Model for Elastic Modulus of Multi-ConstituentParticulate Composites

D. Veazie & J.L. Jordan & J.E. Spowart &B.W. White & N.N. Thadhani

Received: 19 September 2012 /Accepted: 15 January 2013# Society for Experimental Mechanics (outside the USA) 2013

Abstract In this paper, a methodology has been developedto accurately predict the elastic properties of multi-constituent particulate composites by accounting for irre-versible effects, such as energy loss that arises due to inter-nal friction. The complex dependence on loading densityand particle properties (i.e., size, shape, morphology, etc.) isinvestigated in terms of their effects on the effective elasticmodulus of the composite. Confirmed by experimental datafrom the compression loading of individual Ni and Alparticles dispersed in an epoxy matrix, it is believed thatthis approach captures the effects of internal friction, con-sequently providing a more accurate and comprehensiverepresentation for predicting and understanding the materialbehavior of multi-constituent particulate reinforced compo-sites. The present methodology provides a model to directlycompare the elastic modulus from an uncomplicated test,such as dual-cantilever beam loading in dynamic mechani-cal analysis (DMA), to the modulus obtained by other morecomplex experimental methods such as quasi-static com-pression. The model illustrates an efficient method to incor-porate input data from DMA to represent realistic elastic

moduli, hence promising for the characterization and designof particulate composites.

Keywords Multi-constituent particle-reinforcedcomposites . Internal friction . Compression modulus .

Dynamic mechanical analysis . Irreversible energy loss

Introduction

Large-scale acceptance of advanced particulate compositesfor military and civilian use is increasing because thesematerials possess high specific strength and stiffness, flexi-bility in design, and continually advancing manufacturingtechniques. Several particulate composite structures arepresently in service, however, most multi-constituent partic-ulate composite components have been limited, partly be-cause the expansive database of mechanical properties varywidely depending upon the experimental technique used.Laboratory testing of particulate composites that containevery combination of particulate material, volume fraction,and particle shape and size for static and dynamic mechan-ical behavior is impractical. Recent research has shown, asfor the case of particulate composites of aluminum andnickel powders in an epoxy binder prepared by casting,the yield stress measured by quasi-static and dynamic testto have a complex dependence on loading density andparticle properties (i.e., size, shape, morphology, etc.) [1].Since multi-constituent particulate composites contain real-istic complex microstructures (i.e., they are composed oftwo or more reinforcement particles, a binder, and the rein-forcement/binder interphase regions), detailed analyses andmodels that incorporate microstructural particulate features,such as particle interactions and friction, must be developedto provide a means of predicting critical engineering prop-erties from the properties of the constituents.

D. VeazieSchool of Engineering, Southern Polytechnic State University,Marietta, GA 30060, USA

J.L. Jordan (*)Air Force Research Laboratory, AFRL/RWME, Eglin AFB,FL 32542, USAe-mail: jennifer.jordan@eglin.af.mil

J.E. SpowartAir Force Research Laboratory, AFRL/RXBC, Wright-PattersonAFB, FL 45433, USA

B.W. White :N.N. ThadhaniSchool of Materials Science and Engineering, Georgia Instituteof Technology, Atlanta, GA 30332, USA

Experimental MechanicsDOI 10.1007/s11340-013-9722-9

Several authors have developed methods to predict theeffective elastic properties (elastic moduli or stiffness orcompliance) of particulate composites in terms of the elasticmoduli of the constituent materials [2–7]. These analysesare generally called micromechanics models, and are divid-ed into one of two methodologies, 1) a mechanics of materi-als approach and 2) an elasticity approach. Many of themethods formulated by the mechanics of materials approachembody the usual concepts of vastly simplifying assump-tions regarding the hypothesized behavior of the mechanicalsystem. In order to obtain a satisfactory, effective solutionthat agrees with experimental results, these simplifyingassumptions, if in fact not 100 % phenomenological, mustbe at least plausible. Elasticity approaches are usually char-acterized by more rigorous satisfaction of physical laws(equilibrium, deformation, continuity and compatibility,and constitutive or stress–strain relations) than in mechanicsof materials [8]. The effective elastic responses of particu-late composites have been obtained by directly applyingconventional micromechanical composite models such asthe Mori-Tanaka method [9] and the self-consistent method[10, 11]. Because of its implications that involve energyconcepts, including the principles of minimum complemen-tary energy and minimum potential energy, it is the latterelasticity approach which will form the basis of the model-ing in this paper.

