March  2020, 19(3): 1747-1793. doi: 10.3934/cpaa.2020072

Bending-torsion moments in thin multi-structures in the context of nonlinear elasticity

1. 

King Abdullah University of Science and Technology, 4700 KAUST, CEMSE Division, Thuwal 23955-6900, Saudi Arabia

2. 

Università degli Studi di Salerno, Dipartimento di Ingegneria Industriale, Via Giovanni Paolo II, 132 Fisciano (SA), Italy

* Corresponding author

Received  April 2019 Revised  July 2019 Published  November 2019

Here, we address a dimension-reduction problem in the context of nonlinear elasticity where the applied external surface forces induce bending-torsion moments. The underlying body is a multi-structure in $\mathbb{R}^3$ consisting of a thin tube-shaped domain placed upon a thin plate-shaped domain. The problem involves two small parameters, the radius of the cross-section of the tube-shaped domain and the thickness of the plate-shaped domain. We characterize the different limit models, including the limit junction condition, in the membrane-string regime according to the ratio between these two parameters as they converge to zero.

Citation: Rita Ferreira, Elvira Zappale. Bending-torsion moments in thin multi-structures in the context of nonlinear elasticity. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1747-1793. doi: 10.3934/cpaa.2020072
References:
[1]

E. AcerbiG. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string, J. Elasticity, 25 (1991), 137-148.  doi: 10.1007/BF00042462.  Google Scholar

[2]

E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal., 86 (1984), 125-145.  doi: 10.1007/BF00275731.  Google Scholar

[3]

J.-F. BabadjianE. Zappale and H. Zorgati, Dimensional reduction for energies with linear growth involving the bending moment, J. Math. Pures Appl., 90 (2008), 520-549.  doi: 10.1016/j.matpur.2008.07.003.  Google Scholar

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D. Blanchard and G. Griso, Junction between a plate and a rod of comparable thickness in nonlinear elasticity, J. Elasticity, 112 (2013), 79-109.  doi: 10.1007/s10659-012-9401-6.  Google Scholar

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M. Bocea and I. Fonseca, A Young measure approach to a nonlinear membrane model involving the bending moment, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 845-883.  doi: 10.1017/S0308210500003516.  Google Scholar

[6]

G. BouchittéI. Fonseca and M. L. Mascarenhas, Bending moment in membrane theory, J. Elasticity, 73 (2003), 75-99.  doi: 10.1023/B:ELAS.0000029996.20973.92.  Google Scholar

[7]

G. BouchittéI. Fonseca and M. L. Mascarenhas, The Cosserat vector in membrane theory: a variational approach, J. Convex Anal., 16 (2009), 351-365.   Google Scholar

[8]

R. BunoiuG. Cardone and S. Nazarov, Scalar problems in junctions of rods and a plate. Ⅱ. self-adjoint extensions and simulation models, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), 481-508.  doi: 10.1051/m2an/2017047.  Google Scholar

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G. CaritaJ. MatiasM. Morandotti and D. R. Owen, Dimension reduction in the context of structured deformations, Journal of Elasticity, 133 (2018), 1-35.  doi: 10.1007/s10659-018-9670-9.  Google Scholar

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P. G. Ciarlet, Plates and Junctions in Elastic Multi-structures, vol. 14 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris; Springer-Verlag, Berlin, 1990, An asymptotic analysis.  Google Scholar

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S. Conti and B. Schweizer, Rigidity and gamma convergence for solid-solid phase transitions with SO(2) invariance, Comm. Pure Appl. Math., 59 (2006), 830-868.  doi: 10.1002/cpa.20115.  Google Scholar

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I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer Monographs in Mathematics, Springer, New York, 2007.  Google Scholar

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I. FonsecaS. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients, SIAM J. Math. Anal., 29 (1998), 736-756.  doi: 10.1137/S0036141096306534.  Google Scholar

