Research Papers

Fiber Diameter-Dependent Elastic Deformation in Polymer Composites—A Numerical Study

[+] Author and Article Information
Nitin Garg

Department of Mechanical Engineering,
University of Wyoming,
Laramie, WY 82071
e-mail: enitingarg@gmail.com

Gurudutt Chandrashekar

Department of Mechanical Engineering,
University of Wyoming,
Laramie, WY 82071
e-mail: chandrashekarg@trine.edu

Farid Alisafaei

Department of Mechanical Engineering,
University of Wyoming,
Laramie, WY 82071
e-mail: alisafae@seas.upenn.edu

Chung-Souk Han

Lawrence Livermore National Laboratory,
7000 East Avenue,
Livermore, CA 94550
e-mail: chungsouk.han@gmail.com

1Present address: Trine University, One University Avenue, Angola, IN 46703.

2Present address: Department of Material Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104.

3Present address: 9824 Peters Ranch Way, Elk Grove, CA 94550.

4Corresponding author.

Contributed by the Materials Division of ASME for publication in the Journal of Engineering Materials and Technology. Manuscript received March 8, 2018; final manuscript received April 22, 2019; published online June 4, 2019. Assoc. Editor: Erdogan Madenci.

J. Eng. Mater. Technol 142(1), 011002 (Jun 04, 2019) (7 pages) Paper No: MATS-18-1065; doi: 10.1115/1.4043766 History: Received March 08, 2018; Accepted April 24, 2019

Microbeam bending and nano-indentation experiments illustrate that length scale-dependent elastic deformation can be significant in polymers at micron and submicron length scales. Such length scale effects in polymers should also affect the mechanical behavior of reinforced polymer composites, as particle sizes or diameters of fibers are typically in the micron range. Corresponding experiments on particle-reinforced polymer composites have shown increased stiffening with decreasing particle size at the same volume fraction. To examine a possible linkage between the size effects in neat polymers and polymer composites, a numerical study is pursued here. Based on a couple stress elasticity theory, a finite element approach for plane strain problems is applied to predict the mechanical behavior of fiber-reinforced epoxy composite materials at micrometer length scale. Numerical results show significant changes in the stress fields and illustrate that with a constant fiber volume fraction, the effective elastic modulus increases with decreasing fiber diameter. These results exhibit similar tendencies as in mechanical experiments of particle-reinforced polymer composites.

