Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure: the limiting regimes. (English) Zbl 07930854
Summary: We identify effective models for thin, linearly elastic and perfectly plastic plates exhibiting a microstructure resulting from the periodic alternation of two elastoplastic phases. We study here both the case in which the thickness of the plate converges to zero on a much faster scale than the periodicity parameter and the opposite scenario in which homogenization occurs on a much finer scale than dimension reduction. After performing a static analysis of the problem, we show convergence of the corresponding quasistatic evolutions. The methodology relies on two-scale convergence and periodic unfolding, combined with suitable measure-disintegration results and evolutionary \(\Gamma \)-convergence.
MSC:
| 74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |
| 74G65 | Energy minimization in equilibrium problems in solid mechanics |
| 74K20 | Plates |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
Keywords:
perfect plasticity; periodic homogenization; dimension reduction; quasistatic evolution; rate-independent processes; \( \Gamma \)-convergenceReferences:
| [1] | L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford University, New York, 2000. · Zbl 0957.49001 |
| [2] | A. Braides, Approximation of Free-Discontinuity Problems, Lecture Notes in Math. 1694, Springer, Berlin, 1998. · Zbl 0909.49001 |
| [3] | D. Breit, L. Diening and F. Gmeineder, On the trace operator for functions of bounded \mathbb{A}-variation, Anal. PDE 13 (2020), no. 2, 559-594. · Zbl 1450.46017 |
| [4] | L. Bufford, E. Davoli and I. Fonseca, Multiscale homogenization in Kirchhoff’s nonlinear plate theory, Math. Models Methods Appl. Sci. 25 (2015), no. 9, 1765-1812. · Zbl 1329.35044 |
| [5] | M. Bukal and I. Velčić, On the simultaneous homogenization and dimension reduction in elasticity and locality of Γ-closure, Calc. Var. Partial Differential Equations 56 (2017), no. 3, Paper No. 59. · Zbl 1370.74018 |
| [6] | M. Bužančić, K. Cherednichenko, I. Velčić and J. Žubrinić, Spectral and evolution analysis of composite elastic plates with high contrast, J. Elasticity 152 (2022), no. 1-2, 79-177. · Zbl 1513.74114 |
| [7] | M. Bužančić, E. Davoli and I. Velčić, Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure, preprint (2022), https://arxiv.org/abs/2212.02116. |
| [8] | D. Caillerie, Thin elastic and periodic plates, Math. Methods Appl. Sci. 6 (1984), no. 2, 159-191. · Zbl 0543.73073 |
| [9] | L. Carbone and R. De Arcangelis, Unbounded Functionals in the Calculus of Variations: Representation, Relaxation, and Homogenization, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 125, Chapman & Hall/CRC, Boca Raton, 2019. |
| [10] | M. Cherdantsev and K. Cherednichenko, Bending of thin periodic plates, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 4079-4117. · Zbl 1343.35225 |
| [11] | F. Christowiak and C. Kreisbeck, Homogenization of layered materials with rigid components in single-slip finite crystal plasticity, Calc. Var. Partial Differential Equations 56 (2017), no. 3, Paper No. 75. · Zbl 1375.49015 |
| [12] | F. Christowiak and C. Kreisbeck, Asymptotic rigidity of layered structures and its application in homogenization theory, Arch. Ration. Mech. Anal. 235 (2020), no. 1, 51-98. · Zbl 1437.49025 |
| [13] | G. Dal Maso, A. DeSimone and M. G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal. 180 (2006), no. 2, 237-291. · Zbl 1093.74007 |
| [14] | A. Damlamian and M. Vogelius, Homogenization limits of the equations of elasticity in thin domains, SIAM J. Math. Anal. 18 (1987), no. 2, 435-451. · Zbl 0614.73012 |
| [15] | E. Davoli, Linearized plastic plate models as Γ-limits of 3D finite elastoplasticity, ESAIM Control Optim. Calc. Var. 20 (2014), no. 3, 725-747. · Zbl 1298.74145 |
| [16] | E. Davoli, Quasistatic evolution models for thin plates arising as low energy Γ-limits of finite plasticity, Math. Models Methods Appl. Sci. 24 (2014), no. 10, 2085-2153. · Zbl 1350.74011 |
| [17] | E. Davoli, R. Ferreira and C. Kreisbeck, Homogenization in BV of a model for layered composites in finite crystal plasticity, Adv. Calc. Var. 14 (2021), no. 3, 441-473. · Zbl 1467.49044 |
| [18] | E. Davoli, C. Gavioli and V. Pagliari, A homogenization result in finite plasticity, preprint (2022), https://arxiv.org/abs/2204.09084. |
| [19] | E. Davoli, C. Gavioli and V. Pagliari, Homogenization of high-contrast media in finite-strain elastoplasticity, preprint (2022), https://arxiv.org/abs/2301.02170. |
| [20] | E. Davoli and C. Kreisbeck, On static and evolutionary homogenization in crystal plasticity for stratified composites, Research in Mathematics of Materials Science, Assoc. Women Math. Ser. 31, Springer, Cham (2022), 159-183. · Zbl 1502.74009 |
