Composite Structures 93 (2011) 2655–2662

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Composite Structures

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Incremental damage theory of particulate-reinforced compositeswith a ductile interphase

Hui Yang a, Puhui Chen b, Yunpeng Jiang c,⇑, Keiichiro Tohgo d

a School of Jincheng, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, Chinab Faculty of Aerospace, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, Chinac Department of Engineering Mechanics, Hohai University, Nanjing 210098, Chinad Department of Mechanical Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu 432-8561, Japan

a r t i c l e i n f o

Article history:Available online 5 May 2011

Keywords:Particulate-reinforced composite (PRC)Ductile interphaseFinite element method (FEM)Debonding damage

0263-8223/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruct.2011.04.033

⇑ Corresponding author. Tel./fax: +81 25 84231645E-mail address: ypjiang@nuaa.edu.cn (Y. Jiang).

a b s t r a c t

This paper deals with a new micromechanics model of particulate-reinforced composites (PRCs) describ-ing the evolution of debonding damage, matrix plasticity and particle size effect on the deformation. Aductile interphase was considered in the frame of incremental damage theory to analyze the dependenceof elastic–plastic–damage behavior on particle size. Progressive debonding damage was controlled by acritical energy criterion for particle–matrix interfacial separation. The equivalent stresses of the matrixand interphase were determined by field fluctuation method. The influences of progressive debondingdamage, particle size and interphase properties on the overall stress–strain response of PRC wereexplained simultaneously. Due to the existence of a ductile interphase, stress transfer and plastic initia-tion in PRC become very complicated, and thus a unit-cell (UC) based FEM was used to simulate their evo-lutions and demonstrate the role of the interphase. Finally, particle size effect on the mechanicalbehaviors of composites was interpreted.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Particulate-reinforced composites (PRCs) have become moreand more attractive in the modern industry. They exhibit reduceddensity, higher elastic modulus, higher electrical resistivity, higherthermal conductivity, improved optical properties, superiormagnetic properties, increased strength/hardness, improved duc-tility/toughness, higher resistance to crack propagation includingcreep cracking, etc. Generally, many properties of PRCs are influ-enced by particle size, which attributes to the significant modifica-tion of microstructures by the introduction of inorganic particles.In explaining particle size effect, the interphase is undoubtedlyone of the most important factors [1]. Quite a number of researcheshave been published to date for studying the impact of the inter-phase, and the readers can refer our previous work [2]. Jianget al. have systematically investigated the effects of the interphaseon the overall stiffness, elastic–plastic and damage behaviors ofPRCs by using FEM, theoretical and experimental methods [2–6].In a recently published work, Jiang et al. have extended Tohgo’sincremental damage (ID) theory to three-phase case for interpret-ing particle size effect on the damage progression [2]. For the sakeof simplicity, the interphase is assumed to be brittle, and thus onlylinear elastic deformation of the interphase is considered during

ll rights reserved.

.

the macroscopic deformation. However, such a piece of assump-tion is very hard to be satisfied in many actual cases. On the con-trary, the elastic–plastic deformation of an interphase was foundto have a great influence on the mechanical behavior of PRCs.Ruiz-Navas et al. [7] modified Ti5Si3 ceramic particles with theCu coating, and found that Al + Ti5Si3–Cu composites exhibit supe-rior mechanical properties compared to Al + Ti5Si3 composites.Wang et al. [8] used FEM to study the effect of an interphase oncomposite toughness, and their results showed that a thin and highstrength interphase results in efficient stress transfer between par-ticle and matrix and leads to the deflection and propagation of thecrack within the matrix. Lauke [9] numerically analyzed the inter-facial adhesion strength between a coated particle and a polymermatrix material, and indicated the influence of a ductile interphaseon the local stress field. Wang and Yang [10] have employed FEManalysis to simulate the behavior of energy dissipation for the PRCswith a ductile interphase. Paskaramoorthy [11], Wu [12] andPapanicolaou [13] et al. also investigated the effect of an interphaseon the mechanical behaviors of composites. Ding and Weng [14]developed a homogenization theory to determine the overall elas-toplastic behavior of a PRC with a ductile interphase. Furthermore,the effect of a ductile interphase on the damage behaviors of PRC isexpected to be important. For the PRCs, Tohgo and Itoh [15] andTohgo and Weng [16] proposed an ID theory of PRCs taking into ac-count the plasticity of a matrix and progressive debonding damageof particles based on Eshelby’s equivalent inclusion method [17]

2656 H. Yang et al. / Composite Structures 93 (2011) 2655–2662

and Mori–Tanaka’s mean field concept [18]. In order to fully studythe effect of an interphase, it is necessary to extend Tohgo’s ID the-ory to the three-phase case with a ductile interphase.