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Fu, S. Y., Feang, X. Q., Lauke, B., and Mai, Y. W., 2008, “Effects of Particle Size, Particle/Matrix Interface Adhesion and Particle Loading on Mechanical Properties of Particulate–Polymer Composites,” Compos. Part B, 39(6), pp. 933–961. [CrossRef]
Nakamura, Y., Yamaguchi, M., Okubo, M., and Matsumoto, T., 1992, “Effects of Particle Size on Mechanical and Impact Properties of Epoxy Resin Filled With Spherical Silica,” J. Appl. Polym. Sci., 45, pp. 1281–1289. [CrossRef]
Adachi, T., Osaki, M., Araki, W., and Kwon, S. C., 2008, “Fracture Toughness of Nano- and Micro-Spherical Silica-Particle-Filled Epoxy Composites,” Acta Mater., 56(9), pp. 2101–2109. [CrossRef]
Jancar, J., Hoy, R. S., Jancarova, E., and Zidek, J., 2015, “Effect of Temperature, Strain Rate and Particle Size on the Yield Stresses and Post-Yield Strain Softening of PMMA and Its Composites,” Polymer, 63, pp. 196–207. [CrossRef]
Spanoudakis, J., and Young, R. J., 1984, “Crack Propagation in a Glass Particle Filled Epoxy-Resin. 1. Effect of Particle-Volume Fraction and Size,” J. Mater. Sci., 19(2), pp. 473–486. [CrossRef]
Tjernlund, J. A., Gamstedt, E. K., and Gudmundson, P., 2006, “Length-Scale Effects on Damage Development in Tensile Loading of Glass-Sphere Filled Epoxy,” Int. J. Solids Struct., 43(24), pp. 7337–7357. [CrossRef]
Kalfus, J., and Jancar, J., 2008, “Reinforcing Mechanisms in Amorphous Polymer Nano-Composites,” Compos. Sci. Technol., 68(15–16), pp. 3444–3447. [CrossRef]
Keddie, J. L., and Jones, R. A. L., 1995, “Glass Transition Behavior in Ultra-Thin Polystyrene Films,” ISR Chem. Soc., 35, pp. 21–26. [CrossRef]
Lam, D. C. C., Yang, F., Chong, A. C. M., Wang, J., and Tong, P., 2003, “Experiments and Theory in Strain Gradient Elasticity,” J. Mech. Phys. Solids, 51(8), pp. 1477–1508. [CrossRef]
McFarland, A. W., and Colton, J. S., 2005, “Role of Material Microstructure in Plate Stiffness With Relevance to Microcantilever Sensors,” J. Micromech. Microeng., 15, pp. 1060–1067. [CrossRef]
Liebold, C., and Muller, W. H., 2016, “Comparison of Gradient Elasticity Models for the Bending of Micromaterials,” Comput. Mater. Sci., 116, pp. 52–61. [CrossRef]
Lam, D. C. C., and Chong, A. C. M., 2000, “Effect of Cross-Link Density on Strain Gradient Plasticity in Epoxy,” Mater. Sci. Eng. A, 281(1–2), pp. 156–161. [CrossRef]
Zhang, T. Y., and Xu, W. H., 2002, “Surface Effects on Nanoindentation,” J. Mater. Res., 17(7), pp. 1715–1720. [CrossRef]
Han, C. S., and Nikolov, S., 2007, “Indentation Size Effects in Polymers and Related Rotation Gradients,” J. Mater. Res., 22(6), pp. 1662–1672. [CrossRef]
Han, C. S., 2010, “Influence of the Molecular Structure on Indentation Size Effect in Polymers,” Mater. Sci. Eng. A, 527(3), pp. 619–624. [CrossRef]
Briscoe, B. J., Fiori, L., and Pelillo, E., 1998, “Nanoindentation of Polymeric Surfaces,” J. Phys. D, 31(19), pp. 2395–2405. [CrossRef]
Chong, A. C. M., and Lam, D. C. C., 1999, “Strain Gradient Plasticity Effect in Indentation Hardness of Polymers,” J. Mater. Res., 14(10), pp. 4103–4110. [CrossRef]
Balta Calleja, F. J., Flores, A., and Michler, G. H., 2004, “Microindentation Studies at the Near Surface of Glassy Polymers: Influence of Molecular Weight,” J. Appl. Polym. Sci., 93, pp. 1951–1956. [CrossRef]
Alisafaei, F., Han, C. S., and Lakhera, N., 2014, “Characterization of Indentation Size Effects in Epoxy,” Polym. Test., 40, pp. 70–78. [CrossRef]
Xu, W. H., Xiao, Z. Y., and Zhang, T. Y., 2005, “Mechanical Properties of Silicone Elastomer on Temperature in Biomaterial Application,” Mater. Lett., 59(17), pp. 2153–2155. [CrossRef]
Seltzer, R., Kim, J. K., and Mai, Y. W., 2011, “Elevated Temperature Nanoindentation Behaviour of Polyamide 6,” Polym. Int., 60, pp. 1753–1761. [CrossRef]
Alisafaei, F., and Han, C. S., 2015, “Indentation Depth Dependent Mechanical Behavior in Polymers,” Adv. Condens. Matter Phys., 2015, p. 391579. [CrossRef]
Dutta, A. K., Penumadu, D., and Files, B., 2004, “Nanoindentation Testing for Evaluating Modulus and Hardness of Single-Walled Carbon Nanotube-Reinforced Epoxy Composites,” J. Mater. Res., 19(1), pp. 158–164. [CrossRef]
Shen, L., Liu, T., and Lv, P., 2005, “Polishing Effect on Nanoindentation Behavior of Nylon 66 and Its Nanocomposites,” Polym. Test., 24(6), pp. 746–749. [CrossRef]
Chandrashekar, G., and Han, C. S., 2016, “Length Scale Effects in Epoxy: The Dependence of Elastic Moduli Measurements on Spherical Indenter Tip Radius,” Polym. Test., 53, pp. 227–233. [CrossRef]
Alisafaei, F., Han, C. S., and Sanei, S. H. R., 2013, “On Time and Depth Dependence of Hardness, Dissipation, and Stiffness in the Indentation of Polydimethylsiloxane,” Polym. Test., 32(7), pp. 1220–1228. [CrossRef]
Bucsek, A. J., Alisafaei, F., Han, C. S., and Lakhera, N., 2016, “Effect of Crosslink Density on the Indentation Size Effect in Polydimethylsiloxane,” Polym. Bull., 73(3), pp. 763–772. [CrossRef]
Chandrashekar, G., Alisafaei, F., and Han, C. S., 2015, “Length Scale Dependent Deformation in Natural Rubber,” J. Appl. Polym. Sci., 132, p. 42683. [CrossRef]
Alisafaei, F., Han, C. S., and Garg, N., 2016, “On Couple Stress Elasto-Plastic Constitutive Frameworks for Glassy Polymers,” Int. J. Plast., 77, pp. 30–53. [CrossRef]
Nikolov, S., Han, C. S., and Raabe, D., 2007, “On the Origin of Size Effects in Small-Strain Elasticity of Solid Polymers,” Int. J. Solids Struct., 44(5), pp. 1582–1592 (Corrigendum: Int. J. Solids Struct., 2007, 44, p. 7713). [CrossRef]
Garg, N., and Han, C. S., 2013, “A Penalty Finite Element Approach for Couple Stress Elasticity,” Comput. Mech., 52(3), pp. 709–720. [CrossRef]
Nix, W. D., and Gao, H., 1998, “Indentation Size Effects in Crystalline Materials: A Law for Strain Gradient Plasticity,” J. Mech. Phys. Solids, 46(3), pp. 411–425. [CrossRef]
Yang, F., Chong, A. C. M., Lam, D. C. C., and Tong, P., 2002, “Couple-Stress Based Strain Gradient Theory for Elasticity,” Int. J. Solids Struct., 39(10), pp. 2731–2743. [CrossRef]
Hadjesfandiari, A. R., and Dargush, G. F., 2011, “Couple Stress Theory for Solids,” Int. J. Solids Struct., 48(18), pp. 2496–2510. [CrossRef]
Ghosh, S., Kumar, A., Sundararaghavan, V., and Waas, A. M., 2013, “Non-Local Modeling of Epoxy Using an Atomistically-Informed Kernel,” Int. J. Solids Struct., 50(19), pp. 2837–2845. [CrossRef]
Han, C. S., Ma, A., Roters, F., and Raabe, D., 2007, “A Finite Element Approach With Patch Projection for Strain Gradient Plasticity Formulations,” Int. J. Plast., 23(4), pp. 690–710. [CrossRef]
Lee, M. G., and Han, C. S., 2012, “Patch Projection Based Explicit Finite Element Approach for Strain Gradient Plasticity Formulations,” Comput. Mech., 49(2), pp. 171–183. [CrossRef]
Petera, J., and Pittman, J. F. T., 1994, “Isoparametric Hermite Elements,” Int. J. Numer. Methods Eng., 37, pp. 3489–3519. [CrossRef]
Fischer, P., Klassen, M., Mergheim, J., Steinmann, P., and Müller, R., 2011, “Isogeometric Analysis of 2D Gradient Elasticity,” Comput. Mech., 47(3), pp. 325–334. [CrossRef]
Garg, N., and Han, C. S., 2015, “An Axisymmetric Finite Element Formulation for Couple Stress Elasticity,” Arch. Appl. Mech., 85(5), pp. 587–600. [CrossRef]
Garg, N., Han, C. S., and Alisafaei, F., 2016, “Length Scale Dependence in Elastomers—Comparison of Indentation Experiments With Numerical Simulations,” Polymer, 98, pp. 201–209. [CrossRef]
Huang, Z. M., 2001, “Micromechanical Prediction of Ultimate Strength of Transversely Isotropic Fibrous Composites,” Int. J. Solids Struct., 38(22–23), pp. 4147–4172. [CrossRef]
Bazant, Z. P., and Christensen, M., 1972, “Analogy Between Micropolar Continuum and Grid Frameworks Under Initial Stress,” Int. J. Solids Struct., 8(3), pp. 327–346. [CrossRef]
Leonetti, L., Greco, F., Trovalusci, P., Luciano, R., and Masiani, R., 2018, “A Multiscale Damage Analysis of Periodic Composites Using a Couple-Stress/Cauchy Multidomain Model: Application to Masonry Structures,” Composites B, 141, pp. 50–59. [CrossRef]
Bacca, M., Dal Corso, F., Veber, D., and Bigoni, D., 2013, “Anisotropic Effective Higher-Order Response of Heterogeneous Cauchy Elastic Materials,” Mech. Res. Commun., 54, pp. 63–71. [CrossRef]
Xue, Z., Huang, Y., and Li, M., 2002, “Particle Size Effect in Metallic Materials: A Study by the Theory of Mechanism-Based Strain Gradient Plasticity” Acta Mater., 50(1), pp. 149–160. [CrossRef]
Chandrashekar, G., and Han, C. S., 2018, “Numerical Evaluation of the Size-Dependent Elastic Properties of Cellular Polymers,” ASME J. Eng. Mater. Technol., 140(1), p. 011004. [CrossRef]


