Asymptotic model of a nonlinear adaptive elastic rod

Abstract

In this paper we apply the asymptotic expansion method to obtain a nonlinear adaptive elastic rod model. We first consider the model derived in [2, 3] with the modifications proposed in [5], with a remodeling rate equation depending nonlinearly on the strain field and for a thin rod whose cross section is a function of a small parameter. Based on the asymptotic expansion method for the elastic case [6], we prove that, when the small parameter tends to zero the solution of the nonlinear adaptive elastic rod model converges to the leading term of its asymptotic expansion. Moreover, we show that this term is also the solution of a well-known simplified adaptive elastic model, with generalized Bernoulli-Navier equilibrium equations and a remodeling rate equation whose driving mechanism is the strain energy per unit volume, in good agreement with some of the models used in practice.