Contrast Expansion Method for Elastic Incompressible Fibrous Composites

Contrast parameter expansion of the elastic fields for 2D composites is developed by Schwarz’s method and by the method of functional equations for the case of circular inclusions. A computationally efficient algorithm is described and implemented in symbolic form to compute the local fields in 2D elastic composites and the effective shear modulus for macroscopically isotropic composites. The obtained new analytical formula contains high-order terms in the contrast parameter and explicitly demonstrates dependence on the location of inclusions. As a numerical example, the hexagonal array is considered.


Introduction
The general potential theory of mathematical physics yields methods of integral equation to numerically solve various boundary value problems.Integral equations for plane elastic problems were constructed by Muskhelishvili [1], first, extended to doubly periodic problems in [2], and developed in [3][4][5] and papers cited therein.The obtained results were applied to computations of the effective properties of the elastic media.Integral equations are efficient for the numerical investigation of nondilute composites when interactions of inclusions have to be taken into account.Another type of integral equation based on the generalized alternating method of Schwarz was proposed by Mikhlin [6] and developed in [7].
Consider a problem on the plane R 2 isomorphic to the complex plane C. Let  fl C ∪ {∞} \ (⋃  =1   ∪   ) where the boundary   of   is a simple closed Lyapunov curve oriented in clockwise sense.Schwarz's method for a multiplyconnected domain  = Ĉ \ ⋃  =1 (  ∪   ) is based on the separate solutions of the simple boundary value problems for simply connected domains Ĉ \ (  ∪   ) ( = 1, 2, . . ., ).It was demonstrated in [7] that Schwarz's method can be realized as the iterative schemes constructed on contrast and on concentration parameters considered as small perturbation parameters and precisely described below.Convergence was proved for Laplace's equation for the contrast expansion.
The main advantage of Schwarz's method consists in analytical solution to the problems discussed when the physical parameters are presented in symbolic form in the final exact or approximate formulae.Such formulae were recently obtained in [8] for biharmonic functions which describes elastic materials for a circular multiply-connected domain.The results of [8] are based on the concentration expansion.
In the present paper, we apply Schwarz's method based on the contrast parameter expansion for a circular multiplyconnected domain.Schwarz's method is used in the form of the functional equations method [7][8][9].Elastic isotropic materials are described through two independent moduli.In order to make the presentation clear, we simplify the problem by consideration of incompressible materials when Poisson's ratio ] is equal to 0.5.Then, we introduce only one contrast parameter: where  1 and  denote the shear modulus of inclusions   and matrix , respectively.
In the present paper, we deduce a computationally efficient algorithm implemented in symbolic form to compute the local fields in 2D elastic composites and the effective shear modulus   for macroscopically isotropic composites.The obtained new analytical formula is valid up to ( 3 ) and explicitly demonstrates dependence on the location of inclusions.Such an approach has advantages over pure numerical methods when dependencies of the effective constants on the mechanical properties of constitutes and on geometrical structure are required.
The numerical examples from Section 6 give sufficiently accurate values of   for all admissible , that is, for || ≤ 1 and for  not exceeding 0.4.

Contrast Expansion Method
Consider a finite number  of inclusions as mutually disjoint simply connected domains   ( = 1, 2, . . ., ) in the complex plane C ≡ R 2 .It is worth noting that the number  is given in a symbolic form with an implicit purpose to pass to the limit  → ∞ later.
This is a boundary value problem for the multiply-connected domain  on the functions  (1)  0 (),  (1)  0 () analytic in  and twice continuously differentiable in the closure of  including infinity.The described above step is the first step of the iterative scheme when we pass from  (0) 0 () and  (0) 0 () to  (1)  0 () and  (1)  0 ().The step ( + 1) consists in the solution to the problems for every domain   : (20) Therefore, the conjugation problem (8)-( 9) is reduced to the sequence of the problems separately for the domains   ( = 1, 2, . . ., ) and .The described iterative scheme is computationally effective if the inclusions   have simple shape.The next section is devoted to its explicit realization for circular inclusions.
The functional-differential equations ( 22)-( 23) include the meromorphic functions Φ  () not belonging to H (2,2) .They were written as on the vector-function () = (  (),   ())  introduced in all   by substitution (21) in the space H (2,2) × H (2,2) .The operator A and the given vector-function  are determined by ( 22)-( 23).Equation ( 28) was explicitly written in [11] as the system of functional-differential equations It was proved in [11] that the operator A is compact in 2) .One can see that the contrast parameter  plays the role of the spectral parameter; hence,  can be written in the form of power series in .This implies that the method of successive approximations applied to (29) converges in H (2,2) × H (2,2) for sufficiently small .Let  *  (),  *  () ( = 1, 2, . . ., ) be a solution of (29).This unique solution belongs to H (2,2) × H (2,2) .The given vector-function  is twice differentiable in the closures of   .Hence, the pumping principle [9, page 22] can be applied as follows.The shifts in composition operators of the right part of (29) are the shift strictly into domains.Hence, if we substitute  *  (),  *  () into (29), we obtain that  *  (),  *  () are twice differentiable in the closures of   .The equivalent method of successive approximations can be applied to ( 22)-(23) more conveniently in computations.

