Predictionofstaticanddynamicbehaviorof ... · PDF filePredictionofstaticanddynamicbehaviorof powertrainsuspensionrubbercomponents Robin Öhrn Master Thesis ... garding the physical

Prediction of static and dynamic behavior ofpowertrain suspension rubber components

Robin Öhrn

Master ThesisDepartment of Management and Engineering

LIU-IEI-TEK-A–15/02356−SE

Prediction of static and dynamic behavior ofpowertrain suspension rubber components

Master Thesis in Mechanical EngineeringDivision of Solid Mechanics

Department of Management and EngineeringLinköping University

by

Robin Öhrn

LIU-IEI-TEK-A–15/02356−SE

Supervisors: Kjell SimonsonLinköping University

Daniel Högberg & Magnus WickströmVolvo Cars Corporation

Examiner: Jonas StålhandLinköping University

Linköping, 19 August, 2015

AbstractThe Department of Powertrain Suspension at Volvo Car Corporation constructsengine mounts which are mainly made of natural filled rubbers and aluminum.Volvo feels a growing need of gaining a better understanding and knowledge re-garding the physical behavior of the rubber components. This due to a desireto reduce the lead time with the supplier and decrease the development phase ofengine mounts in the near future. The engine mount designs are based on severalrequirements from different attribute areas such as noise, vibration and harsh-ness (NVH), driveability and ride comfort. Currently, the engineers at PowertrainSuspension are trying to meet all requirements by predicting static and dynamicstiffness. Today, Volvo relies on external suppliers to deliver engine mounts thatfulfill the desired requirements which in most cases are impossible and often thesuggested component size will be be too large in the engine compartment.

The purpose of this thesis is to develop a Finite Element (FE) model of an en-gine mount and to determine its static and dynamic stiffness. Volvo believes thatif they are able to perform in-house finite element analysis (FEA) of the rubbercomponents, the supplier interaction can be reduce which will result in decreasedlead time. To study the feasibility of in-house FEA of rubber component, thiswork has focused on the left lower tie bar (LLTB). The main task has been tounderstand and implement the rubber behavior in a commercial FE-software.

Natural filled rubbers are mainly dependent on frequency, strain rate and strainamplitude. The mechanical behavior of rubber is generally modeled as a combina-tion of hyperelastic, viscoelastic and elastoplastic parts. For the quasi-static case,only a hyperelastic model is needed to describe the static stiffness and to achieveaccurate correlation with measured data, provided by the supplier. For the dy-namical load cases, however, it is a necessary requirement to be able to describeall three parts of the mechanical behavior. The method used in this thesis is the,so called, the overlay model. The overlay model has shown a good correlation withexperimental data in previous studies and thereby is sufficient for capturing thefrequency and amplitude dependent behavior. The analyse have been carried outfor frequencies up to 80 Hz and a displacement amplitude of 4 mm. The resultthat only has a few percentage of average deviation with respect to measured data.

iii

Acknowledgments

This master thesis has been carried out at the Department Powertrain Suspensionat Volvo Car Corporation and in cooperation with the Division of Solid Mechanicsat Linköping University.I would like to take opportunity to express my gratitude and appreciation to thepeople that have been involved and their support during this project. A specialthanks go to my co-worker Dr. Rajesh Moolam for your endless support, I wouldnot have come this far without your help and advice. My supervisors at Volvo,Magnus Wickström and Daniel Högberg, that always had an answer to my ques-tions and guided me through the project, and Anton Wahnstöm, for his help withdifferent graphical tools and layout. Thanks to the rest of the colleagues at Power-train Suspension for making me feel welcomed and for all the assistance I neededduring this project. Also, thank you to Dr. Magnus Alvelid for his commitmentto this work.

I would also like to show my appreciation to my supervisor at Linköping Uni-versity, Professor Kjell Simonsson, for his support and enormous dedication inimproving both the report and the methods used in this thesis. Also, a thank youto my examiner Associate Professor Jonas Stålhand.

Finally, thanks to my family and friends for your support.

The thesis ends my studies for the Degree of Master of Science in MechanicalEngineering.

Gothenburg, June, 2015

Robin Öhrn

v

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 LLTB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.8 Additional consideration . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Presentation of rubber 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Vulcanization . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Damping and Dynamic Modulus . . . . . . . . . . . . . . . 132.2.3 Dynamic stiffness . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Amplitude and frequency dependency . . . . . . . . . . . . . . . . 152.3.1 Rate dependent . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.2 The Fletcher-Gent effect . . . . . . . . . . . . . . . . . . . . 162.3.3 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Constitutive description 193.1 Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.1 Strain energy function . . . . . . . . . . . . . . . . . . . . . 213.1.2 Neo-Hooke material . . . . . . . . . . . . . . . . . . . . . . 233.1.3 Mooney-Rivlin material . . . . . . . . . . . . . . . . . . . . 233.1.4 Yeoh material . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Viscoelastic response . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 Elastoplastic response . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 The overlay model . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

vii

4 Finite element model setup 314.1 Work procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1.1 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 Material models and parameters . . . . . . . . . . . . . . . 374.1.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Modification of the Abaqus input file . . . . . . . . . . . . . . . . . 424.2.1 Implementation of the overlay model . . . . . . . . . . . . . 424.2.2 Definition of amplitude curve . . . . . . . . . . . . . . . . . 42

4.3 Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Result 455.1 Quasi-static load case . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1.1 Tuning of material parameters and computational time . . 495.2 Dynamic load case . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6 Discussion and Conclusions 556.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.1.1 Quasi static load case . . . . . . . . . . . . . . . . . . . . . 556.1.2 Dynamic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7 Future studies 59

A Definition of amplitude curve 61

B Material input 65

Contents 1

List of Figures1.1 Placement of the engine and the engine mounts . . . . . . . . . . . 31.2 Graph of the static force-displacement behavior . . . . . . . . . . . 51.3 A overview of the LLTB, blue and grey represents rubber respec-

tively aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 The structure of vulcanized carbon-black-filled rubbers. Showingthe carbon particles, polymer chains and cross-links (dashed lines) 12

2.2 Measuring procedure for determination of SHORE/IRHD value andthe relationship between the shear modulus and hardness’s . . . . 13

2.3 An example of the hysteresis loop in harmonic shear . . . . . . . . 132.4 Frequency relationship for the shear modules and damping . . . . 162.5 Strain amplitude dependence of dynamic shear modulus and damping 162.6 Temperature dependence of dynamic shear modulus (a) and phase

angle (b) for a filled natural rubber. Influence of frequency is alsoshown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Simple one dimensional model describing the behavior of naturalfilled rubber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Characteristics behavior of the Neo-Hookean model . . . . . . . . . 233.3 Characteristic behavior of the M-R model . . . . . . . . . . . . . . 243.4 Characteristic behavior of the Yeoh model . . . . . . . . . . . . . . 253.5 The Zener model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.6 The relaxations modulus for the Zener model . . . . . . . . . . . . 273.7 Elastoplastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.8 One dimensional model describing the overlay model . . . . . . . . 29

4.1 A double symmetry FE-model of the LLTB. . . . . . . . . . . . . . 314.2 Working scheme of the development process of the FE-model. . . . 334.3 Overview of the mesh . . . . . . . . . . . . . . . . . . . . . . . . . 354.4 Deformation and penetration of elements during contact . . . . . . 364.5 Principle diagram of the overlay method in FE program. . . . . . . 394.6 Frequency step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.7 The *ELCOPY keyword in the Abaqus input file . . . . . . . . . . 424.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1 Force Displacement plot for the quasistatic case . . . . . . . . . . . 465.2 A closer view of the non-linear force . . . . . . . . . . . . . . . . . 475.3 Static stiffness for the quasi static case . . . . . . . . . . . . . . . . 485.4 Static stiffness-Force . . . . . . . . . . . . . . . . . . . . . . . . . . 495.5 Tuning of M-R parameters . . . . . . . . . . . . . . . . . . . . . . . 505.6 Computational time of the three different hyperelastic models . . . 515.7 Dynamical Force Displacement for LFLA . . . . . . . . . . . . . . 525.8 Dynamical stiffness Kd for LFLA . . . . . . . . . . . . . . . . . . . 535.9 Dynamical stiffness Kd for LFHA . . . . . . . . . . . . . . . . . . . 535.10 Kd as a function of frequency of and strain amplitude . . . . . . . 54

Contents xi

A.1 Outgoing displacement and reaction force using the Periodic step . 61A.2 Outgoing displacement and reaction force using the Smooth step . 62A.3 Zoomed in view of the red marked circle in Figure A.1b and Fig-

ure A.2b, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 62A.4 Dynamic stiffness for Smooth step and Periodic step . . . . . . . . 63

B.1 Implementation of the material parameter in the Abaqus input file 66

xii Contents

List of Tables1.1 Requirements for the engine mounts. For global coordinate system

see Figure 1.1a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

4.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Load cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.1 Difference between measured data and FEA for the force-displacementcurve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2 Difference Ks for the hyperelastic models . . . . . . . . . . . . . . 495.3 Difference of the dynamical stiffness for LFLA . . . . . . . . . . . . 545.4 Difference of the dynamical stiffness for LFHA . . . . . . . . . . . 54

Chapter 1

Introduction

This chapter provides the background to the work presented in this Master Thesis,a problem description, the methodology, and, finally the objectives of the work.

1.1 BackgroundThe Department of Powertrain Suspension at Volvo Car Corporation in Gothen-burg, develops suspension systems where the functionally vital parts are made ofnatural rubber. The rubber parts of the suspension systems are designed, man-ufactured and delivered by external suppliers today. The product developmentcycle for the rubber part is based on a series of design loops of computer aideddesign (CAD) generated design proposals followed by computer aided engineering(CAE) verification of these proposals, and, finally, testing of physical prototypes.The aim is to, as close as possible, reach the functional requirements set by Pow-ertrain Suspension at VCC for each part of the suspension system.

In new vehicle programs, the need to save time and to get the necessary design(component) space (in the tightly packed engine compartment) for each part ofthe suspension system, is always present. As a consequence of this, Powertrainsuspension has felt an ever increasing need to better understand the relationshipbetween the characteristics and the physical shape of the rubber parts. It wouldbe ideal for every new vehicle program if VCC could use CAE to develop a basicdesign for each rubber part that would satisfy the most fundamental functionalrequirements, and that would fit the available space in the engine compartment.

If this state could be reached, it would speed up the early development stagesof the rubber parts considerably, and would strengthen VCC’s position in the co-operation with its suppliers. The elasticity and damping properties of elastomers,such as natural rubber, is indeed very helpful when it comes to isolate the driverfrom the engine vibrations, but it poses a challenge when it comes to CAE analysisin terms of modeling the elasticity and damping properties to get agreement withmeasurements.

1

2 Introduction

1.2 Problem descriptionThe internal combustion process, and rotating and reciprocating parts in the carengine produce vibration and radiate sound. To prevent most of the vibrationsto reach the driving compartment, the engine is normally suspended by naturalrubber mounts, see Figure 1.1. The drawback of using this technique is that rigidbody movements relative to the car body are introduced. This affects the drive-ability of the car as well as the driving experience.

