Influence of filler content and interphase properties on large deformation micromechanics of particle filled acrylics

Mechanics of Materials 57 (2013) 134–146

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Influence of filler content and interphase properties on large deformation micromechanics of particle filled acrylics Eray Gunel, Cemal Basaran ⇑ Department of Civil, Structural and Environmental Engineering, University at Buffalo, SUNY, United States

a r t i c l e

i n f o

Article history: Received 15 February 2012 Received in revised form 17 October 2012 Available online 21 November 2012 Keywords: Particle filled composites Interphase properties Interparticle distance Viscoplastic model Particle debonding Unit cell model

a b s t r a c t Large deformation response of alumina trihydrate (ATH) filled poly(methyl metacrylate) PMMA over a wide temperature range was investigated. A multi-particle unit cell model of PMMA/ATH composite with different volume fractions was employed to study the influence of interphase properties and interparticle distance. The model constitutes of amorphous polymer matrix, rigid fillers and interphase between filler and matrix. Polymer matrix was modeled using a thermodynamically consistent dual-mechanism viscoplastic model, whereas fillers were assumed as linear elastic. In numerical simulations, the interphase between filler and matrix was modeled through a user defined interphase model and matrix material was modeled through a user defined material model. Damage evolution in PMMA/ATH at large deformations due to particle debonding was studied based on a critical stress criterion. Composite elastic modulus results from finite element analysis of unit cell model were compared with analytical model predictions (Mori–Tanaka, Eshelby, Lielens models) at different temperatures for various volume fractions of fillers. The influence of interphase characteristics, interparticle distance and temperature on large deformation response of PMMA/ATH was examined through multi-particle unit cell model. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Composite materials are increasingly used in aerospace, aircraft, armor, automotive, biomedical, energy, infrastructure, marine and sports industry. Demand on high performance materials and inherent complex nature of composites yield several problems to be solved. One major problem in particle filled composites (PFCs) is the inadequate interfacial bonding between filler and matrix. Chemical bond and mechanical friction between matrix and filler normally provide material integrity in PFCs. Incompatible chemical characteristics of filler and matrix cause poor interfacial adhesion which may result in severe problems during manufacturing and service life. Composite materials with adequate interfacial bonding strength between

⇑ Corresponding author. E-mail address: [email protected] (C. Basaran). 0167-6636/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmat.2012.10.012

filler phase and matrix phase have superior characteristics and properties in comparison to monolithic constituents. PFCs consist of various small sized particles which are randomly embedded in matrix material. Random dispersion of particles within matrix results in a homogenous structure and isotropic-like characteristics at macro-scale. As a consequence, number of mechanical and thermal properties to be determined from experimental studies is reduced and constitutive modeling of PFCs at macro-scale is significantly simplified. On the other hand, heterogeneous structure at micro-scale requires investigation of complex deformation behavior at filler and matrix boundaries. Rigid or soft particles in PFCs primarily act as stress raisers promoting energy absorbing mechanisms, and also act as boundaries entrapping crack growth. PFCs including rigid fillers exhibit different failure mechanisms depending on particle size and concentration (Serenko et al., 2002; Hornsby and Premphet, 1997; Suwanprateeb, 2000; Dasari et al., 2006; Finnigan et al., 2005), interfacial bonding

E. Gunel, C. Basaran / Mechanics of Materials 57 (2013) 134–146

strength (Basaran et al., 2006; Lazzeri et al., 2005), relative matrix and filler strength (Yuan and Misra, 2006; Dasari and Misra, 2004; Dasari et al., 2004). The influence of particle size on mechanical properties of PFCs can be attributed to specific surface of particulates. For the case of PFCs with small particles, large specific surface favors higher degree of bonding between constituents and improves reinforcement effect. However, very small particles may agglomerate leading to non-uniform dispersion and weak regions which significantly reduce desired mechanical properties of PFCs (Dasari et al., 2006; Finnigan et al., 2005). Different failure mechanisms are observed in PFCs depending on the interfacial strength which also controls stress distribution in the vicinity of filler. In general, strong interfacial strength improves composite strength and stiffness in the expense of ductility. Weak interfacial strength causes reduction in strength and stiffness along with improved toughness (Ranade et al., 2006). Surface treatment on particles in the form of chemical agents can be used to improve adhesion between matrix and particles. It was observed that chemical agents promotes strong bond between filler and matrix resulting in uniform particle dispersion and increase in yield stress and stiffness (Lazzeri et al., 2005). In a recent study, effect of surface treatment on tensile deformation behavior of alumina trihydrate (ATH) filled poly(methyl methacrylate) (PMMA) was investigated (Nie, 2005). Adhesion promoting agents resulted in higher tensile strength but lower toughness in tensile testing of PMMA/ATH. Micro-deformation mechanisms in PFCs also depend on relative stiffness and strength of constituent materials. If both constituent materials have material properties in the same order, particle cracking occurs. If embedded particles are much stiffer and stronger than matrix, particle debonding or matrix tearing becomes major damage mode (Basaran et al., 2006). Temperature is another significant factor controlling micro-deformation mechanism and material response for the case of PFCs with polymer matrix. For PMMA/ATH with adhesion promoting agents, particle cracking was observed as the dominant deformation mechanism at low temperatures while particle disintegration and interfacial failure were observed at high temperatures (Gunel and Basaran, 2009). The nature of deformation in polymer matrix was also observed to be significantly dependent on temperature. The transition in mode of micro-deformation was found to be around glass transition temperature (hg) of PMMA, above which deformation in polymer was more severe and flow-like. Particle failure in PMMA/ATH was attributed to large micro-deformation fields around ATH agglomerates caused by flow-like polymer matrix deformation (Gunel and Basaran, 2009). A great number of researches are available in the literature dedicated for investigation and modeling of failure mechanisms in PFCs and prediction of effective thermal and mechanical properties of PFCs. These studies can be classified into two major categories. The first group consists of macroscopic constitutive models of heterogeneous materials using a local description of micro-structural behavior (Ju and Lee, 2000, 2001; Lee and Pyo, 2008; Nie and Basaran, 2005; Basaran and Nie, 2007; Matous and Geubelle, 2006; Ju and Sun, 2001). In these models, a

