IOP Conference Series: Materials Science and Engineering PAPER OPEN ACCESS A Comparative Study of Analytical and Numerical Evaluation of Elastic Properties of Short Fiber Composites To cite this article: Babu Reddy and K Badari Narayana 06 IOP Conf. Ser.: Mater. Sci. Eng. 49 0089 iew article online for updates and enhancements. This content was downloaded from IP address 48.5.3.83 on 9/0/08 at 0:
A Comparative Study of Analytical and Numerical Evaluation of Elastic Properties of Short Fiber Composites Babu Reddy and Badari Narayana K Assistant Professor/Research Scholar, TU PG Centre, Kalaburagi, Karnataka, 5850, India Professor, BMSIT & M, Bengaluru, Karnataka, 560064, India E-mail: babureddy.dh@gmail.com, bn.kanti@gmail.com Abstract: Unlike case of continuous fiber composites, prediction of elastic properties of short fiber composites using corresponding elastic properties of constituents is not a straight forward task. Many authors have attempted to predict properties using completely eir by analytical or by experimental methods or a combination of both leading to empirical solutions. The current trend is to use well known numerical solution Finite element method (FEM) to model short fiber composite to predict ir properties. In this paper, a RE (Representative olume Element) approach is used to model, with appropriate boundary and loading conditions and application of homogenization process to estimate elastic properties. The present values are compared with available experimental and analytical solutions. The methods that best match with current FE solutions are highlighted.. Introduction The advantages of traditional long fiber composites such as high strength and high stiffness are well known[]. However, one common problem with se composites is ir inability to manufacture intrinsic and complicated products using simple manufacturing techniques. In addition, achieving isotropy at lamina level is virtually impossible. Though closer to isotropy is possible at laminate level, it s again a contrived process. As against this, plastics are very good in ir mold-ability and by nature are isotropic. But plastics are not strong enough to be used in primary structures. A relative compromise between se two ends is a short fiber composite (SFC). The SFCs are also called as discontinuous fiber composites. The SFC is a composite made usually of plastic as matrix phase and chopped short fibers as reinforcing phase. The length of fiber is usually shorter than few millimeters []. An SFC is better in strength than a typical plastic and yet it retains isotropy and mold-ability of plastic. The SFCs can be classified into two types based on ir randomness i.e. Aligned Short Fiber composites [ASFCs] and Randomly Distributed Short Fiber Composites [RDSFCs]. In ASFCs fibers have orientations along a particular direction in matrix (Figure ). Usually it is not possible to perfectly align se fibers in a particular direction, but a majority of m can at least be made to orient in a particular direction. Hence se are also known as preferentially oriented short fiber composites. These aligned fibers are usually transversely isotropic requiring at least 5 independent elastic constants are need to define m. The RDSFCs on or hand have fibers randomly distributed in all possible directions. Furr, randomness of short fiber composites is of two types i.e. 3D Randomness and D Randomness []. The first type consists of short fibers distributed completely random in three dimensional spaces, usually in an isotropic polymer matrix. This means every orientation in matrix has an equal chance Content from this work may be used under terms of Creative Commons Attribution 3.0 licence. Any furr distribution of this work must maintain attribution to author(s) and title of work, journal citation and DOI. Published under licence by Ltd
