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Mechanics of Materials

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Uniaxial compression model for a metal-matrix/hollow-microspherecomposite synthesized by pressure infiltration

S.V. Shil'koa,⁎, D.A. Chernousa, Qiang Zhangb, Yingfei Linc, Heeman Choed

a V.A. Belyi Metal-Polymer Research Institute of National Academy of Sciences of Belarus, Gomel 246050, Belarusb School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, ChinacGuangdong Provincial Key Laboratory for Technology and Application of Metal Toughening, Guangdong Institute of Materials and Processing, Guangzhou 510650, Chinad School of Materials Science and Engineering, Kookmin University, Seoul 02707, Republic of Korea

A R T I C L E I N F O

Keywords:Pressure infiltrationPorous metalAluminum alloyGlass microsphereThree-phase micromechanical model

A B S T R A C T

A new calculation technique is proposed for determining the deformation-strength characteristics of compositesmade of a porous metal matrix and hollow spherical inclusions. It is based on a three-phase micromechanicalmodel of a dispersion-filled composite with taking into account the presence of an interfacial layer. For vali-dation/exemplification purposes, this method is used to describe the initial pressing stage of a porous aluminumalloy containing hollow glass microspheres; the simulated mechanical characteristics and axial stress peak of thetarget composite are compared with experimental data and the results by a simplified alternative model.Additionally, the developed method is able to provide acceptable prediction accuracy for the elastic moduli ofthe composite and the peak axial stress. The proposed methodology is capable of predicting not only the elasticparameters but also strength characteristics of heterogeneous materials under consideration, which is demon-strated using three composites synthesized by pressure infiltration technology.

1. Introduction

Microporous composites based on metal and ceramic matrices arebecoming increasingly widespread in various technological branches.The addition of micron-size hollow spherical particles (microspheres) asdispersed fillers allows the adjustment of their physical and mechanicalcharacteristics and the reduction of the final density. In several works,Al2O3 microspheres (Wang et al., 2013, Alizadeh and Mirzaei-Aliabadi, 2012), SiC hollow spheres (Luong et al., 2013, Cox et al.,2014), fly ash (Rohatgi et al., 2006, 2002, Balch et al., 2005), glasscenospheres, and other ceramic hollow particles (Tao et al., 2009,Xia et al., 2014) have been used as fillers for various aluminum alloys.Their properties make syntactic foams an attractive choice for appli-cations in energy and sound absorbers, impact dampers, and light hulland shell materials with core structures for the aeronautic and auto-motive industries (Goel et al., 2015). Compared to common aluminumfoams with gas porosity, syntactic foams containing a hollow particles-filled aluminum matrix exhibit higher mechanical characteristics(strength and elasticity modulus), more effective thermal insulation,and lower (near zero) thermal expansion coefficient, corresponding tobetter dimensional stability for structural elements and machine parts.

Such advanced composites are mainly fabricated by pressing

powder mixtures. In particular, pressure infiltration technique for aporous aluminum alloy with glass microspheres has been analyzed indetail (Lin et al., 2017, 2016). A comparison was made in Ref.Lin et al. (2017) using the results of the calculation method developedwith the data obtained on the basis of those from the Kiser et al. (1999),Wu et al. (2007), Mondal et al. (2009), and Ferguson et al. (2013). Itwas shown that the technique developed in Ref. Lin et al. (2017) pro-vided more accurate estimates, in comparison with other alternativemodels for the analysis on the critical axial stress of the compositeunder consideration.

Lin et al. (2017) proposed a structural interpretation of thestress–strain diagram characterizing this process, which showed apronounced maximum stress peak corresponding to the destruction ofthe microspheres; they also developed a calculation method for a moreaccurate prediction, compared to previously reported mathematicalmodels containing the axial stress amplitude at which the microsphereintegrity was violated. However, Lin et al.’s methodology relies on thesimplified approach of averaging the tensile strengths of the targetcomposite components and requires additional verification.

