A. Suwannachit and U. Nackenhorst
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
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A. Suwannachit and U. Nackenhorst
Contents
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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
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Motivation
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Conventional approach for thermomechanical analysis of rolling tires
from [Whicker et al., 1981]
temperature distribution
Deformation
module
deformed
geometry
Tires are assumed
to be elastic !
thermoviscoelastic
Dissipation
module
energy
dissipation
Empirical models
Linear viscoelasticity
Thermal
module
Large deformations or
complicated properties
like damage etc.?
Goal
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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)
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Thermoviscoelastic constitutive model
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Helmholtz free energy function [Simo&Holzapfel, 1996]
: right Cauchy Green tensor
: absolute temperature
thermoelasticy
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rate-dependent response
: strain-like internal variables
Uncoupled kinematics (volumetric-isochoric split)
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Evolution law of internal variables
shear modulus
viscosity
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Thermal sensitivity of viscosities and shear moduli [Johlitz et al., 2010]
temperature-independent evolution equations !
relaxation time
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Thermodynamic consistency
2nd law of thermodynamics
2nd Piola-Kirchhoff stress :
viscous dissipation :
entropy :
Fourier’s law of heat conduction :
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Isentropic operator-split scheme
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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
A. Suwannachit and U. Nackenhorst
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Numerical test on constitutive modeling
• Pure shear loading conditions
• Fixed temperature at bottom
• Tube model for time-infinity
response
Steady-state responses
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f =10Hz
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Arbitrary-Lagrangian-Eulerian (ALE) relative kinematics
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Mesh points are neither fixed to material particles nor fixed in space
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Material velocity is split into a relative and convective part
=0, in case of stationary rolling
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Balance equations in time-independent form [Nackenhorst, 2004]
centrifugal force
internal force
impulse flux over boundary
external volume and surface loads
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Local mesh refinement in contact region
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Challenging task: treatment of inelastic material behavior
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Treatment of inelastic properties
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Problem: evolution law of internal variables is affected by convective terms
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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
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Solution strategy for thermomechanical analysis
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A three-phase staggered scheme
(neglecting convective part)
penalty contact constraint
(frictionless)
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ω = 50 rad/s
Numerical examples
(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
ω
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(II) Application with car tires
 ≈ 45000 DOF
 15 material groups in cross-section
 thermoelastic/thermoviscoelastic material
 bilinear approach for cords
30mm
ω
303K
 fixed temperature at rim contact 303K
 outside air 303K, contained air 318K
318K
 internal pressure ≈ 0.2 MPa
 rolling speed ≈ 80 km/h
 vertical displacements 30mm at rim strip
303K
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Contact pressure distribution
Steady-state response (reaction forces ≈ 4.81kN)
no rotation (reaction forces ≈ 4.61kN)
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temperature distribution
local dissipation
?
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Internal strains
ω
radial components
circumferential components
von Mises stress
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Conclusion
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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
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Parameter identification and model validation
Frictional heating
slip velocities and
circumferential contact shear stress
[Ziefle&Nackenhorst, 2008]