This article presents a critical review and evaluation of several mechanics models
which are commonly used for elastic stiffness predictions of polymer/clay
nanocomposites (PCNs). In particular, the assumptions and limitations of these models
and formulas were examined. Stiffness predictions were compared to published data in
order to assess their applicability and effectiveness to real PCNs. Whilst a more
complex model can achieve better agreement with experimental data, more parameters,
which must be obtained by separate independent tests, are always required. This may
lead to increasing errors for model applications. It is shown that meso-mechanics
models due to Mori—Tanaka, Ji et al. and Luo—Daniel give the best overall
predictions, with due consideration of the dispersion and orientation effects of the
platelets embedded in a polymer matrix.
Usuki, A., Kojima, Y., Kawasumi, M., Okada, A., Fukushima, Y., Kurauchi, T., et al. (1993). Synthesis of Nylon 6-Clay
Hybrid, Journal of Materials Research,
8(5):
1179-1184.
2.
Kojima, Y., Usuki, A., Kawasumi, M., Okada, A., Fukushima, Y., Kurauchi, T., et al. (1993). Mechanical Properties of Nylon
6-Clay Hybrid , Journal of Materials Research,
8(5):
1185-1189.
3.
Wang, Z. and Pinnavaia, T.J. (1998). Nanolayer Reinforcement of Elastomeric
Polyurethane , Chemistry of Materials,
10(12):
3769-3771.
4.
Masenelli-Varlot, K., Reynaud, E., Vigier , G. and Varlet, J. (2002). Mechanical Properties of Clay-Reinforced
Polyamide, Journal of Polymer Science Part B: Polymer
Physics, 40(3):
272-283.
5.
Messersmith, P.B. and Giannelis , E.P. (1995). Synthesis and Barrier Properties of
Poly(-Caprolactone)-Layered Silicate Nanocomposites ,
Journal of Polymer Science Part A: Polymer Chemistry,
33(7):
1047-1057.
6.
Oriakhi, C. (1998). Nano Sandwiches,
Chemistry in Britain, 34(11):
59-62.
7.
Vaia, R.A. and Giannelis, E.P. (1997). Polymer Melt Intercalation in
Organically-Modified Layered Silicates: Model Predictions and
Experiment, Macromolecules ,
30(25):
8000-8009.
8.
Cho, J.W. and Paul, D.R. (2001). Nylon 6 Nanocomposites by Melt
Compounding, Polymer,
42(3):
1083-1094.
9.
Yu, Z.-Z. , Yang, M., Zhang , Q., Zhao, C. and Mai , Y.-W. (2003). Dispersion and Distribution of
Organically Modified Montmorillonite in Nylon-66 Matrix,
Journal of Polymer Science Part B: Polymer Physics ,
41(11):
1234-1243.
10.
Kornmann, X. , Berglund, L.A., Sterte , J. and Giannelis, E.P. (1998). Nanocomposites Based on Montmorillonite
and Unsaturated Polyester, Polymer Engineering and
Science, 38(8):
1351-1358.
11.
Vaia, R.A. and Giannelis, E.P. (1997). Lattice Model of Polymer Melt
Intercalation in Organically-Modified Layered Silicates,
Macromolecules, 30(25):
7990-7999.
12.
Ginzburg, V.V. , Singh, C. and Balazs, A.C. (2000). Theoretical Phase Diagrams of
Polymer/Clay Composites: The Role of Grafted Organic Modifiers,
Macromolecules, 33(3):
1089-1099.
13.
Bicerano, J. , Douglas, J.F. and Brune, D.A. (1999). Model for the Viscosity of Particle
Dispersions, Journal of Macromolecular Science - Reviews
in Macromolecular Chemistry and Physics,
C39(4):
561-642.
14.
Fornes, T.D. , Yoon, P.J., Hunter , D.L., Keskkula, H. and Paul, D.R. (2002). Effect of Organoclay Structure on Nylon 6
Nanocomposite Morphology and Properties ,
Polymer, 43(22):
5915-5933.
15.
Fornes, T.D. , Yoon, P.J., Keskkula , H. and Paul, D.R. (2001). Nylon 6 Nanocomposites: The Effect of
Matrix Molecular Weight, Polymer,
42(25):
9929-9940.
16.
Sheng, N., Boyce, M.C., Parks, D.M., Rutledge, G.C., Abes, J.I. and Cohen, R.E. (2004). Multiscale Micromechanical Modeling of
Polymer/Clay Nanocomposites and the Effective Clay Particle,
Polymer, 45(2):
487-506.
