Analysis of 3D Random Al18B4O33 Whisker Reinforced Mg Composite

Using FEM and Random Sequential Adsorption

Wook Jin Lee1, Jae Hyoung Son1, Ik Min Park1, Jeong-Jung Oak2,Hisamichi Kimura2 and Yong Ho Park1

1School of Materials Science and Engineering, Pusan National University,San 30 Jangjeon-dong, Geumjeong-gu, Busan 609-735, Korea2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

In order to understand the deformation behavior of randomly orientated ceramic whisker reinforced composite materials, threedimensional (3D) finite element models were developed. The actual distributions of the whiskers in the composite materials were reconstructedfor the representative volume element of the composite, using a random sequential adsorption algorithm. The samples were random Al18B4O33

whisker reinforced magnesiummatrix composites with a volume fraction of 15%. After modeling, the role of the random ceramic whisker in thedeformation behavior of the magnesiummatrix composite was investigated by the finite element method (FEM). The elastic modulus and stress-strain behaviors of the composite predicted by the microstructure-based model correlated well with the experimental results.[doi:10.2320/matertrans.M2009208]

(Received June 18, 2009; Accepted March 26, 2010; Published May 19, 2010)

Keywords: finite element method, composite, whisker, metal matrix composite (MMC), deformation behavior

1. Introduction

Randomly orientated ceramic whisker reinforced compo-sites have evolved into appealing alternatives to the materialsused traditionally for structural applications in a wide rangeof engineering fields, due to their high thermal stiffness,strength and isotropic mechanical properties.1–3) In order tofacilitate the development and design of whisker reinforcedcomposites, constitutive relationships must be developed thatpredict their effective elastic modulus and overall stress-strain response. Although some analytical and semi analyt-ical models have been developed to evaluate the effectivematerial properties of fiber and particle reinforced compo-sites based on homogenization techniques,4–6) they have notbeen able to be extended to composites with irregularreinforcement geometries and complex microstructures.

On the other hand, numerical models seem to be wellsuited to describe the behaviors of these composites, becausethere are no restrictions on the geometry, material propertiesor number of phases in these models. Therefore, the finiteelement method (FEM) has been extensively employed todetermine the effective material properties of short fiber andparticle reinforced composites.7–12) Numerous studies havebeen reported on the effective modulus and overall stress-strain responses of these composites using representative unitcell7,8) and 2D/3D microstructure9–12) models. Most of thesemodels were constructed using representative volume ele-ments (RVEs), which were selected to reflect the actualmicrostructures of the composite. Recently, Gusev et al.9)

developed a Monte Carlo13) based simulation for an alignedfiber reinforced composite, in which the actual distributionsof the fiber aspect ratio were fully represented by 100 non-overlapping fibers. Kari et al.10) employed FE analysis toinvestigate the effect of the volume fraction and fiber aspectratio on a random short SiC fiber reinforced metal matrixcomposite (MMC). Boehm et al.11) and Duschbauer et al.12)

studied an MMC reinforced by random short fibers using a

periodic unit cell model, which was developed by the randomsequential adsorption (RSA)14) scheme. The results of thesestudies showed the capability and advantage of FE analysis inthe simulation of composite materials, not only for predictingthe effective modulus, but also for the local field character-ization and overall stress-strain response analysis. However,in spite of the vast number of studies that have beenconducted on the FE modeling of short fiber and particlereinforced composites, these approaches cannot be directlyapplied to random whisker reinforced composites. Becauseof their periodic boundary conditions, these models sufferfrom a geometric restriction and have been restricted to dilutefiber/whisker composites or to fibers with low aspect ratios.

The objective of this paper was to develop a numericalmodel for the evaluation of the effective material propertiesand micromechanics of random whisker reinforced compo-sites. To overcome the limitations of the conventionalmodeling techniques, non-periodic RVE models were devel-oped for a composite containing 15 vol% of randomwhiskers. In order to determine the influence of the size ofthe RVE on the model conformability, 4 different volumes ofRVE were modeled and simulated by the RSA algorithm and3D FEM. The numerical predictions were compared withthe experimental results obtained for a squeeze infiltratedAl18B4O33/Mg random whisker composite.

