Asymptotic variational analysis of incompressible elastic strings

Abstract

Starting from three-dimensional nonlinear elasticity under the restriction of incompressibility, we derive reduced models to capture the behavior of strings in response to external forces. Our Γ-convergence analysis of the constrained energy functionals in the limit of shrinking cross sections gives rise to explicit one-dimensional limit energies. The latter depend on the scaling of the applied forces. The effect of local volume preservation is reflected either in their energy densities through a constrained minimization over the cross-section variables or in the class of admissible deformations. Interestingly, all scaling regimes allow for compression and/or stretching of the string. The main difficulty in the proof of the Γ-limit is to establish recovery sequences that accommodate the nonlinear differential constraint imposed by the incompressibility. To this end, we modify classical constructions in the unconstrained case with the help of an inner perturbation argument tailored for 3d-1d dimension reduction problems. MSC (2010): 49J45 (primary) • 74K05

Theories for incompressible rods: a rigorous derivation via $Γ$-convergence

“…In [13, Proposition 5.1], Conti & Dolzmann established a first lemma of this type for 3d-2d reductions in the context of incompressible membranes. Recently, the authors tailored the statement for 3d-1d reductions in their work on incompressible strings [19]. Sections 4 and 5 invoke [19, Lemma 2.3] in the following formulation.…”

“…in blood vessels, tires, seat belts, etc. From an analytical point of view, there is recent work on membranes [13], plates in the Kirchhoff [14] and von Kármán regime [9,31], hyperelastic shells [3], and strings [19]. The challenge in the mathematical analysis of these models lies in the non-convex constraint imposed on the elastic energy functionals to guarantee local volume-preservation; precisely, the Jacobian determinant of admissible deformation fields has to be constant and equal to one, cf.…”

“…The scaling regimes α ∈ [0, 2), which give rise to (degenerate) models for incompressible strings, are studied in [19]. In this paper, we focus on the cases α ≥ 2 to deduce a hierarchy of theories for incompressible rods.…”

“…With (1.10) at hand, an inner perturbation argument in the spirit of [13], adapted for 3d-1d reductions in [19,Lemma 2.3] (see Lemma 3.1 below), allows us to replace (y ε ) ε by a sequence that satisfies the incompressibility constraint exactly. We remark that this auxiliary result is only applicable if the cross-section variables of y ε are decoupled, for which we extend y ε suitably to a cuboid containing Ω. Technically, this task reduces to finding a divergence-free extension (see e.g.…”