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December  2013, 6(6): 1569-1586. doi: 10.3934/dcdss.2013.6.1569

Modeling a hard, thin curvilinear interface

Frédéric Lebon 1, and  Raffaella Rizzoni 2,

1. 

LMA, CNRS UPR 7051, Aix-Marseille University, Centrale Marseille, F 13402 Marseille Cedex 20, France
2. 

Dipartimento di Ingegneria, Universitá di Ferrara, I 44122 Ferrara, Italy

Received  June 2012 Revised  September 2012 Published  April 2013

In this paper, some results obtained on the asymptotic behavior of hard, thin curvilinear interfaces i.e., in cases where the interphase and adherents have comparable rigidities, are presented. The case of hard interfaces is investigated in terms of cylindrical coordinates and some analytical examples are presented.
Keywords: curved interface., contact mechanics, elasticity, imperfect interface, Asymptotic analysis.
Mathematics Subject Classification: Primary: 74M15, 74K35, 74B05; Secondary: 74Q1.
Citation: Frédéric Lebon, Raffaella Rizzoni. Modeling a hard, thin curvilinear interface. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1569-1586. doi: 10.3934/dcdss.2013.6.1569
References:
[1]

F. Ascione and G. Mancusi, Curve adhesive joints, Composite Structures, 94 (2012), 2657-2664. doi: 10.1016/j.compstruct.2012.03.024.

[2]

Y. Benveniste, A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media, Journal of the Mechanics and Physics of Solids, 54 (2006), 708-734. doi: 10.1016/j.jmps.2005.10.009.

[3]

Y. Benveniste and T. Miloh, Imperfect soft and stiff interfaces in two-dimensional elasticity, Mechanics of Materials, 33 (2001), 309-323. doi: 10.1016/S0167-6636(01)00055-2.

[4]

K. Bertoldi, D. Bigoni and W. Drugan, Structural interfaces in linear elasticity. Part I. Nonlocality and gradient approximations, Journal of the Mechanics and Physics of Solids, 55 (2007), 1-34. doi: 10.1016/j.jmps.2006.06.004.

[5]

K. Bertoldi, D. Bigoni and W. Drugan, Structural interfaces in linear elasticity. Part II. Effective properties and neutrality, Journal of the Mechanics and Physics of Solids, 55 (2007), 35-63. doi: 10.1016/j.jmps.2006.06.005.

[6]

D. Bigoni and A. Movchan, Statics and dynamics of structural interfaces in elasticity, International Journal of Solids and Structures, 39 (2002), 4843-4865. doi: 10.1016/S0020-7683(02)00416-X.

[7]

N. Challamel and U. A. Girhammar, Boundary-layer effect in composite beams with interlayer slip, Journal of Aerospace Engineering, 24 (2011), 199-209. doi: 10.1061/(ASCE)AS.1943-5525.0000027.

[8]

J. Cognard, R. C. Hcadec, L. Sohier and P. Davies, Analysis of the nonlinear behavior of adhesives in 2 bonded assemblies - comparison of tast and arcan tests, International Journal of Adhesion and Adhesives, 28 (2008), 393-404.

[9]

J. Cognard, P. Davies, L. Sohier and R. Créac'hcadec, A study of the non-linear behaviour of adhesively-bonded composite assemblies, Composite Structures, 76 (2006), 34-46. doi: 10.1016/j.compstruct.2006.06.006.

[10]

I. Doghri, "Mechanics of Deformable Solids. Linear, Nonlinear, Analytical and Computational Aspects," Springer-Verlag, Berlin, 2000.

[11]

V. A. Duong, A. D. Diaz, S. Chataigner and J.-F. Caron, A layerwise finite element for multilayers with imperfect interfaces, Composite Structures, 93 (2011), 3262-3271. doi: 10.1016/j.compstruct.2011.05.001.

[12]

W. Eckhaus, "Asymptotic Analysis of Singular Perturbations," Studies in Mathematics and its Applications, 9, North-Holland Publishing Co., Amsterdam-New York, 1979.