17.
Hu, N., Fukunaga, H., Lu, C., Kameyama, M. and Yan, B. (2005). Prediction of Elastic Properties of
Carbon Nanotube Reinforced Composites, Proceedings of the
Royal Society of London - A, 461(2058):
1685-1710.
18.
Fertig III, R.S. and Garnich, M.R. (2004). Influence of Constituent Properties and
Microstructural Parameters on the Tensile Modulus of a Polymer/Clay
Nanocomposite, Composites Science and Technology
, 64(16):
2577-2588.
19.
Hbaieb, K., Wang, Q.X., Chia, Y.H.J. and Cotterell, B. (2007). Modelling Stiffness of Polymer/Clay
Nanocomposites , Polymer,
48(3):
901-909.
20.
Hill, R. (1952). The Elastic Behaviour of a Crystalline
Aggregate , Proceedings of the Physical Society of London
Section A , 65(389):
349-355.
21.
Hashin, Z. and Shtrikman, S. (1963). A Variational Approach to the Theory of
the Elastic Behaviour of Multiphase Materials, Journal of
the Mechanics and Physics of Solids,
11(2):
127-140.
22.
Walpole, L.J. (1966). On Bounds for the Overall Elastic Moduli
of Inhomogeneous Systems-II, Journal of the Mechanics and
Physics of Solids, 14(5):
289-301.
23.
Walpole, L.J. (1969). On the Overall Elastic Moduli of
Composite Materials, Journal of the Mechanics and Physics
of Solids, 17(4):
235-251.
24.
Iwakuma, T. and Koyama, S. (2005). An Estimate of Average Elastic Moduli of
Composites and Polycrystals, Mechanics of
Materials, 37(4):
459-472.
25.
Takayanagi, M., Minami, S. and Uemura, S. (1964). Application of Equivalent Model Method to
Dynamic Rheo-Optical Properties of Crystalline Polymer,
Journal of Polymer Science Part C-Polymer Symposium,
5(1):
113-122.
26.
Eshelby, J.D. (1957). The Determination of the Elastic Field of
an Ellipsoidal Inclusion, and Related Problems, Proceedings of the Royal Society
of London. Series A, Mathematical and Physical
Sciences, 241(1226):
376-396.
27.
Hill, R. (1965). A Self-Consistent Mechanics of Composite
Materials , Journal of the Mechanics and Physics of
Solids, 13(4):
213-222.
28.
Dai, L.H. , Huang, Z.P. and Wang, R. (1999). Explicit Expressions for Bounds for the
Effective Moduli of Multi-Phased Composites by the Generalized Self-Consistent
Method, Composites Science and Technology ,
59(11):
1691-1699.
29.
Halpin, J.C. and Kardos, J.L. (1976). The Halpin-Tsai Equations: A
Review, Polymer Engineering and Science,
16(5):
344-352.
30.
Hermans, J.J. (1967). Elastic Properties of Fiber Reinforced
Materials When Fibers Are Aligned, Koninklijke Nederlandse
Akademie Van Weteschappen-Proceedings Series B-Physical Sciences ,
70(1):
1-9.
31.
Mori, T. and Tanaka, K. (1973). Average Stress in Matrix and Average
Elastic Energy of Materials with Misfitting Inclusions,
Acta Metallurgica , 21(5):
571-574.
32.
Benveniste, Y. (1987). A New Approach to the Application of
Mori-Tanaka's Theory in Composite Materials, Mechanics of
Materials, 6(2):
147-157.
33.
Ji, X.L. , Jing, J.K., Jiang , W. and Jiang, B.Z. (2002). Tensile Modulus of Polymer
Nanocomposites, Polymer Engineering and Science,
42(5):
983-993.
34.
Luo, J.-J. and Daniel, I.M. (2003). Characterization and Modeling of
Mechanical Behavior of Polymer/Clay Nanocomposites,
Composites Science and Technology,
63(11):
1607-1616.
35.
Fornes, T.D. and Paul, D.R. (2003). Modeling Properties of Nylon 6/Clay
Nanocomposites using Composite Theories, Polymer,
44(17):
4993-5013.
36.
Wang, J. and Pyrz, R. (2004). Prediction of the Overall Moduli of
Layered Silicate-Reinforced Nanocomposites - Part II: Analyses,
Composites Science and Technology,
64(7-8):
935-944.
37.
