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Prediction of effective elastic modulus for glass microspheres loaded polymer composites
Materials Today: Proceedings xxx (xxxx) xxx
Contents lists available at ScienceDirect
Materials Today: Proceedings journal homepage: www.elsevier.com/locate/matpr
Prediction of effective elastic modulus for glass microspheres loaded polymer composites P. Jeyaprakash ⇑, A. Devaraju Department of Mechanical Engineering, Adhi College of Engineering & Technology, Kanchipuram, Tamilnadu, India
a r t i c l e
i n f o
Article history: Received 20 May 2019 Received in revised form 2 August 2019 Accepted 4 August 2019 Available online xxxx Keywords: Particle filled composites Elastic modulus Glass microspheres Mechanical properties Finite element analysis
a b s t r a c t Particle filled polymers are used in very large quantities in advance composite materials. Although investigations have done to evaluate the mechanical properties of particle filled composites, Finite Element Analysis (FEA) analysis have not done so for. Therefore, in this work, the elastic modulus of randomly distributed particle filled composite materials are computed by finite element method and verified with the already published experimental data. Spherically shaped solid and hollow glass particles and epoxy are used as fillers and matrix materials respectively. The FEA is applied to a single representative volume element model. The failure theories and mathematical models are verified with the numerical values. Ó 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the International Conference on Materials Engineering and Characterization 2019.
1. Introduction Particle filled Composite materials are leading classes of engineering materials. They have been developed since the 1960 due to their remarkable properties i.e. high strength and stiffness, less weight in comparison with metals and high chemical resistance. composites have special structural, energetic and mechanical properties i.e. higher Young‘s modulus, highly resistant and exceptional stiffness etc. [1,2]. Thus it is alluring to compute the effective elastic properties from the information of the structure of the composite material. The concept of using microspheres as filler material comes from the late 1970s when hollow glass spheres were used in combination with a resin system for lightweight plastics in marine applications [3,4]. As the automotive industry looks for stronger, lighter and inexpensive alternatives to sheet metal, microspheres with diameters ranging from 10 mm to 100 mm are becoming readily available in a variety of materials and can provide the potential weight saving required. Ranging in weight from 0.13 g/cm3 to 0.54 g/cm3, microspheres can lower the standard weight of SMC parts from 1.5 g/cm3 to 1.3 g/cm3 or lower. Since to the spheres come in a selection of sizes and materials, the overall strength of the spheres can be controlled by the spheres outer diameter, density and material composition [5].
⇑ Corresponding author. E-mail address: [email protected] (P. Jeyaprakash).
Currently, microspheres are being used in a wide range of applications, which include everything from bowling balls to sport and leisure boats. The effective elastic properties are one of the most significant properties that decide the mechanical performance of a material [6,7]. The problem of computing the effective mechanical properties of a composite material with complex microstructures is a moving assignment to scientists and researchers due to its fundamental and innovative significance in almost every area of material science. This problem keeps on being the focus of intense research, from design as well as theoretical point of view. The essential data required for the assessment of the effective moduli is the volume fractions and elastic moduli, of each phase. It is evident that most of the studies, the effective elastic modulus is determined by experimental analysis. The prediction of effective elastic modulus of composites by numerical analysis is scarcely reported. Hence an attempt is made in this study to predict the effective elastic modulus of polymeric composites using finite element analysis. 2. Materials and methods The sample composites used in this study consist of an epoxy matrix filled with various volumes of solid and hollow glass filler. The epoxy used was based upon Diglycidyl ether of Bisphenol-A (DGEBA). Fig. 1 shows the chemical structure for epoxy. The size of hollow and solid microspheres is 20 mm and 100 mm respectively. The modulus and Poisson’s ratio of solid glass microspheres
https://doi.org/10.1016/j.matpr.2019.08.064 2214-7853/Ó 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the International Conference on Materials Engineering and Characterization 2019.
Please cite this article as: P. Jeyaprakash and A. Devaraju, Prediction of effective elastic modulus for glass microspheres loaded polymer composites, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.064
2
P. Jeyaprakash, A. Devaraju / Materials Today: Proceedings xxx (xxxx) xxx
Fig. 1. Chemical structure of Epoxy (DGEBA).
Vm ðSm Sf Þ Su ¼ Sf 1 þ Sf þ Vf ðSm Sf Þð1 gsf Þ
ð6Þ
Vf ðSm Sf Þ Sl ¼ Sm 1 þ Sm Vm ðSm Sf Þð1 gsm Þ
ð7Þ
where the parameters Sf and Sm (i.e. Kf, Km, Gf and Gm) are defined by are stated in literature [8] to have values of 76 GPa and 0.23 respectively. The hollow glass spheres had a density of 0.37 g/cm3. The modulus and Poisson’s ratio for the hollow glass spheres are 1.99 GPa and 0.21 respectively. Solid or hollow glass spheres were added in various volume fractions to this initial epoxy combination. For the solid/hollow filler combinations, the hollow and solid glass fillers are added to the matrix epoxy at volume ratios of 50:50 as gives the better results. The elastic modulus of spheres embedded in a continuous and isotropic elastic matrix is computed through the finite element analysis of the two-dimensional unit cells. 3. Theoretical formulation The Voigt and Reuss models [9] are the two most fundamental models used to develop the properties of a two-phase composite system comprising of a spherical filler addition surrounded in a continuous matrix material. The Voigt model, which is defined by Eqs. (1) and (2), assumes that the particulate addition and matrix material are exposed to constant strain.
