Advanced Search
Article Contents
Article Contents

Numerical approximation of von Kármán viscoelastic plates

  • * Corresponding author: Martin Kružík

    * Corresponding author: Martin Kružík 

This paper is dedicated to Alexander Mielke in the occasion of his 60th birthday

Abstract / Introduction Full Text(HTML) Figure(4) Related Papers Cited by
  • We consider metric gradient flows and their discretizations in time and space. We prove an abstract convergence result for time-space discretizations and identify their limits as curves of maximal slope. As an application, we consider a finite element approximation of a quasistatic evolution for viscoelastic von Kármán plates [44]. Computational experiments exploiting C1 finite elements are provided, too.

    Mathematics Subject Classification: Primary: 74D05, 74D10, 35A15, 35Q74, 49J45; Secondary: 49S05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  A rectangular mesh with Gauss integration points: midpoints are used for evaluation of Q1 elements (left) and four points for evaluation of Bogner-Fox-Schmit elements (right)

    Figure 2.  Time sequence of energy minimizers $ (u, v) $ in Benchmark Ⅰ. To emphasize the mesh deformation, the nodes displacement is magnified by a factor of 4

    Figure 3.  Time sequence of energy minimizers $ (u, v) $ in Benchmark Ⅱ. To emphasize the mesh deformation, the nodes displacement is magnified by a factor of 7

    Figure 4.  Example of $ C^1 $ approximation by the Bogner-Fox-Schmit rectangular elements: the function $ v_0 = (1-x_1^2)^2(1-x_2^2)^2 $ from Benchmark 1 is represented by its value (A), gradient components (C), (D) and the second mixed derivative (B) in all mesh nodes.

