Stress Distribution in Composites with Co-Phase Periodically Curved Two Neighboring Hollow Fibers

In this paper, stress distribution is examined in the case where infinite length co-phase periodically curved two neighboring hollow fibers are contained by an infinite elastic body. The midline of the fibers is assumed to be in the same plane. Using the three-dimensional geometric linear exact equations of the elasticity theory, research is carried out by use of the piecewise homogeneous body model. Moreover, the body is assumed to be loaded at infinity by uniformly distributed normal forces along the hollow fibers. On the inter-medium between the hollow fibers and matrix surfaces, complete cohesion conditions are satisfied. The boundary form perturbation method is used to solve the boundary value problem. In this investigation, numerical results are obtained by considering the zeroth and first approximations to calculate the self-equilibrium shear stresses and normal stress at the contact surfaces between the hollow fibers and matrix. Numerous numerical results have been obtained and interpreted about the effects of the interactions between the hollow fibers on this distribution.

As noted in [4][5][6], one of the main factors that determines the strength of unidirectional fibrous composite is the fibers' curvature. Curvature can occur as a result of design or as a result of a technological process and is called periodic or local. Practical applications of these composite materials require determination of the stress-strain state considering the curl of the fibers. Selfbalanced stresses arise due to the curvature of the fibers, and these stresses can cause the fibers to separate from the matrix under uniaxial tension or stress along the fibers [5,7,8]. This separation leads to the formation of macro cracks, whose accumulation can significantly alter the strength and stiffness properties of the composites [9]. In addition, the initial minor curling of reinforcing fibers is taken as a model for investigating various stability loss or fracture problems of unidirectional composites [10]. Therefore, creating the mechanics of composite materials containing curved structures is important both in terms of fundamental advances in the mechanics of rigid deformable bodies and in modern engineering that uses special composite components. In [11], to achieve this aim, using the three-dimensional exact equations of the elasticity theory together with the piecewise homogenous body model, a method was developed for investigating the stress-strain state produced in unidirectional fibrous composite materials. Fiber curving was considered periodic in that study. Reviews of the numerical results obtained by using this method are given in [5].
The method discussed in [11] is proposed for the case where the concentration of fibers is so small that the interactions between them are negligible. In [12], the method is developed to be used in the problem of two neighboring periodically inclined fibers, and numerical results are given taking into account the interaction between the fibers. In [13], this method is extended to the nonlinear geometric state, and from here, the numerical results obtained for stresses in a composite material containing one and two neighboring periodically inclined fibers are presented. Studies on the loss of stability problems corresponding to these conditions are reviewed in [14]. In [15][16][17][18], this approach is developed for a periodically located row of fibers embedded in an infinite matrix and the obtained numerical results are presented. In [19], the stability loss of the related problem is studied. In addition, some similar studies in the local curvature case are presented in publications [20][21][22][23].
However, in all the investigations given above, it is assumed that the fibers embedded in the matrix are traditional materials. In [24], the reinforcing element is taken as a double-walled carbon nanotube (DWCNT), and the loss of internal stability in composite materials containing a straight infinitely long DWCNT is examined. In [25], the stress distribution is studied in an infinite body containing an infinite length periodically curved hollow fiber with low concentration. In this study, to calculate the effect of the interactions between the hollow fibers on the stresses, the same situation is developed for the two neighboring hollow fiber cases. The problem is the stress distribution in the infinite body containing infinite length, periodically curved two neighboring hollow fibers. The investigations are made by using the three-dimensional geometric linear exact equations of the elasticity theory with the piecewise homogeneous body model. In addition, it is assumed that uniformly distributed normal forces are acting on the body at infinity along the hollow fibers.

Mathematical Formulation of the Problem
Infinite length, periodically curved two neighboring hollow fibers embedded in an infinite elastic body are taken into account. We assume that the perpendicular sections of the hollow fibers are circles with radius R and thickness H and this does not change throughout the hollow fibers. We also note that the midline of the hollow fibers is in the same plane and has co-phase initial periodic tilting with respect to each other. In the aforementioned model, it is thought that there are normal forces with intensity p that are evenly distributed in the longitudinal direction of the hollow fibers.