In general, all micromechanical models have inherentlimitations, such as assumptions of a void free matrix orperfectly bonded reinforcements, which usually can beaccounted for in the representative volume element (RVE)by various means, including analytical resolution or finiteelement methods. The difficulty here lies in the ability tosolve the RVE for realistic material behavior, such as non-linear material behavior, kinetic friction between the partic-ulates, friction between the particulates and the matrix, etc.It can be shown that for a RVE under classical boundaryconditions, the macro strains and stresses are equal to thevolume averages over the RVE of the unknown micro strainand stress fields inside the RVE. In linear elasticity, relatingthose two mean values gives the effective or overall stiffnessof the composite at the macro scale. When linear elasticityassumptions are not plausible, the RVE problem can usuallybe solved very accurately by the well-known finite elementmethod, or by a mean-field homogenization (MFH) method,which is based on assumed relations between volume aver-ages of stress or strain fields in each phase of a RVE.Compared to the direct finite element method and, in reality,to all other existing scale transition methods, MFH is boththe easiest to use and the fastest in terms of computationtime. However, two shortcomings of MFH are that it isunable to give detailed strain and stress fields in each phaseand it is restricted to ellipsoidal inclusion shapes [12]. Thesolution for the effective properties of an inhomogeneous

material containing a single ellipsoidal inclusion in an infi-nite matrix was determined analytically by Eshelby [13] in alandmark paper, which is the cornerstone of MFH models.For non-dilute inclusions, Mori-Tanaka type methods[14–16] allow an efficient evaluation of the effective prop-erties covering a wide variety of inclusion shapes and dis-tributions. Higher order methods have been developed onthe basis of statistical considerations [17] which give betterpredictions for the stress fields and the effective properties.

Due to irreversible effects, such as energy loss that arisesdue to kinetic friction, the goal of this research is to developa framework to account for these effects to accurately pre-dict the elastic properties of multi-constituent particulatecomposites, and in turn, establish a correlation between theelastic modulus measured by quasi-static compression andthe modulus measured by double-cantilever beam loading indynamic mechanical analysis (DMA). The research dis-cussed here provides a model to directly compare the elasticmodulus from an uncomplicated, non-destructive test (i.e.,DMA) to the modulus obtained by other more complexexperimental methods.

Materials and Testing Procedures

Composites of aluminum and nickel powders in an epoxybinder (Epon 826), varying aluminum particle size andaluminum and nickel volume fraction, were prepared bycasting. The specific processing techniques and parametersare explained in Jordan, et al., [1]. The aluminum particlediameter was varied between 5 μm (Valimet, H5 aluminum)and 50 μm (Valimet, H50 aluminum). The H5 aluminum,shown in Fig. 1(a), was found to have an average particlediameter of 5.43 μm with spherical smooth particle mor-phology. The H50 aluminum particles, Fig. 1(b), were alsosmooth and nominally spherical with an average particlediameter of 51.91 μm. The nickel particles (Micron Metals)had rougher surface texture and more irregular shape, asshown in Fig. 1(c), with an average particle diameter of47.45 μm. However, there was also a small fraction ofparticles in the nickel powder that had an average particlediameter of 97.44 μm. The material properties (constants) ofthe constituents are shown in Table 1.

Each particulate composite was prepared as a single largeblock of material consisting of aluminum and nickel asdescribed in Table 2. From each block, several samplesapproximately 12.7 mm in diameter and 25.4 mm in heightfor quasi-static compression tests and samples approximate-ly 60 mm×12.5 mm×3.2 mm for dynamic mechanicalanalysis were machined, and, subsequently, three sampleswere tested using the DMA and five samples were tested inquasi-static compression. The calculated void content (fromparticle density and composite total density measured using

Exp Mech

ASTM D-3574 as a guide) for each composite is plotted inFig. 2 as a function of the composites’ decreasing particlevolume fraction.