Marzari, N. and Ferrari, M. (1992). Textural and Micromorphological Effects
on the Overall Elastic Response of Macroscopically Anisotropic
Composites , Journal of Applied Mechanics-Transactions of
the ASME, 59(2):
269-275.
38.
Price, C.D. , Hine, P.J., Whiteside , B., Cunha, A.M. and Ward, I.M. (2006). Modelling the Elastic and Thermoelastic
Properties of Short Fibre Composites with Anisotropic Phases ,
Composites Science and Technology ,
66(1):
69-79.
39.
Mura, T. (1982). Micromechanics of Defects in Solids,
Martinus Nijhoff Publishers.
Dordrecht.
40.
Qiu, Y.P. and Weng, G.J. (1990). On the Application of Mori-Tanaka's
Theory Involving Transversely Isotropic Spheroidal Inclusions ,
International Journal of Engineering Science,
28(11):
1121-1137.
41.
Tandon, G.P. and Weng, G.J. (1984). The Effect of Aspect Ratio of Inclusions
on the Elastic Properties of Unidirectionally Aligned Composites,
Polymer Composites, 5(4):
327-333.
42.
Wang, J. and Pyrz, R. (2004). Prediction of the Overall Moduli of
Layered Silicate-Reinforced Nanocomposites - Part I: Basic Theory and
Formulas, Composites Science and Technology,
64(7-8):
925-934.
43.
Norris, A.N. (1990). The Mechanical Properties of Platelet
Reinforced Composites, International Journal of Solids and
Structures , 26(5-6):
663-674.
44.
Berryman, J.G. (1980). Long-Wavelength Propagation in Composite
Elastic Media. 2, Ellipsoidal Inclusions, Journal of the Acoustical Society of America,
68(6):
1820-1831.
45.
Brune, D.A. and Bicerano, J. (2002). Micromechanics of Nanocomposites:
Comparison of Tensile and Compressive Elastic Moduli, and Prediction of Effects of
Incomplete Exfoliation and Imperfect Alignment on Modulus,
Polymer, 43(2):
369-387.
46.
Chen, T., Dvorak, G.J. and Benveniste, Y. (1992). Mori-Tanaka Estimate of the Overall
Elastic Moduli of Certain Composite Materials, Journal of
Applied Mechanics , 59:
539-546.
47.
Yoon, P.J., Hunter, D.L. and Paul, D.R. (2003). Polycarbonate Nanocomposites. Part 1.
Effect of Organoclay Structure on Morphology and Properties, Polymer , 44(18):
5323-5339.
48.
Yoon, P.J., Hunter, D.L. and Paul, D.R. (2003). Polycarbonate Nanocomposites: Part 2.
Degradation and Color Formation, Polymer,
44(18):
5341-5354.
49.
Oshinski, A.J. , Keskkula, H. and Paul, D.R. (1996). The Role of Matrix Molecular Weight in
Rubber Toughened Nylon 6 Blends: 1 . Morphology,
Polymer, 37(22):
4891-4907.
50.
Tyan, H.-L. , Wei, K.-H. and Hsieh, T.-E. (2000). Mechanical Properties of Clay-Polyimide
(Btda-Oda) Nanocomposites Via Oda-Modified Organoclay,
Journal of Polymer Science Part B: Polymer Physics ,
38(22):
2873-2878.
Bauer, C.L. and Farris, R.J. (1989). Determination of Poisson's Ratio for
Polyimide Films, Polymer Engineering and Science
, 29(16):
1107-1110.
53.
Christensen, R.M. (1990). A Critical Evaluation for a Class of
Micromechanics Models, Journal of the Mechanics and
Physics of Solids, 38(3):
379-404.
54.
Kwon, P. and Dharan, C.K.H. (1995). Effective Moduli of High Volume Fraction
Particulate Composites, Acta Metallurgica et
Materialia, 43(3):
1141-1147.
55.
Advani, S.G. and Tucker III, C.L. (1987). The Use of Tensors to Describe and
Predict Fiber Orientation in Short Fiber Composites ,
Journal of Rheology, 31(8):
751-784.
56.
O'Regan, M. , Akay, D.F. and Meenan, B. (1999). A Comparison of Young's Modulus
Predictions in Fibre-Reinforced-Polyamide Injection Mouldings,
Composites Science and Technology,
59(3):
419-427.
57.
Dray, D., Gilormini, P. and Regnier, G. (2007). Comparison of Several Closure
Approximations for Evaluating the Thermoelastic Properties of an Injection Molded
Short-Fiber Composite, Composites Science and
Technology, 67(7-8):
1601-1610.
