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Advances in Mechanical Engineering
Volume 2010 (2010), Article ID 954841, 14 pages
The Self-Balanced Shear Stresses in the Elastic Body with a Locally Curved Covered Fiber
1Department of Mathematical Engineering, Faculty of Chemistry and Metalurgy, Yildiz Technical University, 34010 Istanbul, Turkey
2Institude of Mathematics and Mechanics of National Academy of Science of Azerbaijan, 37041 Baku, Azerbaijan
3Department of Mechanical Engineering, Faculty of Mechanical Engineering, Yildiz Technical University, Yildiz Campus, Beşiktas, 34349 Istanbul, Turkey
Received 20 November 2009; Revised 10 June 2010; Accepted 4 July 2010
Academic Editor: Hirosi Noguchi
Copyright © 2010 Kadriye Simsek Alan and Surkay D. Akbarov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Within the framework of the piecewise homogenous body model, with the use of the three-dimensional geometrically nonlinear exact equations of the theory of elasticity, the method developed for the determination of the stress distribution in the composites with unidirectional locally curved covered fibers is used for investigation of the shear stresses acting along the fibers. All the investigations are carried out for an infinite elastic body containing a single locally curved covered fiber, for the case where there exists the bond covering cylinder with constant thickness between fiber and matrix material are considered. It is assumed that the considered material is loaded at infinity by uniformly distributed normal forces in the fiber lying direction. Under formulation and mathematical solution of the boundary value problem, the boundary form perturbation method is used. The numerical results related to stress distribution in considered body and the influence of geometrical nonlinearity to this distribution are presented and interpreted.
It is well known that in the structure of the unidirectional fibrous composites in many cases fibers have an initial curving caused by design factors or caused by action of various factors during technological processes [1–5]. Therefore, the theoretical investigations of the self-balanced stresses arising as a result of fiber curving have a great significance in the viewpoint of the theoretical and application sense [1–5]. For this purpose, within the framework of a piecewise homogenous body model, by using the exact three-dimensional equations of elasticity theory, Akbarov and Guz  presented a method for investigation of the stress state in unidirectional composites. Akbarov and Guz  applied this method to fibers with periodic curvatures. Akbarov and Guz  listed the results in  in detail. Fiber concentration was assumed by Akbarov and Guz  to be so small that the interaction between the fibers was neglected. Kosker and Akbarov  developed this method for two neighbouring periodically curved fibres and also some corresponding numerical results presented. The method used by Akbarov and Guz  and Kosker and Akbarov  is developed in  for geometric nonlinear statement, and numerical results for one and two neighboring periodically curved fibers are presented. That is all as far as the above-mentioned periodical curving case is concerned. There are also few investigations, such as [10–12], which regard the local curving case. But these investigations have been carried out in the case where the concentration of the fibres is small enough to ignore any interaction between the fibres. Moreover, it is by the use of the linear theory of elasticity that these investigations have been carried out. According to the well-known mechanical considerations and to the results obtained in , taking the geometrical nonlinearity into account influences significantly the values of the self-balanced stresses caused by fibers’ curving. Akbarov et al.  developed the investigations [10–12] for geometrical nonlinear statement and they presented numerical results. However, the results remained within the limits of the zeroth and first approximations. Kosker and Simsek [14, 15] analysed only the normal stresses. Kosker and Simsek  developed the method in  in such way as to obtain normal stress values up to the second approximation in the fiber matrix interface; Kosker and Simsek  analysed the normal stresses of an infinite elastic body containing a single locally curved covered fiber, for the case where there exists the bond covering cylinder with constant thickness between fiber and matrix materials. However, these results remained within the limits of the first approximation. It is obvious that it is necessary to increase the number of approximation added to the solution, so as to increase the sensitivity of the numerical results.
In this study, within the framework of the piecewise homogeneous body model, with the use of the three dimensional geometrically nonlinear exact equations of the theory of elasticity, the method developed for the determination of the stress distribution in the composites with unidirectional locally curved covered fibers is used for investigation of the shear stresses acting along the fibers. The case is considered where a covered single locally curved fiber of infinite length is located in an infinite elastic body with a low concentration of fibers. The interaction between the fibers is neglected. It is assumed that there exists the bond covering cylinder with constant thickness between fiber and matrix material. The numerical results related to stress distribution in considered body and influence of geometrical nonlinearity to this distribution are presented and interpreted.
