Abstract
We consider a class of models for nonlinearly elastic surfaces in this work. We have in mind thin, highly deformable structures modeled directly as two-dimensional nonlinearly elastic continua, accounting for finite membrane and bending strains and thickness change. We assume that the stored-energy density is polyconvex with respect to the second gradient of the deformation, and we require that it grow unboundedly as the local area ratio approaches zero. For sufficiently fast growth, we show that the latter is uniformly bounded away from zero at an energy minimizer. With this in hand, we rigorously derive the weak form of the Euler-Lagrange equilibrium equations.
Similar content being viewed by others
References
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Anicic, S.: Polyconvexity and existence theorems for nonlinearly elastic shells. J. Elast. 132, 161–173 (2018)
Antman, S.S.: Nonlinear Problems of Elasticity, 2nd edn. Springer, New York (2005)
Balaban, M.M., Green, A.E., Naghdi, P.M.: Simple force multipoles in the theory of deformable surfaces. J. Math. Phys. 8, 1026–1036 (1967)
Ball, J.M., Currie, J.C., Olver, P.J.: Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41, 135–174 (1981)
Cohen, H., DeSilva, C.N.: On a nonlinear theory of elastic shells. J. Méc. 7, 459–464 (1968)
Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd edn. Springer, New York (2008)
Evans, L.C.: Partial Differential Equations, 2nd edn. Am. Math. Soc., Providence (2010)
Healey, T.J., Li, Q., Cheng, R.-B.: Wrinkling behavior of highly stretched rectangular elastic films via parametric global bifurcation. J. Nonlinear Sci. 23, 777–805 (2013)
Healey, T.J., Krömer, S.: Injective weak solutions in second-gradient nonlinear elasticity. ESAIM Control Optim. Calc. Var. 15, 863–871 (2009)
Hilgers, M.G., Pipkin, A.C.: Bending energy of highly elastic membranes. Q. Appl. Math. 50, 389–400 (1992)
Hilgers, M.G., Pipkin, A.C.: Bending energy of highly elastic membranes II. Q. Appl. Math. 54, 307–316 (1996)
Li, Q., Healey, T.J.: Stability boundaries for wrinkling in highly stretches elastic sheets. J. Mech. Phys. Solids 97, 260–274 (2016)
Müller, I., Strehlow, P.: Rubber and Rubber Balloons. Springer, Berlin (2004)
Pipkin, A.C.: Relaxed energy densities for large deformations of membranes. IMA J. Appl. Math. 52, 297–308 (1994)
Rektorys, K.: In: Reidel, D. (ed.) Variational Methods in Mathematics, Science and Engineering, 2nd edn. Reidel Dordrecht (1980)
Steigmann, D.J., Ogden, R.W.: Elastic surface-substrate interactions. Proc. R. Soc. Lond. A 455, 437–474 (1993)
Acknowledgements
This work was supported in part by the National Science Foundation through grant DMS-2006586.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Millard Beatty on the occasion of his 90th Birthday
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Healey, T.J. Existence of Weak Solutions for Non-Simple Elastic Surface Models. J Elast 151, 47–57 (2022). https://doi.org/10.1007/s10659-021-09840-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-021-09840-w