Voigt model Ec ¼ Em Vm þ Ef Vf
ð1Þ
Em Ef Em Vf þ Ef Vm
ð2Þ
Reuss model Ec ¼
gKj ¼
1 þ #j 3ð1 #j Þ
ð8Þ
gGj ¼
2ð4 5#j Þ 15ð1 #j Þ
ð9Þ
where j = matrix or the filler; c = Phase continuity parameter
c¼
2uc 1
uc
ð10Þ
uc ¼Critical volume fraction The Eqs. (11)–(14) shows the C-Combining Rule. This rule is applicable for the case of a matrix filled with a more compliant filler, such as a matrix epoxy loaded with hollow glass spheres (i.e., Ef < Em).
Pc ¼ Vm Pu þ Vf Pl þ kp Vf Vm ðPu Pl Þ
ð11Þ
where, kp = phase continuity parameter and P = bulk modulus (K) or shear modulus (G).
Vm ðPm Pf Þ Pl ¼ Pf 1 þ Pf Vf ðPm Pf ÞðgPf Þ
ð12Þ
Some more theoretical models are also given in the Eqs. (3) and (4)
Neilson0 s model Ec ¼ Em ð1 þ 2:5Vf Þ 0
Einstein s model Ec ¼ Em ð1 þ Vf Þ
ð3Þ ð4Þ
The conditions of constant stress or strain over the filler-matrix boundaries are thermodynamically impossible; hence, more rigorous models have been proposed in an effort to increase the prescient exactness of the Voigt and the Reuss models utilizing bounding connections or self-consistent field theories [10]. In this research, The S-Combining Rule is also used to model the glass filled composites for predicting the modulus properties (Eqs. (5)–(10)). S-Combining rule
1 Vm Vf 1 1 ¼ þ þ cVf Vm Sc Sl Su Sl Su
Fig. 2. Geometric Model of Particle filled epoxies.
ð5Þ
Fig. 3a. Finite Element Analysis Loading Condition of SGM.
Fig. 3b. Finite Element Analysis Loading Condition of HGM.
Please cite this article as: P. Jeyaprakash and A. Devaraju, Prediction of effective elastic modulus for glass microspheres loaded polymer composites, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.064
P. Jeyaprakash, A. Devaraju / Materials Today: Proceedings xxx (xxxx) xxx
Pu ¼ Pm 1
kp ¼
Vf ðPm Pf Þ Pm Vm ðPm Pf ÞðgPm Þ
2u c 1 uc ½1 2uc ð1 uc ÞgPm
ð13Þ
ð14Þ
All the parameters used in the S and C combining rule are well known and easily found in the literature. The tensile strength of particle filled composites is computed by following analytical models (15) and (16).
Tavmans model
rc ¼ rm ð1 bVf2=3 Þ
3
ð15Þ
b = 1.21 for Spherical particles
Wongs model
rc ¼ rm ð1 bV2=3 f Þ
ð16Þ
b = 0.5 for lower concentration of fillers
Fig. 4. Stress distribution for RVE model of SGM filled epoxies (Vf = 0.1).
Fig. 5. Displacement for RVE model of SGM filled epoxies (Vf = 0.1).
Please cite this article as: P. Jeyaprakash and A. Devaraju, Prediction of effective elastic modulus for glass microspheres loaded polymer composites, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.064
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P. Jeyaprakash, A. Devaraju / Materials Today: Proceedings xxx (xxxx) xxx
4. Numerical analysis
ANSYS WOI
12
ANSYS WI
In this study, using the finite-element software ANSYS Structural analysis is carried out in solid glass microspheres (SGM), hollow glass microspheres (HGM) and solid/hollow mixed glass microspheres filled epoxy ternary system for determine the effective Elastic modulus. In order to perform this analysis, a single representative volume element (RVE) as shown in Fig. 2 and NODE 182 element is considered for analysis [11]. The geometry took advantage of symmetry and therefore was modeled as onequarter of a two dimensional unit cell with a uniaxial pressure applied away from the top of the unit cell as shown in Figs. 3a and b. The finite element model is used to simulate the microstructure of composite materials for nine different particle loading concentrations. Furthermore, the effect of interface between the particle and matrix on the effective elastic modulus is numerically determined for various loading concentrations. 5. Results and discussion The structural analysis of solid and hollow glass microspheres loaded epoxies is carried out by single RVE model at different filler
Elastic modulus (GPa)
10
EXPERIMENTAL NEILSON MODEL
8 S-COMB REUSS MODEL
6 4 2
Volume fraction of filler
0 0
0.1
0.2
0.3
0.4
0.5
Fig. 7. Effective Elastic modulus of SGM filled epoxies.
quantities. The effect of volume fraction on effective elastic modulus of SGM filled epoxy is computed. The different volume fractions varying from 0.05 to 0.45 of the glass microsphere of 100 lm is considered in this study. Figs. 4 and 5 show the stress distribution and the displacement of SGM filled epoxy for the given boundary
Fig. 6. Stress distribution for RVE model of SGM filled epoxies with interface (Vf = 0.1).