[13]

S. Kumar and J. P. Scanlan, Stress analysis of shaft-tube bonded joints using a variational method, Journal of Adhesion, 86 (2010), 369-394. doi: 10.1080/00218461003704329.

[14]

M. Kumar and Parul, Methods for solving singular perturbation problems arising in science and engineering, Mathematical and Computer Modelling, 54 (2011), 556-575. doi: 10.1016/j.mcm.2011.02.045.

[15]

F. Lebon and R. Rizzoni, Asymptotic analysis of a thin interface: The case involving similar rigidity, International Journal of Engineering Science, 48 (2010), 473-486. doi: 10.1016/j.ijengsci.2009.12.001.

[16]

F. Lebon and R. Rizzoni, Asymptotic behavior of a hard thin linear elastic interphase: An energy approach, International Journal of Solids and Structures, 48 (2011), 441-449. doi: 10.1016/j.ijsolstr.2010.10.006.

[17]

F. Lebon and R. Rizzoni, Asymptotic analysis of an adhesive joint with mismatch strain, European Journal of Mechanics, A Solids, 36 (2012), 1-8. doi: 10.1016/j.euromechsol.2012.02.005.

[18]

F. Lebon, R. Rizzoni and S. Ronel-Idrissi, Numerical analysis of two non-linear soft thin layers, Lecture Notes in Applied and Computational Mechanics, 61 (2012), 299-308. doi: 10.1007/978-3-642-24638-8_20.

[19]

F. Lebon and S. Ronel, First order numerical analysis of linear thin layers, Journal of Applied Mechanics, 74 (2007), 824-828. doi: 10.1115/1.2424716.

[20]

C. Licht, A. Léger and F. Lebon, Dynamics of elastic bodies connected by a thin adhesive layer, in "Ultrasonic Wave Propagation in Non Homogeneous Media" (eds. M. Duchamp and A. Léger), Springer Proceedings in Physics, 128, Springer, Berlin-Heidelberg, (2009), 99-110. doi: 10.1007/978-3-540-89105-5_9.

[21]

C. Licht and G. Michaille, A modelling of elastic adhesive bonded joints, Advances in Mathematical Sciences and Applications, 7 (1997), 711-740.

[22]

R. Rizzoni and F. Lebon, Asymptotic analysis of an adhesive joint with mismatch strain, European Journal of Mechanics, A Solids, 36 (2012), 1-8. doi: 10.1016/j.euromechsol.2012.02.005.

[23]

F. Zaittouni, F. Lebon and C. Licht, Theoretical and numerical study of the behaviour of bonded plates, [Etude théorique et numérique du comportement d'un assemblage de plaques], Comptes Rendus - Mecanique, 330 (2002), 359-364.

show all references

References:
[1]

F. Ascione and G. Mancusi, Curve adhesive joints, Composite Structures, 94 (2012), 2657-2664. doi: 10.1016/j.compstruct.2012.03.024.

[2]

Y. Benveniste, A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media, Journal of the Mechanics and Physics of Solids, 54 (2006), 708-734. doi: 10.1016/j.jmps.2005.10.009.

[3]

Y. Benveniste and T. Miloh, Imperfect soft and stiff interfaces in two-dimensional elasticity, Mechanics of Materials, 33 (2001), 309-323. doi: 10.1016/S0167-6636(01)00055-2.

[4]

K. Bertoldi, D. Bigoni and W. Drugan, Structural interfaces in linear elasticity. Part I. Nonlocality and gradient approximations, Journal of the Mechanics and Physics of Solids, 55 (2007), 1-34. doi: 10.1016/j.jmps.2006.06.004.

[5]

K. Bertoldi, D. Bigoni and W. Drugan, Structural interfaces in linear elasticity. Part II. Effective properties and neutrality, Journal of the Mechanics and Physics of Solids, 55 (2007), 35-63. doi: 10.1016/j.jmps.2006.06.005.

[6]

D. Bigoni and A. Movchan, Statics and dynamics of structural interfaces in elasticity, International Journal of Solids and Structures, 39 (2002), 4843-4865. doi: 10.1016/S0020-7683(02)00416-X.