of having a fiber along it. This type of randomness (or at least randomness close to it) is observed in products manufactured using injection molding machines. In second type, fibers are distributed randomly in a plane, meaning every direction in plane has an equal probability of having a fiber along it. This type of randomness is observed in RDSFC specimens made by Handlayup process. In such specimens, even though out of plane fibers do exit, angle is usually very small [3]. In both types, matrix is normally isotropic in nature. But, fibers can be isotropic as well as transversely isotropic, though former is quite common. For example, Glass fibers are isotropic while carbon fibers are transversely isotropic. Aligned Short Fiber composites Randomly Distributed Short Fiber Composites Figure. Types of Short Fiber Composites. Prediction of Mechanical properties of SFCs Estimation of mechanical properties of SFC s is not a straight forward task, because properties of SFC s are controlled by various parameters. For example, stiffness and strength of SFC s depend, inter alia, on [] Properties of constituent materials (fibers, matrix, binder, filler etc.), Relative proportions of each constituents (i.e. olume Fractions), Orientations of fibers, Aspect ratios of fibers (l/d ratios), and Interaction between fiber and matrix. Owing to se numerous factors involved in determination of stiffness and strength properties, physical testing of SFC component is not always economical. Besides, test data can easily become invalid if any of governing parameters change. Hence, attempts have been made to predict properties of SFCs analytically and numerically by using properties of fibers, matrix and ir relative proportions... Analytical Methods Researchers have approached problem of prediction for properties of composite materials from various angles. A set of approaches known as Bounding value methods predict bounds (maximum and minimum values) of elastic properties sought rar than value itself[4-8]. Here idea is, if bounds are close enough, n bounds mselves are solutions. The Self-Consistent methods [9-4] use traditional elastic solution of an elliptical inclusion in an infinite media. Selfconsistency here means orientation average of inclusion stress or strain is made equal to overall stress or strain. The popular Halpin-Tsai equations come under what are known as Semi-Empirical Methods [5-8]. In se methods, part of solution is derived from oretical work and or part from experimental works... Numerical Methods to Predict Elastic properties With advent of sophisticated Finite Element software, researchers are focusing ir attention on FEM approaches in predicting properties of SFCs. Owing to its numerical nature, FE methods can also be used to conduct parametric studies. Bohm [9] made FEM analysis of Spherical particles in a matrix and compared those with Hashin s [6] Bounds and found a close agreement. Well-documented
results on FE based methods can be found in Gusev [0]. Recently in a well cited article Kari et. al. [] have, made extensive study on RDSFCs and transversely distributed composite materials. This was followed by Yi Pan [3] where various aspects of modeling an RE were explored. But, when it comes to modeling of RDSFCs no final word is said yet, still a lot many numerical experiments are being carried out to simulate behavior as close as possible to reality. Thus, owing to a large number of variables that control behaviors of RDSFCs, a lot more needs to be done. 3. Methodology of Modelling Two major approaches were attempted in this work. The first one was Analytical method where properties of composite were predicted using some analytical equations. In second method, Numerical modeling using commercial Finite Element codes was tried. Three elastic parameters studies were Young s Modulus (E), Poisson s Ratio ( ) and Shear Modulus (G). Though not attempted in resent work, or properties which are equality important for any structure design are σ s, K s, and α s. The details of methods used in current work are given below 3.. Analytical methods In analytical method, three different equations available in literature were chosen to compare results of present work. The choice of se equations was based on relevance of model to our work. For analytical methods, wherever an explicit equation for shear modulus was not available it was obtained from usual isotropic relation E = G(+ ). 3.. Christensen Model [] In this model properties of aligned SFRCs are used in predicting properties of Random distributed short fibers. The aligned short fiber composites can be considered to be transversely isotropic and hence have five independent elastic constants. Christensen used se 5 independent constants already available in literature and calculated E 4 8 4 4 4 6 3E 8 7 K3 3 E K E K 3 3 3 3D () E 4 6 6 K 4 K 3 3 3D 4E 6 4 4 3 4 3 () Here, K 3 is Plain Strain Bulk modulus, in direction, and is major Poisson s ratio. is Shear modulus in ij plane, ij E is Young s modulus 3.. Manera s Approach [] Manera used Puck s equation and made some benign simplifications and assumptions which led to simple utility functions for prediction of Elastic properties of glass fiber composites. In addition, Manera also conducted tests on glass fiber specimens and found out utility value of those equations. The equations are 3
