Biomedical Engineering Reference
In-Depth Information
The continuum mechanics approach gives the possibility to deal anisotropic
effects, which hardly can be handled with classical mechanics like simple tension or
compression tests. However, three issues can be pointed out.
•
First, the boundary value method, respectively FEM, provides only an apparent
estimate in common use, since the Hill-Condition cannot be fulfilled completely.
This issue can be solved by increasing the RVE size and by regarding the conver-
gence of the estimate.
•
Secondly, the boundary conditions exhibit a big influence and should be chosen
carefully. In this study a RVE size of 50 elements per edge in combination with the
PMUBC set of Zysset and Pahr established as good compromise with respect to the
computational resources. Pure displacement or stress conditions are not working.
•
Since each RVE represents a random window of the structure, the randomness
can be averaged by using the Monte-Carlo procedure at last. It could be shown
that multiple simulations show a normal distribution and simple averaging of the
moduli works.
An additional scientific issue is to extract isotropic moduli out of an anisotropic
stiffness matrix since there are unlimited possible combinations. This work proposes
a procedure based on a Voigt and Reuss approximation. These approximations give
the lower and upper bounds, representing the softest and the stiffest combinations of
the moduli. While knowing that the effective moduli have to be in-between, simple
averaging leads to the effective moduli in practice. Both bounds fall together for a
sufficient RVE size. A RVE size of 50 elements per edge gives very close bounds.
The effective moduli are used to build an isotropic stiffness matrix, which can be
compared to the original anisotropic matrix by the Euclidian error measure.
The influences of percentage amount of initial cells and volume fraction were
studied. Since the number of initial cells controls the structure an empirical regression
equation could be found to model this behavior. A specific number of initial cells
could be determined to reach amaximal stiffnesswith respect to each volume fraction.
The algorithm is not made to model real bone structure, however, the empirical
equation works well to model real bone measurements. This model predicts different
stiffness for same volume fraction by varying the structure parameter. In principle,
the stiffness variation that is observed in measurements can be explained by structure
only.
References
1. Ashman, R.B., Corin, J.D., Turner, C.H.: Elastic properties of cancellous bone: measurement
by an ultrasonic technique. J. Biomech.
20
, 979-986 (1987)
2. Ashman, R.B., Rho, J.Y.: Elastic modulus of trabecular bone material. J. Biomech.
21
, 177-181
(1988)
3. Rho, J.Y., Ashman, R.B., Turner, C.H.: Youngs modulus of trabecular and cortical bone mate-
rial: ultrasonic and microtensile measurements. J. Biomech.
26
, 111-119 (1993)