The further modified model presented in this study is consideredsuperior to the technique described in Ref. Lin et al. (2017) in terms ofthe accuracy of predicting the critical axial stress. According to the

https://doi.org/10.1016/j.mechmat.2020.103349Received 16 October 2019; Received in revised form 27 December 2019; Accepted 4 February 2020

⁎ Corresponding author.E-mail address: Shilko_mpri@mail.ru (S.V. Shil'ko).

Mechanics of Materials 144 (2020) 103349

Available online 05 February 20200167-6636/ © 2020 Elsevier Ltd. All rights reserved.

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abovementioned works, this study aims to develop a model based onthe micromechanical theory of porous and dispersion-filled materials(Pleskachevskii et al., 2003, Choi et al., 2017, Pan'kov, 1998,Christensen, 1979, Shil'ko et al., 2013) as well as to improve the cal-culation method for the deformation-strength characteristics of aporous metal matrix filled with microspheres.

2. Experimental and calculation methods

Fig. 1 schematizes the pressure infiltration process for the pre-paration of the target composite. The preformed microspheres were putin a steel mold and a vertical pressure of ~0.5 MPa was applied to forcethe complete infiltration of molten aluminum; the pressure was main-tained for about 5 min, until solidification.

Fig. 2 shows the resulting composite microstructure, revealing mi-crospheres randomly distributed in the Al matrix and a homogeneousmacroscopic structure. Fig. 3 illustrates the typical initial part of thestress–strain diagram of the composite under quasi-static compressiontest, which was carried out according to the ISO 13314 standard forporous metal compression testing, at constant crosshead speed and astrain rate of 10−3 s−1. The axial stress σ is the ratio between thevertical force acting on the punch and its cross-sectional area, while thelongitudinal deformation ε is the ratio between the punch verticaldisplacement and the initial height of the sample.

Experimental diagrams for various composites compositions withdifferent sizes of glass spheres as hollow inclusions have been pre-viously obtained and analyzed (Lin et al., 2017).

The σ-ε diagram clearly shows the initial quasi-static compressionsection corresponding to the elastic deformation of the composites. Itcan be seen that when the composite foam is compressed, it first de-forms in a linear-elastic behavior, followed by non-linear elasticity nearthe peak stress. The compression deformation of the microporouscomposite is a result of a large number of unit deformations composedof the microspheres and aluminum matrix. During the compressionprocess, the loading force is transmitted from the aluminum matrix tothe microspheres, resulting in compressive stress on the upper and

lower end walls of the microspheres, and tensile and shearing force nearthe equator. In the earlier stage of elastic deformation, the matrix andmicrospheres are elastically deformed synchronously, and the corre-sponding compression curve shows linear elastic deformation. In thevicinity of the peak stress where the load is relatively high, part ofmicrospheres with weak compressive capacity buckle, leading to theinitiation of microcracks at the equator of the microsphere, as describedin Ref. Lin et al. (2016). The composite units in these regions appear toexperience local plastic yield, which causes the elastic deformationcompression curve of the overall composite to be non-linear.

The deformation range of the composite foam at the elastic de-formation stage normally does not exceed 4%; this circumstance allowsus to neglect the compaction and the associated change in the

Fig. 1. Schematic of pressure infiltration unit for the fabrication of the porous aluminum composite containing glass microspheres.

Fig. 2. Optical microstructural image of the porous aluminum/glass micro-sphere composite.

S.V. Shil'ko, et al. Mechanics of Materials 144 (2020) 103349

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deformation characteristics of the porous metal matrix when analyzingthe overall linearly elastic deformation. The mechanical properties ofthe matrix material are characterized by the Young modulus Em and thePoisson ratio νm. The material forming the microsphere wall (i.e., glass)can be considered as isotropic linearly elastic and is characterized bythe Young modulus Eg, the Poisson ratio νg, and the tensile strength σTg.The structural parameters of the composite are the volume fraction cand wall relative thickness q of the microspheres, which is equal to theratio of the thickness to the average radius of microspheres.

