On the Role of Curvature in the Elastic Energy of Non-Euclidean Thin Bodies

Abstract

We prove a relation between the scaling h β of the elastic energies of shrinking non-Euclidean bodies S h of thickness h → 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].

Derivation of a Homogenized Bending–Torsion Theory for Rods with Micro-Heterogeneous Prestrain

“…In general, we expect that the type of limiting model does not only depend on α but also on other properties of the prestrain. Indeed, in a situation without homogenization it is shown in [8,36] that bending and von Karman type plate models arise in the case α = 0 depending on the geometry of the prestrain (see also [37] for related results in the case of rods and [31,32] for recent results for plates and shells beyond the von Karman regime). It is an interesting question if these results are stable with respect to (small) rapidly oscillating perturbations.…”

Scaling of the elastic energy of small balls for maps between manifolds with different curvature tensors

“…Similar to the reasoning in [8], the proof of Theorem 1.2 relies on a corresponding Γ-convergence result where the notion of convergence of sequences of maps u h : B h (p) → M incorporates a blow-up which reveals the map f . One key additional difficulty for non-flat targets is that maps u h with small energy need not be continuous.…”

“…In [8] it is shown that the quadratic quantity I R(p) is actually induced by a scalar product and in particular I R(p) = 0 if and only if R(p) = 0. Recall that by Gauss' theorema egregium, a small ball B h (p) in M can be mapped into R n with zero energy E B h (p) if and only if R ≡ 0 on B h (p).…”

“…Motivated by experiments and formal asymptotic expansions in the physics literature [1], Maor and Shachar [8] studied the behaviour of a model elastic energy of maps between manifolds with incompatible metrics. For thin objects they analysed the scaling of the minimal elastic energy as a function of the thickness.…”

Energy Minimising Configurations of Pre-strained Multilayers

“…There is, by now, a considerable body of work on thin plates with pre-strain. We mention, with no attempt to be exhaustive, [12,24,25,28,29,38,39] and refer to our companion paper [11] for further discussion and references. As our primary interest is to model multilayers, our set-up is quite different from those of the aforementioned contributions where the pre-strain depends only on the in-plane variables.…”