135

homogenization step is necessary to relate local variables to overall variables and a more complicated localization step to impose the influence of overall controlled quantities to corresponding local quantities. Though interfacial bond between fillers and matrix has a significant effect on mechanical properties, most analytical and numerical models assume a perfect bonding condition and utilize traction and displacement continuity at the filler-matrix boundary. Recently, Basaran and Nie (2007) developed a thermodynamics based damage coupled viscoplastic model for PFCs with imperfect interphase based on the work of Ju and Chen (1994a,b). In these models, three governing micro-mechanical ensemble-volume average field equations were considered to relate ensemble-volume averaged stresses, strains, volume fractions, eigenstrains, particle shapes and orientations and elastic properties of constituents. The ensemble-volume averaged stress definition was used for viscoplastic modeling of PMMA/ATH while effective thermal and mechanical properties were obtained based on composite spherical assemblage approach and generalized self-consistent scheme model (Christensen and Lo, 1979; Hashin, 1962). Since interparticle distance is another important factor controlling material response due to interparticle interactions, explicit pair-wise interparticle interactions were also exploited for both elastic and viscoplastic response. In this model, damage evolution in PMMA/ATH was considered only in terms of entropy production during deformation while source of damage was assumed as degradation in material stiffness and strength due to formation and coalescence of cracks and/ or voids (Basaran and Nie, 2007). Regnier showed that viscoelastic behavior of spherical glass beads reinforced amorphous poly(ethylene) terephthalate cannot be truly predicted by classical homogenization micromechanical models, which depend only on the constituent behavior, particle shape and distribution (Cruz et al., 2009). The reason for the difference between experimental and model results was attributed to the lack of the implementation of fillers’ effect on the matrix viscoelastic behavior which changes in both glassy and rubbery regions. Another problem in micromechanical homogenization models is the difficulty of determination of effective compliance tensors for the case of imperfect interphase conditions. Damage evolution in the PFC due to particle debonding requires continuous update of stiffness of composite by monitoring and prediction of the state of bonding between particle and matrix which is conveyed through some phenomenological characterization of particle debonding (Dai and Huang, 2001). Progressive debonding damage generally imposed by transformation of some percent of perfectly bonded particles to debonded particles through some averaged stress and strain criterion (Tohgo and Weng, 1994; Chen et al., 2003) or critical energy criterion (Tohgo et al., 2010; Chen et al., 2010). Though particle debonding is implicitly implemented and actual stress fields around and inside the fillers remain as unknown, micromechanical homogenization models are very powerful tools that can be used both for material modeling of PFCs and prediction of effective material properties of PFCs. The other class of PFC modeling is based on representative unit volume cells which consist of one or several par-

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ticles embedded in a finite matrix volume. In literature, finite element simulations of representative volume elements (RVE) of PFCs including only one particle have been commonly used to investigate fracture or debonding of particles (Needleman, 1987; Bao, 1992; Finot et al., 1994; Brockenbrough and Zok, 1995; Wang et al., 2008; Ghassemieh, 2002). The unit-cell models are advantageous over micromechanical homogenization models in providing details of progressive debonding damage process. Yet detailing of mesh geometry in unit cell models and strong mesh dependency of results make them impractical and computationally expensive. In order to take into account the different damage stages simultaneously occurring in several particles, multi-particle cell models (Llorca and Segurado, 2004; Eckschlager et al., 2002; Segurado and Llorca, 2005,2006; Sun et al., 2007) and damage cell models (Bao, 1992; Brockenbrough and Zok, 1995) were proposed. Multi-particle cell models are also useful for the investigation of the influence of interparticle interactions on material response. Wang investigated the influence of interphase thickness and strength on toughness and failure mechanisms of PFCs through a unit cell model incorporating a single particle in polymer matrix resin assumed as plastic (Wang et al., 2008). Random unit cell models are also used for prediction of elastic properties of PFCs. The principle in random unit cell models is the equivalency of a RVE with critical size (below which equivalency diminishes) and an actual particle reinforced composite. The random unit cell models provide a narrower range between the bounds of elastic properties with respect to three dimensional single or two particle unit cells and analytical models of Mori and Tanaka (1973) and Hashin and Shtrikman (1963) over a large domain of particle fractions. It was also mentioned that random unit cells require more computational resources and consume more computational time in comparison to single-particle and two-particle unit cells (Sun et al., 2007). Brockenbrough and Zok studied flow characteristics of particle-reinforced metal matrix composites in a unit cell model employing matrix constitutive model within the context of small strain deformation plasticity. The numerical simulation results from unit cell model were fitted to simple analytical functions to explicitly formulate a constitutive law for composites undergoing progressive damage during tensile straining (Brockenbrough and Zok, 1995). Analytical models dedicated for the prediction of effective elastic properties of PFCs rely on description of complicated strain fields occurring inside and outside of an inclusion enclosed in an infinite matrix. The general assumption on linear elastic matrix and filler limits their applicability to small deformations. In two-phase models including only matrix and filler constituents, the influence of interphase characteristics on effective properties is neglected. In this category, the first work is the solution of Eshelby for the ellipsoidal inclusion in an infinite matrix (Eshelby, 1957). Later, Mori and Tanaka incorporated theory average stress in matrix into Eshelby solution (Mori and Tanaka, 1973). In the case of three phase models, imperfect interphase characteristic are incorporated through interphase strength and thickness. Gao obtained a closed-form solution of Eshelby problem with general