A Volvo engine can weigh up to 300 kg, the heavy load from the engine whiledriving affects the car in several different ways. Noise, Vibration and Harshness(NVH) and Driveability and Ride Comfort are attribute areas which depend onthe outcome of Powertrain Suspension result. Each of them has their individualneeds, specifications and requirements on the engine mounts. All the demands aresometimes very difficult to satisfy at the same time, and often there has to be acompromise among attribute areas, see Table 1.1.

Desired requirementsStiffness Attribute areaLow frequency and high amplitude (z-direction) Ride ComfortLow frequency and high amplitude (x-direction) DriveabilityLow frequency and low amplitude (xyz-direction) NVHHigh frequency and low amplitude (xyz-direction) NVH

DampingLow Freq High amplitude z Ride Comfort

Table 1.1: Requirements for the engine mounts. For global coordinate system seeFigure 1.1a

The engine mount’s fundamental properties from a Powertrain Suspensionssystem point of view, is the weight carrying capability and damping ability of rigidbody transient movement in z- and x-direction. There are two main categories ofcomponents in the powertrain suspension system, namely, mounts and tie-bars.The placement of the engine, the coordinate system, and the engine mounts andtie-bars can be seen in Figure 1.1b.

1.2 Problem description 3

(a) Placement of the engine (b) Placement of engine mounts

Figure 1.1: Placement of the engine and the engine mounts [1].

where specifically the LHM (Left Hand Mount), RHM (Right Hand Mount),RUTB (Right Upper Tie Bar), LLTB (Left Lower Tie Bar) and RLTB (RightLower Tie Bar) are shown. For more details regarding the engine mounts, seeSection 1.3.

With the technical specifications taken in consideration, the engine mounts needto be studied at a basic level in order to gain knowledge of how different designsand geometries can affect the end result. Elastomeric materials, e.g. rubber, haveunique material properties. Their advantages have been used in several engineer-ing applications and are commonly used as a mounting for stiff structures suchas vibration insulators, shock absorbers, and suspensions Austrell [2]. Using FEAthe mechanical behavior of rubber components with complex geometries can bedetermined during both static and dynamic conditions. Modeling and predictingof dynamic behavior is widely regarded the more difficult of the two, and a lowerdegree of accuracy than for the static case is normally to be expected.

In industrial applications, a reinforcing filler is added to the rubber, commonlycarbon black. The purpose of adding carbon black as a filler is to increase thetear strength, stiffness, and fatigue resistance Austrell [2]. Unfortunately, the me-chanical behavior of filled rubber is complex, with a strong temperature and strainamplitude dependency, and a weak frequency dependency Österlöf [3]. Further-more, the main driver attribute for rubber suspending the powertrain, NVH, willsuffer from the increased loss of damping introduced by the carbon black, leadingto a higher dynamic stiffness.

Computerized analysis of rubber has grown significantly over the last years and ithas provided engineers with an important tool to understand how rubber behaves.Constitutive relations which capture the major mechanical aspects of natural filledrubber, such as visco-elasticity and visco-plasticity, as well as specific and correctmaterial data for these relations, are of highest importance. The knowledge re-garding rubber materials is limited and the complexity of its structure makes it

4 Introduction

difficult to use general rules and design tools. A solid understanding of the basicattributes of natural filled rubber is necessary to be able to choose the correctmaterial or model(s) available in commercial software that, with respect to theirspecific weaknesses and strengths, best suit the analysis case in question. There-fore, it is important to establish good models in order to achieve results thatcorrelate well with, for instance, measured data.

In order to fully understand the problem given in the background, it is impor-tant to understand how the engine mounts works regarding requirements that areplaced on, both, the system and the component level. On the system level, thepowertrain can generally be treated as a rigid body with a certain total mass, typi-cally 300 kg, a certain center of gravity, and with certain inertia properties. Theseproperties, together the with position and individual stiffnesses and damping prop-erties of the components of the powertrain suspension system, fully determines therigid body movement of the powertrain.

A system level requirement that comes early in new car programs, is the staticvertical load carrying capability of the engine mounts. The static stiffness of theleft and right engine mounts are initially chosen in such a way that geometricalcollapse caused by creep is avoided. This gives a lower limit vertical static stiffness.The next system level requirement that needs to be fulfilled includes choosing atotal system vertical stiffness and damping that pushes the vertical eigenmodes ofthe rigid body powertrain away from the eigenfrequency range 4-12 Hz for humaninner organ. The attribute area that develops and place these requirements onthe powertrain suspension system is called Ride Comfort. It may be noted thatthe energy driving such vertical powertrain movements comes from vertical roaddisturbances.

The second system level requirement comes from the attribute area DriveAbil-ity and aims for preventing unwanted longitudinal movements of the center ofgravity of the powertrain relative to the car body. These movements would, oth-erwise, be transferred to the car body via the elastic suspension. The energydriving such longitudinal movements comes from drive shaft torque fluctuationsin combination with coupled rotational and translational rigid body movements ofthe powertrain, or longitudinal road disturbances. Fulfilling the requirements ofDriveAbility incorporate:

• Finding a combination of individual stiffnesses for the mount system com-ponents, that decouples the translational and rotational rigid body modesof the powertrain from each other as much as possible.

• Achieving a balance of individual component stiffnesses, that gives a minimallongitudinal movement of the powertrain center of gravity, when drive shafttorque is applied.

The last of the major three attribute areas that the powertrain suspension systemis systematically designed to fulfill, is NVH. The system level requirements placed

1.2 Problem description 5

by NVH on the powertrain suspension differs from the previous ones since it aimsat minimizing the effects of both rigid body movements and flexible modes of thepowertrain. Fulfilling the requirements of NVH incorporate:

• Minimizing dynamic stiffness and damping of each rubber component of thepowertrain suspension system, and to minimize the amount of vibrationsthat are transferred from the powertrain to the car body.

• Minimal stiffnesses of all system components is often not possible to reachin a world of contradictory requirements. Focus must then be put on thesuspension points were the car body have the highest sensitivities for noiseand vibration transfer.

All these system level requirements boil down to simple, but often contradic-tionary, component level requirements: Static characteristics: a prescribed forcevs displacement curve with the slope at zero load commonly used for its staticstiffness. Dynamic characteristics; dynamic stiffness and damping at prescribedpreloads, amplitudes and frequencies. Basically, the stiffness is considered to be adesign variable where an increased/decreased stiffness can affect the driveabilityand the driving experience in many different ways. Changing the stiffness andpredicting the consequences of it is a complex and expensive process, and the re-sult can in some cases be very hard to predict. At VCC the most common wayto determine the stiffness is to analysis statically measured data in the form offorce-displacement curves provided by the supplier. A typical data plot from thesupplier can be seen in Figure 1.2

Figure 1.2: Graph of the static force-displacement behavior [1].

The static data plot is of high importance since VCC have several in-housemethods in order to extract necessary parameters used in product development

6 Introduction

stage. An experience based approximation commonly used at VCC to determinatethe dynamic stiffness Kd, from the static stiffness, Ks, is to take

Kd = 1.5Ks (1.1)

Within VCC components, the dynamic stiffness normally varies in a range of1.1 ≤ Kd ≤ 5.

1.3 LLTBIn this thesis the focus will be on the LLTB, see Figure 1.3, that are found in aVolvo V40. The tie-bar, as the name reveals, takes care of the drive shaft torque,but also absorb forces and restrict powertrain movements in the longitudinal di-rection of the car together with the engine mounts.

1

2 3 4

5

Figure 1.3: A overview of the LLTB, blue and grey represents rubber and alu-minum respectively [1]. 1) Main rubber element, 2) Back snubber/bumpstop, 3)Insert, 4) Front snubber/bumpstop and 5) Outer frame.

1.4 Previous work 7

1.4 Previous workThe theory and methods that have been used in this thesis are mainly based on thework by Austrell [2]. He investigated the characteristics and behavior of naturalfilled rubber and developed different experimental methods to extract hyperelasticparameters to material models that are implemented in commercial FE software.A method of modeling the dynamic properties of filled rubbers is also providedtherein. His experiments showed that the constitutive models which were avail-able in the commercial software at that time, where not capable of modeling thebehavior of filled rubbers in dynamic applications. He suggested to add a rateindependent friction element in order to the describe the viscoplastic behavior.This was the embryo to what is known as the ’overlay model’ today. The overlaymodel has been adapted in this work in order to model the mechanical behavior ofnatural filled rubber. The FE software has improved enormously during the yearswith more advanced material models. This made it possible to continue furtherwith Austrell work, see e.g. Olsson [4] and Karlsson & Petterson [5] which are twoexamples of many publications based on the work by Austrell.

Olsson [4] presented a method to model the rate and amplitude dependent be-havior of rubber components exposed to dynamic loading. Using a standard finiteelement code, he showed that how rubber can be modeled using an overlay of vis-coelastic and elastoplastic finite element models without a need to define new con-stitutive relations. The harmonic shear tests were performed on thirteen differentrubber materials and these tests were done for different amplitude and frequencyranges to capture the amplitude and frequency dependency.The viscoelastic elasto-plastic model was also successfully fitted to experiments. The dynamic responseof the finite element model was then compared to measurements of a real testspecimen, thus, verifying the entire procedure from material test to finite elementmodel Olsson [4].

The overlay model is described in Gracia et al [6]. Therein, two industrial compo-nents made from natural filled rubber were subjected to several load cases usingthe overlay model for the FEA. By comparing FEA results for different param-eters in the overlay model and experimental data, they found that the overlaymodel improves the result if several layers of meshes with perfectly plastic mate-rial are used instead if of a single mesh with multilinear elastoplastic behavior. Thedrawback is that for low strain-energy cases where less accuracy is required, thecomputational cost (simulation time) becomes unnecessary high. In this specificarticle, other material models were found to be more suitable since they requiredless simulation time while still providing reasonable results.

1.5 ObjectivesThe objective of this work is to evaluate the possibilities of using FEA for mod-elling of primarily static, and secondarily dynamic, behavior of rubber parts of

8 Introduction

Powertrain Suspension systems.

Focus will initially be on finding a method to predict the force-displacement be-havior, as described in the Problem description section, finding a constitutiverelationship for describing the behavior of the rubber part under static (or quasi-static) conditions, and to achieve close correlation to measured force-displacementdata from physical testing.

If good correlation between the results for the CAE-model and correspondingmeasured physical data is reached for different standard suspension components,the thesis work is to be extended to incorporate a well defined dynamic response.

The following objectives are identified for this thesis.

1. Model the rubber elastic properties that are frequency dependent and am-plitude dependent using the overlay method.

2. Calculation of force-displacement curves for the LLTB using the FEM solverABAQUS/Standard for static and dynamic loads.

3. Evaluate different hyperelastic material models for the LLTB to calculatethe static stiffness and force-displacement.

4. Calculation for the components static stiffness curves, dynamic stiffness andcomparison with measurements from the supplier.

5. To draw conclusions from the overlay method implementations for CAE cal-culations of rubbers parts.

1.6 MethodologyThe Master thesis been divided into the following steps:

• Literature survey of rubber in order to obtain a fundamental knowledgeregarding natural filled rubber behavior and different material models.

• Develop an FE-model of the LLTB.

• Post-processing and review of the results.

