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A Finite Element Material Modeling Technique for the Hysteretic Behaviour of Reinforced Rubber

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A Finite Element Material Modeling Technique for the Hysteretic Behaviour of Reinforced Rubber DEGREE PROJECT IN VEHICLE ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2017 A finite element material modeling technique for the hysteretic behaviour of reinforced rubber PETTER LIND KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES i A finite element material modeling technique for the hysteretic behaviour of reinforced rubber by Petter LIND Degree project in Solid Mechanics Second level, 30.0 HEC Stockholm, Sweden 2017 Abstract Reinforced rubber is thanks to its elastic and dissipative properties found in industrial applications such as isolators, flexible joints and tires. Its dissipative propertied comes from material related losses which have the effect that energy invested when deform- ing the material is not retained when returning it back to its initial state. The material losses are in turn caused by interactions in the material on a level below the micro scale. These interaction forms a macro stress strain response that is dependent on both strain amplitude, strain rate and temperature. It is thus a challenge to accurately model components made of reinforced rubber and and features of interest related to them, such as the rolling resistance for a tire. It is also difficult to device general design guide lines for such components due to rubbers many and complex dependencies and a simple accurate phenomenological model for modeling these properties are highly sought for in industry today. This thesis presents a method for modeling the strain amplitude and strain rate be- havior for cyclically loaded rubber along with a method of choosing its material param- eters. The proposed modeling technique results in a model with the same frequency dependency over all strain rates. An approximation which is shown to be valid over a few decades of strain amplitudes and rates and is believed useful for many industrial applications. The material model presented can in addition be implemented in commercial FE- softwares by using only pre-defined material models. This was achieved by implementa- tion of the overlay method. The thesis also presents a method for how to implement the modeling technique in simulations with purpose to determine the rolling resistance of a truck tyre. ii En materialmodell för att modellera hysteresberoendet för förstärkt gummi Petter LIND Examensarbete i Hållfasthetslära Avancerad nivå, 30 hp Stockholm, Sverige 2017 Sammanfattning Förstärkt gummi används, tack vare sina elastiska och dissipativa egenskaper, i in- dustriella komponenter som exempelvis bussningar i drivlinan, däck och flexibla gummi- kopplingar. Dissipationen orsakas av materialförluster som i sin tur orsakas av interak- tioner på längdskalor kortare än micro-nivå i materialet. Dessa Interaktioner resulterar i ett material som mekaniskt kan klassificeras som ett ickelinjärt material beroende av töjningsamplitud, töjningshastighet och temperatur. Det är därför en utmaning att göra modeller som på ett korrekt sätt förutsäger be- teendet för gummikomponenter och egenskaper relaterade till dessa, som exempelvis rullmotståndet i ett däck. Det är även svårt att ge generella design riktlinjer för dessa komponenter på grund gummits många materialberoenden och enkla användvändbara fenomenologiska modeller som kan underlätta vid sådana processer är därför högt efter- frågade av industrin idag. I denna rapport presenteras en materialmodell för att modellera töjningsamplitud- och töjningshastighetsberoendet för gummi under cyklisk last samt en metod för att välja dess materialparametrar. Den föreslagna materialmoddeleringstekniken resulterar i en modell med samma töjningshastighetsberoende för alla töjningsamplituder. En approx- imation som är användbar inom ett antal decader av töjningsamplituder och töjning- shastigheter vilket borde vara tillräckligt för de flesta industriella tillämpningar idag. Den föreslagna materialmodellen kan dessutom implementeras i kommersiella FE- programvaror genom att endast använda i programmet inbyggda materialmodeller. Detta sker genom tillämpning av overlay-metoden. I rapporten presenteras även en metod för hur modelleringstekniken kan implementeras genom en tillämpning i simuleringar med syfte att bestämma rullmotståndet för ett lastbilsdäck. iii Acknowledgements I would first like to thank my excellent supervisor at Scania, Rickard Österlöf, for his guidance throughout this whole project. It was always possible to ask questions and get some helping advices whenever I ran into a trouble spot. I would also like to thank all co-workers at RTCC and at the neighboring group RTLC. You have all been warm, kind and supportive. You helped me with softwares, scripts, installations and even fixing my bike! I am very grateful. An extra thank you goes to my fellow students and friends Jonas Barrskog and Aron Ingi Ingvason for making my work so much easier. Finally, I would like to thank Carl Dahlberg at KTH. Carl provided excellent support on how to structure the text in this thesis. I am also very grateful for his valuable remarks regarding the text during the last phase of this project. Petter Lind Stockholm, 7th June 2017 iv Contents Abstract i Sammanfattning ii Acknowledgements iii 1 Introduction 1 1.1 Background .................................... 