Compression experiments at quasi-static strain rates (ap-proximately 1×10−4 and 1×10−3s−1) were conducted withan MTS 810 testing system with a 100 KN test frame and aconstant crosshead displacement rate, as described in J.L.Jordan, et al., [1]. The strain in the sample was determinedthrough multiple virtual strain gauges marked on the sampleand tracked using digital image correlation software. Theelastic modulus was determined by fitting a straight line tothe initial, linear part of the stress–strain curve, as shown inFig. 3. To reduce experimental error, the average of thevalues from approximately six replicate tests, each per-formed on different specimens, was reported.

The elastic (storage) modulus, loss modulus and tan δfrom DMA were determined by the deformation of rectan-gular specimens in a double cantilever configuration using aTA Q800. The DMA tests were carried out in air at fixedfrequency modes of 1, 5, 10, 50 and 100 Hz over a temper-ature range from −100 °C to 100 °C at a heating rate ofapproximately 2 °C/min. The initial strain level was approx-imately 0.3 %. The typical storage moduli, as well as tan δ,were plotted as a function of temperature, as depicted inFig. 4.

DMA Parameters

The elastic (storage) modulus from DMA is the stiffness of aviscoelastic material and is proportional to the energy storedduring a loading cycle. The storage modulus for the

composites in this study, at room temperature and 1 Hz, isplotted as a function of decreasing volume fraction of par-ticles in Fig. 5. This plot shows that the elastic modulus ofthe composite is relatively constant for composites contain-ing large particles (both Al and Ni) at high volume fractions(MNML-1 and MNML-2). The modulus decreases sharplyas the particle size decreases, as evidenced in the compositesthat contain only large or small aluminum particles with nonickel particles (MNML-5 and MNML-6). DMA test resultsshow the composite modulus decreasing almost linearly asthe volume fraction of the composite decreases from 40 % to30 %, regardless of the particle size (MNML-6, MNML-5,MNML-4, and MNML-3). Finally, the storage modulusdecreases at lower volume fractions, and does not show adependence of particle size at the lower volume fractions(MNML-7 and MNML-8).

Another important material property determined fromDMA is the loss modulus, E″, defined as being proportionalto the energy dissipated during one loading cycle. This lossmodulus, which is influenced by the internal friction in thecomposite, was plotted as a function of decreasing volumefraction of particles at room temperature and 1 Hz in Fig. 6.The loss modulus decreases almost linearly with decreasing

Fig. 1 (a) 5 μm aluminum particles, (b) 50 μm aluminum particles, and (c) nickel particles

Table 1 Summary of constituent properties

Material E [GPa] ν [GPa] G [GPa] μ

Epoxy 3.9 0.39 1.4 0.57

Aluminum 70.0 0.33 25.6 1.3

Nickel 200.0 0.31 76.3 1.05

Table 2 Summary of constituent and composite materials

Material Vf % Al Vf % Ni Density [g/cm3]

Epoxy - - 1.190

H5 Aluminum - - 2.688

H50 Aluminum - - 2.749

Nickel - - 8.728

Epoxy–40H50–10 Ni 40 10 2.576

Epoxy–40H5–10Ni 40 10 2.568

Epoxy–20H50–10 Ni 20 10 2.299

Epoxy–20H5–10Ni 20 10 2.128

Epoxy–40H50 40 0 1.811

Epoxy–40H5 40 0 1.800

Epoxy–20H50 20 0 1.458

Epoxy–20H5 20 0 1.505

Exp Mech

particle volume fraction and particle size. This loss modulussubstantiates the irreversible kinetic friction assertion thatenergy loss due to internal friction would tend to be higherin composites that contain larger particles and higher volumefractions.

The loss factor, tan δ, is the ratio of loss modulus tostorage modulus, and represents a measure of the energylost (mechanical damping). In essence, tan δ indicateshow ‘effectively’ the material loses energy to internalfriction or molecular rearrangement [17]. Tan δ, which isproportional to the energy loss in a multi-constituent par-ticulate composite, was plotted as a function of decreasingvolume fraction and particle size in Fig. 7. This plotshows an almost linear correlation between internal fric-tion or molecular rearrangement and decreasing particlesize at higher (MNML-1 and MNML-5) or lower(MNML-4 and MNML-8) particle volume fractions. Like-wise, at similar volume fractions, tan δ tends to decrease as theparticle sizes decrease.

