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A numerical investigation of the effect of particle clustering on the mechanical properties of composites

A numerical investigation of the effect of particle clustering on the mechanical properties of composites
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  Acta Materialia 51 (2003) 2355– A numerical investigation of the effect of particle clusteringon the mechanical properties of composites J. Segurado, C. Gonza´lez, J. LLorca  ∗  Department of Materials Science, Polytechnic University of Madrid, E. T. S. de Ingenieros de Caminos, 28040 Madrid, Spain Received 10 December 2002; accepted 16 January 2003 Abstract The effect of the reinforcement spatial distribution on the mechanical behavior was investigated in model metal-matrix composites. Homogeneous microstructures were made up of a random dispersion of spheres. The inhomogeneousones were idealized as an isotropic random dispersion of spherical regions—which represent the clusters—with thespherical reinforcements concentrated around the cluster center. The uniaxial tensile stress-strain curve was obtainedby finite element analysis of three-dimensional multiparticle cubic unit cells, which stood as representative volumeelements of each material, with periodic boundary conditions. The numerical simulations showed that the influence of reinforcement clustering on the macroscopic composite behavior was weak, but the average maximum principal stressin the spheres—and its standard deviation—were appreciably higher in the inhomogeneous materials than in the homo-geneous ones (up to 12 and 60%, respectively). The fraction of broken spheres as a function of the applied strain werecomputed from experimental values of the Weibull parameters for the strength of the spheres, and the local stresscomputed in the simulations. It was found that the presence of clustering greatly increased (by a factor between 3 and6) the fraction of broken spheres, leading to a major reduction of the composite flow stress and ductility. ©  2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords:  Finite element simulations; Composites; Clustering; Fracture 1. Introduction The optimisation of the mechanical properties of composites is based on the knowledge of therelationship between the microstructure and themacroscopic response. This has been achieved bythe development of micromechanical models, ∗ Corresponding author: Tel.: ( + 34) 913 365 375; fax: ( + 34)915 437 845.  E-mail address: (J. LLorca). 1359-6454/03/$30.00  ©  2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.doi:10.1016/S1359-6454(03)00043-0 which initially considered only the matrix andreinforcement properties and their respective vol-ume fractions. It was soon evident that this infor-mation was not sufficient for accurate prediction of many properties, so more refined models, whichincluded the effect of particle shape, size andorientation, were elaborated. There is, however,unmistakable experimental evidence (particularlyin the field of metal-matrix composites) that thespatial distribution of particles determines, or atleast influences significantly, several importantmechanical properties such as the yield strength  2356  J. Segurado et al. / Acta Materialia 51 (2003) 2355–2369 [1], the onset of damage [2 – 4], the ductility [5],the fracture toughness [6] and the threshold of fatigue crack growth [7 – 8].Random but non-homogeneous particle distri-butions arise during composite processing for sev-eral reasons. In cast metal-matrix composites, par-ticle clustering is due to the combined effect of reinforcement settling and the rejection of thereinforcement particles by the matrix dendriteswhile these are growing into the remaining liquidduring solidi fi cation [9]. The spatial distribution of the particles is normally more homogeneous inpowder-metallurgy composites but clustering stilloccurs on a smaller scale because of static chargeson particle surfaces or due to geometrical reasonswhen there is a large difference between the matrixand the reinforcement particle size [10].Although there is a qualitative understanding of the effects of clustering on the mechanical proper-ties of composites, a  quantitative  assessment can-not be made in the absence of a detailedmicromechanical modelling. Two different stra-tegies were used in the past to study the in fl uenceof non-homogeneous particle distribution on thecomposite properties. In the  fi rst one [11 – 13], thenon-homogeneous composite was represented by athree-dimensional network of cells, each one rep-resenting one composite region with a differentlocal volume fraction of reinforcement. The consti-tutive equation for each cell was obtained from thetraditional models developed for homogeneouscomposites ( fi nite element analysis of a periodicunit cell or mean- fi eld approximations), and theoverall composite behaviour was computed byaveraging the behaviour of all cells.This approximation to the problem was veryclever, and led to sensible results of overall com-posite properties, but it cannot give accurate infor-mation about the local stress and strain  fi eldswithin the clustered regions because the complexinteraction between matrix and particles is replacedby a homogeneous material with higher reinforce-ment content. A detailed modelling of the clusterswas achieved in two-dimensions by discretizingthe actual microstructure of the composite withstandard [14 – 16] or Voronoi-cell  fi nite elements[3,17], and