Based on the previous studies [2–6], a ductile interphase wasintroduced and studied in the frame of ID theory [15], and theequivalent stresses of the matrix and interphase during the dam-age progression were determined by field fluctuation method.Numerical computations of elastic–plastic–damage stress–strainrelations under uniaxial tension were carried out for differentmicrostructures. Influences of debonding damage, interphaseproperties, particle size and particle volume fraction on the overallstress–strain response of PRCs were studied systematically. Addi-tionally, a unit-cell (UC) based FEM was used to understand the de-tailed stress transfer and plastic initiation induced by a ductileinterphase.

2. Incremental damage theory of PRC with a ductile interphase

The adopted composite system consists of three phases of par-ticle, matrix and interphase between them. The interphase concen-tration fI is related to that of particles fP by

fI=fP ¼ ð1þ 2t=dPÞ3 � 1 ð1Þ

Here dP and t are particle diameter and interphase thickness,respectively. The initial volume fraction of interphasefI0 ¼ fP0½ð1þ 2t=dPÞ3 � 1�, and fP0 is the initial volume fraction ofparticles. The evolution of interphase content can be expressed interms of the surrounded particles.

2.1. Constitutive relations of the constituents

The elastic incremental stress–strain relations of an isotropicmatrix, interphase and particles follow as:

dri ¼ CiðEi; miÞ : dei; i ¼M; I or P ð2Þ

where the symbol ‘‘:’’ is contraction product, dri and dei are theincremental stress and strain, respectively, and Ci(Ei, mi) is the stiff-ness tensor. The above equations replaced i with M, I and P stand forthe isotropic matrix, interphase and particles. Ei and mi are Young’smodulus and Poisson’s ratios of constituents, respectively. The elas-tic–plastic deformation of the matrix, interphase and particle is de-scribed by the Prandtl–Ruess equation (the J2-flow theory), which isapproximated by the following isotropic relation:

dri ¼ CiðE0i; m0iÞ : dei; i ¼M; I or P ð3Þ

where

E0i ¼Ei

1þ Ei=H0i; m0i ¼

mi þ Ei=ð2H0iÞ1þ Ei=H0i

; i ¼M; I or P ð4Þ

E0i and m0i represent tangent Young’s moduli and tangent Poisson’sratios of the constituents under elastic–plastic deformation. H0ishows the work-hardening ratio of each phase,

H0i ¼ drie= depl

e

� �i; i ¼M; I or P ð5Þ

where

rie ¼

32

r0kl

� �i r0kl

� �i� �1=2

; deple

� �i

¼ 23

deplkl

� �idepl

kl

� �i� �1=2

; i ¼ M; I or P ð6Þ

Here rie and depl

e

� �iare the von Mises equivalent stress and incre-

mental equivalent plastic strain, respectively. r0kl

� �i is the deviatoricstress component, and depl

kl

� �iis the incremental plastic strain. Eq.

(3) is strictly valid in the case of monotonic proportional loading.

In the composite system, the stress and strain of particles, inter-phase and matrix are denoted with superscripts ‘‘P’’, ‘‘I’’ and ‘‘M’’,respectively, and without any superscripts for the composite.