Let us select O m x m1 x m2 x m3 cartesian and O m r m θ m z m cylindrical coordinate sets for the centerline of each hollow fiber as the Lagrange coordinates (Fig. 1). Here, m = 1, 2, shows the first and second hollow fibers, respectively. As can be seen from Fig. 1, between these coordinates the following relations are satisfied: Assuming that the midlines of the hollow fibers are in the x 12 = x 22 = 0 plane, the equations of these lines are written as follows: Here, L is the bending amplitude of the hollow fibers and is the period of bending. Assuming that L is much smaller than , let us define the ε = L/ parameter (0 < ε << 1). The degree of initial defect of the hollow fibers is characterized by this parameter.
In the following, we will show the values belonging to the first and second hollow fibers with the superscript (21) and (22), respectively and the values belonging to the matrix (infinite elastic medium) with the superscript (1). Let us write the following governing field equations to be satisfied in each hollow fiber and infinite elastic medium: Let us denote the inner surface of the hollow fibers with S 0k (k = 1, 2) and the outer surface (the contact surface with the matrix) with S k (k = 1, 2). We write the contact conditions below, assuming that the inner radii of the hollow fibers remain constant and that there are ideal contact conditions between the hollow fibers and the matrix.
In the case discussed, the following conditions are also provided: Tensor notation is used in the formulae given above. We should state that the sum cannot be made according to the underlined indices.
Thus, the formulation of the addressed problem is completed. The problem is reduced to the solution of Eq. (3) within the framework of (4) contact and (5) boundary conditions.

Solution of the Problem
We can write the equations of the surfaces S k and S 0k from the cross-sectional form condition of the hollow fibers.
Here, t 3 ∈ (−∞, +∞) is a parameter. Using Eq. (6) and doing some known operations, we obtain the following expressions for the components of the unit external normals of S k surfaces: The boundary-form perturbation method given in [5] will be used to examine this problem. According to this method, all the expressions are searched in series form in terms of the small parameter defined above.
Expressions (6) and (7) are also written serially according to ε: The a qk θ q , which are the coefficients of ε k , can be easily obtained from (6)- (8). From (3), the governing field equations provided separately for each approach in (9) are obtained. If we use (10), we open each approach in (9) to a series around ( . If we substitute these last statements in (4) and use the expressions of n qr , n qθ , n qz in (10), then as a result of some long but known operations, the contact conditions provided in r k = R + H and r k = R, for each approach in (9) are obtained. In this case, the k-th contact condition includes the sizes of all the previous k-1 approaches.
We will deal with the case where the nonlinear terms in the equations related to the zeroth approximation can be omitted, and since ∇ n u (k)j, 0 << 1, δ j n in the first and other approaches can replace (g j n + ∇ n u (k)j, 0 ). In addition, we will assume that the σ (k), 0 (ij) (ij) = zz stresses in the zeroth approximation can be neglected as well as the σ (k), 0 zz stresses [15]. According to this assumption, the governing field equations and contact conditions of the zeroth approach are obtained as follows: Thus, the zeroth approach is reduced to the solution of Eq. (11) within the framework of contact conditions (12).
For the first approach, we can write the governing field equations as follows: For this approach, we obtain the contact conditions as follows: In this section, we will obtain the solutions of the boundary-value problems of the zeroth and first approaches formulated above. For simplicity, we will assume that both fiber materials are the same and that Poisson's ratio of ν (21) = ν (22) = ν (2) (ν (2k) k-th fiber to Poisson's ratio) is equal to Poisson's ratio of the matrix material which we will show as ν (1) .
In this case, we get the following solution for the zeroth approach: E (1) and E (2) in (18) are the elasticity moduli of the matrix and fiber materials, respectively.
