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TECHNICAL PAPERS

# A Hybrid Numerical-Analytical Method for Modeling the Viscoelastic Properties of Polymer Nanocomposites

[+] Author and Article Information
Hua Liu

Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208

L. Catherine Brinson1

Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208cbrinson@northwestern.edu

$Tg$ is normally designated as the temperature where the tan $δ$ curve peaks. The $Tg$ broadening in the present study refers to the broadening of the tan $δ$ peak.

Quasicontinuum (QC) method was developed based on the Cauchy-Born rule to simulate deformation in solids. The Cauchy-Born rule is a local approximation of the strain energy and gives the strain energy at a given point by using the strain energy associated with a crystal subjected to the same homogeneous deformation as exists at that point. By using this rule, the continuum stress tensor and tangent stiffness can be derived directly from the interatomic potential by differentiating the potential with respect to local deformation gradient once and twice, respectively.

Note that in Eshelby’s analysis, an inclusion is defined as a finite domain embedded in the matrix that has the same properties as the surrounding matrix, while an inhomogeneity is defined as a finite domain embedded in the matrix that has different properties to its surrounding matrix. The term inclusion in this article actually refers to the inhomogeneity.

Dilute refers to the fact that there is only a single inclusion embedded in the infinite matrix.

The strains were applied to the matrix. As the inclusions see the matrix as infinite body, they see these strains as farfield strains.

1

Author to whom correspondence should be addressed.

J. Appl. Mech 73(5), 758-768 (Dec 05, 2005) (11 pages) doi:10.1115/1.2204961 History: Received February 18, 2005; Revised December 05, 2005

## Abstract

In this paper, we present a novel hybrid numerical-analytical modeling method that is capable of predicting viscoelastic behavior of multiphase polymer nanocomposites, in which the nanoscopic fillers can assume complex configurations. By combining the finite element technique and a micromechanical approach (particularly, the Mori-Tanaka method) with local phase properties, this method operates at low computational cost and effectively accounts for the influence of the interphase as well as in situ nanoparticle morphology. A few examples using this approach to model the viscoelastic response of nanotube and nanoplatelet polymer nanocomposite are presented. This method can also be adapted for modeling other behaviors of polymer nanocomposites, including thermal and electrical properties. It is potentially useful in the prediction of behaviors of other types of nanocomposites, such as metal and ceramic matrix nanocomposites.

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## Figures

Figure 1

SEM image of fracture surface of SWNT/PMMA nanocomposites; note that the SWNTs are highly curved in situ (2)

Figure 10

Storage and loss Young’s moduli of nanocomposites with flat/half-annulus nanoplatelets; nanoplatelet content 0.5vol%

Figure 9

Two-dimensional finite element model (mesh) for numerical dilute strain concentration tensor calculation in nanoplatelet/PC nanocomposite (top); isolated mesh of the interphase (bottom)

Figure 8

Schematics of two-dimensional configuration of a nanoplatelet: length L, thickness t, and radius of curvature R

Figure 4

Experimentally measured complex Young’s modulus of PC and its Prony series approximation

Figure 3

Schematics of composite configurations studied in this paper: (a) nanotube/PC nanocomposite: nanotube fractions (vNT) 2.5% and 5%; nanotube/PMMA nancomposites: vNT 0.5%; interphase fractions (vint) each a factor of 8 larger; (b) nanoplatlet/PC nanocomposite when the nanoplatelets are flat; and (c) nanoplatlet/PC nanocomposite when the nanoplatelets are curved. For both (b) and (c), nanoplatelet volume fraction vNP is 0.5% and interphase vint 24%

Figure 2

Schematics of difference between the real three-phase composite (a) and the virtual counterpart (b) if classical Mori-Tanaka method is applied

Figure 12

Temperature domain response at 1Hz, prediction for the nanotube/PMMA nanocomposite (nanotube content 0.5vol%) versus the bulk PMMA

Figure 11

Experimentally measured complex Young’s modulus of PMMA and its Prony series approximation

Figure 7

Predicted storage and loss shear moduli of the nanocomposite (nanotube content 2.5 and 5vol%) against those of bulk PC and the interphase

Figure 6

Normal distribution curves of the complex shear stain fields inside the inclusion (a) real part; (b) imaginary part; and the interphase (c) real part; and (d) imaginary part

Figure 5

Two-dimensional finite element model (mesh) for numerical dilute strain concentration tensor calculation in nanotube/PC nanocomposite (left); close up on the nanotube and interphase (right)

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