A Naghdi Type Nonlinear Model for Shells with Little Regularity
Tóm tắt
In this paper a new nonlinear two-dimensional shell model of Naghdi’s type is formulated for shells which middle surface is parameterized by a $W^{1, \infty}$ function. Therefore the model inherently contains undeformed geometries with corners, so the model also includes models of junctions of nonlinear shells. Deformation of the shell is described by a pair $(\boldsymbol{\psi}, {\mathbf{R}})$ of independent unknowns, where $\boldsymbol{\psi}$ is the deformation of the middle surface and ${\mathbf{R}}$ is a function with value in rotations that describes rotation of the shell cross-section. The model is formulated as the minimization problem for the total energy functional that includes flexural, membrane, shear and drill energies differently scaled with respect to the thickness of the shell. We relate the new model for smooth enough undeformed geometry to the known shell models in two ways. First we restrict the proposed model on two particular subsets of admissible functions and obtain exactly the flexural shell model and a perturbation of the Koiter shell model. More important, we consider asymptotics, using $\Gamma$ –convergence, of the proposed model with respect to the thickness as a small parameter in the membrane and flexural regime and obtain exactly the nonlinear membrane and flexural shell model obtained as $\Gamma$ –limits starting from nonlinear three–dimensional elasticity. In that way we link rigorously the proposed model with the nonlinear three–dimensional elasticity.
Tài liệu tham khảo
citation_journal_title=J. Elast.; citation_title=Polyconvexity and existence theorem for nonlinearly elastic shells; citation_author=S. Anicic; citation_volume=132; citation_publication_date=2018; citation_pages=161-173; citation_id=CR1
citation_journal_title=Discrete Contin. Dyn. Syst., Ser. S; citation_title=Existence theorem for a first-order Koiter nonlinear shell model; citation_author=S. Anicic; citation_volume=12; citation_publication_date=2019; citation_pages=1535-1545; citation_id=CR2
citation_journal_title=Math. Methods Appl. Sci.; citation_title=The infinitesimal rigid displacement lemma in Lipschitz co-ordinates and application to shells with minimal regularity; citation_author=S. Anicic, H. Dret, A. Raoult; citation_volume=27; citation_issue=11; citation_publication_date=2004; citation_pages=1283-1299; citation_id=CR3
citation_title=Nonlinear Problems of Elasticity; citation_publication_date=1995; citation_id=CR4; citation_author=S.S. Antman; citation_publisher=Springer
citation_journal_title=Math. Mech. Solids; citation_title=Refined dimensional reduction for isotropic elastic Cosserat shells with initial curvature; citation_author=M. Bîrsan, I.D. Ghiba, R. Martin, P. Neff; citation_volume=24; citation_publication_date=2019; citation_pages=4000-4019; citation_id=CR5
citation_journal_title=Math. Mech. Solids; citation_title=Existence of minimizers in the geometrically non-linear 6-parameter resultant shell theory with drilling rotations; citation_author=M. Bîrsan, P. Neff; citation_volume=19; citation_publication_date=2014; citation_pages=376-397; citation_id=CR6
citation_title=On the dislocation density tensor in the Cosserat theory of elastic shells; citation_inbook_title=Adv. Struct. Mater.; citation_publication_date=2016; citation_pages=391-413; citation_id=CR7; citation_author=M. Bîrsan; citation_author=P. Neff; citation_publisher=Springer
citation_journal_title=Int. J. Eng. Sci.; citation_title=Shells without drilling rotations: a representation theorem in the framework of the geometrically nonlinear 6-parameter resultant shell theory; citation_author=M. Bîrsan, P. Neff; citation_volume=80; citation_publication_date=2014; citation_pages=32-42; citation_id=CR8
citation_journal_title=Asymptot. Anal.; citation_title=A simplified model for elastic thin shells; citation_author=D. Blanchard, G. Griso; citation_volume=76; citation_publication_date=2012; citation_pages=1-33; citation_id=CR9
