MULTISCALE HOMOGENIZATION METHOD TO PREDICT FILLER SIZE-DEPENDENT - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS MULTISCALE HOMOGENIZATION METHOD TO PREDICT FILLER SIZE-DEPENDENT THERMOELASTIC PROPERTIES OF POLYMER NANOCOMPOSITES S. Chang 1 , S. Yang 1, S. Yu 1 , M. Cho 1 * 1 Division of WCU Multiscale Mechanical Design, School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea * Corresponding author(mhcho@snu.ac.kr) Keywords : Homogenization, Nanocomposites, Thermoelastic, Size effect, Molecular dynamics, Finite element analysis 1 Introduction numerically obtained using the homogenization When nano-sized particles are added into a polymer method via finite element analysis (FEA). The matrix, the arrangement of the polymer chains near results are compared with those of the previous the particle is changed and critically immobilized [1]. results obtained from the micromechanics-based This structural change is related to the mechanical bridging method. Then, using the thermoealstic and thermal properties of nanocomposites and their properties of the interphase, the overall elastic particle size dependency. Thus, it is important to stiffness and coefficient of thermal expansion (CTE) consider the particle-size effect in the multi-scale of nanocomposites that has perturbations in radii of modeling of nanocomposites. the reinforced particles are estimated for stochastic In order to describe the particle-size effect, Yang analysis of real nanocomposites. and Cho [2] have suggested a sequential scale bridging method that combines molecular dynamics 2 Homogenization theory (MD) simulations and micromechanics model. In the 2.1 Fundamental formulation scale bridging method, an additional effective interphase is defined between the particle and matrix. The homogenization method is a numerical tool to Then, the property of the interphase is obtained with describe the mechanical and thermal behaviors of a the results of MD simulations. By bridging the MD heterogeneous material using a rigorous simulation results to the micromechanics model, mathematical foundation. The main purpose of the repeated MD simulation procedure is replaced by a homogenization method is to find equivalent simple linear algebraic equation to estimate the macroscopic homogeneous properties of the effective properties of the nanocomposites. microscopically heterogeneous structures such as Together with the micromechanics models, the composites. In this method, the displacement field is mathematical homogenization method has been expressed using the asymptotic expansion as efficiently applied to estimate the effective follows: properties of heterogeneous structures. Compared    ( ) 0 ( , ) 1 ( , )... (1) u X u x y u x y with the micromechanics models, the mathematical   (2) / homogenization method is more advantageous to x y consider complicated shape of the heterogeneity and where the non-dimensional parameter, ε , represents the ratio of the microscopic representative length to is more rigorous to capture high volume fraction conditions. the macroscopic one. Considering the microstructure, The present study utilizes the homogenization the governing equation for thermoelastic problem can be described as follow: method to describe the particle-size effect on the thermoelastic properties of nanocomposites that are obtained from MD simulations. For this task, the thermoelastic properties of the effective interphase that is adopted to describe the particle size effect are

   By substituting Eq. (13) into Eq. (5) and Eq. (6), the u u      ... k i C d t u dA tensor, χ and  are obtained from: ijkl   i i t V x x A X t l j (3)      [ ] [ ][ ][ ] B T C B χ [ ] [ ] B T C (14) u dV dV     ( ) i C T T dV y y C C V V 0 ijkl kl  X y y V x X   j      [ ] [ ][ ][ ] B T C B [ ] [ ] B T C (15) dV dV After substituting Eq.(1) into Eq. (3), the governing y y C C V V y y equation is arranged by the power of the parameter ε , which can be expressed as a simple algebraic as below: expression given by: 1 1  [ K χ ][ ] [ ] Q       (16) 2 (4) (1/ ) (1/ ) (1) 0 O O O  2   [ K ][ ] [ ] P (17) As Eq. (4) is always satisfied regardless of the where, parameter ε , following equations are easily derived,   [ K ] [ ] [ ][ ] B T C B (18)     1   1 kl dV u u    (5) y m i i C C dV C dV V y    ijml y ijml y   V y y V y y y n j j [ ] Q [ ] [ ] B T C (19) dV     1   1 y C u u V   y   (6) k i i C dV C dV and    ijkl y ijkl kl y V y y V y y y   l j j    [ ] P [ ] [ ] B C . (20) T dV    u u d   y    0  C C ... H k i V t u d y   ijkl i i   x x In the same manner, Eq. (8) and Eq. (9) can be t l j (7) expressed as shown below.   