2.2. Incremental damage theory with a ductile interphase

Fig. 1 shows the states before and after an incremental defor-mation of a representative volume element (RVE) during the dam-age process, where a constant macroscopic stress r and itsincrement dr are applied on the boundary of RVE. All the debond-ing damage is supposed to occur between particle and interphase.The states before the incremental deformation are described interms of the intact particle content fP and damaged particle contentf dP . In the very beginning, the initial damaged particle content f d

P isassumed to be zero, and fP equals to fP0. Let dfP represent the vol-ume fraction of particles debonded in an incremental deformation,so the states after deformation can expressed by the intact parti-cles loading fP � dfP and damaged particles loading f d

P þ dfP. There-fore, a composite undergoing the damage process contains theintact and debonded particle, and particles to be damaged duringan incremental deformation. In order to solve this problem analyt-ically, some necessary assumptions are firstly given:

(1) All the constituents and composites are isotropic at the mes-oscopic scale.

(2) The debonding damage is controlled by the critical stress ofparticles.

(3) After the damage, the particle stress reduces to zero.(4) The progressive damage of the composites is described by a

decrease in the intact particle content and an increase in thedebonded particle concentration.

The constitutive equations of an isotropic PRC are expressed inthe form of the hydrostatic (dekk–drkk) and deviatoric components

de0ij—dr0ij� �

. So, the incremental strain de dekk; de0ij� �

– stressdr drkk; dr0ij� �

relation of the composite is given by,

dekk ¼1

3jtdrkk þ

13jd

rPkkdfP; de0ij ¼

12lt

drij þ1

2ldr0Pij dfP ð7Þ

where

jt ¼ jM 1� cj

1þ acj

� jd ¼ jMð1� aÞ½1� ð1� aÞcj�

lt ¼ lM 1�cl

1þ bcl

!

ld ¼ lMð1� bÞ½1� ð1� bÞcl�

ð8Þ

and

cj ¼ �fPðjP � jMÞ

jM þ ðjP � jMÞa� fI0ðjI � jMÞ

jM þ ðjI � jMÞaþ f d

P

ð1� aÞ

cl ¼ �fPðlP � lMÞ

lM þ ðlP � lMÞb� fI0ðlI � lMÞ

lM þ ðlI � lMÞbþ f d

P

ð1� bÞ

a ¼ 13

1þ mM

1� mM; b ¼ 2

154� 5mM

1� mM

ð9Þ

Here jP, jI, jM and lP, lI, lM are bulk and shear modulus of threeconstituents, respectively, which are related to Young’s modulus EP,EI, EM and Poisson’s ratio mP, mI, mM by

ji ¼Ei

3ð1� 2miÞ; li ¼

Ei

2ð1þ miÞ; i ¼M; I or P ð10Þ

The incremental stresses of the three phases drP, drM and drI

are expressed by

I IInterphase

, σσ ε

P PIntact Particle

, σ ε

DamagedParticle

M MMatrix

, σ ε

P

P

Intact Particle VolumeFraction: Damaged ParticleVolume Fraction:

(a)

d

f

f

P P

P P

Intact Particle VolumeFraction: Damaged ParticleVolume Fraction:

(b)

d

f df

f df

−

+

σ d+σ σ

Fig. 1. The states of composite undergoing damage process before and after incremental deformation, dfP is a volume fraction of the reinforcements damaged in theincremental deformation. The initial volume fraction of interphase fI0 is fP0 � [(1 + 2t/dP)3 � 1], where dP is particle diameter, t is interphase thickness, and fP0 is the initialvolume fraction of particles. (a) State before the incremental deformation. (b) State after the incremental deformation.

H. Yang et al. / Composite Structures 93 (2011) 2655–2662 2657

drPkk ¼

jP

½1� ð1� aÞcj�½jM þ ðjP � jMÞa�drkk þ rP

kkdfP� �

dr0Pij ¼lP

½1� ð1� bÞcl�½lM þ ðlP � lMÞb�dr0ij þ r0Pij dfP

� �

drMkk ¼

11� ð1� aÞcj

drkk þ rPkkdfP

� �dr0Mij ¼

11� ð1� bÞcl

dr0ij þ r0Pij dfP

� �

drIkk ¼

jI

½1� ð1� aÞcj�½jM þ ðjI � jMÞa�drkk þ rP

kkdfP� �

dr0Iij ¼lI

½1� ð1� bÞcl�½lM þ ðlI � lMÞb�dr0ij þ r0Pij dfP

� �

ð11Þ

0 0.02 0.04 0.06 0.08 0.10

100

200

300

Debonding

PorousCompositesStre

ss σ

xx (

MPa

)