citation_journal_title=SIAM J. Math. Anal.; citation_title=An up-to-the-boundary version of Friedrichs’s lemma and applications to the linear Koiter shell model; citation_author=A. Blouza, H. Dret; citation_volume=33; citation_publication_date=2001; citation_pages=877-895; citation_id=CR10
citation_journal_title=Q. Appl. Math.; citation_title=Existence and uniqueness for the linear Koiter model for shells with little regularity; citation_author=A. Blouza, H. Dret; citation_volume=57; citation_publication_date=1999; citation_pages=317-337; citation_id=CR11
citation_journal_title=J. Elast.; citation_title=Nagdhi’s shell model: existence, uniqueness and continuous dependence on the midsurface; citation_author=A. Blouza, H. Dret; citation_volume=64; citation_issue=2–3; citation_publication_date=2001; citation_pages=199-216; citation_id=CR12
citation_title=The Mathematical Theory of Finite Element Methods; citation_publication_date=1994; citation_id=CR13; citation_author=S.C. Brenner; citation_author=L.R. Scott; citation_publisher=Springer
citation_title=Mathematical Elasticity. Vol. II. Theory of Plates; citation_publication_date=1997; citation_id=CR14; citation_author=P.G. Ciarlet; citation_publisher=North-Holland
citation_title=Mathematical Elasticity. Vol. III. Theory of Shells; citation_publication_date=2000; citation_id=CR15; citation_author=P.G. Ciarlet; citation_publisher=North-Holland
citation_journal_title=C. R. Acad. Sci. Paris, Ser. I; citation_title=Un modèle bi-dimentionnel non linéaire de coques analogue à celui de W.T. Koiter; citation_author=P.G. Ciarlet; citation_volume=331; citation_publication_date=2000; citation_pages=405-410; citation_id=CR16
citation_journal_title=Arch. Ration. Mech. Anal.; citation_title=Asymptotic analysis of linearly elastic shells. III. Justification of Koiter’s shell equations; citation_author=P.G. Ciarlet, V. Lods; citation_volume=136; citation_publication_date=1996; citation_pages=191-200; citation_id=CR17
citation_journal_title=Math. Models Methods Appl. Sci.; citation_title=An existence theorem for a two-dimensional nonlinear shell model of Koiter’s type; citation_author=P.G. Ciarlet, C. Mardare; citation_volume=28; citation_publication_date=2018; citation_pages=2833-2861; citation_id=CR18
citation_journal_title=C. R. Math. Acad. Sci. Paris; citation_title=A nonlinear shell model of Koiter’s type; citation_author=P.G. Ciarlet, C. Mardare; citation_volume=356; citation_publication_date=2018; citation_pages=227-234; citation_id=CR19
citation_journal_title=Anal. Appl. (Singap.); citation_title=An intrinsic formulation of the Kirchhoff-Love theory of linearly elastic plates; citation_author=P.G. Ciarlet, C. Mardare; citation_volume=16; citation_publication_date=2018; citation_pages=565-584; citation_id=CR20
citation_title=An introduction to shell theory; citation_inbook_title=Differential Geometry: Theory and Applications; citation_publication_date=2008; citation_pages=94-184; citation_id=CR21; citation_author=P.G. Ciarlet; citation_author=C. Mardare; citation_publisher=Higher Education Press
citation_journal_title=C. R. Acad. Sci. Paris; citation_title=Asymptotic justification of the intrinsic equations of Koiter’s model of a linearly elastic shell; citation_author=P.G. Ciarlet, C. Mardare; citation_volume=357; citation_publication_date=2019; citation_pages=99-110; citation_id=CR22
citation_journal_title=Chin. Ann. Math., Ser. B; citation_title=Justification of a two-dimensional nonlinear shell model of Koiter’s type; citation_author=P.G. Ciarlet, A. Roquefort; citation_volume=22; citation_publication_date=2001; citation_pages=129-144; citation_id=CR23
citation_journal_title=Arch. Ration. Mech. Anal.; citation_title=Confining thin elastic sheets and folding paper; citation_author=S. Conti, F. Maggi; citation_volume=187; citation_publication_date=2008; citation_pages=1-48; citation_id=CR24
citation_journal_title=SIAM J. Math. Anal.; citation_title=Rigorous derivation of Föppl’s theory for clamped elastic membranes leads to relaxation; citation_author=S. Conti, F. Maggi, S. Müller; citation_volume=38; citation_publication_date=2006; citation_pages=657-680; citation_id=CR25
Cosserat, E., Cosserat, F.: Théorie des corps déformables, Librairie Scientifique A. Hermann et Fils. (English translation by D. Delphenich, 2007, PDF available at
