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 of Cowin and Hegedus with later modifications, and 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, 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.

Conical differentiability for bone remodeling contact rod models

“…The existence and uniqueness of solution to the family of bone remodeling rod models defined by (1.9) or (1.18) can be proved using the same arguments of Figueiredo and Trabucho [5] and also Monnier and Trabucho [9]. The proof of existence relies on Schauder's fixed point theorem together with the Cauchy-Lipschitz-Picard theorem (used to solve the remodeling rate equation, for a fixed dispacement), the Stampacchia theorem (that is necessary to guarantee the existence of solution to the variational inequality, for a fixed change of volume fraction) and regularity results.…”

“…We observe that we could have considered in (1.1) a remodeling rate equation depending nonlinearly on e 33 (u s ), that is (cf. Figueiredo and Trabucho [5])…”

“…These three sets represent, respectively, the lateral boundary of the rod Ω s and its extremities. We assume that the boundary ∂ω s is divided into two nonempty disjoint parts denoted by ∂ω sc and ∂ω sg and consequently we denote [5].…”

“…it is an element of the matrix b ijkl (d s ) which is the inverse of the matrix composed of the three-dimensional elastic coefficients of the rod Ω s , as explained in Figueiredo and Trabucho [5]). The element L ds is defined by…”

“…Moreover, the bone remodeling model that we adopt in this paper, can be mathematically justified by the asymptotic expansion method as in Figueiredo and Trabucho [5] (cf. also Trabucho and Viãno [13], for an explanation of the asymptotic expansion method applied to elastic rod contact models), and is defined by the following system, formulated in the set Ω × [0, T ] independent of s (cf.…”