Multiscale Approach to Understand the Multiphysics Phenomena in Bone Adaptation - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 53

Table 1 Inventory of the unknowns and equations present in the multiphysics poro-elastic model
at the macroscopic scale
Unknowns Equations

Physical quantity Number Macroscopic equations Number

Solid displacement u ½ 0 3 Macroscopic balance equation ( 24 ) 3

Total stress tensor S tot 6 Modified Biot constitutive law equation ( 23 ) 6

Darcy velocity V 3 Modified Darcy law equation ( 22 ) 3

Ionic concentration n b ½ 0 1 Electro-diffusive Nernst-Planck equation ( 20 )2

Fluid pressure p b ½ 0 1 Mass conservation equation ( 25 ) 1

Streaming potential W b ½ 0 1 Fluid mass conservation equation ( 26 ) 1

Porosity g f 1

Total of scalar unknowns 16 Total of scalar equations 16

The coupled Biot equation is derived from the mass conservation and the Darcy
law:

ot e X u ½ 0 þ ! o

V ¼ a : o r X

ot p b ½ 0 þ - :

ð 25 Þ

Again, in addition to the classical Biot formulation involving the Biot tensor a
[ 8 ] and the number ! which are explicitly derived through the upscaling proce-
dure, another electro-chemical effect - has to be also taken into account. This
quantity, which is due to the electrically induced pores deformation is explicitly
given in Lemaire et al. [ 74 ].
Finally, it can be shown that the fluid mass conservation could be expressed in
terms of the porosity, the Darcian velocity and the displacement [ 102 ]:

o g f
ot þr X

o V þ g f r X

ot u ½ 0 ¼ 0 :

ð 26 Þ

3.4 Summary of the Macroscopic Description of Bone Tissue

A problem dealing with studying the multiphysics evolution of a poroelastic
material is defined by a macroscopic set of time-space partial differential equations
over the macroscopic domain involving different physical unknowns. In addition,
convenient macroscopic boundary conditions have to be provided. To check that
our multiscale strategy results in a macroscopic problem that is convenient with
the number of the unknowns, we recapitulate in Table 1 the final macroscopic
description of bone tissue.
It is important to indicate that, in addition to these macroscopic laws, the purely
microscopic cellular problems have to be solved on the periodic cell to express all
the homogenized parameters. Moreover, the double layer potential u ; solution of