We prove a relation between the scaling $h^β$ of the elastic energies of shrinking non-Euclidean bodies $S_h$ of thickness $h\to 0$, and the curvature along their mid-surface $S$. This extends and generalizes similar results for plates [BLS16, LRR] to any dimension and co-dimension. In particular, it proves that the natural scaling for non-Euclidean rods with smooth metric is $h^4$, as claimed in [AAE+12] using a formal asymptotic expansion. The proof involves calculating the $Γ$-limit for the elastic energies of small balls $B_h(p)$, scaled by $h^4$, and showing that the limit infimum energy is given by a square of a norm of the curvature at a point $p$. This $Γ$-limit proves asymptotics calculated in [AKM+16].
Universal bounds on the electrical and elastic response of two-phase bodies and their application to bounding the volume fraction from boundary measurements
On Asymptotic Rigidity and Continuity Problems in Nonlinear Elasticity on Manifolds and Hypersurfaces
University of Oxford（牛津大学）；Shanghai Jiao Tong University（上海交通大学）；New York University – Shanghai（纽约大学上海分校）；NYU-ECNU Institute of Mathematical Sciences（纽约大学-华东师范大学数学科学研究所）；University of Wisconsin, Madison（威斯康星大学麦迪逊分校）
Inverse Elastic Scattering for Multiscale Rigid Bodies with A Single Far-field Pattern
Weierstrass Institute（韦伊尔斯特拉斯研究所）；South University of Science and Technology of China（南方科技大学）；University of North Carolina, Charlotte（北卡罗来纳大学夏洛特分校）；University of Graz（格拉茨大学）
