Biomedical Engineering Reference
In-Depth Information
RVE Procedure for Estimating the Elastic
Properties of Inhomogeneous Microstructures
Such as Bone Tissue
Tanja Blöß and Michael Welsch
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
andmicrostructure. Besides several measurement methods, themethod of representa-
tive 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 constitu-
tive 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.
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Keywords
Cancellous bone
Boundary conditions
Elastic properties
FEM
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Homogenization
Voigt and Reuss approximation