solving numerically the boundary-valueproblem. These models gave valuable insights intothe deformation patterns and the onset of damagein two-dimensional clusters, but the differences inthe reinforcement spatial arrangement between twoand three-dimensions, as well as the discrepanciesin the stress states, limited the validity of theirquantitative predictions [18 – 19].In this paper, the three-dimensional simulationsof the elastic [20] and elasto-plastic deformation[21] of composites reinforced with a homogeneousand isotropic distribution of spheres are extendedto address the effect of inhomogeneous particledistribution. A new algorithm to generate non-homogeneous microstructures was developed, andtheir mechanical behaviour is studied by  fi niteelement analysis of three-dimensional multiparticleunit cells. The macroscopic response and the localstress and strain  fi elds were computed for micro-structures with different degrees of clustering, andcompared with those of homogeneous compositeswith either random or regular particle distributions.Finally, the progress of damage by particle fracturewas determined for each microstructure assumingthat the particle strength follows the Weibull stat-istics. 2. Microstructure generation Among the various kinds of inhomogeneous par-ticle distributions experimentally reported, a verysimple one was selected for this study. The micro-structure of the clustered material was idealized asan isotropic, random dispersion of sphericalregions of radius  R cl  which represent the clusters.The spherical reinforcements were concentratedaround the center of each cluster, within a concen-tric sphere of radius  r  cl  (Fig. 1). The three para-meters controlling the microstructure were theaverage sphere volume fraction,  x , the local spherevolume fraction at the center of each cluster,  x cl ,and the number of spheres which belong to eachcluster,  n cl . This particle distribution is representa-tive of the actual microstructure of many com-posites [5 – 6] and, in addition, the degree of inhom-ogeneity can be controlled by reducing  r  cl  (and,thus increasing  x cl ).As in previous investigations [20 – 21], the mech-anical response of the composite was obtained by  2357  J. Segurado et al. / Acta Materialia 51 (2003) 2355  –  2369 Fig. 1. Bidimensional representation of the inhomogeneous microstructure showing the randomly distributed spherical regions whichrepresent the clusters. The spheres are concentrated at the center of each cluster. the  fi nite element analysis of a cubic unit cell of side  L , which stood as a representative volumeelement of the material, with periodic boundaryconditions. The  fi nite size of the unit cell leads toa new parameter in the generation, namely the totalnumber of clusters,  N  , in the cell. The correspond-ing particle dispersions in the cubic unit cell werecreated with an algorithm which comprised twoparts, the generation of the center of the clustersand the dispersion of the spheres around the clustercenter. In the  fi rst one, the coordinates of a randomdispersion of   N   points,  >  X  i , representing the centersof the  N   clusters, were generated within the cubicunit cell by the modi fi ed random sequential adsorp-tion algorithm (MRSA) developed previously [20].All the clusters had the same radius  R cl  andbehaved as impenetrable, non-overlappingspheres.  R cl  was chosen so that the volume occu-pied by the spherical cluster regions in the unit cellwas maximum. The packing limit for a random dis-persion of hard spheres is 0.64, but this magnitudewas approached by the MRSA algorithm onlywhen the number of clusters large enough. Itreached only 52% for the small number of clusters(7) used here.Once the coordinates of the cluster centers wereknown, the positions of the  n cl  spheres within eachcluster were generated. The sphere diameter,  D ,was determined from the total number of spheres(  Nn cl ) and the average sphere volume fraction ( x )in the unit cell, and the sphere center positionswere created randomly and sequentially with therandom sequential adsorption (RSA) algorithm[22]. Sphere  i  was accepted if the distance betweenthis sphere and all the spheres previously accepted  j  =  1 ,...,  i  1 exceeded a minimum value,1.035  D , imposed by the feasibility of creating anadequate  fi nite element mesh. Of course, if the sur-face of sphere  i  cuts any of the cubic unit cell sur-faces, this condition has to be checked with the  2358  J. Segurado et al. / Acta Materialia 51 (2003) 2355  –  2369 particles near the opposite surface because themicrostructure of the composite is periodic. If   >  x i stands for the center coordinates of sphere  i  theseconditions are expressed bymin{||  >  x i  >  x  j   > h ||}  1.035  D  (1)for any value of   > h  =  ( k  , l , m ) where  k  ,  l , and  m  cantake the values 0,  L ,    L , leading to 27 conditionswhich should be checked for each pair of particles.Moreover, to avoid distorted  fi nite elements dur-ing meshing the sphere surface should not be veryclose to the cubic unit cell faces, and this led to||  x ik   r  ||   0.045  D  ;  k   1,2,3||  x ik   r    L ||   0.045  D  ;  k   1,2,3(2)In addition, the sphere  j  which belongs to thecluster  i  should be contained inside a sphere of radius  r  cl  centered at the point  >  X  i so that the localvolume fraction at the center of the cluster is  x cl .This condition is imposed on the sphere centercoordinates bymin  ||  >  x  j  (  >  X  i   > h )||   D 