Poro-Micromechanics of Bone: Impact Loading and Wave Propagation
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Poro-Micromechanics of Bone: Impact Loading and Wave Propagation Christian Hellmich1 and Franz-Josef Ulm2 1
Vienna University of Technology (TU Wien), Institute for Mechanics of Materials and Structures, A-1040 Vienna, Austria 2 Massachusetts Institute of Technology, Department of Civil and Environmental Engineering, Cambridge, MA 02139, USA ABSTRACT Based on ‘universal’ mechanical building blocks inherent to all different bones, this paper shows the integration of two classically separated fields of bone biomechanics, namely bone micromechanics [1] on the one hand, and bone poromechanics [2] on the other. Indeed, despite the complex hierarchical organization of bone, it was recently possible to identify such elementary components at the micro and nanolevel of the material for the explanation of the diversity of macroscopic (poro-)elastic properties of different bones. The mechanical properties (i.e. elasticity) of these elementary components are (within limits of experimental scatter) the same for all bones; they are ‘universal’, i.e. independent of tissue type, species, and anatomical location. The mechanical interaction between these elementary components (mechanical morphology) and the dosages of these components in different tissues determine the macroscopic material properties. Continuum micromechanics has turned out as well-suited theoretical tool to derive tissue-dependent elastic properties at different length scales from upscaling the elasticity of ‘universal’ building blocks from the nanometer scale. Once respective micromechanics models are developed and validated, poro-micromechanics allows for the quantification of poroelastic properties such as the Biot and Skempton coefficients, as functions of the volume fractions of mineral, collagen, and the micropore space (Haversian and Volkmann canals, and the inter-trabecular space). Skempton coefficients, related to undrained conditions, allow for quantification of the marrow pressure rise due to impact loading, as can be shown by model predictions of non-destructive impact experiments. Moreover, the aforementioned poroelastic properties enter the governing equations for wave propagation in anisotropic porous media. They allow for the prediction of undrained, fast and slow waves, as is verified by comparison of model results with experimental findings.
INTRODUCTION - FUNDAMENTALS OF CONTINUUM MICROMECHANICS [3,4] A central concept of continuum micromechanics is that, for the study of deformations of a solid, the observer must physically label or mark constitutive material elements (representative volume elements - RVE). The characteristic length of such an RVE, l, must be both considerably larger than the dimensions of heterogeneities within the RVE, d, and significantly smaller than the characteristic lengths of geometry or loading of a structure built up by the material defined on the RVE, L; in mathematical terms, this separation-of-scales requirement reads as: d«l«L. In general, the microstructure within each RVE is so complicated that it cannot
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