They can be expressed in terms of the equivalent densities of thecomposite ρk, porous matrix ρm and filler ρf:

= =−−

= = − ⎛

⎝⎜ −

⎞

⎠⎟c

VV

ρ ρρ ρ

q tR

ρρ

, 1 1fk

k m

f m

f

g

11/3

(1)

where Vk and Vf are the total composite volume and the volume of themicrospheres, respectively, R and t1 are the average radius and wallthickness, correspondingly, of the inclusions, and ρg = 2540 kg m–3 isthe density of the glass walls.

To predict the effective elastic characteristics of composites dis-persion-filled with hollow or composite inclusions, various versions ofthe self-consistency method can be used (Pan'kov, 1998). However, thistechnique cannot provide the determination of the stress and straindistributions in a detached inclusion and near-boundary matrix volumeunder given conditions of composite loading, reducing the accuracy ofthe derived values of the strength parameters. For reliable estimation ofthe axial stress corresponding to the beginning of the microsphere de-struction during the pressing process, a modified version of the three-phase micromechanical model for materials filled with spherical in-clusions has been proposed (Christensen, 1979, Shil'ko et al., 2013).The structural unit of the target composite is a layered spherical shellplaced in an elastically deformable space (Fig. 4a). The inner layer,bounded by two spherical surfaces with radii a and R (inner and outersurface, respectively), is formed by the material of the inclusion walls;the outer layer, bounded by two spherical surfaces with radii R and b(inner and outer surface, respectively), is formed by the matrix mate-rial. The “composite phase k” is defined as a macroscopic quasi-homogeneous media whose properties correspond to the properties of

the disperse-reinforced composite under consideration; therefore, thisconditional phase represents the average behavior of the compositematerial around an inclusion. The structural parameters R, a, b can bederived from the structural parameters of the composite as follows:

= − = − = ⎛⎝

⎞⎠

⇒ = −a R t R q c Rb

b R c(1 ), ·13

1/3(2)

The deformation of this structural unit is described by the sphericalcoordinates r, θ, and φ, whose beginning coincides with the center ofthe shell. The axes 1, 2, and 3 in Fig. 4a correspond to the principal axesof the macroscopic strain tensor; the composite loading is modeled byplacing the components of this tensor at an infinite distance from thecoordinate origin (r → ∞). The effective elastic characteristics of thecomposite and its stress–strain state are determined by considering twocharacteristic loading modes: volumetric deformation and simple shear.

In the pressing process, the composite under consideration is in astate of uniaxial macroscopic deformation: ε1 = ε3 = 0, ε2 = ε ≠ 0.The relation between the axial stress σ2 = σ and the longitudinal de-formation is as follows:

= = ⎛⎝

+ ⎞⎠

σ C ε K G ε43k k22 (3)

where C22 is the component of the elastic modulus tensor of the com-posite, which determines σ2 in the state of uniaxial deformation, and Kkand Gk are the effective volume and shear moduli of the composite to bedetermined, respectively.

Let us represent the given deformation state as a combination ofvolumetric and shear components as follows:

= = − =e ε e ε e ε3

,3

, 230 12 23 (4)

where е0 is the average axial deformation and е12 and е23 are the sheardeformations in the 1,2 and 2,3 planes, respectively.

Further mathematical computations are similar to those reported by'Shil'ko et al. (2013) for continuous spherical inclusions. The centralsymmetry of this model allows the determination of the elastic dis-placements for each composite component as follows:

= + + −= + +

= −

u V r V r θ ϕ V r θ ϕ θu V r θ θ ϕ V r θ θ ϕ

u V r θ ϕ V r θ ϕ ϕ

( ) ( )sin cos 2 ( )(sin sin cos ) ,( )sin cos cos 2 ( )sin cos (1 sin ) ,

( )sin sin 2 ( )sin sin cos ,

r r r r

θ θ θ

ϕ ϕ ϕ

(0) (12) 2 (23) 2 2 2

(12) (23) 2

(12) (23)

(5)

where V r( )iρ( ) indicates the r functions describing the ith elastic dis-

placement (i = r, θ, φ) for a strain eρ (ρ = 0, 12, 23) at infinity. Todetermine these functions, we consider independently three deforma-tion types, for each of which one strain is given (e0, e12, or e23).

