In this paper, we consider a family of linearly elastic shells in normal compliance contact with a deformable foundation. We obtain error estimates for the approximation of the solution to a three-dimensional (3D) problem with a two-dimensional (2D) limit solution in terms of the thickness ε of the shell. The proof of the main result relies on a corrector method. We also provide a strong formulation for the 2D obstacle boundary value problem associated.
LionsJL. Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Lecture notes in mathematics. Cham: Springer-Verlag, 1973.
2.
CiarletPGDestuynderP. Justification of the two-dimensional linear plate model. J Mecanique1979; 18(2): 315–344.
3.
CiarletPG. Mathematical elasticity. volume II: Theory of plates. Amsterdam: Academic Press, Elsevier, 1997.
4.
DestuynderP. Sur une justification des modèles de plaques et de coques par les méthodes asymptotiques. PhD Thesis, Université Pierre et Marie Curie, Paris, 1980.
5.
BermúdezAViañoJM. Une justification des équations de la thermoélasticité des poutres à section variable par des méthodes asymptotiques. RAIRO Analyse numerique1984; 18(4): 347–376.
6.
Rodríguez-ArósAViañoJM. Mathematical justification of viscoelastic beam models by asymptotic methods. J Math Anal Appl2010; 370(2): 607–634.
7.
ViañoJMRodríguez-ArósA. Mathematical justification of Kelvin–Voigt beam models by asymptotic methods. Z Angew Math Phys2012; 63(3): 529–556.
8.
TrabuchoLViañoJM. Mathematical modelling of rods. In: Finite element methods (part 2), numerical methods for solids (part 2), Handbook of Numerical Analysis, vol. 4. Elsevier, 1996, pp. 487–974.
9.
ViañoJMFigueiredoJRibeiroC, et al. A model for bending and stretching of piezoelectric rods obtained by asymptotic analysis. Z Angew Math Phys2015; 66(3): 1207–1232.
10.
IragoHViañoJRodríguez-ArósA. Asymptotic derivation of frictionless contact models for elastic rods on a foundation with normal compliance. Nonlinear Anal Real World Appl2013; 14(1): 852–866.
11.
Rodríguez-ArósAViañoJM. A bending–stretching model in adhesive contact for elastic rods obtained by using asymptotic methods. Nonlinear Anal Real World Appl2015; 22: 632–644.
12.
ViañoJMRodríguez-ArósASofoneaM. Asymptotic derivation of quasistatic frictional contact models with wear for elastic rods. J Math Anal Appl2013; 401(2): 641–653.
13.
CiarletPG. Mathematical elasticity, volume III: Theory of shells. Amsterdam: North-Holland Publishing Co., 2000.
14.
CastiñeiraGRodríguez-ArósA. Linear viscoelastic shells: an asymptotic approach. Asympt Anal2018; 107(3–4): 169–201.
15.
CastiñeiraGRodríguez-ArósA. On the justification of viscoelastic elliptic membrane shell equations. J Elasticity2018; 130(1): 85–113.
16.
CastiñeiraGRodríguez-ArósA. On the justification of viscoelastic flexural shell equations. Comput Math Appl2019; 77(11): 2933–2942.
17.
Cao-RialMTCastiñeiraGRodríguez-ArósA, et al. Asymptotic analysis of elliptic membrane shells in thermoelastodynamics. J Elasticity2021; 143(2): 385–409.
18.
XiaoL. Asymptotic analysis of dynamic problems for linearly elastic shells–justification of equations for dynamic membrane shells. Asympt Anal1998; 17(2): 121–134.
19.
XiaoL. Asymptotic analysis of dynamic problems for linearly elastic shells – justification of equations for dynamic Koiter shells. Chinese Ann Math2001; 22(03): 267–274.
20.
Rodríguez-ArósA. Mathematical justification of an elastic elliptic membrane obstacle problem. C R Mecanique2017; 345(2): 153–157.
21.
Rodríguez-ArósA. Models of elastic shells in contact with a rigid foundation: an asymptotic approach. J Elasticity2018; 130(2): 211–237.
22.
Cao-RialMTRodríguez-ArósA. Asymptotic analysis of unilateral contact problems for linearly elastic shells: error estimates in the membrane case. Nonlinear Anal Real World Appl2019; 48: 40–53.
23.
Rodríguez-ArósACao-RialMT. Asymptotic analysis of linearly elastic shells in normal compliance contact: convergence for the elliptic membrane case. Z Angew Math Phys2018; 69(5): 115.
24.
Cao-RialMTCasti neiraGRodríguez-ArósA, et al. Asymptotic analysis of a problem for dynamic thermoelastic shells in normal damped response contact. Commun Nonlinear Sci Numer Simul2021; 103: 105995.
25.
CiarletPGMardareCPiersantiP. An obstacle problem for elliptic membrane shells. Math Mech Solids2019; 24(5): 1503–1529.
26.
PiersantiP. Asymptotic analysis of linearly elastic flexural shells subjected to an obstacle in absence of friction. J Nonlinear Sci2023; 33(4).
27.
CiarletPGPiersantiP. A confinement problem for a linearly elastic Koiter’s shell; [un problème de confinement pour une coque de koiter linéairement élastique]. C R Math2019; 357(2): 221–230.
28.
MardareC. Asymptotic analysis of linearly elastic shells: error estimates in the membrane case. Asympt Anal1998; 17: 31–51.
29.
HanWSofoneaM. Quasistatic contact problems in viscoelasticity and viscoplasticity. AMS/IP studies in advanced mathematics. Providence, RI: American Mathematical Society, 2002.
30.
GeneveyK. A regularity result for a linear membrane shell problem. Model Math Anal Numer1996; 30(4): 467–488.
31.
PiersantiP. On the improved interior regularity of a boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle. Discrete Contin Dyn Syst2022; 42(2): 1011–1037.
32.
MeixnerAPiersantiP. Numerical approximation of the solution of an obstacle problem modelling the displacement of elliptic membrane shells via the penalty method. Appl Math Optim2024; 89(2): 45.
33.
PiersantiP. On the justification of the frictionless time-dependent Koiter’s model for thermoelastic shells. J Differ Equ2021; 296: 50–106.
34.
PiersantiPShenX. Numerical methods for static shallow shells lying over an obstacle. Numer Algorithms2020; 85: 623–652.