Strain εxx

I II III

Fig. 2. The stress–strain relation of the composite is classified into three pieces I, IIand III. I represents the composites, II is the debonding process and III stands for theporous material. Based on the microstructures in three different states, the

2.3. The critical stress of debonding damage

The particle encounters debonding damage when the micro-scopic stress of particle reaches a critical value rP

cr . Since elasticstrain energy stored in the particle is released and the void surfaceis created by full debonding damage, the following relation is ob-tained from energy balance during the debonding process of aparticle:

rPcr

� �2

EPd3 / Cd2 ð12Þ

where C is the specific interface energy and EP is the Young’s mod-ulus of the particle. Eq. (12) gives a proportional relationship be-tween rP

cr andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEPC=dP

p[19]. In the present investigation, the

following relation is used for the sake of simplicity.

rPcr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEPC=dP

qð13Þ

Furthermore, C means the critical strain energy release rate for par-ticle–matrix interfacial debonding from a viewpoint of fracturemechanics. By introducing a critical stress intensity factor KC de-fined by C ¼ K2

C=EP, Eq. (13) is described by

rPcr ¼ KC=

ffiffiffiffiffidP

pð14Þ

C or KC is uniquely given for a combination of constituent materialsin composites, and they would be basically obtained by fracturetoughness tests for the interface between constituent materials.Eq. (14) describes the dependence of the critical stress on particlesize.

2.4. Equivalent stress of the matrix and interphase

The equivalent stress can be used to indicate the elastic–plasticstate of a ductile material, and Tohgo et al. determined the equiv-alent stress of the matrix by the energy method in the ID theory.For the present problem, the equivalent stresses of two phasesneed to be given simultaneously, and unfortunately they combinedin the same equation. So, field fluctuation (FF) method [20] shouldbe adopted here to solve this dilemma. As known from the previ-ous study [2], the microstructure under study will experience threestages during the damage progression. They are the composite,debonding and porous material, respectively. Therefore, the uniax-ial stress–strain relation of a composite shown in Fig. 2 can be clas-sified into I, II and III stages. ‘‘I’’ represents the composites, ‘‘II’’ thedebonding process and ‘‘III’’ the porous material. The equivalent

equivalent stresses of the matrix and interphase are determined.

R

R

z

r

particle

interphase

matrix

U

Fig. 3. FE mesh and two-dimensional problem, here, dP = 30 lm, t = 1.5 lm andf = 6.3%.

2658 H. Yang et al. / Composite Structures 93 (2011) 2655–2662

stresses of the matrix and interphase are separately determinedaccording to the corresponding microstructures in the three stages.

For the first stage, the composite contains particles, matrix andinterphase. An initial equivalent stress rr

e (r = I, M) of the matrixand interphase before plastic deformation and damage is given by

rre

� �2 ¼ 3l2r r2

xx

frE2

dEdlr

; r ¼ I;M ð15Þ

and their increments are given by

drre ¼

3l2r

frrreE2

dEdlr

rxxdðrxxÞ; r ¼ I;M ð16Þ

Here rxx is the far field uniaxial stress. The detailed deduction pro-cess for Eq. (15) is explained in Appendix-1, and expressions for dE/dlr in Eq. (16) are in Appendix-2.

For the third stage, the composite reduces to a porous material,and contains voids, matrix and interphase. The increments of theequivalent stress can be also determined by Eq. (16), but theexpression of the overall modulus E must reduce to a porousmaterial.

For the second stage, the composite contains voids, particles,matrix and interphase, and the boundary conditions continuouslychanges since the stress incrementally decreases induced by thedebonding damage. Therefore, the prerequisite to use field fluctu-ation method does not strictly hold. According to Tohgo’s energymethod [16], the combined energy equation in the matrix andinterphase is given as,

dU � dR ¼ fME0M6 1þ m0M� �rM

e drMe þ

fIE0I

6 1þ m0I� �rI

edrIe þ

fM

jMrM

m drMm

þ fI

jIrI

mdrIm þ fPr

PdeP � 12

dfPrPeP ð17Þ

where rimði ¼M; IÞ denotes the hydrotic stress of the constituents,

dU is the incremental energy of composites and dR is the energy re-leased by debonding damage

