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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. Contents.

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

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**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**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**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**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)**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]**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]**Numerical test on constitutive modeling**f =10Hz • Pure shear loading conditions • Fixed temperature at bottom • Tube model for time-infinity response Steady-state responses**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**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]**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**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**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**Internal strains**ω radial components circumferential components temperature distribution local dissipation von Mises stress ?**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

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