Multiscale Approach to Understand the Multiphysics Phenomena in Bone Adaptation - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 46

structure of bone described as a set of nested porosities like a set of Russian nested
dolls. This idea is developed using a phenomenological approach in Cowin et al.
[ 27 ], Gailani and Cowin [ 39 ] and a homogenization procedure in Rohan et al.
[ 132 ]. As a result, the interstitial fluid transport outcomes are governed by the
phenomena at the smallest porous level.

3.1 Equations at the Microscale of the Lacuno-Canalicular
Structure

Neglecting the magnetic effects, at the microscale, the multiphysics description
integrate the solid matrix electro-mechanics, the interstitial fluid electrokinetics
and the solid-fluid interfacial phenomena.

3.1.1 Piezo-Electricity in the Solid Matrix of Bone Tissue

Due to the piezo-electric property of the collagen matrix of bone, this phase can be
considered as a sort of dielectric material characterized by a permittivity tensor e s
and exhibiting a quasi-permanent space charge q s : The electric potential in the
solid / s is thought to occur when a number of collagen molecules are stressed in
the same way in response to a gradient of the displacement vector field u ; moving
charge carriers from the inside to the surface of the specimen. As a result, the
constitutive laws for the stress tensor S s and the electrical displacement vector field
D s in the solid involve both the displacement and the electrical potential effects,
the piezo-electric coupling being quantified by the the piezo-electric third-order
tensor P Z :

: e ð u Þþ P Z r / s ; S s ¼ C ð 1 Þ

D s ¼ P Z : r u e s r / s ; ð 2 Þ

represents the fourth-order elasticity tensor of the solid, r is the gradient
operator, I T stands for the transpose operator, e ð I Þ¼ 1 = 2 ðr I þr I T Þ is the
operator that gives the symmetric part of the gradient of the quantity I : Here e ð u Þ
represents the symmetric part of the gradient of the displacement of the solid phase
(i.e. the infinitesimal strain second-order tensor). By neglecting the body forces,
the mechanical equilibrium and the Maxwell-Gauss equations in the solid phase
are:

where C

r S s ¼ 0 ; ð 3 Þ

r D s ¼ q s ; ð 4 Þ

where r designates the divergence operator,