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January  2021, 14(1): 299-319. doi: 10.3934/dcdss.2020322

Numerical approximation of von Kármán viscoelastic plates

Manuel Friedrich 1, , Martin Kružík 2,, and  Jan Valdman 3,

1. 

Institute for Computational and Applied Mathematics, University of Münster, Einsteinstr. 62, D-48149 Münster, Germany
2. 

Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vodárenskou vĕží 4, CZ-182 00 Praha 8, Czechia, Faculty of Civil Engineering, Czech Technical University, Thákurova 7, CZ-166 29 Praha 6, Czechia
3. 

Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vodárenskou vĕží 4, CZ-182 00 Praha 8, Czechia, Institute of Mathematics, University of South Bohemia, Branišovská 1760, CZ-370 05 České Budĕjovice, Czechia

* Corresponding author: Martin Kružík

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

Received  March 2019 Revised  October 2019 Published  April 2020

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.

Keywords: Viscoelasticity, metric gradient flows, $ \Gamma $-convergence, dissipative distance, minimizing movements, numerical approximation.
Mathematics Subject Classification: Primary: 74D05, 74D10, 35A15, 35Q74, 49J45; Secondary: 49S05.
Citation: Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 299-319. doi: 10.3934/dcdss.2020322
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces (2nd ed), Elsevier, Amsterdam, 2003.  Google Scholar

[2]

L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 19 (1995), 191–246.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

S. S. Antman, Physically unacceptable viscous stresses, Z. Angew. Math. Phys., 49 (1998), 980-988.  doi: 10.1007/s000330050134.  Google Scholar

[6]

S. S. Antman, Nonlinear Problems of Elasticity, Springer, New York, 2005.  Google Scholar

[7]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1977), 337-403.  doi: 10.1007/BF00279992.  Google Scholar

[8]

J. M. Ball, J. 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.  Google Scholar

[9]

R. C. Batra, Thermodynamics of non-simple elastic materials, J. Elasticity, 6 (1976), 451-456.  doi: 10.1007/BF00040904.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

I. Bock, J. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[15]

G. Capriz, Continua with latent microstructure, Arch. Ration. Mech. Anal., 90 (1985), 43-56.  doi: 10.1007/BF00281586.  Google Scholar

[16]

V. Casarino and D. Percivale, A variational model for nonlinear elastic plates, J. Convex Anal., 3 (1996), 221-243.   Google Scholar

[17]

P. G. Ciarlet, Mathematical Elasticity, Vol. I: Three-dimensional Elasticity, North-Holland, Amsterdam, 1988.  Google Scholar

[18]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719208.  Google Scholar

[19]

G. Dal Maso, An Introduction to $\Gamma$-convergence, Birkhäuser, Boston $\cdot$ Basel $\cdot$ Berlin, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[20]

E. De Giorgi, A. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

M. Friedrich and M. Kružík, Derivation of von Kármán plate theory in the framework of three-dimensional viscoelasticity, preprint, arXiv: 1902.10037. Google Scholar

[24]

G. Friesecke, R. 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.  Google Scholar

[25]

G. Friesecke, R. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[31]

A. Mielke, C. 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.  Google Scholar

[32]

A. Mielke and T. Roubíček, Rate-Independent Systems - Theory and Application, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

P. Podio-Guidugli, Contact interactions, stress, and material symmetry for nonsimple elastic materials, Theor. Appl. Mech., 28/29 (2002), 261-276.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

R. A. Toupin, Elastic materials with couple stresses, Arch. Ration. Mech. Anal., 11 (1962), 385-414.  doi: 10.1007/BF00253945.  Google Scholar

[42]

R. A. Toupin, Theory of elasticity with couple stress, Arch. Ration. Mech. Anal., 17 (1964), 85-112.  doi: 10.1007/BF00253050.  Google Scholar

[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.  Google Scholar

[44]

T. von Kármán, Festigkeitsprobleme im maschinenbau in encyclopädie der mathematischen wissenschaften, Leipzig, 4 (1910), 311-385.   Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces (2nd ed), Elsevier, Amsterdam, 2003.  Google Scholar

[2]

L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 19 (1995), 191–246.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

S. S. Antman, Physically unacceptable viscous stresses, Z. Angew. Math. Phys., 49 (1998), 980-988.  doi: 10.1007/s000330050134.  Google Scholar

[6]

S. S. Antman, Nonlinear Problems of Elasticity, Springer, New York, 2005.  Google Scholar

[7]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1977), 337-403.  doi: 10.1007/BF00279992.  Google Scholar

[8]

J. M. Ball, J. 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.  Google Scholar

[9]

R. C. Batra, Thermodynamics of non-simple elastic materials, J. Elasticity, 6 (1976), 451-456.  doi: 10.1007/BF00040904.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

I. Bock, J. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[15]

G. Capriz, Continua with latent microstructure, Arch. Ration. Mech. Anal., 90 (1985), 43-56.  doi: 10.1007/BF00281586.  Google Scholar

[16]

V. Casarino and D. Percivale, A variational model for nonlinear elastic plates, J. Convex Anal., 3 (1996), 221-243.   Google Scholar

[17]

P. G. Ciarlet, Mathematical Elasticity, Vol. I: Three-dimensional Elasticity, North-Holland, Amsterdam, 1988.  Google Scholar

[18]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719208.  Google Scholar

[19]

G. Dal Maso, An Introduction to $\Gamma$-convergence, Birkhäuser, Boston $\cdot$ Basel $\cdot$ Berlin, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[20]

E. De Giorgi, A. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

M. Friedrich and M. Kružík, Derivation of von Kármán plate theory in the framework of three-dimensional viscoelasticity, preprint, arXiv: 1902.10037. Google Scholar

[24]

G. Friesecke, R. 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.  Google Scholar

[25]

G. Friesecke, R. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[31]

A. Mielke, C. 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.  Google Scholar

[32]

A. Mielke and T. Roubíček, Rate-Independent Systems - Theory and Application, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

P. Podio-Guidugli, Contact interactions, stress, and material symmetry for nonsimple elastic materials, Theor. Appl. Mech., 28/29 (2002), 261-276.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

R. A. Toupin, Elastic materials with couple stresses, Arch. Ration. Mech. Anal., 11 (1962), 385-414.  doi: 10.1007/BF00253945.  Google Scholar

[42]

R. A. Toupin, Theory of elasticity with couple stress, Arch. Ration. Mech. Anal., 17 (1964), 85-112.  doi: 10.1007/BF00253050.  Google Scholar

[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.  Google Scholar

[44]

T. von Kármán, Festigkeitsprobleme im maschinenbau in encyclopädie der mathematischen wissenschaften, Leipzig, 4 (1910), 311-385.   Google Scholar

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)
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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
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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
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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.
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