ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

= − ∙
∙ + − 1
= − 2
= ∙
Moreover, the mechanical part of the model consists of evolution equations for the
degree of cure and the mechanical material function (cf. eq. (4)). The material
function specifies the dependencies of the mechanical behaviour on the degree of cure
and the temperature.
= , , = , , (4)
The second part of the system of equations is given by an equation of transient heat
conduction (eq. (5)) where we take the Fourier’s law as ansatz for the heat flux vector
on the reference configuration .
− Div + + − = 0 = −θ det ∙ Grad θ (5)
Therein is the heat capacity density. The functions and are rates of
dissipated energy due to the thermo-chemical process and the inelastic behaviour of the
Maxwell-elements, respectively. The function gives a thermo-elastic coupling. The
heat capacity density and the three mentioned functions of eq. (5) are fully determined
by the free energy function.
The material model has been fitted to one specific acrylic bone cement using different
experimental techniques and inverse methods for parameter identification [1].
Furthermore, the system of equations (3)-(5) has been implemented into the user
subroutine HYPELA2 of MSC.MARC ® and can be used for finite element simulations.
4. PARAMETRIC FE-MODEL OF A VERTEBRAL BODY
The representative geometry of a vertebral body is modelled by using a parametric
approach with an analytical description based on equations provided in [2]. The
underlying shape of the cross section of a vertebral body is given by an ellipse with the
angle depending radius .
= ∙ / ∙ sin + ∙ cos (6)
Therein the constants A and B describe the half diameters of the elliptical cross section.
In order to approximate the typical shape of a vertebral body, a modified radius ,
is defined as follows:
, = ∙ 1 + ∙ ∙
+ ∙ ∙
(3)
∙ 1 − ∙ cos ∙
. (7)
The parameters , , and , , are employed to represent the shapes of the
anterior part and the vertebral foramen, respectively. The concavity of the vertebral
body cortex is modelled by a dependency of the radius on the -coordinate. The
parameter defines the magnitude of the concavity and the parameter H specifies the
half of the height of the vertebral body.