Simulation of Thermomechanical Couplings of Viscoelastic Materials

Transcription

1 Simulation of Thermomechanical Couplings of Viscoelastic Materials Frank Neff 1, Thomas Miquel 2, Michael Johlitz 1, Alexander Lion 1 1 Institute of Mechanics Faculty for Aerospace Engineering Universität der Bundeswehr München 2 École Polytechnique, Paris October, 14th 2016

2 Contents 1 Introduction 2 Modelling 3 Implementation in COMSOL Multiphysics 4 Results and Conclusions

3 Contents 1 Introduction 2 Modelling 3 Implementation in COMSOL Multiphysics 4 Results and Conclusions

4 Motivation Motivation Technical applications of elastomers, e.g. bearings tires sealings etc. CSR New Material Technologies GmbH

5 Motivation Motivation Technical applications of elastomers, e.g. bearings tires sealings etc. CSR New Material Technologies GmbH Properties of elastomers viscoelastic material behaviour glass transition aging (physical, chemical) Mullins-Effect Payne-Effect self-heating swelling

6 Motivation Self-heating dynamic mechanical loads and finite strains energy dissipation causes increase in temperature

7 Motivation Self-heating dynamic mechanical loads and finite strains energy dissipation causes increase in temperature Need to consider self-heating temperature influences the whole stress-strain relation dynamic mechanical loads cause change in temperature complex structures exhibit a complex temperature field

8 Motivation Challenges finite viscoelastic material behaviour (isothermal) coupling between dissipated energy and temperature temperature dependent change of material parameters time efficient computation Shear test 3d

9 Motivation Challenges finite viscoelastic material behaviour (isothermal) coupling between dissipated energy and temperature temperature dependent change of material parameters time efficient computation Our contributions model for non-linear viscoelastic materials model for heat transfer multiphysics coupling thermal expansion multiphysics coupling dissipation following: Johlitz 2015 Dippel 2015 Dippel and Johlitz 2014 Shear test 3d

10 Contents 1 Introduction 2 Modelling 3 Implementation in COMSOL Multiphysics 4 Results and Conclusions

11 Kinematics Deformation Gradient F F IVC ˆF ˆF i RC EIC CC ˆF e F (Johlitz, Dippel, and Lion 2015) Separate volumetric and isocoric deformation volumetric deformation F due to thermal expansion elastomers are considered as nearly incompressible referring to mechanical loads isochoric deformation ˆF due to mechanical loadings multiplicative split of the deformation gradient (Lee 1969) F = ˆF F

12 Kinematics Deformation Gradient F F IVC ˆF ˆF i RC EIC CC ˆF e F (Johlitz, Dippel, and Lion 2015) Separate volumetric and isocoric deformation volumetric deformation F due to thermal expansion elastomers are considered as nearly incompressible referring to mechanical loads isochoric deformation ˆF due to mechanical loadings multiplicative split of the deformation gradient (Lee 1969) F = ˆF F Volumetric deformation gradient F F = J 1 3 I det ( F) = α (θ θ0 ) coefficient of thermal expansion α

13 Kinematics Deformation Gradient F F IVC ˆF ˆF i RC EIC CC ˆF e F (Johlitz, Dippel, and Lion 2015) Separate volumetric and isocoric deformation volumetric deformation F due to thermal expansion elastomers are considered as nearly incompressible referring to mechanical loads isochoric deformation ˆF due to mechanical loadings multiplicative split of the deformation gradient (Lee 1969) F = ˆF F Volumetric deformation gradient F F = J 1 3 I det ( F) = α (θ θ0 ) coefficient of thermal expansion α Isochoric deformation gradient ˆF ˆF = J 1 3 F ) det(ˆf = 1

14 Kinematics Finite viscoelasticity η ε c 10 c e 10 Modelling finite viscoelasticity equilibrium spring: hyperelastic material model (e.g. Neo-Hooke) no time dependence ε ε in ε el 3-parameter-model σ σ neq Maxwell-element: spring is modelled as Neo-Hookean-spring time dependent behaviour of the damper σ eq t strain-time relation / stress-time relation of the 3-parameter-model t

15 Kinematics Finite viscoelasticity η ε c 10 c e 10 Modelling finite viscoelasticity equilibrium spring: hyperelastic material model (e.g. Neo-Hooke) no time dependence ε ε in ε el 3-parameter-model σ σ neq Maxwell-element: spring is modelled as Neo-Hookean-spring time dependent behaviour of the damper Elastic-inelastic split of ˆF σ eq t strain-time relation / stress-time relation of the 3-parameter-model t elastic part ˆF e inelastic part ˆF i multiplicative split of the isochoric deformation gradient ˆF = ˆF e ˆF i

