Multiscale Approach to Understand the Multiphysics Phenomena in Bone Adaptation - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 50

for each non-dimensional number; ii. expanding all the fields in terms of the small
parameter g and collecting the terms corresponding to each power of g (cascade of
equations); iii. identifying the slow variables that do not vary at the microscopic
scale and proposing closure problems for the supplementary fluctuating terms; iv.
deriving the modelling at the macroscale by averaging the remaining quantities at
the microscale over the representative cell.

Remark The obtention of the macroscopic equations requires to average quantities
of the microscale over the representative cell Y : Thus, we define hi ¼ 1

j Y j R Y dV
the averaging operator on the representative cell Y : When focussing on the fluid
phase occupying the representative cell, the average over the fluid domain Y f is:
hi f ¼ 1

j Y f j R Y f dV f : Finally, hi int ¼ 1 j Y j R oY int dS represents the solid-fluid inter-

face averaging operator.

When focussing on the multiphysics behaviour of bone tissue, the homogeni-
zation procedure has been carried out for the Poisson-Boltzmann ( 8 ) and the fluid
flow ( 10 ) equations in a previous study [ 86 ], whereas the upscaling of the Nernst-
Planck equations ( 13 ) and its associated ionic exchanges has been summarized in
Kaiser et al. [ 57 ] and Lemaire et al. [ 76 ]. Furthermore, when neglecting the per-
icellular matrix and the ionic exchanges at the solid-fluid interface, the homoge-
nization of the piezo-poro-mechanics model of bone tissue is extensively presented
in [ 74 ].

3.3 Consequences at the Scale of Bone Tissue

Thanks to the upscaling procedure, it is possible to obtain the model at the
macroscale that will mimic the macroscopic behaviour of bone tissue.

3.3.1 Macroscopic Unknowns

First, the homogenization procedure allows to identify the variables purely mac-
roscopic (slow variables). So, the solid displacement u ½ 0 ; the bulk concentration
n b ½ 0 ; the streaming potential W b ½ 0 and the fluid pressure p b ½ 0 are slow variables that
do not vary at the microscopic scale [ 74 ].

Remark All these quantities are indexed [0] since they correspond to the first term
in the expansion of the corresponding physical quantity into a sequence of the
small parameter g :

As summarized in Table 1 , the macroscopic unknowns are the Darcy velocity V ;
the solid displacement u ½ 0 ; the total stress tensor S tot ¼ \S s ½ 0 [ þ \S f ½ 0 [

;