  • [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces (2nd ed), Elsevier, Amsterdam, 2003.
    [2] L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 19 (1995), 191–246.
    [3] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures Math. ETH Zürich, Birkhäuser, Basel, 2005.
    [4] I. Anjam and J. Valdman, Fast MATLAB assembly of FEM matrices in 2D and 3D: edge elements, Appl. Math. Comput., 267 (2015), 252-263.  doi: 10.1016/j.amc.2015.03.105.
    [5] S. S. Antman, Physically unacceptable viscous stresses, Z. Angew. Math. Phys., 49 (1998), 980-988.  doi: 10.1007/s000330050134.
    [6] S. S. Antman, Nonlinear Problems of Elasticity, Springer, New York, 2005.
    [7] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1977), 337-403.  doi: 10.1007/BF00279992.
    [8] J. M. BallJ. C. Currie and P. L. Olver, Null Lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal., 41 (1981), 135-174.  doi: 10.1016/0022-1236(81)90085-9.
    [9] R. C. Batra, Thermodynamics of non-simple elastic materials, J. Elasticity, 6 (1976), 451-456.  doi: 10.1007/BF00040904.
    [10] I. Bock, On Von Kármán equations for viscoelastic plates, J. Comp. Appl. Math., 63 (1995), 277-282.  doi: 10.1016/0377-0427(95)00082-8.
    [11] I. Bock and J. Jarušek, Solvability of dynamic contact problems for elastic von Kármán plates, SIAM J. Math. Anal., 41 (2009), 37-45.  doi: 10.1137/080712179.
    [12] I. BockJ. Jarušek and M. Šilhavý, On the solutions of a dynamic contact problem for a thermoelastic von Kármán plate, Nonlin. Anal.: Real World Appl., 32 (2016), 111-135.  doi: 10.1016/j.nonrwa.2016.04.004.
    [13] F. K. Bogner, R. L. Fox and L. A. Schmit, The generation of inter-element compatible stiffness and mass matrices by the use of interpolation formulas, Proceedings of the Conference on Matrix Methods in Structural Mechanics, (1965), 397–444.
    [14] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.
    [15] G. Capriz, Continua with latent microstructure, Arch. Ration. Mech. Anal., 90 (1985), 43-56.  doi: 10.1007/BF00281586.
    [16] V. Casarino and D. Percivale, A variational model for nonlinear elastic plates, J. Convex Anal., 3 (1996), 221-243. 
    [17] P. G. Ciarlet, Mathematical Elasticity, Vol. I: Three-dimensional Elasticity, North-Holland, Amsterdam, 1988.
    [18] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719208.
    [19] G. Dal Maso, An Introduction to $\Gamma$-convergence, Birkhäuser, Boston $\cdot$ Basel $\cdot$ Berlin, 1993. doi: 10.1007/978-1-4612-0327-8.
    [20] E. De GiorgiA. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve, Att. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 68 (1980), 180-187. 
    [21] J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Ration. Mech. Anal., 88 (1985), 95-133.  doi: 10.1007/BF00250907.
    [22] M. Friedrich and M. Kružík, On the passage from nonlinear to linearized viscoelasticity, SIAM J. Math. Anal., 50 (2018), 4426-4456.  doi: 10.1137/17M1131428.
    [23] M. Friedrich and M. Kružík, Derivation of von Kármán plate theory in the framework of three-dimensional viscoelasticity, preprint, arXiv: 1902.10037.
    [24] G. FrieseckeR. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (2002), 1461-1506.  doi: 10.1002/cpa.10048.
    [25] G. FrieseckeR. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence, Arch. Ration. Mech. Anal., 180 (2006), 183-236.  doi: 10.1007/s00205-005-0400-7.
    [26] P. Harasim and J. Valdman, Verification of functional a posteriori error estimates for an obstacle problem in 2D, Kybernetika, 50 (2014), 978-1002.  doi: 10.14736/kyb-2014-6-0978.
    [27] S. Krömer and J. Valdman, Global injectivity in second-gradient nonlinear elasticity and its approximation with penalty terms, Mathematics and Mechanics of Solids, 24 (2019), 3644-3673.  doi: 10.1177/1081286519851554.
    [28] M. Lecumberry and S. Müller, Stability of slender bodies under compression and validity of the von Kármán theory, Arch. Ration. Mech. Anal., 193 (2009), 255-310.  doi: 10.1007/s00205-009-0232-y.
    [29] H. Le Dret and A. Raoult, The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., 74 (1995), 549-578. 
    [30] H. Le Dret and A. Raoult, The membrane shell model in nonlinear elasticity: A variational asymptotic derivation,, J. Nonl. Sci., 6 (1996), 59–84. doi: 10.1007/BF02433810.
    [31] A. MielkeC. Ortner and Y. Şengül, An approach to nonlinear viscoelasticity via metric gradient flows, SIAM J. Math. Anal., 46 (2014), 1317-1347.  doi: 10.1137/130927632.
    [32] A. Mielke and T. Roubíček, Rate-Independent Systems - Theory and Application, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.
    [33] A. Mielke and T. Roubíček, Rate-independent elastoplasticity at finite strains and its numerical approximation, Math. Models & Methods in Appl. Sci., 26 (2016), 2203-2236.  doi: 10.1142/S0218202516500512.
    [34] C. Ortner, Two Variational Techniques for the Approximation of Curves of Maximal Slope, , Technical report NA05/10, Oxford University Computing Laboratory, Oxford, UK, 2005.
    [35] O. Pantz, On the justification of the nonlinear inextensional plate model, Arch. Ration. Mech. Anal., 167 (2003), 179-209.  doi: 10.1007/s00205-002-0238-1.
    [36] J. Y. Park and J. R. Kang, Uniform decay of solutions for von Karman equations of dynamic viscoelasticity with memory, Acta Appl. Math., 110 (2010), 1461-1474.  doi: 10.1007/s10440-009-9520-7.
    [37] P. Podio-Guidugli, Contact interactions, stress, and material symmetry for nonsimple elastic materials, Theor. Appl. Mech., 28/29 (2002), 261-276. 
    [38] T. Rahman and J. Valdman, Fast MATLAB assembly of FEM matrices in 2D and 3D: Nodal elements, Appl. Math. Comput., 219 (2013), 7151-7158.  doi: 10.1016/j.amc.2011.08.043.
    [39] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.  doi: 10.1002/cpa.20046.
    [40] S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. Ser. A, 31 (2011), 1427-1451.  doi: 10.3934/dcds.2011.31.1427.
    [41] R. A. Toupin, Elastic materials with couple stresses, Arch. Ration. Mech. Anal., 11 (1962), 385-414.  doi: 10.1007/BF00253945.
    [42] R. A. Toupin, Theory of elasticity with couple stress, Arch. Ration. Mech. Anal., 17 (1964), 85-112.  doi: 10.1007/BF00253050.
    [43] J. Valdman, MATLAB Implementation of C1 finite elements: Bogner-Fox-Schmit rectangle, In: Wyrzykowski R., Dongarra J., Deelman E., Karczewski K. (eds) Parallel Processing and Applied Mathematics. PPAM 2019, LNCS, Springer, 12044 (2020), 256–266. doi: 10.1007/978-3-030-43222-5_22.
    [44] T. von Kármán, Festigkeitsprobleme im maschinenbau in encyclopädie der mathematischen wissenschaften, Leipzig, 4 (1910), 311-385. 
  • 加载中



Article Metrics

HTML views(2039) PDF downloads(314) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint