For the therapy of arthrosis, three-dimensional scaffold-free cartilage constructs were developed by fzmb GmbH [1]. The cartilage constructs were investigated and characterised by biomechanical, biochemical and histological analysis. The biomechanical properties are an essential indicator for the functional characterisation of tissue. In this contribution, the elastic and relaxation moduli are calculated for 17 cartilage constructs.
IntroductionThe biomechanical properties hold a lot of relevant information for the functional characterisation of cartilage tissue [2]. In the process the modulus of elasticity E is typically evaluated for a defined pressure applied to the cartilage tissue. This single observation neglects the dependence of the biomechanical properties according to time. In this case the modulus of relaxation R with the help of experimental force-way-compression and force-timerelaxation data by means of an elastic material and generalized Maxwell model is calculated. The aptitude of the models is evaluated by the comparison with experiments and simulations.
Linear Elastic ModelThe general stress-strain relationship for linear conditions readswhere σ is the stress, D the elasticity matrix and ε is the strain. The constitutive equation also includes initial stress and strain, initial pressure and thermal effects. But in our linear elastic model these values are zero. The linear elastic model is calculated as an isotropic material model. The following equation defines the elasticity matrix D Gl.2 where E is the modulus of elasticity and v is Poisson's ratio, which defines the contraction in the perpendicular direction [3]. For a detailed description, refer to the technical literature.
Generalized Maxwell ModelThe generalized Maxwell model is the most general form of the viscoelasticity models. The general linear dependence of the stress deviator on the strain history can be expressed by the hereditary integralwhere the function R(t) is called the relaxation modulus function. This function can be found by measuring the stress development over time t when the material is held at a constant strain. R(t) is often approximated in a Prony series
Gl.4This could be interpreted as a sum of i Maxwell models (branch) connected in parallel. The Maxwell model consists of a purely viscous damper and a purely elastic spring connected in series. The parameter τ i is the relaxation time constant of the spring -dashpot pairs and E i represents the stiffness of the spring in branch i. Parameter E 0 represents the stiffness of the spring parallel to the Maxwell models [3,4].
MethodsIn the following chapters the experimental setup and the simulation methods, linear elastic and generalized Maxwell model are explained.
Experimental SetupIn the experiment, the force-way-charts and force-timecharts were obtained by applied pressure to the cartilage constructs by means of compression discs. The dimensions of the discs were 10 mm in diameter. The experiment was
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