# Thermoelasticity of elastomeric constructions with initial stresses

## DOI:

https://doi.org/10.32347/2410-2547.2020.104.299-308## Keywords:

finite element method, elastomer, thermoelasticity, dissipative warming, initial stresses## Abstract

The article presents an algorithm for solving linked problems of thermoelasticity of elastomeric structural elements on the basis of a moment scheme of finite elements. To model the processes of thermoelastic deformation of structures with initial stresses the incremental theory of a deformed solid is used. At each step of deformation, the stiffness matrix is adjusted using an incremental geometric stiffness matrix. The use of triple approximation of displacements, deformations and volume change function allows to consider the weak compressibility of elastomers. The components of the stress tensor are calculated according to the Duhamel-Neumann law. To solve the problem of thermal conductivity, a thermal conductivity matrix considering the boundary conditions on the surface of a finite element is constructed. A sequential approximation algorithm is used to solve the thermoelasticity problem. At each stage of the solution, the characteristics of the thermal stress state are calculated. Based on the obtained components of stress and strain tensors, the intensity of internal heat sources is calculated as the dissipative energy averaged over the load cycle. To calculate the dissipative characteristics of the viscoelastic elastomer the parameters of the Rabotnov’s relaxation nucleus are used. Solving the problem of thermal conductivity considering the function of internal heat sources allows you to specify the heating temperature of the body. At each cycle of the algorithm, the values of the physical and mechanical characteristics of the thermosensitive material are refined. This approach to solving thermoelastic problems is implemented in the computing complex "MIRELA+". Based on the considered approach, the solutions of a number of problems are obtained. The results obtained satisfactorily coincide with the solutions of other authors. Considering the effect of preload and the dependence of physical and mechanical properties of the material on temperature leads to significant adjustments to the calculated values.

## References

Illiushyn A. A. Fundamentals of the mathematical theory of thermoviscoelasticity / A. A. Illiushyn, B. E. Pobedria. – M.: Nauka, 1970. 280p.

Pobedria B. E. Linked problems of tthermoelasticity / B. E. Pobedria // Mechanics of Polymers. -1969.- №3. – P. 415 – 421.

Kovalenko A. D. Fundamentals of thermoelasticity / A. D. Kovalenko. – K.: Naukova dumka, 1970. – 307 p.

Karnaukhov V. H. Linked problems of thermoelasticity theory of plates and shells / V. H. Karnaukhov, Y. F. Kyrychok. – K.: Naukova dumka, 1986. – 221 p.

Karnaukhov V. H. Linked problems of thermoviscoelaticity / V. H. Karnaukhov. – K.: Naukova dumka, 1982. –280 p.

Zhuk Ya. A. Linked thermomechanical behavior of a three-layer viscoplastic beam under harmonic loading / Ya. A. Zhuk, Y. K. Senchenkov// Applied mechanics.– 2001.– T.37, №1. – P. 93– 99.

Rabotnov Yu. N. Elements of hereditary mechanics of solids / Yu. N. Rabotnov. – M.: Nauka, 1977.– 384 p.

Kyrychevskyi V. V. Nonlinear problems of thermomechanics of constructions from weakly compressible elastomers / V. V. Kyrychevskyi, A. S. Sakharov. – Kyev: Budivelnyk, 1992. – 216 p.

The finite element method in the design of transport structures / A. C. Horodetskyi, V. Y. Zavorotskyi, A. Y. Lantukh-Liashchenko, A. O. Rasskazov. – M.: Transport, 1981. –143 p.

Finite Element Method: Theory, Algorithms, Implementation / V. A. Tolok, V. V. Kyrychevskyi, S. Y. Homeniuk, S. N. Hrebeniuk, D. P. Buvailo. – K.: Naukova dumka, 2003. – 316 p.

Novatskyi V. Dynamic problems of thermoelasticity / V. Novatskyi. –M.: Myr, 1975.– 256 p.

Vashizu K. Variational methods in elasticity and plasticity / K. Vashizu.– M.: Myr, 1987. – 542 p.

Dokhniak B. M. Calculation of prestressed elastomer constructions / B. M. Dokhniak, Yu. H. Kozub // Proc. 13 Int. Symposium “Problems of tire and rubber-cord composites”. – M: SRC NIISP. – October 14-18, 2002. – P. 119-123.

The finite element method in the computing complex "MIRELA+" / V. V. Kyrychevskyi, B. M. Dokhniak, Yu. H. Kozub, S. I. Homeniuk, R. V. Kyrychevskyi, S. N. Hrebeniuk.– K.: Naukova dumka, 2005. – 402p.

Khoroshun L. P. Determination of the axisymmetric stress-strain state of thermosensitive shells of revolution by the method of spline collocation / L. P. Khoroshun, S. V. Kozlov, I. Yu. Patlashenko // Applied mechanics. – 1988.– T.24, №6.–P. 56 – 63.

## Downloads

## Published

## Issue

## Section

## License

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.