1. A novel approach for thermomechanical analysis of stationary rolling tires within an ALE-kinematic framework
A. Suwannachit and U. Nackenhorst
Institute of Mechanics and Computational Mechanics (IBNM)
Leibniz Universität Hannover, Germany
Akron, September 13, 2011

2. Contents Motivation & Goal Thermoviscoelastic constitutive model
Isentropic operator-split scheme
ALE-relative kinematics & treatment of inelastic properties
Solution strategy for thermomechanical analysis
Numerical examples
Conclusion & Outlook

3. temperature distribution
Motivation
Conventional approach for thermomechanical analysis of rolling tires from [Whicker et al., 1981]
temperature distribution
deformed geometry
energy dissipation
Deformation module
Dissipation module
Thermal module
Tires are assumed to be elastic !
Empirical models
Linear viscoelasticity
Large deformations or complicated properties like damage etc.?
thermoviscoelastic
Goal
Description of dissipative rolling behavior with constitutive model at finite-strain
Energy loss derived from 2nd law of thermodynamics
Special care on constitutive description of rubber components (large deformations, viscous hysteresis, dynamic stiffening, internal heating, temperature dependency)

4. Thermoviscoelastic constitutive model
Helmholtz free energy function [Simo&Holzapfel, 1996]
: right Cauchy Green tensor
: absolute temperature
thermoelasticy
rate-dependent response
: strain-like internal variables
Uncoupled kinematics (volumetric-isochoric split)
Evolution law of internal variables
shear modulus
viscosity

5. Thermodynamic consistency
Thermal sensitivity of viscosities and shear moduli [Johlitz et al., 2010]
temperature-independent evolution equations !
relaxation time
Thermodynamic consistency
2nd law of thermodynamics
2nd Piola-Kirchhoff stress :
entropy :
viscous dissipation :
Fourier’s law of heat conduction :

6. Isentropic operator-split scheme
A fractional-step approach to solve the coupled thermomechanical problems in two sequential steps [Armero&Simo, 1992]
fixed motion
fixed entropy, but varying temperature
Advantages:
Avoid large non-symmetric tangent operator by simultaneous solution
unconditionally stable solutions

7. Numerical test on constitutive modeling f =10Hz
Pure shear loading conditions
Fixed temperature at bottom
Tube model for time-infinity response
Steady-state responses

8. Arbitrary-Lagrangian-Eulerian (ALE) relative kinematics
Mesh points are neither fixed to material particles nor fixed in space
Material velocity is split into a relative and convective part
=0, in case of stationary rolling
Balance equations in time-independent form [Nackenhorst, 2004]
centrifugal force
impulse flux over boundary
internal force
external volume and surface loads
Local mesh refinement in contact region
Challenging task: treatment of inelastic material behavior

9. Treatment of inelastic properties
Problem: evolution law of internal variables is affected by convective terms
Solution: a separate treatment of relative and convective terms [Ziefle&Nackenhorst, 2008]
Euler-step:
Advection-type equations
Solve by using Time Discontinuous Galerkin method
Lagrange-step:
Neglect convective parts
Solve equilibrium equations in Lagrangian kinematics

10. Solution strategy for thermomechanical analysis
A three-phase staggered scheme
(neglecting convective part)
penalty contact constraint (frictionless)
Advection-type equations
Solve by using Time Discontinuous Galerkin method

11. Numerical examples (I) Rolling viscoelastic rubber wheel ω = 50 rad/s
13200 DOF
constitutive parameters from previous example
compute with 5 different angular velocities (ω = 5,10,20,50,100 rad/s)
fixed temperature at inner ring Θ=293K
no heat exchange with ambient air
dynamic stiffening
temperature rise depending on excitation frequency

12. (II) Application with car tires
≈ DOF
15 material groups in cross-section
thermoelastic/thermoviscoelastic material
bilinear approach for cords
fixed temperature at rim contact 303K
outside air 303K, contained air 318K
internal pressure ≈ 0.2 MPa
rolling speed ≈ 80 km/h
vertical displacements 30mm at rim strip
30mm
303K ω
318K
303K
Contact pressure distribution
Steady-state response (reaction forces ≈ 4.81kN)
no rotation (reaction forces ≈ 4.61kN)

13. ? Internal strains temperature distribution local dissipation
von Mises stress ?
Internal strains
radial components
circumferential components ω

14. Conclusion
Thermoviscoelastic constitutive model (large deformations, viscous hysteresis, dynamic stiffening, internal heating, temperature dependency)
Solution of thermomechnical coupled problems with isentropic operator-split scheme
Three-phase computational approach for thermomechanical analysis
Numerical tests with viscoelastic rolling wheel and car tires
Outlook
Parameter identification and model validation
Frictional heating
slip velocities and circumferential contact shear stress [Ziefle&Nackenhorst, 2008]