1 1.1.1 Purpose .................................. 3 1.1.2 Previous work ............................... 3 1.2 Outline of thesis .................................. 5 2 On tyres and rubber 6 2.1 The tyre ....................................... 6 2.1.1 Construction and terminology ..................... 6 2.1.2 Rolling resistance ............................. 7 2.2 Reinforced rubber ................................. 9 2.2.1 Hysteresis ................................. 10 2.2.2 Storage- and loss modulus ........................ 10 3 Rheological model 13 3.1 Hyperelastic material definition ......................... 14 3.2 Viscoelastic material definition ......................... 15 3.2.1 Logarithmic distribution of time constants over the frequency axis 16 3.2.2 Power-function distribution of time constants over the frequency axis 17 3.3 Plastic material definition ............................ 19 4 Finite element modeling 21 4.1 The overlay method ................................ 21 4.2 Virtual material test ................................ 22 4.3 Tyre modeling ................................... 22 4.3.1 preprocessing ............................... 22 4.4 Simulation steps .................................. 24 4.5 Simplifications ................................... 25 4.6 Postproccesing ................................... 26 5 Results 27 5.1 Virtual material test ................................ 27 5.1.1 Log distribution method ......................... 28 5.1.2 Power distribution method ....................... 30 5.2 Rolling resistance ................................. 32 5.2.1 Mesh study ................................ 33 5.2.2 Contribution from different parts .................... 33 5.2.3 Solution accuracy ............................. 34 v 6 Discussion and conclusions 36 6.1 Discussion ..................................... 36 6.2 Conclusions .................................... 36 6.3 Further work .................................... 37 6.4 Best practice for tyre simulation ......................... 37 A Derivations 39 A.1 Ziegler hardening law (1D) ........................... 39 A.2 Stress response in one maxwell element during cyclic pertubation ..... 41 B Details 43 B.1 Test matrix ..................................... 43 B.2 Python code .................................... 44 B.3 Simulation data .................................. 45 References 47 vi List of Figures 2.1 tyre structure ................................... 6 2.2 Free body diagram of a steady state rolling tyre ................ 8 2.3 Stress-strain response for reinforced rubber [9] ................ 10 2.4 Schematic representation of a stress strain response for illustrating param- eters for calculation of the storage- and loss modulus ............ 11 3.1 Rheological representation of the material model ............... 13 3.2 Stress strain response for the a generalized maxwell element, plastic and combined material model respectively. ..................... 14 3.3 Storage modulus for ten different maxwell elements using the log distri- bution method ................................... 16 3.4 Storage modulus comparison using the log distribution method ...... 17 3.5 Loss modulus comparison using the log distribution method ........ 17 3.6 Storage modulus for ten different maxwell elements using the power dis- tribution method ................................. 18 3.7 Storage modulus comparison using the power distribution method .... 18 3.8 Loss modulus comparison using the power distribution method ...... 19 3.9 One dimensional schematic solutions for different parameters in the Ziegler hardening law. The orange line shows the expected behavior. The Green line corresponds to C and blue line to C 0 .............. 20 !1 ! 4.1 Principle of the overlay method ......................... 21 4.2 Virtual FE-specimen in (a) undeformed and (b) deformed shape ...... 22 4.3 Comparison between the mesh and a real tyre cross section ......... 23 4.4 FE-mesh for the tyre ............................... 23 4.5 Simulation setup ................................. 25 5.1 Stress strain response from all virtual test using the log distribution method 28 5.2 Material response for different load frequencies using the log distribution method ....................................... 29 5.3 Storage modulus for material using
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