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D. D. FoxA. Raoult and J. C. Simo, A justification of nonlinear properly invariant plate theories, Arch. Rational Mech. Anal., 124 (1993), 157-199.  doi: 10.1007/BF00375134.  Google Scholar

[22]

L. Freddi, M. G. Mora and R. Paroni, Nonlinear thin-walled beams with a rectangular cross-section–Part I, Math. Models Methods Appl. Sci., 22 (2012), 1150016, 34. doi: 10.1142/S0218202511500163.  Google Scholar

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L. FreddiM. G. Mora and R. Paroni, Nonlinear thin-walled beams with a rectangular cross-section–Part Ⅱ, Math. Models Methods Appl. Sci., 23 (2013), 743-775.  doi: 10.1142/S0218202512500595.  Google Scholar

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G. FrieseckeR. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (2002), 1461-1506.  doi: 10.1002/cpa.10048.  Google Scholar

[25]

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G. Gargiulo and E. Zappale, A remark on the junction in a thin multi-domain: the non convex case, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 699-728.  doi: 10.1007/s00030-007-5046-8.  Google Scholar

[27]

A. GaudielloB. GustafssonC. Lefter and J. Mossino, Asymptotic analysis for monotone quasilinear problems in thin multidomains, Differential Integral Equations, 15 (2002), 623-640.   Google Scholar

[28]

A. GaudielloB. GustafssonC. Lefter and J. Mossino, Asymptotic analysis of a class of minimization problems in a thin multidomain, Calc. Var. Partial Differential Equations, 15 (2002), 181-201.  doi: 10.1007/s005260100114.  Google Scholar

[29]

A. GaudielloR. MonneauJ. MossinoF. Murat and A. Sili, On the junction of elastic plates and beams, C. R. Math. Acad. Sci. Paris, 335 (2002), 717-722.  doi: 10.1016/S1631-073X(02)02543-8.  Google Scholar

[30]

A. GaudielloR. MonneauJ. MossinoF. Murat and A. Sili, Junction of elastic plates and beams, ESAIM Control Optim. Calc. Var., 13 (2007), 419-457.  doi: 10.1051/cocv:2007036.  Google Scholar

[31]

A. Gaudiello and A. Sili, Asymptotic analysis of the eigenvalues of an elliptic problem in an anisotropic thin multidomain, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 739-754.  doi: 10.1017/S0308210510000521.  Google Scholar

[32]

A. Gaudiello and E. Zappale, Junction in a thin multidomain for a fourth order problem, Math. Models Methods Appl. Sci., 16 (2006), 1887-1918.  doi: 10.1142/S0218202506001753.  Google Scholar

[33]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001  Google Scholar

[34]

I. Gruais, Modeling of the junction between a plate and a rod in nonlinear elasticity, Asymptotic Anal., 7 (1993), 179-194.   Google Scholar

[35]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993, Oxford Science Publications.  Google Scholar

[36]

W. Laskowski and H. T. Nguyen, Effective energy integral functionals for thin films with bending moment in the Orlicz-Sobolev space setting, in Function Spaces X, vol. 102 of Banach Center Publ. doi: 10.4064/bc102-0-10.  Google Scholar

[37]

W. Laskowski and H. T. Nguyen, Effective energy integral functionals for thin films with three dimensional bending moment in the Orlicz-Sobolev space setting, Discuss. Math. Differ. Incl. Control Optim., 36 (2016), 7-31.  doi: 10.7151/dmdico.1179.  Google Scholar

[38]

H. Le Dret, Problèmes variationnels dans les multi-domaines, vol. 19 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1991, Modélisation des jonctions et applications. [Modeling of junctions and applications].  Google Scholar

[39]

H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., 74 (1995), 549-578.   Google Scholar

[40]

H. Le Dret and A. Raoult, Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results, Arch. Ration. Mech. Anal., 154 (2000), 101-134.  doi: 10.1007/s002050000100.  Google Scholar

[41]

J. Matos, Young measures and the absence of fine microstructures in a class of phase transitions, European J. Appl. Math., 3 (1992), 31-54.  doi: 10.1017/S095679250000067X.  Google Scholar

