thermo-mechanical analysis of stationary rolling tires
Final Report Abstract
In this project a multi-scale finite element framework for detailed tire calculations has been developed. This framework accounts for fast stationary rolling motion, large deformations, temperature dependent, inelastic material behavior of fibre-reinforced rubber compounds, thermomechanical coupling, air inflation, tractive rolling using phenomenological rubber friction models and stochastically averaged, homogenized meso-scale tread-road interaction. By means of this framework, the tire’s thermo-mechanical response as well as rolling resistance can be evaluated for stationary accelerated, free rolling, braked and cornering situations. It furthermore enables a quantification of the influence of surface roughness and tread design on rolling resistance. The insight obtained by these calculations can yield more detailed knowledge on sources of rolling resistance, which is essential for the development of low rolling resistance and more sustainable tire designs. In this framework the macro-scale stationary rolling motion of tires is described exploiting the benefits of arbitrary Lagrangian Eulerian (ALE) kinematics, which allows for a time-independent formulation of balance equations and therefore an efficient direct calculation of the steady state. Here, the strong thermomechanial coupling induced by the severe temperature dependency and the structural heating by internal dissipation of rubber compounds is treated with an isentropic operator split scheme. The evolution of material history to describe inelastic material behavior is realized with a staggered approach, which separates the solution of the evolution equation into a local part and an advective transport. The present algorithm solves the balance equations in a three phased staggered scheme. In the first isentropic mechanical phase, the balance of linear momentum is solved including inertia effects of the tractive stationary rolling motion. Here, the evolution of the internal material history variables is calculated locally. Note that the assumption of constant entropy requires the local calculation of an equivalent isentropic temperature. In the second thermal phase, the stationary heat conduction problem is solved accounting for structural heating by internal heating and heat transfer by convection into the surrounding air and by conduction into the ground. In the third phase the advective transport of material history is solved using a time discontinuous Galerkin method. To date, the capabilities of the developed framework were demonstrated using a thermo-viscoelastic material formulation, which can be extended to arbitrarily complex material formulations. Due to the high level of detail, finite element models with a large number of elements are required to obtain an accurate solution. In order to allow for an efficient treatment, the calculation of element stiffness matrices and the solution of the global stiffness matrix are performed in parallel. In order to include meso-scale tread-road interaction (penetration of tread block by largest asperities) into this framework, the stochastic average contact response of single tread blocks was computed by solving a random Signorini problem using a quasi-Monte Carlo method. This contact behavior is homogenized by introducing an energy-equivalent uniaxial compression test, which allows for the formulation of constitutive contact law based on the tread’s bulk material model. Note that the description of inelastic behavior in the contact area requires the introduction of internal variables. These internal variables need to be treated analogous to those in the bulk within a staggered scheme of local evolution and advective transport. By means of this framework, a difference of 10% rolling resistance in between smooth and rough asphalt surfaces has been computed, which is in good agreement with experimental results reported in literature.
Publications
- Finite Element Analysis of Tires in Rolling Contact, GAMM-Mitteilungen, 37: 27–65, (2014)
U. Nackenhorst
(See online at https://doi.org/10.1002/gamm.201410003) - Thermomechanical contact of rubber-like solids on rough surfaces, Proc. Appl. Math. Mech., 14: 231–232, (2014)
R. Beyer, U. Nackenhorst
(See online at https://doi.org/10.1002/pamm.201410103) - A multilevel adaptive sparse grid stochastic collocation approach to the non-smooth forward propagation of uncertainty in discretized problems
R. L. Gates, M. R. Bittens
- Homogenization of thermomechanical tread-road contact for the investigation of tire rolling resistance. Proc. Appl. Math. Mech. 16,1, Special Issue: Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker‐Vereinigung (DMV), Braunschweig 2016; October 2016, Pages 513-514
R. Beyer, U. Nackenhorst
(See online at https://doi.org/10.1002/pamm.201610245)
