Effective properties for single size, rigid spherical inclusions in an elastic matrix
Introduction
One of the most basic cases in heterogeneous media science and composite materials technology has remained unresolved. This is the problem of determining the effective mechanical properties for a single size distribution of rigid spherical particles in an elastic matrix phase, over the full range of particle concentration from dilute up to the percolation limit. This negative situation has not been for lack of effort, but rather likely is because of the nature of the problem. That this basic problem is analytically difficult is probably obvious as it is somewhat akin to the many body problem of classical mechanics. And, bounding techniques are of no use on this problem. A new approach is needed.
Applications of composite combinations of materials of the type of interest here are pervasive for both solids and fluids, and have always been so. It all began with water born geological slurries and now crosses over into the entire materials processing industry. At the product end of the process, solids are often formed which still retain the basic heterogeneous architecture, with use properties dominated by this form. Although interest here is primarily with the solids case, there are critically useful fluids results which can be brought to bear upon the solids problem. The classical result in this regard is the Einstein [6] formula for the effective viscosity of a dilute suspension of rigid spherical inclusions in a Newtonian viscous fluid. Other critical fluids results will be stated later as needed. In the solids context the first credible micromechanics model was the Composites Spheres Model of Hashin [9]. Thereafter followed a many years rush of model building, some useful, some with only a tenuous grasp upon physical reality. Closely related to the Composite Spheres Model was the Generalized Self Consistent Method Model, GSCM, of Christensen and Lo [3] and Christensen [5]. There is no need to summarize the many other models, this has been amply done elsewhere. To come up to the present time two typical efforts will be noted. In recent years there has been much interest in porous, cellular materials. Roberts and Garboczi [12] have used the finite element method to study in great detail the porous materials case, containing an impressive range of different cellular, morphological forms. Garboczi and Berryman [8] have used what appears to be a hybrid combination of the Differential Method and the GSCM to study concrete properties. Suffice to say, none of the models, past or present, have been successfully applied to obtain theoretical forms for the fundamental problem of concern here.
The plan is as follows. There is no periodic packing order for single size spheres that has the symmetry which can guarantee isotropy. Accordingly a brief study will be made of the random (isotropic) packing of rigid spheres. With a specific result for this case, then the mechanical properties problem can be formulated. First, the case of an incompressible matrix phase is considered. A broad range of packing distributions are considered with that for single size spheres at one extreme and at the other extreme is a distribution which necessarily involves a gradation in sphere sizes, as in the GSCM, which in the limit can become volume filling. Finally these results will be generalized in one sense at the cost of a restriction in another. The generalization involves letting the elastic matrix phase become compressible, but this can only be accomplished here for the case of single size spherical inclusions, which was the original objective.
As already mentioned, new micromechanics models are continuously being formulated. Any proposed two phase micromechanics model that claims generality might do well to recover the GSCM at one extreme and the results to be presented here, or some future result for the current problem, at the other extreme. Such forms may be of special importance in the nano-composites area where the packing of nearly rigid inclusions is a primary variable. These situations arise in considering the incorporation of nano-size particles into the resin phase of composites for marine structures.
Section snippets
Random packing of spheres
The closest packing for single size, rigid spheres that is possible is that of a hexagonal arrangement having the maximum volume fraction of spheres, thought of here as perfect packing. This volume fraction is
Unfortunately this hexagonal arrangement does not have the symmetry necessary to assure 3D isotropy of mechanical properties. This situation has been understood for a very long time, and it becomes necessary to investigate the random packing of spheres. This also has been known
Incompressible matrix case, a spectrum of inclusion size distributions
The problem here is that of rigid spherical inclusions in an isotropic elastic matrix phase. In determining the effective shear modulus, generality will be given to various types of size distributions of the spherical inclusions, ranging from single size spheres at one extreme up to a distribution of sizes that can become volume filling at the limit, for the other extreme. The latter case corresponding to that in effect with the GSCM and, indeed, with nearly all micromechanics models. For a
Compressible matrix case, single size inclusions
The results in Section 3 show that the case of single size spherical inclusions has profoundly different effects than those of other cases having a distribution of sizes of inclusions. The major limitation of the work of Section 3 was that of the incompressibility of the matrix phase. Now an approach will be taken to remove that condition and constraint, in the case of single size inclusions, cM=1/2.
The starting form is the incompressible result (18). In considering the generalization of (18)
Final remarks
As discussed in Section 4 the Poisson's ratio value of νm=1/5 does apparently convey some special characteristics in this particular problem. This same special value of νm=1/5 has been noticed in other problems as well. In the case of elastic materials containing voids, Roberts and Garboczi [12] have shown such effects. The case of voids will be briefly stated next for direct comparison with the previous case of rigid inclusions.
The classical solution for dilute inclusions of spherical voids in
Acknowledgements
This work was performed under the auspices of the US Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48. This work was partially supported by the Office of Naval Research, Program on Solid Mechanics, Dr Y.D.S. Rajapakse, Program Manager.
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