Biomedical Engineering Reference
In-Depth Information
with the fibrous case is that the elastic bone coefficients
E
t
(
)and
E
n
(
) depend
on the relative density of the bone attached to the considered material point of the
interface (Eq. 2.7). The parameter
ρ
ρ
η
is equivalent to the penalization parameter of
the SIMP model.
In what follows, the outlines of the periprosthetic remodeling model is stated. As
previously said, a principle of global optimum behavior was chosen as a remodeling
engine.
2.3.3
Model for Periprosthetic Adaptation
Consider three bodies in the tridimensional Euclidian space: the femur, occupying
the open region
s
, and the interface body
defined in the region denoted by
i
(Figure 2.6). The surface
u
is the part of the
boundary
b
, the femoral shaft occupying region
t
is submitted
to surface forces
t
. Both the bone and the shaft are modeled by elastic material. Let
us also denote
σ
, the Cauchy stress tensor, and
ε
(
·
)
=∇
where Dirichlet boundary conditions are applied, while
s
(
·
), the linearized Green
strain tensor. The constitutive behavior of bone follows the equation
σ
= C
ρ
ε
with the homogenized elastic tensor
C
ρ
given by Eq. (2.2). This implies a linear
dependence on displacements and a nonlinear dependence on the relative density
ρ
. The interface region
i
allows a nonlinear constitutive dependence on strains,
depending also on
ρ
, due to its attachment to
b
.Theshaftin
s
has a linear
material behavior and is independent of
.
The equilibrium condition is stated by the variational problem: find the displace-
ment field
u
∈
U
ρ
such that
b
σ
(
u
)
· ε
(
λ
)d
+
i
σ
(
u
)
· ε
(
λ
)d
b
i
+
s
σ
(
u
)
·
ε
(
λ
)d
−
l
(
λ
)
=
0
∀
λ
∈
V
(2.8)
s
H
1
(
H
1
(
where
define the set
of admissible displacements and admissible variations, respectively. The three first
terms of Eq. (2.8) represent the virtual work of the internal forces in each region.
U
:
={
u
∈
):
u
|
u
=
0
}
and
V
:
={
λ
∈
):
λ
|
u
=
0
}
t
Γ
t
Ω
b
Ω
i
Ω
s
b
b
b
Γ
u
Figure 2.6
Bone, stem, and interface domains.
