The Concept of Equivalent Homogeneity

As we have understood, at sufficiently small scale all materials are heterogeneous in nature. In such a situation, one would like to start at atomistic or molecular level. This will lead to an intractable situation for engineering materials. Hence, the continuum hypothesis is invoked in such situations. In this hypothesis, a statistical averaging process is considered. Further, the actual constituents and their structures are idealized in such a way that resulting material is considered to be a continuum. Once we establish the existence of continuum hypothesis then the concept of equivalent homogeneity can be developed based on the structure of the material.

In general, the heterogeneity can be divided into two types. In the first type, the heterogeneity occurs as an idealized continuous variation of properties with the position and in the second type there is an abrupt change in properties across the interfaces of the constituents. In unidirectional fibrous composites, in cross sectional planes, we get the second type of inhomogeneity. Hence, the second type of inhomogeneity is of our concern in this micromechanical study. However, within the constituent we assume that the constituents are homogeneous and orthotropic, transversely isotropic or isotropic in nature.

Now we will introduce the characteristic dimension of inhomogeneity based on the constituent arrangement and nature. We will consider an idealized system of fibres and matrix in a composite as shown in Figure 7.7. For this system the characteristic dimension of inhomogeneity is the mean distance between fibres,  as shown. Now there also exits a length scale over which the properties can be averaged in some meaningful way, that is, . The length scale of averaging must be a dimension much larger than that of the characteristic dimension of the inhomogeneity, that is, . When this condition is satisfied the material can be idealized as being effectively homogeneous and the analysis of such a body can be done using the average properties associated with length scale .

The condition mentioned above is called the effective or equivalent homogeneity. The other terms used are macroscopic homogeneity and statistical homogeneity.

Concept of Volumetric Averaging

The effective or the average properties of the composite can be given through the relations between the average stress and average strain in the composite. Here, we introduce the concept of volumetric averaging.

Let us consider an RVE with dimension of inhomogeneity of  and averaging dimension of  (see Figure 7.7).  Let  be the volume of the RVE. Let us further assume that this RVE is subjected to macroscopically homogeneous stress or deformation field. Let us define the average stress as

(7.74)

and the average strain as

(7.75)

where,   is the infinitesimal strain tensor at . Now, the effective, linear stiffness tensor   is given by the relation 

(7.76)

The process of volume averaging may look very simple at the first sight. However, the averaging of stresses and strains involves a significant amount of task. The exact stress and strain fields, that is, and  in heterogeneous material are needed.

The inverse of the relation in Equation (7.76) can be given as

(7.77)

where,   is the effective compliance tensor.