Table 1 Effect of filler content in SGM filled the epoxies on Elastic Modulus. Filler
Neat Solid
Volume fraction (Vf)
– 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Elastic Modulus (GPa)
Theoretical modulus (GPa)
Ansys
Experimental [5]
Neilsons model
Einsteins theory
Voigt model
Reuss model
Wongs model
S-combination Rule
2.45 3.13 – 3.9 – 4.99 5.7 – – 10.1
2.45 2.76 3.06 3.37 3.67 3.98 4.29 4.59 4.90 5.20
2.45 2.57 2.69 2.82 2.94 3.06 3.18 3.31 3.43 3.55
2.45 6.13 9.8 13.48 17.16 20.83 24.51 28.19 31.87 35.55
2.45 2.57 2.71 2.87 3.04 3.23 3.45 3.7 4.00 4.34
2.45 2.51 2.57 2.63 2.69 2.75 2.82 2.88 2.94 3.00
2.45 2.72 3.04 3.41 3.85 4.36 4.97 5.71 6.6 7.69
WOI
WI
– 2.93 3.33 3.79 4.34 5.03 5.98 7.22 8.59 11.82
– 2.79 3.1 3.44 3.76 4.21 4.71 5.27 5.52 –
Please cite this article as: P. Jeyaprakash and A. Devaraju, Prediction of effective elastic modulus for glass microspheres loaded polymer composites, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.064
5
P. Jeyaprakash, A. Devaraju / Materials Today: Proceedings xxx (xxxx) xxx Table 2 Effect of filler content in SGM filled the epoxies on Tensile Strength. Filler
Volume fraction
Max. stress (MPa)
Theoretical stress (MPa)
Ansys
Neat Solid
– 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
WOI
WI
57.73 57.81 57.23 56.03 55.02 54.58 52.37 36.19 24.33
57.82 56.25 55.9 55.07 53.5 53.14 40.49 32.03 0
70
Tensile Strength (MPa)
60 50 40 30
Ansys WOI Ansys WI
20
Experimental Tavmans model
10
Wongs model
Volume fraction of filler
0 0
0.1
0.2
0.3
0.4
0.5
Fig. 8. Tensile Strength of SGM filled epoxies.
conditions of a Representative Volume Element model. The effect of interface between SGM and epoxy is shown in Fig. 6. The Table 1 shows the effect of SGM filler content in the epoxies on elastic modulus. The theoretical models as discussed in Section 2 are also
Experimental [4]
Tavmanmodel
Wongmodel
58 57.34 57.14 56.92 56.59 56.18 55.65 55.06 – –
58 48.47 42.88 38.19 34 30.15 26.55 23.14 19.9 16.79
58 54.06 51.75 49.81 48.08 46.49 45 43.59 42.26 40.97
computed. The numerical results are compared with the literature experimental data [12]. Fig. 7 shows the impact of volume fraction on the effective elastic modulus of Solid Glass Microspheres filled epoxy composite. It is construed that the volume fraction of filler increases as effective elastic modulus increases. The numerical results gives closure values to experimental values. Superior accuracy is observed for S-Combining rule while comparing with other models [2,4]. The effective elastic modulus is increased significantly for the case of solid glass filler. The tensile strength of SGM filled epoxy is computed numerically and compared with experimental data [5] shown in Table 2. Also the analytical models Tavman and Wongs [2] are calculated. Fig. 8 shows tensile strength of SGM loaded epoxies. It is seen that with addition of SGMs, tensile strength of the composite decreases and this decrement is a function of the SGM content. This is confirmed by experimental values. The stress distribution of HGM filled epoxy for the given boundary conditions of a RVE model at Vf = 0.1 is shown in Fig. 9. The Table 3 shows the effect of HGM filler content in the epoxies on elastic modulus. The composite modulus is decreased significantly for the case of Hollow glass filler. Fig. 10 shows the effect of volume fraction on the effective elastic modulus of HGM filled epoxy composite.
Fig. 9. Stress distribution for RVE model of HGM filled epoxies (Vf = 0.1).
