Image source: based on [1]
Development of a user-defined material model for a multiphysics simulation of short fibre reinforced plastic gears
Plastic gears are now used not only in simple actuators, but also increasingly in power transmission drives (e.g. e-bikes, light vehicles). This is due to their low-cost injection moulding, low weight and the possibility of dry running. Despite growing demand, reliable design methods that consider all relevant plastic-specific properties and damage mechanisms are still lacking. Compared to steel gears, the design of plastic gears presents challenges in modelling the complex material behaviour. The modelling approach presented below aims to capture time-dependent effects such as load speed, load duration and creep behaviour in the simulation of short fibre reinforced plastic gears.
Figure 1: CT-scan of a short fibre-reinforced plastic gear (by the Leibniz Institute for Composite Materials (IVW)) [1] and abstraction of the problem to a single fibre reinforced volume element
Multiphysics simulation for short fibre reinforced plastic gears
Currently, there are no applicable design methods available for plastic gears that cover all relevant plastic-specific properties and damage mechanisms. Even advanced simulation methods are still unable to fully considering the interactions between all damage mechanisms. In particular, the creep behavior of plastic gears and the influence of time-dependent material behavior on mechanical loading represent existing gaps in current research.
As part of a previous research project at the Chair for Machine Elements, Gears, and Tribology (MEGT), a multiphysics simulation for short fibre reinforced plastic gears made of PA46GF30 was developed, with the aim of enabling the calculation of interactions between various damage mechanisms. The simulation results were intended to provide practical guidance for gear design. The material model used in this simulation previously considered anisotropic linear-elastic and isotropic plastic material behavior. [1]
The time-dependent, viscoelastic material behavior of the short fibre reinforced plastic could not be represented at that time. As a result, viscoelastic deformation behavior and time-dependent effects such as creep, fatigue under cyclic loading were not taken into account, preventing a comprehensive representation of all relevant damage mechanisms. Since the standard material library in Abaqus does not provide a suitable material model capable of decomposing the material behavior into the underlying constituents of the composite material (fibre and matrix), an ongoing research project focuses on implementing the time-dependent, viscoelastic material behavior of PA46GF30 as a user-defined material model (UMAT), which is integrated into Abaqus as a subroutine.
User-defined material model for short fibre reinforced plastic gears
The mechanical behavior of PA46GF30 is determined by a large number of influencing factors. The main basis for the stiffness and strength behavior of the short fibre reinforced plastic lies in the specific composition, in particular in the matrix type polyamide 46, the glass fibres and the fibre content of 30 wt%. In addition, microstructural characteristics such as fibre orientation, the distribution of the fibres and their length play a central role in the stiffness and strength behavior.
As shown in Figure 1, based on the CT-scan of a short fibre reinforced plastic gear, the short glass fibres are highly dispersed and exhibit varying orientations. This pronounced anisotropy poses challenges for the implementation of a user-defined material model. For this reason, the complexity of the geometry and high anisotropy was excluded in a first step and the implementation of the material model was tested on a simple volume element with one fibre (cf. Figure 1). This model is primarily used for verification of the material model and was deliberately kept as simple as possible in order to better understand the effects and behavior of the fibre within the material model.
Figure 2: Decomposition of the stress tensor and assignment of stress components to the characteristics of the fibre and matrix based on [2]
The mathematical model used for this work is a mechanism-based, three-dimensional, viscoelastic material model for unidirectional fibre reinforced plastic composites. The special feature of this material model lies in the decomposition of the material behavior into the underlying components of the composite material (fibre and matrix). [2] When subjected to external stress, the fibres absorb most of the load and are considered elastic. In contrast, the thermoplastic matrix behaves viscoelastic. [2, 3] Viscoelastic, time-dependent material behaviour is only applied to parts of the decomposition corresponding to the thermoplastic matrix's material behaviour. [2]
The behaviour of the viscoelastic material is described using the generalised Maxwell model (cf. Figure 3), which is then applied to the relevant stress and strain decomposition terms [2]. In conventional finite element method (FEM) programmes, the stress and strain fields are solved when the test specimen is subjected to external loading by specifying the node displacements. The strains are then determined from these displacements. These strains then serve as input variables for the user-defined material model, which is used to calculate the corresponding stresses. The parallel-connected, isolated spring of the generalised Maxwell model represents the long-term behaviour of the material stiffness after all stresses in the material have been relieved (cf. Figure 3) [2].
Figure 3: Generalized Maxwell rheological model according to [2, 4]
The user-defined material routine is implemented in the programming language Fortran. To integrate the user-defined material routine into Abaqus, a single Fortran file is required. The user-defined material routine is divided into two main subprograms, namely ORIENT and UMAT. The ORIENT subprogram is used for local coordinate transformation to specify the fibre orientation. The UMAT contains all the mathematical equations that define the material behavior for the fibre and the Matrix.
Results and Conclusion
Figure 4: Stress relaxation for constant strain in the main directions (xx- direction left, yy-direction middle and zz-direction right)
To verify the implemented material model, plausibility tests were carried out on the volume element, including creep tests and stress relaxation tests. The material behavior was examined in all loading directions. Figure 4 shows for example the stress relaxation behavior under constant strain in the main directions.
The results of the plausibility tests showed that, in a first step, the implemented material model quantitatively captures the expected material behaviour of the fibre and the matrix. The mathematical model used in this work thus forms a solide initial basis for the future extension of the material model to short fibre reinforced plastic composites. In the further course of development, the user-defined material model will be expanded to include the anisotropy of the fibres and the plastic material behaviour and tested on the gear geometry as part of the multiphysics simulation.
Dipl.-Ing. Victoria Schröder, Research Assistant at Chair for Machine Elements, Gears and Tribology (MEGT), RPTU University of Kaiserslautern-Landau
Prof. Dr.-Ing. Oliver Koch, Head of Chair for Machine Elements, Gears and Tribology (MEGT), RPTU University of Kaiserslautern-Landau
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References
[1] Kassem, W.; Gebhard, A.; Schmidt, S.; Oehler, M.; Hausmann, J.; Koch, O.; Breuer, U.: Auslegung spritzgegossener Kunststoffzahnräder. Abschlussbericht zum FVA-Forschungsvorhaben 856 I (Heft 1491), IGF-Nr. 20379 N. Forschungsvereinigung Antriebstechnik e.V., Frankfurt/Main, 2022
[2] Gennaro, L.; Daghia, F.; Olive, M.; Jacquemin, F.; Espinassou, D.: A new mechanism-based temperature-dependent viscoelastic model for unidirectional polymer matrix composites based on Cartan decomposition. in: European Journal of Mechanics / A Solids 90, 104364. 2021.
[3] Stommel, M.; Stojek, M.; Korte, W.: FEM zur Berechnung von Kunststoff- und Elastomerbauteilen. 2., neu bearbeitete und erweiterte Auflage. München: Hanser, 2018.
[4] Grellmann, W.; Seidler, S.: Mechanische Eigenschaften von Kunststoffen. in: Kunststoffprüfung. 3. Aufl. Carl Hanser Verlag, München, 2015. Kap. 4, S. 79–246.