Let us now consider the solution of the problem (13)- (17) belonging to the first approach. Considering the above assumptions and the solution of the zeroth approach (18), the Eq. (13) are obtained as follows: These equations coincide with the linearized three-dimensional elasticity equations. Similarly, Eq. (14) become: Considering the solution obtained in the zeroth approach, the contact conditions of the first approach (16) are obtained as follows: For the solution of Eq. (21), let us use the following representation given in [5], taking into account (19): The ψ (k) and χ (k) functions here provide the following equations: (23) are fixed as follows: Thus, the solution of Eq. (23) is obtained by considering the expressions on the right side of Eq. (21) as follows: In (25) and (26), α = 2π/ and I n (x), K n (x) are the Bessel functions with imaginary arguments and Macdonald functions, respectively. A n , and F (2k) n unknowns are complex constants and provide the following relationships: In order to write the (r 2 , θ 2 ) coordinates in (r 1 , θ 1 ) coordinates or, on the contrary, (r 1 , θ 1 ) in (r 2 , θ 2 ) coordinates, we will make use of the summation theorem [26] and the relations between cylindrical coordinate sets. r m exp iθ m = R mn exp iφ mn + r n exp iθ n Thus, an infinite dimensional system of algebraic equations is obtained from (21) in terms of unknown constants (27) by using (22)- (26). If this system is solved using the convergence criterion and the unknowns are determined, then the desired stress values are determined. Let us show that the convergence criterion is satisfied in order for this system to be replaced by a finite one. Let us introduce the following notations: (2)q rs (n) , q = 1, 2; k = 1, 2, 3; r; s = 1, 2, 3, The system of infinite algebraic equations in terms of defined notations is written as: n . Thus, from (31) and (33) we get the following: The infinite system of algebraic Eq. (34) must be approximated by the finite one to obtain numerical results. To make this displacement possible, the determinant of the algebraic infinite system of equations must be of normal type [27]. This is achieved if we show that the following series is convergent: The functions I n (x) and K n (x) are satisfied in the following asymptotic estimates: Since the hollow fibers do not touch each other, the following inequalities are achieved: From (36)-(37), and analyzing W (1)2 nm , we show the convergence of the series (35) as follows: Similar proof is made in [11,12]. Thus, the infinite algebraic system can be replaced by a finite one. The number of equations the finite system will consist of, will be decided by the convergence of the numerical results.

Numerical Results
In this study, the numerical results have been obtained using the zeroth and first approximations. Next approaches can only contribute quantitatively to the results. In addition, Poisson ratios are taken as ν (1) = ν (2) = 0.3, and the dimensionless parameter ρ = R 12 / (R + H), which shows the contribution of the interaction of hollow fibers to the stress values, is defined. Furthermore, by examining the contribution of the parameters E = E (2) /E (1) , γ 1 = 2π(R + H)/ = α (R + H) and γ 2 = H/(R + H) to the stress distribution, the ratio of elasticity modulus and the effect of the thickness and radius of the hollow fibers will be calculated. While calculating the stresses, since they have maximum values, θ = 0, αt 3 = π/2 for σ nn , θ = 0, αt 3 = 0 for σ ne and θ = 0, αt 3 = 0 for σ nτ are used.   In Figs. 2-4, the dependencies between σ nn / |p|, σ nτ / |p|, σ ne / |p| and γ 1 (a), γ 2 (b), respectively are given where E = 50 and ε = 0.015, and γ 2 = 0.3 in (a) and γ 1 = 0.5 in (b). According to Figs. 2 and 3, as the hollow fibers approach each other, the values of σ nτ / |p| are increasing, but the absolute values of σ nn / |p| are decreasing. Although the relationship between the absolute values of these stresses and fiber thickness is monotone, the relationship between these values and the outer radius of the hollow fiber is non monotone. From Fig. 4, it is observed that while the absolute value of σ ne / |p| increases as the hollow fibers approach each other up to about γ 1 = 0.35, these values then decrease. The same change exists between the fiber thickness and the absolute values of σ ne / |p|. In this case, the change occurs around γ 2 = 0.12. In addition, it is seen that the σ nτ / |p| shear stress values, as absolute values, are higher than the other stress values. According to these results, attention should be paid to the σ nτ / |p| shear stress.
With the increase of the distance between the hollow fibers, the interaction between the fibers disappears and thus the same values are reached with the stress values obtained in case [25]. This shows the accuracy of both the algorithms used and the programming made by the authors with the FTN77 programming language.