• Validation of the final FE-model of the LLTB by comparing the results tomeasurements carried out by the supplier.

1.7 Limitations 9

1.7 LimitationsIn order to keep focus on the main tasks and complete the assignment in time,several limitations have been introduced.

• In commercial Finite Element codes there are several material models forhandling rubber material. This master thesis will only use and study theimpact of three different hyperelastic material models: Mooney-Rivlin, Neo-Hooke and Yeoh.

• Material parameters will be taken from Austrell [2].

• The FEA will be performed for existing geometries only, and no modificationsof the geometries will be studied.

• The FEA will focus on the LLTB.

• The Mullins effect will not be taken into consideration.

• Temperature effects on the behavior of the rubber material will not be takeninto consideration.

1.8 Additional considerationThe work presented in this thesis does not raise any questions regarding gender,sustainable society or ethical related topics. The aim of this work is to create amethod for rubber calculations with the goal to improve in-house FEA for VCC.

10 Introduction

Chapter 2

Presentation of rubber

This chapter provides a basic presentation of the molecular structure of vulcanizedfilled elastomers and material properties will be presented. For a more detaileddescription, the reader is referred to read the work by Austrell [2].

2.1 IntroductionThe word rubber is a collective term of vulcanized elastomer. All rubber materialsare consider to be elastic polymers and therefore the name "elastomer" have beenadapted. Rubber is thought of as an elastic and incompressible material but inreality there is no such thing as a purely elastic rubber. However, approximatingrubber as elastic can sometimes be very beneficial and provide quite good results.This is often used for natural rubbers under dynamic loading and for filled rubbersubjected to quasi-static loads. For unfilled rubbers, the deviation in the hysteresisis often quite small and can be neglected. These rubbers are of limited use inpractice, however. Under static loading it is also a good approximation to treatrubber as elastic Austrell [2], and good results can be obtained by fitting an elasticmodel to an experimental loading curve if ignoring the unloading curve Olsson [4].

2.1.1 VulcanizationVulcanization is the process when the raw plastic elastomeric material transformsinto a solid and elastic consistence Austrell [2]. The long molecular chains arelinked together during this chemical process and form a more solid molecularstructure with increased stability. The cross links, as seen in Figure 2.1, are estab-lished by adding a small amount of sulfur. The mixture is heated up to 170◦C, atthis moment the cross-links are connecting to the molecular chains Gil and Jesus[7].

As mentioned earlier, the fillers is added into the raw mixture before vulcan-ization in order to increase the stiffness. The particles of the carbon-black filleris very small and each particle size varies approximately between 20-50µm. The

11

12 Presentation of rubber

filler and the elastomer are connected by the cross-links and exist into two separatephases in the vulcanized rubber. The rubber phase forms a continuous network,and the filler material lumps inside the rubber network. This makes it a two-phasematerial made from constituents with completely different mechanical propertiesAustrell [2].

2.2 Material propertiesA polymer is a very long macromolecule, composed of many smaller subunits, ormonomers. The structural properties of the polymer chain depends on the specificmonomer, the branching and the amounts of crosslinking Österlöf [3]. Elastomersare made of organic compounds with a wide variations, but all are polymers.The raw material can be divided into two main categories, natural and synthetic.The sap of the so called "rubber trees" solidifies into thin sheets and are thencompressed into bales which is the raw material in natural rubber. The reason whythe consistency of raw natural rubber is soft and plastic is that there is no chemicalbounds between the molecular chains. Synthetic rubber on the other hand ismanufactured elastomers, made of petroleum products by polymerizing differentmonomers. The biggest advantages with synthetic elastomers is their capabilityof customization for specific purposes. Compared with natural rubber, syntheticrubber is better for designing against high temperatures, oils and abrasion butlack in fatigue resistance c.f. Österlöf [3]. This makes the natural rubber the firstchoice in many industrial applications compared to synthetic rubber.

Figure 2.1: The structure of vulcanized carbon-black-filled rubbers. Showing thecarbon particles, polymer chains and cross-links (dashed lines) [2].

2.2.1 HardnessToday, in the modern industry, the hardness or stiffness of natural filled rubbersis often defined by its SHORE or International Rubber Hardness Degree (IRHD)value. A Brinell test for measuring the hardness is illustrated in Figure 2.2. A

2.2 Material properties 13

constant force is applied and the indentation depth is measured. This methodgives an indirect measure of the elastic modulus and sometimes this is the onlyvalue available for estimating the shear modulus of the material.

F=constant

Δ

IRDH or SHORE units

30 40 50 60 70

2.5

2

1.5

1

0.5

0

G

[M

Pa

]

Figure 2.2: Measuring procedure for determination of SHORE/IRHD value andthe relationship between the shear modulus and hardness [2].

2.2.2 Damping and Dynamic Modulus

It is fundamental to have a correct description of the dynamic modulus in the FEmodel when dealing with dynamic processes. There are several ways do describeand determine the damping and dynamic modulus. One way is to describe thecomplex modulus in terms of an absolute value (dynamic modulus) and a phaseangle. An illustration of the hysteresis loop during a harmonic shear can be seenin Figure 2.3.

2κ0

κ

2τ0Uc

τ

Figure 2.3: An example of the hysteresis loop in harmonic shear.


For cyclic loads, the dynamic shear modulus is defined by

Gdyn = τ0

κ0(2.1)

where τ0 is the amplitude of the shear stress and κ0 is the amplitude of the shearstrain. The Uc is the energy loss per unit volume for one cycle is Uc = πτ0κ0sin(δ).For viscoelastic materials, the hysteresis is attributed the phase angle δ as d =sin(δ). For a material with elastoplastic properties, the phase angle is not welldefined. In this thesis, the damping d is defined by

d = Ucπκ0τ0

(2.2)

see below and Alkhatib [8].

2.2.3 Dynamic stiffnessThe dynamic stiffness, Kd, depends on amplitude and frequency of the applieddisplacement and is further discussed in Section 2.3. The dynamic stiffness isalways larger than the static stiffness, Ks, defined according to.

Kd = ηKs (2.3)

where η is the dynamic-to-static coefficient which is > 1 and becomes larger forincreased frequencies.

The complex stiffness for an elastomer that is exposed to a sinusoidal displace-ment xi(t) with the corresponding output force f0(t) is described as [8]:

K∗ = F ∗0Xi

= F0

Xicosδ + j

F0

Xisinδ (2.4)

The displacement and the output force ,respectively, are given by xi(t) = Xiejωt

and f0(t) = F0ej(ωt+δ), where F ∗0 = F0e

jωt.

Xi is the displacement amplitude, F0 is the force amplitude and δ is the phase anglebetween the input displacement and the output force, and ω is the input frequency.

A more general form of Eq (2.4) can be expressed as:

K∗ = K′+ jK

′′(2.5)

whereK

′= F0

Xicosδ and K

′′= F0

Xisinδ (2.6)

The dynamic stiffness Kd is the magnitude of K∗

Kd = |K∗| =√

(K ′)2 + (K ′′)2 = F0

Xi(2.7)

2.3 Amplitude and frequency dependency 15

The loss factor β is used for determining the damping and the hysteresis of theengine mount. It is defined as

β = tanδ = K′′

K ′ (2.8)

2.3 Amplitude and frequency dependencyThe dynamic properties of filled rubbers are dependent of frequency, temperature,static preload and amplitude. It is also highly time dependent. The dynamicstiffness is dependent on the strain rate, and during a rapid process it increasessignificantly, a behavior best described by as a viscoelastic process Gil and Jesus[7]. The major part of the relaxation occurs in a very short time. Rubber is a nearincompressible material, with a ratio between the bulk and shear modulus on theorder of 1000-2000 [4].

The stiffness will increase with an increased frequency but with an increased am-plitude the stiffness will be lower Austrell [2]. The unfilled rubber shows a smallhysteresis. As it usually follows practically the same path during loading andunloadinga and the approximation of pure elasticity can be adapted. For filledrubber, the loading path is considerably different from the unloading path. Thisis due to the phenomena called Mullins effect where the rubber undergo strain-induced stress-softening. This means that the stiffness decreases as a function ofstrain, which can sometimes be identified as a damage. Thus, a material which isexposed to a certain level of strain will the second time, at the same level of strain,have a lower stress compared with the first stretch. This is due to the a breakdown of the cross-links in the molecular chains, see the dashed lines in Figure 2.1.During cyclic loading with constant strain amplitude the material will decrease instiffness the first few load cycles and after three to five cycles a steady state will bereached and the stiffness will be constant. This is called "mechanical conditioning"or just "conditioning" Austrell [2]. However, if the material is unloaded and put torest, the cross-links may, re-establish i.e. the material will receive some recoveryGil and Jesus [7].

2.3.1 Rate dependentThe load rate is crucial for the dynamic properties of rubber, as shown in Fig-ure 2.4. The shear modulus will increase with the applied frequencies while theloss factor β will increase for low frequency, reach a maximum, and then decreaseat very high frequencies. In this thesis the FEA will only be carried out for fre-quencies from 1 to 80 Hz. The measurements might not show a decrease in theloss factor therefore. Nevertheless, the models presented are capable of modellingthis behavior as well. The rate dependent loss is commonly attributed to theresistance in reorganizing the polymeric chains during loading. Since this reorga-nization cannot occur instantaneously, the loss of energy will be rate dependentOlsson [4].


10 0 10 2 10 4 10 6

f [Hz]

10 0 10 2 10 4 10 6

f [Hz]

G [MPa] Damping d

Figure 2.4: Frequency relationship for the shear modules and damping [4].

2.3.2 The Fletcher-Gent effectThe Fletcher-Gent effect (known as the Payne effect) describes the strain ampli-tude dependency of the shear modulus and damping, see Figure 2.5. An increasein amplitude will result in a decrease in modulus while the loss factor will showalmost the same behavior as for the frequency dependency [4].

10 -3 10 -2 10 -1 10 0

G [MPa] Damping d

10 -3 10 -2 10 -1 10 0

κ0

κ0

Figure 2.5: Strain amplitude dependence of dynamic shear modulus and damping[4].

The amplitude dependence is associated to the breakdown and reforming ofthe filler structure Olsson [4]. However, more recent research suggests that theamplitude dependence is caused by changes in the weak bonds between the fillerstructure and the polymeric chains. As the rubber is deformed, these bonds willmove along the surface of the filler, resulting in a rate-independent energy lossOlsson [4].

2.3.3 TemperatureThe temperature is also an important parameter. For temperatures over 0◦Cand below the vulcanization temperature, the stiffness is relatively temperature

2.3 Amplitude and frequency dependency 17

independent. For temperatures, the stiffness will be remarkably higher. Below-60◦C to -80◦C the rubber will be in a glassy state, as illustrated in Figure 2.6.

-80 -60 -40 -20 0 20 40 60

0.8

0.6

0.4

0.2

109

108

107

106

Glasssy

Transition

Rubbery

f

Rubbery

f

T [ °C]

δ [rad]

Gdyn

[Pa]

Figure 2.6: Temperature dependence of dynamic shear modulus (a) and phaseangle (b) for a filled natural rubber. Influence of frequency is also shown [5].


