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

one cements as well as its behaviour in solid state. The parameters of the material
model have been fitted to one specific acrylic bone cement. Furthermore, the model has
been implemented into a commercial finite element code. The capability of our material
model will be demonstrated by the help of finite element simulations using a parametric
model of a vertebral body of the lumbar spine.
3. MATERIAL MODEL FOR ACRYLIC BONE CEMENTS
The material model presented in [1] is based on the standard concepts of nonlinear
continuum mechanics, e.g. the application of strain and stress measures at large
deformations and the evaluation of the laws of thermodynamics to obtain a
thermodynamically consistent material behaviour. The material response is assumed to
be nearly incompressible combined with viscoelastic behaviour of the isochoric part. A
formulation of a chain of Maxwell-elements connected in parallel is employed to define
the viscoelastic behaviour. Furthermore, a degree of cure is introduced to describe the
progress of the chemical process of polymerization. This variable is treated as an
internal variable within the material model. Finally, the consideration of the temperature
θ within different material functions as well as the evaluation of the transient heat
equation leads to a fully thermochemical-mechanically coupled material model.
The underlying kinematics is described by the deformation gradient . To separate
different sources of deformation, a multiplicative decomposition of the deformation
gradient is applied. Firstly, thermo-chemical volume changes due to heat expansion and
chemical shrinkage are separated from pure mechanical deformations according to eq.
(1). Therein the thermo-chemical volume changes are assumed to be isotropic and
therefore can be described by a scalar material function .
= ∙
= = , (1)
Next, the mechanical deformation gradient is split into a pure volumetric part ,
which depends on the determinant of the mechanical deformation gradient, and a
remaining volume preserving (isochoric) deformation gradient (cf. eq. (2)). The
latter part is employed for the formulation of the viscoelastic material behaviour.
= ∙
=
= det = 1/ In order to fulfil the laws of thermodynamics and to ensure a thermodynamically
consistent material behaviour, a free energy function has been formulated. It is
additively decomposed into three parts. The first part represents a pure elastic
material behaviour due to volumetric changes. Secondly, a volume preserving,
viscoelastic part = ∑ , is modelled, which consist of a sum of free energy
functions for = 1 … Maxwell-elements. The last part is a thermo-chemical part describing thermally stored amounts as well as initially stored energy which is getting
released due to the exothermal chemical reaction.
An evaluation of the chosen ansatz for the free energy function within the laws of
thermodynamics leads to a coupled system of equations which describes the thermomechanically
coupled material behaviour. Among others, it consists of an equation for
the 2 nd Piola-Kirchhoff stress (eq. (3)) including a bulk modulus as well as stiffness
parameters and relaxation times = /2 for each of the Maxwell-Elements.
(2)