[42]

M. G. Mora and S. Müller, Derivation of the nonlinear bending-torsion theory for inextensible rods by Γ-convergence, Calc. Var. Partial Differential Equations, 18 (2003), 287-305.  doi: 10.1007/s00526-003-0204-2.  Google Scholar

[43]

M. G. Mora and S. Müller, Derivation of a rod theory for multiphase materials, Calc. Var. Partial Differential Equations, 28 (2007), 161-178.  doi: 10.1007/s00526-006-0039-8.  Google Scholar

[44]

F. Murat and A. Sili, Comportement asymptotique des solutions du système de l'élasticité linéarisée anisotrope hétérogène dans des cylindres minces, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 179–184. doi: 10.1016/S0764-4442(99)80159-1.  Google Scholar

[45]

F. Murat and A. Sili, Effets non locaux dans le passage 3d–1d en élasticité linéarisée anisotrope hétérogène, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 745–750. doi: 10.1016/S0764-4442(00)00232-9.  Google Scholar

[46]

L. Scardia, Asymptotic models for curved rods derived from nonlinear elasticity by Γ-convergence, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 1037-1070.  doi: 10.1017/S0308210507000194.  Google Scholar

[47]

L. Trabucho and J. Viano, Mathematical modelling of rods, in Handbook of Numerical Analysis, Vol. IV, Handb. Numer. Anal., Ⅳ, North-Holland, Amsterdam, 1996,487–974.  Google Scholar

[48]

V. Šverák, On the problem of two wells, in Microstructure and Phase Transition, vol. 54 of IMA Vol. Math. Appl., Springer, New York, 1993,183–189. doi: 10.1007/978-1-4613-8360-4_11.  Google Scholar

show all references

References:
[1]

E. AcerbiG. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string, J. Elasticity, 25 (1991), 137-148.  doi: 10.1007/BF00042462.  Google Scholar

[2]

E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal., 86 (1984), 125-145.  doi: 10.1007/BF00275731.  Google Scholar

[3]

J.-F. BabadjianE. Zappale and H. Zorgati, Dimensional reduction for energies with linear growth involving the bending moment, J. Math. Pures Appl., 90 (2008), 520-549.  doi: 10.1016/j.matpur.2008.07.003.  Google Scholar

[4]

D. Blanchard and G. Griso, Junction between a plate and a rod of comparable thickness in nonlinear elasticity, J. Elasticity, 112 (2013), 79-109.  doi: 10.1007/s10659-012-9401-6.  Google Scholar

[5]

M. Bocea and I. Fonseca, A Young measure approach to a nonlinear membrane model involving the bending moment, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 845-883.  doi: 10.1017/S0308210500003516.  Google Scholar

[6]

G. BouchittéI. Fonseca and M. L. Mascarenhas, Bending moment in membrane theory, J. Elasticity, 73 (2003), 75-99.  doi: 10.1023/B:ELAS.0000029996.20973.92.  Google Scholar

[7]

G. BouchittéI. Fonseca and M. L. Mascarenhas, The Cosserat vector in membrane theory: a variational approach, J. Convex Anal., 16 (2009), 351-365.   Google Scholar

[8]

R. BunoiuG. Cardone and S. Nazarov, Scalar problems in junctions of rods and a plate. Ⅱ. self-adjoint extensions and simulation models, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), 481-508.  doi: 10.1051/m2an/2017047.  Google Scholar

[9]

G. CaritaJ. MatiasM. Morandotti and D. R. Owen, Dimension reduction in the context of structured deformations, Journal of Elasticity, 133 (2018), 1-35.  doi: 10.1007/s10659-018-9670-9.  Google Scholar

[10]

N. Chaudhuri and S. Müller, Rigidity estimate for two incompatible wells, Calc. Var. Partial Differential Equations, 19 (2004), 379-390.  doi: 10.1007/s00526-003-0220-2.  Google Scholar

[11]

