On the Justification of Viscoelastic Elliptic Membrane Shell Equations
Source abstract
Abstract
We consider a family of linearly viscoelastic shells with thickness (2varepsilon), clamped along their entire lateral face, all having the same middle surface (S=oldsymbol{ heta}(ar{omega})subset mathbb{R}3), where (omegasubsetmathbb{R}2) is a bounded and connected open set with a Lipschitz-continuous boundary (gamma). We make an essential geometrical assumption on the middle surface (S), which is satisfied if (gamma) and (oldsymbol{ heta}) are smooth enough and (S) is uniformly elliptic. We show that, if the applied body force density is (O(1)) with respect to (varepsilon) and surface tractions density is (O(varepsilon)), the solution of the scaled variational problem in curvilinear coordinates, (oldsymbol{u}( varepsilon)), defined over the fixed domain (varOmega=omega imes (-1,1)) for each (tin[0,T]), converges to a limit (oldsymbol{u}) with (ualpha(varepsilon) ightarrow ualpha) in (W1,2(0,T,H 1(varOmega))) and (u3(varepsilon) ightarrow u3) in (W1,2(0,T,L2(varOmega))) as (varepsilon o0). Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average (ar{oldsymbol{u}}= frac{1}{2}int-11 oldsymbol{u}dx3), which belongs to the space (W1,2(0,T, VM( omega))), where $$VM(omega)=H10(omega) imes H10(omega) imes L 2(omega), $$ satisfies what we have identified as (scaled) two-dimensional equations of a viscoelastic membrane elliptic shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.
Source: expertdata · Quality: candidate unverified · Source record ↗
Research context
OpenAlex topics: Advanced Mathematical Modeling in Engineering · Contact Mechanics and Variational Inequalities · Elasticity and Material Modeling
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (34)
Show source-provided references
1. Lions, J.-L.: Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal. Lecture Notes in Mathematics, vol. 323. Springer, Berlin (1973)10.1007/BFb0060528 ↗
2. Ciarlet, P.G., Destuynder, P.: A justification of the two-dimensional linear plate model. J. Méc. 18(2), 315–344 (1979)
3. Destuynder, P.: Sur Une Justification des Modèles de Plaques et de Coques Par les Méthodes Asymptotiques. Université Pierre et Marie Curie, Paris (1980). Ph.D. thesis
4. Ciarlet, P.G., Lods, V.: On the ellipticity of linear membrane shell equations. J. Math. Pures Appl. 75, 107–124 (1996)
5. Ciarlet, P.G., Lods, V.: Asymptotic analysis of linearly elastic shells. Justification of membrane shell equations. Arch. Ration. Mech. Anal. 136, 119–161 (1996)10.1007/BF02316975 ↗
6. Ciarlet, P.G.: Mathematical Elasticity. Vol. III: Theory of Shells. Studies in Mathematics and Its Applications, vol. 29. North-Holland, Amsterdam (2000)
7. Li-ming, X.: Asymptotic analysis of dynamic problems for linearly elastic shells—justification of equations for dynamic membrane shells. Asymptot. Anal. 17, 121–134 (1998)
8. Li-ming, X.: Asymptotic analysis of dynamic problems for linearly elastic shells—justification of equations for dynamic flexural shells. Chin. Ann. Math. 22B, 13–22 (2001)
9. Li-ming, X.: Asymptotic analysis of dynamic problems for linearly elastic shells—justification of equations for dynamic koiter shells. Chin. Ann. Math. 22B, 267–274 (2001)
10. Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)10.1007/978-3-642-66165-5 ↗
11. Lemaitre, J., Chaboche, J.L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1990)10.1017/CBO9781139167970 ↗
12. Pipkin, A.C.: Lectures in Viscoelasticity Theory. Applied Sciences. Springer, New York (1972)10.1007/978-1-4615-9970-8 ↗