http://www.uni-due.de/~hm0014/Cosserat_files/Cosserat09_eng.pdf
), Reprint 2009, Paris, 1909
citation_title=Direct Methods in the Calculus of Variations; citation_publication_date=2008; citation_id=CR27; citation_author=B. Dacorogna; citation_publisher=Springer
citation_title=Convex Analysis and Variational Problems; citation_publication_date=1999; citation_id=CR28; citation_author=I. Ekeland; citation_author=R. Témam; citation_publisher=SIAM
citation_journal_title=J. Elast.; citation_title=Local symmetry group in the general theory of elastic shells; citation_author=V.A. Eremeyev, W. Pietraszkiewicz; citation_volume=85; citation_publication_date=2006; citation_pages=125-152; citation_id=CR29
citation_journal_title=Arch. Ration. Mech. Anal.; citation_title=A justification of nonlinear properly invariant plate theories; citation_author=D.D. Fox, A. Raoult, J.C. Simo; citation_volume=124; citation_publication_date=1993; citation_pages=157-199; citation_id=CR30
citation_journal_title=Commun. Pure Appl. Math.; citation_title=A theorem on geometric rigidity and the derivation of nonlinear plate theory from the three-dimensional elasticity; citation_author=G. Friesecke, R.D. James, S. Müller; citation_volume=55; citation_publication_date=2002; citation_pages=1461-1506; citation_id=CR31
citation_journal_title=Arch. Ration. Mech. Anal.; citation_title=A hierarchy of plate models derived from nonlinear elasticity by
–convergence; citation_author=G. Friesecke, R.D. James, S. Müller; citation_volume=180; citation_publication_date=2006; citation_pages=183-236; citation_id=CR32
citation_journal_title=C. R. Math. Acad. Sci. Paris; citation_title=Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence; citation_author=G. Friesecke, R.D. James, M.G. Mora, S. Müller; citation_volume=336; citation_publication_date=2003; citation_pages=697-702; citation_id=CR33
Ghiba, I.D., Bîrsan, M., Lewintan, P., Neff, P.: The isotropic Cosserat shell model including terms up to
$O(h^{5})$
. Part I: derivation in matrix notation. J. Elast. (2020). Published online in
https://doi.org/10.1007/s10659-020-09796-3
Ghiba, I.D., Bîrsan, M., Lewintan, P., Neff, P.: The isotropic Cosserat shell model including terms up to
$O(h^{5})$
. Part II: existence of minimizers. J. Elast. (2020). Published online in
https://doi.org/10.1007/s10659-020-09795-4
citation_journal_title=Comput. Methods Appl. Mech. Eng.; citation_title=On drilling degrees of freedom; citation_author=T.J.R. Hughes, F. Brezzi; citation_volume=72; citation_publication_date=1989; citation_pages=105-121; citation_id=CR36
citation_journal_title=Comput. Struct.; citation_title=Numerical assessment of some membrane elements with drilling degrees of freedom; citation_author=T.J.R. Hughes, A. Masudg, I. Harari; citation_volume=55; citation_publication_date=1995; citation_pages=297-314; citation_id=CR37
citation_journal_title=Proc. K. Ned. Akad. Wet., Ser. B, Phys. Sci.; citation_title=On the nonlinear theory of thin elastic shells. I; citation_author=W.T. Koiter; citation_volume=69; citation_publication_date=1966; citation_pages=1-17; citation_id=CR38
citation_journal_title=Proc. K. Ned. Akad. Wet., Ser. B, Phys. Sci.; citation_title=On the nonlinear theory of thin elastic shells. II; citation_author=W.T. Koiter; citation_volume=69; citation_publication_date=1966; citation_pages=18-32; citation_id=CR39
citation_journal_title=Proc. K. Ned. Akad. Wet., Ser. B, Phys. Sci.; citation_title=On the nonlinear theory of thin elastic shells. III; citation_author=W.T. Koiter; citation_volume=69; citation_publication_date=1966; citation_pages=33-54; citation_id=CR40
citation_journal_title=Math. Mech. Solids; citation_title=Nonlinear bending-torsion model for curved rods with little regularity; citation_author=M. Kosor, J. Tambača; citation_volume=22; citation_publication_date=2017; citation_pages=708-717; citation_id=CR41
citation_journal_title=Nonlinear Anal.; citation_title=The Kirchhoff theory for elastic pre-strained shells; citation_author=H. Li; citation_volume=78; citation_publication_date=2013; citation_pages=1-16; citation_id=CR42