In Tab. 1, the values of σ nn / |p|, σ nτ / |p| and σ ne / |p| are given for γ 1 = 0.6, γ 2 = 0.4 and various E, ρ, ε. From this table, it is seen that the values of the stresses are increasing with the parameters ε and E which define the degree of defect and modulus of elasticity ratio, respectively. Also, when γ 1 = 0.6, γ 2 = 0.4, as the hollow fibers approach each other, the values of σ nn / |p| and the absolute values of σ ne / |p| decrease, but the absolute values of σ nτ / |p| increase.

Conclusions
In this study, the normal and shear stresses at the fiber-matrix interface are studied in the case of two neighboring hollow fibers with infinite length (at least ten times the bending amplitude) with the same phase periodic curvature embedded in an infinite elastic medium. It is thought that the object in question has uniformly distributed normal forces acting along the hollow fibers at infinity, and the midlines which pass through the centers of the fibers have the same plane and phase curvature. The case in which the thicknesses and outer radii of the fibers are the same and where these values do not change along the fibers is discussed. The hollow fibers may be close to each other, but they do not come into contact. The research has been carried out using the piecewise-homogeneous body model and the three-dimensional geometric linear exact equations of elasticity theory. Thus, it is aimed to obtain better quality results than the numerical results obtained using approximate theories.
Within the framework of the above assumptions, the governing field equations provided separately in the hollow fibers and in the matrix are written in accordance with the piecewisehomogeneous body model. Added to these are the conditions provided by the inner surfaces of the fibers and the ideal contact conditions provided at the surfaces where the fibers come into contact with the matrix. Thus, the problem has been formulated as a boundary-value problem which has been solved by using the boundary form perturbation method. According to this method, all expressions related to the surface equations, hollow fibers and matrix are serialized in terms of the small parameter defining the bending degree of the fibers. When these series are used in the governing field equations and boundary conditions, the boundary-value problems provided separately for each approach are obtained. Naturally, the k-th boundary-value problem includes the magnitudes of the previous boundary value problems. Numerical results have been obtained by solving the boundary value problems of the zeroth and first approach. Since the solution of the boundary value problems belonging to the next approaches will affect the numerical results only quantitatively, and not qualitatively, the solutions up to the first approach are considered to be sufficient. With some simple assumptions, the boundary-value problem of the zeroth approach can be solved analytically, while the solution of the boundary-value problem of the first approach is brought to the infinite system of algebraic equations. It has been shown that this infinite system of algebraic equations can be replaced by a finite one by using the convergence criterion. Numerical results are produced by taking a sufficient number of equations according to this criterion.
The obtained numerical results consist of the calculations made at the points where the normal stress of σ nn and the values of σ nτ and σ ne shear stresses are maximum. Dimensionless parameters related to the distance between the hollow fibers, the ratio of elasticity constants and fiber bending amplitude are defined, and the effects of these parameters on the stresses are discussed. Accordingly, the following conclusions have been made: • The relationship between the σ nn normal stress and the inner radius of the hollow fibers is non-monotone. In addition, as the fibers approach each other, the values of the σ nn normal stress decrease. • The relationship between the σ nτ shear stress and the inner radius of the hollow fibers is also non-monotone. However, as the fibers approach each other, the absolute values of the σ nτ shear stress increase. This increase is remarkable. • The relationship between the σ ne shear stress and the inner radius of the hollow fibers is also non-monotone. However, the approach of the hollow fibers to each other increases the absolute values of these stress values up to a certain value of the inner radius of the fiber and decreases after this value. • The relationship between the stresses under consideration and the thickness of the hollow fibers is monotone. • The increase in the bending degree and in the ratio of elasticity constants increases the values of the stresses as absolute values. • The numerical results obtained when the distance between the hollow fibers is increased so that there is no interaction between them, coincides with those obtained in [25]. In addition, the numerical results obtained by taking the inner space of fibers to zero are consistent with the results of [12], in which problem the fibers are filled. This demonstrates the accuracy of the numerical results.
The numerical results obtained should be taken into account in the production of materials which have the properties discussed here, when used as structural elements. In addition, for certain values of the parameters (given in [24]), instead of hollow fibers, carbon nanotubes can be considered, and stress distribution and stability problems can be examined.