J. Cent. South Univ. Technol. (2008) 15(s1): 573-576

DOI: 10.1007/s11771-008-424-x

Modeling of nanoplastic by asymptotic homogenization method

ZHANG Wei-min(张为民), HE Wei(何 伟), LI Ya(李 亚), ZHANG Ping(张 平), ZHANG Chun-yuan(张淳源)

(College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105, China)

Abstract: The so-called nanoplastic is a new simple name for the polymer/layered silicate nanocomposite, which possesses excellent properties. The asymptotic homogenization method (AHM) was applied to determine numerically the effective elastic modulus of a two-phase nanoplastic with different particle aspect ratios, different ratios of elastic modulus of the effective particle to that of the matrix and different volume fractions. A simple representative volume element was proposed, which is assumed that the effective particles are uniform well-aligned and perfectly bonded in an isotropic matrix and have periodic structure. Some different theoretical models and the experimental results were compared. The numerical results are good in agreement with the experimental results.

Key words: effective properties; homogenization; polymer/layered silicate nanocomposites; nanoplastic

1 Introduction

In recent years, nanoplastic (NP) attracted great interest since the work of USUKI et al^[1], because only a few volume fraction of clay crystals can remarkably enhance the elastic modulus and increase the strength and heat resistance, etc. The enhanced properties strongly depend on particular features of the second- phase particles. In particular, the volume fraction of particle (φ_p), the aspect ratio of particle (l/t) and the ratio of particle modulus to that of the matrix. Several theoretical and numerical predictions^[2-9] of the overall moduli of composites or nanocomposites were given. The deviations of the theoretical and numerical predictions and the experimental data are different between the moduli of the silicate of nanometer scale and the polymer matrices and the wide range of variation of the aspect ratio and the non-aligned orientation of the silicate. Therefore, it is necessary and important to develop new appropriate theoretical and numerical methods to predict the mechanical properties.

In this paper, we tried to model the structure of NP by assuming the material with periodic structure and apply the asymptotic homogenization method (AHM) to determining the overall moduli of NP, a proper RVE was chosen. AHM is a mathematically rigorous technique for predicting both the local and global properties of this kind of inhomogeneous media. In section 2, a concise description of AHM was reviewed. A mechanical model was proposed and a simple representative volume element was chosen to descript NP. The expression for the homogenized elastic modulus tensor was given. In section 3, the results of different models and the experimental results are compared. The results of the present paper were in good agreement with the experimental results. Finally, some concluding remarks are given in section 4.

2 Model description and method of AHM

Generally, in NP, the particles are random dispersed and orientated in the polymer matrix. The material structure rigorously possesses periodicity. However, it is the randomicity, the overall elastic properties can be estimated by means of AHM through choosing the material with periodic structure.

Let L and l be the macro- and micro-length scales, respectively (see Fig.1). A relative length scale parameter e can be defined by ε=l/L<<1. In order to deal with two different length scales, the global coordinate is denoted by x and the local coordinate is denoted by y=x/ε. A binary nanocomposite occuping the region B is considered here. The matrix and particles are assumed to be elastic and isotropic. Particles may be the exfoliated silicate platelets, the intercalated silicate stacks and the original aggregated silicate particles, which are considered effective particles, the same meaning used by SHENG et al^[7]. Particles are assumed to be uniform, well-aligned and perfectly bonded in the matrix. Assuming B is a subset of R³, it has a smooth boundary of B: ?B=?B₁?B₂, where ?B₁ is the displacement- prescribed boundary, and ?B₂ is the traction-prescribed boundary.

Fig.1 Schematic diagram of boundary value for periodic microstructure and unite cell

A typical RVE (φ_p=0-0.1, l/t=0-1 000) is shown in Fig.1. From Fig.1, the structure is assumed to be periodic both in the direction of particle alignment and its perpendicular direction. We focused our attention on calculation of the influence of the RVE representation with various particle volume factions (φ_p), particle aspect ratios (l/t) and the ratios of particle modulus to that of the matrix on the overall modulus of the nanocomposite. In order to change the volume fraction, we can increase or decrease the size of Y₁, Y₂ and the ratio of Y₁ to Y₂, the size and the relative position of the effective particles and the aspect ratio of l to t are unchanged, where l is the length, t is the thickness of the effective particles. The equations of the problem can be expressed as follows.

 in B                         (1)

 on                            (2)

 on                      (3)

              (4)

                       (5)

here, superscript ε emphasizes that the variation of these variables is measured at the relative scale of ε. The elastic modulus tensor  is Y-periodic in local coordinate y and satisfies the symmetry and positivity conditions.