citation_journal_title=Anal. Appl. (Singap.); citation_title=Well-posedness for Koiter and Naghdi shells with a G1-midsurface; citation_author=H. Dret; citation_volume=2; citation_issue=4; citation_publication_date=2004; citation_pages=365-388; citation_id=CR43
citation_journal_title=J. Nonlinear Sci.; citation_title=The membrane shell model in nonlinear elasticity: a variational asymptotic derivation; citation_author=H. Dret, A. Raoult; citation_volume=6; citation_publication_date=1996; citation_pages=59-84; citation_id=CR44
citation_journal_title=J. Math. Pures Appl.; citation_title=The nonlinear membrane model as variational limit of nonlinear three–dimensional elasticity; citation_author=H. Dret, A. Raoult; citation_volume=74; citation_publication_date=1995; citation_pages=549-578; citation_id=CR45
citation_journal_title=Proc. R. Soc. Edinb., Sect. A; citation_title=The quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function; citation_author=H. Dret, A. Raoult; citation_volume=125; citation_publication_date=1995; citation_pages=1179-1192; citation_id=CR46
citation_journal_title=Arch. Ration. Mech. Anal.; citation_title=Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results; citation_author=H. Dret, A. Raoult; citation_volume=154; citation_publication_date=2000; citation_pages=101-134; citation_id=CR47
citation_journal_title=ESAIM Control Optim. Calc. Var.; citation_title=A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry; citation_author=M. Lewicka; citation_volume=17; citation_publication_date=2009; citation_pages=493-505; citation_id=CR48
citation_journal_title=Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5); citation_title=Shell theories arising as low energy gamma–limit of 3d nonlinear elasticity; citation_author=M. Lewicka, M.G. Mora, R. Pakzad; citation_volume=9; citation_publication_date=2010; citation_pages=1-43; citation_id=CR49
citation_journal_title=Arch. Ration. Mech. Anal.; citation_title=The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells; citation_author=M. Lewicka, M.G. Mora, R. Pakzad; citation_volume=200; citation_publication_date=2011; citation_pages=1023-1050; citation_id=CR50
citation_journal_title=ESAIM Control Optim. Calc. Var.; citation_title=Scaling laws for non-Euclidean plates and the
isometric immersions of Riemannian metrics; citation_author=M. Lewicka, M.R. Pakzad; citation_volume=17; citation_publication_date=2011; citation_pages=1158-1173; citation_id=CR51
citation_journal_title=Fields Inst. Commun.; citation_title=The infinite hierarchy of elastic shell models: some recent results and a conjecture; citation_author=M. Lewicka, R. Pakzad; citation_volume=64; citation_publication_date=2009; citation_pages=407-420; citation_id=CR52
citation_journal_title=ESAIM Proc. Surv.; citation_title=Thin structures with imposed metric; citation_author=M. Lewicka, A. Raoult; citation_volume=62; citation_publication_date=2018; citation_pages=79-90; citation_id=CR53
Ljulj, M.: Three-dimensional elastic body and a Naghdi type shell interaction modelling. PhD thesis, University of Zagreb (2020)
citation_journal_title=Acta Math. Appl. Sin. Engl. Ser.; citation_title=Nonlinear shell models of Kirchhoff-Love type: existence theorem and comparison with Koiter’s model; citation_author=C. Mardare; citation_volume=35; citation_publication_date=2019; citation_pages=3-27; citation_id=CR55
citation_journal_title=Rev. Roum. Math. Pures Appl.; citation_title=On the derivation of nonlinear shell models from three-dimensional elasticity; citation_author=C. Mardare; citation_volume=53; citation_publication_date=2008; citation_pages=499-522; citation_id=CR56
citation_title=The Theory of Shells and Plates; citation_publication_date=1972; citation_id=CR57; citation_author=P.M. Naghdi; citation_publisher=Springer
citation_journal_title=Contin. Mech. Thermodyn.; citation_title=A geometrically exact Cosserat-shell model including size effects, avoiding degeneracy in the thin shell limit. Part I: formal dimensional reduction for elastic plates and existence of minimizers for positive Cosserat couple modulus; citation_author=P. Neff; citation_volume=16; citation_publication_date=2004; citation_pages=577-628; citation_id=CR58
