We consider a mathematical model which describes the equilibrium of two elastic membranes fixed on their boundary and attached to an adhesive body, say a glue. The variational formulation of the model is in a form of an elliptic quasivariational inequality for the displacement field. We prove the unique weak solvability of the model, and then we state and prove a convergence result, for which we provide the corresponding mechanical interpretation. Next, we consider two associated optimization problems for which we provide existence results. Finally, we the present numerical simulation which validates our convergence result. We end this paper with some concluding remarks and an Appendix, in which we present the preliminary material needed in this paper.
AtanackovicTMOparnicaLPilipovicS.On a model of viscoelastic rod in unilateral contact with a rigid wall. IMA J Appl Math2006; 71: 1–13.
2.
BelgacemFBBernardiCBlouzaA, et al. On the unilateral contact between membranes, part 1: finite element discretization and mixed reformulation. Math Model Nat Phenom2009; 4: 21–43.
3.
BelgacemFBBernardiCBlouzaA, et al. A finite element discretization of the contact between two membranes. Math Model Numer Anal2009; 43: 33–52.
4.
GoldbergNO’ReillyO.On contact point motion in the vibration analysis of elastic rods. J Sound Vib2020; 487: 115579.
5.
HanCWriggersP.On the error indication of shells in unilateral frictionless contact. Comput Mech2002; 28: 169–176.
6.
PiersantiPShenX.Numerical methods for static shallow shells lying over an obstacle. Numer Algorithms2020; 85: 623–652.
7.
SofoneaMMateiA.Variational inequalities with applications. A study of antiplane frictional contact problems (advances in mechanics and mathematics), vol. 18. New York: Springer, 2009.
8.
Rodrguez-ArósAViañoJMSofoneaM.Numerical analysis of a frictional contact problem for viscoelastic materials with long-term memory. Numer Math2007; 108: 327–358.
9.
ViañoJM.Análisis numérico de un método numérico con elementos finitos para problemas de contacto unilateral sin roazimiento en elasticidad: aproximaciòn y resoluciòn de los problemas discretos. Rev Internac Métod Numér Cálc Diseño Ingr1986; 2: 63–86.
10.
AlnaesMSLoggAØlgaardKB, et al. Unified form language: a domain-specific language for weak formulations of partial differential equations. ACM Trans Math Software2014; 40: 1–37.
11.
BarattaIADeanJPDokkenJS, et al. DOLFINx: the next generation FEniCS problem solving environment. Zenodo [preprint], 2023. .
12.
ScroggsMDokkenJSRichardsonC, et al. Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes. ACM Trans Math Software2022; 48(2): 1–23.
13.
ScroggsMWBarattaIARichardsonCN, et al. Basix: a runtime finite element basis evaluation library. J Open Source Softw2022; 7(73): 3982.
14.
GlowinskiR.Numerical methods for nonlinear variational problems. New York: Springer-Verlag, 1984.
15.
GlowinskiRLionsJLTrémolièresR.Numerical analysis of variational inequalities. Amsterdam: North-Holland, 1981.
16.
HanWReddyB.Plasticity: mathematical theory and numerical analysis. New York: Springer-Verlag, 1999.
17.
HanWSofoneaM.Quasistatic contact problems in viscoelasticity and viscoplasticity (studies in advanced mathematics), vol. 30. Providence, RI: American Mathematical Society; Somerville, MA: International Press, 2002.
18.
SofoneaMMigórskiS.Variational-hemivariational inequalities with applications (pure and applied mathematics). Boca Raton, FL; London: Chapman & Hall/CRC Press, 2018.
19.
Rodríguez-ArósA.Mathematical justification of the obstacle problem for elastic elliptic membrane shells. Appl Anal2018; 97: 1261–1280.
20.
Rodríguez-ArósACao-RialM.Asymptotic analysis of linearly elastic shells in normal compliance contact: convergence for the elliptic membrane case. Z Angew Math Phys2018; 69. 115.
21.
SofoneaMMateiA.Mathematical models in contact mechanics (London mathematical society lecture note series), vol. 398. Cambridge: Cambridge University Press, 2012.
