A. Suwannachit and U. Nackenhorst Institute of Mechanics and Computational Mechanics (IBNM)

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. Motivation Conventional approach for thermomechanical analysis of rolling tires from [Whicker et al., 1981] Empirical models Large deformations or complicated propertieslike damage etc.? Tires are assumed to be elastic ! Linear viscoelasticity 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) temperature distribution energy dissipation deformed geometry Deformation module Dissipation module Thermal module

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

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

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

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

9. Lagrange-step: Euler-step: • Neglect convective parts • Solve equilibrium equations in Lagrangian kinematics • Advection-type equations • Solve by using Time Discontinuous Galerkin method 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]

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. ω= 50 rad/s Numerical examples ω (I) Rolling viscoelastic rubber wheel • 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. Contact pressure distribution Steady-state response (reaction forces ≈ 4.81kN) no rotation (reaction forces ≈ 4.61kN) • ≈ 45000 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 (II) Application with car tires ω 318K 303K

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

14. Outlook • Parameter identification and model validation • Frictional heating slip velocities and circumferential contact shear stress [Ziefle&Nackenhorst, 2008] 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