Equilibrium of thin shells under large strains without through-the-thickness shear and self-penetration of matter

Paolo Maria Mariano; Domenico Mucci; Paolo Maria Mariano; Domenico Mucci

doi:10.3934/mine.2023092

Mathematics in Engineering

2023, Volume 5, Issue 6: 1-21. doi: 10.3934/mine.2023092

Previous Article Next Article

Research article Special Issues

Equilibrium of thin shells under large strains without through-the-thickness shear and self-penetration of matter

Paolo Maria Mariano ^{1
,
,},
Domenico Mucci ^{2
,
,}

1.
DICEA, Università di Firenze, via Santa Marta 3, I-50139 Firenze, Italy
2.
DSMFI, Università di Parma, Parco Area delle Scienze 53/A, I-43134 Parma, Italy

Received: 13 August 2022 Revised: 24 May 2023 Accepted: 24 May 2023 Published: 07 June 2023

We consider elastic thin shells without through-the-thickness shear and depict them as Gauss graphs of parametric surfaces. (We use the term shells to include plates and thin films therein.) We consider an energy depending on the first derivative of the Gauss map (so, it involves curvatures) and its second-rank minors. For it we prove existence of minimizers in terms of currents carried by Gauss graphs. In the limiting process we adopt sequences of competitors that satisfy a condition that prevents self-penetration of matter.
- shells,
- plates,
- thin films,
- curvature effects,
- geometric measure theory,
- Gauss graphs
Citation: Paolo Maria Mariano, Domenico Mucci. Equilibrium of thin shells under large strains without through-the-thickness shear and self-penetration of matter[J]. Mathematics in Engineering, 2023, 5(6): 1-21. doi: 10.3934/mine.2023092

Related Papers:

Abstract

We consider elastic thin shells without through-the-thickness shear and depict them as Gauss graphs of parametric surfaces. (We use the term shells to include plates and thin films therein.) We consider an energy depending on the first derivative of the Gauss map (so, it involves curvatures) and its second-rank minors. For it we prove existence of minimizers in terms of currents carried by Gauss graphs. In the limiting process we adopt sequences of competitors that satisfy a condition that prevents self-penetration of matter.

References

[1]	L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford: Oxford University Press, 2000.
[2]	S. S. Antman, Nonlinear problems of elasticity, New York, NY: Springer, 1995. https://doi.org/10.1007/978-1-4757-4147-6
[3]	S. S. Antman, J. L. Ericksen's work on Cosserat theories of rods and shells, J. Elast., in press. https://doi.org/10.1007/s10659-022-09914-3
[4]	G. Anzellotti, Functionals depending on curvatures, Rend. Sem. Mat. Univ. Politec. Torino, Fascicolo Speciale (1989), 47–62.
[5]	G. Anzellotti, R. Serapioni, I. Tamanini, Curvatures, functionals, currents, Indiana Univ. Math. J., 39 (1990), 617–669.
[6]	P. G. Ciarlet, J. Nečas, Unilateral problems in nonlinear three-dimensional elasticity, Arch. Rational Mech. Anal., 97 (1987), 171–188. https://doi.org/10.1007/BF00250807 doi: 10.1007/BF00250807
[7]	E. Davoli, M. Kružik, P. Piovano, U. Stefanelli, Magnetostatic thin films at large strain, Continuum Mech. Thermodyn., 33 (2021), 327–341. https://doi.org/10.1007/s00161-020-00904-1 doi: 10.1007/s00161-020-00904-1
[8]	J. L. Ericksen, C. A. Truesdell, Exact theory of stress and strain in rods and shells, Arch. Rational Mech. Anal., 1 (1957), 295–323. https://doi.org/10.1007/BF00298012 doi: 10.1007/BF00298012
[9]	D. D. Fox, A. Raoult, J. C. Simo, A justification of nonlinear properly invariant plate theories, Arch. Rational Mech. Anal., 124 (1993), 157–199. https://doi.org/10.1007/BF00375134 doi: 10.1007/BF00375134
[10]	G. Friesecke, R. D. James, S. Müller, A Hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence, Arch. Rational Mech. Anal., 180 (2006), 183–236. https://doi.org/10.1007/s00205-005-0400-7 doi: 10.1007/s00205-005-0400-7
[11]	M. Giaquinta, G. Modica, J. Souček, Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 106 (1989), 97–159. https://doi.org/10.1007/BF00251429 doi: 10.1007/BF00251429
[12]	M. Giaquinta, G. Modica, J. Souček, Erratum and addendum to "Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity", this archive, volume 106 (1989): 97–159, Arch. Rational Mech. Anal., 109 (1990), 385–392. https://doi.org/10.1007/BF00380382 doi: 10.1007/BF00380382
[13]	M. Giaquinta, G. Modica, J. Souček, Cartesian currents in the calculus of variations I: Cartesian currents, Berlin, Heidelberg: Springer, 1998.
[14]	M. Giaquinta, G. Modica, J. Souček, Cartesian currents in the calculus of variations II: Variational integrals, Berlin: Springer, 1998. https://doi.org/10.1007/978-3-662-06218-0
[15]	M. Giaquinta, D. Mucci, Maps into manifolds and currents: area and $W^{1, 2}$-, $W^{1/2}$-, $BV$-energies, Pisa: Edizioni della Normale, 2007.
[16]	H. Le Dret, A. Raoult, Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results, Arch. Rational Mech. Anal., 154 (2000), 101–134. https://doi.org/10.1007/s002050000100 doi: 10.1007/s002050000100
[17]	P. M. Mariano, Equilibrium configurations of mixture thin films undergoing large strains, Math. Method. Appl. Sci., 41 (2018), 479–489. https://doi.org/10.1002/mma.4667 doi: 10.1002/mma.4667
[18]	D. Mucci, On the curvature energy of Cartesian surfaces, J. Geom. Anal., 31 (2021), 8460–8519. https://doi.org/10.1007/s12220-020-00601-0 doi: 10.1007/s12220-020-00601-0
[19]	P. Neff, A geometrically exact planar Cosserat shell-model with microstructure: existence of minimizers for zero Cosserat couple modulus, Math. Mod. Meth. Appl. Sci., 17 (2007), 363–392. https://doi.org/10.1142/S0218202507001954 doi: 10.1142/S0218202507001954
[20]	P. Neff, M. Bîrsan, F. Osterbrink, Existence theorem for geometrically nonlinear Cosserat micropolar model under uniform convexity requirements, J. Elast., 121 (2015), 119–141. https://doi.org/10.1007/s10659-015-9517-6 doi: 10.1007/s10659-015-9517-6
[21]	O. Pantz, On the justification of the nonlinear inextensional plate model, Arch. Rational Mech. Anal., 167 (2003), 179–209. https://doi.org/10.1007/s00205-002-0238-1 doi: 10.1007/s00205-002-0238-1
[22]	J. C. Simo, D. D. Fox, On a stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization, Comput. Method. Appl. Mech. Eng., 72 (1989), 267–304. https://doi.org/10.1016/0045-7825(89)90002-9 doi: 10.1016/0045-7825(89)90002-9
[23]	J. C. Simo, J. E. Marsden, P. S. Krishnaprasad, The Hamiltonian structure of nonlinear elasticity: the material and convective representations of solids, rods, and plates, Arch. Rational Mech. Anal., 104 (1988), 125–183. https://doi.org/10.1007/BF00251673 doi: 10.1007/BF00251673

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)