The DMA measurements for this research were made indual cantilever beam mode. For this loading mode (bending),it is important to ensure that the theory used to calculatethe corresponding strains from the applied loads is correctand representative of the physical properties. If theBernoulli-Euler Beam Theory [18] is used, pure bending(no shear or rotary inertia) is assumed. However, if theTimoshenko Beam Theory [19] is applied, the effects ofshear or rotary inertia are included, which becomes im-portant due to the fact that transverse shear effects aremore pronounced at higher frequencies (or shorter speci-men lengths) [20]. In general, the decay length for endeffects in dynamic testing is much greater than predictedby St. Venant’s Principle. A comparison of the plots of E″and tan δ should show that these quantities are constantwith increasing dynamic frequency of the DMA tests formaterials that possess similar damping characteristics [21].However, a plot of the maximum E″ difference (increasingdynamic frequency from 1 Hz to 100 Hz) as a function ofdecreasing volume fraction of particles shows considerably

Fig. 2 Calculated void content for each composite

Fig. 3 Representative stress–strain curve showing method for deter-mination of elastic modulus [26]

Fig. 4 Typical storage moduli and tan δ from DMA

Fig. 5 Storage (elastic) moduli as a function of decreasing volumefraction of particles

Exp Mech

high E″ difference for multi-constituent particulate compo-sites that have higher volume fractions (MNML-1 andMNML-2), whereas the maximum E″ difference is virtu-ally constant at the lower composite volume fractions (seeFig. 8).

Modeling Approach and Discussion

In this section, a model based on internal friction is pre-sented to account for the difference between DMA storagemodulus and quasi-static compression modulus. The inter-nal friction model shows the dependence of particle density(loading), particle size, particle ‘roughness’ (related to fric-tion coefficients), and elastic properties (E, G and ν) in amulti-constituent particulate composite. In multi-constituentparticulate composites with very low particle volume frac-tions (< 20 %), internal friction or molecular rearrangementin the matrix material usually dominates. In these cases, the

friction effects on the effective elastic modulus are charac-terized straightforwardly by the loss modulus (E″, as ameasure of energy lost to internal friction or molecularrearrangement). Friction from particle-particle and particle-matrix interactions is accounted for by incorporating a localinterpretation of the classical Coulomb model [22] for ki-netic friction and the irreversible friction effects from load-ing and contact hysteresis. Internal friction from thesemodels are explicit functions of the applied loads (bothnormal and shear), and the ‘weighting’ of the effects fromthese models is introduced via the loss modulus.

For modeling friction that involves deformable solids,as in the case of multi-constituent particulate compositesthat possess a higher volume fraction of particles, a localCoulomb friction model can be introduced. As describedby Savkoor [23], when deformable solids are pressedtogether, a finite area of contact C is formed and thecontact force is generally distributed nonuniformly insidethe contact area. The local loading on a surface elementin the contact can be defined in terms of the normal andshear components of traction. In this analysis, the normaladhesion between surfaces is ignored and the action ofnormal traction in the contact is assumed unilateral. Theshear traction at any point in the contact arising fromfriction depends on the normal traction and the relativemotion between the surfaces at this point. A simple andeffective way of defining this relationship is based on thepointwise application of the global Coulomb model. Therelationship is the local Coulomb friction model thatapplies inside the contact of deformable solids. Whetheror not relative motion can occur at any point inside thecontact depends on the ratio of the tangential and thenormal traction at that point. Therefore, the macroscopicfrictional behavior of the solids, in particular the transi-tion from static to kinetic friction of solids, essentiallydepends on the distribution of normal and shear tractionin the contact [24].