Please cite this article as: P. Jeyaprakash and A. Devaraju, Prediction of effective elastic modulus for glass microspheres loaded polymer composites, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.064
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P. Jeyaprakash, A. Devaraju / Materials Today: Proceedings xxx (xxxx) xxx
Table 3 Effect of filler content in HGM filled the epoxies on Elastic modulus. Filler
Volume fraction
Neat Hollow
– 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Elastic modulus (GPa)
Theoretical modulus (GPa)
Ansys
Experimental [4]
Neilsons model
Einsteins theory
Voigt model
Reuss model
Wongs model (low)
C-Combination Rule
2.45 – – 2.7 – 3.07 – 2.61 – 2.67
2.45 2.75 3.06 3.37 3.67 3.98 4.29 4.59 4.9 5.2
2.45 2.57 2.69 2.82 2.94 3.06 3.18 3.31 3.43 3.55
2.45 2.43 2.4 2.38 2.36 2.33 2.31 2.29 2.26 2.43
2.45 2.42 2.39 2.37 2.34 2.32 2.29 2.26 2.24 2.22
2.45 2.51 2.57 2.63 2.69 2.76 2.82 2.88 2.94 3.00
2.45 2.43 2.41 2.38 2.36 2.34 2.32 2.3 2.28 2.26
Woi
Wi
– 2.323 2.318 2.316 2.313 2.31 2.306 2.303 2.299 2.298
– 2.323 2.319 2.316 2.313 2.319 2.306 2.303 2.299 2.298
Table 4 Effect of filler content in SGM/HGM filled the epoxies on Elastic modulus.
3.5
Filler
Elastic Modulus (Gpa)
3
Volume fraction
2.5
Elastic modulus (GPa) Ansys
Theoretical modulus (GPa) Neilsons
Einstein
– 3 3.63 4.498
2.45 2.75 3.06 3.37
2.45 2.51 2.57 2.63
2 Neat Solid/Hollow
1.5
Ansys Experimental C-Comb
1
– 0.05 0.1 0.15
Wongs model 2.51 2.57 2.63
0.5 Volume fraction of filler 0 0
0.1
0.2
0.3
0.4
0.5
Fig. 10. Effective Elastic modulus of HGM filled epoxies.
It is inferred that the volume fraction of filler decreases as effective elastic modulus increases. The numerical results give closure values to experimental values. In analytical models C-Combining
rule confirms the above results. Binary filler mixtures of solid and hollow glass were added to the matrix epoxy to yield ternary composite systems. For these blended filler systems, the volume fraction of filler was differed. As mentioned in Section 2 the solid and hollow glass fillers were added to the matrix epoxy at volume proportion of 50:50. The Table 4 shows the effect of SGM/HGM filler content in the epoxies on elastic modulus. The composite modulus is increased significantly for the case of Solid/Hollow glass filler. Fig. 11 shows the stress distribution of SGM/HGM filled epoxy at volume fraction of 0.1 and Fig. 12 shows the Effective
Fig. 11. Stress distribution for RVE model of SGM/HGM filled epoxies (Vf = 0.1).
Please cite this article as: P. Jeyaprakash and A. Devaraju, Prediction of effective elastic modulus for glass microspheres loaded polymer composites, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.064
P. Jeyaprakash, A. Devaraju / Materials Today: Proceedings xxx (xxxx) xxx
6. Conclusion
5 4.5 Elastic modulus (Gpa)
7
In this study, the effective elastic modulus of a glass microspheres filled epoxy composites is computed numerically. Effective Elastic modulus is decreases with increase of HGM content in epoxy but increase with increase of SGM content. It is increases as a function of volume fraction in SGM/HGM ternary composite system. The numerical results conforms (95%) the literature experimental data. This is investigated with S-Combining Rule and Ccombining rules give the closure values (98%) of effective elastic modulus.
4 3.5 3 2.5 Ansys
2
Neilsons model
1.5
Einsteins model 1
Wong model
0.5
References
Volume fraction of filler
0 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Fig. 12. Effective Elastic modulus of SGM/HGM filled epoxies.
SGM
ELASTIC MODULUS(GPA)
5 4.5
HGM SGM/HGM
4 3.5 3 2.5 2 1.5 1 0.5 0
0.05
0.1
0.15
VOLUME FRACTION
Fig. 13. Comparison of Effective Elastic modulus with various fillers.
elastic modulus of SGM/HGM filled epoxy at various volume fractions. Effective elastic modulus comparison of SGM, HGM and SGM/HGM ternary system is shown in Fig. 13.
