RVE procedure for estimating the elastic properties of inhomogeneous microstructures such as bone tissue

Abstract

Cancellous bone can roughly be seen as a two-phase material consisting of the bone tissue reinforcement and the interstitial bone marrow matrix. Thus, for a computer-aided mechanical stress analysis of bones a constitutive law is required, which can predict the inhomogeneous elasticity depending on the local bone density and microstructure. Besides severalmeasurement methods, the method of representative volume element (RVE) in combination with the finite element solution technique has been established for this purpose. This work investigates this method in detail. Therefore, random but statistical equivalent RVEs are created to have unlimited access to different structures. Generally, an apparent and not an effective stiffness is obtained due to the RVE method. However, a very close solution can be achieved if several issues are considered carefully. These issues can be divided into the set of boundary conditions, the RVE size and averaging the randomness. The influences are investigated accurately. A new approach is proposed to deduce an isotropic constitutive law from the anisotropic stiffness matrix. There are unlimited possible solutions in theory. However, the Voigt and Reuss approximations give the possible bounds. A method is described, which allows to obtain the effective stiffness by merging these bounds. A structural analysis is performed with different RVEs and the effective stiffness is estimated for varying parameters. An empirical equation is introduced, which covers the whole stiffness range. Therein, the microstructure is modelled with a single parameter. Real bone measurements can be fitted with this equation as well.