A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 11/14/14

A novel approach for thermomechanical analysis of A novel approach for thermomechanical analysis of stationary rolling tires within an ALE-kinematic stationary rolling tires within an ALE-kinematic frameworkframework

A. SuwannachitA. Suwannachit and U. Nackenhorst and U. NackenhorstInstitute of Mechanics and Computational Mechanics (IBNM)Institute of Mechanics and Computational Mechanics (IBNM)Leibniz Universität Hannover, GermanyLeibniz Universität Hannover, Germany

Akron, September 13, 2011Akron, September 13, 2011

A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 22/14/14

ContentsContents

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

A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 33/14/14

MotivationMotivation Conventional approach for thermomechanical analysis of rolling tires Conventional approach for thermomechanical analysis of rolling tires

from from [Whicker et al., 1981] [Whicker et al., 1981]

GoalGoal 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)

Deformation module

Dissipation module

Thermal module

deformed geometry

energy dissipation

temperature distribution

Tires are assumed to be elastic !

Empirical models

Linear viscoelasticity

Large deformations or complicated properties like damage etc.?

thermoviscoelastic

A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 44/14/14

Thermoviscoelastic constitutive modelThermoviscoelastic constitutive model Helmholtz free energy function Helmholtz free energy function [Simo&Holzapfel, 1996][Simo&Holzapfel, 1996]

Uncoupled kinematics (volumetric-isochoric split)

thermoelasticy rate-dependent response

Evolution law of internal variables

shear modulus

viscosity

: right Cauchy Green tensor

: absolute temperature

: strain-like internal variables

A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 55/14/14

Thermal sensitivity of viscosities and shear moduli [Johlitz et al., 2010]

22ndnd Piola-Kirchhoff stress : Piola-Kirchhoff stress :

entropy :entropy :

Thermodynamic consistency

2nd law of thermodynamics

viscous dissipation :viscous dissipation :

Fourier’s law of heat conduction :Fourier’s law of heat conduction :

temperature-independent evolution equations !

relaxation time

A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 66/14/14

Isentropic operator-split schemeIsentropic operator-split scheme A fractional-step approach to solve the coupled thermomechanical problems in

two sequential steps [Armero&Simo, 1992]

Advantages:• Avoid large non-symmetric tangent operator by simultaneous solution• unconditionally stable solutions

fixed entropy, but varying temperature

fixed motion

A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 77/14/14

Numerical test on constitutive modelingNumerical test on constitutive modeling

• Pure shear loading conditions• Fixed temperature at bottom• Tube model for time-infinity response

f =10Hz

Steady-state responses

A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 88/14/14

Arbitrary-Lagrangian-Eulerian (ALE) relative kinematicsArbitrary-Lagrangian-Eulerian (ALE) relative kinematics

Material velocity is split into a relative and convective partMaterial velocity is split into a relative and convective part

=0, in case of stationary rolling

• Local mesh refinement in contact region

Balance equations in time-independent form [Nackenhorst, 2004]

external volume and surface loadsinternal force

centrifugal force impulse flux over boundary

• Challenging task: treatment of inelastic material behavior

Mesh points are neither fixed to material particles nor fixed in spaceMesh points are neither fixed to material particles nor fixed in space

A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 99/14/14

Treatment of inelastic propertiesTreatment 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]

Lagrange-step:

• Neglect convective parts• Solve equilibrium equations in Lagrangian kinematics

Euler-step:

• Advection-type equations• Solve by using Time Discontinuous Galerkin method

A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 1010/14/14

Solution strategy for thermomechanical analysisSolution strategy for thermomechanical analysis A three-phase staggered schemeA three-phase staggered scheme

penalty contact constraint(frictionless)

(neglecting convective part)

A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 1111/14/14

Numerical examplesNumerical examples((II) Rolling viscoelastic rubber wheel) 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

ω = 50 rad/s

ω

temperature rise depending on excitation frequency

dynamic stiffening

A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 1212/14/14

((IIII) Application with car tires) Application with car tires

ω

≈ 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

Contact pressure distribution

Steady-state response (reaction forces ≈ 4.81kN) no rotation (reaction forces ≈ 4.61kN)

303K

318K

303K

30mm

A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 1313/14/14

temperature distribution local dissipation von Mises stress

?

Internal strains

radial components circumferential components

ω

A. Suwannachit and U. NackenhorstA. Suwannachit and U. Nackenhorst 1414/14/14

Conclusion 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

OutlookOutlook Parameter identification and model validation Frictional heating

slip velocities and circumferential contact shear stress [Ziefle&Nackenhorst, 2008]