2   r  cl  (3)where  > h  takes the 27 values indicated above sincethe spheres in the cluster may be located near toopposite faces of the cubic cell, given the periodicmicrostructure of the composite. Obviously, x cl  n cl   D 2 r  cl  3 (4)Clusters with a moderate local volume fraction of spheres can be generated by the RSA algorithm incombination with Eqs (1), (2) and (3). To reachhigher volume fractions, a new algorithm wasdeveloped. The standard RSA methodologydescribed above was used to accommodate thespheres within a spherical region of radius  r  0cl   r  cl , ensuring that the conditions expressed by Eqs(1) and (2) were ful fi lled, and the spherical regionwas compressed step by step. Assuming that >  x  j N   1  stands for the coordinates of the center of sphere  j  (which belongs to cluster  i ) after the(  N   1) compression step, the sphere is attractedtowards the cluster center and the new coordinatesof its center,  >  x  j N  , are given by >  x  j N    >  x  j N   1  (1  s c )[min{(  >  X  i   > h )  >  x  j N   1 }] (5)where  s c (  1) is the compression factor and thevector  > h  takes any of the 27 values indicated aboveto include the case when the cluster intersects oneface of the cubic cell and — due to the periodicmicrostructure — the center of the cluster and of sphere  j  are located near opposite cell faces.If the new sphere positions satisfy the conditionsimposed by Eqs (1) and (2), the local sphere vol-ume fraction in the cluster has increased to x  N  cl   1 s 3c x  N   1cl  (6)and the compression process is repeated until thedesired value of   x cl  is reached. Otherwise, thespheres which do not ful fi ll Eq. (1) or (2) are trans-lated randomly to a new position inside the currentcluster region (a sphere of radius  r   N  cl ) until Eqs (1)and (2) are veri fi ed, and a new compression stepbegins.In addition to these isotropic and inhomo-geneous microstructures, two other materials withdifferent particle distributions were used for com-parison. The  fi rst one was an isotropic and homo-geneous microstructure made up of a random dis-persion of spherical particles. The correspondingcubic unit cells were generated with the RSAalgorithm and the conditions provided by Eqs (1)and (2), as explained elsewhere [20]. The secondwas representative of a regular distribution of par-ticles in space because the minimum interparticledistance is maximum for these microstructures.The orthotropic material provided by a body-cent-ered cubic (BCC) regular arrangement of spheresin space was chosen because its behavior is theclosest to that of an isotropic material among thevarious regular particle distributions [18]. 3. Finite element simulations Four different types of particle arrangementswere generated with the algorithms presented inthe previous section. The average sphere volumefraction in the cell,  x  , was 15% in all cases. Threemicrostructures were representative of inhomo-geneous composites with local sphere volume frac-tions at the center of the clusters of   x cl  =  20, 30and 40%. Each cluster was formed of seven  2359  J. Segurado et al. / Acta Materialia 51 (2003) 2355  –  2369 spheres, and seven clusters were included in theunit cell, giving a total of 49 spheres. The fourthmicrostructure stood for the homogeneous com-posite and was made up by a random dispersionof only 30 spheres because it was demonstratedthat this number was large enough to capture accu-rately the elasto-plastic response of the composite[21]. The differences in the particle arrangementamong the materials are shown in Figs. 2(a), (b)and (c), with three unit cells representative of thehomogeneous composite and of inhomogeneousmaterials with  x cl  =  20 and 40%, respectively.Four sphere center distributions within a cubicunit cell were generated for each microstructure.Spheres were generated from the particle centresand diameter, and those intersecting the cube faceswere split into an appropriate number of parts andcopied on the opposite sides of the cube. Threefaces of the cube were meshed with quadratic tri-angles, and the meshes were copied on the oppositesides. The pairing of the nodes on opposite cubefaces was necessary to apply the periodic boundaryconditions. The model volume (matrix andspheres) was meshed using modi fi ed 10-nodetetrahedra with integration at four Gauss points andhourglass control. The modi fi ed elements inAbaqus [23] exhibit minimal volumetric clockingduring plastic straining, and this was checked byrunning one analysis using hybrid elements, wherethe hydrostatic pressure is treated as an inde-pendent variable. The results of the models withmodi fi ed and hybrid tetrahedra were almost ident-ical and the latter were used as they are more econ-omical from the computational viewpoint.Approximately 500 elements were used to rep-resent each sphere, ensuring that the sphere volumefraction in the discretized models was within 0.1%of the theoretical value. In addition, an adaptativemeshing technique was used to reduce the elementsize in the matrix within the cluster zone, coarsen- Fig. 2. Sphere distribution within the unit cells. (a) Homo-geneous composite (b) Inhomogeneous composite with sevenclusters containing seven spheres,  x cl  =  20%. (c) Inhomo-geneous composite with seven clusters containing sevenspheres,  x cl  =  40%. All the spheres belonging to one cluster in(b) and (c) are marked in dark gray to show that the microstruc-ture is periodic.

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