Multiscale Approach to Understand the Multiphysics Phenomena in Bone Adaptation - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 49

3.1.3 Solid-Fluid Interface Conditions

The solid-fluid interface oY fs is characterized by its normal unit vector n directed
toward the solid domain. Concerning the surface charge density, the presence of
the Stern layer and the ionic exchanges quantified by the coefficients a (homo-
geneous to lengths [ 76 ]) may generate an electrical jump r fs by crossing the
interface from the solid part to the fluid phase. As a result, two different electric
surface charges are introduced, r f and r s ¼ r f þ r fs ; referring respectively to the
surface charge density seen from the fluid or the solid. The set of interface con-
ditions is then given by:

/ f ¼ / s ; ð 14 Þ

n ¼ r f ;

e f r / f ð 15 Þ

ð P Z : r u e s r / s Þ n ¼ r s ; ð 16 Þ

v ¼ du
dt

; ð 17 Þ

J n ¼ a o n b
ot

; ð 18 Þ

S s n ¼ S f n : ð 19 Þ

They correspond, respectively, to the electric potentials continuity and the
electric flux conditions at the interface, to the no-slip condition, to the ionic
exchange property of this interface and to the continuity of the normal stress.

3.2 Upscaling Procedure

An asymptotic periodic homogenization process [ 7 , 102 ] is carried out to propa-
gate our microscopic description of the phenomena at the upper scale. The prin-
ciple of this procedure is to consider a periodic representative cell Y or
representative volume element (RVE), whose size must be large enough to contain
all relevant microscopic heterogeneities and small enough to ensure the macro-
scopic homogeneity. Let ' and L be characteristic lengths of the micro- and macro-
scales respectively, and x and X the associated coordinate systems. In order to keep
the independence between these two scales, the ratio g ¼ '= L must be small in
comparison with 1 : In our problem, the length ' associated with the microscale is
typically the pore size length and the length L is typically of the order of the
cortical tissue size.
The general procedure of the periodic homogenization consists in: i. writing the
equations at the microscale in a non-dimensional fashion and giving scaling laws