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

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

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.

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 MasoA. 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 MasoA. 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

[1]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318

[2]

Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, 2021, 20 (2) : 547-558. doi: 10.3934/cpaa.2020280

[3]

Bimal Mandal, Aditi Kar Gangopadhyay. A note on generalization of bent boolean functions. Advances in Mathematics of Communications, 2021, 15 (2) : 329-346. doi: 10.3934/amc.2020069

[4]

Andreas Koutsogiannis. Multiple ergodic averages for tempered functions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1177-1205. doi: 10.3934/dcds.2020314

[5]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034

[6]

Tomáš Bodnár, Philippe Fraunié, Petr Knobloch, Hynek Řezníček. Numerical evaluation of artificial boundary condition for wall-bounded stably stratified flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 785-801. doi: 10.3934/dcdss.2020333

[7]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[8]

Xinpeng Wang, Bingo Wing-Kuen Ling, Wei-Chao Kuang, Zhijing Yang. Orthogonal intrinsic mode functions via optimization approach. Journal of Industrial & Management Optimization, 2021, 17 (1) : 51-66. doi: 10.3934/jimo.2019098

[9]

Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2020  doi: 10.3934/jcd.2021006

[10]

Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020378

[11]

Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $ (n, m) $-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117

[12]

Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015

[13]

Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020106

[14]

Meenakshi Rana, Shruti Sharma. Combinatorics of some fifth and sixth order mock theta functions. Electronic Research Archive, 2021, 29 (1) : 1803-1818. doi: 10.3934/era.2020092

[15]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331

[16]

Kalikinkar Mandal, Guang Gong. On ideal $ t $-tuple distribution of orthogonal functions in filtering de bruijn generators. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020125

[17]

Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 2021, 15 (2) : 227-240. doi: 10.3934/amc.2020055

[18]

Tomasz Szostok. Inequalities of Hermite-Hadamard type for higher order convex functions, revisited. Communications on Pure & Applied Analysis, 2021, 20 (2) : 903-914. doi: 10.3934/cpaa.2020296

[19]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395

[20]

Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020053

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (113)
  • HTML views (176)
  • Cited by (0)

Other articles
by authors

[Back to Top]