Fig. 6 Loss moduli as a function of decreasing volume fraction ofparticles

Fig. 7 Tan δ as a function of decreasing particle size and volumefraction of particles

Fig. 8 Maximum E″ difference (1 Hz to 100 Hz) as a function ofdecreasing volume fraction of particles

Exp Mech

The local model of Coulomb friction applies at any pointinside the contact C, as shown in Fig. 9. It determineswhether or not the local shear traction (tx, ty) due to frictionis sufficiently large to initiate sliding, depending upon thelocal normal pressure p and the coefficient of friction μ. At apoint in the contact interface where t≤μp, static frictionprevents relative motion of surface points in contact (localslip). If slip occurs at any contact point, then the local sheartraction equals t=μp. The model rules imply that the distri-bution of normal pressure has a strong influence on therelative motion of points in contact. The model also requiresthat at any point the traction and slip vectors are collinearand oppositely directed [24].

When particle-particle interactions are dominant, typical-ly observed in multi-constituent particulate composites athigh particle volume fractions (>40 %), the size and shapeof the contact area C, and the normal pressure p due tonormal load P can be determined from the knowledge ofthe geometry of the undeformed solids and their bulk mate-rial properties. Because the multi-constituent particulatecomposite considered here, consisting of individual Ni andAl particles dispersed in an epoxy matrix, possessed nomi-nally spherical particles, the contact area C is assumed acircular area of radius ac. Thus, from the Hertz Theory ofcontact for solids [25] involving the contact between twoelastic solids of revolution with radii R1 and R2 and withmaterial constants E1, υ1 and E2, υ2 under a normal load P,the contact results in a circular area of radius ac and a centraldeflection α as:

ac ¼ 3

4

PR*

E*

� �1 3=

and a ¼ ac2

R*

� �¼ 9P2

16R*E�2

� �1 3=

ð1Þ

The geometric and elastic properties of the two solids arecombined into two constants, an equivalent radius R* and anequivalent modulus E* as:

1

R*¼ 1

R1þ 1

R2and

1

E*¼ 1� u21

E1þ 1� u22

E2ð2Þ

In multi-constituent particulate composites at lower par-ticle volume fractions (40 %<Vf<20 %), particle-matrixinteractions are usually dominant. The friction generated isthus characterized by the contact area between the particlesand the surrounding matrix with elastic constants E1, υ1, E2,υ2 and Em,, υm, respectively, under load P. The surfacecontact area of revolution with radius as is modified as:

as ¼ P

4pE*t R

*� ��1

where1

E*t¼ 1� u21

E1þ 1� u22

E2þ 1� u2m

Em

ð3ÞTo obtain the storage and loss moduli, as well as other

parameters for the input into the model, load was applied tothe multi-constituent particulate composite in the dual can-tilever beam mode by DMA. In this mode, both normal andshear stresses (including shear or rotary inertia) are intro-duced, resulting in internal friction effects that are highlydependent upon the particle volume fraction. Consequently,the particle volume fraction of the multi-constituent partic-ulate composite affords insight into the ratio of the particle-particle contact area radius, ac, to the particle-matrix surfacecontact area radius, as. Using SEM image analyses of multi-constituent particulate composite cross-sections as depictedin Fig. 10 [26], and the Savitzky-Golay smoothing proce-dure [27] to approximate friction contact as a function ofparticle volume fraction Vf, a Tailed Extreme Value FourParameter (Amplitude) expression [28] is derived for thetotal quantitative value of the internal friction contact arearadius, a, as:

a ¼ A0 exp�Vf þ A1 þ A2 � A2A3 exp � Vf þA2 ln A3ð Þ�A1

A2

� �A3A2

24

35

ð4Þ

Fig. 9 Hertz Theory of contact for solidsFig. 10 SEM micrograph of a multi-constituent particulate compositecross-section (Epoxy 50 % Al-40 % Ni-10 %) [From Reference 26]

Exp Mech

with the four parameters being the fitted values A0=0.97, A1=73.9, A2=6.62 and A3=99.57. The Savitzky-Golay procedureis a time-domain method of smoothing based on least squaresquartic polynomial fitting that relies on the linearity of anunweighted polynomial model, thus enabling the procedureto be quite efficient with large data sets. The functionapproaches maximum particle-particle contact area radius,a→ac at the highest particle volume fractions and proportion-al particle-matrix surface contact area radius, a→as at lowerparticle volume fractions, as shown in Fig. 11.

