Research ArticleOn the Effect of Un

Research Article
On the Effect of Unit-Cell Parameters in Predicting the Elastic
Response of Wood-Plastic Composites
Fatemeh Alavi,1 Amir Hossein Behravesh,1 Abbas S.Milani,2 and Davoud Karimi1
1 Department of Mechanical Engineering, Tarbiat Modares University, Tehran 14115-111, Iran
2 School of Engineering, University of British Columbia, Kelowna, BC, Canada V1V 1V7
Correspondence should be addressed to Amir Hossein Behravesh; amirhb@modares.ac.ir
Received 27 November 2012; Accepted 8 February 2013
Academic Editor: Toshio Hattori
Copyright © 2013 Fatemeh Alavi et al.This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper presents a study on the effect of unit-cell geometrical parameters in predicting elastic properties of a typical wood
plastic composite (WPC). The ultimate goal was obtaining the optimal values of representative volume element (RVE) parameters
to accurately predict the mechanical behavior of the WPC. For each unit cell, defined by a given combination of the above
geometrical parameters, finite element simulation in ABAQUS was carried out, and the corresponding stress-strain curve was
obtained. A uniaxial test according to ASTM D638-02a type V was performed on the composite specimen. Modulus of elasticity
was determined using hyperbolic tangent function, and the results were compared to the sets of finite element analyses.Main effects
of RVE parameters and their interactions were demonstrated and discussed, specially regarding the inclusion of two adjacent wood
particles within one unit cell of the material. Regression analysis was performed tomathematicallymodel the RVE parameter effects
and their interactions over themodulus of elasticity response.Themodel was finally employed in an optimization analysis to arrive
at an optimal set of RVE parameters that minimizes the difference between the predicted and experimental moduli of elasticity.
1. Introduction
It is well known that particles arrangement in a matrix
affects the local stress/strain field in the ensuing composite,
and in turn influences the macrolevel behavior of the
material. Fortunately, effectivemechanical properties of some
heterogeneous materials rely on the average response of
their microstructures and properties of their individual
constituents [1]; hence, basicmicromechanics theories can be
sufficiently used in analyzing such materials.The connection
between the macro- and mesolevel studies is traditionally
viewed via the concept of the representative volume element
(RVE). However, today little quantitative knowledge is
available about minimum RVE sizes for various engineering
materials. Several attempts have been made to determine the
optimum size of an RVE [2]. Numerical-statistical, analytical
approaches, and experimental observations are three methods
applied by researchers to determine this size.
In the numerical-statistical approach, multiple realizations,
finite element simulations of materials unit cells,
and appropriate statistical procedures are employed. Kanit
et al. [3] proposed a quantitative definition of RVE size
which was associated with a given precision in estimating
the desired overall properties and the number of realizations
for a given volume of microstructure. Eventually, they
demonstrated how a minimal volume size for the computation
of effective properties can be determined depending
on the chosen precision and number of realizations.
Trias et al. [4] analyzed the random distribution of fibers
by means of optical microscopy, and the obtained images
were used to generate realizations of statistical representative
volume elements (SRVEs) at microscale. They solved
finite element models of real-microstructure SRVEs with
arbitrary boundary conditions to obtain probability density
functions of stress, strain, and dilatational energy density
and then related the results to stress tensor at any point in
the macroscale by means of a two-scale approach. Other
numerical-statistical approaches based on setting a tolerance
for the scatter in results are given by Vinogradov [5].
Monte-Carlo simulation is a related approach which was
successfully applied by Ostoja-Starzewski [6] and Gusev [7]
2 Journal of Engineering
to predict the overall elastic constants of the studied periodic
composite.
Among analytical approaches, Drugan and Willis [8]
employed an explicit nonlocal constitutive equation by considering
averaged strain fields varying with the position of
material points. A micromechanics model was also proposed
by Jiang et al. [1] for studying the effective elasticmodulus of
composites containing regularly distributed sphere particles.
Three typical particle arrangements in the forms of simple
cubic lattice, body-centered cubic lattice, and face-centered
cubic lattice were investigated. In the case of irregular-shaped
particles, Li and Wongsto [9] derived new unit cells capable
of dealing with problems involving reinforcing particles of
irregular geometries and local imperfections such as debonding
and microcracks in the matrix. Boundary conditions
for their proposed unit cells were derived from appropriate
considerations of the conditions resulting from translational
symmetry transformations. Giraud et al. [10] investigated
the arbitrarily oriented ellipsoidal inhomogeneities to determine
the macroscopic poroelastic properties of transversely
isotropic geomaterials or rock-like composites. They mainly
dealt with separating the effect of matrix anisotropy and of
inhomogeneities in fiber orientation distribution and shape.
In the RVEs based on experimental observations, experimental
analysis often involves the selection of particular
sample geometries for mechanical testing and subjecting
specimens to image analysis after testing. Graham and Yang
[11] and Shan and Gokhale [12] employed this approach for
an instance. In the case of wood-plastic composites, analytical
macromodels have been employed by Hugot and Cazaurang
[13] to predict theWPC effective properties.
Based on the above review, RVE modeling has been
proven as an efficient approach to represent mechanical properties
of composites including WPCs, as micromechanical
modeling of a whole material structure would be computationally
costly and often infeasible. On the other hand,
the determination of optimum RVE parameters including
its dimensions, particle size, and orientation is of major
concern during such an analysis. Having appropriate RVE
parameters, the analysis can further proceed to predict a
composite’s effective properties.Majority of themicromodels
in the current literature of WPCs encompass only one particle,
which precludes the interaction effect of particles at the
microscale. In this investigation, for the first time to address
this issue, two particles of different geometrical specifications
are included in a WPC RVE. Particles are modeled using
elliptical geometries, the sizes ofwhich together with the RVE
size, location, and the angles of the ellipses center lines are
considered as study parameters. Main and interaction effects
of the parameters on the ensuing macrolevel modulus of
elasticity (MOE) are investigated using finite element analysis.
The Taguchi method of DOE (design of experiments) is
employed to determine effects of RVE parameters. Regression
analysis is performed to mathematically model the MOE as
a function of RVE parameters. Finally, sequential quadratic
programming is employed to optimize the RVE parameters to
bestmatch themodeledMOEwiththe preparedWPCsample
with experimentally determined mechanical properties.
Figure 1: A typicalwood plastic composite sample.