two-dimensional eigenstrain and uniform tension for the case of imperfect interface. In this solution, possible normal displacement overlapping at the interface was observed and conditions for nonoverlapping were proposed (Gao, 1995). Teng extended Mori–Tanaka approach to formulate stiffness of particulate composites for the case of debonded particle–matrix interphase (Teng, 2010). In another study, the central role of interphase properties and particle debonding conditions on composite yield strength and stiffness was studied (Vörös et al., 1997). Basaran and Nie also proposed a modification of Ju and Chen (1994a,b) two phase model for PFCs to incorporate imperfect interfacial bonding conditions (Nie and Basaran, 2005). In order to overcome the deficiency in incorporating the true influence of imperfect interphase and interfacial debonding, several upper and lower bound models were proposed for elastic properties of PFCs (Wu et al., 2004; Zhong et al., 2004). Weng proposed a bounding method based on the work of Hashin and Shtrikman (1963) by establishment of a comparison between PFC and reference material (Weng, 1992). For lower bound, reference material was chosen as matrix. For the upper bound, filler was chosen as the reference material. Later, Lielens further improved the model by incorporating the influence of volume fraction of particles and implicitly the influence of interparticle interactions (Lielens et al., 1998). For PMMA/ATH, Lielens model predictions for elastic modulus were observed to be in good agreement with tensile test results for a wide range of temperatures below glass transition temperature (hg) of polymer matrix (Stapountzi et al., 2009). In this work, random and heterogeneous micro-structure of particle filled acrylics were studied using a multiparticle cell model with different volume fractions over a wide temperature range spanning from well-below to well-above glass transition temperature of matrix. The amorphous polymer matrix was modeled using a thermodynamically consistent dual-mechanism viscoplastic model. ATH agglomerates were assumed as perfectly intact so that no particle cracking was allowed while a linear elastic model was used for constitutive behavior. The influence of interparticle interactions on composite properties was conveyed through different volume fractions. The imperfect interphase was investigated by considering different interphase thicknesses together with temperature dependent interphase strength and stiffness. Progressive damage in particle filled acrylics due to particle debonding was exploited through a critical stress-criterion. Finite element analysis with elastic matrix assumption was compared with analytical models for validation of multi-particle unit cell (MPUC) model. In numerical simulations of multi-particle unit cell model including a viscoplastic material model for amorphous polymer matrix, interparticle characteristics, interparticle distance and temperature effects on progressive damage by particle debonding at large deformations were investigated.

2. Constitutive modeling of amorphous polymer matrix Large deformation behavior of amorphous polymers can be studied through dual decomposition of material re-

E. Gunel, C. Basaran / Mechanics of Materials 57 (2013) 134–146

sponse into two parallel working mechanisms of intermolecular structure and molecular network structure (Palm et al., 2006; Boyce et al., 2000; Ames et al., 2009; Richeton et al., 2006, 2007) as depicted in Fig. 1. More recently, Anand proposed a trial-mechanism including a single mechanism for intermolecular structure and two submechanisms for molecular network structure (Srivastava et al., 2010). Both dual- and trial-mechanism models are proven to be successful in describing large deformation behavior of amorphous polymers for isothermal test conditions. Different mechanisms in dual- or trial-models are not always active and contributions may vary significantly depending on temperature and rate. At high temperatures, material response is dominated by the molecular network resistance (Sweeney and Ward, 1996), while intermolecular resistance controls the deformation at low temperatures (Srivastava et al., 2010). In this study, an improved version of dual-decomposition model is considered for non-isothermal stretching of amorphous polymers based on the work of Anand (Ames et al., 2009; Srivastava et al., 2010; Anand and Su, 2005; Anand et al., 2009). Further details in constitutive modeling, material property determination and validation of dual- and trial-mechanism models of amorphous polymers can be found in Ames et al. (2009), Srivastava et al. (2010), Anand et al. (2009). According to Fig. 1, it is observed that the deformation in intermolecular resistance mechanism (FI) and molecular network mechanism (FM) are equal to each other and equal to total deformation (F) (Eq. (1)) while total stress (T) is the summation of stresses due to intermolecular interactions (TI) and molecular network interactions (TM) (Eq. (2)).

F ¼ FI ¼ FM

ð1Þ

T ¼ TI þ TM

ð2Þ

Initial elastic response to deformation due to intermolecular resistance is governed by van der Waals interactions with surrounding molecules. Using a Helmholtz free energy in reference configuration for constitutive relation describing intermolecular resistance which was developed by Anand and On (1979), Anand (1986); Cauchy stress (TI), Kirchhoff stress ðSeI Þ and Mandel stress ðMeI Þ can defined as:

TI ¼ J 1 FeI SeI FeT I

137

A simple constitutive relation for rubber networks developed by Gent (1996) is used to describe strain hardening due to polymer chain-stretching and constitutive relation in molecular network. Second Piola–Kirchhoff stress ðSeM Þ, Mandel stress ðMeM Þ and Cauchy stress (TM) can be derived from Gent free energy in reference configuration (Gent, 1996) as:

 1 I1  3 1 TM ¼ J1 FeM SeM FeT dev ððBeM Þd Þ M ¼ J lM 1  IM

ð4Þ

Details on elasto-viscoplastic model for matrix material can be found in the Appendix section. 3. Numerical simulations 3.1. Multi-particle unit cell (MPUC) model The unit cell model primarily consists of amorphous polymer matrix, hard fillers and imperfect interphase. For numerical simulations, finite element package ABAQUS was used. Solid brick elements (C3D8) were used for matrix and fillers while interphase was modeled on filler-matrix boundaries through a user defined interface model (UINTER). Average particle size of ATH fillers is known as 35 lm (Gunel and Basaran, 2010). Accordingly, ATH agglomerates were considered as perfect spherical inclusions with a diameter of 35 lm in MPUC model. A total of eight identical particles were included in RVE of composite which is assumed to be located at the corner of mid-section of actual composite as depicted in Fig. 2. There needs to be at least eight particles to provide interparticle interactions in all directions. The spherical particles were assumed to be equally spaced in all directions and interparticle distances were taken as 2.5, 5, 10 and 20 lm in unit cell models which are designated as SP1, SP2, SP3 and SP4, respectively. An interparticle distance which is smaller than 2.5 lm was

ð3Þ

Molecular Network Resistance (M)

ηM χI ηI

μM μI

Intermolecular Resistance (I) Fig. 1. Schematic representation of material model.