Chapter 3

Constitutive description

In this chapter the basic constitutive description of rubber will be presented. Fora more comprehensive and detailed description, the reader is referred to the liter-ature cited in this chapter.

Commercial FE programs are today powerful tools for predicting the behaviorof rubber components. Rubber is often escribed by using a hyperelastic materialmodel, which is a good approximation when considering only the elastic behavior.For the application considered herein, time dependent properties such as hystere-sis, and frequency and amplitude dependencies must also be accounted for.The characteristic behavior of natural filled rubbers to a uniaxial cyclic load is de-scribed in Figure 3.1. The elastic part is represented by a spring which is assumedto be nonlinear, the damping behavior is described by a rate dependent viscousdamper and a rate independent friction element.

σ

σ

ε

ε

Figure 3.1: Simple one dimensional model describing the behavior of natural filledrubber [2]. Where σ is the stress and ε is the strain respectively.

For this simple model, the elastic, viscous and friction elements acts in paralleland the total stresses becomes,

19

20 Constitutive description

σ = σe + σc + σf (3.1)

where σe is the elastic stress, σv is viscous stress and σf friction stress.

The behavior for a loading and unloading sequence can be illustrated as in theleft of Figure 3.1. The characteristics of the nonlinear spring is represented bythe dashed line. This is a good approximation for unfilled rubbers. For filled rub-bers there will always be a hysteresis present and the loading and unloading pathsmust differ. Hence, there is a need for a rate dependent, or viscous, element. Inaddition, a friction element is also required in order to impose the correct behavior.

The rate dependent and the elastic stresses are related to the network of polymersdescribed in Chapter 2. While the friction stress is related to the fillers Austrell [2].

The effects of static preload, amplitude dependence, and frequency dependenceon the dynamic modulus can be modeled using the simple generalized overlaymodel in Figure 3.1. Furthermore the hysteresis loop is to some extent capturedby using the overlay model Austrell [2]. Properties like stress relaxation, creep,and permanent set behavior of elastomers are not described by the overlay modelOlsson [4].

3.1 Hyperelasticity

The constitutive behavior of a hyperelastic material governed by a strain energyfunction. Hyperelastic materials can be incompressible or very nearly so. Hence,mixed hybrid formulations can be used effectively [9]. Rivlin and Mooney [10][11] developed the first hyperelastic models, the Neo-Hookean and Mooney-Rivlinsolids. Many other hyperelastic models have since been developed. Other widelyused hyperelastic material models include the Yeoh model [12], Ogden model andArruda-Boyce model. However, this thesis will focus on the Yeoh, Neo-Hookeanand Mooney-Ravelin models. The Neo-Hooke and Yeoh models depend only onthe first strain invariant Gil and Jesus [7]. This single invariant dependence givesthe advantage of more robust models than, for instance, the Mooney-Rivlin modelwhich also depends on the second strain invariant Olsson [4]. A Mooney-Rivlinmodel fitted to a uniaxial test may behave very non-physically when loaded ina different direction. In contrast, the Neo-Hookean and Yeoh models will oftenyield in a physically correct behavior in all directions, as long as they are correctlyfitted for one direction Austrell [2]. The Neo-Hookean model is a simplified modelof Yeoh. The main difference between the two models is the inability of the Neo-Hooke model to capture the increase in stiffness of rubber during large tensilestrains Austrell [2]. The Neo-Hooke model is also incapable of representing themodest non-linear behavior during shear Olsson [4].

3.1 Hyperelasticity 21

3.1.1 Strain energy functionThe strain-energy function represents a constitutive statement which is used todescribe hyperelastic materials. It may be defined as shown below for many ma-terials. The equations used in this section are mainly taken from Gil and Jesus [7]and Spencer [13]. For a hyperelastic material, the strain-energy is given by

W (Eij) =Eij∫0

Sij(Eij)dEij (3.2)

where W is the strain-energy function, Eij is the Green-Lagrange strain tensor andSij is the second Piola-Kirchhoffs stress tensor, i = j = 1, 2, 3. Furthermore, thesecond Piola-Kirchhhoff stress is obtained by differentiation of the strain energywith respect to the Lagrangian strain Gil and Jesus [7]

Sij = ∂W

∂Eij= 2 ∂W

∂Cij(3.3)

where the Cauchy-Green deformation tensor C and the Green-Lagrange straintensor E [13] are defined as:

C = FTF E = 12(C− I) (3.4)

where F is the deformation gradient and I is the identity tensor. The deformationgradient in Cartesian coordinates can be represented by the matrix

[F] =

∂x∂X

∂x∂Y

∂x∂Z

∂y∂X

∂y∂Y

∂y∂Z

∂z∂X

∂z∂Y

∂z∂Z

(3.5)

The principal stretches λi (i = 1, 2, 3), which can be calculated from the eigen-values of the left Cauchy-Greens deformation tensor B = FFT , can be used tocalculate the strain invariants Ii Austrell [2].

I1 = tr(B) (3.6)

I2 = 12(tr(B)2 − tr(B2)) (3.7)

I3 = det(B) (3.8)

For a hyperelastic, isotropic and incompressible material, the Cauchy stress tensorσij , is given by [2]:

σ = 2(∂W∂I1

+ I1∂W

∂I2)B− 2∂W

∂I2B2 + pI (3.9)


where p an indeterminate reaction stress which enforces the incompressibility

In most FE program the hyperelastic materials are described by a polynomialform of the strain-energy function [2], such as

W =∞∑

i=0,j=0Cij(I1 − 3)i(I2 − 3)j (3.10)

where Cij are unknown constants. The sum is formally written to infinity but inreality only a few terms are used.

3.1 Hyperelasticity 23

3.1.2 Neo-Hooke materialThe Neo-Hookean (N-H) model is polynomial of first order with one constant,according to below.

W = C10(I1 − 3) (3.11)

For low shear values the Neo-Hookean model gives a good comparison with ex-perimental data, even though it is only described with one parameter Austrell[2]. For a simple compression/tension (CT) test and shear test, Håkansson [14]showed the characteristics of the Neo-Hookean model, as seen in Figure 3.2. Thestiffness increases during compression, while the stiffness decreases in the case oftension [14], illustrated in Figure 3.2a. The shear stress is completely linear dur-ing deformation as shown in Figure 3.2b. The drawback with this model is that itlacks giving a relevant description of the rubber at large deformation Håkansson[14]. This gives a poor correlation with experimental testing, mainly because theNeo-Hookean model only depends on one constant.

λ

σ

σ

σ

λ= ΔL/L

(a) Tension curve

τ

τκ

τ

κ

(b) Shear curve

Figure 3.2: Characteristics behavior of the Neo-Hookean model [14]

3.1.3 Mooney-Rivlin materialThe Mooney-Rivlin (M-R) model is also derived from (3.10). It is similar to theNeo-Hookean model but is based on both the first and second strain invariant withtwo constants, and is defined as

W = C10(I1 − 3) + C02(I2 − 3)2 (3.12)


This model has been used in industrial applications for natural rubber with goodresults and correlates well with experimental data. However, for filled rubbersthis model is too simple in order to obtain a good correlation with measurementsAustrell [2]. A similar CT-test and shear test as for the Neo-Hookean model wascarried out by Austrell [2]. It is easy to see in Figure 3.3 that the M-R model isnot accurate enough.

Exp

M-R

σ

σ

λ= ΔL/L

σ

λ

(a) Tension curve

Exp

M-R

τ

τκ

τ

κ

(b) Shear curve

Figure 3.3: Characteristic behavior of the M-R model [2].

For high values of stretching the M-R model shows a linear tension behaviorbut the corresponding result of the carbon black rubber is highly progressive.For the simple shear test the M-R model also deviates from the experimentalresult in a similar way as for the C-T test Austrell [2]. Compared with the Neo-Hookean model the M-R model can be adjusted to achieve a better result due toits additional constant term Håkansson [14].

3.1.4 Yeoh materialAccording to Austrell [2], the Yeoh model is more suitable for carbon black filledrubbers. The model has a good combination of capability of describing the rubberbehavior and mathematical simplicity. For compressible rubbers the Yeoh modelis defined as:

W =n∑i=1

Ci0(I1 − 3)i +n∑k=1

Ck1(J − 1)2k (3.13)

where J= detF, if J= 1 the Yeoh model for incompressible rubbers is obtainedHåkansson [14]. It is common practice to taken i = 3 for incompressible situations.

3.2 Viscoelastic response 25

Eq (3.13) then gives

W =3∑i=1

Ci0(I1 − 3)i = C10(I1 − 3) + C20(I1 − 3)2 + C30(I1 − 3)3 (3.14)

The stress-stretch curve becomes more adjustable when using higher order polyno-mials Håkansson [14]. The Yeoh model depends only on the first strain invariantand three material constants. According to Håkansson [14] the material constantscan be approximated as factors of the shear modulus according to

C10 = G

2 , C20 = −G20 , C30 = G

200 (3.15)

The behavior of the Yeoh model is illustrated in Figure 3.4 for a CT and a sim-ple shear test by Håkansson [14]. The Yeoh model is preferred for filled rubbers.However the disadvantage is the higher number of constants that need to be de-termined, which is a complex process Austrell [2].

λ

σ

σ

σ

λ= ΔL/L

(a) Tension curve

τ

τκ

τ

κ

(b) Shear curve

Figure 3.4: Characteristic behavior of the Yeoh model, see [14].

3.2 Viscoelastic responseThe remaining sections in this chapter will give the reader a brief overview ofthe simplest and most general models for describing viscoelastic and elastoplasticmodelling. For additional information it is suggested to read the work by Karlsson& Persson [5].


Hypereleastic material models are useful for describing the elastic properties offilled rubbers during static loading. However, as mentioned previously, there areother features of the filled rubbers that cannot be described using hyperelasticity,such as hysteresis and damping during dynamical load cases. A common way todescribe the viscoelastic behavior is with the Zener model, as seen in Figure 3.5.

G∞

τη G

Figure 3.5: The Zener model, see [4].

The Zener model consists of an elastic spring which is connected in parallelwith a dashpot and a linear spring (the Maxwell model). The Zener model isthe simplest viscoelastic model Austrell [2]. The two spring elements have theshear modulus G∞ and G, respectively. The dashpot element has a viscosity co-efficient η. The frequency dependent damping of rubber can be represented bythe Zener model and its response behavior with respect to the dynamic modulusand damping is quite similar in comparison with the rubber properties Austrell [2].

As mentioned earlier the dynamic modulus increases with an increased frequencyand the damping reaches a maximum value and then decreases with increasingfrequency Austrell [2]. This model is purely viscoelastic and independent of theamplitude. Thus, the damping and the dynamic modulus depends only on thefrequency Olsson [4].The stress for a one-dimensional load case can be determined as sum of stress fora time t2 [14], such as

σ(t2) =t2∫

−∞

Er(t2 − t)dε

dtdt (3.16)

where Er is the relaxation modulus, illustrated in Figure 3.6 and can be derivedsuch as

Er(t) = E∞ + (E0 − E∞)e−t/tr (3.17)

where tr is the relaxation time. If additional Maxwell elements are added to theZener model a more general form of the viscous model is obtain. This will includemore parameters that need to be determined but the chance of having highercorrelation with measured data is increased Austrell [2]. This general form of therelaxation modulus is describe by the Prony series in Eq (3.18) Håkansson [14].