2. Formulation of the Problem
A schematic view of the infinite elastic body with a locally curved covered fibre is given in Figure 1.
In the present study, it was assumed that the sections vertical to the fiber surface were R-radius circles (constant radius), that thickness of the transition material was constant along the fiber, and that the body was under loading of the uniformly distributed normal forces with intensity along the axis direction. With the middle line of the nanofiber, we associate Lagrangian rectilinear and cylindrical system of coordinates (Figure 1). The fiber, transition material, and the matrix material are homogeneous, isotropic, and linearly elastic. The initial imperfection form of the fiber is selected as follows: Assuming that is smaller than we introduce a small parameter .
The contact surfaces between the fiber and the bond covering cylinder and between the bond covering cylinder and matrix material are denoted by and respectively; the following equations of these surfaces and the components of their normal vectors are obtained, using the condition provided by nanofibercross section. where Fiber and transition material values are represented by superscripts () and (), respectively, and the matrix values (the infinite elastic body) are represented by the superscript (). The following governing field equations are obtained for fiber, transition material, and infinite elastic body. Also, perfect contact conditions are assumed at the interfaces and For the analyzed case, the conditions are also assumed. In (4) and (5), the conventional tensor notation is used, and the subscripts in parantheses denote the physical components of the corresponding tensors, namely, where , and . are Lamé parameters; are the covariant componentes of the unit normal to surfaces and
Hereinafter, parenthesis will not be used for the physical characteristics so as so simplify the expressions. Tensor notation will also be used in the rest of the study.
3. Solution Method
For the analysis of this problem, the boundary form perturbation method given in  is used. According to this method, the unknown values are presented as series in the small parameter : In addition, the quantities , and are also presented in the following series forms: where is a parameter and .
’s, are coefficients included in these expressions. Field equations for each approximation in (8) were obtained in (4). When these last expressions are placed in (5) and expressions in (9) are used, contact conditions provided in for each approximation in (8) are obtained, following long but well-known operations. In this case, . contact condition included all the, quantities of the previous approximations. Expression of the appropriate equation sets and contact conditions obtained for the zeroth, first, and second approximations will be discussed.
As given in , the following solution was obtained in this case for the zeroth approximation: and in (10) represent the elasticity modules of the matrix, fiber, and the transition material, respectively.
Taking into consideration the fact that the solution obtained for the zeroth approximation, other assumptions and the right-side function to be obtained for the second approximation would have values small enough to be neglected, governing equations for the first and second approximation can be defined as follows in terms of the physical components of the tensors and vectors: The mechanical and geometrical relations for the foregoing approximation are which coincide with the equations of the Three-Dimensional Linearised Theory of Elasticity (TDLTE) . It should be noted that the homogeneous parts of the equations obtained for the second and the subsequent approximations also coincide with the equations of the TDLTE.
Now we consider the contact conditions for each approximation, which are derived from (5). For this purpose, we substitute the expressions (8) and (9) into (5) and expand the quantities involved in (8) into Taylor series in vicinity of a point
Then, grouping the terms with identical powers of the parameter and taking into account the foregoing assumptions, we derive the contact conditions for each approximation: (i)for the zeroth approximation (ii)for the first approximation (iii)for the second approximation where From (13)–(15), by replacing (i) with , and z, we obtain explicit forms of corresponding contact conditions in the approximations considered. In (14) and (15), the following notation is used:
Similar contact conditions can also be obtained for subsequent approximations. Note using instead of when the foregoing relations are written for the interface
To solve the problem (11) and (12), we employ the representations : functions satisfy the following equations: where, is given in the following equations: We apply the exponential Fourier transform with respect to that is, the right-side functions of the related contact conditions were considered; the differential equations given in (18) and transformed to Fourier form were solved so as to obtain and functions, which were used when calculating the Fourier transforms of these approximations. where are the Bessel Functions of purely imaginary argument and are the Macdonald functions. Moreover, was used in (21) and in (22) as Fourier transform parameter. Functions (21) and (22) were used in boundary-value problems of the related approximations. Solution of the targeted linear equations set thereby enabled calculation of the values to be used in obtaining the Fourier transformed forms of the first and second approximations of the targeted stresses. To obtain real stress values, the following inverse Fourier transform given (e.g.) for was applied: The subsequent approximations are found in a similar way.