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Fig. 1

Indentation hardness HI versus indentation depth h redrawn from Refs. [1315,23,24]

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Fig. 2

Degrees-of-freedom for a four-node element with full and reduced integration points

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Fig. 3

Epoxy beam loading–unloading data illustrating elastic behavior (adapted from Ref. [9])

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Fig. 4

Bending rigidity versus beam thickness h (adapted from Ref. [31] with experimental data of Ref. [9])

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Fig. 5

Hexagonal configuration subjected to prescribed displacement

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Fig. 6

σp with (a) and without (b) rotation gradients for d = 8 μm, r = 5 μm, and Vf = 32.65%

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Fig. 7

σpmax versus r for Vf = 32.65% (hexagonal configuration)

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Fig. 8

Eeff over r for Vf = 32.65%

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Fig. 9

Eeff versus Vf and r with and without rotation gradients χ

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Fig. 10

ϕ3 for Vf = 60.13%

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Fig. 11

σVMmax versus r for Vf = 32.65% in the hexagonal configuration

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Fig. 12

Average shear stress τave versus r with and without rotation gradients χ

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Fig. 13

Square configuration with prescribed displacement

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Fig. 14

Eeff for Vf = 30.68%

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Fig. 15

Eeff versus Vf and r in the square configuration with and without rotation gradients χ

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Fig. 16

Effect of rotation gradients on σpmax (left) and Eeff (right) where Eeff* and Eeffo are effective elastic moduli with and without rotation gradients (Vf = 44.1%, r = 6 μm, E = 3.8 GPa, v = 0.355)

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Fig. 17

Eeff of glass–epoxy (left) and graphite–epoxy (right) composites in the square configuration over r for varying Vf



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