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Dynamic frictional thermoviscoelastic Gao beams

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Abstract

In this work, we provide mathematical and numerical analyses for a thermoviscoelastic nonlinear beam model with Coulomb friction dry law. Since the dynamic frictional conditions are nonsmooth, a regularization technique with smoothing parameters is applied to approximate a nonlinear variational formulation. We prove the existence of weak solutions satisfying the regularized variational formulation based on a priori estimates and results for a pseudomonotone operator. Then, we pass to limits, as the smoothing parameters tend to be zero, in order to show convergence results for the regularized formulation. We propose the fully discrete numerical schemes in which a guarded Newton method is used to compute fully discrete numerical approximations of a nonlinear system at each time step. Numerical experiments are performed with selected data to present numerical simulations as well.

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References

  1. Ahn, J.: A viscoelastic Timoshenko beam with Coulomb law of friction. Appl. Math. Comput. 218, 7078–7099 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Ahn, J.: Damageable elastic buckling bodies with a crack. ZAMP 71(66), 1–27 (2020)

    MathSciNet  MATH  Google Scholar 

  3. Ahn, J.: Nonlinear thermoviscoelastic Timoshenko beams with dynamic frictional contact. Appl. Anal. (2021)

  4. Ahn, J., Kuttler, K.L., Shillor, M.: Dynamic contact of two Gao beams. Electron. J. Differ. Equ. 2012, 1–42 (2012)

    Article  MathSciNet  Google Scholar 

  5. Ahn, J., Lee, S., Park, E.-J.: \({C}^0\) interior penalty methods for a dynamic nonlinear beam model. Appl. Math. Comput. 339, 685–700 (2018)

    MathSciNet  MATH  Google Scholar 

  6. Ali, E.J., Gao, D.Y.: On SDP method for solving canonical dual problem in post buckling of large deformed elastic beam. Commun. Math. Sci. 16(5), 1225–1240 (2018)

    Article  MathSciNet  Google Scholar 

  7. Andrews, K.T., M’Bengue, M.F., Shillor, M.: Vibrations of a nonlinear dynamic beam between two stops. Discrete Contin. Dyn. Syst. Ser. B 12(1), 23–38 (2009)

  8. Andrews, K.T., Shillor, M., Wright, S.: On the dynamic vibration of an elasticbeam in frictional contat with a rigid obdtacle. J. Elast. 42, 1–30 (1996)

    Article  Google Scholar 

  9. Andrews, K.T., Dumont, Y., M’Bemgue, M.F., Purcell, J., Shillor, M.: Analysis and simulations of a nonlinear elastic dynamic Beam. ZAMP 63, 1005–1019 (2012)

  10. Bajkowski, J., Kuttler, K., Shillor, M.: A thermoviscoelastic beam model for brakes. Eur. J. Appl. Math. 15(2), 181–202 (2004)

    Article  MathSciNet  Google Scholar 

  11. Bernardi, C., Copetti, M.I.M.: Finite element discretization of a thermoelastic beam. SeMA 64, 41–49 (2014)

    Article  MathSciNet  Google Scholar 

  12. Bernardi, C., Copetti, M.I.M.: Discretization of a nonlinear dynamic thermoviscoelastic Timoshenko beam model. ZAMM 97(5), 532–549 (2017)

    Article  MathSciNet  Google Scholar 

  13. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley Interscience, New York (1983)

    MATH  Google Scholar 

  14. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Provvidence, RI (1991)

    Google Scholar 

  15. Gao, D.Y.: Nonlinear elastic beam theory with application in contact problems and variational approaches. Mech. Res. Commun. 23(1), 11–17 (1996)

    Article  MathSciNet  Google Scholar 

  16. Gao, D.Y.: Finite deformation beam models and triality theory in dynamical post-buckling analysis. Int. J. Non-linear Mech. 35, 103–131 (2000)

    Article  MathSciNet  Google Scholar 

  17. Gu, R.J., Kuttler, K., Shillor, M.: Frictional wear if a thermoelastic beam. J. Math. Anal. Appl. 242, 212–236 (2000)

    Article  MathSciNet  Google Scholar 

  18. Huang, H., Han, W., Zhou, J.: The regularization method for an obstacle problem. Numer. Math. 69, 155–166 (1994)

    Article  MathSciNet  Google Scholar 

  19. Kuttler, K.: Modern Analysis. CRC Press, Boca Raton (1988)

    MATH  Google Scholar 

  20. Kuttler, K.: Time dependent implicit evolution equations. Nonlinear Anal. Theory Methods Appl. 10(5), 447–463 (1986)

    Article  MathSciNet  Google Scholar 

  21. Kuttler, K.L., Li, J., Shillor, M.: Existence for dynamic contact of a Stocastic viscoelastic Gao beam. Nonlinear Anal. Real World Appl. 22, 568–580 (2015)

    Article  MathSciNet  Google Scholar 

  22. Kuttler, K., Renard, Y., Shillor, M.: Models and simulations of dynamic frictional contact of a beam. Comput. Methods Appl. Mech. Eng. 177, 259–272 (1999)

    Article  MathSciNet  Google Scholar 

  23. Kuttler, K., Shillor, M.: Set-valued pseudomonotone maps and degenerate evolution equations. Commun. Contemp. Math. 1(1), 87–123 (1999)

    Article  MathSciNet  Google Scholar 

  24. Kuttler, K., Shillor, M.: Dynamic bilateral contact with discontinuous friction coefficient. Nonlinear Anal. 45, 309–327 (2001)

    Article  MathSciNet  Google Scholar 

  25. Kuttler, K., Shillor, M.: Dynamic contact with normal compliance wear and discontinuous friction coefficient. SIAM J. Math. Anal. 34(1), 1–27 (2002)

    Article  MathSciNet  Google Scholar 

  26. Kuttler, K., Shillor, M.: Dynamic contact with Signorini’s condition and slip rate dependent friction. Electron. J. Differ. Equ. 2004(83), 1–21 (2004)

  27. Lions, J.L.: Quelques Methods de Resolution des Problémes aux Limites Non Linéaires. Springer, Dunod (1969)

    MATH  Google Scholar 

  28. Machalová, J., Netuka, H.: Solution of Contact problems for nonlinear Gao beams and obstacle. J. Appl. Math. 2015: Article ID 420649 (2015)

  29. Machalová, J., Netuka, H.: Control variational method approach to bending and contact problems for Gao beams. Appl. Math. 62(6), 661–667 (2017)

    Article  MathSciNet  Google Scholar 

  30. Renardy, M., Rogers, R.C.: An introduction to partial differential equations. In: Texts in Applied Mathematics, 13 (2 edn). Springer, Berlin (2004)

  31. Simon, J.: Compact sets in the space \({L}^p(0,{T};{B})\). Ann. Mat. Pura. Appl. 146, 65–96 (1987)

    Article  MathSciNet  Google Scholar 

  32. Washizu, K.: Variational Methods in Elasticity and Plasticity. Pergamon Press, New York (1968)

    MATH  Google Scholar 

  33. Zeidler, E.: Nonlinear Functional Analysis and its Applications II/B. Springer, Berlin (1989)

    Google Scholar 

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  1. Department of mathematics and statistics, Arkansas State University, Jonesboro, AR, 72464, USA

    Jeongho Ahn

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  1. Jeongho Ahn
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Ahn, J. Dynamic frictional thermoviscoelastic Gao beams. Z. Angew. Math. Phys. 72, 194 (2021). https://doi.org/10.1007/s00033-021-01632-5

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