P. G. Ciarlet, Mathematical Elasticity: Three-dimensional Elasticity, vol. Ⅰ, North-Holland, Amsterdam, 1988.  Google Scholar

[12]

P. G. Ciarlet, Plates and Junctions in Elastic Multi-structures, vol. 14 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris; Springer-Verlag, Berlin, 1990, An asymptotic analysis.  Google Scholar

[13]

P. G. Ciarlet, Theory of Plates. Mathematical Elasticity, vol. Ⅱ, North-Holland, Amsterdam, 1997.  Google Scholar

[14]

S. Conti and B. Schweizer, Rigidity and gamma convergence for solid-solid phase transitions with SO(2) invariance, Comm. Pure Appl. Math., 59 (2006), 830-868.  doi: 10.1002/cpa.20115.  Google Scholar

[15]

G. Dal Maso, An Introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[16]

C. De Lellis and L. Székelyhidi, Simple proof of two-well rigidity, C. R. Math. Acad. Sci. Paris, 343 (2006), 367-370.  doi: 10.1016/j.crma.2006.07.008.  Google Scholar

[17]

R. Ferreira, Redução Dimensional em Elasticidade Não Linear Através da Γ-Convergência (Dimensional Reduction in Non-linear Elasticity via Γ-Convergence), Master's thesis, Faculty of Sciences of the University of Lisbon (FCUL), 2006. Google Scholar

[18]

I. FonsecaD. Kinderlehrer and P. Pedregal, Energy functionals depending on elastic strain and chemical composition, Calc. Var. Partial Differential Equations, 2 (1994), 283-313.  doi: 10.1007/BF01235532.  Google Scholar

[19]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer Monographs in Mathematics, Springer, New York, 2007.  Google Scholar

[20]

I. FonsecaS. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients, SIAM J. Math. Anal., 29 (1998), 736-756.  doi: 10.1137/S0036141096306534.  Google Scholar

[21]

D. D. FoxA. Raoult and J. C. Simo, A justification of nonlinear properly invariant plate theories, Arch. Rational Mech. Anal., 124 (1993), 157-199.  doi: 10.1007/BF00375134.  Google Scholar

[22]

L. Freddi, M. G. Mora and R. Paroni, Nonlinear thin-walled beams with a rectangular cross-section–Part I, Math. Models Methods Appl. Sci., 22 (2012), 1150016, 34. doi: 10.1142/S0218202511500163.  Google Scholar

[23]

L. FreddiM. G. Mora and R. Paroni, Nonlinear thin-walled beams with a rectangular cross-section–Part Ⅱ, Math. Models Methods Appl. Sci., 23 (2013), 743-775.  doi: 10.1142/S0218202512500595.  Google Scholar

[24]

G. FrieseckeR. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (2002), 1461-1506.  doi: 10.1002/cpa.10048.  Google Scholar

[25]

G. FrieseckeR. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal., 180 (2006), 183-236.  doi: 10.1007/s00205-005-0400-7.  Google Scholar

[26]

G. Gargiulo and E. Zappale, A remark on the junction in a thin multi-domain: the non convex case, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 699-728.  doi: 10.1007/s00030-007-5046-8.  Google Scholar

[27]

A. GaudielloB. GustafssonC. Lefter and J. Mossino, Asymptotic analysis for monotone quasilinear problems in thin multidomains, Differential Integral Equations, 15 (2002), 623-640.   Google Scholar

[28]

A. GaudielloB. GustafssonC. Lefter and J. Mossino, Asymptotic analysis of a class of minimization problems in a thin multidomain, Calc. Var. Partial Differential Equations, 15 (2002), 181-201.  doi: 10.1007/s005260100114.  Google Scholar

[29]

A. GaudielloR. MonneauJ. MossinoF. Murat and A. Sili, On the junction of elastic plates and beams, C. R. Math. Acad. Sci. Paris, 335 (2002), 717-722.  doi: 10.1016/S1631-073X(02)02543-8.  Google Scholar