dU ¼ rde ð18Þ

dR ¼ 12rded ð19Þ

Following Qiu and Weng’s definition [21] of the equivalentstress,

rðrÞe

� �2 ¼ 1Vr

ZVr

32r0ðrÞij ðxÞr

0ðrÞij ðxÞdV ; r ¼ I;M ð20Þ

rðrÞe þ drðrÞe

� �2 ¼ 1Vr

ZVr

32

r0ðrÞij ðxÞ þ dr0ðrÞij ðxÞ� �

� r0ðrÞij ðxÞ þ dr0ðrÞij ðxÞ� �

dV ; r ¼ I;M ð21Þ

Neglecting the higher order terms of the increments, oneobtains

rðrÞe drðrÞe ¼32

r0ðrÞij dr0ðrÞij þ1fr

ZVr

32r0ptðrÞ

ij ðxÞdr0ptðrÞij ðxÞdV

� ; r

¼ I;M ð22Þ

where r0ðrÞij is the average deviatoric stress of the rth phase, which isdefined by Eq. (11). It is very difficult to know the distribution ofr0ptðrÞ

ij ðxÞ in the materials. An approximation method is adopted here,and a ratio f is defined by

f ¼ drIe

drMe

ð23Þ

drðrÞe ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32

r0ðrÞij þ dr0ðrÞij

� �r0ðrÞij þ dr0ðrÞij

� �r�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32r0ðrÞij r0ðrÞij

rð24Þ

Based on the above relation, the equivalent stresses of the duc-tile constituents can be solved. Since the detailed microscopicstress rij(x) in the composites fluctuates over its mean value de-fined by Eq. (11), the above equation shows that the perturbedstress fields rpt

ij ðxÞ over the constituents should change with thesame ratio as the average stresses. This strategy partially accountsfor the fluctuation effect by the inhomogeneity from Eq. (15), andthe following equation is reached,

drMe ¼

dU � dR� fMjM

rMm drM

m þfIjIrI

mdrIm þ fPrPdeP � 1

2 dfPrPeP� �

fME0M6 1þm0Mð Þr

Me þ

fIE0I

6 1þm0Ið ÞrIef

� ð25Þ

3. FEM

Due to the existence of a ductile interphase, the stress field inthe composite becomes too complicated. For better understandingthe elastic–plastic behavior at the micro-scale, axi-symmetric FEanalyses were performed [22].

The behavior of PRC is examined by using a unit-cell with theassumption of a periodical microstructure. The computation modelshown in Fig. 3 is an approximation of a periodical distribution ofthree-dimensional hexagonal cells, here, dP = 30 lm, t = 1.5 lmand fP0 = 6.3%. Only 1/4 of a RVE is used in the simulation by con-sidering the symmetry conditions. Periodical boundary conditionsare imposed on the RVE, and the other boundary conditions at theedges of a RVE are,

uz ¼ 0 along the axis z ¼ 0 ð26Þur ¼ 0 along the axis r ¼ 0 ð27Þuz ¼ D along the axis z ¼ U ð28Þ

where uz and ur are displacements along the directions z and r,respectively. On the upper boundary (z = U) a uniform displacementU was prescribed in z-direction.

FE mesh of an axisymmetric RVE is shown in Fig. 3. A perfectbonding is assumed between three phases. A relatively finer meshis used around the interphase to account for high stress gradient.Element size was checked so that numeric results are independenton the mesh density.

4. Numerical results and discussion

Numerical analyses were carried out on SiC particle reinforcedaluminum (Al) alloy composite (SiC/A356-T4) under uniaxial ten-sion [23]. The developed model is employed to predict the relation-ship between the stress (rxx) and strain (exx) in the tensile directionof the composites under uniaxial tension. The stress–strain relationof Al-alloy matrix is given by Ramburg–Osgood relation as follows:

P0

0 0.02 0.04 0.06 0.08 0.10

100

200

300

400

500

600

301515

75.7

MatrixID

increasing dP (μm)

7

brittle ductile interphase

5.7

fP0

=0.15, t=2.5μm

Strain εxx

Stre

ss σ

xx (

MPa

)

Fig. 5. Particle size effects on the stress–strain relations of the composites. The IDrepresents Tohgo–Chou–Weng’s model without particle size effects.

