# ELASTIC-REINFORCED PIPE FROM THREE LAYERS WITH RING FIBERS UNDER THE EXPOSURE OF INTERNAL PRESSURE

### Abstract

A numerical solution of the problem of deforming a three-layer pipe with annular square fibers under the influence of internal pressure with large displacements and deformations is presented by a model of a piecewise-homogeneous medium. The pipe was modeled as an assembly of ring elements. Such elements are square rings made of binder material, including ring fibers as their reinforcing core. The pipe design was accepted as a long cylindrical shell, which is axially deformable when loaded with pressure, when the central and extreme sections of the ring elements move in the planes of their original position.

The study of the deformation of soft composite structures, when their initial configuration changes significantly, remains one of the problems of the mechanics of composite materials. As one of these structures, we considered a long thin-walled tube of elastic layers with ring fibers of a more rigid elastic material. Pipes of this structure can be used to create flexible gas and air ducts, in order to transport substances in a spray form, to collect environmentally hazardous waste.

The most common approach in the study of the bodies of a fibrous structure is based on the use of a model of a piecewise-homogeneous medium, when the matrix and fibers are considered as contacting bodies. A numerical solution of the problem according to this model of the deformation of a pipe of three layers with annular square fibers under the influence of internal pressure during large displacements and deformations is presented. The pipe was modeled as an assembly of ring elements. Such elements are square rings made of binder material, including ring fibers as their reinforcing core. The pipe design was accepted as a long cylindrical shell, which is axially deformable when loaded with pressure, when the extreme and central sections of the ring elements move in the planes of their original position.

The boundary problem for assembling the ring elements of the shell was formulated on the basis of the equations of the nonlinear theory of elasticity for the matrix and the fibers in it. The problem was solved using the finite-difference method, first-order derivatives of the main quantities with respect to the axial and radial coordinates, and approximated using second-order finite-difference relations. The discrete analogue of the problem was solved on the basis of the Newton method procedure. The uniqueness of the solution of the boundary value problem was ensured by the continuation of the pressure solution in the pipe. As a result of solving the boundary value problem at a finite pressure value, the nodal values of displacements, strains, and stresses for the matrix () and fibers () were determined.

The boundary surfaces of the pipe acquire a wave-like shape (corrugated) with a period along the generatrix equal to the reinforcement period. Arrows of deflection (double amplitudes of wave formation) in the inner and outer surfaces of the pipe are not very different from each other. The middle surface of the pipe, due to its thin-walled state, remains in the deformed configuration almost cylindrical, when its deflections are small compared to the deflections in the boundary surfaces. The matrix material is extruded from the regions of the fiber matrix layer between the fibers into the region of the matrix layer. The bounding cylindrical surfaces of the annular fibers in the deformable pipe become convex, and their end surfaces become concave.

### References

2. Киричевский В. В. Метод конечных элементов в механике эластомеров. Київ: Наук. дум-ка, 2002. 655 с.

3. Черных К. Ф. Нелинейная теория упругости в машиностроительных расчетах. Ленинград: Машиностроение, 1986. 336 с.

4. Holzapfel G. A., Gasser T. C., Ogden R. W. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. of Elasticity. 2000. Vol. 61. P. 1–48.

5. Holzapfel G. A., Gasser T. C., Stadler M. A ctructural model for the viscoelastic behavior of arterial walls: Continuum formulation and finite element analysis. European J. of Mechanics ASolids. 2002. Vol. 21. P. 441–463.

6. Green A. E., Adkins J. E. Large elastic deformations and non–linear continuum mechanics. Oxford: Аt the Clarendon Press, 1960.

7. Akhundov V. M. Analysis of elastomeric composites based on fiber-reinforced systems. 1. Development of design methods for composite materials. Mechanics of Composite Materials. 1998. Vol. 34, No. 6. Р. 515–524.

8. Korn G. A. and Korn T. M. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems and Formulas for Reference and Review. New-York: General Publ. Company, 2000. 1151 p.

9. Ортега Дж., Рейнболдт В. Итерационные методы решения нелинейных систем уравнений со многими неизвестными. Москва: Мир, 1975. 558 с.

10. Levinson M. and Burgess I. W. A comparison of some simple constitutive relations for slightly compressible rubber–like materials. Int. J. Mech. Sci. 1971. Vol. 13. P. 563–572.

11. Blatz P. J. and Ko W. L. Application of finite elastic theory to the deformation of rubber mate-rials. Trans. Soc. Rheology. 1962. Vol. 7, No 6. P. 223–251.

12. Энциклопедия полимеров: у 3 т. / под ред. В. А. Кабанова и др. Москва: Советская энциклопедия, 1977. 1044 с.

13. Akhundov V. M., Kostrova M. M., Naumova I. Ju. Graphic Visualization of Deformed Fibre-Reinforced Materials. Metallurgical and Mining Industry. 2017. No. 2. P. 52–58.

*Bulletin of Zaporizhzhia National University. Physical and Mathematical Sciences*, (1), 4-13. Retrieved from http://journalsofznu.zp.ua/index.php/phys-math/article/view/231