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July  2019, 18(4): 1783-1826. doi: 10.3934/cpaa.2019084

Existence and regularity of solutions for an evolution model of perfectly plastic plates

P. Gidoni 1, , G. B. Maggiani 2, and  R. Scala 3,

1. 

Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy
2. 

Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy
3. 

Dipartimento di Matematica Guido Castelnuovo, Università degli Studi di Roma "La Sapienza", Piazzale Aldo Moro 5, 00185, Roma, Italy

Received  June 2018 Revised  November 2018 Published  January 2019

We continue the study of a dynamic evolution model for perfectly plastic plates, recently derived in [19] from three-dimensional Prandtl-Reuss plasticity. We extend the previous existence result by introducing non-zero external forces in the model, and we discuss the regularity of the solutions thus obtained. In particular, we show that the first derivatives with respect to space of the stress tensor are locally square integrable.

Keywords: Dynamic evolution, perfect plasticity, Prandtl-Reuss plasticity, thin plates, functions of bounded deformation, functions of bounded hessian.
Mathematics Subject Classification: 74C05, 74K20, 49J45.
Citation: P. Gidoni, G. B. Maggiani, R. Scala. Existence and regularity of solutions for an evolution model of perfectly plastic plates. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1783-1826. doi: 10.3934/cpaa.2019084
References:
[1]

H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984.  Google Scholar

[2]

J. F. Babadjian and M. G. Mora, Stress regularity in quasi-static perfect plasticity with a pressure dependent yield criterion, Journal of Differential Equations, 264 (2018), 5109-5151.  doi: 10.1016/j.jde.2017.12.034.  Google Scholar

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.  Google Scholar

[4]

A. Bensoussan and J. Frehse, Asymptotic behaviour of the time-dependent Norton Hoff law in plasticity theory and $H^1$ regularity, Comment. Math. Univ. Carolinae, 37 (1996), 285-304.   Google Scholar

[5]

H. Brezis, Opérateurs maximaux monotones et semi-groupes de constractions dans les espaces de Hilbert, American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[6]

P. Ciarlet, Mathematical Elasticity. Vol II. Theory of Plates, Studies in Mathematics and its Applications, 27. North-Holland Publishing Co., Amsterdam, 1997.  Google Scholar

[7]

G. Dal Maso, A. DeSimone and M. G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal., 180 (2006), 237-291.  doi: 10.1007/s00205-005-0407-0.  Google Scholar

[8]

E. Davoli and M. G. Mora, A quasistatic evolution model for perfectly plastic plates derived by $\Gamma$-convergence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 615-660.  doi: 10.1016/j.anihpc.2012.11.001.  Google Scholar

[9]

E. Davoli and M. G. Mora, Stress regularity for a new quasistatic evolution model of perfectly plastic plates, Calc. Var. Partial Differential Equations, 54 (2015), 2581-2614.  doi: 10.1007/s00526-015-0876-4.  Google Scholar

[10]

F. Demengel, Fonctions à hessien borné, Ann. Inst. Fourier (Grénoble), 34 (1984), 155-190.  Google Scholar

[11]

A. Demyanov, Regularity of stresses in Prandtl-Reuss plasticity, Calc. Var. Partial Differential Equations, 34 (2009), 23-72.  doi: 10.1007/s00526-008-0174-5.  Google Scholar

[12]

A. Demyanov, Quasistatic evolution in the theory of perfectly elasto-plastic plates. Ⅰ. Existence of a weak solution, Math. Models Methods Appl. Sci., 19 (2009), 229-256.  doi: 10.1142/S0218202509003413.  Google Scholar

[13]

A. Demyanov, Quasistatic evolution in the theory of perfectly elasto-plastic plates. Ⅱ. Regularity of bending moments, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2137-2163.  doi: 10.1016/j.anihpc.2009.01.006.  Google Scholar

[14]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics Appl. Math., vol. 28, SIAM, Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.  Google Scholar

[15]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[16]

R. V. Kohn and R. Temam, Dual spaces of stresses and strains, with application to Hencky plasticity, Appl. Math. Optim., 10 (1983), 1-35.  doi: 10.1007/BF01448377.  Google Scholar

[17]

J.Lubliner, Plasticity Theory, Macmillan Publishing Company, New York, 1990. Google Scholar

[18]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.  doi: 10.1007/s00526-004-0267-8.  Google Scholar

[19]