H. Yang et al. / Composite Structures 93 (2011) 2655–2662 2659

eMe ¼

rMe

EMþ k

ry

EM

rMe

ry

� 1=n

ð29Þ

where EM is the Young’s modulus, ry is the yield strength, n is thestrain hardening exponent, and k is a constant. They are EM = 70GPa,mM = 0.33, ry = 86 MPa, n = 0.212 and k = 3/7. The material proper-ties of SiC particles are EP = 427GPa and mP = 0.17. The materialproperties of interphase are EI = 250GPa and mI = 0.33. The stress–strain relation of the ductile interphase also obeys the formula ofEq. (27), only ry and n will be changed. KC ¼ 2:6MPa

ffiffiffiffiffimp

[2] wasused for interfacial debonding between particles and matrix.

4.1. Analytical model

4.1.1. Comparison between FF method and ID theoryIn our previous work [2], the accuracy of ID theory has been al-

ready confirmed through the comparison with the experiments.The FF method in obtaining the equivalent stress needs to be ver-ified firstly by comparing with the ID theory. Therefore, the overallstress–strain relations for a porous material, composite with andwithout damage effect are calculated by the FF and ID methodssimultaneously. Fig. 4 shows the comparison between the stress–strain relations of a composite, debonding process and porousmaterial without (a) and with (b) an interphase predicted by FFmethod and ID theory. Since the plastic behavior of a ductile inter-phase couldn’t be considered with the ID theory, a brittle inter-phase was assumed in the numerical computation. As clearlyseen from the comparison, all the results predicted by FF methodare in good agreement with those by ID theory. The comparisonindicated that FF method and the adopted strategy is very reason-able to solve the equivalent stress of the ductile constituents.

4.1.2. Role of particle sizeFig. 5 demonstrates the particle size effects on the stress–strain

relations of the composites with no damage. The ID denotes Tohgo-Chou-Weng’s model without particle size effects. Here, two caseswith a brittle and ductile interphase were studied for the directcomparison, and the interphase thickness is supposed to be

0 0.02 0.04 0.060

50

100

150

200

250 (a)

fp0

=15%, t=0, dp=50μm

Porous material

Debonding

Composites

IDT FFM

Stre

ss σ

xx (

MPa

)

Strain εxx

0 0.02 0.04 0.060

100

200

300 (b)

Porous material

fp0

=15%, t=10μm, dp=50μm

Debonding

Composites

IDT FFM

Stre

ss σ

xx (

MPa

)

Strain εxx

Fig. 4. Comparison between the stress–strain relations of the composite, porousmaterial and debonding progression without (a) and with (b) an interphasepredicted by FF method and ID theory.

2.5 lm. The overall stresses of the composite with a ductile inter-phase are remarkably lower than those with a brittle one, which iscaused by the decrease of stress transfer capacity in due to theyielding deformation of a ductile interphase. As the particle sizeis larger than 30 lm, the stress–strain curves converge to a con-ventional result by the ID theory without particle size effects. How-ever, as the particle size is smaller than 15 lm, the stress–strainrelations become size-dependent evidently.

Fig. 6 shows the effect of particle size on the stress–strain rela-tions of the composites with debonding damage. After the debond-ing damage, the stress–strain relation of the composite is almostconsistent with a porous material. The debonding damage is de-layed much more with smaller particle size, which is easily ex-plained from Eq. (12).

4.1.3. Role of interphase plasticityThe interphase plasticity is expected to greatly affect the overall

stress–strain relations. In our preceding studies [2–6], the effects ofPoisson ratio, Young’s modulus and thickness of the interphasehave been systematically analyzed. Therefore, only yieldingstrength and hardening exponent of the interphase are to be inves-tigated here. Fig. 7 depicts the dependence of the stress–strainrelations of composites on hardening exponent of an interphaseat a constant thickness with no damage, and rI

y ¼ rMy . For the hard-

ening exponent n in the range of 0.01–1.0, the overall stress–straincurves for different values of n are very close, which indicates thatthe overall mechanical response is insensitive to hardening expo-nent n. As n of the interphase is higher than that of the matrix,the overall stress–strain curves of the composites nearly overlaptogether.