16 Kinetics Entropy balance Clausius-Duhem-Inequality (CDI) (Haupt 2002) ρ 0 ψ + T : Ė ρ 0s θ q 0 θ Gradθ 0 ψ: Helmholtz free energy T: 2nd Piola-Kirchhoff stress tensor E: Green-Lagrangean strain tensor

17 Kinetics Entropy balance Clausius-Duhem-Inequality (CDI) (Haupt 2002) ρ 0 ψ + T : Ė ρ 0s θ q 0 θ Gradθ 0 ψ: Helmholtz free energy T: 2nd Piola-Kirchhoff stress tensor E: Green-Lagrangean strain tensor Additive formulation of the Helmholtz free energy ψ = ψ vol eq (J,θ) + ψiso eq (Ĉ) + ψ iso neq (Ĉe )

18 Kinetics Entropy balance Clausius-Duhem-Inequality (CDI) (Haupt 2002) ρ 0 ψ + T : Ė ρ 0s θ q 0 θ Gradθ 0 ψ: Helmholtz free energy T: 2nd Piola-Kirchhoff stress tensor E: Green-Lagrangean strain tensor Additive formulation of the Helmholtz free energy ψ = ψ vol eq (J,θ) + ψiso eq Helmholtz free energy and CDI ( ) ( ψeq vol p + ρ 0 J J ( 1 2 J ˆT ψ eq iso ρ 0 Ĉ ρ 0 ψeq vol ρ 0 s + ρ 0 θ ˆF 1 i ψiso neq T Ĉe ˆF i (Ĉ) + ψ iso neq ) θ q 0 θ Gradθ + ) (Ĉe ) : Ĉ ψ neq iso + ρ 0 Ĉe : (Ĉe ˆL i + ˆL ) i Ĉe 0

19 Material modelling Material modelling Approach for Helmholtz free energy density ρ 0 ψ vol eq = 1 [(J 2 K 1) 2 + (lnj) 2] K α (J 1)(θ θ 0 ) ρ 0 c(θ) ρ 0 ψ iso ( ) eq = c 10 IĈ 3 ρ 0 ψ iso ( ) neq = ce 10 IĈe 3

20 Material modelling Material modelling Approach for Helmholtz free energy density ρ 0 ψ vol eq = 1 [(J 2 K 1) 2 + (lnj) 2] K α (J 1)(θ θ 0 ) ρ 0 c(θ) ρ 0 ψ iso ( ) eq = c 10 IĈ 3 ρ 0 ψ iso ( ) neq = ce 10 IĈe 3 Evaluation of the constitutive equations p = K [ (J 1) + lnj J ] + K α (θ θ 0 ) s = 1 ( c(θ) K α (J 1) + ρ 0 ρ 0 θ ) ˆT = 2J 1 c 10 I + 2J 1 c e 10 Ĉ 1 i 2 ( 3 J 1 c 10 trĉ + ce 10 trĉe )Ĉ 1

21 Material modelling Final equations 2nd Piola-Kirchhoff stress tensor T = pj C 1 + 2J 2 3 c 10 ( I 1 3 tr(ĉ)ĉ 1) + 2J 2 3 c e 10 (Ĉ 1 i 1 3 tr (Ĉ 1 )Ĉ 1 ) i Ĉ

22 Material modelling Final equations 2nd Piola-Kirchhoff stress tensor T = pj C 1 + 2J 2 3 c 10 ( I 1 3 tr(ĉ)ĉ 1) + 2J 2 3 c e 10 Evolution equation Ĉ i = 2ce 10 (Ĉ η(θ) 1 3 tr (Ĉ Ĉ 1 ) i )Ĉi (Ĉ 1 i 1 3 tr (Ĉ 1 )Ĉ 1 ) i Ĉ

23 Material modelling Final equations 2nd Piola-Kirchhoff stress tensor T = pj C 1 + 2J 2 3 c 10 ( I 1 3 tr(ĉ)ĉ 1) + 2J 2 3 c e 10 Evolution equation Ĉ i = 2ce 10 (Ĉ η(θ) 1 3 tr (Ĉ Ĉ 1 ) i )Ĉi Heat transfer equation K αθj + ρ 0 (A + B θ) θ λ θ Div(Grad(θ)) c e 10 Ĉ 1 i Ĉ Ĉ 1 i (Ĉ 1 i 1 3 : Ĉ i = 0 tr (Ĉ 1 )Ĉ 1 ) i Ĉ