For the given e0 in Eq. (5),= = = = = =V V V V V V 0r θ ϕ r θ ϕ

(12) (12) (12) (23) (23) (23) . Let us substitute theresulting displacement expressions into the equilibrium equations foran elastically deformable body in spherical coordinates(Novozhilov, 1961); the solution of the equation obtained is(Christensen, 1979)

= +V A r Arr

(0)1

22 (6)

where A1 and A2 are the constants for each component of the three-phase model and are derived from the boundary conditions, homo-genization principle, and deformations at infinity.

The Eshelby principle of homogenization (Eshelby, 1956) can beapplied for A2k = 0. The volumetric strain e0 is specified at an infinitedistance from the origin (r → ∞) where =A ek1 0. Thus, two constantsfor the inclusion shell (A1g, A2g) and other two for the matrix (A1m, A2m)are unknown. For their determination, we compose a system of fiveequations, considering the absence of the radial component of the stresson the inner shell surface (r = a), and the continuity of the radial

Fig. 3. Typical initial section of the stress–strain diagram of the porous alu-minum/glass microsphere composite (сp-Al/S38) under quasi-static compres-sion.

S.V. Shil'ko, et al. Mechanics of Materials 144 (2020) 103349

3

displacement ur and the radial component of the stress tensor at theinclusion/matrix (r = R) and matrix/composite interfaces (r = b). Thefifth unknown parameter is the effective bulk modulus of the compo-site, Kk.

For the given e12 in Eq. (5), then = = = =V V V V 0r r θ ϕ(0) (23) (23) (23) . The

use of the equilibrium equations for the displacements of an elasticmaterial in spherical coordinates provides a system of differentialequations to determine V V and V, ,r θ ϕ

(12) (12) (12). The solution of thissystem is (Pan'kov, 1998):

= −−

+ + −−

= − −−

− + = −

V A r νν

A r Ar

νν

Ar

V A r νν

A r Ar

Ar

V V

61 2

3 5 41 2

,

7 41 2

2 2 , ,

r

θ ϕj

θj

(12)3 4

3 54

62

(12)3 4

3 54

62

( ) ( )(7)

where ν is the Poisson's ratio of the components (inclusion walls, ma-trix, and composite k); A3,…, A6 are the constants for each component,derived from the boundary conditions, homogenization principle, anddeformations at infinity.

The fulfillment of the Eshelby principle reduces to the equality:A6k = 0. Additionally, the boundedness of strains and stresses at in-finity requires A4k = 0. As an external load on the composite, we set theshear strain e12 at infinity (r → ∞); moreover, A3k = e12.

There are nine unknown constants in the general solution of theequations of the elasticity theory: four for the inclusion walls (A3g,…,A6g), four for the matrix (A3m,…, A6m), and one for the composite (A5k).They can be determined by composing ten equations and taking intoaccount the absence of the radial and shear components of the stresstensor (σrr and shear σrθ, respectively) on the inner surface of the in-clusion walls (r= a) along with the continuity of the radial and angulardisplacements (ur and angular uθ, respectively) and the radial and shearcomponents of the stress tensor at the inclusion/matrix and matrix/composite interfaces. The tenth value to be determined is the shearmodulus of the composite, Gk.