Fig. 8 shows the dependence of stress–strain relations of thecomposites on the yield strength of an interphase with a giventhickness without debonding damage. The higher the ratio rI

y=rMy

is, the earlier the particle detaches from the matrix. Since the inter-phase with higher yielding strength transfers the applied load to

0 0.02 0.04 0.06 0.08 0.10

100

200

300

400

500

80μm50μm

porousmatrix

40μm30μm

20μm

dP=10μm

fP0

=0.15, t=2.5μm

σI

y=1.5σM

y

Stre

ss σ

xx(M

Pa)

Strain εxx

Fig. 6. Particle size effect on the stress–strain relations of the composites withdebonding damage.

0 0.02 0.04 0.06 0.080

100

200

300

0.10.20.515

MatrixσI

y/σM

y=10

fP0

=0.15, t=2.5μm, dP=40μm

Strain εxx

Stre

ss σ

xx(M

Pa)

Fig. 8. Dependence of the stress–strain relations of composites on the yieldstrength of an interphase with a constant thickness without debonding damage.

0 0.02 0.04 0.06 0.08 0.10 0.120

100

200

300

400

500fP0

=0.15, t=1.5μm

Prediction with σI

y=3σM

y

Lloyd's results

Strain εxx

Stre

ss σ

xx (

MPa

)

0

0.03

0.06

0.09

0.12

0.15

matrixfP

d-16

16μm f Pd

Fig. 9. Comparison of the stress–strain relations between the predictions andLloyd’s testing results for a composite (15 vol.% SiC/A356-T4).

0 0.02 0.04 0.06 0.08 0.10

100

200

300

400

Composite Interphase

0.5&1.0

0.212 0.01

1.00.5

0.212

n=0.01

fP0=0.15, t=2.5μm, dP=40μm

Stre

ss σ

xx (

MPa

)

Strain εxx

Fig. 7. Dependence of the stress–strain relations of composites on the hardeningexponent n of an interphase with a constant thickness without debonding damage.

2660 H. Yang et al. / Composite Structures 93 (2011) 2655–2662

the reinforcements more efficiently, the particle stress would in-crease up to the critical stress more rapidly. In summary, the yieldstrength of an interphase significantly influences the damage pro-cess and deformation behavior of the composites.

4.1.4. Verification of analytical modelComparisons will be conducted between Lloyd’s experimental

results [23] and numerical results for the stress–strain relationsof PRCs under uniaxial tension. Number frequency of particleswas assumed to follow the lognormal distribution,

pðdÞ ¼ 1ffiffiffiffiffiffiffi2pp

ddexp �ðln d� /Þ2

2d2

" #ð30Þ

where d is the standard deviation and the mean particle diameter �dP

is given by

�dP ¼ expð/þ d2=2Þ ð31Þ

In Eq. (30), d and / were set as 0.4 and 2.693, respectively. Fig. 9shows the comparison of stress–strain relations between the pre-

Fig. 10. Stress distribution in the axisymmetric RVE subjected to tensi

dictions and Lloyd’s testing results for a composite (15 vol.% SiC/A356-T4), and progress of debonding damage represented by voidvolume fraction f d

P as well. Here, the average particle size is16 lm, and the interphase thickness was assumed to be 1.5 lm. Itis found that the experimental stress–strain relation of SiC/A356-T4 composites can be well described by the present model takingaccount of the interphase, particle size distribution and debondingdamage.

4.2. FE analysis

4.2.1. Stress transferIn order to investigate the effect of interphase yield strength on

the local stress distribution around the particle, FE analyses wereperformed. Fig. 10 depicts the stress distribution in the axi-sym-metric RVE subjected to tensile loading: (a) rI

y ¼ 5r My and (b)

rIy ¼ 0:5rM

y . The local stress in the particle shown in Fig. 10b islower relative to that in Fig. 10a for high yield strength. So, aninterphase with higher yield strength is much more efficient inincreasing the stress transfer in the particle. The present resultsare also helpful to explain the variation tendency of stress–strainrelation shown in Fig. 8. An interphase with higher yielding stresscould transfer higher stress to the particle, so the particle stress be-come much higher, and exceed to the critical stress defined by Eq.(14) earlier.

