) Square Matrix with triangular fibres (SMTF). The corresponding fibre volume fraction for each of these is: RMCF= ; RMHF= ; RMTF= ; SMCF= ; SMHF= ; SMTF= Figure 4: Geometry configurations of matrix and fibre (b=2 for RM and b=3 for SM) Figure 4 shows the geometry configuration of the unit cell used. It is obvious from Figure 1 that: a) total area is 6 for RM and 9 for SM; and the fibre s area is πr 2 = for circular fibre, 3 3/2 t 2 /2=3 3/ /2= for hexagonal fibre, and 0.5 for triangular fibre. The fibre volume fraction listed above is obtained based on these data. The finite element meshes used in the calculation are shown in Figure

6 a) SMCF b) SMHF c) SMTF Figure 5: Finite element meshes for square matrix with different fibre geometries The F results obtained are listed in Tables 3-10 and also shown in Figures 6-8. A comparison of the results between FA and MMA for 1 yield quite similar results. The average error between these results is within 3%, for all results, in determination of the 1. It indicates that MMA can provide acceptable accurate results for 1. Figure 6 shows a comparison between the results of varying fibre and matrix geometries. Figures 7 and 8 list the comparison in results between 2 and G 12, respectively, derived through MMA and FA. The difference of these results is within 8%, between all the results collected for 2 and G 12. As revealed by the graph the difference in results is not always constant. The square matrix with circular fibres, for example, has a smaller difference than the other models between MMA and FA results. This highlights the non-linear nature of FA modelling software. Another point to note is that the values obtained for 1 using FA are lower than those from MMA, while the values obtained for 2 and G 12 using FA are higher than those from MMA. This is a result of the model thickness or the length in the fibre direction and will be discussed later in this paper. Table 3: Composites properties for carbon fibre with poxy Resin Properties 1 (GPa) 2 (GPa) G 12 (GPa) Approach MMA FA MMA FA MMA FA RMCF RMHF RMTF SMCF SMHF SMTF Table 4: Composites properties for carbon fibre with Polyester Resin Properties 1 (GPa) 2 (GPa) G 12 (GPa) Approach MMA FA MMA FA MMA FA RMCF RMHF RMTF SMCF SMHF SMTF

7 Properties Table 5: Composites properties for Kevlar49 with poxy Resin 1 (GPa) 2 (GPa) G 12 (GPa) 1(GPa) Approach MMA FA MMA FA MMA FA RMCF RMHF RMTF SMCF SMHF SMTF Table 6: Composites properties for Kevlar49 with Polyester Resin Properties 1 (GPa) 2 (GPa) G 12 (GPa) 1(GPa) Approach MMA FA MMA FA MMA FA RMCF RMHF RMTF SMCF SMHF SMTF Properties Table 7: Composites properties for -glass with poxy Resin 1 (GPa) 2 (GPa) 1(GPa) G 12 (GPa) Approach MMA FA MMA FA MMA FA RMCF RMHF RMTF SMCF SMHF SMTF

8 Properties Table 8: Composites properties for -glass with Polyester Resin 1 (GPa) 2 (GPa) G 12 (GPa) 1(GPa) Approach MMA FA MMA FA MMA FA RMCF RMHF RMTF SMCF SMHF SMTF Properties Table 9: Composites properties for S-glass with poxy Resin 1 (GPa) 2 (GPa) 1(GPa) G 12 (GPa) Approach MMA FA MMA FA MMA FA RMCF RMHF RMTF SMCF SMHF SMTF Properties Table 10: Composites properties for S-glass with Polyester Resin 1 (GPa) 2 (GPa) G 12 (GPa) 1(GPa) Approach MMA FA MMA FA MMA FA RMCF RMHF RMTF SMCF SMHF SMTF