3.3 Elastoplastic response 27

E0

ER(t)

t

E∞

Figure 3.6: The relaxations modulus for the Zener model

The Prony has been used in the setup of the FE-model for describing the viscousproperties, this is further explained in the next chapter.

Er(t) = E∞ +n∑j=1

Eje−t/trj (3.18)

3.3 Elastoplastic response

An elastoplastic model, similar to the Zener model, is obtained by replacing thedashpot with a frictional element. This model is able to describe the rate inde-pendent damping behavior of natural filled rubbers Austrell [2].

As illustrated in Figure 3.7, the elastic response is modelled by two parallel springswith the elastic shear modulus G respectively G∞, respectively. This capture themechanical response for an elastoplastic material with linear kinematic hardening.For this particular model the stress independent of the strain rate Olsson [4].

G∞

τGτy

Figure 3.7: Elastoplastic model [4].


τepj ={Gepj κ, if elastic0, otherwise

3.4 The overlay model

In a linear viscoelastic model the dynamic mechanical properties depend only ontemperature and frequency and are independent of the type of deformation, suchas constant strain, constant stress or constant energy. This means for instancethat if a sinusoidal strain is applied, a sinusoidal stress will be the response Graciaet al [6].

However, for filled rubbers the situation is different, as the filled rubbers cor-respond to non-linear viscoelastic solids which, in addition, the dynamic mechan-ical properties are dependent on the dynamic strain amplitude Gracia et al [6].The overlay model represents a constitutive approach based on rheological modelsfor modelling the mechanical behavior and specifically the mechanical hystere-sis shown by filled elastomers when these are submitted to cyclic loads Olsson[4]. Filled rubbers can be modelled by a combination of viscoelastic and elasto-plastic models, resulting in a material model able to sum up the elastic, viscousand frictional material responses Gracia et al [6]. For a one-dimensional case, seeFigure 3.8.

3.4 The overlay model 29

G∞

τ

τyM

Gep

1τy

2

τy1

Gep

2

Gep

M

Gve2

Gve1

GveN

tr1

tr2

trN

κ

Viscoelastic

part

Elastoplastic

part

Elastic

part

Figure 3.8: One dimensional model describing the overlay model [4].

As previously mentioned, the total stress can be determined as a summation ofthe stress contributions from all parallel elements, which is easily seen in Figure 3.8.According to Olsson [4], one could use the same method for the three-dimensionalcase. Thus, the total stress tensor can be written as a summation of the stresstensors from all parallel contributions, as below.

τ = τe + τve + τep = τe +M∑i=1

τvei +N∑j=1

τepj (3.19)

where τe, τve and τep is elastic shear stress respectively viscoelastic and elasto-plastic.


Chapter 4

Finite element model setup

In this chapter the FE-model used in the simulations will be presented.

It is of high importance to have an accurate model representation, and a workingmethod that is easy to follow and works effectively during the simulations in orderto minimize the computational cost (simulation time). One way of reducing thesimulation time is to use of geometrical asymmetries in the FEM setup. For theLLTB part, there exists a double geometric symmetry i.e symmetry in XY-planeand symmetry in XZ-plane. Therefore a double symmetry model has been usedin the simulations, representing a quarter of the LLTB component to be analyzed,see Figure 4.1. The complete FE-model of the LLTB should resemble the testsetup used by the supplier as much as possible. Furthermore, it is also very im-portant that the mechanical behavior of the rubber is well described in order toobtain an accurate model which provides reliable results that are comparable tomeasurements.

The aim of this chapter is to give the reader an insight in the most importantsteps that are taken during the FE-modeling, with the main focus on representingrubber properties such as damping and dynamic modulus.

Figure 4.1: A double symmetry FE-model of the LLTB.

31

32 Finite element model setup

4.1 Work procedureThis section gives a brief overview of the process of building the FE-model, c.f.Figure 4.2.

The work has been divided into three main steps, each containing a number ofsub-steps, as illustrated in Figure 4.2. Before the first step (the pre-processingwork) can start, it is necessary to import the geometry of the LLTB to ANSA[15]. This is done by using an already existing CAD model, in CATIA V5 [16],provided by the rubber supplier. After the LLTB CAD model had been convertedinto ANSA files, it is then necessary to perform a few modifications and clean upof the CAD model in order to create a high quality mesh. ANSA is an advancedmultidisciplinary CAE pre-processing tool, which provides all the necessary func-tionality for full-model build up, from importing CAD data to ready-to-run solverinput file, in a single integrated environment [15].

The pre-process work is divided into three sub-steps, where the first step is tocreate and apply an appropriate mesh with a suitable type of element. Certainareas were required to have a higher mesh resolution, e.g. areas where contact canoccur during static loading, and areas where certain boundary conditions (BCs)are applied. In the second step, the material properties for the rubber model’sand aluminum parts are assigned. The hyperelastic material model, viscoelasticproperties for the rubber, and hardening behavior for the aluminum are specifiedin FE-model. In the third and final step of the preprocess, the BCs are defined.This FE-model includes three different types of BCs: contact conditions, loads,and constraints.

When the pre-process work is finished and the setup of the FE-model is com-plete the solving step is initiated. The program used for solving the problemis Abaqus/Standard v6.14-1 [17]. Both quasi-static and dynamic load cases areanalyzed using an implicit integration scheme available in Abaqus. If a simula-tion does not converge due to element distortion or contact over-closure betweenaluminum frame and rubber, a remake of the mesh was preformed together withmodifications of the definition of the amplitude curve for the prescribed displace-ment, and a decreased step size (increment size). When a solution is obtained,post-processing process starts and the results are exported from Abaqus viewerto Matlab [18] for comparison with the supplier’s measured data. The two finalsteps in the working scheme represents an iterative process where the tuning ofboth hyperelastic, viscoelastic, and elasto-plastic material parameters is necessaryin order to obtain a good correlation between FEA and measured data.

Each step in the working procedure is described in more detail in Figure 4.2 below.

4.1 Work procedure 33

Import of CAD geometry

Clean up and mesh

Material properties and

material models

Boundary conditons

Pre

process

Modi!cation of the

Abaqus keyword !le (Implementation of the overlay model)

Final FE-model

Pre

process

Post

proccesing

Solver

Validation of the result

END

OK

NOT OK

Figure 4.2: Working scheme of the development process of the FE-model.

4.1.1 MeshAs previously described the geometry is imported to ANSA where all preprocess-ing work is done. The complete mesh of the LLTB can be seen in Figure 4.3.

A few modifications on the original model were performed. This did not changethe outer geometry, only the orientation and size of the mesh surfaces, which pro-vided a smoother transition between the mesh layers and made it possible to havea detailed mesh where required. The focus was on cleaning up the original surfacesin order to have a good quality mesh around the "snubber" which is the criticalarea during contact interaction and at higher loads.

It is well known that a finer mesh, containing a larger number of elements, of-


ten gives more accurate results. However, despite this fact, some critical locationsshowed during the analysis that it was necessary to have a coarse mesh in orderto reach convergence. This is due to large deformations when the snubber c.f.Figure 4.3 part is exposed to a contact with the aluminum frame, causing the ele-ments to distort, c.f. Figure 4.4. At a critical level of displacement, some elementswill become too distorted, penetrate other elements, causing negative volume andresulting in convergence issues. This problem was solved by increasing the elementsize, which made it possible to reach a convergence. This is because the largerelements allowed bigger deformation without becoming too distorted. This ap-proach will give a lower accuracy of the results. However, when comparing the FEresults with measurement data, see below, the difference was still found to be small.

The type of element is an important factor in order to obtain accurate resultsfrom FEA. In Abaqus user guide [9] it is suggested to use solid continuum hy-brid elements for incompressible hyperelastic materials with initial Poisson’s ratiogreater than 0.495. This is to minimize the risk of possible convergence problems.In the final FE-model, the tetrahedral element C3D10 element type was used foraluminium frame, and the tetrahedral element C3D4H used for the rubber.


Figure 4.3: Overview of the mesh


Figure 4.4: Deformation and penetration of elements during contact


4.1.2 Material models and parametersThe LLTB part is made of two materials: aluminum and natural filled rubber.The stress-strain relationship for aluminum is linear for low values of strain. How-ever, if the strain is increased the aluminum starts to yield. When the yield limitis reached the corresponding behavior of the stress-strain relationship becomesnon-linear and irreversible specially for a low strain rates. In the FE-model thematerial properties of aluminum have been assigned by adopting an elastic piece-wise linear plasticity model.

Since rubber shows a non-linear behavior, c.f. Chapter 3, the linear elastic stress-strain relationship with a constant Young’s modulus does not apply. Instead themost suitable model to describe the rubber behavior, i.e. hyperelasticity withviscoelastic and elastoplastic properties, has been utilized in this work. Three hy-perelastic material model has been tested, Neo-Hooke, Mooney-Rivlin and Yeoh,which have been previously discussed in Chapter 3. In the analysis, the assump-tion was to treat rubber as an incompressible material, thus the definition of theYeoh model given in Eq (3.14) have been adapted.

Unfortunately no experimental tests were carried out for extracting material pa-rameters for the rubber. This is because the experimental testing is highly complexand very time consuming. Instead, the material parameters were taken from Aus-trell [2], Olsson [4] and Ali et al [19]. However, those parameters were extractedfrom different rubber compound mixtures which have different IRHD numbers.So, it was necessary to interpolate and/or tune the material parameters to achievegood correlations (inverse modeling). All the material parameters used in thesimulations are given in Table 4.1.


RubberDensity 1.20E+03 [kg/m3]Young’s modulus 1.101 [MPa]Poisson’s ratio 0.4995 [-]

AluminumDensity 2700 [kg/m3]Young’s modulus 70 [GPa]Poisson’s ratio 0.3 [-]Proof Stress 290 [MPa]Tensile Strength 340 [MPa]

Hyperelasticmaterial parametersYeoh [MPa] C10 C20 C30

0.6803 -0.0982 0.0188

M-R [MPa] C10 C012.368 3.114

Neo-Hooke [MPa] C101.065

Viscoelastic material parametersG∞ 1.65 [MPa]Gve1 0.3461 [MPa]Gve2 0.4054 [MPa]Gve3 0.2408 [MPa]tr1 0.00328 [s]tr2 0.0128 [s]tr3 0.189 [s]Elastoplastic material parametersGep1 1.01 [MPa]Gep2 2.041 [MPa]Gep3 2.541 [MPa]τy1 0.0287 [MPa]τy2 0.0121 [MPa]τy3 0.00121 [MPa]

Table 4.1: Material properties

To the author’s knowledge, in commercial FE-software’s, there is not any con-stitute model for handling filled elastomers. Instead of implementing a new con-stitutive model, an approach of combining already existing models have been usedin this thesis. This procedure which is referred to as the overlay model, has been


previously discussed in Section 3.4, and is obtained by an overlay of meshes usingstandard material descriptions, see Figure 4.5. The foundation of the overlay modelis to create a hyperelastic, a viscoelastic and an elasto plastic FE-model, where allthree layers have identical element meshes. A suitable number of viscoelastic orelastoplastic FE-models are connected in parallel by assembling different layers ofelements to the same nodes Olsson [4].

