Modified formulation of the iterative algorithm for solving linear viscoelasticity problems based on separation of time and spice variables

M. S. Pavlov ; A. A. Svetashkov ; N. A. Kupriyanov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 61 (2019), pp. 82-94

Résumé

Designing of the structures made of polymeric and composite viscoelastic materials requires development of the efficient and cost-effective methods for calculating stress-strain state. This paper proposes an iterative algorithm for solving such problems. The advantages of the algorithm over existing methods are the following: firstly, there is a possibility to parallelize the calculations of spatial and time components of the stress-strain state; secondly, this algorithm eliminates the need to integrate the history of stress and displacement variation in time. Moreover, the formulated iterative algorithm allows one to obtain the parameters of the stress-strain state of a viscoelastic solid without using integral operator inverse to the relaxation operator. The proposed method involves the following concept. The integral operators of the shear and volume relaxation are replaced by some values of the elastic shear and volume moduli. The identity of the obtained elastic problem with initially stated viscoelastic problem is ensured by supplementing right-hand sides of the equilibrium equations and boundary conditions with the corresponding residuals. In addition, each residual involves the result of viscoelastic operator effect on the required parameters, and, therefore, it cannot be found directly. The numerical implementation assumes the iteration process to be built, in which the residuals on the current step are calculated using the solutions obtained on the previous one. The paper describes the formulated iterative algorithm, as well as its application in conjunction with commercial or free computer software employed for a finite element analysis. The paper also includes an example of the model problem solution.

Détail

Keywords: linear viscoelasticity, integral operators, auxiliary constitutive equations, iterative algorithm.
Mots-clés : convergence

Bibliographie

[1] R. Lakes, Viscoelastic Materials, Cambridge University Press, N.Y., 2009, 461 pp. | DOI

[2] M. Pavlov et. al., “Mathematical model of composite fibre-glass aramide-wired cord rheological properties”, 13th International Conference of Students and Young Scientists on Prospect of Fundamental Sciences Development, PFSD 2016 (Tomsk, Russian Federation, 26–29 April 2016), AIP Conference Proceeding, 1772, American Institute of Physics, 2016, 6 pp. | DOI

[3] S. Doubal, J. Doubal, Theory of Viscoelasticity Handbook, Delter, 2014, 81 pp.

[4] A. R. Rzhanitsyn, Creep theory, Izdatel'stvo literatury po stroitel'stvu, M., 1968

[5] J. Cl. Maxwell, “On the Dynamical Theory of Gases”, Philosophical Transactions, 157 (1867), 49–88 | DOI

[6] L. Boltzman, “Zur Theorie der Elastischen Nachwirkung”, Wiener Berichte, 70 (1874), 275–306 | DOI

[7] V. Volterra, Lecons sur les fonctions de lignes, Gautierr Villars, Paris, 1912, 230 pp.

[8] V. Volterra, Theory of Functionals and of Integral and integrodifferential Equations, Blackie Son Limited, London–Glasgow, 1930, 226 pp. | Zbl

[9] A. A. Adamov, V. P. Matveenko, N. A. Trufanov, I. N. Shardakov, Applied viscoelasticity methods, UrO RAN, Ekaterinburg, 2003

[10] A. A. Il'yushin, B. E. Pobedrya, Foundations of the mathematical theory of thermal viscoelasticity, Nauka, M., 1970 | MR

[11] J. N. Reddy, An Introduction to Continuum Mechanics, Cambridge University Press, N.Y., 2008, 449 pp. | DOI | MR

[12] A. C. Pipkin, Lectures of Viscoelasticity Theory, Springer Verlag, N.Y., 1986, 181 pp. | DOI

[13] W. Flugge, Viscoelasticity, Blaisdell Press, N.Y., 1967, 187 pp. | MR

[14] R. M. Cristensen, Theory of Viscoelasticity: An Introduction, Academic, N.Y., 1980, 364 pp.

[15] M. A. Koltunov, V. P. Mayboroda, V. G. Zubchaninov, Strength calculations for the products made of polymeric materials, Vysshaya shkola, M., 1983

[16] R. A. Schapery, “On the characterization of nonlinear viscoelastic materials”, Polymer Engineering and Science, 9:4 (1969), 295–310 | DOI

[17] Y. Tang, T. Li, X. Ma, “Creep and recovery behavior analysis of space mesh structures”, Acta Astronautica, 2016, no. 128, 455–463 | DOI

[18] K. Kwok, S. Pellegrino, “Micromechanics Models for Viscoelastic Plain-Weave Composite Tape Springs”, AIAA Journal, 55:1 (2017), 309–321 | DOI

[19] R. L. Taylor, K. S. Pister, G. L. Goudreas, “Thermochemical analysis of viscoelastic solids”, Int. J. for Numerical Methods in Engineering, 2 (1970), 45–59 | DOI | Zbl

[20] J. C. Simo, “On fully three-dimensional finite strain viscoelastic damage model: Formulation and computational aspects”, Computer. Methods In Applied Mechanics and Engineering, 60:2 (1987), 153–173 | DOI | Zbl

[21] R. A. Schapery, “Analysis of viscoelastic composite materials”, J. Composite Materials, 1:3 (1967), 228–267 | DOI

[22] V. I. Malyy, N. A. Trufanov, “Methods of quasi-constant operators in the theory of viscoelasticity of the anisotropic materials”, Izvestiya Akademii Nauk SSSR. Mekhanika tverdogo tela, 1987, no. 6, 148–154

[23] R. Haj-Ali, A. Muliana, “Numerical finite element formulation of the schapery non-linear viscoelastic material model”, Int. J. for Numerical Methods in Engineering, 2004, no. 59, 25–45 | DOI | Zbl

[24] I. Tsukrov et al, “Numerical modeling of nonlinear elastic components of mooring systems”, IEEE J. Oceanic Engineering, 30:1 (2005), 37–46 | DOI

[25] R. G. Kulikov, N. A. Trufanov, “Iterative method for solving quasistatic nonlinear viscoelastic problems”, Computating Continuum Mechanics, 2:3 (2009), 44–56

[26] S. P. Barba, “Method of elastic solutions to the problem of transient creep”, Uchenye zapiski TSAGI, 21:5 (1990), 112–123

[27] S. M. Pavlov, A. A. Svetashkov, “Iteration method for solving linear viscoelasticity problems”, Russian Physics Journal, 36:4 (1993), 129–136 | DOI | MR

[28] A. Svetashkov, N. Kupriyanov, K. Manabaev, “Modification of the iterative method for solving linear viscoelasticity boundary value problems and its implementation by finite element method”, Acta Mechanica, 229:6 (2018), 2539–2559 | DOI | MR | Zbl

[29] A. A. Svetashkov, Applied problems of mechanics of viscoelastic materials, Izdatel'stvo Tomskogo politekhnicheskogo universiteta, Tomsk, 2012

[30] B. E. Pobedrya, Numerical methods in the theory of elasticity and plasticity, Izdatel'stvo MGU, M., 1995

[31] Yu. N. Rabotnov, Elements of hereditary mechanics of solids, Nauka, M., 1977