9 Figure 6: Comparison of 1 results from FA with those from MMA for the case of Carbon Fibre/poxy Resin Figure 7: Comparison of 2 results from FA with those from MMA for the case of Carbon Fibre/poxy Resin Figure 8: Comparison of 2 results from FA with those from MMA for the case of Carbon Fibre/poxy Resin 800

10 To study the effect of fibre and matrix materials on the overall material properties, several common matrix and fibre materials were selected for analysis and compared to investigate the effect material properties on the overall composite. It can be seen from Tables 3-10 that the material properties merely translate the corresponding values of 1, 2, and G 12 depending on the material property of components used. For example, polyester resin/matrix has a higher Young s modulus than epoxy resin, so the effect of matrix properties, compared to epoxy, is that it will increase the overall material properties 1 of the composite in a linear manner. Table 11: ariation of 1 due to variation of thickness in the 1 axis (fibre direction) 1-axis thickness (mm) Stress(1-axis) Displacement (1-axis) Strain (GPa) Table 11 lists the 1 values obtained with respect to different thickness values of the model. That is, the thickness of the material in the fibre direction. It can be used to examine the effect of thickness on 1 when using FA. For the F results to converge towards the results obtained through MMA, a large thickness is needed to minimize end effects in the model. As the strain is a function of displacement over the original length, the strain values converge, that is, they become more uniform as the thickness increases. Nonetheless the thickness value also increases towards convergence, having the net effect of increasing the value of the 1. It could be thought that the displacement would decrease in a linear manner to the increase in thickness so that the expression L/L is held constant, this is however not the case. From this discussion it can be concluded that the ratio between thickness (fibre direction length) and the transverse direction length of the model needs to be at least 5:1 to minimize end effects in determination of the 1. For the 2 the converse ratio applies. The greater the ratio is, the more accurate the results obtained through FA are. 4. Conclusions From the analysis conducted, it can be seen that FA provides a sufficient means to calculate the overall material properties of the composite material and achieves results with acceptable accuracy. In the case of varying fibre geometries, triangular fibres exist in practice but are not common. Hexagonal fibres, not to be confused with a hexagonal array, are even less common. Nonetheless, FA also provides a method in which manufacturing of the constituent lamina of a composite is not necessary in order to begin initial testing and can therefore predict geometries that even do not exist in practice. Laboratory testing is still required before the final manufacture of composite materials, due to the inclusion of voids and matrix- fibre interface. While FA provides a powerful tool to obtain pre-manufacturing analysis in order to determine if the manufacture of composites for specific applications is worthwhile in practice. The error rates, between the results obtained using MMA and FA is within 5% on average, which is an acceptable value. This is due to the fact that the 2 values had an approximate variance of 8% from the MMA values, whereas 1 and G 12 only had a variance of less than 3%. This error could be attributable to the fact that a composite 801

