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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 |
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.
References:
[1] |
E. Acerbi, G. 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. |
[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. |
[3] |
J.-F. Babadjian, E. 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. |
[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. |
[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. |
[6] |
G. Bouchitté, I. Fonseca and M. L. Mascarenhas,
Bending moment in membrane theory, J. Elasticity, 73 (2003), 75-99.
doi: 10.1023/B:ELAS.0000029996.20973.92. |
[7] |
G. Bouchitté, I. Fonseca and M. L. Mascarenhas,
The Cosserat vector in membrane theory: a variational approach, J. Convex Anal., 16 (2009), 351-365.
|
[8] |
R. Bunoiu, G. 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. |
[9] |
G. Carita, J. Matias, M. 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. |
[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. |
[11] |
P. G. Ciarlet, Mathematical Elasticity: Three-dimensional Elasticity, vol. Ⅰ, North-Holland, Amsterdam, 1988. |
[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. |
[13] |
P. G. Ciarlet, Theory of Plates. Mathematical Elasticity, vol. Ⅱ, North-Holland, Amsterdam, 1997. |
[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. |
[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. |
[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. |
[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. |
[18] |
I. Fonseca, D. 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. |
[19] |
I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer Monographs in Mathematics, Springer, New York, 2007. |
[20] |
I. Fonseca, S. 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. |
[21] |
D. D. Fox, A. 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. |
[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. |
[23] |
L. Freddi, M. 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. |
[24] |
G. Friesecke, R. 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. |
[25] |
G. Friesecke, R. 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. |
[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. |
[27] |
A. Gaudiello, B. Gustafsson, C. Lefter and J. Mossino,
Asymptotic analysis for monotone quasilinear problems in thin multidomains, Differential Integral Equations, 15 (2002), 623-640.
|
[28] |
A. Gaudiello, B. Gustafsson, C. 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. |
[29] |
A. Gaudiello, R. Monneau, J. Mossino, F. 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. |
[30] |
A. Gaudiello, R. Monneau, J. Mossino, F. Murat and A. Sili,
Junction of elastic plates and beams, ESAIM Control Optim. Calc. Var., 13 (2007), 419-457.
doi: 10.1051/cocv:2007036. |
[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. |
[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. |
[33] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001 |
[34] |
I. Gruais,
Modeling of the junction between a plate and a rod in nonlinear elasticity, Asymptotic Anal., 7 (1993), 179-194.
|
[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. |
[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. |
[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. |
[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]. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
E. Acerbi, G. 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. |
[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. |
[3] |
J.-F. Babadjian, E. 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. |
[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. |
[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. |
[6] |
G. Bouchitté, I. Fonseca and M. L. Mascarenhas,
Bending moment in membrane theory, J. Elasticity, 73 (2003), 75-99.
doi: 10.1023/B:ELAS.0000029996.20973.92. |
[7] |
G. Bouchitté, I. Fonseca and M. L. Mascarenhas,
The Cosserat vector in membrane theory: a variational approach, J. Convex Anal., 16 (2009), 351-365.
|
[8] |
R. Bunoiu, G. 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. |
[9] |
G. Carita, J. Matias, M. 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. |
[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. |
[11] |
P. G. Ciarlet, Mathematical Elasticity: Three-dimensional Elasticity, vol. Ⅰ, North-Holland, Amsterdam, 1988. |
[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. |
[13] |
P. G. Ciarlet, Theory of Plates. Mathematical Elasticity, vol. Ⅱ, North-Holland, Amsterdam, 1997. |
[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. |
[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. |
[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. |
[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. |
[18] |
I. Fonseca, D. 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. |
[19] |
I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer Monographs in Mathematics, Springer, New York, 2007. |
[20] |
I. Fonseca, S. 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. |
[21] |
D. D. Fox, A. 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. |
[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. |
[23] |
L. Freddi, M. 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. |
[24] |
G. Friesecke, R. 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. |
[25] |
G. Friesecke, R. 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. |
[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. |
[27] |
A. Gaudiello, B. Gustafsson, C. Lefter and J. Mossino,
Asymptotic analysis for monotone quasilinear problems in thin multidomains, Differential Integral Equations, 15 (2002), 623-640.
|
[28] |
A. Gaudiello, B. Gustafsson, C. 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. |
[29] |
A. Gaudiello, R. Monneau, J. Mossino, F. 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. |
[30] |
A. Gaudiello, R. Monneau, J. Mossino, F. Murat and A. Sili,
Junction of elastic plates and beams, ESAIM Control Optim. Calc. Var., 13 (2007), 419-457.
doi: 10.1051/cocv:2007036. |
[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. |
[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. |
[33] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001 |
[34] |
I. Gruais,
Modeling of the junction between a plate and a rod in nonlinear elasticity, Asymptotic Anal., 7 (1993), 179-194.
|
[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. |
[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. |
[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. |
[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]. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |

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