24 Material modelling Final equations 2nd Piola-Kirchhoff stress tensor T = pj C 1 + 2J 2 3 c 10 ( I 1 3 tr(ĉ)ĉ 1) + 2J 2 3 c e 10 Evolution equation Ĉ i = 2ce 10 (Ĉ η(θ) 1 3 tr (Ĉ Ĉ 1 ) i )Ĉi Heat transfer equation K αθj + ρ 0 (A + B θ) θ λ θ Div(Grad(θ)) c e 10 Ĉ 1 i Ĉ Ĉ 1 i (Ĉ 1 i 1 3 : Ĉ i = 0 tr (Ĉ 1 )Ĉ 1 ) i Ĉ Williams-Landel-Ferry equation (WLF) ( ) η(θ) = η 0 exp C 1(θ θ G ) C 2 +θ θ G

25 Contents 1 Introduction 2 Modelling 3 Implementation in COMSOL Multiphysics 4 Results and Conclusions

26 Physics interfaces Mechanical behaviour Physics interface: Finite Viscoelasticity Domain feature Boundary conditions: Fixed Constraint Predescribed Displacement Predescribed Load

27 Physics interfaces Mechanical behaviour Physics interface: Finite Viscoelasticity Domain feature Boundary conditions: Fixed Constraint Predescribed Displacement Predescribed Load Domain feature: Finite Viscoelasticity 1 equilibrium part: Neo-Hooke incompressibility: Penalty-Formulation Lagrange-Formulation number of Maxwell-Elements (0-5)

28 Physics interfaces Thermal behaviour Physics interface: Heat Transfer Domain feature Boundary conditions: Temperature Adiabat

29 Physics interfaces Thermal behaviour Physics interface: Heat Transfer Domain feature Boundary conditions: Temperature Adiabat Domain feature: Heat Transfer 1 Fourier heat transfer

30 Multiphysics coupling Couplings Multiphysics coupling: Thermal Expansion coupling of: Finite Viscoelasticity Heat-Transfer thermal expansion coefficient

31 Multiphysics coupling Couplings Multiphysics coupling: Thermal Expansion coupling of: Finite Viscoelasticity Heat-Transfer thermal expansion coefficient Multiphysics coupling: Dissipation self-heating WLF-Parameters for each Maxwell-Element could be combined with thermal expansion

32 Contents 1 Introduction 2 Modelling 3 Implementation in COMSOL Multiphysics 4 Results and Conclusions

33 Results Temperature influence of viscoelasticity Static shear test shear angle 20 temperature 293K and 363K chosen academic material parameters

34 Results Temperature influence of viscoelasticity Static shear test shear angle 20 temperature 293K and 363K chosen academic material parameters Dynamic shear test shear angle 20 and 40 (amplitude) sinus cycle with 4 Hz frequency temperature 363 K chosen academic material parameters

35 Results Cyclic deformation of an hourglass sample Cyclic test of an hourglass sample tension 6mm sinus cycle with 4 Hz frequency (tension only) temperature 333 K chosen academic material parameters material parameters could be identified by DMA experiment

36 Conclusions Conclusions a model for finite viscoelastic material behaviour is proposed and implemented temperature dependence of mechanical behaviour is guaranteed by WLF-approach for the relaxation times coupling of dissipated mechanical energy and temperature field coupling temperature and volumetric expansion modular setup leads to flexibility in application

37 Conclusions Literature Dippel, B. (2015). Experimentelle Charakterisierung, Modellierung und FE-Berechnung thermomechanischer Kopplung. Ph. D. thesis, Universität der Bundeswehr München. Dippel, B. and M. Johlitz (2014). Thermo-mechanical couplings in elastomers - experiments and modelling. ZAM Zeitschrift für Angewandte Mathematik und Mechanik. Haupt, P. (2002). Continuum Mechanics and Theory of Materials. Springer Verlag. Johlitz, M., B. Dippel, and A. Lion (2015). Dissipative heating of elastomers: a new modelling approach based on finite and coupled thermomechanics. Continuum Mechanics and Thermodynamics, Lee, E. (1969). Elastic-plastic deformatin at finite strain. Jounal of Applied Mechanics 36, 1 6.