A solution for the strain of e23 can be derived from that for e12. Thispassage requires a rotation of the coordinate axes and the necessaryrecalculations. The nature of the dependence of V V and V, ,r θ ϕ

(23) (23) (23)

Fig. 4. (a) Three- and (b) four-phase (with an additional interphase layer) models for a composite containing hollow inclusions; g indicates the material of theinclusion walls (glass), m means the matrix material (porous metal), l is the interfacial layer, and k represents a composite with the desired characteristics.

S.V. Shil'ko, et al. Mechanics of Materials 144 (2020) 103349

4

on r is similar to that of V r V r and V r( ), ( ), ( )r θ ϕ(12) (12) (12) ; they differ only

because the determination of the constants needs A3k = e23.By calculating all the constants A in Eqs. (7) and (6), we obtain the

elastic displacement formulas (Eq. (5)) with respect to the sphericalcoordinates r, θ, and φ. Then, we can determine the dependence on thecoordinates of the strains and stresses in each component of the three-phase model.

Based on the stress tensor components, we can then define theequivalent von Mises stress as follows:

= − + − + − + + +

σ r θ ϕ

σ σ σ σ σ σ σ σ σ

( , , )12

( ) ( ) ( ) 6( )

u

rr θθ rr ϕϕ θθ ϕϕ rθ rϕ θϕ2 2 2 2 2 2

(8)

The equivalent stress is highest in the glass at the inclusion/matrixinterface under the considered composite load. This material, formingthe inclusion walls, is characterized by brittle fracturing. In this regard,the Mises criterion (Shil'ko et al., 2013) can be represented in the formof equality:

=σ σu gmax T

where σumax is the maximum equivalent von Mises stress at r = R andσTg is the tensile strength of the inclusion walls.

The achievement of the value σTg does not lead to glass destruction,although microcracks begin to appear on the microsphere surface; inthe diagram, this is expressed in violation of the σ(ε) dependence lin-earity. As a criterion for the destruction of a hollow microsphere, wetake the tensile strength of the wall material averaged over the outermicrosphere surface by the equivalent stress:

∫ ∫=σ π σ R θ ϕ θdθdϕ1

4( , , )sinu

π π

u0

2

0 (9)

The value of the axial stress corresponding to the microsphere de-struction during the pressing process is determined as follows:

=σ σu gT (10)

The validity of the choice of the equivalent stress averaged over theinterface is confirmed as the main strength parameter of the composite,in particular, by the results of (Christensen, 1979, Fedotov, 2011).

The proposed calculation procedure can be summarized in the fol-lowing steps.

1) For a given value of ε, the volume e0 and shear e12, e23 componentsof the strained state in Eq. (4) are determined.

2) V r( )iρ( ) is found for each strain (e0, e12, and e23), composing the

system of continuity equations, boundary conditions, and homo-genization criterion. The effective volumetric and shear moduli ofthe composite are also obtained when considering the volumetricand shear deformations, respectively.

3) The known distribution of elastic displacements (Eq. (5)) allows thecalculation of the distributions of the stress tensor components andthe equivalent stress via Eq. (8).

4) By averaging over the angular coordinates at r = R according toEq. (9), the average deviator intensity over the microsphere surfaceis derived. Due to the linear stress dependence on the strain for all

the composite components at the considered loading stage, the re-sulting 〈σu〉 value is proportional to the strain as follows:

=σ Qεu (11)

The implementation of steps 1–4 provides the coefficient Q.

5) By equating the average value of the equivalent stress to the tensilestrength of the inclusion wall material (Eq. (10)), the axial stress σukcorresponding to the microsphere destruction during pressing isdetermined as

= ⎛⎝

+ ⎞⎠

σ K GσQ

43uk k k

gT

(12)

The σuk value corresponding to the peak stress in the σ–ε diagram isconsidered as the composite strength characteristic. The explicit formsof the obtained analytical dependencies are not provided here due totheir length.