2. Modeling Strategy and Viewpoint

The mechanical and physical properties of the compositematerials are always regarded as a small-scale microstructurein nature. The main idea of the method for modeling RVEto represent a real composite behavior is to find a globallyhomogeneous medium equivalent to the original composite,where the strain energy stored in both systems is approx-imately the same.10) As pointed out by several research-ers,9–12) a RVE should be typical of the whole compositemicrostructure and contains a sufficient number of inclusions

Materials Transactions, Vol. 51, No. 6 (2010) pp. 1089 to 1093#2010 The Japan Institute of Metals

for the apparent overall properties to be effectively inde-pendent of the boundary condition. In this point of view, thisstudy focused on the statistical homogeneity of the effectiveelastic modulus (Young’s modulus) of the model to bestatistically representative of the composite. The size ofRVE, which should be sufficiently large to represent a realcomposite, was carefully selected by checking the model tosatisfy above condition.

3. Experimental Procedure

The elastic behavior and overall stress-strain response ofthe random whisker reinforced composite were obtained bythe FE analysis of a cubic RVE with a volume of L3

consisting of non-overlapping whiskers. The RVE wasgenerated using the RSA algorithm, which was modified toprovide for a user specified minimum distance between thewhiskers. The analyzed material was a squeeze infiltratedAl18B4O33/Mg random whisker composite with 15 vol%.15)

The matrix alloy was AS52 magnesium alloy (4.1–5.3%Al,2.37%Si, 0.2%Sb and balance Mg) and the reinforcementwas Al18B4O33 whiskers with a diameter of 0.5–1.0 mm andlength of 10–30 mm (Grade M12, Shikoku Chemicals Co.,Japan). The optical micrograph of the composite is shown inFig. 1. In the microstructure, the Al18B4O33 whisker appearsdark and matrix appears light. In the matrix, polygonal shapeMg2Si secondary phases were also observed. Because thisstudy mainly focused on the micromechanical effects of thewhisker, the effect of Mg2Si phases was considered withinthe properties of the matrix material in the finite elementanalysis, by using the experimental tensile test data of theAS52 alloy which prepared by the same process of thecomposite.

The RSA algorithm used for the generation of the RVEsof the composite consisted of adding whiskers sequentiallyto a cubic space by randomly generating the center point(Cðx; y; zÞ), radius (r) and length (l) of each whisker, withinthe range of values of the actual Al18B4O33 whiskers (Fig. 2).To designate the direction of each whisker in 3D space,two Euler angles (’; �) were determined randomly by aquaternion method16) to afford a uniform distribution of the

orientation of all of the whiskers. During the RSA procedure,newly generated candidate whiskers were deleted if theyoverlaid any whiskers that had been generated previously.The minimum distance between the whiskers was set to0.3 mm, which was imposed by the practical limitations ofcreating an adequate finite element mesh in the matrixbetween the whiskers. The flowchart of the RSA procedure isshown in Fig. 3. The RSA algorithm with the combination ofthe above conditions was used to generate the RVE modelsof the composites up to the desired volume fraction of 15%.

All of the finite element calculations were performed withcommercial ANSYS software.17) The matrix and whiskerswere meshed with ten-noded quadratic tetrahedral elements(SOLID 187, structural solid element, ANSYS), which weregenerated by sweeping the corresponding 2D meshes on thetop surface of each whisker. In order to evaluate the influenceof the RVE volume on the model conformability, fourdifferent cubic RVE lengths (L ¼ 5, 10, 15 and 20 mm) wereconsidered.

Three different models were generated and analyzed foreach volume of RVE. The typical model geometries andmeshes of the RVE models of the Al18B4O33/Mg compositeare shown in Fig. 4. The typical number of elements andnode points were about 23,500 and 9,000 for L ¼ 5, 77,400and 30,700 for L ¼ 10, 219,200 and 93,300 for L ¼ 15,and 424,300 and 189,300 for L ¼ 20, respectively. The

Fig. 1 An optical micrograph of squeeze infiltrated Al18B4O33/Mg

random whisker composite. (�1000)

Fig. 2 Modeling a whisker as a cylinder in 3D space.

Fig. 3 A flow chart of the random sequential adsorption (RSA) procedure.

