Numerical behavior for quasi static thermoelasticity without positive definite elasticity

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Authors

  • J. Baldonedo CINTECX, Departamento de Ingeniería Mecánica, Universidade de Vigo, Spain
  • J.R. Fernández Departamento de Matemática Aplicada I, Universidade de Vigo, Spain
  • R. Quintanilla Departamento de Matemáticas, E.S.E.I.A.A.T.-U.P.C., Spain

Abstract

This paper presents a numerical study of the energetic behavior of some quasi-static thermoelastic problems in one- and two-dimensional settings. Firstly, we describe the two-dimensional thermoelastic problem decomposing the elastic tensor into two parts: the first one is positively defined for the first component of the displacement field, and the second one is negatively defined for the second component. The variational formulation is also derived. Restricting ourselves to the one-dimensional setting and assuming that the elastic coefficient is negative, we prove that the exponential energy decay follows if the coupling coefficient is smaller than the square root of the product between the heat capacity and the elastic coefficient. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. Some numerical simulations are performed: in a first onedimensional example, we show the decay of the discrete energy depending on the value of the coupling coefficient and the heat diffusion. Secondly, two dimensional studies are considered depending on the expression of the elastic tensors, including diagonal matrices with the same eigenvalue, diagonal matrices with different eigenvalues and full matrices.

Keywords:

thermoelasticity without positive definite elasticity, exponential energy decay, finite elements, discrete energy decay, numerical simulations

References


  1. N. Bazarra, J.R. Fernández, R. Quintanilla, On the approximate problem for the incremental thermoelasticity, Journal of Thermal Stresses, 44, 619–633, 2021.

  2. B.A. Boley, J.H. Weiner, Theory of Thermal Stresses, John Wiley, New York, 1960.

  3. D.E. Carlson, Linear thermoelasticity, Flugge Handbuch der Physik, Vol VI a/2 (ed. C. Truesdell), Springer, Berlin, 297–345, 1972.

  4. P.G. Ciarlet, Basic error estimates for elliptic problems, [in:] Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions [eds.], 17–351, 1993.

  5. W.A. Day, Justification of the uncoupled and quasi-static approximation in the problem of dynamic thermoelasticity, Archive for Rational Mechanics and Analysis, 77, 387–396, 1981.

  6. W.A. Day, Further justification of the uncoupled and quasi-approximations in thermoelasticity, Archive for Rational Mechanics and Analysis, 79, 85–95, 1982.

  7. A.H. England, A.E. Green, Steady-state thermoelasticity for initially stressed bodies, Philosophical Transactions of the Royal Society of London Series A, Mathematical and Physical Sciences, 253, 517–542, 1961.

  8. B.F. Esham, R.J. Weinacht, Singular perturbations and the coupled/quasi-static approximation in linear thermoelasticity, SIAM Journal on Mathematical Analysis, 25, 1521–1536, 1994.

  9. B.F. Esham, R.J. Weinacht, Limitation of the coupled/quasi-static approximation in multi-dimensional linear thermoelasticity, Applicable Analysis, 73, 72–87, 1999.

  10. M.D. Gilchrist, B. Rashid, J.G. Murphy, G. Saccomandi, Quasi-static deformations of biological soft tissue, Mathematics and Mechanics of Solids, 18, 6, 622–633, 2013.

  11. A.E. Green, Thermoelastic stresses in initially stressed bodies, Proceedings of the Royal Society of London A, 266, 119, 1962.

  12. D. Ieşan, Incremental equations in thermoelasticity, Journal of Thermal Stresses, 3, 41–56, 1980.

  13. D. Ieşan, A. Scalia, Thermoelastic Deformations, Kluwer Academic Publishers, Dordrecht, 1996.

  14. R.J. Knops, On the Quasi-Static Approximation to the Initial Traction Boundary Problem of Linear Elastodynamics, [in:] New Achievements in Continuum Mechanics and Thermodynamics, B.E. Abali et al. [eds.], Advanced Structured Materials, 108, Springer Nature, Switzerland, 2019.

  15. R.J. Knops, R. Quintanilla, On quasi-static approximations in linear thermoelastodynamics, Journal of Thermal Stresses, 41, 1432–1449, 2018.

  16. R.J. Knops, R. Quintanilla, On the quasi-static approximation in the initial boundary value problem of linearised elastodynamics, Journal of Engineering Mathematics, 126, 11, 2021.

  17. R.J. Knops, R. Quintanilla, On existence and uniqueness for a quasi-static thermoelastic problem, in preparation, 2023.

  18. R.J. Knops, E.W. Wilkes, Theory of elastic stability, Flugge Handbuch der Physik, Vol VI a/3, C. Truesdell [ed.], Springer Verlag, Berlin, 125–302, 1973.

  19. A. Logg, K.-A. Mardal, G. Wells, Automated Solution of Differential Equations by the Finite Element Method, Springer, 2012.

  20. C.B. Navarro, R. Quintanilla, On existence and uniqueness in incremental thermoelasticity, Zeitschrift füe Angewandte Mathematik und Physik, 35, 206–215, 1984.

  21. E. Pucci, G. Saccomandi, On a special class of nonlinear viscoelastic solids, Mathematics and Mechanics of Solids, 15, 803–811, 2010.

  22. E. Pucci, G. Saccomandi, On the nonlinear theory of viscoelasticity of differential type, Mathematics and Mechanics of Solids, 17, 624–630, 2012.

  23. G. Saccomandi, L. Vergori, Large time approximation for shearing motions, SIAM Journal of Applied Mathematics, 78, 1964–1983, 2016.