[1] Lukwai Lap, Models for some mechanical properties of particulate filled composite materials, The Hong Kong Polytechnic University, 2012. [2] A. Krishnakumari, A. Devaraju, M. Saravanan, Evaluation of mechanical properties of hybrid root fibre reinforced polymer composites, Mater. Today Proc. 5 (2018) 14560–14566. [3] Steven E. Taylor, Mechanical Characterization of Particulate Filled Vinyl Ester Composite Material, The University of Tulsa, The Graduate School, 2016. [4] P. Jeyaprakash, V. Prabakaran, A. Devaraju, Experimental and numerical analysis of carbon epoxy fibre composite under buckling load, Mater. Today Proc. 5 (2018) 14526–14530. [5] Matthew J. Quesenberry, Phillip H. Madison, Robert E. Jensen, Characterization of Low Density Glass Filled Epoxies, Army Research Laboratory, 2013. [6] Debasmitamishra, A study on Thermal and Dielectric Characteristics of Solid Glass Microsphere Filled Epoxy Composites, National Institute of Technology, Rourkela (India), 2014. [7] R. Harikumar, A. Devaraju, ‘‘Fabrication and experimental analysis of copper wire embedded with GFRP composites, Mater. Today Proc. 5 (2008) 14327– 14332. [8] A. Devaraju, P. Sivasamy, Comparative analysis of mechanical characteristics of sisal fibre composite with and without nano particles, Mater. Today Proc. 5 (2018) 14362–14366. [9] H. Ku, J. Epaarachchi, M. Trada, P. Wong, Modelling of tensile properties glass powder/epoxy composites post-cured in an oven and in microwaves, J. Reinforced Plast Compos (2013). [10] Jae-soon Jang, Particle Size Effect on Mechanical and Thermal Properties of SiO2 Particulate Polymer Composites, University of Delaware, 2012. [11] V.K. Srivastava, U. Gabbert, H. Berger, M. Shraddha Singh, Analysis of particles loaded fiber composites for the evaluation of effective material properties with the variation of shape and size, Int. J. Eng. Sci. Technol. 3 (1) (2011) 52–68. [12] Anupama Upadhyay, Ramvir Singh, Prediction of effective elastic modulus of biphasic composite materials, Modern Mech. Eng. 2 (2012) 6–13.
Please cite this article as: P. Jeyaprakash and A. Devaraju, Prediction of effective elastic modulus for glass microspheres loaded polymer composites, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.064
Contents lists available at ScienceDirect
Materials Today: Proceedings journal homepage: www.elsevier.com/locate/matpr
Prediction of effective elastic modulus for glass microspheres loaded polymer composites P. Jeyaprakash ⇑, A. Devaraju Department of Mechanical Engineering, Adhi College of Engineering & Technology, Kanchipuram, Tamilnadu, India
a r t i c l e
i n f o
Article history: Received 20 May 2019 Received in revised form 2 August 2019 Accepted 4 August 2019 Available online xxxx Keywords: Particle filled composites Elastic modulus Glass microspheres Mechanical properties Finite element analysis
a b s t r a c t Particle filled polymers are used in very large quantities in advance composite materials. Although investigations have done to evaluate the mechanical properties of particle filled composites, Finite Element Analysis (FEA) analysis have not done so for. Therefore, in this work, the elastic modulus of randomly distributed particle filled composite materials are computed by finite element method and verified with the already published experimental data. Spherically shaped solid and hollow glass particles and epoxy are used as fillers and matrix materials respectively. The FEA is applied to a single representative volume element model. The failure theories and mathematical models are verified with the numerical values. Ó 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the International Conference on Materials Engineering and Characterization 2019.
1. Introduction Particle filled Composite materials are leading classes of engineering materials. They have been developed since the 1960 due to their remarkable properties i.e. high strength and stiffness, less weight in comparison with metals and high chemical resistance. composites have special structural, energetic and mechanical properties i.e. higher Young‘s modulus, highly resistant and exceptional stiffness etc. [1,2]. Thus it is alluring to compute the effective elastic properties from the information of the structure of the composite material. The concept of using microspheres as filler material comes from the late 1970s when hollow glass spheres were used in combination with a resin system for lightweight plastics in marine applications [3,4]. As the automotive industry looks for stronger, lighter and inexpensive alternatives to sheet metal, microspheres with diameters ranging from 10 mm to 100 mm are becoming readily available in a variety of materials and can provide the potential weight saving required. Ranging in weight from 0.13 g/cm3 to 0.54 g/cm3, microspheres can lower the standard weight of SMC parts from 1.5 g/cm3 to 1.3 g/cm3 or lower. Since to the spheres come in a selection of sizes and materials, the overall strength of the spheres can be controlled by the spheres outer diameter, density and material composition [5].
⇑ Corresponding author. E-mail address: [email protected] (P. Jeyaprakash).
Currently, microspheres are being used in a wide range of applications, which include everything from bowling balls to sport and leisure boats. The effective elastic properties are one of the most significant properties that decide the mechanical performance of a material [6,7]. The problem of computing the effective mechanical properties of a composite material with complex microstructures is a moving assignment to scientists and researchers due to its fundamental and innovative significance in almost every area of material science. This problem keeps on being the focus of intense research, from design as well as theoretical point of view. The essential data required for the assessment of the effective moduli is the volume fractions and elastic moduli, of each phase. It is evident that most of the studies, the effective elastic modulus is determined by experimental analysis. The prediction of effective elastic modulus of composites by numerical analysis is scarcely reported. Hence an attempt is made in this study to predict the effective elastic modulus of polymeric composites using finite element analysis. 2. Materials and methods The sample composites used in this study consist of an epoxy matrix filled with various volumes of solid and hollow glass filler. The epoxy used was based upon Diglycidyl ether of Bisphenol-A (DGEBA). Fig. 1 shows the chemical structure for epoxy. The size of hollow and solid microspheres is 20 mm and 100 mm respectively. The modulus and Poisson’s ratio of solid glass microspheres
https://doi.org/10.1016/j.matpr.2019.08.064 2214-7853/Ó 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the International Conference on Materials Engineering and Characterization 2019.