To analyze the static to kinetic friction by partial slip, alocal interpretation of the classical Coulomb Model is used,as depicted in Fig. 12 [29]. In the case of a circular Hertziancontact, the simpler problem is where slipping is prevented(infinite friction) across the entire contact area. From tan-gential forces, T, the result of the linear elastic analysis forthe shear traction τ and the shift δ is given as:

d ¼ T1

8a; where 1 ¼ 2� u1ð Þ=G1 þ 2� u2ð Þ=G2½ �; ð5Þ

and tðrÞ ¼ T

2paa2 � r2� ��1 2=

; r < a: ð6Þ

For tangential forces 0≤T≤μP, static friction preventsslip inside a circular “locked region” of radius c, whereasslipping occurs in an annular region c≤r≤a. The relativedisplacement of the solids in the direction of T is the shift δ.The results for the circular locked region of radius c, and theshear traction both in the locked (r<c) and the annular slipregion (c<r<a), (see Fig. 12) are

c

a¼ 1� T

μP

� �1 3=

; ð7Þ

whereas tðrÞ ¼ 3μP2pa2

1� r2 a2�� �1 2=

� c a=ð Þ 1� r2 a2�� �1 2=

in 0 � r � c;

ð8Þ

and tðrÞ ¼ 3μP2pa2

1� r2 a2�� �1 2=

in c � r � a: ð9Þ

The expressions for the tangential shift δ and the tangen-tial compliance are

d ¼ 31μP8a

1� 1� T

μP

� �2 3=( )

anddddT

¼ 1

4a1� T

μP

� ��1 3=

: ð10Þ

Kinetic friction that arises in the region of slip isessentially an irreversible process [24]. If two solids areunder a normal load P, and subjected to a tangential loadT, such that T<μP, and the bodies are tangentiallyunloaded by reducing the tangential force to 0, theforce-displacement curve generated encloses an areawhich represents the energy dissipation. The energy dissi-pated is given by

Ud ¼ 9 μPð Þ2λ10a

1� 1� T

μP

� �5 3=

� 5

6

T

μP1� 1� T

μP

� �2 3=( )" #

:

ð11ÞFig. 11 Internal friction contact area radius approximated by a TailedExtreme Value 4 Parameter (Amplitude) expression as a function ofparticle volume fraction

Fig. 12 Coulomb Model where slipping occurs in an annular region

Exp Mech

Because this analysis is based on the assumption of aconstant coefficient of friction, the slipping in the an-nular region generates heat (accounted for in the lossmodulus, E″) and causes damage that may continuouslymodify the value of μ. Figure 13 shows the results offriction energy dissipated in the multi-constituent partic-ulate composite system as a function of particle size andvolume fraction of particles. Multi-constituent particulatecomposites that contain higher particle volume fractions(> 40 %) where particle-particle interactions are domi-nant, show higher friction energy than those compositesat lower particle volume fractions (40 %<Vf<20 %),where particle-matrix interactions are dominant. Compo-sites that contained only one type of particle (Al) athigh particle volume fraction (40 %) showed the highestfriction energy.

As abovementioned, the basis for the determination ofthe Young’s modulus of a multi-constituent particulate com-posite material presented here utilizes the elasticity ap-proach where U is the strain energy of any compatiblestate of strain εx, εy, εz, γxy, γyz, γzx that satisfies thespecified displacement boundary conditions, i.e., an ad-missible strain field. The expression for the total strainenergy is

U ¼ 1

2

ZV

σx"x þ σy"y þ σz"z þ txygxy þ tyzgyz þ tzxgzx� �

dV

ð12Þ

where σx, σy, σz, τx, τy, τz represents the average stressesthat correspond to the applied load at the boundaries. By

Fig. 14 Stress measured from DMA as a function of volume fractionof particles

Fig. 15 Strain measured from DMA as a function of volume fractionof particles

Fig. 16 Tangential stiffness measured from DMA as a function ofvolume fraction of particles

Fig. 13 Friction energy in the multi-constituent particulate compositesystem as a function of particle size and volume fraction of particles

Exp Mech

definition of the ratio of elastic stress to strain, and rearrangingthe above equation for the total strain energy comprised of the

strain energy from DMA data, UDMA, and the energy frominternal friction, UFriction, the Young’s modulus E may beexpressed as

