A Riemannian approach to the membrane limit in non-Euclidean elasticity

作者：Kupferman Raz^*; Maor Cy

来源：Communications in Contemporary Mathematics, 2014, 16(5): 1350052.

DOI：10.1142/S0219199713500521

• Riemannian manifolds
• nonlinear elasticity
• incompatible elasticity
• membranes
• gamma-convergence

Non-Euclidean, or incompatible elasticity, is an elastic theory for pre-stressed materials, which is based on a modeling of the elastic body as a Riemannian manifold. In this paper we derive a dimensionally reduced model of the so-called membrane limit of a thin incompatible body. By generalizing classical dimension reduction techniques to the Riemannian setting, we are able to prove a general theorem that applies to an elastic body of arbitrary dimension, arbitrary slender dimension, and arbitrary metric. The limitingmodel implies the minimization of an integral functional defined over immersions of a limiting submanifold in Euclidean space. The limiting energy only depends on the first derivative of the immersion, and for frame-indifferent models, only on the resulting pullback metric induced on the submanifold, i.e. there are no bending contributions.

• 出版日期2014-10