6 8 E ( E E ) E 45 9 f f m m 3 G ( E E ) E 5 4 3 3 f f m m E G ( ) (3) (4) (5) (6) Where, f = Fiber volume fraction, E f, E m are Young s Moduli of fiber and matrix respectively. 3..3 Pan s Modified Rule of Mixture [3] Pan showed that fact in aligned FRP s: volume fraction is same as area fraction, cannot be applied by default in RDSFCs. Pan suggested a relation between A f and f and obtained final relation as 3 f D f Ec E f Em( ) 3D f f c f m( ) (7) (8) Where, f = Fiber volume fraction, E f, E m are Young s Moduli of fiber and matrix respectively. f, m are Poisson s Ratios of fiber and matrix respectively. 3.. Numerical Modeling and Analysis The numerical modeling involved series of MATLAB [4] programs and Python [5] scripts that interact with Abaqus [6]. The following major steps are involved. 3.. Generation of RE An RE is a sample part of composite whose properties are expected to be same as that of composites as a whole. For this expectation to be valid, RE must be big enough to accommodate sufficiently large number of constituents. In addition, it should also have orientation of fibers which also is representative of orientation seen in composite. In present work generation of RE was done using Modified Random Sequential Algorithm [7]. The orientations of fibers to be placed in RE are generated using MATAB software, owing its crisp and intuitive vector operational capabilities. Typical REs generated for three volume fractions are shown in figure 4
a b c Figu ure. REs at various volume fractions (a) f =% (b) f = 6% and (c) f =0= 0% 3.. Importing RE into com mmercial FE software for analysiss The automation of modeling and mesh hing in Abaqus is achieved via its scrip pting language whichh is based on open source programming language Python. In present case e, fiber orientations are generated in MATLAB and rest of modeling was done using this scripting language. The script used could accommodate variation of fiber length,, aspect ratio, min nimum gap between fibers etc. The script coul ld call Abaqus and have problem solved. 3..3 Homogenization Schemee The RE so generated is subjected to homogeneous load and displacement boundary conditions [8]. In applying homogenization scheme, following volume averagingg techniquess are useu ed. When homogeneous boundary conditions are applied to RE, average stre ess and averagee strains are defined by ij ijd ij d ij (9) Homogeneous boundary conditions applied on surface of a homogeneous body that will l produce a hom mogeneous field [8]. The equation for homogeneous bou undary conditions are given by u i 0 S ij x j wheree are constantt strains. As far as load is concerned, a uniform pressure scheme is applied. The pressure is applied in three e faces in each of three perpendicular directions. 3..4 Post Proc cessing In addition, anor script is written exclusively for f post processing. The script could open binary results file of Abaqus (i.e. *.odb file) generated by analysis of prev vious steps and a calculate average values stress and strain and elastic moduli in each direction. A typ pical displacement plo ot is shown in Figure (b) ). The computed values are written in a text file which is use ed for post processing and plotting using Matlab. (0) 4. Results and discussions Most of work in field of short fiber composite is directed towards D Randomly Distributed Short Fiber Composites (RDSFCs). How wever, literature in 3D RDSFCs is rapidly growing. In 5
this work an attempt is made to analyze 3D RDSFCs using FEM. It is well known that 3D RDSFCs are isotropic in nature and hence only two elastic constants are sufficient to describe its complete elastic behavior. This work is refore restricted to analysis and to estimate three important elastic properties namely Young s modulus, Shear modulus and Poisson s ratio. Before estimating elastic properties of RDSFCs isotropic nature of model is to be verified, same is discussed below. 