E ¼ 1

2

ZV

σ2

UDMA þ UFrictiondV : ð13Þ

The strain energy from DMA data is rather straight-forwardly derived from the measured stress and strain;shown in Figs. 14 and 15 respectively, as a function ofdecreasing particle volume fraction in the multi-constituentparticulate composite materials. The measured tangentialstiffness from DMA, as elucidated in Eq. (10), providesthe means to introduce the energy dissipated from in-ternal friction, Ud, for the total friction energy, UFriction.The tangential stiffness is averaged over the individualparticle volume fractions as depicted in Fig. 16, en-abling the ‘weighting’ of the effects of the energydissipated from internal friction via the loss modulus(tan δ) as

UFriction ¼ 9 μPð Þ2 l10a

1� 1� T

μP

� �5 3=

� 5

6

T

μP1� 1� T

μP

� �2 3=( )" #

E0 0

E0

4a

l1� T

μP

� �3" #

ð14Þ

The results from the constituent model developed inthis research using DMA data and internal frictionaleffects agree well with the experimental data for a multi-constituent particulate composite containing aluminum andnickel powders in an epoxy binder [1], as shown inFig. 17. A key enabling technology in this research isthe establishment of the correlation between the mechan-ical and structural properties of the constituents and themechanics-based, mechanical response in multi-constituentparticulate composites.

Conclusion

In this research, a model is developed to accurately predictthe elastic properties of multi-constituent particulate compo-sites by accounting for irreversible effects, such as energyloss that arises due to internal friction. The model shows thedependence of particle density (loading), particle size, par-ticle ‘roughness’ (related to friction coefficients), and elasticproperties (E, G and ν) in a multi-constituent particulatecomposite. The internal friction introduced in this model isan explicit function of the applied loads, and the ‘weighting’of the effects from this models is introduced via the DMAloss modulus, thus providing a method to account for the

difference between DMA storage modulus and quasi-staticcompression modulus.

The multi-constituent particulate composites in this studywere composed of aluminum and nickel powders in anepoxy binder, varying aluminum particle size and aluminumand nickel volume fraction. Compression experiments atquasi-static strain rates were conducted to determine theelastic modulus, and DMA tests were performed to deter-mine the elastic (storage) modulus, loss modulus and tan δ.The elasticity approach of a micromechanics model, alongwith the pointwise application of the global Coulomb modeland the Hertz theory of contact, was employed to determinethe energy dissipated from internal friction. Results from theinternal friction model showed very good agreement withquasi-static experimental data, and established a correlationbetween the properties of the constituents and the mechanics-based, mechanical response of a multi-constituent particulatecomposite.

This investigation of multi-constituent particulate com-posites has identified the need to address the effects ofparticle density (loading), particle size, and particle ‘rough-ness’ on the elastic properties. Although the results pre-sented here represent a coupled effect of the particle sizeand loading density on the energy dissipated from internalfriction, and are specific to the materials, loading direction

Fig. 17 Young’s modulus comparison for the multi-constituent partic-ulate composite showing predictions of Young’s modulus from DMAfriction model, which compare well with experimental (quasi-static)compression data

Exp Mech

and loading modes, and the environments examined, thesensitivity of the internal friction to the elastic modulus fromDMA implies that these effects must be accounted for intesting methods or predictive schemes to estimate the mechan-ical response of multi-constituent particulate composites.

References

1. Jordan JL, White B, Spowart JE, Thadhani NN, Richards DW. AStatistical Approach to Static and Dynamic Testing of Epoxy-Based Multi-Constituent Particulate Composites. Submitted forpublication

2. Paul B (1960) Prediction of Elastic Constants of Multiphase Mate-rials. Transactions of the Metallurgical Society of AIME, 36–41

3. Hashin Z (1962) The Elastic Moduli of Heterogeneous Materials.Journal of Applied Mechanics, 143–150

4. Hashin Z, Shtrikman S (1963) A Variational Approach to theTheory of Elastic Behaviour of Multiphase Materials. Journal ofthe Mechanics and Physics of Solids, 127–140

5. Cho J, Joshi MS, Sun CT (2006) Effect of Inclusion Size onMechanical Properties of Polymeric Composites with Micro andNano Particles. Compos Sci Technol 66:1941–1952

