The University of Arizona
Please note that this event has ended!

Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models

Modeling, Computation, Nonlinearity, Randomness and Waves Seminar

Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models
Series: Modeling, Computation, Nonlinearity, Randomness and Waves Seminar
Location: MATH 402
Presenter: Marta Lewicka, Department of Mathematics, University of Pittsburgh

We propose results that relate the following two contexts: (i) Given a Riemann metric G on a thin plate, we study the question of what is the infimum of the averaged pointwise deficit of an immersion from being an orientation-preserving isometric immersion of G over all weakly regular immersions. This deficit is measured by the non-Euclidean energies E^h, which can be seen as modifications of the classical nonlinear three-dimensional elasticity. 

 

(ii) We perform the full scaling analysis of E^h, in the context of dimension reduction as the plate's thickness h goes to 0, and the derivation of Gamma-limits of h^{-2n}E^h for all n. We show the energy quantization, in the sense that the even powers 2n of h are indeed the only possible ones (all of them are also attained).  For each n, we identify conditions for the validity of the scaling h^{2n}, in terms of the vanishing of Riemann curvatures of G up to appropriate orders, and in terms of the matched isometry expansions.  We also establish the singular asymptotic behaviour of the minimizing immersions.

 

The problems that we discuss arise from the description of elastic materials displaying heterogeneous incompatibilities of strains that may be associated with growth, swelling, shrinkage, plasticity, etc. Our results and methods display the interaction of nonlinear pdes, geometry and mechanics of materials in the prediction of patterns and shape formation.