4.2.2. Yield initiationThe yielding strength of the interphase is expected to change

yield initiation in the composites. Fig. 11 illustrates the plasticstrain distribution in the axisymmetric RVE with different inter-phase subjected to tensile loading. All the parameters are the sameas those used in Section 4.2.1. For a stronger interphase withrI

y ¼ 5rMy (Fig. 11a), the matrix firstly yields in which a certain

le loading (loading direction "): (a) rIy ¼ 5rM

y and (b) rIy ¼ 0:5rM

y .

Fig. 11. Plastic strain distribution in the axisymmetric RVE subjected to tensile loading (loading direction "): (a) rIy ¼ 5rM

y and (b) rIy ¼ 0:5rM

y .

H. Yang et al. / Composite Structures 93 (2011) 2655–2662 2661

distance far from the particle pole and along the angle of 45�. How-ever, for a weak interphase shown in Fig. 11b, interphase yieldingwould start at 45�, and plastic strain magnitude is lower. These re-sults also agree with those published in [10].

5. Conclusions

A ductile interphase was considered in the frame of incrementaldamage theory to study the elastic–plastic–damage behavior of thecomposites. Based on the present model, influences of progressivedebonding damage, particle size and interphase properties on theoverall stress–strain response of PRC were discussed. Moreover,FEM was used to understand the role of the interphase. Finally,some important conclusions can be drawn as follows:

(1) The particle size effect on the elastic–plastic–damage behav-ior of the composites can be explained from the viewpoint ofa ductile interphase, which is different from all the existingrelevant researches.

(2) For the composites with a ductile interphase, the equivalentstresses of the ductile interphase and matrix can be deter-mined by field fluctuation method.

(3) The material properties of the ductile interphase do have agreat influence to the overall mechanical behaviors anddamage propagation of the composites.

Acknowledgment

The present work was supported by a Grant-in-Aid for ScientificResearch (No. 08058) from the Japan Society for the Promotion ofScience (JSPS).

Appendix A

A.1. Appendix-1

For an r-phase composite RVE under a far field uniaxial stressrxx, field fluctuation method is adopted, and the following relationholds can be given,

rðrÞe

� �2 ¼ 3l2r r2

xx

frE2

dEdlr

ðA:1Þ

Similarly, for an r-phase composite RVE under a far field uniaxialstress rxx + drxx, the following relation still holds with the variationmethod,

rðrÞe þ drðrÞe

� �2 ¼ 3l2r ðrxx þ drxxÞ2

frE2

dEdlr

ðA:2Þ

Neglecting the higher order terms of the increments, one obtains

drre ¼

3l2r

frrreE2

dEdlr

rxxdðrxxÞ ðA:3Þ

A.2. Appendix-2

gI ¼ lM þ ðlI � lMÞb ðA:4Þ

gP ¼ lM þ ðlP � lMÞb ðA:5Þ

For a composite containing particle, matrix and interphase, theexpressions of dl/dlr are

dl=dlI ¼ lMðfI0ð1� bÞlM=g2I Þð1þ bclÞ

�1

þ lM½1� ð1� bÞcl�ðfI0blM=g2I Þð1þ bclÞ

�2 ðA:6Þ

dl=dlM ¼ ½1� ð1� bÞcl�ð1þ bclÞ�1

þ lMðb� 1Þ fPlP=g2P þ fI0lI=g

2I

� �ð1þ bclÞ

�1

� lM½1� ð1� bÞcl�ð1þ bclÞ�2b fPlP=g

2P þ fI0lI=g

2I

� �ðA:7Þ

where

cl ¼ �fPðlP � lMÞ

lM þ ðlP � lMÞb� fI0ðlI � lMÞ

lM þ ðlI � lMÞbðA:8Þ

By using fp ¼ f dp and lp = 0, the above equations reduce to those for

a porous material, the expressions of dl/dlr are similar to the aboveformulas, but cl is changed to

cl ¼ �fI0ðlI � lMÞ

lM þ ðlI � lMÞbþ fP

ð1� bÞ ðA:9Þ

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