Abstract
This chapter constructs and analyzes a model for the dynamic behavior of nonlinear viscoelastic beam, which is acted upon by a horizontal traction, that may come in contact with a rigid or reactive foundation underneath it. We use a model, first developed and studied by D.Y. Gao, that allows for the buckling of the beam when the horizontal traction is sufficiently large. In contrast with the behavior of the standard Euler–Bernoulli linear beam, it can have three steady states, two of which are buckled. Moreover, the Gao beam can vibrate about such buckled states, which makes it important in engineering applications. We describe the contact process with either the normal compliance condition when the foundation is reactive, or with the Signorini condition when the foundation is perfectly rigid. We use various tools from the theory of pseudomonotone operators and variational inequalities to establish the existence and uniqueness of the weak or variational solution to the dynamic problem with the normal compliance contact condition. The main step is in the truncation of the nonlinear term and then establishing the necessary a priori estimates. Then, we show that when the viscosity of the material approaches zero and the stiffness of the foundation approaches infinity, making it perfectly rigid, the associated solutions of the problem with normal compliance converge to a solution of the elastic problem with the Signorini condition.
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References
Ahn, J., Kuttler, K.L., Shillor, M.: Dynamic contact of two Gao beams. Electron. J. Differ. Equ. 2012(194), 1–42 (2012)
Andrews, K.T., M’Bengue, M.F., Shillor, M.: Vibrations of a nonlinear dynamic beam between two stops. Discrete Continuous Dyn. Syst. Ser. B 12, 23–38 (2009)
Andrews, K.T., Dumont, Y., M’Bengue, M.F., Purcell, J., Shillor, M.: Analysis and simulations of a nonlinear dynamic beam. J. Appl. Math. Phys. (ZAMP) 63, 1005–1019 (2012)
Cai, K., Gao, D.Y., Qin, Q.H.: Post-buckling solutions of hyper-elastic beam by canonical dual finite element method. Math. Mech. Solids (MMS) (2013). doi:10.1177/1081286513482483
Gao, D.Y.: Nonlinear elastic beam theory with application in contact problems and variational approaches. Mech. Res. Commun. 23, 11–17 (1996)
Gao, D.Y.: Bi-complementarity and duality: a framework in nonlinear equilibria with applications to the contact problems of elastoplastic beam theory. J. Appl. Math. Anal. 221, 672–697 (1998)
Gao, D.Y.: Finite deformation beam models and triality theory in dynamical post-buckling analysis. Int. J. Non-Linear Mech. 35, 103–131 (2000)
Gao, D.Y.: Duality Principles in Nonconvex Systems: Theory, Methods and Applications. Kluwer Academic Publishers, Dordrecht (2000)
Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. Studies in Advanced Mathematics. American Mathematical Society/RI and International Press, Providence/Somerville, MA (2002)
Kuttler, K.L.: Time dependent implicit evolution equations. Nonlinear Anal. 10, 447–463 (1986)
Kuttler, K.L.: Dynamic friction contact problems for general normal and friction laws. Nonlinear Anal. 28, 559–575 (1997)
Kuttler, K.L., Shillor, M.: Set-valued pseudomonotone maps and degenerate evolution equations. Commun. Contemp. Math. 1, 87–123 (1999)
Kuttler, K.L., Shillor, M.: Dynamic contact with normal compliance wear and discontinuous friction coefficient. SIAM J. Math. Anal. 34, 1–27 (2002)
Kuttler, K.L., Purcell, J., Shillor, M.: Analysis and simulations of a contact problem for a nonlinear dynamic beam with a crack. Q. J. Mech. Appl. Math. 65, 1–25 (2012)
Kuttler, K.L., Li, J., Shillor, M.: Existence for dynamic contact of a stochastic viscoelastic Gao beam. Nonlinear Analysis Real World Appl. 22, 568–580 (2015)
M’Bengue, M.F.: Analysis of models for nonlinear dynamic beams with or without damage or frictionless contact. Oakland University Doctoral Dissertation (2008)
M’Bengue, M.F.: Analysis of a nonlinear beam in contact with a foundation. J. Appl. Math. Phys. (ZAMP) (2013). doi:10.1007/s00033-013-0348-7
Rochdi, M., Shillor, M., Sofonea, M.: Quasistatic viscoelastic contact with normal compliance and friction. J. Elast. 51, 105–126 (1998)
Russell, D.L., White, L.W.: A nonlinear elastic beam system with inelastic contact constraints. Appl. Math. Optim. 46, 291–312 (2002)
Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Lecture Notes in Physics. Springer, Berlin (2004)
Simon, J.: Compact sets in the space \(L^{p}\left (0,T;B\right )\). Ann. Mat. Pura. Appl. 146, 65–96 (1987)
Zeidler, E.: Nonlinear Functional Analysis and its Applications II/B. Springer, New York (1990)
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Andrews, K.T., Kuttler, K.L., Shillor, M. (2015). Dynamic Gao Beam in Contact with a Reactive or Rigid Foundation. In: Han, W., Migórski, S., Sofonea, M. (eds) Advances in Variational and Hemivariational Inequalities. Advances in Mechanics and Mathematics, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-319-14490-0_9
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