0 u      C ( ) H H i 1 T T d 0       ijkl kl  C H [ ] [ ][ ][ ] C C B χ (21) x dV   j ( ) y C C vol V V y  C are the homogenized elastic y where H and H ijkl kl 1      1            B  H C H [ ] C [ ] stiffness and thermal expansion coefficient of the dV   ( ) y C C vol V V nanocomposite s and given as: y y (22)   1 kl    C H ( C C ) (8) m dY After substituting the tensor, χ and  obtained from  ijkl ijkl ijmn Y Y y n Eq. (16) and Eq. (17) into Eq. (21) and Eq. (22), the   1 1        homogenized elastic stiffness tensor and the ( ) ( ) H H k (9) C C dY ij ijkl pqkl kl Y y coefficient of thermal expansion(CTE) of the Y l nanocomposite can be obtained. 2.2 Finite-element discretization For finite-element discretization, the virtual 3 Characterization of the interfacial properties displacement field is expressed using the shape In this study, the mechanical and thermal properties function as follows:     of matrix, particle, and nanocomposites are obtained  N (10) u u from MD simulations [3]. Then, the thermoelastic and the virtual strain is: properties of the effective interface are numerically   { }   υ B υ (11) obtained from the homogenization method with finite element analysis following the numerical Due to the periodicity of the base-cell, the tensor, scheme that has ever been demonstrated in our   χ x y , has the following symmetry: , earlier work on the elasticity problem [4].    (12) The estimated thermoelastic properties of the ijk ikj effective interphase obtained from the present and is expressed as: multiscale homogenization method are compared           , , , , , (13) with the results obtained from the previous 11 22 33 12 23 13 ijk i i i i i i micromechanics-based bridging method [3] in Fig. 1. As the particle radius increases, both the elastic

PAPER TITLE moduli and the shear moduli gradually decreases 4 Estimation of the nanocomposite properties indicating that the reinforcing effect of larger After obtaining the thermoelastic properties of the particle reinforced cases are superior. At the same interphase, the overall elastic modulus of time, the CTE of the interface gradually increases as nanocomposites is reproduced from Eq. (21) and Eq. the radius of the nanoparticle increases. In all the (22), using the properties of the interface obtained in properties, the results obtained from the present part 3. For establishing the continuum modeling of homogenization method and the previous nanocomposites, it is convenient and useful to micromechanics-based method agree well each other estimate the representative properties of to reproduce the overall thermoelastic properties of nanocomposites with a simplified periodic nanocomposites and their particle size dependency. distribution of the particles, such as simple cubic arrangement. However, in real nanocomposites, the distribution of the reinforcing particles shows more complicated configurations with randomness. As has been mentioned in introduction, the present homogenization method is superior to the micromechanics-based approaches in considering complicated shape of the heterogeneity. Thus, the present homogenization method can be extended to stochastic analysis by considering perturbation of particle radius or volume fractions. In this study, in order to consider stochastic variation of the particle radius, one base-cell having 27 nanoparticles is constructed as shown in Figure 2. The radii of the particles are randomly generated within the limit of 20% perturbations from the radius of 10.90Å. The volume fraction of the whole representative volume element (RVE) is fixed to 5.8%. The maximum and minimum radii of the nanoparticle are 12.78Å and 9.07Å, respectively. Fig.1. Comparison of the thermoelastic proprieties of effective interphase Fig.2. Nanocomposites base-cell composed of 27 nanoparticles with stochastic variations in particle radius. 3