The nonlinear membrane energy: Variational derivation under the constraint “det∇u≠0”

Abstract

Acerbi, Buttazzo and Percivale gave a variational definition of the nonlinear string energy under the constraint "detu>0" (see [E. Acerbi, G. Buttazzo, D. Percivale, A variational definition of the strain energy for an elastic string, J. Elasticity 25 (1991) 137-148]). In the same spirit, we obtain the nonlinear membrane energy under the simpler constraint "detu0".

Brittle membranes in finite elasticity

“…Fix 𝛿 ∈ (0, 1∕2), as in Case 1 we construct Θ 𝛿 ∶ ℝ 2 ⧵ 𝛾 → Ω such that the image set Ω is a Lipschitz domain (depending on 𝛿) and Θ 𝛿 (𝑥) = 𝑥 for every 𝑥 ∈ ℝ 2 ⧵ (𝛾) 𝛿∕𝑀 . Observe that by the construction given in Case 1, see (8), we may assume that ⟨Θ 𝛿 (𝑥), 𝑒 1 ⟩ = 𝑥 1 for every 𝑥 ∈ ℝ 2 ⧵ 𝛾.…”

“…Although such convergence result may be rather expected, the strategy necessary to prove it requires a nontrivial adaption of the arguments of [8, 9, 40], as well as the introduction of new tools. As a first step in our proof we show the density in $$GSBV^{p}(\Sigma , \mathbb {R}^{3})$$ of piecewise affine functions with polyhedral jump set and approximate gradient with maximal rank (see Theorem 3.1).…”

“…We remark that in order to define recovery sequences on the whole domain, we again use the diffeomorphisms introduced in Lemma 3.2. In turn, this requires some refinements of the constructions in [8, 9] (see Lemmas 4.9 and 4.10).…”

“…This paper focuses on the derivation of a reduced model for brittle elastic membranes under non-interpenetration constraints. Our approach is based on Γ-convergence [1], which has been successfully used in the last decades to tackle a number of dimension reduction problems in elasticity for rods and ribbons [2][3][4][5][6][7], membranes [8][9][10][11][12][13][14], and plates [15][16][17][18][19][20], as well as in elastoplasticity [21][22][23][24].…”

“…has been identified in [8,9] and is given by  0 (𝑢) ∶= ∫ Σ 1 𝑊 0 (∇ 𝛼 𝑢) 𝑑𝑥 for 𝑢 ∈ 𝑊 1,𝑝 (Σ 1 , ℝ 3 ) with 𝜕 3 𝑢 = 0, where the reduced elastic energy density 𝑊 0 ∶ 𝕄 3×2 → [0, +∞] is defined as…”