3. Results and discussion

Experimental stress–strain diagrams have been obtained in a pre-vious work (Lin et al., 2017) for the joint pressing process of a porousaluminum alloy containing glass microspheres. Three different matrixcompositions (Table 1) and three types of industrially produced glassmicrospheres (Table 2) were used. The values of the relative wallthickness (q) and tensile strength of the glass material of the micro-sphere walls σTg were taken from (Lin et al., 2017).

An interfacial layer is formed at the microsphere/ metal interface inthe case of Mg-containing alloys (Lin et al., 2017). This layer is made ofMgAl2O4. The characteristic values of this material have been reportedin (Kimberley and Ramesh, 2011): El = 150 GPa, νl = 0.27, andσTl = 3300 MPa. Moreover, the thickness of this layer, t2, for thecomposites under consideration has been previously determined(Lin et al., 2017). As an additional structural parameter of the com-posites containing the MgAl2O4 layer, we use the relative thicknessql = t2/R. Assuming that all the magnesium in the matrix is involved inthe interfacial interaction allows one to take a set of mechanical char-acteristics of Em, = 65.84 GPa, νm = 0.34, and σTm = 55 MPa, which istypical for porous aluminum, such as three different matrices noted inTable 1 (Tret'yakov and Zyuzin, 1973). The difference in the values ofthe tensile strength of the wall material for spheres of different sizes(Table 2) is apparently due to the influence of the technological para-meters for the manufacturing of microspheres on the glass properties(Lin et al., 2017). The noted effect can be neglected and it is assumedthat Eg = 76.25 GPa and νg = 0.23 (Pan'kov, 1998, Christensen, 1979)for the elastic characteristics of glass.

The peak value of the axial stress for a composite containing aninterfacial layer is calculated based on a four-phase model (Fig. 4b).This model differs from the three-phase one (Fig. 4a) by the presence ofan additional spherical shell l bounded by surfaces with radii R andR + t2 (inner and outer surface, respectively); this shell replaces part ofthe matrix volume and is formed by the interfacial layer material. The

Table 1Composition of the aluminum matrices used in the experiments.

Alloy type Composition (mass%)

Cu Mg Mn Fe Si Zn Ti Al

сp-Al 0.1 – 0.1 0.5 0.4 0.1 – 98.85A03 0.1 3.2 0.4 0.5 0.4 0.2 0.1 95.15A06 0.1 6.0 0.5 0.4 0.4 0.2 0.1 92.3

Table 2Structural parameters and strength of the hollow glass microspheres used in theexperiment.

Microsphere type R (μm) q σTg (MPa)

S32 20.47 0.043 1027S38 20.19 0.052 1587K46 20.16 0.064 1788

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calculations for σuk based on the four-phase model practically repeatthose for the case of the three-phase model. The difference is in thepresence of additional constants A1l,…, A6l and six continuity condi-tions at the boundary r = R + t2.

The experimental confidence intervals of the peak stress valuescorresponding to the microsphere destruction during the pressing pro-cess for the nine different composites under consideration are given inTable 3, along with the results of calculating this stress σuk(2) using theproposed technique and σuk(1) using that described in Lin et al. (2017).The experimental and calculated values of the component C22 of theelastic modulus tensor are also shown in Table 3.

The experimental C22 values were derived from the inclinationangle of the initial linear portion of the stress–strain diagram, while thesimulated ones were calculated as a combination of the bulk and shearmoduli according to Eq. (3). Their comparison allows us to concludethat the estimated values of σuk(2) obtained based on the updated cal-culation method and the σuk(1) results obtained by the simplifiedmethod lie in experimental confidence intervals of the axial stress peak.The technique, which implies volume averaging of the componentstrength limits considering the structural factor of spherical inclusions,is simple to implement. At the same time, the refined technique basedon the analysis of the stress–strain state of an individual inclusion andthe boundary volume of the matrix enables us to consider the effect ofall the mechanical characteristics of the composite components on itsinvestigated characteristic (σuk). In addition, with the developedmethodology, not only the strength characteristics of the composite butalso its elastic moduli can be predicted (Table 3).

