Fatigue Life Prediction and Modeling of Elastomeric Components

Transcription

1 A Dissertation Entitled Fatigue Life Prediction and Modeling of Elastomeric Components by Touhid Zarrin-Ghalami Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Engineering Dr. Ali Fatemi, Committee Chair Dr. Efstratios Nikolaidis, Committee Member Dr. Mehdi Pourazady, Committee Member Dr. Vijay Goel, Committee Member Dr. Yong Gan, Committee Member Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo May 2013 i

2 Copyright 2013, Touhid Zarrin-Ghalami This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. ii

3 An Abstract of Fatigue Life Prediction and Modeling of Elastomeric Components by Touhid Zarrin-Ghalami Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Engineering The University of Toledo May 2013 This study investigates constitutive behavior, material properties and fatigue damage under constant and variable amplitude uniaxial and multiaxial loading conditions, with the goal of developing CAE analytical techniques for durability and life prediction of elastomeric components. Such techniques involve various topics including material monotonic and cyclic deformation behaviors, proper knowledge of stress/strain histories, fatigue damage quantification parameters, efficient event identification methods, and damage accumulation rules. Elastomeric components are widely used in many applications, including automobiles due to their good damping and energy absorption characteristics. The type of loading normally encountered by these components in service is variable amplitude cyclic loading. Therefore, fatigue failure is a major consideration in their design and availability of an effective technique to predict fatigue life under complex loading is very valuable to the design procedure. In this work a fatigue life prediction methodology for rubber components is developed which is then verified by means of analysis and testing iii

4 of an automobile cradle mount made of filled natural rubber. The methodology was validated with component testing under different loading conditions including constant and variable amplitude in-phase and out-of-phase axial-torsion experiments. The analysis conducted includes constitutive behavior representation of the material, finite element analysis of the component, and a fatigue damage parameter for life predictions. In addition, capabilities of Rainflow cycle counting procedure and Miner s linear cumulative damage rule are evaluated. Fatigue characterization typically includes both crack nucleation and crack growth. Therefore, relevant material deformation and fatigue properties are obtained from experiments conducted under stress states of simple tension and planar tension. For component life predictions, both fatigue crack initiation approach as well as fatigue crack growth approach based on fracture mechanics are presented. Crack initiation life prediction was performed using different damage criteria. The optimum method for crack initiation life prediction for complex multiaxial variable amplitude loading was found to be a critical plane approach based on maximum normal strain plane and damage quantification by cracking energy density on that plane. The fracture mechanics approach was used for total fatigue life prediction of the component based on specimen crack growth data and FE simulation results. Total fatigue life prediction results showed good agreement with experiments for all of the loading conditions considered. iv

5 This dissertation is dedicated to my dear parents, Jila and Siawash. v

6 Acknowledgements I would like to sincerely thank my dear advisor, Prof. Ali Fatemi, who has supported me throughout this long route with his knowledge, patience and useful hints. This work could not be completed without his guidance, encouragement and advice during long meetings, almost daily, and lots of s. Dr. Yong Gan, Dr. Mehdi Pourazady, Dr. Efstratios Nikolaidis, and Dr. Vijay Goel are highly appreciated for serving on my Ph.D. committee. I would also like to thank Chrysler Group LLC and specifically Dr. Yung-Li Lee for funding the project and Paulstra CRC for providing the components and FE model. The time and effort of Mr. John Jaegly, Mr. Randall Reihing and Mr. Tim Grivanos from the machine shop of the MIME Department for helping in making different fixtures for uniaxial and multiaxial testing of the components is highly acknowledged as well. I would also like to thank my colleagues at the fatigue and fracture research laboratory of the University of Toledo for their help, support and efforts during this period of hard working. I would also like to thank Ms. Debbie Kraftchick and Ms. Emily Lewandowski in the MIME department office for their help in providing the requirements of this study. Finally, I would like to thank my dear parents and sisters for all their support and encouragement throughout my education from the very beginning to doctorate degree. vi

7 Table of Contents Abstract... iii Acknowledgements... vi Table of Contents... vii List of Tables... xi List of Figures... xii List of Abbreviations... xvii List of Symbols... xix 1 Introduction Preview Motivation for the study and objectives Outline Literature Survey Introduction Rubber deformation mechanisms and behavior Fatigue crack initiation and growth approaches Multiaxial fatigue behavior and models Component fatigue testing and life prediction including FE simulations Damage accumulation and cycle counting in variable amplitude loading...23 vii

8 3 Material Characterization and Fatigue Behavior Introduction Material and specimens Testing equipment Specimen test methods and procedures Deformation behavior characterization Constant amplitude crack initiation testing Crack growth testing Variable amplitude crack initiation testing Specimen experimental results and observations Monotonic and cyclic deformation behavior results Constant amplitude crack initiation tests results Crack growth tests results Relationship between crack growth and crack initiation approaches Variable amplitude crack initiation tests analysis methodology Variable amplitude crack initiation tests results and analysis Conclusions Component Finite Element Model and Results Introduction Hyperelastic material deformation characterization Uniaxial FE model definition and specifications...69 viii