4.. Isotropic Material As 3D RDSFCs are isotropic in nature, properties are independent of direction of measurement. To verify this from generated RE model, young s moduli is measured in three perpendicular directions. In Abaqus, three cases were analyzed as separate steps and in each step appropriate boundary conditions are used. The resulting local stresses are n homogenized as per equation (9). Table. gives results obtained for three volume fractions %, 7% and 4%, respectively. As can be seen from table, variation of E among all three directions is less than % validating isotropic nature of RE model. Table. Comparison of E in three perpendicular directions. Fiber olume Fraction E (in GPa) in direction E (in GPa) in direction E3 (in GPa) in direction 3 %ariation from Mean %.3398.3406.3603 0.575 7%.8639.8947.84-0.869 4% 3.6549 3.6599 3.5538 -.9075 4.. Elastic Modulus The properties of fiber and matrix used are given Table Table. Properties of constituents. Constituents E, GPa Reinforcement 73 0.5 Matrix.5 0.4 Young s modulus and Poisson s ratios were directly computed using homogenization technique, whereas shear modulus was derived using isotropic relation E = G(+ ). In this work, fourteen REs with fiber volume fraction varying from to 4% are generated. For each of se REs, three analytical methods mentioned earlier and present FE method are used to compute elastic properties. The results of computed Young s modulus are plotted in Figure 3. Similarly plots for G and are shown in Figures 4 and 5 respectively. The results indicate that at lower volume fractions all four methods show closer values. As volume fraction increases, Manera and Christensen equations tend to give higher values of elastic modulus. The Pan s equation and FE method predict consistently similar values over range of volume fraction. Similar trend can be observed even in case of Shear modulus. However, for case of Poisson s ratio, The Pan s and Manera s predictions show mostly constant values while FE method and Christensen s predictions show a decreasing trend with increase in fiber volume fraction. 6
7 x 09 6.5 6 5.5 Christensen Manera Pan Present FEM Young's Modulus 5 4.5 4 3.5 3.5 0 0.05 0. 0.5 olume Fraction Figure 3. ariation of Young s Modulus with olume fraction.6 x 09.4. Christensen Manera Pan Present FEM Shear Modulus.8.6.4. 0.8 0 0.05 0. 0.5 olume Fraction Figure 4.ariation of Shear Modulus with olume fraction 7
0.5 0.45 Christensen Manera Pan Present FEM 0.4 Poisson's Ratio 0.35 0.3 0.5 0. 0 0.05 0. 0.5 olume Fraction Figure 5. ariation of Poisson s Ratio olume fraction 5. Conclusions In present work a comparative study of analytical and numerical methods for predictions of elastic properties of 3D RDSFCs is conducted. The results show that E values estimated by RE approach are accurate and simulate isotropic behavior well. As expected E and G values increase with increasing volume fraction is well demonstrated by all four methods. However, variation in with volume fraction is almost constant. This can be attributed to fact that addition of fiber in all three directions is nullifying possible increase or decrease in Poisson s ratio. The FE results are much closer to Pan s Model than or two models. For this reason, in 3D RDSFCs, estimation of elastic properties by Pan s model is computationally less expensive than FE estimations. It appears that, whenever a quick and yet fair estimation of elastic properties can be computed by Pan s equations similar to Rule of Mixtures for 3D RDSFCs. However, efforts are on to extended FE analysis over large range of volume fraction to conclusively estimate same. References [] Shao-Yun Fu et al., "Science and engineering of short fibre reinforced polymer composites", CRC Press, 009. [] R.M. Christensen and F.M. Waals, "Effective Stiffness of Randomly Oriented Fiber Composites", Journal of Composite Materials, ol. 6, pp. 58-535, 97. [3] Yi Pan, "Stiffness and progressive damage analysis on random chopped fiber composite using FEM", Ph.D. Thesis, The State University of New Jersey, 00. [4] B. Paul, "Prediction of Elastic Constants of Multiphase Materials", Transactions of ASME, ol. 8, pp.36-4, 960. [5] R. Hill, "Elastic properties of reinforced solids: some oretical principles" J. Mech. Phys. Solids, ol., pp. 357-37, 963. [6] Hashin, Z. and Shtrikman, S. "A variational Approach to ory of Elastic Behavior of Multiphase Materials", Journal of Mechanics and Physics of solids, ol., pp.7-40, 963. [7] Walpole, L. J., "On Bounds for Overall Elastic Moduli of inhomogeneous Systems-I", Journal of Mechanics and Physics of solids, ol. 4, pp.5-6, 966. 8
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