Theories for incompressible rods: A rigorous derivation via Γ-convergence

Article type: Research Article

Authors: Engl, Dominik | Kreisbeck, Carolin^{; *}

Affiliations: Mathematisch Instituut, Universiteit Utrecht, Postbus 80010, 3508 TA Utrecht, The Netherlands. E-mails: D.M.Engl@uu.nl, C.Kreisbeck@uu.nl

Correspondence: [*] Corresponding author. E-mail: C.Kreisbeck@uu.nl.

Abstract: We use variational convergence to derive a hierarchy of one-dimensional rod theories, starting out from three-dimensional models in nonlinear elasticity subject to local volume-preservation. The densities of the resulting Γ-limits are determined by minimization problems with a trace constraint that arises from the linearization of the determinant condition of incompressibility. While the proofs of the lower bounds rely on suitable constraint regularization, the upper bounds require a careful, explicit construction of locally volume-preserving recovery sequences. After decoupling the cross-section variables with the help of divergence-free extensions, we apply an inner perturbation argument to enforce the desired non-convex determinant constraint. To illustrate our findings, we discuss the special case of isotropic materials.

Keywords: Dimension reduction, Γ-convergence, Euler–Lagrange equations, incompressibility, rods

DOI: 10.3233/ASY-201636

Journal: Asymptotic Analysis, vol. 124, no. 1-2, pp. 1-28, 2021