Approximate boundary conditions for a Mindlin-Timoshenko plate surrounded by a thin layer. (English) Zbl 1541.35023
Summary: We consider the model of Mindlin-Timoshenko for a multi-structure composed of an elastic plate surrounded by a thin layer of uniform thickness. From the viewpoint of numerical simulation, the treatment of the behavior of this structure is difficult because of the presence of the thin coating. In order to overcome this difficulty, we use the asymptotic expansion method to identify an approximate model that does not involve the thin layer geometrically but which accounts for its effect through new approximate boundary conditions. These conditions are set on the junction interface between the two sub-structures and depend on the thickness and the physical characteristics of the thin layer. Moreover, we give optimal error estimates between the exact and the approximate solutions of the considered transmission problem, which validate this approximation.
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35J57 | Boundary value problems for second-order elliptic systems |
| 74K20 | Plates |
| 74G10 | Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics |
Keywords:
approximate boundary conditions; asymptotic expansion; asymptotic modeling; Mindlin-Timoshenko plate; thin layerReferences:
| [1] | Argatov, I., An Asymptotic model for a thin bonded elastic layer coated with an elastic membrane, Appl Math Model, 40, 4, 2541-2548, 2016 · Zbl 1452.74067 · doi:10.1016/j.apm.2015.09.109 |
| [2] | Bendali, A.; Lemrabet, K., The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation, SIAM J Appl Math, 56, 6, 1664-1693, 1996 · Zbl 0869.35068 · doi:10.1137/S0036139995281822 |
| [3] | Bendali, A.; Lemrabet, K., Asymptotic analysis of the scattering of a time-harmonic electromagnetic wave by a perfectly conducting metal coated with a thin dielectric shell, Asymptot Anal, 57, 3-4, 199-227, 2008 · Zbl 1145.35464 |
| [4] | Ciarlet PG (1997) Mathematical elasticity, vol. II: Theory of plates. Studies in mathematics and its applications, vol 27. North-Holland Publishing Co., Amsterdam · Zbl 0888.73001 |
| [5] | Furtsev, A.; Rudoy, E., Variational approach to modeling soft and stiff interfaces in the Kirchhoff-Love theory of plates, Int J Solids Struct, 202, 562-574, 2020 · doi:10.1016/j.ijsolstr.2020.06.044 |
| [6] | Geymonat G, Hendili S, Krasucki F, Serpilli M, Vidrascu M (2014) Asymptotic expansions and domain decomposition. Domain decomposition methods in science and engineering XXI (Lecture notes in computational science and engineering), vol 98. Springer, 749-757 · Zbl 1443.65426 |
| [7] | Geymonat, G.; Krasuki, F.; Serpilli, M., Asymptotic derivation of a linear plate model for soft ferromagnetic materials, Chin Ann Math B, 39, 3, 451-460, 2018 · Zbl 1393.74050 · doi:10.1007/s11401-018-0077-5 |
| [8] | Goffi, FZ; Lemrabet, K.; Laadj, T., Transfer and approximation of the impedance for time-harmonic Maxwell’s system in a planar domain with thin contrasted multi-layers, Asymptot Anal, 101, 1-15, 2017 · Zbl 1360.78026 |
| [9] | Milosavljevic, D.; Zmindak, M.; Dekys, V., Approximate phase speed of lamb waves in a composite plate reinforced with strong fibres, J Eng Math, 129, 13, 2021 · Zbl 1502.35167 · doi:10.1007/s10665-021-10147-x |
| [10] | Nawaz, R.; Nuruddeen, RI; Zia, QMZ, An asymptotic investigation of the dynamics and dispersion of an elastic five-layered plate for anti-plane shear vibration, J Eng Math, 128, 9, 2021 · Zbl 1483.35256 · doi:10.1007/s10665-021-10133-3 |
| [11] | Rajagopal, A.; Hodges, DH, Variational asymptotic analysis for plates of variable thickness, Int J Solids Struct, 75, 81-87, 2015 · doi:10.1016/j.ijsolstr.2015.08.002 |
| [12] | Serpilli, M.; Lenci, S., An overview of different asymptotic models for anisotropic three-layer plates with soft adhesive, Int J Solids Struct, 81, 130-140, 2016 · doi:10.1016/j.ijsolstr.2015.11.020 |
| [13] | Serpilli, M., Asymptotic interface models in magneto-electro-thermo-elastic composites, Meccanica, 52, 1407-1424, 2017 · Zbl 1380.74076 · doi:10.1007/s11012-016-0481-4 |
| [14] | Serpilli, M., On modeling interfaces in linear micropolar composites, Math Mech Solids, 23, 4, 667-685, 2018 · Zbl 1395.74022 · doi:10.1177/1081286517692391 |
| [15] | Serpilli, M.; Lebon, F.; Rizzoni, R.; Dumont, S., An asymptotic derivation of a general imperfect interface law for linear multiphysics composites, Int J Solids Struct, 180-181, 97-107, 2019 · doi:10.1016/j.ijsolstr.2019.07.014 |