The hyperelastic and the viscoelastic behavior can be modelled in terms of a singleFE model based on the Prony series in Abaqus [9]. There are several elastoplasticFE-models in Abaqus that can modelled in parallel, unfortunately there exists noelastoplastic model based on hyperelasticity Olsson [4].

Hyperelastic layer

Viscoelastic layer

Elastoplastic layer

Rheological model

FE-model containing -Non-linear elasticity

-Frequency dependence

-Amplitude dependence

Figure 4.5: Principle diagram of the overlay method in FE program [4].

The total deformation gradient, F, and the right Cauchy-Green strain tensorC, are the same for all the mesh layers. The total stress is the weighted averageof the stresses in all the fractions. As the constitutive equation is defined in termsof a stored energy function it is, in this case, this energy function is the sum ofthe different stored energy functions for each layer Gracia et al [6].

The keyword options used in Abaqus for describing the overlay model and theimplementation of the material model parameters, are accordingly to Figure B.1in Appendix B.

4.1.3 Boundary conditionsThe FE-model has several kinds of BCs, which have been applied in order tocapture and resemble the behavior of the LLTB in the car body. In Figure 4.3


it can be seen where the fixed constraints and prescribed displacement are applied.

The fixed boundary represents the bolt connection between the tie-bar and thecar body, and is represented by kinematic coupling which is fixed in all directionsin order to prevent rigid body motion. Since the LLTB is double symmetric, aquarter model has been used, as previously mentioned. In addition to the fixedconstraints, the FE-model is locked in the two symmetry planes, the XY- and XZplane respectively, by using the option YSYMM and ZSYMM in Abaqus.

The prescribed displacement describes the movement of the engine which occursduring driving. This impacts the tie-bar in many different ways and of course inreality there are several or even hundreds of different loading scenarios which canbe tested. However, the work in this thesis only concerns loading cases tested bythe supplier. Thus, for the quasistatic case, the displacement load were defined inthe x-direction with a range of −9.75 to 15.75 mm by using an amplitude curveassociated with a scale factor; for all load cases see Table 4.2. The dynamical sim-ulations were carried out in several steps, assigned with an individual displacementamplitude curve with a corresponding frequency, see Figure 4.6. The incrementsize was adjusted for each step, since a higher frequency requires a smaller timestep size in order to achieve high accuracy and for being able to capture the peaksin the time response.

Time [s]

Stepi

Stepi+1

Stepn

Figure 4.6: Frequency step


quasi-static X-Displacement [mm] Y-Displacement [mm] Z-Displacement [mm]Case 1 -15.75 +9.75 0 0

Dynamic Amplitude [mm] Frequency [Hz] Preload [N]Case 1 0.1 1 - 80 0Case 2 0.2 1 - 80 0Case 3 1 1 - 15 0Case 4 2 1 - 15 0Case 3 4 1 - 15 0

Table 4.2: Load cases

The FE-model consists of two different types of contact conditions. It is quitecomplicated to describe the contact between the rubber and the aluminum. Sinceduring the quasistatic simulations the rubber comes in contact with the aluminumframe and for large deformation the rubber comes contact with itself, the elementsdistort and penetrate other elements. The adopted way of characterise the contactbehavior was to use a simple small sliding contact and global general contact withcorresponding friction µ = 0.4. The purpose of defining global general contactis because of rubber self contact, as it is difficult to anticipate the self contactregions in rubber part before the start of analysis. Thus, it is recommend to useglobal general contact which will take care of any type of contact that may occurduring the simulation. The parts of the rubber and the aluminum that have beenvulcanized together, such as the insert and the arms of the rubber is assigned withthe contact option in Abaqus called Tie definition, which fixes the contact surfacesagainst each other, thus not allowing any relative motion.


4.2 Modification of the Abaqus input file

As previously mentioned, ANSA does not support all Abaqus keyword optionsthus additional manual modifications of the input file were necessary to perform.

For more information regarding the keywords mentioned in this section, the readeris referred to the Abaqus Theory Manual [9].

4.2.1 Implementation of the overlay model

The last step of the preprocessing was to implement the overlay model. Sincethe overlay model does not yet exist in Abaqus, it was found possible to solvethis problem by using the keyword option *ELCOPY in Abaqus, as shown inFigure 4.7.

*ELCOPY, ELEMENT SHIFT=1000000, OLD SET=Rubber, SHIFT NODES=0, NEW SET=Rubber_plas c_1

*SOLID SECTION, ELSET=Rubber_plas c_1, MATERIAL=elastoplas c_rubber_1





Figure 4.7: The *ELCOPY keyword in the Abaqus input file

where *ELCOPY is simplest described as an option used for copying an elementset to create new element sets [9]. In this case, it made it possible to assign theelastoplastic properties to the new element set. Thus elastoplastic properties onnew element sets together with hyperelastic and viscoelastic properties defined onprevious element set define the overlay model, c.f. Figure 4.5.

4.2.2 Definition of amplitude curve

The dynamical displacement loading is represented by a sinusoidal function, asshown in Figure 4.6. During this thesis two alternative methods of defining the am-plitude curve for the dynamical load cases have been used. Both keyword optionsare shown in Figure 4.8a, but unfortunately these keywords are not yet supportedby ANSA. The Periodic Step works for low frequencies. However with an increasedfrequency the dynamic stiffness decreases which is illustrated in Figure 4.8b. Thisis due to the non-zero initial velocity between the shifting frequencies. It wasfound that using the Smooth Step as the definition of the amplitude curve, thatdefines zero initial velocities, gave better results.

The differences between these two options are further explained in Appendix A.

4.3 Solver 43

*AMPLITUDE, NAME=freq_1, DEFINITION = PERIODIC

1, 6.2830, 0, 0

0, 1

*AMPLITUDE, NAME=freq_1_ST, DEFINITION = SMOOTH STEP

0,0

0.25,1

0.75,-1

1,0

(a) Periodic and Smooth Step in Abaqus

Dyn

am

ic s

tiff

ne

ss [

N/m

m]

Frequency [Hz]

Periodic step

Smooth step

Measured data

(b) Dynamic stiffness for Smooth Step andPeriodic Step

Figure 4.8

4.3 SolverThe solver that has been used for both the quasi-static and dynamic case isAbaqus/Standard (implicit) [17], running in the UNIX environment with 64 cores.

There exist many advantages and situations were the Abaqus/Explicit solver mighthave been more beneficial to use. Implicit solutions by Abaqus/Standard is pre-ferred for solving smooth nonlinear problems [9], while the Abaqus/Explicit is bestsuited for problem generated by a blast or impact loading, involving propagationCook et al [20]. Certain problems such as static or quasi-static can be used byboth methods, most common are the problems that normally would be solvedwith Abaqus/Standard but might encounter difficulties to converge due to con-tact or complex material behavior. Abaqus/Standard involves iterations, whilethe explicit solution involves no iterations on the global level. Compared withAbaqus/Standard, the explicit method can be used an as effective alternative insome situations even though it might require a large number of time incrementsif the same analysis in Abaqus/Standard requires many iterations. Furthermore,it requires generally quite less disk space and memory than Abaqus/Standard forthe same simulation [9].

It was decided to proceed with the implicit solver for the applications consideredin this work, as it was anticipated to be more time efficient.


Chapter 5

Result

In this chapter will be present the results that were obtained during this work.A series of simulations have been carried out for the quasi-static load case anddynamic load case, respectively. A comparison of the computation time for thedifferent hyperelastic models has also been performed. All the results presentedin this chapter were obtained using same FE-model setup.

Due to restrictions and confidentiality requirements from VCC, some graphs willhave blank axes.

5.1 Quasi-static load caseAs mentioned earlier, the focus herein has been the task of predicting the force-displacement relationship and the static stiffness Ks for the quasi-static case.The aim through the thesis has been to achieve a good correlation between mea-sured data from the supplier and FEA results. A comparison for both the force-displacement curve and the static stiffness is shown in Figure 5.1 and Figure 5.3,respectively. The hyperelastic models and corresponding parameters that havebeen used in the simulations are listed in Table 4.1.

In Figure 5.1, one can see in the intersection of each plots that for small dis-placements the force response is linear, and that it for larger displacements show anon-linear behavior. The reason why the force is non-linear for larger deformationsis that the snubber comes in contact with the outer frame. This contact is causinga fast increase in the forces due to increasing stiffness of the component. As canbe seen in Figure 5.1 below, the Yeoh and N-H model follow the measured datacurve quite well, while the M-R model gives a large deviation. The Yeoh modelalso has a good correlation in the non-linear region where the N-H model showsa small deflection. The maximum difference between the measured forces and itscalculated forces in the FEA over the displacement range can be seen in Table 5.1.The maximum difference between measured static stiffness and calculated/FEAcounterpart is shown in Table 5.2.

45

46 Result

However, to be able to see the differences for the hyperelastic models, the non-linear part of the positive region of the displacement have been zoomed in andshown in Figure 5.2, where one can easily compare the characteristics of the threemodels. For a further study of parameters optimization for the M-R model, c.f.Section 5.1.1 below.

Fo

rce

[N

]

Displacement [mm]

Measured data

Yeoh, C10

=0.6803 C20

=−0.0982 C30

=0.0188

N−H, C10

=1.065

M−R C10

=2.368 C01

=3.114

Figure 5.1: Force Displacement plot for the quasistatic case

5.1 Quasi-static load case 47

Fo

rce

[N

]

Displacement [mm]

Measured data

Yeoh, C10

=0.6803 C20

= 0.0982 C30

=0.0188

N H, C10

=1.065

M R C10

=2.368 C01

=3.114

Figure 5.2: A closer view of the non-linear force

In Figure 5.2, it can be concluded that the Yeoh model is best suited for thequasi-static case. However, for large displacements (high forces) the N-H model isquite accurate as well. This is probably due to the reason that when the rubber isfully deformed, the stiffness depends mainly by the deformation of the aluminiumframe. In this specific case, the deformation of the aluminum is linear (yield limitnot reached) which makes the force increase continuously with a linear behavior.As already mentioned in Chapter 3, the N-H depends only on one constant (C10)and it is of first order. Thus, the N-H model shows a linear behavior (see inFigure 3.2b) and therefore it correlates well with the measured data for higherloads. However, if larger displacement is applied the aluminum will eventuallyreach its yield limit and plastic deformations occur. The force will once again benon-linear and the N-H will differ with the measured data.

Hyperelastic model min [%] mean [%] max [%]Yeoh 0 5.89 10.93N-H 0 49.69 75.92M-R 0 335.79 623.45

Table 5.1: Difference between measured data and FEA for the force-displacementcurve

48 Result

The reason to the high value of the maximum deviation presented in Table5.1 is due to a rapid force increase at the upper end of the displacement values.When the displacement is -15.50 and +9.50 mm (only 0.25 mm from the maxi-mum displacement), the deviation for the Yeoh model is 2% and 3 %, respectively.This is to be compared with the maximum displacement 15.75 mm at which themaximum deviation is 23.5 %. Although the deviation is large for the maximumdisplacement, it is not considered to be of high importance due to the normalworking region of lower displacements, in which case the deviation is much lower.

