Multiscale Approach to Understand the Multiphysics Phenomena in Bone Adaptation - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 52

D ¼ g f D # 1

:

ð 21 Þ

The explicit definition of # is obtained during the homogenization process and
is detailed in Kaiser et al. [ 59 ].

3.3.4 Modified Darcy Law

As obtained in clayey materials [ 81 , 102 ], the fluid flow, in addition to the
hydraulic pressure gradient, may be governed by supplementary driving phe-
nomena: the electro-osmotic seepage flow induced by the streaming potential
fluctuations and the osmotic flow in response to the chemical gradient. As a result,
the macroscopic fluid flow is described through a modified Darcy law including, in
addition to the pressure gradient induced flow (indexed by P), chemo-osmosis
(indexed by C) and electro-osmosis (indexed by E):

V ¼ V P þ V C þ V E ¼ K P r X p b ½ 0 K C r X n b ½ 0 K E r X W b ½ 0 : ð 22 Þ

The macroscopic permeability tensors K k (k ¼ P ; C ; E) are obtained through the
homogenization process as detailed in Lemaire et al. [ 74 , 85 , 86 ].

3.3.5 Coupled Biot Problem

Bone fluid flow is generated by the strain of the solid matrix. Classically, in bone
biomechanics, the calculation of the hydraulic velocities caused by the mechanical
loading are based on the poro-elasticity theory [ 15 ]. Here, a Biot-like constitutive
relation derived from our microscale analysis is obtained [ 74 ]:

: e X ð u ½ 0 Þ a p b ½ 0 þ s :

S tot ¼ C ð 23 Þ

In this equation, e X corresponds to the macroscopic part of the operator e ; that is
to say built from r X : Thus, e X ð u ½ 0 Þ corresponds to the macroscopic strain tensor.
Furthermore, the homogenized fourth-order elasticity tensor

and the homog-
enized Biot second-order tensor a are obtained following the classical treatment
of poro-elasticity as proposed by Auriault and Sanchez-Palencia [ 8 ]. Moreover, the
macroscopic electro-chemical tensor s representing the macroscopic effects of
the fluid electro-chemical phenomena is similar to the one previously obtained for
the multiphysical description of clayey materials [ 81 , 102 ]. This tensor accounts
for: i. the spherical Donnan pressure effect; ii. the action of the Maxwell tensor;
iii. the electro-chemical effects occurring at the solid-fluid interface [ 74 ].
As a result, by neglecting the body forces, the macroscopic equilibrium equa-
tion simply reads:

C

r X S tot ¼ 0 : ð 24 Þ