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

Numerical approximation of von Kármán viscoelastic plates

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  January 2021 Early access  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.

Citation: Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete and 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.

[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. 

show all references

References:
[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. 

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.
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