Please cite this article as: P. Jeyaprakash and A. Devaraju, Prediction of effective elastic modulus for glass microspheres loaded polymer composites, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.064
2
P. Jeyaprakash, A. Devaraju / Materials Today: Proceedings xxx (xxxx) xxx
Fig. 1. Chemical structure of Epoxy (DGEBA).
Vm ðSm Sf Þ Su ¼ Sf 1 þ Sf þ Vf ðSm Sf Þð1 gsf Þ
ð6Þ
Vf ðSm Sf Þ Sl ¼ Sm 1 þ Sm Vm ðSm Sf Þð1 gsm Þ
ð7Þ
where the parameters Sf and Sm (i.e. Kf, Km, Gf and Gm) are defined by are stated in literature [8] to have values of 76 GPa and 0.23 respectively. The hollow glass spheres had a density of 0.37 g/cm3. The modulus and Poisson’s ratio for the hollow glass spheres are 1.99 GPa and 0.21 respectively. Solid or hollow glass spheres were added in various volume fractions to this initial epoxy combination. For the solid/hollow filler combinations, the hollow and solid glass fillers are added to the matrix epoxy at volume ratios of 50:50 as gives the better results. The elastic modulus of spheres embedded in a continuous and isotropic elastic matrix is computed through the finite element analysis of the two-dimensional unit cells. 3. Theoretical formulation The Voigt and Reuss models [9] are the two most fundamental models used to develop the properties of a two-phase composite system comprising of a spherical filler addition surrounded in a continuous matrix material. The Voigt model, which is defined by Eqs. (1) and (2), assumes that the particulate addition and matrix material are exposed to constant strain.
Voigt model Ec ¼ Em Vm þ Ef Vf
ð1Þ
Em Ef Em Vf þ Ef Vm
ð2Þ
Reuss model Ec ¼
gKj ¼
1 þ #j 3ð1 #j Þ
ð8Þ
gGj ¼
2ð4 5#j Þ 15ð1 #j Þ
ð9Þ
where j = matrix or the filler; c = Phase continuity parameter
c¼
2uc 1
uc
ð10Þ
uc ¼Critical volume fraction The Eqs. (11)–(14) shows the C-Combining Rule. This rule is applicable for the case of a matrix filled with a more compliant filler, such as a matrix epoxy loaded with hollow glass spheres (i.e., Ef < Em).
Pc ¼ Vm Pu þ Vf Pl þ kp Vf Vm ðPu Pl Þ
ð11Þ
where, kp = phase continuity parameter and P = bulk modulus (K) or shear modulus (G).
Vm ðPm Pf Þ Pl ¼ Pf 1 þ Pf Vf ðPm Pf ÞðgPf Þ
ð12Þ
Some more theoretical models are also given in the Eqs. (3) and (4)
Neilson0 s model Ec ¼ Em ð1 þ 2:5Vf Þ 0
Einstein s model Ec ¼ Em ð1 þ Vf Þ
ð3Þ ð4Þ
The conditions of constant stress or strain over the filler-matrix boundaries are thermodynamically impossible; hence, more rigorous models have been proposed in an effort to increase the prescient exactness of the Voigt and the Reuss models utilizing bounding connections or self-consistent field theories [10]. In this research, The S-Combining Rule is also used to model the glass filled composites for predicting the modulus properties (Eqs. (5)–(10)). S-Combining rule
1 Vm Vf 1 1 ¼ þ þ cVf Vm Sc Sl Su Sl Su
Fig. 2. Geometric Model of Particle filled epoxies.
ð5Þ
Fig. 3a. Finite Element Analysis Loading Condition of SGM.
Fig. 3b. Finite Element Analysis Loading Condition of HGM.
Please cite this article as: P. Jeyaprakash and A. Devaraju, Prediction of effective elastic modulus for glass microspheres loaded polymer composites, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.064
P. Jeyaprakash, A. Devaraju / Materials Today: Proceedings xxx (xxxx) xxx
Pu ¼ Pm 1
kp ¼
Vf ðPm Pf Þ Pm Vm ðPm Pf ÞðgPm Þ
2u c 1 uc ½1 2uc ð1 uc ÞgPm
ð13Þ
ð14Þ
All the parameters used in the S and C combining rule are well known and easily found in the literature. The tensile strength of particle filled composites is computed by following analytical models (15) and (16).
Tavmans model
rc ¼ rm ð1 bVf2=3 Þ
3
ð15Þ
b = 1.21 for Spherical particles
Wongs model
rc ¼ rm ð1 bV2=3 f Þ
ð16Þ
b = 0.5 for lower concentration of fillers
Fig. 4. Stress distribution for RVE model of SGM filled epoxies (Vf = 0.1).
Fig. 5. Displacement for RVE model of SGM filled epoxies (Vf = 0.1).