For additional testing of the developed calculation procedure, weobtain the estimates of the axial stress limit for the composites studied(Ferguson et al., 2013, Santa Maria et al., 2013). In these composites,the aluminum alloy is filled with ceramic hollow microspheres. Themicrosphere stacks are formed by alumina (Al2O3). The values of themechanical characteristics of this material were taken from Ref.Gu et al. (2004): Eg = 140 GPa and νg = 0.32. The tensile strengthgiven in Ref. Ferguson et al. (2013) is σTg = 2944 MPa; here, threevariants of the Al-A206 matrix aluminum alloy were considered (as-cast, T4, and T7), which differ in their processing parameters. We didnot establish the values of the elastic characteristics of these alloys inFerguson et al. (2013) and Santa Maria et al. (2013) and others in lit-erature. This circumstance allowed us to accept the values of theYoung's moduli of the matrix material based on the condition of themaximum correspondence of the slope of the initial elastic section (lessthan 2% strain) of the experimental compression diagram of the com-posite specimens calculated for the value of the effective Young's

modulus Ek calculated by our methodology. The experimental com-pression diagrams are given in Santa Maria et al. (2013), and the Ekvalue is expressed in terms of the effective shear modulus and the bulkmodulus of the composite:

=+

E K GK G9

3kk k

k k (13)

The value of the Poisson's ratio for all variants of the matrix is as-sumed to be the same and coincides with the value for the previouslyconsidered porous aluminum νm = 0.34. The Table 4 shows the valuesof the Young's modulus adopted in this method and the values of theyield strength of the matrix material taken from Ref. Santa Mariaet al. (2013). It can be noted that the values of Young's modulus fordifferent variants of the matrix alloy differ slightly. The presence of aninterfacial layer for composites studied in Ferguson et al. (2013) andSanta Maria et al. (2013) is not taken into account.

The experimental compression diagram of a composite is con-structed for a uniaxial stress state of a composite in Ferguson et al.,2013, Santa Maria et al., 2013) and for a state of uniaxial strain inLin et al. (2017), respectively. Therefore, when obtaining a calculatedestimate of the peak stress for the composites studied inFerguson et al. (2013) and Santa Maria et al. (2013), the value of theultimate stress calculated by formula ((12) should be recalculated asfollows:

= −−

σ σ vv

1 21

.ukσ ukεk

k (14)

Here σukε is the value of the peak stress calculated by formula (12)for the state of uniaxial strain, σukσ is the peak stress value for uniaxialstress state and νk is the calculated value of the effective Poisson's ratioof the composite. The quantity νk is expressed in terms of the effectiveshear modulus and the bulk modulus of the composite:

= −+

v K GK G

3 26 2

.k k kk k (15)

Thus, the differences in calculating the axial stress for a compositefilled with ceramic microspheres Ferguson et al., 2013) include thevalues of the characteristics of the components, the absence of an in-terfacial layer, and recalculation of the ultimate stress according toformula ((14).

Table 5 presents a comparison of the results of the developed cal-culation technique σuk(2) with experimental data σukexp and calculatedestimates σuk(1) given in Ferguson et al. (2013). It can be noted that theaccuracy of the prediction using the developed technique is not inferiorto the accuracy of the estimated assessments given inFerguson et al. (2013). The maximum relative deviation of σuk(1) fromσukexp was 13.5% and was averaged to be 4.7% over all the data pre-sented. For the results of the developed technique (σuk(2)), the max-imum deviation was 9.3% and the average deviation was 5.4%. Thesomewhat overestimated estimates for the as-cast alloy variant are dueto the fact that, within the framework of the developed technique, theplastic strain of the matrix material is not taken into account. Ac-counting for this process is the subject of further development of the

Table 3Experimental and calculated values of the peak axial stress and elastic moduli of the aluminum matrices used in the experiments.