4. Numerical Results and Discussion
Solution of the linear equations system related to the first approximation gave the Fourier transformed values (based on Fourier parameter) of the targeted stress values. integral in inverse Fourier transform, which had to be used to obtain real values, turned into , depending on the odd or even of the quantities.
This integral was solved via the following approximation: and values were the parameters determined on the basis of convergence criteria and were used as . In addition, a 10-point Gauss-Legendre numerical integration method was used for the numerical calculation of integral. Contact conditions related to the second approximation included the values related to the first approximation. To obtain these values, the inverse Fourier transform given in (23) was used. These contact conditions applied in the Fourier transform given in the previous part. Taking into consideration the odd or even characteristic of the functions, a linear equation system—having one or two-fold integrals on the right side—was obtained. solution of this equation system gave the unknown to be used in the calculation of the Fourier transform of the targeted values. Inverse Fourier transform had to be used to obtain the real values of the targeted values in the second approximation. Therefore, an integration calculation was required at this point. This process resulted in calculation of three-fold integration. This calculation is given in detail in . See Figure 2.
Stress distribution analyses were made in the scope of the calculation and discussion of the numerical values of shear stresses. These stresses were in the direction of the n normal vector and , e tangential vectors on surfaces, which constitute the fiber, the material covering the fiber and the matrix intersection surfaces. In case, which corresponds to the case when the local curvature was neglected, stresses coincided with respectively.
and parameters were defined for the numerical results and we assume that , , and . The parameter was used to show the effect of geometric nonlinearity on the stress distribution. for and for .
Figures 3 and 4 show the change in stresses according to parameter. In these figures, , , , and . In addition, forand for . The intermittently lines in these figures represent the lines obtained in the problem of “uncovered, locally curved, unidirectional fiber in an infinite elastic body in the same parameter values” . Here, it can be concluded that when the thickness of the transition material is increased, it is possible to obtain the same result as those obtained in the case the locally curved uncovered fiber in the same parameter values.
Figures 5 and 6 show the distributions of and with respect to , for the case where , and in addition, these graphs illustrate the influence of geometrical nonlinearity on the distributions of the stresses considered. As expected, the numerical values obtained at coincided with the values obtained for the problem of the locally curved uncovered fiber . The same figures also show that the increase in the value of the m parameter results in an increase in the shear stress values (as absolute value) in the interfaces. In addition, as a result of the geometric nonlinearity, the increase in brings about an increase in tension status and a decrease in compression status in the shear stress values of both contact surfaces. At compression status, values—which were smaller than the related stability loss values —were used.
Figures 7 and 8, showing the change in stress values according to parameter, also show the effect of geometric nonlinearity on the relationship between and stress values and at and value. The graphs reveal that monotonically decrease in results in a decrease in the stress values of the contact surfaces of the transition material at compression status and an increase in the same at tension status. According to the graphs, the maximum absolute values of the stresses increase monotonically with .
shear stress values in Table 1 are given for different , and parameters at values. It can be concluded from this table that an increase in results in larger stress values (absolute value) on the transition material and matrix interface, and that an increase in value results in smaller stress values on the transition material and matrix interface. In addition, an increase in the value of brings about an increase in the stress values.
Table 2 gives the first and second approximation values of , stress values at and values according to different parameters. Here, for and for . This table shows that an increase in results in smaller stress values on the transition material and matrix interface and smaller stress values.
In the present paper, in the framework of the piecewise homogeneous body model with the use of the three-dimensional geometrically nonlinear exact equations of the theory of elasticity, the method developed for the determination of the stress distribution in the unidirectional fibrous composites with locally curved fibers has been used to investigate shear stresses acting along the fibers for the case where there exists the bond covering cylinder with constant thickness between fiber and matrix materials are considered. All the investigations were made for an infinite elastic body with a locally curved covered fiber.
Numerical results were obtained for the self-balanced shear stresses which operate on the fiber-transition material and transition material-matrix material interface and are caused by the locally curved covered fiber. From analyses of the numerical results, the following conclusions can be drawn.(i)The increase in the value of the parameter results in an increase in the absolute values of the shear stress in the interfaces.(ii)As a result of the geometric nonlinearity, the absolute values of the shear stresses decrease (increase) in compression (tension) of the composite in the fiber direction.(iii)An increase in results in an increase in the absolute values of the shear stresses on the transition material and matrix material interface.(iv)An increase in value results in a decrease in stress on the transition material and matrix material interface.
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