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4. Numerical analysis
ANSYS WOI
12
ANSYS WI
In this study, using the finite-element software ANSYS Structural analysis is carried out in solid glass microspheres (SGM), hollow glass microspheres (HGM) and solid/hollow mixed glass microspheres filled epoxy ternary system for determine the effective Elastic modulus. In order to perform this analysis, a single representative volume element (RVE) as shown in Fig. 2 and NODE 182 element is considered for analysis [11]. The geometry took advantage of symmetry and therefore was modeled as onequarter of a two dimensional unit cell with a uniaxial pressure applied away from the top of the unit cell as shown in Figs. 3a and b. The finite element model is used to simulate the microstructure of composite materials for nine different particle loading concentrations. Furthermore, the effect of interface between the particle and matrix on the effective elastic modulus is numerically determined for various loading concentrations. 5. Results and discussion The structural analysis of solid and hollow glass microspheres loaded epoxies is carried out by single RVE model at different filler
Elastic modulus (GPa)
10
EXPERIMENTAL NEILSON MODEL
8 S-COMB REUSS MODEL
6 4 2
Volume fraction of filler
0 0
0.1
0.2
0.3
0.4
0.5
Fig. 7. Effective Elastic modulus of SGM filled epoxies.
quantities. The effect of volume fraction on effective elastic modulus of SGM filled epoxy is computed. The different volume fractions varying from 0.05 to 0.45 of the glass microsphere of 100 lm is considered in this study. Figs. 4 and 5 show the stress distribution and the displacement of SGM filled epoxy for the given boundary
Fig. 6. Stress distribution for RVE model of SGM filled epoxies with interface (Vf = 0.1).
Table 1 Effect of filler content in SGM filled the epoxies on Elastic Modulus. Filler
Neat Solid
Volume fraction (Vf)
– 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Elastic Modulus (GPa)
Theoretical modulus (GPa)
Ansys
Experimental [5]
Neilsons model
Einsteins theory
Voigt model
Reuss model
Wongs model
S-combination Rule
2.45 3.13 – 3.9 – 4.99 5.7 – – 10.1
2.45 2.76 3.06 3.37 3.67 3.98 4.29 4.59 4.90 5.20
2.45 2.57 2.69 2.82 2.94 3.06 3.18 3.31 3.43 3.55
2.45 6.13 9.8 13.48 17.16 20.83 24.51 28.19 31.87 35.55
2.45 2.57 2.71 2.87 3.04 3.23 3.45 3.7 4.00 4.34
2.45 2.51 2.57 2.63 2.69 2.75 2.82 2.88 2.94 3.00
2.45 2.72 3.04 3.41 3.85 4.36 4.97 5.71 6.6 7.69
WOI
WI
– 2.93 3.33 3.79 4.34 5.03 5.98 7.22 8.59 11.82
– 2.79 3.1 3.44 3.76 4.21 4.71 5.27 5.52 –
Please cite this article as: P. Jeyaprakash and A. Devaraju, Prediction of effective elastic modulus for glass microspheres loaded polymer composites, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.064
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P. Jeyaprakash, A. Devaraju / Materials Today: Proceedings xxx (xxxx) xxx Table 2 Effect of filler content in SGM filled the epoxies on Tensile Strength. Filler
Volume fraction
Max. stress (MPa)
Theoretical stress (MPa)
Ansys
Neat Solid
– 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
WOI
WI
57.73 57.81 57.23 56.03 55.02 54.58 52.37 36.19 24.33
57.82 56.25 55.9 55.07 53.5 53.14 40.49 32.03 0
70
Tensile Strength (MPa)
60 50 40 30
Ansys WOI Ansys WI
20
Experimental Tavmans model
10
Wongs model
Volume fraction of filler
0 0
0.1
0.2
0.3
0.4
0.5
Fig. 8. Tensile Strength of SGM filled epoxies.
conditions of a Representative Volume Element model. The effect of interface between SGM and epoxy is shown in Fig. 6. The Table 1 shows the effect of SGM filler content in the epoxies on elastic modulus. The theoretical models as discussed in Section 2 are also
Experimental [4]
Tavmanmodel
Wongmodel
58 57.34 57.14 56.92 56.59 56.18 55.65 55.06 – –
58 48.47 42.88 38.19 34 30.15 26.55 23.14 19.9 16.79
58 54.06 51.75 49.81 48.08 46.49 45 43.59 42.26 40.97
computed. The numerical results are compared with the literature experimental data [12]. Fig. 7 shows the impact of volume fraction on the effective elastic modulus of Solid Glass Microspheres filled epoxy composite. It is construed that the volume fraction of filler increases as effective elastic modulus increases. The numerical results gives closure values to experimental values. Superior accuracy is observed for S-Combining rule while comparing with other models [2,4]. The effective elastic modulus is increased significantly for the case of solid glass filler. The tensile strength of SGM filled epoxy is computed numerically and compared with experimental data [5] shown in Table 2. Also the analytical models Tavman and Wongs [2] are calculated. Fig. 8 shows tensile strength of SGM loaded epoxies. It is seen that with addition of SGMs, tensile strength of the composite decreases and this decrement is a function of the SGM content. This is confirmed by experimental values. The stress distribution of HGM filled epoxy for the given boundary conditions of a RVE model at Vf = 0.1 is shown in Fig. 9. The Table 3 shows the effect of HGM filler content in the epoxies on elastic modulus. The composite modulus is decreased significantly for the case of Hollow glass filler. Fig. 10 shows the effect of volume fraction on the effective elastic modulus of HGM filled epoxy composite.