1090 W. J. Lee et al.

Al18B4O33 whiskers were modeled as linear elastic, with aYoung’s modulus of 392GPa and a Poisson’s ratio of 0.24.The experimental stress-strain curve of the squeeze castmonolithic AS52 alloy15) with the Multilinear KinematicHardening (MKIN) option of ANSYS was used in describingthe plasticity of the matrix. The Von-mises yield criterionwas used to determine the start point of the plastic strain.18)

The Young’s modulus and Poisson’s ratio of the AS52 alloywere 44.7GPa and 0.35, respectively. The effect of thethermal residual stress, which was induced by the coolingprocess after casting, was also modeled using �T ¼ 625�C(650� 25�C).

To predict the x-axis elastic modulus (E11) and overallelastic-plastic response, a 4% uniaxial tensile strain (theexperimentally measured fracture strain of the composite15))was applied to each model by fixing the displacement of they-z plane in the x-direction, and applying an x-directiondisplacement on the opposite side of the model. With thesame procedures, the E22 and E33 values of the models werecalculated from the FE analysis results. In the same manner,the shear modulus, G, was evaluated by applying a uniaxialshear strain to the models along all three directions.

4. Results and Discussion

4.1 Generation of random whiskers in 3D spaceCompared with the non-periodic models, the periodic unit

cell and RVE models exhibited good isotropic properties andproper accuracy with relatively small volumes. For thesereasons, the modeling of composites with a periodic bounda-ry condition has been intensively studied in the case ofparticle and short fiber reinforced composites.10–12) However,using periodic conditions and the RSA algorithm, it wasconfirmed that the volume fraction of random whiskers in theRVE could not exceed about 9%, because of the geometrical

jamming limit. Therefore, the present study adopted non-periodic boundary conditions to describe the complex micro-structures of the high aspect ratio random whiskers. Tominimize the inherent heterogeneity of the non-periodicRVE, a sufficiently large volume of the RVE was consideredin the present models. With the present non-periodicboundary conditions and modified RSA algorithm, the RVEsof the random whisker composite were successfully repro-duced up to a volume fraction of 15%. The number ofwhiskers which was generated in each RVE model is shownin Table 1. The average number of whiskers increasedexponentially with increasing L and, for L ¼ 20, about 320individual whiskers were fully or partially generated in a20� 20� 20 mm3 cubic space of the RVE. In this model, the3D orientation, aspect ratio and distribution of the randomwhiskers were fully represented.

4.2 Influence of the size of the RVE on the effectivematerial properties

One of the important parts of the modeling RVE approachis defining a proper representative volume to be modeled,which is required to give appropriate properties of amacroscopic composite structure. If the volume of the RVEconsidered is less than the minimum volume required, it maylead to a wrong prediction of the material properties.However, limited computational resources restrict the RVEvolume, because as the RVE volume becomes larger, themore computational resources are required. To the authors’best of knowledge, the question of the proper RVE size is stillopen to discuss and the answer can depend on the overallproperty of interest and on the type of considered micro-structure.10,19,20) Therefore, a reasonable compromise be-tween the accuracy and required CPU time is necessary.

Figure 5(a) shows the evolution of the ensemble averages,E11, E22 and E33, of the RVE models, which were predictedby the FEM. In order to determine the effect of RVE volumeon model conformability, the standard deviations of theelastic modulus around the mean value were calculated usingthe results of the three different RVE models. In the case ofsmall RVE volume, the calculated elastic modulus exhibitedhigh deviation values. With increasing RVE volume, thedeviations were decreased and the calculated results con-verged and stabilized. The standard deviation was less than2% when L ¼ 20. The evolution of the Poisson’s ratio (�) andshear modulus (G) of the models is shown in Figs. 5(b) and(c), respectively. Similarly to the results in Fig. 5(a), thevariations of the values of � and G were very small and thestandard deviations were less than 2%. Although the largestcubic RVE length (20 mm) in this study was shorter thanthe range of the length of whiskers (10–30 mm), a reasonable

(a) (b)

(c) (d)

Fig. 4 Typical model geometry and meshes of 3D random whisker

reinforced composite with various cubic RVE length of L. (a) L ¼ 5mm,

(b) L ¼ 10 mm, (c) L ¼ 15 mm and (d) L ¼ 20mm.

Table 1 The number of whiskers in RVEs.