11 thickness of 10mm was used in contrast to the length of the model being 60mm. As discussed earlier in the results, the thickness has a direct effect on the overall material properties until convergence occurs. The ratio of thickness to length (see Table 11) needs to be considered to reduce end effects, and thus produce more accurate results. For convergence to occur with the 2, quite possibly a lesser thickness of 1-2mm could have been taken, or a greater length of material taken, or a combination of both to reduce the ratio of thickness to length, whereas with 1 the converse applies. It was found that the hexagonal fibres yielded the greatest results in overall material properties of the composite lamina. It is also noted that the hexagonal fibres had the highest fibre volume fraction of the varying geometries. It can therefore be stated from the results of this paper that the overall material properties are mainly dependant on the fibre and matrix volume fractions and are not dependant on geometry. Geometry will only define how the internal stresses and strains are dispersed within the material but, as the average strain was determined, this has no net effect on the overall material properties. In practice, however, the internal stresses due to varying the fibre-matrix geometries will have an impact on the fibre-matrix interface and will invariably affect the overall material properties and should not be overlooked. 5. References 1. Antoniou, A.., Kensche, C. and Philippidis, T. P., Mechanical behavior of glass/epoxy tubes under combined static loading. Part II: alidation of FA progressive damage model. Composites Science and Technology, 69(13), pp Bao, G., Hutchinson, J. W. and McMeeking, R. M., Particle reinforcement of ductile matrices against plastic-flow and creep. Acta Metallurgica t Materialia, 39(8), pp Feng, X. Q., Mai, Y. W. and Qin, Q. H., A micromechanical model for interpenetrating multiphase composites. Computational Materials Science, 28(3), pp Gasik, M. M., Micromechanical modelling of functionally graded materials. Computational Materials Science, 13(1-3), pp Grassi, M., Zhang, X. and Meo, M., Prediction of stiffness and stresses in z-fibre reinforced composite laminates. Composites Part a-applied Science and Manufacturing, 33(12), pp Levy, A. and Papazian, J. M., Tensile properties of short fiber-reinforced sic/al composites.2. Finite-element analysis. Metallurgical Transactions a-physical Metallurgy and Materials Science, 21(2), pp Li, Z. H., Schmauder, S., Wanner, A. and Dong, M., xpressions to characterize the flow behavior of particle-reinforced composites based on axisymmetrical unit-cell models. Scripta Metallurgica t Materialia, 33(8), pp Qin, Q. H., 2004a. Material properties of piezoelectric composites by BM and homogenization method. Composite structures, 66(1), pp Qin, Q. H., 2004b. Micromechanics-B solution for properties of piezoelectric materials with defects. ngineering analysis with boundary elements, 28(7), pp Qin, Q. H., Micromechanics-BM Analysis for Piezoelectric Composites. Tsinghua Science & Technology, 10(1), pp

12 11. Qin, Q. H., Mai, Y. W. and Yu, S. W., ffective moduli for thermopiezoelectric materials with microcracks. International Journal of Fracture, 91(4), pp Qin, Q. H. and Swain, M.., A micro-mechanics model of dentin mechanical properties. Biomaterials, 25(20), pp Qin, Q. H. and Yang, Q. S. (2008). Macro-Micro Theory on Multifield Coupling Behaivor of Hetereogenous Materials. Beijing, Higher ducation Press and Springer. 14. Qin, Q. H. and Yu, S. W. (1997). Using Mori-Tanaka method for effective moduli of cracked thermopiezoelectric materials. ICF 9-Sydney, Australia Qin, Q. H. and Yu, S. W., ffective moduli of piezoelectric material with microcavities. International Journal of Solids and Structures, 35(36), pp Tvergaard,., Analysis of tensile properties for a whisker-reinforced metal matrix composite. Acta Metallurgica t Materialia, 38(2), pp Wang, Y. and Qin, Q. H., A generalized self consistent model for effective elastic moduli of human dentine. Composites science and technology, 67(7), pp Wang, Y. and Qin, Q. H., Micromechanics for determining effective material properties of dentine composites. Advances in ngineering Mechanics, 1, pp Xu, L. M., Li, C., Fan, H. and Wang, B., lastic property prediction by finite element analysis with random distribution of materials for tungsten/silver composite. Journal of Materials Science, 43(17), pp Yang, Q. S. and Qin, Q. H., Fiber interactions and effective elasto-plastic properties of short-fiber composites. Composite structures, 54(4), pp Yang, Q. S. and Qin, Q. H., Modelling the effective elasto-plastic properties of unidirectional composites reinforced by fibre bundles under transverse tension and shear loading. Materials Science and ngineering: A, 344(1), pp Yang, Q. S. and Qin, Q. H., Micro-mechanical analysis of composite materials by BM. ngineering Analysis with Boundary lements, 28(8), pp Yu, S. W. and Qin, Q. H., Damage analysis of thermopiezoelectric properties: Part II. ffective crack model. Theoretical and Applied Fracture Mechanics, 25(3), pp Zahl, D. B. and McMeeking, R. M., The influence of residual-stress on the yielding of metal matrix composites. Acta Metallurgica t Materialia, 39(6), pp

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