Homogenization for the multiple scale analysis of musculoskeletal mineralized tissues

Using multiscale models and homogenization methods the elastic properties of two important musculoskeletal mineralized tissues, the mineralized turkey leg tendon (in short mineralized tendon) and the osteon, are modeled and simulated at different length scales. Our first aim is to find homogenization methods which predict the apparent elastic properties of the investigated tissues numerical accurate as well as are computationally efficient. Our second aim is to find the key parameters determining the elastic properties of the investigated tissues. After a short introduction (first chapter), we present the background required for this work (second chapter). Our models are based on the boundary value problems of static linear elasticity. We state different boundary value problems and recall existence and uniqueness results of them. In the third chapter we introduce the homogenization methods employed in this thesis. These are: the Mori-Tanaka method, the self-consistent method, some homogenization methods for periodic materials, and the representative volume element based homogenization method with displacement and traction boundary conditions. We describe the implementation of these homogenization methods in the fourth chapter. In order to predict the coarse-scale elastic properties of the mineralized tendon we employ the MoriTanka and the self-consistent method. In the fifth chapter we perform various numerical tests for the building unit of the mineralized tendon to clarify the numerical accuracy and the computational efficiency of the employed homogenization methods. We show that the numerical accuracy of the Mori-Tanaka method improves about one order of magnitude, if we decrease the tool parameter of the Mori-Tanaka method about one order of magnitude. Similar applies to the self-consistent method. Furthermore, we fix tool parameters of the homogenization methods such that the predicted coarse-scale elastic properties are numerical accurate. In the sixth chapter we present our multiscale model of the mineralized tendon. Performing a global sensitivity analysis (Elementary Effects method) and a parametric study of our model we investigate the essential parameters influencing the elastic properties of the mineralized tendon. These are: the microporosity and different parameters, describing the shape and the volume fraction of the mineral within the mineralized tendon. Finally, we compare our model elastic properties with experimentally derived elastic properties, given by our project partners from the Charité Berlin. We find a very good agreement, we have small relative errors of 6-8%. In the seventh chapter we develop a multiscale model for the osteon. We employ the RVE-based homogenization method with displacement boundary conditions. We perform a convergence analysis of our method as well as compare different homogenization methods with each other. Performing a parametric study of the osteon model we determine the key parameters influencing the apparent elastic properties of the osteon. These are: the type of the circular lamellar units contained in an osteon and the numbers of the circular lamellar units. Since there is, in contrast to the mineralized tendon, no experimental data available by our colleagues, we compare our model with data found in the literature. Our predicted elastic properties agree well with the data found in the literature. In the last chapter of this work we draw conclusions.