ABSTRACT of the Doctoral Thesis
It has been recognized for a long time that bone tissue has the ability to change its shape and internal architecture according to its functional requirements i.e. loading environment it is exposed to. Changes in the actual stress or strain pattern within the bone tissue will tend to stimulate pronounced cell activity resulting in a resorption of bone material in regions of low loading levels and vice versa in a deposition of new material in highly stressed zones, thus, giving rise to a new equilibrium state. The development of methods for predicting such adaptive changes in bone can be of great benefit for clinical practice since for unfavorable configurations of bone/implant systems this functional adaptation process may lead to runaway bone resorption and the loss of the implants.
Since the physical and biochemical background of the phenomenon of functional adaptation of bone is not fully understood at present (Some known facts are briefly summarized in section 2.3), scientific investigations have been focused on the development of phenomenologically based numerical simulation tools which are connecting some mechanical stimulus (e.g. strain energy density etc.) to the actual changes in the parameters characterizing the material behavior via simple mathematical expressions (A comprehensive overview of the relevant literature is presented in section 2.5). The present study uses such remodeling rules for the numerical simulation and prediction of stress induced bone remodeling changes (both internal and surface remodeling effects are addressed). The basically orthotropic characteristics of bony tissue are taken into account by introducing a unified bone material model which is based on suitable micromechanical models of trabecular bone material. The Finite Element Method is applied to calculate the stress and strain fields within the bone. The applicability of the method is shown by some illustrative examples, which showed good agreement between the numerical predictions and actual clinical results.
The mathematical description of the remodeling behavior leads to a system of nonlinear coupled differential equations, which has to be integrated with respect to time (The present study applies a simple Euler forward time integration scheme) in order to follow the adaptational changes. The stability of such a remodeling system is studied by introducing linear stability theory to a simple example for which closed analytical solutions are known and by employing "numerical experiments" on more complex models.
The concept of functional adaptation can be considered a "natural implementation" of a heuristic design improvement method based on optimality criteria. It can be applied advantageously for solving various structural design problems such as the optimization of the topology and the shape of structures, as is shown in several examples.
Moreover, the remodeling procedure presented in this study offers an interesting approach for the design improvement of structures built of advanced composite materials (e.g. functionally graded materials) which allow a microstructural gradation over macroscopic and/or microscopic distances. The present study addresses the problem of design improvement with respect to a special class of such materials, particle or fiber reinforced composites. A Mori-Tanaka mean-field approach is used in order to obtain analytical expressions for the linear elastic material parameters in dependence of the particle volume fraction and the inclusion aspect ratio (the particles are assumed to be aligned and to have a spheroidal shape). Several examples are given which show the applicability of the proposed method for improvement of the design of composite structures with respect to optimal particle distributions and particle orientations.