Fig. 2. Multi-particle unit cell model for a volume fraction of 31.3% with an interparticle distance of 10 lm.

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not selected due to meshing difficulties. Different interparticle distances in unit cell models lead to different volume fractions which were calculated from the geometry as 9% for SP4, 19.2% for SP3, 31.3% for SP2 and 34% for SP1. In actual PMMA/ATH composites, filler volume fraction is around 48% (Gunel and Basaran, 2010). A higher volume fraction was not possible without allowing particle overlapping due to the lack of available free volume of matrix. A similar limit on maximum volume fraction of fillers without particle overlapping was found to be 35% (Sun et al., 2007). On the free surfaces of RVE (the left and top surface in Fig. 2), fillers are exposed and traction free boundary conditions were employed. A layer of matrix material with a thickness equal to interparticle distance was used to enclose the fillers forming the remaining surfaces of RVE. On these matrix surfaces other than free surfaces on the left and the top of RVE, displacement boundary conditions were imposed perpendicular to loading direction such that displacements in 1-direction is restrained on the back surface, displacements in 2-direction is restrained on the centerline of right surface and displacements in 3-direction is restrained on the centerline of bottom surface of RVE (Fig. 2). Displacement controlled loading was employed on the front surface and 1-direction was taken as longitudinal axis of RVE. The displacement rate imposed on the front surface of RVE was calculated by multiplying longitudinal direction length of unit cell model with a constant strain rate of 0.018 Hz. This constant strain rate was obtained from isothermal simulations of PMMA matrix model by using the relative displacements between two points at the middle of the samples separated by 500 lm. In all finite analysis of MPUC model, isothermal conditions were assumed. 3.2. Material parameters and constitutive models Material properties for PMMA constitutive model were obtained by conducting isothermal tests at different

temperatures (both above and below glass transition temperature) and at different rates while viscoplastic model related material parameters were obtained from several simulations in a one-dimensional version of numerical algorithm of material model developed in MatLAB. The effect of particle filler content on glass transition temperature of particle filled composite are not included in the simulation models. The effect of particle filled content on the overall properties of particle filled composite was captured by modeling constituents separately in MPUC model which only reflects direct particle–matrix physical–mechanical interactions but not the influence of particle inclusion on glass transition temperature. Material parameters involved in constitutive model of PMMA are presented in Table 1. The parameters presented herein are only for elasto-viscoplastic model of PMMA (matrix) and are assumed to be independent of particle inclusion. The constitutive model for PMMA was implemented numerically based on staggered method in user defined material subroutine (UMAT). ATH fillers were assumed as linear elastic and the elastic modulus and Poisson’s ratio of ATH fillers were taken as temperature independent with values of 70 GPa and 0.24, respectively (Nie and Basaran, 2005). ATH properties were assumed to be constant in the temperature range (30–125 °C) where testing and simulations were conducted, because melting temperature of ATH is around 300 °C. Perfect interphase condition in finite element analysis was imposed through tie-constraint between matrix and filler elements. The imperfect interphase was characterized by different interphase thicknesses (d) while interphase strength and stiffness was taken as 10% of temperature dependent yield strength and elastic modulus of matrix. Interphase thickness was selected as 10%, 1%, 0.1% and 0.01% of particle radius which are designated as C1, C2, C3 and C4, respectively. Details on analytical models for spherical inclusion problems can be found in the Appendix.

Table 1 Material parameters for PMMA. Parameters

Parameters 373

href g ðKÞ cg1 ðKÞ cg2

32.58 83.5

c1 (K)

9

Parameters

Parameters 2.72  1010

0.5

X gM ð1=KÞ

0.001

hl (K)

5

ð1=KÞ

0

b2 (1/K)

–4.58  102

Dl (K) sl

14

ðMPaÞ

35

b3

4.04  102

0.03

X rM SgM SrM

ðMPaÞ

0.2

hS (K)

5

mpref ðs1 Þ moI ðs1 Þ

2.43  1012 1.56  1019

lrM ðMPaÞ

g

b1 (MPa)

0.001

c2

116

X l ðMPa=KÞ

0.4

mref ðs1 Þ

1.73  102

X rl ðMPa=KÞ

0

DS (K)

5

QI (J/K)

Eg (MPa)

500

0.001

X gS ðMPa=KÞ

0.01

V (m3)

1.39  1027

Er (MPa)

1.5

ug ug

0

0

ap

0.21

hE (K) DE (K) sE

5 15

h/ (K) D/ (K)

0 10

10 2.15

nI hI

2.17 40.42

0.06

X gu ð1=KÞ

4.5  106

X rS ðMPa=KÞ Bg (MPa) X gB ðMPa=KÞ DhB (K)

5

X gE ðMPa=KÞ

20.05

X ru ð1=KÞ

0

gg

7.97

c (MPa) moM ðs1 Þ

4.33  109

X rE ðMPa=KÞ mg mr

0

IgM

4.8

gr

9.55

QM (J/K)

1.30  1019

0.35 0.49 9

IrM hM (K) DM (K)

6.67 10 15

Dhg ðKÞ hg (K)

5 5 0.07

hM nM ms (g/mol)

14.43 5 100.13

lgM ðMPaÞ

X gg ðK1 Þ

60

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E. Gunel, C. Basaran / Mechanics of Materials 57 (2013) 134–146

(a) 6

θ=30°C

5

θ=50°C

4.0

θ =75°C (this wo rk) Non-Interacting

3.5

Lielens

Elastic Modulus (GPa)

3.0

Mori-Tanaka (Lower Bound) Upper Bound

2.5

4 θ=75°C

3 θ=90°C

2

1

0 0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

Volume Fraction of Fillers (%)

