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.

Composite materials have increased in importance as they have superior properties than the materials they are made of. Today, composites have numerous application areas such as energy, sports, military, automotive, marine, aerospace, civil engineering, biomedical and even the music industry [

As noted in [

The method discussed in [

However, in all the investigations given above, it is assumed that the fibers embedded in the matrix are traditional materials. In [

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

Assuming that the midlines of the hollow fibers are in the

Here,

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

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

We can write the equations of the surfaces

Here,

The boundary-form perturbation method given in [

Expressions

The

We will deal with the case where the nonlinear terms in the equations related to the zeroth approximation can be omitted, and since

Thus, the zeroth approach is reduced to the solution of

For the first approach, we can write the governing field equations as follows:

For this approach, we obtain the contact conditions as follows:

where

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

In this case, we get the following solution for the zeroth approach:

Let us now consider the solution of the problem

These equations coincide with the linearized three-dimensional elasticity equations. Similarly,

Considering the solution obtained in the zeroth approach, the contact conditions of the first approach

For the solution of

The

Thus, the solution of

In

In order to write the

Thus, an infinite dimensional system of algebraic equations is obtained from

The system of infinite algebraic equations in terms of defined notations is written as:

It is obtained from

The infinite system of algebraic

The functions

Since the hollow fibers do not touch each other, the following inequalities are achieved:

From

Similar proof is made in [

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

10 | 2.1 | 0.010 | 0.0142 | −0.4253 | −0.0432 |

0.015 | 0.0214 | −0.6380 | −0.0648 | ||

0.020 | 0.0285 | −0.8507 | −0.0865 | ||

2.2 | 0.010 | 0.0196 | −0.3296 | −0.0455 | |

0.015 | 0.0294 | −0.4944 | −0.0683 | ||

0.020 | 0.0393 | −0.6593 | −0.0911 | ||

5.0 | 0.010 | 0.0594 | −0.1463 | −0.0487 | |

0.015 | 0.0892 | −0.2195 | −0.0730 | ||

0.020 | 0.1189 | −0.2926 | −0.0974 | ||

50 | 2.1 | 0.010 | 0.0437 | −1.7108 | −0.1841 |

0.015 | 0.0656 | −2.5662 | −0.2762 | ||

0.020 | 0.0874 | −3.4216 | −0.3682 | ||

2.2 | 0.010 | 0.0541 | −1.0815 | −0.1951 | |

0.015 | 0.0812 | −1.6223 | −0.2927 | ||

0.020 | 0.1083 | −2.1631 | −0.3903 | ||

5.0 | 0.010 | 0.1937 | −0.3005 | −0.2116 | |

0.015 | 0.2906 | −0.4508 | −0.3175 | ||

0.020 | 0.3875 | −0.6010 | −0.4233 | ||

100 | 2.1 | 0.010 | 0.0614 | −2.5159 | −0.2667 |

0.015 | 0.0922 | −3.7739 | −0.4001 | ||

0.020 | 0.1229 | −5.0319 | −0.5335 | ||

2.2 | 0.010 | 0.0724 | −1.5016 | −0.2791 | |

0.015 | 0.1086 | −2.2525 | −0.4186 | ||

0.020 | 0.1448 | −3.0033 | −0.5582 | ||

5.0 | 0.010 | 0.2661 | −0.3806 | −0.3024 | |

0.015 | 0.3992 | −0.5709 | −0.4537 | ||

0.020 | 0.5323 | −0.7612 | −0.6049 | ||

150 | 2.1 | 0.010 | 0.0712 | −2.9623 | −0.3118 |

0.015 | 0.1069 | −4.4434 | −0.4677 | ||

0.020 | 0.1425 | −5.9246 | −0.6236 | ||

2.2 | 0.010 | 0.0819 | −1.7238 | −0.3234 | |

0.015 | 0.1229 | −2.5857 | −0.4852 | ||

0.020 | 0.1639 | −3.4476 | −0.6469 | ||

5.0 | 0.010 | 0.3040 | −0.4223 | −0.3502 | |

0.015 | 0.4561 | −0.6334 | −0.5253 | ||

0.020 | 0.6081 | −0.8446 | −0.7004 |

In

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 [

In

Stress | Number of equations | ||||||
---|---|---|---|---|---|---|---|

87 | 96 | 114 | 123 | 132 | 177 | 186 | |

0.0663 | 0.0663 | 0.0658 | 0.0657 | 0.0656 | 0.0655 | 0.0656 | |

−2.4423 | −2.4781 | −2.5222 | −2.5354 | −2.5450 | −2.5648 | −2.5662 | |

−0.2764 | −0.2764 | −0.2762 | −0.2762 | −0.2762 | −0.2762 | −0.2762 |

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 piecewise-homogeneous 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

The relationship between the

The relationship between the

The relationship between the

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 [

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 [