Neff, P.: Geometrically exact Cosserat theory for bulk behaviour and thin structures. Modelling and mathematical analysis. Signatur HS 7/0973. Habilitationsschrift, Universitäts- und Landesbibliothek, Technische Universitat Darmstadt, Darmstadt, 2004
citation_journal_title=Math. Models Methods Appl. Sci.; citation_title=A geometrically exact planar Cosserat shell-model with microstructure: existence of minimizers for zero Cosserat couple modulus; citation_author=P. Neff; citation_volume=17; citation_publication_date=2007; citation_pages=363-392; citation_id=CR60
citation_journal_title=J. Elast.; citation_title=Existence theorem for geometrically nonlinear Cosserat micropolar model under uniform convexity requirements; citation_author=P. Neff, M. Bîrsan, F. Osterbrink; citation_volume=121; citation_publication_date=2015; citation_pages=119-141; citation_id=CR61
citation_journal_title=Interfaces Free Bound.; citation_title=A geometrically exact Cosserat shell-model for defective elastic crystals. Justification via Gamma-convergence; citation_author=P. Neff, K. Chelmiński; citation_volume=9; citation_publication_date=2007; citation_pages=455-492; citation_id=CR62
citation_journal_title=Math. Mech. Solids; citation_title=Large elastic deformation of micromorphic shells. Part I: variational formulation; citation_author=A. Norouzzadeh, R. Ansari, M. Darvizeh; citation_volume=24; citation_publication_date=2019; citation_pages=3920-3956; citation_id=CR63
citation_journal_title=Arch. Ration. Mech. Anal.; citation_title=On the justification of the nonlinear inextensional plate model; citation_author=O. Pantz; citation_volume=167; citation_publication_date=2003; citation_pages=179-209; citation_id=CR64
citation_title=On a description of deformable junction in the resultant nonlinear shell theory, (English summary) Advanced methods of continuum mechanics for materials and structures; citation_inbook_title=Adv. Struct. Mater.; citation_publication_date=2016; citation_pages=457-468; citation_id=CR65; citation_author=W. Pietraszkiewicz; citation_publisher=Springer
citation_journal_title=J. Math. Pures Appl.; citation_title=Des lois géometriques qui regissent les déplacements d’ un systéme solide dans l’ espace, et de la variation des coordonnées provenant de ces déplacement considérées indépendant des causes qui peuvent les produire; citation_author=O. Rodrigues; citation_volume=5; citation_publication_date=1840; citation_pages=380-440; citation_id=CR66
citation_journal_title=J. Elast.; citation_title=On structured surfaces with defects: geometry, strain incompatibility, stress field, and natural shapes; citation_author=A. Roychowdhury, A. Gupta; citation_volume=131; citation_publication_date=2018; citation_pages=239-276; citation_id=CR67
citation_journal_title=J. Elast.; citation_title=Koiter’s shell theory from the perspective of three-dimensional nonlinear elasticity; citation_author=D. Steigmann; citation_volume=111; citation_publication_date=2013; citation_pages=91-107; citation_id=CR68
citation_journal_title=Adv. Math. Sci. Appl.; citation_title=A note on the “flexural” shell model for shells with little regularity; citation_author=J. Tambača; citation_volume=16; citation_publication_date=2006; citation_pages=45-55; citation_id=CR69
citation_journal_title=J. Elast.; citation_title=A new linear shell model for shells with little regularity; citation_author=J. Tambača; citation_volume=117; citation_issue=2; citation_publication_date=2014; citation_pages=163-188; citation_id=CR70
citation_journal_title=Appl. Math. Model.; citation_title=A new linear Naghdi type shell model for shells with little regularity; citation_author=J. Tambača, Z. Tutek; citation_volume=40; citation_publication_date=2016; citation_pages=10549-10562; citation_id=CR71
citation_journal_title=ESAIM Control Optim. Calc. Var.; citation_title=Existence theorem for nonlinear micropolar elasticity; citation_author=J. Tambača, I. Velčić; citation_volume=16; citation_publication_date=2010; citation_pages=92-110; citation_id=CR72