6. Zhu LJ, Cai WZ, Tu ST (2009) Computational Micromechanics ofParticle-Reinforced Composites: Effect of Complex Three-Dimensional Microstructures. J Comput 4:1237–1242

7. Boutaleb S, Zairi F, Mesbah A, Nait-Abdelaziz M, Gloaguen JM,Boukharouba T, Lefebre JM (2009) Micromechanics-based Mod-eling of Stiffness and Yield Stress for Silica/Polymer Nanocompo-sites. Int J Solid Struct 46:1716–1726

8. Jones RM (1999) Mechanics of Composite Materials, 2nd edn.Taylor & Francis, LLC., New York

9. Mori T, Tanaka K (1973) Average Stress in the Matrix and AverageElastic Energy of Materials with Misfitting Inclusions. Acta MetallMater 21:571–574

10. Hill R (1965) Theory of Mechanical Properties of Fibre-Strengthened Materials—III. Self-Consistent Model. J Mech PhysSolid 13:213

11. Budiansky B (1965) On the Elastic Moduli of Some Heteroge-neous Materials. J Mech Phys Solid 13:223–227

12. Odorizzi S (2009) Multi-Scale Modeling of Composite Materialsand Structures with DIGIMAT to ANSYS. Newsletter Engin Soft6:24–27

13. Eshelby JD (1957) “The Determination of the Elastic Field of anEllipsoidal Inclusion and Related Problems. Proc Roy Soc LondA241:376–396

14. Benveniste Y (1987) A New Approach to the Application of Mori–Tanaka’s Theory in Composite Materials. Mech Mater 6:147–157

15. Pettermann HE, Böhm HJ, Rammerstorfer FG (1997) Some Di-rection Dependent Properties of Matrix-Inclusion Type Compo-sites with Given Reinforcement Orientation Distributions. ComposB 28B:253–265

16. Torquato S (1991) Random Heterogeneous Media: Microstructureand Improved Bounds on Effective Properties. Appl Mech Rev44:37–75

17. Menard KP (2008) Dynamic Mechanical Analysis: A PracticalIntroduction, 2nd edn. CRC Press, Boca Raton, FL

18. Timoshenko S (1953) History of Strength of Materials. McGraw-Hill, New York

19. Timoshenko SP (1921) On the Correction Factor for Shear of theDifferential Equation for Transverse Vibrations of Bars of UniformCross-section. Philosophical Magazine 744

20. Mindlin RD (1951) Influence of rotatory inertia and shear defor-mation on flexural motions of isotropic elastic plates. ASMEJournal of Applied Mechanics 18:31–38

21. Nakao T, Okano T, Asano I (1985) Theoretical and experimentalanalysis of flexural vibration of the viscoelastic Timoshenko beam.J Appl Mech 52(3):728–731

22. Drucker DC (1953) Coulomb Friction, Plasticity, and Limit Loads,Brown University Providence RI Division of Applied Mathemat-ics, Providence, RI

23. Savkoor AR (1992) Models of Friction Based on Contact andFracture Mechanics. In: Singer IL, Pollock HM, NATO ASI SeriesE (eds) Fundamentals of Friction Macroscopic Processes. KluwerAcademic Press, Dordrecht, pp 111–133

24. Savkoor AR (2001) Models of Friction. In: Lemaitre J (ed) Hand-book of Materials Behavior Models. Academic, San Diego, CA

25. Johnson KL (1985) Contact Mechanics. Cambridge UniversityPress, Cambridge, U. K

26. White BW (2011) Microstructure and Strain Rate Effects on theMechanical Behavior of Particle Reinforced Epoxy-Based Reac-tive Materials,” PhD Dissertation, Georgia Institute of Technology

27. Savitzky A, Savitzky A, Golay MJE (1964) Smoothing and differ-entiation of data by simplified least squares procedures. AnalChem 36:1627–1639

28. Christopoulos A, Lew MJ (2000) Beyond Eyeballing: FittingModels to Experimental Data. Crit Rev Biochem Mol Biol 35(5):359–391

29. Mindlin RD (1949) Compliance of Elastic Bodies in Contact.Transact ASME J Appl Mech 16:259–268

Exp Mech