(b)

300 14

250 Elastic Modulus (MPa)

Charalambides showed that Lielens model is a good candidate for stiffness prediction of PMMA-ATH (Stapountzi et al., 2009) in a wide temperature range of 0–94 °C for different volume fractions of fillers. In this section, finite element analysis results of MPUC model will be compared with Lielens model and other analytical models in order to verify whether the model in this study may be considered as a ‘‘representative’’ of actual composite. In these simulations, perfect bonding between fillers and matrix was assumed while matrix was considered as linear elastic. Effective elastic stiffness predictions from theoretical analytical models are only valid at early stages of deformation, because these models do not incorporate the influence of imperfect interphase conditions or progressive damage by particle debonding. Elastic modulus results from MPUC model was obtained by dividing average longitudinal stress at a cross-section perpendicular to loading direction to the longitudinal strain in the RVE. In Fig. 3, MPUC model simulation results for different volume fractions at 75 °C are presented with analytical model predictions. Eshelby solution for spherical inclusion in infinite matrix can be considered as the non-interacting solution. Mori–Tanaka solution coincides with lower bound for effective elastic modulus proposed by Weng (1992). Finally, Lielens model predictions are presented as the main comparison basis for simulation results. Elastic modulus predictions by analytical models at low volume fractions are close to each other (except for Upper Bound predictions) and asymptotically approach to elastic modulus of matrix at zero volume fraction. At higher volume fractions, Lielens model was shown to be superior to other analytical models and semi-empirical models Stapountzi et al. (2009). Since MPUC model results and experimentally validated Lielens model predictions are in good agreement, it can be concluded that MPUC model is indeed a representative volume element of actual composite. It should be also noted that though same material type was used in this study and in Stapountzi et al. (2009), experimental elastic modulus measurements under flexural loading in Stapountzi et al. (2009) are higher than finite

Elastic Modulus (GPa)

4.1. MPUC model with linear elastic matrix

12

200

θ=100 °C

10 8

150

6

100

θ=125 °C

4

50

2

0 0%

Elastic Modulus (MPa)

4. Results and discussion

5%

10%

15%

20%

25%

30%

35%

40%

0 45%

Volume Fraction of Fillers (%)

Fig. 4. Comparison of the MPUC model results to predictions by Lielens model in terms of elastic modulus of PMMA/ATH composites at various particle volume fractions and temperatures.

element simulation results under tensile loading in this study. The reason for this difference was attributed to testing conditions, mode of loading and different PMMA mechanical properties. Composite elastic modulus predictions by MPUC model are indeed in good agreement with Lielens model predictions over the entire temperature range considered as depicted in Fig. 4a and b. Since there is a substantial difference in elastic modulus values at temperatures below and above hg of matrix which directly influences elastic modulus of PMMA/ATH composite especially at low volume fractions, results are separately presented for glassy (Fig. 4a) and rubbery (Fig. 4b) regions of material. Since ATH properties are assumed to be constant in this temperature range, temperature dependence of elastic modulus of composite is can be completely attributed to matrix phase which will diminish with increasing volume fraction of fillers. A 34% ATH inclusion increases the stiffness by 2.3–3 times and a temperature increase from 30 °C to 125 °C leads to nearly 3 order decrease in stiffness.

SP1

2.0

SP2

4.2. MPUC model with viscoplastic matrix

SP3

1.5

SP4

1.0 0.5 0.0 0%

10%

20%

30%

40%

50%

60%

Volume Fraction of Fillers (%)

Fig. 3. Comparison of the MPUC model results to predictions by analytical models in terms of elastic modulus of PMMA/ATH composites at various particle volume fractions.

In Section 4.1, it was shown that MPUC model can be considered as a representative volume of PMMA/ATH composite. MPUC model with elastic matrix was shown to be a good candidate for stiffness prediction of the composite at early stages of deformation. In this section, applicability of MPUC model will be extended to large deformation response of PMMA/ATH composite by modeling polymer matrix as a viscoplastic material, incorporating

E. Gunel, C. Basaran / Mechanics of Materials 57 (2013) 134–146

(a) 90

T=30 °C-perfect

80

T=50 °C-perfect

70 60 Stress (MPa)

the imperfect interphase conditions and including progressive damage due to particle debonding. Dual- and trial-mechanism viscoplastic constitutive models for amorphous polymers have been proven to be useful in prediction of material response at large deformation under different loading conditions over a wide temperature and strain range (Palm et al., 2006; Boyce et al., 2000; Ames et al., 2009; Richeton et al., 2005, 2006, 2007; Srivastava et al., 2010; Anand and Su, 2005; Anand et al., 2009; Dupaix and Boyce, 2007; Arruda et al., 1995; Hasan et al., 1993). In MPUC model, ductile amorphous PMMA matrix is modeled by improved dual-mechanism viscoplastic model and hard brittle ATH fillers are assumed to be linear elastic. ATH agglomerates are actually clusters of perfect ATH crystals held together by some chemical bond forming an imperfect interphase within ATH agglomerates. In this study, ATH fillers are idealized as spherical shaped inclusions embedded in polymer matrix at periodic interparticle distance while imperfect interphase between ATH crystals within agglomerates is not included in the model. Numerical simulations of MPUC model were implemented in finite element program Abaqus/Standard through user defined material subroutine for viscoplastic constitutive model of polymer matrix and user defined interphase subroutine for imperfect interphase between matrix and fillers. Details on MPUC model with different volume fractions (interparticle distance) and interphase properties were discussed in Section 3.1. Experimental results of isothermal tensile testing on PMMA/ATH presented in Nie (2005) was primarily considered for validation of MPUC model results, however, volume fraction of fillers in MPUC model is only 34% at most densely packed form while volume fraction of fillers in actual composite is around 48%. Due to this remarkable difference in volume fractions, direct comparison of MPUC model with experimental results is not possible. However, this is not a deficiency for MPUC model; it is possible to achieve higher volume fractions by allowing particle overlapping. In Fig. 5a and b, stress–strain curves obtained from finite element analysis of MPUC model are presented. Stress histories in MPUC model are obtained by averaging stress values at a cross-section perpendicular to loading direction while strain histories correspond to longitudinal strain of REV. It is clear from Fig. 5a and b that temperature has a profound effect on PMMA/ATH composite response similar to the previous experimental observations in Basaran and Nie (2007). MPUC model simulation results are parallel with the common observation of decreasing yield strength and stiffness with increasing temperature in polymeric materials. Decrease in resistance to deformation becomes more remarkable as temperature gets closer to glass transition of polymer matrix, while a significant increase in ductility of composite is observed at the same time. The failure in composites is predicted by level of stretch in polymer matrix. Numerical simulations were terminated when limited chain extensibility (Im) is exceeded at any integration point of a matrix element. In simulations at 90 °C and 100 °C, no failure was predicted for composites up to the deformation limit defined. Limited chain extensibility is a temperature dependent material property which was formulated from