G. B. Maggiani and M. G. Mora, A dynamic evolution model for perfectly plastic plates, Math. Models Methods Appl. Sci., 26 (2016), 1825-1864.  doi: 10.1142/S0218202516500469.  Google Scholar

[20]

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

[21]

R. T. Rockafellar, Convex Integral Functionals and Duality, in Contributions to Nonlinear Functional Analysis, Academic Press, (1971), 215-236.  Google Scholar

[22]

P. M. Suquet, Sur le équations de la plasticité: existence et regularité des solutions, J. Mécanique, 20 (1981), 3-39.  Google Scholar

[23]

R. Temam, Mathematical Problems in Plasticity, Gauthier-Villars, Paris, 1985.  Google Scholar

show all references

References:
[1]

H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984.  Google Scholar

[2]

J. F. Babadjian and M. G. Mora, Stress regularity in quasi-static perfect plasticity with a pressure dependent yield criterion, Journal of Differential Equations, 264 (2018), 5109-5151.  doi: 10.1016/j.jde.2017.12.034.  Google Scholar

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.  Google Scholar

[4]

A. Bensoussan and J. Frehse, Asymptotic behaviour of the time-dependent Norton Hoff law in plasticity theory and $H^1$ regularity, Comment. Math. Univ. Carolinae, 37 (1996), 285-304.   Google Scholar

[5]

H. Brezis, Opérateurs maximaux monotones et semi-groupes de constractions dans les espaces de Hilbert, American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[6]

P. Ciarlet, Mathematical Elasticity. Vol II. Theory of Plates, Studies in Mathematics and its Applications, 27. North-Holland Publishing Co., Amsterdam, 1997.  Google Scholar

[7]

G. Dal Maso, A. DeSimone and M. G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal., 180 (2006), 237-291.  doi: 10.1007/s00205-005-0407-0.  Google Scholar

[8]

E. Davoli and M. G. Mora, A quasistatic evolution model for perfectly plastic plates derived by $\Gamma$-convergence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 615-660.  doi: 10.1016/j.anihpc.2012.11.001.  Google Scholar

[9]

E. Davoli and M. G. Mora, Stress regularity for a new quasistatic evolution model of perfectly plastic plates, Calc. Var. Partial Differential Equations, 54 (2015), 2581-2614.  doi: 10.1007/s00526-015-0876-4.  Google Scholar

[10]

F. Demengel, Fonctions à hessien borné, Ann. Inst. Fourier (Grénoble), 34 (1984), 155-190.  Google Scholar

[11]

A. Demyanov, Regularity of stresses in Prandtl-Reuss plasticity, Calc. Var. Partial Differential Equations, 34 (2009), 23-72.  doi: 10.1007/s00526-008-0174-5.  Google Scholar

[12]

A. Demyanov, Quasistatic evolution in the theory of perfectly elasto-plastic plates. Ⅰ. Existence of a weak solution, Math. Models Methods Appl. Sci., 19 (2009), 229-256.  doi: 10.1142/S0218202509003413.  Google Scholar

[13]

A. Demyanov, Quasistatic evolution in the theory of perfectly elasto-plastic plates. Ⅱ. Regularity of bending moments, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2137-2163.  doi: 10.1016/j.anihpc.2009.01.006.  Google Scholar

[14]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics Appl. Math., vol. 28, SIAM, Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.  Google Scholar

[15]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[16]

R. V. Kohn and R. Temam, Dual spaces of stresses and strains, with application to Hencky plasticity, Appl. Math. Optim., 10 (1983), 1-35.  doi: 10.1007/BF01448377.  Google Scholar

[17]

J.Lubliner, Plasticity Theory, Macmillan Publishing Company, New York, 1990. Google Scholar

[18]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.  doi: 10.1007/s00526-004-0267-8.  Google Scholar

[19]

G. B. Maggiani and M. G. Mora, A dynamic evolution model for perfectly plastic plates, Math. Models Methods Appl. Sci., 26 (2016), 1825-1864.  doi: 10.1142/S0218202516500469.  Google Scholar

[20]

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

[21]

R. T. Rockafellar, Convex Integral Functionals and Duality, in Contributions to Nonlinear Functional Analysis, Academic Press, (1971), 215-236.  Google Scholar

[22]

P. M. Suquet, Sur le équations de la plasticité: existence et regularité des solutions, J. Mécanique, 20 (1981), 3-39.  Google Scholar

[23]

R. Temam, Mathematical Problems in Plasticity, Gauthier-Villars, Paris, 1985.  Google Scholar

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