The most important postulate in AHM is , which has a multi-scale asymptotic expansion.

                                    (6)

where   is Y-periodic in y and independent of e. Substituting the above equations into Eqns.(4) and (5), we can obtain the asymptotic expansions of  and .

It is proposed that the solution of the perturbation displacement takes the form

           (7)

where   is a constant of integration independent of y_i; the third-order tensor  is Y-periodic with respect to y. It can be shown that  is the solution to the auxiliary variational problem with the periodic boundary conditions.

     (8)

The function  is often called the elastic corrector and has the following symmetry .

The homogenized elastic modulus tensor  is determined by

      (9)

We can also solve the global boundary value of  using the homogenized properties  subject to boundary conditions, and compute the micro-strains and the micro-stresses.

Two-dimensional plane strain simulations of well-oriented particle distributions are subject to small-strain axial tensile loading. The effective elastic tensor or the homogenized elastic tensor , especially , can be calculated numerically by Eqn.(9).

3 Comparison of results of different models and experiment

Among the existing models, Halpin–Tsai equations give reasonable estimates for effective modulus, and the Mori–Tanaka type models give the best results for large-aspect-ratio fillers. Here, we focused on predictions of the longitudinal stiffness E₁₁.

The Halpin–Tsai equation and a closed-form solution based on the Mori–Tanaka model are given in Eqns.(10) and (11), respectively.

                       (10)

    (11)

where  E_p is the elastic modulus of the particle; E_m is the elastic modulus of the matrix;  is the Poisson’s ratio of the matrix; A and A_i are constants, A and A_i can be calculated from the matrix/particle properties and components of the Eshelby tensor, which depends on the particle aspect ratio (l/t) and dimensionless elastic constants of the matrix; φ_p is volume fraction.

The plane strain AHM simulation results of E₁₁/E_m are compared with the uniaxial analytical Mori-Tanaka model solution and the Halpin-Tsai equation solution in Fig.2. The modulus of elasticity and the Poisson ratios of the matrix and silicate platelets are taken to be E_m=     4 GPa, n_m= 0.35, n_p = 0.2, respectively. From Fig.2, we can see that AHM predictions are lie between those of two theoretical models.

Fig.3 shows the comparison of plane strain AHM results with the reported experimental data^[10]. It can be seen that the plane strain AHM results agree well with the reported experimental data.

Fig.2 Comparison of plane strain AHM results with uniaxial results of Mori-Tanaka model and H-T model: (a), (b) Dependence of E₁₁/E_m on φ_p; (c), (d) Dependence of E₁₁/E_m on E_p/E_m

Fig.3 Comparison of plane strain AHM results with experimental data^[10] (Dependence of E₁₁ on mass fraction of montmorillonite w_p 1.2%, E_p=250 GPa, l=100 nm, t=5 nm^[7]):  (a) For MMW (medium molecular mass grade) nylon 6-clay nanocomposites at room temperature (E_m=2.71 GPa); (b) For HMW (high molecular mass grade) nylon 6-clay nanocomposites at room temperature (E_m=2.75 GPa)

4 Conclusions

The asymptotic homogenization method is applied to predicting the overall effective modulus of the nanoplastics. Comparison of the Halpin–Tsai equations and the Mori–Tanaka type models, experimental results agree well with the AHM simulation results, and the random micro-structure of NP can be simulated by periodic structure when a proper RVE is chosen. This provides not only a new approach for the prediction of the overall effective moduli of NP, but also a measurement to solve the boundary value problems of bodies made from the nanoplastics.

References

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(Edited by LI Yan-hong)

Foundation item: Project(10672138, 10372087) supported by the National Natural Science Foundation of China; Project(07QDZ19) supported by the Scientific Research Foundation of Xiangtan University for the Doctors

Received date: 2008-06-25; Accepted date: 2008-08-05

Corresponding author: ZHANG Wei-min, Professor; Tel: +86-732-8292933; E-mail: zhangwm933@yahoo.com.cn