50

T=30 °C-imperfect

40

T=50 °C-imperfect

30

T=75 °C-perfect

20 T=75 °C-imperfect

10 0

0

0.02

0.04

0.06

0.08

0.1

Strain

(b) 35 30

T=75°C-perfect

T=90°C-perfect

25 Stress (MPa)

140

20 T=100°C-perfect

15

T=75°C-imperfect

10 T=90°C-imperfect

5 T=100°C-imperfect

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Fig. 5. Stress–strain curves of PMMA/ATH composites with perfect interphase and imperfect interphase with a thickness of d/r = 0.1% at various temperatures (particle volume fraction of 31.3%).

isothermal tensile tests on PMMA. In simple terms, limited chain extensibility is the stretch level in polymer chains without failure. Therefore, the ‘‘ultimate failure’’ of composite in MPUC model is predicted by monitoring the stretch in polymer matrix. It should be noted that the final stage of failure in rigid particle reinforced composites constitutes tearing/breaking of ductile matrix regardless of origin of primary failure (particle cracking or interphase failure). Progressive damage in MPUC model prior to matrix failure is due to interphase failure which is assumed to be controlled by a critical stress criterion. When the stress at any interphase point exceeds temperature dependent interphase strength, particle debonding will take place which will continuously increase with stretch level through redistribution of stresses in the vicinity of debonded regions. Perfect interphase cases result in higher yield strength and stiffness values at all temperatures in comparison to imperfect interphase cases (Fig. 5a and b). This observation on the influence of interphase on material response is parallel to experimental studies on PMMA/ATH composites with different interphase characteristics (Basaran et al., 2006, 2008). In addition to improved strength and stiffness observed in cases with perfect interphase bonding, a slight reduction in ductility was also observed at all temperatures. This can be attributed to improved reinforcement effect of fillers in the case of a perfect bonding between matrix and filler which restrain the free movement of polymer chains and increasing the degree of localized plastic deformations in the matrix. Though overall deformation in the composite is relatively small, localized plastic defor-

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mations in the matrix initiate failure in the composite and such failure mechanisms were experimentally observed as matrix tearing/cracking in PMMA/ATH composites with adhesion promoting agents (Basaran et al., 2006). In the case of imperfect interphase conditions, reinforcement effect of fillers is much smaller and fillers do not restrain polymer chain movement and deformation in the composite can extend to large stretch levels without any failure in the matrix. After interphase failure takes place and particles are separated from matrix, debonded particles can be considered as large voids in the matrix, as their contribution to overall load carrying capacity is minimal. Continuous debonding of fillers from the matrix gradually decreases composite stiffness and strength. At large deformations, matrix dominates the response in composites with imperfect interphase and strain hardening is not observed at 90 °C and 100 °C (Fig. 5b). Neat PMMA display some strain hardening at large deformations due to the resistance in molecular network, yet the stiffness reduction due to debonded particles counter acts against strain hardening in matrix and causes further reduction in stress. On the other hand, perfect bonding leads to continuous stress increase at large deformations and strain softening observed at 90 °C for imperfect interphase is also diminished due to the reinforcement effect of fillers. Progressive damage due to particle debonding is an important factor that affects stiffness and strength of composite at early stages of deformation and strain softening/ characteristics of composite in the post-yield region. Particle debonding histories corresponding to the simulations with imperfect interphase conditions in Fig. 5a and b are presented in Fig. 6. Time domain in particle debonding histories is considered in terms of normalized strain with respect to ultimate strain at failure of composite ðeðtÞ=ef Þ. Particle debonding is observed to take place in two distinctive regimes. In the first regime, particle debonding occurs at a faster rate leading to a maximum equilibrium particle debonding percentage around 90% of total interphase area and this percentage remains nearly constant in the second regime where matrix dominates materials response. At low temperatures (30 and 50 °C), composite failure was predicted to take place before this equilibrium particle debonding le-

vel. At higher temperatures, the equilibrium debonding level was reached at early stages of deformation, i.e. particles are fully debonded from matrix at small deformations displaying an imperfect interphase characteristic. Particle debonding results in continuous degradation in stiffness of composite especially in this early stage of deformations. Viscoplastic behavior of matrix coupled with stiffness degradation leads to different levels of nonlinearity in response of composite as depicted in Fig. 5a and b. In MPUC model, particle debonding was observed to start at the pole of particles where stress concentration is highest and continuously proceed towards the equator of particles while the width of debonded regions along longitudinal direction also increases with increasing stretch levels. Fluctuations in percentage debonding during the second regime were caused by Poisson’s effect. Transverse direction deformation in matrix leads to closure of matrix-filler interphase in some regions which reopen at higher levels of stretch (Fig. 6). In addition to interphase characteristics and temperature, filler content has also a strong influence on material response as shown in Fig. 7. The reinforcement effect of fillers in composites with perfect interfacial bonding was observed as larger stiffness and higher yield strength with respect to composites with imperfect interphase (Fig. 5a and b). It is clear from Fig. 7 that reinforcement effect become more remarkable with increasing filler content. The increase in filler content from 9.01% in SP4 to 34% in SP1 results in 1.5-fold increase in yield strength and 2-fold increase in initial stiffness of composite. In the post-yield region, level of strain hardening also increases with increasing filler content (Fig. 7). Therefore, overall response of composite with perfect interphase characteristics is improved by filler inclusion while the effectiveness increases with increasing filler content. On the other hand, composites with imperfect interphase show significant decrease in yield strength and stiffness of composite in comparison to neat PMMA. The reason for these reductions was explained as evolution of a porous-like structure in matrix through particle debonding. Interphase failure eliminates any possible contribution from particles to load carrying capacity of composite. Once particles are separated from matrix, load cannot be transferred to fillers and matrix material carries all applied load

