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Existence for dynamic contact of a stochastic viscoelastic Gao beam. (English) Zbl 1326.74099
Summary: This work presents and analyzes a model for the vibrations of a viscoelastic Gao Beam, which may come in contact with a deformable random foundation and allows for stochastic inputs. The body force involves a stochastic integral that includes Brownian motion. In addition, the gap between the beam and the foundation is a stochastic process, which is one of the novelties in the paper, and contact is described with the normal compliance condition. The existence and uniqueness of strong solutions to the model is established and it is shown that the solutions are adapted to the filtration determined by a given Wiener process for the stochastic force noise term.

##### MSC:
 74M15 Contact in solid mechanics 35Q74 PDEs in connection with mechanics of deformable solids 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 35R60 PDEs with randomness, stochastic partial differential equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 74H20 Existence of solutions of dynamical problems in solid mechanics 35D35 Strong solutions to PDEs
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##### References:
 [1] 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) · Zbl 0971.74059 [2] D.Y. Gao, D.L. Russell, A finite element approach to optimal control of a “smart” beam, in: P.K.K. Lee, L.G. Tham and Y.K. Cheung (Eds.) Int. Conf. Computational Methods in Structural and Geotechnical Engineering, Hong Kong, December 12-14, 1994, pp. 135-140. [3] Gao, D. Y., Duality principles in nonconvex systems: theory, methods and applications, (2000), Kluwer Academic Publishers Dordrecht [4] Ahn, J.; Kuttler, K. L.; Shillor, M., Dynamic contact of two Gao beams, Electron. J. Differential Equations, 2012, 194, 1-42, (2012) · Zbl 1302.74116 [5] Andrews, K. T.; M’Bengue, M. F.; Shillor, M., Vibrations of a nonlinear dynamic beam between two stops, Discrete Contin. Dyn. Syst., 12, 1, 23-38, (2009) · Zbl 1167.74019 [6] 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., 63, 6, 1005-1019, (2012) · Zbl 1261.35093 [7] Kuttler, K. L.; Purcell, J.; Shillor, M., Analysis and simulations of a contact problem for a nonlinear dynamic beam with a crack, Quart. J. Mech. Appl. Math., 65, 1, 1-25, (2012), First published online November 2, 2011, http://dx.doi.org/10.1093/qjmam/hbr018 · Zbl 1248.74032 [8] M’Bengue, M’Bagne F.; Shillor, Meir, Regularity result for the problem of vibrations of a nonlinear beam, Electron. J. Differential Equations, 2008, 27, 1-12, (2008) · Zbl 1137.74027 [9] K.T. Andrews, K.L. Kuttler, M.F. M’Bengue, M. Shillor, Nonlinear dynamic Gao beam in contact with a rigid or reactive foundation, Preprint 2014. [10] Prévôt, C.; Röckner, A concise course on stochastic partial differential equations, (Lecture notes in Mathematics, (2007), Springer) · Zbl 1123.60001 [11] Kuttler, K.; Li, J., Implicit stochastic equations, J. Differential Equations, 257, 816-842, (2014) · Zbl 1296.60168 [12] Krylov, N. V.; Rozowskii, B. L., Stochastic evolution equations, (Current Problems in Mathematics, vol. 14, (1979), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchin. i Tekhn. Informatsii Moscow), 71-147, 256. MR MR570795 (81m:60116) [13] Krylov, N. V., On kolmogorov’s equations for finite dimensional diffusions, (Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998), Lecture Notes in Math, vol. 1715, (1999), Springer Berlin), 1-63, MR MR1731794 (2000k:60155) · Zbl 0943.60070 [14] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge · Zbl 0761.60052 [15] Karatzas; Shreve, Brownian motion and stochastic calculus, (1991), Springer Verlag · Zbl 0734.60060 [16] Kallenberg, O., Foundations of modern probability, (2003), Springer [17] Simon, J., Compact sets in the space $$L^p(0, T; B)$$, Ann. Mat. Pura. Appl., 146, 65-96, (1987) · Zbl 0629.46031 [18] Lions, J. L., Quelques methods de resolution des problemes aux limites non lineaires, (1969), Dunod Paris · Zbl 0189.40603 [19] Kikuchi, N.; Oden, J. T., Contact problems in elasticity: A study of variational inequalities and finite element methods, (1988), SIAM Philadelphia · Zbl 0685.73002 [20] Shillor, M.; Sofonea, M.; Telega, J. J., Models and analysis of quasistatic contact, (Lecture Notes in Physics, (2004), Springer Berlin) · Zbl 1180.74046 [21] Bensoussan, A.; Temam, R., Equations stochastiques de type Navier-Stokes, J. Funct. Anal., 13, 195-222, (1973) · Zbl 0265.60094
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