[7]

N. Challamel and U. A. Girhammar, Boundary-layer effect in composite beams with interlayer slip, Journal of Aerospace Engineering, 24 (2011), 199-209. doi: 10.1061/(ASCE)AS.1943-5525.0000027.

[8]

J. Cognard, R. C. Hcadec, L. Sohier and P. Davies, Analysis of the nonlinear behavior of adhesives in 2 bonded assemblies - comparison of tast and arcan tests, International Journal of Adhesion and Adhesives, 28 (2008), 393-404.

[9]

J. Cognard, P. Davies, L. Sohier and R. Créac'hcadec, A study of the non-linear behaviour of adhesively-bonded composite assemblies, Composite Structures, 76 (2006), 34-46. doi: 10.1016/j.compstruct.2006.06.006.

[10]

I. Doghri, "Mechanics of Deformable Solids. Linear, Nonlinear, Analytical and Computational Aspects," Springer-Verlag, Berlin, 2000.

[11]

V. A. Duong, A. D. Diaz, S. Chataigner and J.-F. Caron, A layerwise finite element for multilayers with imperfect interfaces, Composite Structures, 93 (2011), 3262-3271. doi: 10.1016/j.compstruct.2011.05.001.

[12]

W. Eckhaus, "Asymptotic Analysis of Singular Perturbations," Studies in Mathematics and its Applications, 9, North-Holland Publishing Co., Amsterdam-New York, 1979.

[13]

S. Kumar and J. P. Scanlan, Stress analysis of shaft-tube bonded joints using a variational method, Journal of Adhesion, 86 (2010), 369-394. doi: 10.1080/00218461003704329.

[14]

M. Kumar and Parul, Methods for solving singular perturbation problems arising in science and engineering, Mathematical and Computer Modelling, 54 (2011), 556-575. doi: 10.1016/j.mcm.2011.02.045.

[15]

F. Lebon and R. Rizzoni, Asymptotic analysis of a thin interface: The case involving similar rigidity, International Journal of Engineering Science, 48 (2010), 473-486. doi: 10.1016/j.ijengsci.2009.12.001.

[16]

F. Lebon and R. Rizzoni, Asymptotic behavior of a hard thin linear elastic interphase: An energy approach, International Journal of Solids and Structures, 48 (2011), 441-449. doi: 10.1016/j.ijsolstr.2010.10.006.

[17]

F. Lebon and R. Rizzoni, Asymptotic analysis of an adhesive joint with mismatch strain, European Journal of Mechanics, A Solids, 36 (2012), 1-8. doi: 10.1016/j.euromechsol.2012.02.005.

[18]

F. Lebon, R. Rizzoni and S. Ronel-Idrissi, Numerical analysis of two non-linear soft thin layers, Lecture Notes in Applied and Computational Mechanics, 61 (2012), 299-308. doi: 10.1007/978-3-642-24638-8_20.

[19]

F. Lebon and S. Ronel, First order numerical analysis of linear thin layers, Journal of Applied Mechanics, 74 (2007), 824-828. doi: 10.1115/1.2424716.

[20]

C. Licht, A. Léger and F. Lebon, Dynamics of elastic bodies connected by a thin adhesive layer, in "Ultrasonic Wave Propagation in Non Homogeneous Media" (eds. M. Duchamp and A. Léger), Springer Proceedings in Physics, 128, Springer, Berlin-Heidelberg, (2009), 99-110. doi: 10.1007/978-3-540-89105-5_9.

[21]

C. Licht and G. Michaille, A modelling of elastic adhesive bonded joints, Advances in Mathematical Sciences and Applications, 7 (1997), 711-740.

[22]

R. Rizzoni and F. Lebon, Asymptotic analysis of an adhesive joint with mismatch strain, European Journal of Mechanics, A Solids, 36 (2012), 1-8. doi: 10.1016/j.euromechsol.2012.02.005.

[23]

F. Zaittouni, F. Lebon and C. Licht, Theoretical and numerical study of the behaviour of bonded plates, [Etude théorique et numérique du comportement d'un assemblage de plaques], Comptes Rendus - Mecanique, 330 (2002), 359-364.

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