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Fig. 6. Particle debonding histories in PMMA/ATH composites with an interphase thickness of d/r = 0.1% and particle volume fraction of 31.3% at various temperatures.

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Fig. 7. Stress–strain curves of neat PMMA and PMMA/ATH composites with perfect interphase and imperfect interphase with a thickness of d/ r = 1% at 90 °C.

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(a) 90 80 Percent Debonding (%)

while presence of fillers actually adversely affects performance of composite. Degree of severity of interphase failure becomes more remarkable at higher particle volume fractions supporting the porous-like structure of matrix explanation. Another result of damage due to particle debonding is observed as the disappearance of strain hardening characteristic of composite which was attributed to stiffness reduction in the post-yield region due to particle debonding. It is clear that particle inclusion in a bulk volume matrix material can be beneficial only if adequate bonding between filler and matrix is provided. In the absence of proper bonding between particles and matrix, it is possible to obtain a weaker composite with respect to its matrix material. In order to prevent such a deficiency in the composite, adhesive promoting agents are used to assure a strong bond between constituents of composite (Basaran et al., 2006). The perfect and imperfect interphase conditions presented in this study are probably two extreme cases of interphase characteristics. In actual cases, interphase conditions will yield an intermediate level of performance which could be easily investigated by employing different interphase properties in user defined interphase model.

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4.3. Effect of interparticle distance on particle debonding The influence of interparticle distance on effective mechanical properties and large deformation response of PMMA/ATH composites was studied in terms of particle volume fraction in previous sections. It was observed that yield strength and stiffness increase with increasing filler content in composites with perfect interfacial bonding. For the case of imperfect interphase, increase in filler content was found to have a reverse effect on composite performance which was attributed to progressive damage due to particle debonding. In this section, the influence of interparticle distance on damage evolution in particle filled composites is presented. Interparticle distance is an important factor that determines the state of stress in matrix and interphase causing different modes of deformation in the matrix (crazing, shear yielding, dilatational band formation). A critical interparticle distance which depends on interfacial adhesion and relative stiffness of matrix and fillers is frequently used to asses ductile-to-brittle transition related to toughening mechanisms observed in particle filled composites (Thio et al., 2002; González et al., 2008). If interparticle distance is larger than the critical value, crack bridging without plastic deformation in matrix is observed. Fillers in particle filled composites essentially promote toughening mechanisms by acting as stress raisers facilitating matrix deformation. The preliminary microdeformation mechanism that triggers localized plastic deformation in the matrix can be particle debonding, rubber cavitation or particle failure which depends on relative strength and stiffness of constituents, particle and interphase characteristics. Imperfect interphase in MPUC model was characterized through different interphase thicknesses with temperature dependent strength and stiffness. It was assumed that particle debonding will take place when normal stress in the interphase exceeds a critical stress level. Particle debonding was observed to initiate at the pole of

0.02

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0.12

0.14

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0.18

Longitudinal Strain (%)

Fig. 8. The influence of interparticle distance on particle debonding in PMMA/ATH composites with imperfect interphase for the cases of interphase thickness of d/r = 1% at 75 °C (a) and the influence of interparticle distance on particle debonding in PMMA/ATH composites with imperfect interphase for the cases interphase thickness of d/r = 0.1% at 100 °C.

particles and propagate towards the equator of particles with increasing width along the loading direction. In Fig. 8a and b, progressive particle debonding during stretching of PMMA/ATH composite with imperfect interphase is presented. In Fig. 8a and b, percentage debonding is calculated with respect to total surface of particles and presented relative to longitudinal strain in the RVE. In MPUC model simulations, particle debonding was observed to initiate at early stage of deformation and continuously increase approaching to a constant level. The rate of particle debonding is much higher at the beginning and gradually decreases after a certain level of deformation which depends on temperature. At constant interphase thickness, the influence of interparticle distance is different in the glassy (Fig. 8a) and rubbery (Fig. 8b) regimes of PMMA matrix. At low temperatures below hg of PMMA, matrix is stiff and strong enough to resist to deformation at a higher extent. Temperature dependent interphase strength was also assumed to be strong in this temperature range. Therefore, particle debonding initiates at higher stretch levels at low temperatures (Fig. 8a) in comparison to temperatures above hg (Fig. 8b). The influence of interparticle distance is also more obvious at low temperatures. Smaller interparticle distance yields to overlapping of stress fields around particles which increases stress values in the interphase resulting in higher particle debonding concentration relative to larger interparticle distance for the same level of

