On a model of a flexural prestressed shell. (English) Zbl 1333.74056

Summary: We derive a linearized prestressed elastic shell model from a nonlinear Kirchhoff model of elastic plates. The model is given in terms of displacement and micro-rotation of the cross-sections. In addition to the standard membrane, transverse shear, and flexural terms, the model also contains a nonstandard prestress term. The prestress is of the same order as flexural effects; hence, the model is appropriate when flexural effects dominate over membrane ones. We prove the existence and uniqueness of the solutions by Lax-Milgram theorem and compare solution with the solution of the standard shell model via numerical examples.


74K25 Shells
74B15 Equations linearized about a deformed state (small deformations superposed on large)


Full Text: DOI


[1] Ciarlet, Studies in Mathematics and its Applications (1997)
[2] Kirchhoff, Über das gleichgewicht und die bewegung einer elastischen scheibe, Journal für die reine und angewandte Mathematik 40 pp 51– (1850) · ERAM 040.1086cj · doi:10.1515/crll.1850.40.51
[3] Ciarlet, Studies in Mathematics and its Applications (2000)
[4] Blouza, Existence and uniqueness for the linear Koiter model for shells with little regularity, Quarterly of Applied Mathematics 57 (2) pp 317– (1999) · Zbl 1025.74020 · doi:10.1090/qam/1686192
[5] Blouza, Nagdhi’s shell model: existence, uniqueness and continuous dependence on the midsurface, Journal of Elasticity and the Physical Science of Solids 64 (2-3) pp 199– (2001) · Zbl 1034.74037 · doi:10.1023/A:1015270504666
[6] Tambača, A new linear shell model for shells with little regularity, Journal of Elasticity 117 (2) pp 163– (2014) · Zbl 1307.74052 · doi:10.1007/s10659-014-9469-2
[7] Ciarlet, An introduction to differential geometry with applications to elasticity, Journal of Elasticity 78 (1-3) pp 1– (2005) · Zbl 1086.74001 · doi:10.1007/s10659-005-4738-8
[8] Lewicka, Scaling laws for non-Euclidean plates and the w2,2 isometric immersions of Riemannian metrics, ESAIM: Control, Optimisation and Calculus of Variations 17 (04) pp 1158– (2011) · Zbl 1300.74028 · doi:10.1051/cocv/2010039
[9] Bhattacharya, Plates with incompatible prestrain, arXiv preprint arXiv:1401.1609 (2014) · Zbl 1382.74081
[10] Lewicka, The Föppl-von Kármán equations for plates with incompatible strains, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 467 (2126) pp 402– (2011) · Zbl 1219.74027 · doi:10.1098/rspa.2010.0138
[11] Lewicka M Raoult A Ricciottim D Plates with Incompatible Prestrain of Higher Order
[12] Lewicka, Models for elastic shells with incompatible strains, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 470 (2165) (2014) · doi:10.1098/rspa.2013.0604
[13] Friesecke, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Communications on Pure and Applied Mathematics 55 (11) pp 1461– (2002) · Zbl 1021.74024 · doi:10.1002/cpa.10048
[14] Friesecke, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Archive for Rational Mechanics and Analysis 180 (2) pp 183– (2006) · Zbl 1100.74039 · doi:10.1007/s00205-005-0400-7
[15] Coutand, Existence d’un minimiseur pour le modèle proprement invariant de plaque en flexion non linéairement élastique, Comptes Rendus de l’Académie des Sciences-Series I-Mathematics 324 (2) pp 245– (1997) · Zbl 0873.73078 · doi:10.1016/S0764-4442(99)80354-1
[16] Paroni, The equations of motion of a plate with residual stress, Meccanica 41 (1) pp 1– (2006) · Zbl 1158.74413 · doi:10.1007/s11012-005-7589-2
[17] Aganović, On the stability of rotating rods and plates, ZAMM-Zeitschrift fur Angewandte Mathematik und Mechanik 81 (11) pp 733– (2001) · doi:10.1002/1521-4001(200111)81:11<733::AID-ZAMM733>3.0.CO;2-8
[18] Gilbarg, Elliptic Partial Differential Equations of Second Order (2001)
[19] Chapelle, The Finite Element Analysis of Shells: Fundamentals (2011) · Zbl 1211.74002 · doi:10.1007/978-3-642-16408-8
[20] Hakula, Scale resolution, locking, and high-order finite element modelling of shells, Computer Methods in Applied Mechanics and Engineering 133 (3) pp 157– (1996) · Zbl 0918.73111 · doi:10.1016/0045-7825(95)00939-6
[21] Pitkäranta, The problem of membrane locking in finite element analysis of cylindrical shells, Numerische Mathematik 61 (1) pp 523– (1992) · Zbl 0768.73079 · doi:10.1007/BF01385524
[22] Ciarlet, The Finite Element Method for Elliptic Problems (1978)
[23] Hecht, New development in freefem++, Journal of Numerical Mathematics 20 (3-4) pp 251– (2012) · Zbl 1266.68090 · doi:10.1515/jnum-2012-0013
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.