Method of Successive Approximations
Application of successive approximations to functional equations is equivalent to Schwarz's method described at the end of Section 2. It follows, for instance, from the uniqueness of the analytic expansion in  near zero.Moreover, each iteration for the functional equations corresponds to an iteration step in Schwarz's method since the coefficients in the series in  are also uniquely determined.It can be also established directly form formulae written below.
Using the series (11) for   () and analogous series for other functions and applying successive approximations to the functional equations we obtain the following iteration scheme.The zeroth approximation is The next approximations for  = 1, 2, . . .are where  , denotes the Kronecker symbol.
Introduce the functions Then, It follows from (31) that () can be written in the following form: where the following expansion is used: The th approximation for () becomes and  (0) () = 0.

Shear Modulus
The averaged shear modulus  ()  of the considered finite composite is introduced as the ratio It is related to the effective shear modulus   for macroscopically isotropic composites by the limit   = lim →∞  ()  .It was demonstrated in [8] that (41) can be transformed into where Here, () is a solution to the problem with  inclusions.
In order to determine , we first calculate the integral ∫ D  ().Equation (37) implies that Application of (35) and Cauchy's integral theorem yield The presented iterative scheme can be easily realized numerically.But we are interested in analytical formulae which can be obtained by symbolic computations.In the next sections, we perform symbolic computations to determine () and the effective shear modulus in the third-order approximation.
It follows from (37) that for the third-order approximation we need the integrals (45) for  = 0, 1, 2. Using the second equation (32), we have The functions  * () =  2 /( −   ) +   are analytic in D  for  ̸ = .Therefore, the integrals with  ̸ =  vanish in (47) by Cauchy's integral theorem and (47) can be calculated by residues In order to use (37) for  = 1 we find from (34) Using the expansion for  ̸ = , we have Along similar lines Subtracting ( 51) and (52) we obtain Using the above formulae we obtain from (49) the exact formula: We are now ready to calculate the integral: Using Cauchy's integral theorem and residues we obtain This yields (58)

Third-Order Approximation.
The third-order approximation requires advanced and long computations presented below.In order to calculate  (3) , we need the following function: It is obtained from (34) for  = 2 by substitution Φ (1)   given by (49) and  (1)   calculated by (33) with  = 1.
Below, we describe general recurrent formulae for an arbitrary th approximation and explicitly write the third-order terms when  = 3.The following double series in  and ( −   ) are used: The coefficients in the power series (61) are presented as series in : Substitution of ( 60)-( 62) into ( 22) and selection of coefficients in the same powers of  and ( −   ) yield ,1 with the zeroth approximation  (0) , = 0. Along similar lines, (23) yields with the zeroth approximation  (0) , =  1, .The expanded form of () follows from (31): Equation ( 65) is in accordance with (37) where Using (66) we calculate the integral The coefficients  () , for  = 0, 1, 2 can be explicitly written by the iterations (64): The limit (43) has to be found.Using (44) and (67) we express  through Taking into account terms up to ( 4 ) we rest  () ,1 for  = 0, 1, 2; hence, The above limits are calculated by use of the formalism developed in [8,12].It was supposed that  equal nonoverlapping disks belong to a parallelogram (fundamental) cell periodically extended to the complex plane by two linearly independent translation vectors.As an example, we shortly present the transformation of the term of  (1)  ,1 given by (68) multiplied by  2 and complexly conjugated in accordance with (71): We are interested in the limit associated with the Eisenstein summation [8,12]: The normalized -sums were introduced in [8,12] by means of the Eisenstein summation:  ( + 1) 2       (1)   +2 where denotes the concentration.The absolute convergence of series in  3 follows from the standard root test (Cauchy's criterion).