| [16] | Serpilli, M., Classical and higher order interface conditions in poroelasticity, Ann Solid Struct Mech, 11, 1-10, 2019 · doi:10.1007/s12356-019-00052-5 |
| [17] | Westbrook, DR, A linear asymptotic theory for anisotropic shells, J Eng Math, 6, 305-312, 1972 · Zbl 0248.73045 · doi:10.1007/BF01535191 |
| [18] | Ammari H, Latiri-Grouz C (1998) Approximate boundary conditions for thin periodic coatings. In: Mathematical and numerical aspects of wave propagation (Golden, CO, 1998), SIAM, Philadelphia, pp 297-301 · Zbl 0963.78015 |
| [19] | Ammari, H.; Latiri-Grouz, C., Approximate boundary conditions for thin periodic layers, RAIRO-M MOD, 33, 4, 673-693, 1999 · Zbl 0944.35008 |
| [20] | Engquist B, Nedelec JC (1993) Effective boundary conditions for acoustic and electro-magnetic scattering in thin layers. Technical report, Ecole polytechnique |
| [21] | Haddar, H.; Joly, P.; Nguyen, HM, Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case, Math Models Methods Appl Sci, 15, 8, 1273-1300, 2005 · Zbl 1084.35102 · doi:10.1142/S021820250500073X |
| [22] | Haddar, H.; Joly, P.; Nguyen, HM, Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: the case of Maxwell’s equations, Math Models Methods Appl Sci, 18, 10, 1787-1827, 2008 · Zbl 1170.35094 · doi:10.1142/S0218202508003194 |
| [23] | Geymonat, G.; Krasuki, F.; Lenci, S., Mathematical analysis of a bonded joint with a soft thin adhesive, Math Mech Solids, 4, 2, 201-225, 1999 · Zbl 1001.74591 · doi:10.1177/108128659900400204 |
| [24] | Le Louër, F., Thin layer approximations in mechanical structures: the Dirichlet boundary condition case, C R Math Acad Sci Paris, 357, 6, 576-581, 2019 · Zbl 1423.35020 · doi:10.1016/j.crma.2019.06.001 |
| [25] | Lemrabet K (1987) Etude de divers problèmes aux limites de Ventcel d’origine physique ou mécanique dans des domaines non réguliers. Thèse de Doctorat d’Etat, U.S.T.H.B |
| [26] | Lemrabet, K., Probleme de Ventcel pour le système de l’élasticité dans un domaine de R3, CR Acad Sci Paris Sér, 1, 304, 151-154, 1987 · Zbl 0624.73066 |
| [27] | Rahmani, L., Ventcel’s boundary conditions for a dynamic nonlinear plate, Asymptot Anal, 38, 3-4, 319-337, 2004 · Zbl 1154.35456 |
| [28] | Rahmani, L., Conditions aux limites approchées pour une plaque mince non linéaire, C R Acad Sci Paris Ser I, 343, 57-62, 2006 · Zbl 1094.74040 · doi:10.1016/j.crma.2006.04.013 |
| [29] | Rahmani, L.; Vial, G., Reinforcement of a thin plate by a thin layer, Math Methods Appl Sci, 31, 3, 315-338, 2008 · Zbl 1132.74026 · doi:10.1002/mma.910 |
| [30] | Rahmani, L., Modelling of the effect of a thin stiffener on the boundary of a nonlinear thermoelastic plate, Math Model Anal, 14, 3, 353-368, 2009 · Zbl 1294.74030 · doi:10.3846/1392-6292.2009.14.353-368 |
| [31] | Rahmani, L., Reinforcement of a Mindlin-Timoshenko plate by a thin layer, Z Angew Math Phys, 66, 3499-3517, 2015 · Zbl 1336.35039 · doi:10.1007/s00033-015-0562-6 |
| [32] | Lagnese, J.; Lions, JL, Modelling analysis and control of thin plates. Research in applied mathematics, 1988, Paris: Masson, Paris · Zbl 0662.73039 |
| [33] | Sare, HDF, On the stability of Mindlin-Timoshenko plates, Q Appl Math, LXVII, 2, 249-263, 2009 · Zbl 1175.35020 · doi:10.1090/S0033-569X-09-01110-2 |
| [34] | Mokhtari, H.; Rahmani, L., Asymptotic modeling of a reinforced plate with a thin layer of variable thickness, Meccanica, 57, 2155-2172, 2022 · Zbl 1524.74329 · doi:10.1007/s11012-021-01467-4 |
| [35] | Aslanyurek, B.; Haddar, H.; Sahinturk, H., Generalized impedance boundary conditions for thin dielectric coatings with variable thickness, Wave Motion, 48, 681-700, 2011 · Zbl 1239.78004 · doi:10.1016/j.wavemoti.2011.06.002 |
| [36] | Costabel, M.; Dauge, M.; Martin, D.; Vial, G., Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer, Asymptot Anal, 50, 121-173, 2006 · Zbl 1136.35021 |
| [37] | Rahmani, L.; Vial, G., Multi-scale asymptotic expansion for a singular problem of a free plate with thin stiffener, Asymptot Anal, 90, 161-187, 2014 · Zbl 1305.74036 |
| [38] | Vial G (2003) Analyse multi-échelle et conditions aux limites approchées pour un probléme avec couche mince dans un domaine à coin. Thèse de doctorat. Université de Rennes I |
| [39] | Vial, G., Efficiency of approximate boundary conditions for corner domains coated with thin layers, C R Acad Sci, 340, 215-220, 2005 · Zbl 1061.65126 |
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