Biomedical Engineering Reference
In-Depth Information
18
Infinitesimal strain elasticity
problems
18.1
Introduction
One of the first applications of the Finite Element Method in biomechanics has
been the analysis of the mechanical behaviour of bone [
2
]. In particular the impact
of prosthesis implants has been investigated extensively. An example of a finite
element mesh used to analyse the mechanical stresses and strain in a human femur
is given in Fig.
18.1
(a). In Fig.
18.1
(b) the femur head is replaced by a prosthesis.
In this case the prosthesis has different mechanical properties compared to the
bone, leading to high stress concentrations at some points in the bone and stress
shielding (lower stresses than normal) in other parts. This normally leads to a
remodelling process in the bone that has to be accounted for when new prostheses
are designed. The purpose of this chapter is to introduce the finite element theory
that forms the basis of these analyses.
18.2
Linear elasticity
Neglecting inertia, the momentum equation, Eq. (
11.9
), may be written as
∇·
σ
+
f
=
0,
(18.1)
where
∇
denotes the gradient operator,
σ
the Cauchy stress tensor and
f
=
ρ
q
a
given distributed volume load. This equation should hold at each position within
the domain of interest
, having boundary
, and must be supplemented with
suitable boundary conditions. Either the displacement field
u
is specified along
u
:
u
=
u
0
at
u
,
(18.2)
or the external load along
p
is specified:
σ
·
n
=
p
at
p
.
(18.3)
The vector
n
denotes the unit outward normal at the boundary
.