Fig. 9. Stress distribution for RVE model of HGM filled epoxies (Vf = 0.1).
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P. Jeyaprakash, A. Devaraju / Materials Today: Proceedings xxx (xxxx) xxx
Table 3 Effect of filler content in HGM filled the epoxies on Elastic modulus. Filler
Volume fraction
Neat Hollow
– 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Elastic modulus (GPa)
Theoretical modulus (GPa)
Ansys
Experimental [4]
Neilsons model
Einsteins theory
Voigt model
Reuss model
Wongs model (low)
C-Combination Rule
2.45 – – 2.7 – 3.07 – 2.61 – 2.67
2.45 2.75 3.06 3.37 3.67 3.98 4.29 4.59 4.9 5.2
2.45 2.57 2.69 2.82 2.94 3.06 3.18 3.31 3.43 3.55
2.45 2.43 2.4 2.38 2.36 2.33 2.31 2.29 2.26 2.43
2.45 2.42 2.39 2.37 2.34 2.32 2.29 2.26 2.24 2.22
2.45 2.51 2.57 2.63 2.69 2.76 2.82 2.88 2.94 3.00
2.45 2.43 2.41 2.38 2.36 2.34 2.32 2.3 2.28 2.26
Woi
Wi
– 2.323 2.318 2.316 2.313 2.31 2.306 2.303 2.299 2.298
– 2.323 2.319 2.316 2.313 2.319 2.306 2.303 2.299 2.298
Table 4 Effect of filler content in SGM/HGM filled the epoxies on Elastic modulus.
3.5
Filler
Elastic Modulus (Gpa)
3
Volume fraction
2.5
Elastic modulus (GPa) Ansys
Theoretical modulus (GPa) Neilsons
Einstein
– 3 3.63 4.498
2.45 2.75 3.06 3.37
2.45 2.51 2.57 2.63
2 Neat Solid/Hollow
1.5
Ansys Experimental C-Comb
1
– 0.05 0.1 0.15
Wongs model 2.51 2.57 2.63
0.5 Volume fraction of filler 0 0
0.1
0.2
0.3
0.4
0.5
Fig. 10. Effective Elastic modulus of HGM filled epoxies.
It is inferred that the volume fraction of filler decreases as effective elastic modulus increases. The numerical results give closure values to experimental values. In analytical models C-Combining
rule confirms the above results. Binary filler mixtures of solid and hollow glass were added to the matrix epoxy to yield ternary composite systems. For these blended filler systems, the volume fraction of filler was differed. As mentioned in Section 2 the solid and hollow glass fillers were added to the matrix epoxy at volume proportion of 50:50. The Table 4 shows the effect of SGM/HGM filler content in the epoxies on elastic modulus. The composite modulus is increased significantly for the case of Solid/Hollow glass filler. Fig. 11 shows the stress distribution of SGM/HGM filled epoxy at volume fraction of 0.1 and Fig. 12 shows the Effective
Fig. 11. Stress distribution for RVE model of SGM/HGM filled epoxies (Vf = 0.1).
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P. Jeyaprakash, A. Devaraju / Materials Today: Proceedings xxx (xxxx) xxx
6. Conclusion
5 4.5 Elastic modulus (Gpa)
7
In this study, the effective elastic modulus of a glass microspheres filled epoxy composites is computed numerically. Effective Elastic modulus is decreases with increase of HGM content in epoxy but increase with increase of SGM content. It is increases as a function of volume fraction in SGM/HGM ternary composite system. The numerical results conforms (95%) the literature experimental data. This is investigated with S-Combining Rule and Ccombining rules give the closure values (98%) of effective elastic modulus.
4 3.5 3 2.5 Ansys
2
Neilsons model
1.5
Einsteins model 1
Wong model
0.5
References
Volume fraction of filler
0 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Fig. 12. Effective Elastic modulus of SGM/HGM filled epoxies.
SGM
ELASTIC MODULUS(GPA)
5 4.5
HGM SGM/HGM
4 3.5 3 2.5 2 1.5 1 0.5 0
0.05
0.1
0.15
VOLUME FRACTION
Fig. 13. Comparison of Effective Elastic modulus with various fillers.
elastic modulus of SGM/HGM filled epoxy at various volume fractions. Effective elastic modulus comparison of SGM, HGM and SGM/HGM ternary system is shown in Fig. 13.
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Please cite this article as: P. Jeyaprakash and A. Devaraju, Prediction of effective elastic modulus for glass microspheres loaded polymer composites, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.064