| [21] | E. Davoli and M. G. Mora, A quasistatic evolution model for perfectly plastic plates derived by Γ-convergence, Ann. Inst. H. Poincaré C Anal. Non Linéaire 30 (2013), no. 4, 615-660. · Zbl 1366.74039 |
| [22] | 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), no. 3, 2581-2614. · Zbl 1327.74038 |
| [23] | F. Demengel, Fonctions à hessien borné, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 2, 155-190. · Zbl 0525.46020 |
| [24] | F. Demengel and R. Temam, Convex functions of a measure and applications, Indiana Univ. Math. J. 33 (1984), no. 5, 673-709. · Zbl 0581.46036 |
| [25] | I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: L^p Spaces, Springer Monogr. Math., Springer, New York, 2007. · Zbl 1153.49001 |
| [26] | G. Francfort and A. Giacomini, On periodic homogenization in perfect elasto-plasticity, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 3, 409-461. · Zbl 1290.74033 |
| [27] | G. A. Francfort and A. Giacomini, Small-strain heterogeneous elastoplasticity revisited, Comm. Pure Appl. Math. 65 (2012), no. 9, 1185-1241. · Zbl 1396.74036 |
| [28] | G. A. Francfort, A. Giacomini and A. Musesti, On the Fleck and Willis homogenization procedure in strain gradient plasticity, Discrete Contin. Dyn. Syst. Ser. S 6 (2013), no. 1, 43-62. · Zbl 1310.74039 |
| [29] | , E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, Rend. Semin. Mat. Univ. Padova 27 (1957), 284-305. · Zbl 0087.10902 |
| [30] | A. Giacomini and A. Musesti, Two-scale homogenization for a model in strain gradient plasticity, ESAIM Control Optim. Calc. Var. 17 (2011), no. 4, 1035-1065. · Zbl 1300.74008 |
| [31] | P. Gidoni, G. B. Maggiani and R. Scala, Existence and regularity of solutions for an evolution model of perfectly plastic plates, Commun. Pure Appl. Anal. 18 (2019), no. 4, 1783-1826. · Zbl 1412.74013 |
| [32] | C. Goffman and J. Serrin, Sublinear functions of measures and variational integrals, Duke Math. J. 31 (1964), 159-178. · Zbl 0123.09804 |
| [33] | H. Hanke, Homgenization in gradient plasticity, Math. Models Methods Appl. Sci. 21 (2011), no. 8, 1651-1684. · Zbl 1233.35017 |
| [34] | M. Heida and B. Schweizer, Non-periodic homogenization of infinitesimal strain plasticity equations, ZAMM Z. Angew. Math. Mech. 96 (2016), no. 1, 5-23. · Zbl 1335.74015 |
| [35] | M. Heida and B. Schweizer, Stochastic homogenization of plasticity equations, ESAIM Control Optim. Calc. Var. 24 (2018), no. 1, 153-176. · Zbl 1393.74014 |
| [36] | P. Hornung, S. Neukamm and I. Velčić, Derivation of a homogenized nonlinear plate theory from 3d elasticity, Calc. Var. Partial Differential Equations 51 (2014), no. 3-4, 677-699. · Zbl 1327.35018 |
| [37] | R. Kohn and R. Temam, Dual spaces of stresses and strains, with applications to Hencky plasticity, Appl. Math. Optim. 10 (1983), no. 1, 1-35. · Zbl 0532.73039 |
| [38] | M. Liero and A. Mielke, An evolutionary elastoplastic plate model derived via Γ-convergence, Math. Models Methods Appl. Sci. 21 (2011), no. 9, 1961-1986. · Zbl 1232.35165 |
| [39] | M. Liero and T. Roche, Rigorous derivation of a plate theory in linear elastoplasticity via Γ-convergence, NoDEA Nonlinear Differential Equations Appl. 19 (2012), no. 4, 437-457. · Zbl 1253.35180 |
| [40] | G. B. Maggiani and M. G. Mora, A dynamic evolution model for perfectly plastic plates, Math. Models Methods Appl. Sci. 26 (2016), no. 10, 1825-1864. · Zbl 1346.74015 |
| [41] | G. B. Maggiani and M. G. Mora, Quasistatic evolution of perfectly plastic shallow shells: A rigorous variational derivation, Ann. Mat. Pura Appl. (4) 197 (2018), no. 3, 775-815. · Zbl 1451.74053 |
| [42] | A. Mielke, T. Roubíček and U. Stefanelli, Γ-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations 31 (2008), no. 3, 387-416. · Zbl 1302.49013 |
| [43] | S. Neukamm and I. Velčić, Derivation of a homogenized von-Kármán plate theory from 3D nonlinear elasticity, Math. Models Methods Appl. Sci. 23 (2013), no. 14, 2701-2748. · Zbl 1282.35039 |
| [44] | G. Panasenko, Multi-Scale Modelling for Structures and Composites, Springer, Dordrecht, 2005. · Zbl 1078.74002 |
| [45] | B. Schweizer and M. Veneroni, Homogenization of plasticity equations with two-scale convergence methods, Appl. Anal. 94 (2015), no. 2, 376-399. |
| [46] | R. Temam, Problèmes mathématiques en plasticité, Math. Methods Inform. Sci. 12, Gauthier-Villars, Paris, 1985. |
| [47] | R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, American Mathematical Society, Providence, 2001. · Zbl 0981.35001 |
| [48] | I. Velčić, On the derivation of homogenized bending plate model, Calc. Var. Partial Differential Equations 53 (2015), no. 3-4, 561-586. · Zbl 1329.35050 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