The results for the static stiffness Ks plotted against displacement and againstforce can be seen in Figure 5.3 and Figure 5.4, respectively. The same deviationfrom measured data as discussed above, is shown even more clearly in Figure 5.4where the rate of increased force at maximum displacement has a big impact ofthe static stiffness.

Ks S

tatic S

tiffness [N

/mm

]

Displacement [mm]

Measured data

Yeoh C10

=0.6803 C20

=−0.0982 C30

=0.0188

N−H, C10

=1.065

M−R C10

=2.368 C01

=3.114

Figure 5.3: Static stiffness for the quasi static case


Ks [N

/mm

]

Force [N]

Measured data

Yeoh

N−H

M−R

Figure 5.4: Static stiffness-Force

Difference min [%] mean [%] max [%]Yeoh 0.07 10.56 64.66N-H 6.18 56.62 99.03M-R 7.22 323.7 612.26

Table 5.2: Difference Ks for the hyperelastic models

5.1.1 Tuning of material parameters and computational timeTo gain more knowledge of how the hyperelastic material parameters affect theresults, a study of tuning the parameters were performed. The result can be seenin the Figure 5.5 below. A comparison of the computational time was performedfor each hyperelastic model and is presented in Figure 5.6.

Note that some of the presented parameters are only considered to be designparameters. This are not real values such as those presented in Table 4.1, whichwere obtained from experiments.

Since the M-R model provided the least accurate result regarding the staticstiffness, it is understandable that the inaccurate material parameters might bea reason for higher deviation in forces and static stiffness. Thus, tuning of theM-R parameters might be needed to get accurate results. The tuning results were

50 Result

performed by keeping one of the material constant (C10 or C01) to a constant valueand change the other to a certain value. The result can be seen in Figure 5.5.

Fo

rce

[N

]

Displacement [mm]

Comparison of M−R parameters

Measured data

C10

=2.368 C01

=3.114

C10

=2.368 C01

=0.65

C10

=2.368 C01

=0.3

C10

=1.21 C01

=0.3

C10

=0.8 C01

=0.3

C10

=0.4 C01

=0.3

Figure 5.5: Tuning of M-R parameters

It is obvious that increasing one or both material constants will decrease thestiffness (the slope in Figure 5.5). The C10 seems to have the largest impact onthe nonlinear part while the C01 contributes to the linear part. To investigate anyfurther differences in hyperelastic material models it was, as mentioned previously,suggested to compare the computational time as an additional information for eachhyperelastic model. The results are presented in Figure 5.6.


Yeoh Neo−Hookean M−R0

500

1000

1500

2000

2500C

PU

tim

e [

s]

Quasi static CPU time

Figure 5.6: Computational time of the three different hyperelastic models

The results were surprisingly similar. The M-R model is about 20 % fastercompared with the Yeoh model. However, since the time difference between theseto models is small and the Yeoh has an higher accuracy according to Table 5.2.It was decided to continue with the Yeoh model as the hyperelastic model for thedynamic analysis.

52 Result

5.2 Dynamic load case

The dynamic simulations have been divided into two main categories; low fre-quency - low amplitude (LFLA) and low frequency - high amplitude (LFHA).

The hysteresis loop for the LFLA case is plotted in Figure 5.7, which can becompared with Figure 2.3 and Figure 3.1. The obtained result for the dynamicload case is more similar to the one Figure 2.3 in as a linear viscoelastic modelwas used for the FEA.

Forc

e [N

]

Displacement [mm]

1 Hz

10 Hz

25 Hz

Figure 5.7: Dynamical Force Displacement for LFLA

As seen in the figure above there is a small change in hysteresis for increasedfrequency. However, if one wishes to plot the hysteresis for the whole frequencyrange the plot becomes a bit unclear and difficult to follow. This is due to theintroduced noised caused by the shifting frequency, which has been previously dis-cussed in Section 4.2 see also Appendix A.

The dynamical stiffness for the LFLA and LFHA case is illustrated in Figure 5.8and Figure 5.9, respectively.

5.2 Dynamic load case 53

Kd [N

/mm

]

Frequency [Hz]

LFLA A=0.1 & A=0.2 [mm]

Measured data A=0.1 mm

FEA A=0.1mm

Measured data A=0.2 mm

FEA A=0.2 mm

Figure 5.8: Dynamical stiffness Kd for LFLA

Frequency [Hz]

Dynam

ic s

tiffness [N

/mm

]

Measured data A=1 mm

FEA A= 1 mm


FEA A= 2 mm


FEA A= 4 mm

Figure 5.9: Dynamical stiffness Kd for LFHA

54 Result

For both cases, the dynamical stiffness obtained by the FEA shows a highervalue and a linear behavior for higher frequencies. Even though the graphs aboveshow a fairly good correlation with measured data, the differences are easy tocompare for LFLA and LFHA in the Table 5.3 and 5.4 respectively.

Difference min[%] mean [%] max [%]A=0.1 mm 1.0418 3.4345 5.7999A=0.2 mm 4.0164 6.9771 9.2443

Table 5.3: Difference of the dynamical stiffness for LFLA

Difference min[%] mean [%] max [%]A= 1 mm 3.0458 5.2043 9.0740A= 2 mm 3.1632 6.2160 10.9339A= 4 mm 0.3976 5.2647 11.0450

Table 5.4: Difference of the dynamical stiffness for LFHA

The deviations seen in the tables above, can be considered to be a satisfying,since it was expected that the deviations would be within a range of ± 30 %. Toocreate a comprehensive and easy understanding of how the dynamic stiffness de-pends on the amplitude and frequency a surface plot was created, c.f. Figure 5.10,where the Kd is presented as a function of the frequency and the amplitude.

Amplitude [mm]Frequency [Hz]

Dynam

ic s

tiffness [N

/mm

]

Figure 5.10: Kd as a function of frequency of and strain amplitude

Chapter 6

Discussion and Conclusions

In this section a brief discussion regarding the methodology used, obtained FEAresults, and main conclusions are given.

6.1 Discussion6.1.1 Quasi static load caseIn the beginning of the project, a full scale model was used instead of the quartermodel which was used later in the work. The full scale model made it possible tohave a much finer mesh compared with the quarter model. As mentioned in Chap-ter 4, it was necessary to use a rough mesh in order to reach convergence. This wasnot necessary for the full scale model and this mainly due to the symmetry planesintroduced for the quarter model. This made the LLTB consist of sharp and flatsurfaces, which did not exist for the full scale model. The convergence problemwith certain elements were due to their large distortion and sharp edges, thus notexperienced in the same way as for the full scale model. The most critical area isaround snubber region which can be seen as zoomed in part in Figure 4.3. This isdue to the large compression when snubber comes in contact with the aluminumframe.

A full scale model with 4 times smaller element size was compared with the quartermodel. The results comparison showed a very small difference, thus the differencewas considered to be negligible. For this reason, the quarter FE-model which cho-sen element size was used in this thesis for both static and dynamic load cases.The results for the quasi static load case; the force displacement curve and staticstiffness curve can be seen in Figure 5.1 and Figure 5.3 respectively. These resultscan be considered sufficiently and accurate with respect to the measurements.Even though the magnitude of the deviation for the static stiffness is large forthe highest corresponding force, it is still accurate within the normal working dis-placement range. The reason why the static stiffness has a large deviation is dueto its definition as the local slope of the force displacement curve, Ks = ∂F/∂u.

55

56 Discussion and Conclusions

When the displacement step is only 0.25 mm and the force increased by almostthe double for one step which is very easy to understand why the ratio becomeshigh. The contributing factor to the rapid rate of force is still to be investigated.However, if one compares the force-displacement curve with measurement data,the same tendency of the fast increased force can be noted but not in the samepace as in the results from the FEA.

Three hyperelastic models were used for the quasi static case and even thoughit was decided to use Yeoh for the dynamic case, it can be a good idea to use theNeo-Hookean model for future simulations. This is because the N-H model hasa good correlation for smaller displacements, which is the most common workingarea for dynamic load cases. The N-H model also has one big advantage com-pared with the other two, namely that it is far less complicated to determine itssingle material constant C10. However, for larger displacements the Yeoh modelis preferred for loading simulations of the type treated in this work.

6.1.2 DynamicThe modeling framework regarding the dynamical behavior and properties forfilled elastomers presented in Chapter 2, seems to capture the amplitude and fre-quency dependency quite well, as can be seen in Section 5.2. Both the LFHAand LFLA case shows a higher stiffness than the measures data. Even thoughthe results are better then expected, it is still important to understand the mostfundamental reasons why the FEA shows a higher dynamical stiffness; especiallyfor future investigations, when VCC wants to run simulations for higher frequen-cies. During this work 15 material parameters have been used (3 hyperelastic, 6viscoelastic and 6 elastoplastic, according the Table 4.1). Among the 15 materialparameters, the viscoelastic and elastoplastic parameters has been tuned from theoriginal values in order to achieve a good correlation for LFLA case. This meansfor instance for the LFHA case, the material parameters are not properly adjustedfor the higher amplitudes. Nevertheless, the obtained results for the LFHA arereasonable, c.f. Table 5.3 and Table 5.4, respectively.

The overlay model that has been adapted for the dynamical simulations seemsto be an efficient tool for describing the characteristics of filled rubbers for lowfrequencies. However, there are some problems that have not been brought upin this thesis. For instance, for frequencies > 100 Hz in combination with higheramplitudes, certain elements are exposed to an extreme level of strain, causingthe convergence issues. This is probably due to the elastoplastic properties thatneed to be adjusted. Regarding the prescribed displacement, it was discovereda bit too late in the project that a the way of defining the amplitude curve ofthe prescribed displacement made an enormous impact of the outgoing result. Byredefining the amplitude curve, the dynamical stiffness increased with increasedfrequency, instead of obtaining a decreasing behavior. For the interested reader,the improvements of the prescribed displacement is further explained in AppendixA.

6.2 Conclusions 57

Unfortunately, in this thesis no result regarding the preload case has been pre-sented. The idea of applying a static preload, followed by an oscillating displace-ment, did not made the overlay model converge due to same issues with highlevel of strain in elastoplastic elements as mentioned above. Several different wayof bypassing this problem were tested. One method that was suggested by theAbaqus support team, was to remove the elastoplastic properties and only use thelinear viscoelastic properties. Although the suggested method did converge, theresults differed more than 50 % and therefore both the method and the resultswere rejected.

6.2 ConclusionsThe purpose of this thesis has been to investigate if there exist a possibility forVCC to perform in-house FEA for certain rubber components as an alternative toonly relying on suppliers suggestions for new designs and measurements. Unfor-tunately, this work has not been able to give a simple answer to that question. Tobe able to provide a reasonable answer to this thesis most fundamental question,one must take several aspects into consideration, which are presented below.

The most important step that needs to be considered is to carry out physicalsexperiments with the purpose of extracting material parameters of the filled rub-ber; the rubber models are strongly depended on its material parameters. Moresuggestions regarding the future work will be presented in the next chapter.

The conclusion regarding the results for the quasi static cases, is that a goodcorrelation can be obtained, even though that the material parameters are ex-tracted from a different compound mixture. It is suggested to VCC to proceedfurther using the Yeoh model with the hyperelastic constants presented in Table4.1.