9 4.4 Stiffness test results comparison with predictions Uniaxial constant amplitude FE simulation results Conclusions Component Uniaxial Fatigue Behavior Introduction Component life predictions Crack initiation life predictions Crack growth-based life predictions Variable amplitude loading life predictions Constant amplitude fatigue experimental program and validation of life predictions Component fatigue tests Damage development Comparison of predicted and experimental fatigue lives Variable amplitude fatigue experimental program and validation of life predictions Component fatigue tests Damage development Comparison of predicted and experimental fatigue lives Conclusions Component Multiaxial Fatigue Behavior Introduction ix

10 6.2 Life prediction methodology for general random and multiaxial loading of components Applications to multiaxial constant amplitude loading Experimental program and results Life predictions and validation of methodology Applications to variable amplitude multiaxial loading Discussion of results Conclusions Summary and Possible Future Research Summary Constitutive behavior Fatigue material relations obtained from specimen experiments Uniaxial constant and variable amplitude component fatigue behavior and life predictions Multiaxial constant and variable amplitude component fatigue behavior and life predictions Possible future research References x

11 List of Tables 3.1 Constant amplitude fatigue crack growth test conditions Crack nucleation test conditions and results Simple tension specimen fatigue test conditions and results with the random loading history used Summary of parameters used for numerical model Summary of component fatigue test results Predicted nucleation and total fatigue lives of the rubber mount component and comparison with experimental fatigue lives Component fatigue test results and predictions under uniaxial variable amplitude loading with R~ Component experimental conditions and test results Summary of component crack initiation and total life experiments and predictions xi

12 List of Figures 3-1 Specimen geometry and dimensions for (a) simple tension, and (b) planar tension. Specimen thickness is 1 mm for both geometries Axial servo-hydraulic Instron frame used for specimen testing Simple tension and planar tension specimens with the corresponding stretch and stress states [36] Crack initiation test setup showing five specimens and grips Maximum strain energy density versus maximum engineering strain obtained from uncracked planar tension specimen (a) Mullins effect showing initial transient softening in planar tension specimen at 133% maximum strain, and (b) stabilization of stress with applied cycles in displacement-controlled incremental step cyclic deformation tests of planar tension specimen at different maximum strain levels (a) Loading history used for random loading tests of simple tension specimens in displacement control and of cradle mounts in load control, and (b) Rainflow cycle count for max/min values of ±100 with relative xii

13 damage distribution from each cycle [63] Superimposed plot of monotonic and stable cyclic curves for simple tension and planar tension specimens Stable cyclic stress-strain loops from incremental step tests at different peak strain levels from (a) simple tension test, and (b) planar tension test Crack nucleation life as a function of peak strain in simple tension tests Equivalent fatigue life as a function of equivalent maximum strain by using Mars-Fatemi R-ratio model Crack length versus cycle linear fits from crack growth test at (a)r T = 0, (b)r T = 0,(c)R T = 0.05, and (d) R T = Fatigue crack growth rate data comparisons at different R-ratios Fatigue crack growth rate data correlations at different R-ratios based on Mars-Fatemi R-ratio model Comparison of fatigue lives obtained from crack initiation and crack growth approaches Experimental versus predicted fatigue lives in blocks to failure for simple tension specimen tests subjected to variable load history Vehicle cradle mount used as illustrative example (a), and rubber mount FE xiii

14 model where due to symmetry half of the model is shown (b) Component mid-life hysteresis loops under axial loading condition for (a) R ~ 0 and, (b) R = Stiffness comparison between FEA simulation and monotonic test of the mount (a) and cyclic tests of the mount (b) Strain (a) and stress (b) histories at the critical element location obtained from FE simulation during one sinusoidal loading cycle between 750 N and 3750 N Maximum principal strain distribution under R =0.2 and load amplitude of 1,500N Maximum R = 0 equivalent engineering strain versus crack depth for load amplitude of 1500 N and R = 0.2 from FEA simulation Component failure definition based on stiffness degradation in fatigue tests Displacement amplitude versus cycles in fatigue tests Experimental failure locations and crack growth direction in component tests Experimental versus predicted fatigue life based on crack initiation approach (a) and based on crack growth approach (b) Component displacement amplitude response as a function of applied blocks xiv

15 of loading under variable amplitude fatigue loading Crack length versus applied load blocks for a test at R ~ 0 and with a load range of 3,600 N Maximum principal strain contour results for R~0 and loading range of 3,000 N obtained from FE simulation Component fatigue testing and predictions for variable amplitude random loading at R~0, both for crack initiation and total component lives Vehicle cradle mount FE model Definition of failure illustrated by variation of axial and rotational displacements in constant amplitude in-phase axial-torsion tests of the cradle mount Experimental response for constant amplitude axial-torsion loading of components for (a) IP and (b) OP tests Axial displacement versus rotation angle of the component for different cycles throughout the CA and A-T tests of (a) IP and (b) 90 OP Axial displacement versus rotation angle of the component for mid-life cycles for CA and A-T IP and 90 OP tests xv

