Numerical simulation of a relaxation test designed to fit a quasi-linear viscoelastic model for temporomandibular joint discs

Abstract

The main objectives of this work are: (a) to introduce an algorithm for adjusting the quasi-linear viscoelastic model to fit a material using a stress relaxation test and (b) to validate a protocol for performing such tests in temporomandibular joint discs. This algorithm is intended for fitting the Prony series coefficients and the hyperelastic constants of the quasi-linear viscoelastic model by considering that the relaxation test is performed with an initial ramp loading at a certain rate. This algorithm was validated before being applied to achieve the second objective. Generally, the complete three-dimensional formulation of the quasi-linear viscoelastic model is very complex. Therefore, it is necessary to design an experimental test to ensure a simple stress state, such as uniaxial compression to facilitate obtaining the viscoelastic properties. This work provides some recommendations about the experimental setup, which are important to follow, as an inadequate setup could produce a stress state far from uniaxial, thus, distorting the material constants determined from the experiment. The test considered is a stress relaxation test using unconfined compression performed in cylindrical specimens extracted from temporomandibular joint discs. To validate the experimental protocol, the test was numerically simulated using finite-element modelling. The disc was arbitrarily assigned a set of quasi-linear viscoelastic constants (c1) in the finite-element model. Another set of constants (c2) was obtained by fitting the results of the simulated test with the proposed algorithm. The deviation of constants c2 from constants c1 measures how far the stresses are from the uniaxial state. The effects of the following features of the experimental setup on this deviation have been analysed: (a) the friction coefficient between the compression plates and the specimen (which should be as low as possible); (b) the portion of the specimen glued to the compression plates (smaller areas glued are better); and (c) the variation in the thickness of the specimen. The specimen’s faces should be parallel to ensure a uniaxial stress state. However, this is not possible in real specimens, and a criterion must be defined to accept the specimen in terms of the specimen’s thickness variation and the deviation of the fitted constants arising from such a variation.