RVE length Number of whiskers (N)

(L/mm) Model #1 Model #2 Model #3 Average

5 15 22 21 19

10 71 73 71 72

15 180 175 178 178

20 321 313 318 317

Analysis of 3D Random Al18B4O33 Whisker Reinforced Mg Composite Using FEM and Random Sequential Adsorption 1091

accuracy and isotropy (error was less than 2%) was observed.From these results, it was expected that the model with acubic RVE length of 20 mm corresponded to reasonablebounds for representing the microstructure of the randomwhisker composite. Moreover, the material properties whichwere obtained using the three coordinate directions wereabout same and the standard deviations were less than 2%for L ¼ 20. These results indicated that proper isotropy andstatistical homogeneity of the non-periodic RVE models wasachieved for L ¼ 20.

4.3 The elastic and overall elastic-plastic response of themodel

Table 2 shows the Young’s modulus of the Al18B4O33/Mgcomposite, which predicted by the RVE models with L ¼ 20

and experimentally. For comparison, the Young’s modulus ofthe composite was calculated theoretically based on empiri-cal predictions made by the H-T model.6) The present modelsreproduced fairly well the experiment findings, while theempirical model underestimated the measured Young’s

modulus of the composite. Since the actual stress fieldduring deformation of the composite is highly heterogeneousin nature and closely related to the geometric parametersbetween the matrix and reinforcement, the theoretical modelsdo not provide good agreement with experimental obser-vation. Unlike in the case of theoretical model, a goodagreement was found between experiment and the resultpredicted by RVE model, in which the geometric parametersof Al18B4O33 whiskers in the composite microstructure weretaken into account. Another advantage of using RVE modelswith computer-aided simulation was that the non-linearresponse of composites, such as stress-strain behavior canbe described by modeling the matrix as an elastic-plasticmaterial. Figure 6 shows the overall stress-strain responsesof the Al18B4O33/Mg composite, predicted by the modelswith L ¼ 20. The three different RVE models showedapproximately identical deformation behavior. Therefore,it can be concluded that the volume of the models wasrepresentative of the composite microstructure and that thenumber of whiskers was sufficient to represent the deforma-tion behaviors of the random whisker composites. Further-more, the experimental tensile behavior corresponded well tothe FEM results. The results in Fig. 6 and Table 2 indicatedthat a sound description of the microstructure of the randomwhisker composite was obtained with the current modelingprocedure. The difference in the Young’s modulus betweenthe present model and experimental results was less than 2%.Moreover, the effect of micromechanical factors, such aswhisker-whisker interactions, plastic flow of the matrix andinternal thermal residual stress, proved to influence signifi-cantly the deformation behavior of the Al18B4O33/Mgcomposite, while the theoretical models could not directlyaccount for these factors. The good agreement between thepresent models and the experimental results reflected that thenon-periodic RVE model, introduced by modified RSAalgorithm, was very effective at predicting the behaviors ofthe random whisker reinforced composite.

(a)

(b)

(c)

Fig. 5 Variation of the mechanical properties with change in RVE size.

(a) elastic modulus, (b) Poisson’s ratio and (c) shear modulus.

Table 2 Elastic modulus predicted by a FE-models and analytical model.

Models Elastic Modulus (E/GPa)

Experiment15Þ 103

FE-analysis 104:9� 1:4

H-T model6Þ 96.2

Fig. 6 Comparison of stress-strain relations obtained from FE-analysis

(L ¼ 20 mm) and the experiment.

1092 W. J. Lee et al.

5. Conclusions

The elastic properties and overall stress-strain response ofthe random Al18B4O33/Mg composite was investigated usingFEM and a RVE model implemented by the RSA algorithm.Using the modified RSA algorithm and non-periodic bound-ary condition, a RVE model of random whisker reinforcedcomposite with 15 vol% was successfully developed. The 3Dorientation, aspect ratio and distribution of the whiskers wereaccurately represented by the present models. Moreover,large representative 3D volumes of the random whiskercomposite were fully reconstructed in the models. From theFEM results, it was shown that the non-periodic RVEmodels achieved reasonable isotropic mechanical propertieswith about 320 whiskers, where the standard deviations ofthe properties in the three coordinate directions were lessthan 2%. The Young’s modulus and overall elastic-plasticresponse of the Al18B4O33/Mg composite correspondedwell to the FEM results. The non-periodic RVE modelingtechnique, using the modified RSA algorithm, was veryeffective in determining the material properties of thewhisker reinforced composite and, therefore, should providea means of developing new composites.

Acknowledgement

This work was supported by the National Core ResearchCenter program (No. R15-2006-022-02001-0) from theKorea Science & Engineering Foundation and a grant-in-aid for the Tohoku University Global COE program.

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