Advanced constitutive relationships for composites

As pointed out in the previous section, if the constituent phases have widely differing moduli, the bounding solutions become less useful. One approach to obtaining more information about such composites is to specify the geometrical structure of the composite. Some exact solutions are available for dilute concentrations of the second phase, notably the results of Eshelby (1957) for ellipsoidal inclusions. These solutions will not be discussed, as they are the limiting case for the self-consistent solutions, to be discussed next. 

The self-consistent (SC) solutions are an approximate approach, in which a particle of one phase of given shape (e.g., spheres, needles or discs) is surrounded by the composite material. This is illustrated in Fig. 3.12, in which a spherical particle is surrounded by an effective medium that represents the elastic properties of the composite. For the spherically shaped particles, one can again use Eq. (3.14) but now H=4pJ i is used for the bulk-modulus calculation and 

The SC solutions appear to run into difficulties when there is a large elastic mismatch between the constituent phases, for example, at high concentrations of a rigid phase in a compliant matrix or of a porous phase in a stiff matrix. The latter situation will be discussed in Section 3.6. One approach to this problem is known as the Generalized Self-Consistent Approach, the concept behind which is illustrated in Fig. 3.14. Instead of a single inclusion in an effective medium, a composite sphere is introduced into the medium. As in the composite sphere assemblage discussed in the last section, the relative size of the spheres reflects the volume fraction, i.e., V = a bf as before. Interestingly, this approach leads to the HS bounds for the bulk modulus. The solution for the shear modulus is complex but can be written in a closed form. 

Two other approaches have gained significant attention the Differential 

Method and the Mori-Tanaka Method (Mori and Tanaka, 1973). In the former technique, the composite is viewed as a sequence of dilute suspensions and, thus, one can use the exact solutions for these cases to determine the effective composite properties. For example, the solution by Eshelby (1957) for ellipsoidal inclusions can be used. The increments of added inclusions are taken to an infinitesimal limit and one obtains differential forms for the bulk and shear moduli, which are then solved. The Mori-Tanaka method involves complex manipulations of the field variables. This approach also builds on the dilute suspension solutions (low F,) and then forces the correct solution as F, - 1. 

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