13. Han, W., Shillor, M., Sofonea, M.: Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage. J. Comput. Appl. Math. 137(2), 377–398 (2001)10.1016/S0377-0427(00)00707-X ↗
14. Han, W., Sofonea, M.: Time-dependent variational inequalities for viscoelastic contact problems. J. Comput. Appl. Math. 136(1–2), 369–387 (2001)10.1016/S0377-0427(00)00627-0 ↗
15. Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplascticity. Studies in Advanced Mathematics. Am. Math. Soc., Providence (2002)10.1090/amsip/030 ↗
16. Jarušek, J.: Contact problems with given time-dependent friction force in linear viscoelasticity. Comment. Math. Univ. Carol. 31(2), 257–262 (1990)
17. Migórski, S., Ochal, A.: Hemivariational inequality for viscoelastic contact problem with slip-dependent friction. Nonlinear Anal., Theory Methods Appl. 61(1–2), 135–161 (2005)10.1016/j.na.2004.11.018 ↗
18. Rodríguez-Arós, A., Sofonea, M., Viaño, J.M.: Numerical Analysis of a Frictional Contact Problem for Viscoelastic Materials with Long-Term Memory. Springer, Berlin (2006)
19. Bock, I., Jarušek, J.: Unilateral dynamic contact of viscoelastic von Kármán plates. Adv. Math. Sci. Appl. 16(1), 175–187 (2006)
20. Bock, I., Lovíšek, J.: On unilaterally supported viscoelastic von Kármán plates with a long memory. Math. Comput. Simul. 61, 399–407 (2003)10.1016/S0378-4754(02)00095-2 ↗
21. Bock, I.: On von Kárman equations for viscoelastic plates. J. Comput. Appl. Math. 63(1), 277–282 (1995)10.1016/0377-0427(95)00082-8 ↗
22. Muñoz-Rivera, J.E., Perla-Menzala, G.: Decay rates of solutions to a von Kármán system for viscoelastic plates with memory. Q. Appl. Math. 57(1), 181–200 (1999)10.1090/qam/1672191 ↗
23. Bock, I.: On large deflections of viscoelastic plates. Math. Comput. Simul. 50(1–4), 135–143 (1999)10.1016/S0378-4754(99)00066-X ↗
24. Rossikhin, Y.A., Shitikova, M.: Analysis of free non-linear vibrations of a viscoelastic plate under the conditions of different internal resonances. Int. J. Non-Linear Mech. 41(2), 313–325 (2006)10.1016/j.ijnonlinmec.2005.08.002 ↗
25. Bock, I., Jarušek, J.: System Modeling and Optimization. Springer, Heidelberg (2013)
26. Quarteroni, A., Formaggia, L.: Mathematical modelling and numerical simulation of the cardiovascular system. In: Handbook of Numerical Analysis. Special Volume Computational Models for the Human Body, vol. XII. Elsevier, Amsterdam (2004)
27. Castiñeira, G., Rodríguez-Arós, Á.: Derivation of models for linear viscoelastic shells by using asymptotic analysis, Unpublished results. arXiv:1604.02280v2
28. Rodríguez-Arós, Á., Viaño, J.M.: Mathematical justification of viscoelastic beam models by asymptotic methods. J. Math. Anal. Appl. 370(2), 607–634 (2010)10.1016/j.jmaa.2010.04.067 ↗
29. Rodríguez-Arós, Á., Viaño, J.M.: Mathematical justification of Kelvin-Voigt beam models by asymptotic methods. Z. Angew. Math. Phys. 63(3), 529–556 (2012)10.1007/s00033-011-0180-x ↗
30. Shillor, M., Sofonea, M., Telega, J.: Models and Analysis of Quasistatic Contact. Lecture Notes in Physics, vol. 655. Springer, Berlin (2004)10.1007/b99799 ↗
31. Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems. Springer, Berlin (2014)
32. Collard, C., Miara, B.: Asymptotic analysis of the stresses in thin elastic shells. Arch. Ration. Mech. Anal. 148, 233–264 (1999)10.1007/s002050050161 ↗
33. Castiñeira, G., Rodríguez-Arós, Á.: On the justification of viscoelastic generalized membrane shell equations. Unpublished results
34. Castiñeira, G., Rodríguez-Arós, Á.: On the justification of viscoelastic flexural shell equations. Unpublished results