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4.4. Effect of interphase thickness on particle debonding Imperfect interphase in MPUC model was characterized by its strength and interphase which were assumed to be temperature dependent. Both strength and stiffness of interphase was assumed to be 10% of matrix yield strength and elastic modulus. Therefore, influence of interphase properties on composite response was investigated by means of different thickness values. Different interphase thickness can be easily obtained by applying different coating agents on fillers which yields different interphase characteristics. In MPUC models with imperfect interphase conditions with different interphase thickness was less remarkable in comparison to interparticle distance. This can be attributed to selection of a weak interphase strength that promotes interfacial debonding at early stages of deformation under critical stress criterion. However, interphase thickness still plays an important in damage evolution due to particle debonding as depicted in Fig. 9a and b. For constant temperature and interparticle distance, interphase thickness controls deformation capacity of interphase and continuation of stress transfer from matrix onto fillers. Since temperature is constant for all cases presented in Fig. 9a and b separately, interphase strength is also constant. On the other hand, interphase stiffness modulus decreases with increasing interphase thickness. A thicker interphase will be more flexible in comparison to a thinner one and will accommodate higher deformation levels without exceeding critical stress level. Therefore, larger interphase thickness will simply delay particle debonding initiation and also evolution of damage will be at a slower rate as depicted in Fig. 9a and b. The influence of

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deformation. In MPUC model simulations with different interparticle distances, stress overlapping between particles was observed for all cases with decreasing effect at smaller particle volume fractions (or larger interparticle distances). There is a remarkable difference in particle debonding histories of smallest (d = 2.5 lm) and largest (d = 20 lm) interparticle distance at 75 °C (Fig. 8a). This difference becomes less noticeable as temperature gets close to hg of matrix as shown in Fig. 8b. In experimental studies on PMMA/ATH composites, it was observed that there is a significant change in polymer deformation around glass transition of PMMA. At low temperatures, matrix and filler phases were observed to deform coherently yielding to particle failure dominantly. At high temperatures, deformation in matrix was observed to be flow-like and more severe. At temperatures above hg, stiffness of matrix diminishes very rapidly and nearly all deformation took in the matrix phase accompanied by some particle failure and interphase failure which was attributed to large deformation fields around fillers (Gunel and Basaran, 2009). MPUC model predictions for high temperatures are in parallel with these observations. Interparticle distance becomes less significant as matrix is nearly in a viscous form deforming around particles more easily while interphase failure took place at very early stages of deformation due to incompatible deformation characteristics of constituents.

50 40 C1 (δ/r=10%)

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0.35

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Fig. 9. The influence of interphase thickness on particle debonding in PMMA/ATH composites with interparticle distance of 20 lm at 30 °C (a) and the influence of interphase thickness on particle debonding in PMMA/ ATH composites with interparticle distance of 20 lm at 30 °C interparticle distance of 2.5 lm at 75 °C.

interphase thickness on particle debonding rapidly vanishes as thickness become smaller than 0.1% of particle radius. Increase in temperature also diminishes the influence of interphase thickness, yet there is a significant difference between the thickest interphase (d/r = 10%) and the thinnest interphase (d/r = 0.01%) considered (Fig. 9b). Though, a critical stress criterion was considered for particle debonding, it is clear that deformation capacity of interphase defined by interphase stiffness modulus determines the initiation and propagation of interphase failure. 5. Conclusions A multi-particle unit cell (MPUC) model was used for investigation of large deformation of particle reinforced acrylics over a wide temperature range. Influence of temperature, interphase thickness and interparticle distance on progressive damage due to particle debonding was studied. In unit cell model, constitutive behavior of amorphous polymer matrix phase was defined through thermodynamically consistent dual-mechanism viscoplastic model, while fillers were assumed as spherical inclusions embedded in matrix at periodic intervals and imperfections within inclusions were neglected. A user interphase subroutine was used to incorporate imperfect interphase condition in finite element analysis of unit cell model. A critical stress criterion was used for detection of particle debonding. In the first part of study, finite element analysis results of MPUC model were compared with analytical

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models in terms of composite elastic modulus. MPUC model predictions were found to be in good agreement with predictions from experimentally validated analytical Lielens model for all volume fractions considered which verified that MPUC model can be considered as a representative volume of actual composite and MPUC model can be confidently used for stiffness predictions of PFCs. In the second part of study, influence of different interphase conditions on composite response at large deformations was studied through MPUC model for various temperatures. Based on finite element simulations, fillers effectively strengthen and stiffen matrix with respect pure PMMA for the composites with perfect interphase conditions where the effect becomes more noticeable with increasing filler content. In the post-yield region, strain softening was not observed for perfect bonding cases while significant amount of strain hardening was observed. Relative to neat polymer material and imperfect interphase cases, there is a slight decrease in ductility which was attributed to increased localized plastic deformation in the matrix due to continuous reinforcement effect of fillers restraining matrix deformation. For imperfect interphase cases, progressive damage due to particle debonding at early stages of deformation yields a remarkable on strength and stiffness of composite. Decrease in performance of composite with increasing filler content was attributed to porous-like structure in composites formed due to particle debonding. Since load carrying capacity of fillers vanishes after particle debonding as stress cannot be transferred to fillers, externally applied loads are nearly carried by only matrix material. In composites with imperfect interphase, there is a noticeable strain softening in post-yield region while no strain hardening was observed due to stiffness reduction due to debonded particles. Interparticle distance was found to have a profound effect on particle debonding. In composites with large volume fractions or equivalently with small interparticle distance, particle debonding was observed to occur faster while interparticle effect gradually diminishes with increasing temperature which was attributed to change in matrix deformation behavior. On the other hand, a thicker interphase was found to postpone initiation of particle debonding and providing some additional margin of ductility to composite relative to thinner interphases with same interphase strength. Therefore, a more flexible (thicker) interphase is more desirable at certain interphase strength while interparticle distance should kept at maximum possible without significantly reducing strength and stiffness of composite. In this work, it was also shown that MPUC modeling can be efficiently used to study particle filled composite behavior under different loading conditions and ambient temperatures and quantify the influence various material characteristics (interphase, filler and matrix characteristics) on material response. MPUC models are also superior to conventional macro level finite element analysis techniques for PFC’s. In MPUC model simulations, damage evolution (in this work, particle debonding followed by composite failure) can be traced and linked to stress/strain levels while finite element simulations of PFC at macro level cannot bring this level of detail reliably.

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