Numerical Simulations
The asymptotic formula (82) is a new theoretical formula obtained in the present paper.The computationally effective formulae and algorithms for the absolutely convergent sums   and   ( > 2) were developed in [13,14].However, the numerical implementation of the conditionally convergent series (74)-(75) for  = 2 and (76) for  = 3 requires further investigations.Formula (42) is similar to the famous Clausius-Mossotti approximation (Maxwell's formula) [15,Section 10.4] applied for calculation of the effective conductivity of dilute composites.It is surprising that the Eisenstein summation and Maxwell's self-consistent formalism are based on the different summation definitions of the conditionally convergent series [16,17].Using an analogy with the conductivity problem we now justify the proper limit value of  1  3 , that is, (76) for  = 3.The Hashin-Shtrikman bounds [15,Ch. 23] for 2D incompressible elastic media become for  < 0.
In the case  > 0, the bounds ℎ + and ℎ − are replaced with each other.One can check that the upper and lower bounds (83) coincide up to ( 3 ): This implies that the coefficients in the term  2  2 of the expressions (82) and (84) must be equal.It follows from (80) that  1 3 must vanish in the framework of the considered Maxwell's formalism.The definition of  2,2 is not essential in the final formulae since the terms with  2,2 are cancelled in (81).The above demonstration is based on the formal pure mathematical arguments.Its physical interpretation has been not clear yet.
Consider a numerical example, the regular hexagonal lattice, when the -sums become the Eisenstein-Rayleigh lattice sums  (1)   =  (1)   and   =   calculated by algorithm presented in [8,18].Only the nonzero values of   and  (1)    are presented in Table 1.
The effective shear modulus for the regular hexagonal lattice becomes  where ( + 1) 2 ( (1) +2 ) 2 )] . (86) The typical dependence of the effective modulus (85) on  is displayed in Figure 1 for  and in Figure 2 for negative .A dependence on  is given in Figure 3.The presented and other graphs demonstrate sufficiently good precision for  < 0.4 and for arbitrary ||.

Conclusion
Schwarz's alternating method is applied to 2D elastic problem for dispersed composites.It is realized for circular inclusions in symbolic form.Exact and approximate formulae for the local fields and for the effective shear modulus are established.
In general, Schwarz's method can be realized as expansion on the concentration  and on the contrast parameter .
The concentration expansion was recently realized in [8].In the present paper, we use the contrast parameter expansion.These two expansions in  and  yield two different computational schemes [7].The effective modulus formulae are the same in the second-order approximations and begin to differ in the third-order terms in  and , respectively.Schwarz's method is used in the form of the functional equations method for circular inclusions [7][8][9].We develop a computationally efficient algorithm implemented in symbolic form to compute the local fields in 2D elastic incompressible composites and the effective shear modulus for macroscopically isotropic composites.The new analytical formula (79) contains the third-order term in  and explicitly demonstrates dependence on the location of inclusions.The theory is supplemented by a numerical example on the hexagonal array of inclusions.Figures 1-3 illustrate the dependence of the effective shear modulus on  and .These numerical examples give sufficiently accurate values of   for all admissible , that is, for || ≤ 1 and for  < 0.4.
One can expect that the precision of   will increase by using of the next approximation terms   ( > 3).As it is noted in Introduction, Schwarz's method can be based on two different expansion, in  used here and in  used in [12].The both expansions contain the locations of inclusions in symbolic form.The expansions in  were held for smaller  but for an arbitrary ] [12].At the present time, we are inclined to use the contrast parameter expansions.However, we suppose that the choice between different variants of Schwarz's method will depend on the further implementation of high-order symbolic-numerical codes.

Figure 1 :
Figure 1: Dependence of the effective shear modulus for the hexagonal array on the concentration  for  = 0.9.The data are computed by (85) (solid line) and its polynomial expansion up to ( 4 ) (dotted line).The Hashin-Shtrikman bounds (83) are shown by dashed lines.
.This means that the rectangle   is normalized by the C-linear transformation  →   = |  | −1/2  to    having the unit area.Therefore,  = 6( 3 / 2 ) 4 .Other are transformed by the same method.As a result we arrive at the following formula:

Table 1 :
The nonzero values of the lattice sums for the hexagonal array.