Algorithm of iterative transformation for effective modules of multicomponent isotropic composite

We consider the effective modules of Voigt, Reiss for isotropic elastic composites. We have reformed the method for constructing iterative transformation of the upper and lower estimates of fork (Voigt-Reuss) towards two-component composite in case of an arbitrary number of components. The method is based on the fact that effective modules of Voigt and Reuss can be regarded as elementary symmetric functions introduced by Gauss. The conditions, which the iteratively – transformed efficient modules must fulfill at every iteration, are shown.


An algorithm for constructing iterative conversions of the effective modules of isotropic multicomponent composite
An algorithm of constructing the new effective modules for the two-component elastic composite.
Iterative transformations of the upper and lower bounds of Hashin-Shtrikman (H-S) modules were constructed in [11].
We will explain the essence of the method using of the effective modules by Voigt and Reuss. Let us suppose that we have the two-component composite with shear modules 1 2 G G > and volume contents 1 2 , γ γ , and 1 2 1 γ γ + = . The expressions of Voigt (V) and Reuss (R) modules have the form: The easily verifiable inequalities take place between modules 1 2 , G G and , v R G G : (3) Thus, model V-R, defined by relations (1), (2), produces modules which are located inside fork 1 2 G G > . The question is as follows: can we get a new compression of the fork by applying iterative transformation? Let us introduce the new modules of such type as Voigt and Reuss: We can easily show that the inequalities are performed as follows: (5) Thus, expressions (4) produce compression already in the two forks: fork 1 2 G G > and fork V R G G > . Relationships (4) can be easily generalized to the case of n steps of iterative transformations: We can prove that the chain of inequalities is performed: The restrictions which are superimposed on the model of iterative transformation. When building type sequences (6), we use an approach that is applied in determining the elementary symmetric Gaussian functions [13]. Let us suppose we have Then, the inequalities of types (5), (7), are valid for ,  The authors of [14] argue that Thus, the tasks of transforming sequences of positive numbers and sequences of effective modules in many ways have common patterns. The difference lies in the fact that the effective modules comprise a quantity of positive numbers each of which is multiplied by some weight multiplier of less than 1. Constructions of convergent sequences have the general requirements: • positivity of each component at each step; • fulfillment of the inequalities of kinds (5) and (7) at each step of iterative transformations.
In this case, obviously, the convergence of sequences effective modules to the limit, which depends only on the initial value of the sequences (from 1 1 2 2 , G G γ γ ) for the case of a two-component composite, can be proved.

An algorithm of constructing the new effective modules for three-, four-, five-component composites
Let us consider a generalization of the mathematical model (6) for the case of multi-component elastic isotropic composites. We will implement all reasoning by the example of the V-R effective modules.
Let us consider the inequality of the form: For a three-component composite, along with the effective modules of Voigt and Reis: We will introduce a module that can be called the geometric mean: (9) Next, we will construct a sequence of the iteratively -reformed effective modules for threecomponent composite. The first iteration will look like: It is easy to verify the fulfillment of inequalities: For n iterations, we have: For the four-component composite, we need the expressions of the two geometric mean modules. Let us suppose that we have: 12) Let us prove the fulfillment of inequalities: We will write down the expressions of the iterated effective modules for the n-th iteration: It is easy to prove the fulfillment of inequalities: We will consider a five-component composite. Let us suppose that we have an environment which is characterized by the five modules of shift: In addition, we introduce three modules (geometrical mean) of the next kind: It is easy to prove the inequalities: For n iterations, we have: Similarly, we can construct the sequences of the iterated effective modules for n components, where 5 n > .

The discussion of the results
The construction of the sequences of iteratively converted effective modules (EM) of the V-R model, carried out in this paper, includes the following necessary steps.
a. The localization of new effective modules having properties similar to the symmetric Gaussian functions. As shown in item 2, the construction of the inertial sequences of two variables requires two symmetric functions. For this, we can take the expressions of effective modules (V and R). And we MEACS2016 IOP Publishing IOP Conf. Series: Materials Science and Engineering 177 (2017) 012094 doi:10.1088/1757-899X/177/1/012094 must have three symmetric functions in case of three variables, etc. Since the tasks of building of the inertial sequences of some positive rational numbers are the same as the problem of constructing iterative sequences of EM, for the case of three-component composite in addition to the EM of V and R, we need to have one more additional effective module ( add G ), which should satisfy the inequality: ≥ For the four-and five-component composites we must have already 2 or 3 additional EM. The modules from the family of the geometric mean act as additional EM. Formulas (10), (11) and (14) give their visual representations.
b. At each stage of the introduction of new additional EM, we have to check the inequalities, which are similar to the inequalities that are performed when constructing of transformations of the symmetric Gaussian functions. These inequalities are shown in the complexes of (11), (13) and (14). The proof of their fulfillment is displayed, too.
The property of convergence of the various sequences to the same limit is one of the most remarkable properties of the iterated effective modules. So, the two-component effective modules of V and R model are reduced to the limit, which can be found by the numerical and analytical ways. The analytical limit of sequences for EM of V-R was found and analyzed by the authors (the mathematical calculations, obtained in deriving the results, are very bulky and this material was not included in this work). It turned out that, first, the limits of analytical and numerical sequences EM V and R coincide completely. Secondly, the calculation of the analytical limit confirmed reasoning of Gauss that the limit of the sequences of symmetric functions and sequences of EM depend only on their initial values (that is, on values which characterize the two-component composite). Third, it was found that the analytical limit of two sequences of Voigt and Reuss EM equals to the geometric mean of V G and R G which are defined by (4). For the case of three or more components, finding the analytical limit of sequences for iterative-converted EM is a more complicated mathematical problem. However, the latter does not mean that its solution cannot be found.

Conclusion
Expressions of the new effective characteristics of multicomponent elastic composites, presented in this paper, are constructed in accordance with the requirements for the iterative transformation of a symmetric mean [14]. They have the following properties: a. The new efficient characteristics are positive for each step of iterative transformations; b. The values all of iterated modules lay within the fork, which is formed by the maximum and minimum elastic shear modules of phases.
Recurrence relations, which determine the number of geometric mean modules needed to build iterative transformations, have been obtained. For the three-component composite , we have to have one geometric mean module for the four-component composite -two geometric mean modules, for the five-component composite -three, and so on. Numerical analysis of convergence for sequences of the iterated effective modules and calculations of the stress-strain state of structures with new effective characteristics are given in the second part of this work.