# Contributions to the Mathematical Foundations of the Theory of Nonlinearly Elastic Shells

Project: Research

## Description

The general principles of Continuum Mechanics and of Elasticity Theory show that thedeformation of a nonlinearly elastic body subjected to applied forces and boundary conditionsis a solution to a nonlinear boundary value problem posed over a bounded open subset ofR3, the closure of which is the reference configuration of the elastic body. This system is ineffect the Euler-Lagrange equations associated with the minimization of a functional, calledthe energy, over an appropriate set of admissible deformations. The existence of solutions tothis minimization problem has been established in a fundamental contribution of J. Ball in1977; as a result, the mathematical foundations of the theory of three-dimensional nonlinearelasticity are now well established.A nonlinearly elastic shell with constant thickness is an elastic body whose referenceconfiguration consists of all points that lie within a distance￡efrom a given surface S inR3, called the middle surface of the shell, the thickness 2e> 0 being thought of as being“small” compared with the dimensions of S. As a result, a shell is most often modeled asa minimization problem “posed over the surface S”, i.e., by means of a “two-dimensional”energy defined over a bounded subset of R2, the coordinates of which are the curvilinearcoordinates used for defining the surface S. This “dimension reduction” is particularlyadapted to numerical simulations, which would be otherwise impracticable should the shellbe modeled as a three-dimensional body.By contrast with the three-dimensional case, where the mathematical model is unambiguouslydefined from the general principles of continuum mechanics, there is no agreementabout what should be a “universal” two-dimensional nonlinear shell model. And indeed,various competing two-dimensional energies have been proposed for modeling nonlinearlyelastic shells. Among these, the nonlinear model proposed by W.T. Koiter in 1966 is widelyused, essentially because one of its virtues is that it seems to well capture the “membraneeffects” and the “flexural effects” that may classically arise in a deformed shell, irrespectiveof the geometry of the middle surface S of the shell and of the boundary conditions imposedon the admissible deformations.No satisfactory mathematical justification and no satisfactory existence theory are as ofnow available for the nonlinear Koiter model, however. The aim of this proposal is tocontribute to the solution of these two open problems.More specifically, our first major aim will be to rigorously justify, by means of ??-convergencetheory, a nonlinear model “of Koiter’s type” proposed by the P.I. in 2000. This aim is intendedto be achieved by showing that the asymptotic behavior of its energy as " approacheszero matches that of the three-dimensional energy, obtained in two successive fundamentalcontributions, first by H. Le Dret and A. Raoult in 1996 for the “membrane case”, secondby G. Friesecke, R.D. James, M.G. Mora and S. Muller in 2003 for the “flexural case”.Our second major aim will be to appropriately modify, either the original Koiter’s energyor that proposed by the P.I., in such a way that the modified energy coincides “to within thefirst order” with Koiter’s energy on the one hand, while on the other hand it also incorporatesthe notions of “polyconvexity on a surface” and “preservation of orientation on a surface”introduced by the P.I., R. Gogu, and C. Mardare in 2013. In this fashion, we should thenbe able to establish the existence of a minimizer of such a modified Koiter’s energy over aspecific set of admissible deformations.?## Detail(s)

Project number | 9042222 |
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Grant type | GRF |

Status | Finished |

Effective start/end date | 1/01/16 → 6/12/19 |