History-Dependent Problems with Applications to Contact Models for Elastic Beams

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• 作者：Krzysztof Bartosz ; Piotr Kalita ; Stanisław Migórski…
• 关键词：Nonlinear inclusion ; Hemivariational inequality ; Euler–Bernoulli beam ; Finite element simulations ; 49J40 ; 74M15 ; 74K10 ; 74G25 ; 74S05
• 刊名：Applied Mathematics and Optimization
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：73
• 期：1
• 页码：71-98
• 全文大小：1,239 KB
• 参考文献：1.Barboteu, M., Bartosz, K., Kalita, P.: An analytical and numerical approach to a bilateral contact problem with nonmonotone friction. Int. J. Appl. Math. Comput. Sci. 23, 263–276 (2013)MATH MathSciNet CrossRef
  2.Barboteu, M., Sofonea, M., Tiba, D.: The control variational method for beams in contact with deformable obstacles. Z. Angew. Mat. Mech. (ZAMM) 92, 25–40 (2012)MATH MathSciNet CrossRef
  3.Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley Interscience, New York (1983)MATH
  4.Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston (2003)CrossRef
  5.Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston (2003)CrossRef
  6.Dumont, Y., Kuttler, K.L., Shillor, M.: Analysis and simulations of vibrations of a beam with a slider. J. Eng. Math. 47, 61–82 (2003)MATH MathSciNet CrossRef
  7.Eck, C., Jarušek, J., Krbeč, M.: Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics, vol. 270. Chapman/CRC Press, New York (2005)
  8.Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, vol. 30. American Mathematical Society, Providence (2002)
  9.Kulig, A., Migórski, S.: Solvability and continuous dependence results for second order nonlinear evolution inclusions with a Volterra-type operator. Nonlinear Anal. 75, 4729–4746 (2012)MATH MathSciNet CrossRef
  10.Kuttler, K.L., Park, A., Shillor, M., Zhang, W.: Unilateral dynamic contact of two beams. Math. Comput. Model. 34, 365–384 (2001)MATH MathSciNet CrossRef
  11.Kuttler, K.L., Shillor, M.: Vibrations of a beam between two stops. Dyn. Contin. Discret. Impuls. Syst. 8, 93–110 (2001)MATH MathSciNet
  12.Migórski, S., Ochal, A., Sofonea, M.: Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact. Math. Models Methods Appl. Sci. 18, 271–290 (2008)MATH MathSciNet CrossRef
  13.Migórski, S., Ochal, A., Sofonea, M.: History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics. Nonlinear Anal. Real World Appl. 12, 3384–3396 (2011)MATH MathSciNet CrossRef
  14.Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemiariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)
  15.Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker Inc., New York (1995)
  16.Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhäuser, Boston (1985)MATH CrossRef
  17.Panagiotopoulos, P.D.: Hemivariational Inequalities. Applications in Mechanics and Engineering. Springer-Verlag, Berlin (1993)MATH CrossRef
  18.Shillor, M., Sofonea, M., Touzani, R.: Quasistatic frictional contact and wear of a beam. Dyn. Contin. Discrete Impuls. Syst. 8, 201–218 (2001)MATH MathSciNet
  19.Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Springer, Berlin (2004)MATH CrossRef
  20.Sofonea, M., Matei, A.: History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22, 471–491 (2011)MATH MathSciNet CrossRef
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• 作者单位：Krzysztof Bartosz (1)
  Piotr Kalita (1)
  Stanisław Migórski (1)
  Anna Ochal (1)
  Mircea Sofonea (2)
  
  1. Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30348, Kraków, Poland
  2. Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860, Perpignan, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Calculus of Variations and Optimal Control
  Systems Theory and Control
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  
• 出版者：Springer New York
• ISSN：1432-0606

文摘

We prove an existence and uniqueness result for a class of subdifferential inclusions which involve a history-dependent operator. Then we specialize this result in the study of a class of history-dependent hemivariational inequalities. Problems of such kind arise in a large number of mathematical models which describe quasistatic processes of contact. To provide an example we consider an elastic beam in contact with a reactive obstacle. The contact is modeled with a new and nonstandard condition which involves both the subdifferential of a nonconvex and nonsmooth function and a Volterra-type integral term. We derive a variational formulation of the problem which is in the form of a history-dependent hemivariational inequality for the displacement field. Then, we use our abstract result to prove its unique weak solvability. Finally, we consider a numerical approximation of the model, solve effectively the approximate problems and provide numerical simulations. Keywords Nonlinear inclusion Hemivariational inequality Euler–Bernoulli beam Finite element simulations