Matrix Inclusion с ql σuk(1) (MPa) σuk(2) (MPa) σukexp (MPa) C22(2) (GPa) C22exp (GPa)

сp-Al S32 0.575 – 63.9 65.2 64.6 ± 1.5 2.11 2.05 ± 0.05S38 0.658 76.4 79.7 78.1 ± 2.3 2.51 2.47 ± 0.07K46 0.653 91.5 90.2 87.8 ± 5.1 2.86 2.78 ± 0.16

5A03 S32 0.605 0.034 95.9 96.1 95.0 ± 1.2 2.65 2.61 ± 0.03S38 0.632 0.035 112.8 111.9 110.5 ± 2.5 4.39 4.35 ± 0.10K46 0.651 0.035 121.1 120.2 119.3 ± 2.1 6.71 6.67 ± 0.12

5A06 S32 0.630 0.039 103.8 103.8 102.1 ± 1.6 3.37 3.34 ± 0.05S38 0.582 0.059 139.9 146.0 142.5 ± 6.5 6.67 6.51 ± 0.30K46 0.568 0.060 149.2 155.6 151.3 ± 7.3 8.02 7.93 ± 0.38

Table 4Mechanical characteristics of aluminum alloys Al-A206.

Alloy grade Em, GPa σTm, MPa

As-Cast 78.5 60T4 76.1 262T7 82.5 345

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technique.

4. Conclusions

We developed a calculation technique to describe the stress–strainstate of a porous metal matrix filled with hollow glass microspheres atthe initial stage (linearly elastic deformation) of the pressing process, atthe mesoscale (detached porous inclusions and boundary volume of thematrix).

We also demonstrated that this method provides acceptable pre-diction accuracy for the elastic moduli of the composite and the peakaxial stress. Additionally, a model taking into account the presence ofan interfacial layer in the composite has also been proposed. Thesomewhat overestimated values of the peak stress are due mainly to thesimplified consideration of the microsphere destruction, which ne-glected plastic deformations of the matrix at the initial loading stage.

CRediT authorship contribution statement

S.V. Shil'ko: Conceptualization, Methodology, Investigation,Writing - original draft, Supervision, Project administration, Fundingacquisition. D.A. Chernous: Methodology, Investigation, Writing -original draft. Qiang Zhang: Investigation, Writing - original draft,Funding acquisition. Yingfei Lin: Investigation, Writing - original draft.Heeman Choe: Writing - original draft, Funding acquisition.

Declarations of Competing Interest

The authors declare that they have no known competing financialinterests or personal relationships that could have appeared to influ-ence the work reported in this paper.

Acknowledgements

This work was supported by the Belarusian Republican Foundationfor Fundamental Research (T18KORG-004) and the National NaturalScience Foundation of China (No. 51771063 and No. 51911530116).Heeman Choe also acknowledges support from the InternationalResearch & Development Program of the National Research Foundationof Korea (2018K1A3A1A39086825).

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Table 5Comparison of experimental and calculated values of peak axial stress foraluminum alloys filled with ceramic microspheres.

Matrix q с σuk(1), MPa σuk(2), MPa σukexp, MPa

As-cast 0.105 0.606 257 259 2400.511 247 248 2270.464 222 225 213

0.06 0.693 172 181 1660.649 167 177 1650.61 160 172 164

T4 0.105 0.56 329 332 3180.54 321 325 3130.49 301 307 310

0.06 0.627 192 179 1860.592 187 170 1810.542 179 166 180

T7 0.105 0.61 346 350 3380.581 342 341 3270.506 331 329 358

0.06 0.61 221 208 2030.6 212 200 2030.596 210 196 185

S.V. Shil'ko, et al. Mechanics of Materials 144 (2020) 103349

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Uniaxial compression model for a metal-matrix/hollow-microsphere composite synthesized by pressure infiltrationIntroductionExperimental and calculation methodsResults and discussionConclusionsCRediT authorship contribution statementmk:H1_6AcknowledgementsReferences