[30]

A. GaudielloR. MonneauJ. MossinoF. Murat and A. Sili, Junction of elastic plates and beams, ESAIM Control Optim. Calc. Var., 13 (2007), 419-457.  doi: 10.1051/cocv:2007036.  Google Scholar

[31]

A. Gaudiello and A. Sili, Asymptotic analysis of the eigenvalues of an elliptic problem in an anisotropic thin multidomain, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 739-754.  doi: 10.1017/S0308210510000521.  Google Scholar

[32]

A. Gaudiello and E. Zappale, Junction in a thin multidomain for a fourth order problem, Math. Models Methods Appl. Sci., 16 (2006), 1887-1918.  doi: 10.1142/S0218202506001753.  Google Scholar

[33]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001  Google Scholar

[34]

I. Gruais, Modeling of the junction between a plate and a rod in nonlinear elasticity, Asymptotic Anal., 7 (1993), 179-194.   Google Scholar

[35]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993, Oxford Science Publications.  Google Scholar

[36]

W. Laskowski and H. T. Nguyen, Effective energy integral functionals for thin films with bending moment in the Orlicz-Sobolev space setting, in Function Spaces X, vol. 102 of Banach Center Publ. doi: 10.4064/bc102-0-10.  Google Scholar

[37]

W. Laskowski and H. T. Nguyen, Effective energy integral functionals for thin films with three dimensional bending moment in the Orlicz-Sobolev space setting, Discuss. Math. Differ. Incl. Control Optim., 36 (2016), 7-31.  doi: 10.7151/dmdico.1179.  Google Scholar

[38]

H. Le Dret, Problèmes variationnels dans les multi-domaines, vol. 19 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1991, Modélisation des jonctions et applications. [Modeling of junctions and applications].  Google Scholar

[39]

H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., 74 (1995), 549-578.   Google Scholar

[40]

H. Le Dret and A. Raoult, Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results, Arch. Ration. Mech. Anal., 154 (2000), 101-134.  doi: 10.1007/s002050000100.  Google Scholar

[41]

J. Matos, Young measures and the absence of fine microstructures in a class of phase transitions, European J. Appl. Math., 3 (1992), 31-54.  doi: 10.1017/S095679250000067X.  Google Scholar

[42]

M. G. Mora and S. Müller, Derivation of the nonlinear bending-torsion theory for inextensible rods by Γ-convergence, Calc. Var. Partial Differential Equations, 18 (2003), 287-305.  doi: 10.1007/s00526-003-0204-2.  Google Scholar

[43]

M. G. Mora and S. Müller, Derivation of a rod theory for multiphase materials, Calc. Var. Partial Differential Equations, 28 (2007), 161-178.  doi: 10.1007/s00526-006-0039-8.  Google Scholar

[44]

F. Murat and A. Sili, Comportement asymptotique des solutions du système de l'élasticité linéarisée anisotrope hétérogène dans des cylindres minces, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 179–184. doi: 10.1016/S0764-4442(99)80159-1.  Google Scholar

[45]

F. Murat and A. Sili, Effets non locaux dans le passage 3d–1d en élasticité linéarisée anisotrope hétérogène, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 745–750. doi: 10.1016/S0764-4442(00)00232-9.  Google Scholar

[46]

L. Scardia, Asymptotic models for curved rods derived from nonlinear elasticity by Γ-convergence, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 1037-1070.  doi: 10.1017/S0308210507000194.  Google Scholar

[47]

L. Trabucho and J. Viano, Mathematical modelling of rods, in Handbook of Numerical Analysis, Vol. IV, Handb. Numer. Anal., Ⅳ, North-Holland, Amsterdam, 1996,487–974.  Google Scholar

[48]

V. Šverák, On the problem of two wells, in Microstructure and Phase Transition, vol. 54 of IMA Vol. Math. Appl., Springer, New York, 1993,183–189. doi: 10.1007/978-1-4613-8360-4_11.  Google Scholar

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