For low frequencies and low amplitudes, there exists a possibility to adjust thematerial parameters in order to capture the required stiffness curve. Nevertheless,it is highly desirable to have the correct material parameters in an early stageof the development process in order to save simulation time (less time spent onfine-tuning the material parameters).

Using the smooth option (mentioned in Section 4.2.1 and also in Appendix A)for the amplitude definition provided good results for the low frequency cases butshows a poor accuracy for frequencies > 100 Hz due to the introduced noise be-tween the shifting frequencies.

The assumed BCs seems to work well, even though for future FEA it is desirableto have a detailed knowledge regarding the measurements. The most interestingand useful information concerns the LLTB mounting procedure in the test rig of

58 Discussion and Conclusions

the rubber components: how it is done and how many dynamic tests cycles areused. Having this kind of information will eventually lead to improved BCs asthey will be applied according to the physical testing. This is especially impor-tant if one wants to determine the stiffnesses in the z-and y-directions, respectively.

ANSA does not support all the required Abaqus keywords options. Even thoughthe developer of ANSA [15], claims that the newer versions from 15.1.0. will haveincreased support for Abaqus keywords it was found not to be sufficient. Thismakes ANSA not to sufficient with respect to the rubber components since, therehas to be manual modifications of the input file in order to implement the overlaymodel. This makes the FE-modeling a time consuming process, where mistakescan easily be made. Instead, it is recommended to use Abaqus/CAE as a pre-processor. However for an experienced CAE engineer, this is probably consideredto be a minor issue.

Based on the conclusions mentioned in the text above, and the suggestions offuture work given in Chapter 7, there exists a possibility for VCC to perform in-house rubber calculations. According to the result presented in this work, it ispossible to determine both the static and the dynamical stiffnesses with a fairlygood accuracy. This new working procedure for VCC have a great potential ofbecoming an effective tool in the nearby future. However, there are several investi-gations and simulations that need to be preformed before VCC can be independentof the suppliers design suggestions and decreasing the lead time by fully relyingon the FEA results.

Chapter 7

Future studies

In this final chapter of the thesis, suggestions regarding improvements of the worktogether with suggestions to make rubber FEA a design tool for VCC, will be pre-sented. Several suggestion have already been discussed in the previous Chapters5 and 6.

In order to improve the method and thereby obtain more accurate results, thefollowing recommendations are made:

1. The results are roughly dependent of two factors: the material parametersand the material model. Therefore the need to determine the material pa-rameters by physical experiments, must be considered to be of the highestpriority. This could preferably be performed by two other thesis topics sinceit is expected to be a quite time consuming project.

2. Perform FEA for other components, e.g. the RHM. The overlay model canbe applicable to any components made of natural filled rubber.

3. The LLTB is exposed mainly to shear forces, which in, this case, is capturedby the hyperelastic material model quite well. However, other componentswill be more exposed to tension and compression forces which makes it nec-essary to investigate other hyperelastic models e.g. the Arruda-Boyce orOgden model, both supported in Abaqus.

4. A further studies how to implement the overlay model in a more efficient wayis necessary. In the current situation it is not possible to perform a simula-tion for the preload case due to the rapid increased strain in the elastoplasticmodel. It is suggested to investigate the use of a simplified multilinear kine-matic hardening model instead. This might reduce the convergence issueand possibly reduce the computational time.

5. Determination of the Ks and Kd for the y-and z-directions in order to gainmore knowledge about the material models and geometric behavior. How-ever, this might be a problem for comparison with measured data since

59

60 Future studies

the measurements are only carried out in the normal working direction (x-direction).

6. For higher frequencies than those investigated in this work, it is recom-mended to explore the possibilities of using the explicit method, as it mighthave a greater potential of capturing rapid dynamical changes. The focuswill be to find a successful way of mass scaling and introducing the dampingto the FE model, in order to increase the critical time step and, thereby,reduce the computational time.

Hopefully, further studies of the type above (it is not necessary to follow the exactorder except for the first step) can contribute to being able to predict the Kd witha assigned preload and for higher frequencies up to 500 Hz in the future.

Below some findings that have not been investigated during this work but, nonethe-less, are of high interest for VCC.

• The method and results presented, had the focus on correlation with themeasured data from the supplier. However, once a reliable method is estab-lished VCC should turn their focus on running real loading scenarios thatoccurs during driving. The prescribed displacement has been carried outfor x-, y- and z-direction only, which is an approximation. More advancedloading cases should be performed. This could contribute to less time spenton the tuning the engine mounts and make the development more efficient

• Since the measurement conditions are unknown, it is assumed that the mea-surements have been carried out in room temperature. If VCC wants tosimulate real loading scenarios one must take the temperature effects intoconsideration. In the engine compartment there can be temperatures of upto 100 ◦C, which can have an impact on the rubber material properties,briefly mentioned in Section 2.3.3.

• The Mullins effect has been neglected, which should be taken into consider-ation for the future. It may be noted that Abaqus has a built in function formodeling it.

• The fatigue properties of filled rubber used in the components should bedetermined, were the focus should be on the ability to resist crack initiationand crack propagation.

Appendix A

Definition of amplitudecurve

In this section the difference of using the Smooth and Periodic Step Abaqus key-word options will be explained.

In Figure A.1a the displacement has been defined as sinusoidal function usingthe periodic option, and the outgoing result can be seen in Figure A.1b.

time [s]

Dis

pla

ce

me

nt

[mm

]

(a) Periodic step

time [s]

Fo

rce

[N

]

(b) Resulting force

Figure A.1: Outgoing displacement and reaction force using the Periodic step

In Figure A.1b the force is increasing with increased frequency until the higherfrequencies is reached, according to the marked red circle. This is totally againstthe theory given in Chapter 2, since the force is expected to increase with higherfrequencies. Instead, by introducing a damping between the shifting frequenciesthis phenomena is no longer present, as seen in Figure A.2. This is due to thevelocities and accelerations causing measurement noise (disturbances) are removed

61

62 Definition of amplitude curve

or heavily damped.

time [s]

Dis

pla

ce

me

nt

[mm

]

(a) Smooth step

time [s]

Fo

rce

[N

]

(b) Resulting force

Figure A.2: Outgoing displacement and reaction force using the Smooth step

It is difficult to observe what really happens in the marked circle, however inFigure A.3a and Figure A.3b one can see the zoomed in image of the marked areafor the Periodic Step and Smooth Step, respectively.

time [s]

Fo

rce

[N

]

(a) Periodic step

time [s]

Fo

rce

[N

]

(b) Smooth step

Figure A.3: Zoomed in view of the red marked circle in Figure A.1b and Fig-ure A.2b, respectively.

As can be observed in Figure A.3 the Periodic Step can not manage the fastshifting frequencies causing the measurement noise as mentioned above. This isthe main reason why the dynamical stiffness decreases; for a comparison betweeenthe results of these two Abaqus options, see Figure A.4.

63

Dyn

am

ic s

tiff

ne

ss [

N/m

m]

Frequency [Hz]

Periodic step

Smooth step

Measured data

Figure A.4: Dynamic stiffness for Smooth step and Periodic step

64 Definition of amplitude curve

65

66 Material input

Appendix B

Material input

** MATERIALS

**Aluminum 6082T6

*MATERIAL, NAME=Aluminum 6082T6

*DENSITY

2.7E-9,

*ELASTIC, TYPE=ISOTROPIC

79237., 0.3

**Hyperelas c_Rubber

*MATERIAL, NAME=Hyperelas c_Rubber

*DENSITY

1.2E-9,

*HYPERELASTIC, YEOH

0.6803, -0.0982, 0.0188

*VISCOELASTIC, TIME=PRONY

0.131, 0., 0.00328

0.15344, 0., 0.0128

0.09114, 0., 0.189

*MATERIAL, NAME=elastoplas c_rubber_1

*DENSITY

1.2E-9,


1.01, 0.4995

*PLASTIC, HARDENING=ISOTROPIC

0.0287, 0., 0.

**elastoplas c_rubber_2


*DENSITY

1.2E-9,


2.041, 0.4995


0.0121, 0., 0.

**elastoplas c_rubber_3


*DENSITY

1.2E-9,


2.541, 0.4995


0.00121, 0., 0.

Figure B.1: Implementation of the material parameter in the Abaqus input file

Bibliography

[1] Volvo Car Corporation, 2015.

[2] P.-E. Austrell. Modeling of elasticity and damping for filled elastomers. Re-port TVSM-1009, Lund University, Division of Structural Mechanics, 1997.

[3] R. Österlöf. Modelling of the Fletcher-Gent effect and obtaining hyperelasticparameters for filled elastomers. Licentiate Thesis, KTH, 2014.

[4] A.K. Olsson. Finite element procedures in modeling the dynamic propertiesof rubber. Doctoral Thesis, Lund University, 2007.

[5] F. Karlsson and A. Persson. Modelling non-linear dynamics of rubber bush-ings - Parameter Identification and Validation. Lund University, 2003.

[6] L.A. Gracia, E. Liarte, J.L. Pelegay, and B. Calvo. Finite element simulationof the hysteretic behaviour of an industrial rubber. Application to design ofrubber components. 2010.

[7] O. Jesus and C. Gil. Finite element modeling of rubber bushing for crashsimulation. Experimental Tests and Validation. Lund Unuveristy, 2006.

[8] F. Alkhatib. Techniques for Engine Mount Modeling and Optimization. Uni-versity of Wisconsin-Milwaukee, 2013.

[9] Dassault Systèmes Simulia Corp. Abaqus Theory Manual, Version 6.13. Prov-idence, RI, USA, 2013.

[10] R.S. Rivlin. Large Elastic Deformations of Isotropic Materials. I. FundamentalConcepts. 1997.

[11] M. Mooney. A theory of large elastic deformation, volume 11. 1940.

[12] O. H. Yeoh. Characterization of Elastic Properties of Carbon-Black-FilledRubber Vulcanizates. 1990.

[13] A. J. M. Spencer. Continuum mechanics. Mineola, N.Y. : Dover, 2004, 2004.ISBN 0486435946.

[14] P. Håkansson. Finite element modelling of rubber block exposed to shockloading. Lund University, 2000.

67

68 BIBLIOGRAPHY

[15] ANSA. version: 15.1.0. BETA CAE Systems SA, 2014.

[16] CATIA. version: 5.19. Dassault Systèmes Simulia Corp, 2014.

[17] Abaqus. version: 6.14-1. Dassault Systèmes Simulia Corp, 2014.

[18] MATLAB. version 8.2.0.701 (R2013b). The MathWorks Inc., Natick, Mas-sachusetts, 2010.

[19] A. Ali, M. Hosseini, and B.B. Shari. A review and comparison on some rubberelaticity models. CSIR, 2010.

[20] R.D. Cook, R.D. Malkus, M.E. Plesha, and R.J. Witt. Concepts and Appli-cations of Finite Element Analysis, 4th Edition. 2001.

Documents

Predictionofstaticanddynamicbehaviorof ... · PDF filePredictionofstaticanddynamicbehaviorof powertrainsuspensionrubbercomponents Robin Öhrn Master Thesis ... garding the physical