16 6-6 Evolution of crack length and depth for constant amplitude axial-torsion (a) in-phase test with N f = 31,290, and (b)out-of-phase test with N f = 21, Crack initiation to total fatigue life ratio for all types of loadings Maximum principal strain location for constant amplitude in-phase loading simulation Experimental versus predicted component initiation life for all loading conditions based on (a) maximum normal strain, (b) critical CED, and (c) CED on MNS plane Experimental versus predicted component total fatigue life for all loading conditions Variable amplitude in-phase and out-of-phase axial-torsion loading history used for component testing Normal strain on maximum normal strain (MNS) plane history of variable amplitude and axial-torsion loading for (a) in-phase, and (b) out-of-phase loading xvi

17 List of Abbreviations A...Axial ASTM...American Society of Testing and Materials A-T...Axial-Torsion CA...Constant Amplitude CAE...Computer Aided Engineering CED...Cracking Energy Density DC...Displacement-Controlled EPDM...Ethylene Propylene Rubber FCG...Fatigue Crack Growth FE...Finite Element FEA...Finite Element Analysis IP...In-Phase LC... Load-Controlled LDR...Linear Damage Rule MNS...Maximum Normal Strain NR...Natural Rubber OP...Out-of-Phase xvii

18 SAE...Society of Automotive Engineers SBR...Styrene-Butadiene Rubber SED...Strain Energy Density SWT...Smith-Watson-Topper VA...Variable Amplitude xviii

19 List of Symbols a..crack length a 0... initial crack length or flaw size a f... final crack length B f blocks to failure B. fatigue crack growth coefficient, number of blocks C. initial Young s modulus, Green deformation tensor c ij... material constants for Rivlin strain energy density function da... change in new crack surface area Du.. change in stored mechanical energy dw c.increment in cracking energy density d.. unit displacement vector E modulus of elasticity, Green-Lagrange strain tensor e max. maximum engineering strain F. deformation gradient tensor, fatigue crack growth exponent F a load amplitude F(R) power-law exponent for Mars-Fatemi model f..... test frequency G shear modulus G(R)... power-law exponent for Mars-Fatemi model h... specimen height I unit matrix I, I I the first, second and third invariants of the stress tensor 1 2, 3 J el elastic volume ratio k. strain dependent material parameter L,L 0 length, initial length of specimen M a... torque amplitude N number of cycles NE... nominal strain N. cycles to failure f Nˆ.. unit vector normal to the plane in space n i principal stretch direction P a... load amplitude R minimum to maximum ratio r, da/dn... crack growth rate r.. critical crack growth rate C xix

20 R M... torque R- ratio R P... load R- ratio R... displacement R- ratio R. strain R-ratio R rotation R- ratio R T energy release rate R-ratio R strain energy density R-ratio W R...unit vector in the undeformed configuration r... unit vector in the current configuration S stress S max maximum engineering stress S ~... 2 nd Piola-Kirchhoff stress T tearing energy, energy release rate, Cauchy stress tensor T C.. critical energy release rate T... traction vector U strain energy density per unit of reference volume U volumetric part of U vol U dev...deviatoric part of U V..left stretch tensor W.strain energy density a displacement amplitude...strain 1 maximum principal strain C.. critical strain for crack initiation...stretch ratio i, i. material constants for Ogden strain energy density function..strain range V.change in volume. mass density a.rotation amplitude xx

21 Chapter 1 Introduction 1.1 Preview The definition of fatigue as currently stated by ASTM [1] is the process of progressive localized permanent structural change occurring in a material subjected to conditions which produce fluctuating stresses and strains at some point or points and which may culminate in cracks or complete fracture after a sufficient number of fluctuations. Fatigue failure is a typical service failure mode resulting from alternating and repeated loads over a period of time. Although elastomeric components are commonly used in many industries and their field of application has broadened in recent years, there is very limited research related to fatigue failure aspects of elastomeric components, as compared to metallic components. As a result, fatigue life evaluation and prediction of rubber components are important topics to investigate to assure their safety and reliability [2]. Elastomers are highly nonlinear elastic materials with large deformations. These characteristics along with high energy absorption ability and dynamic damping capacity as well as low cost of manufacturing rubber parts make this material ideal for many 1

22 diverse engineering applications. This includes the automotive industry (e.g. car tires, bumpers, engine mount), piping (e.g. gaskets, bearings, hoses, seals) and as vibration isolators in many other applications. However, the behavior of rubber materials is complex due to high deformability, quasi incompressibility, stress softening, and time-dependent effects. Therefore, a correct description of the behavior of elastomers must include both geometric and material non-linearities as well as incompressibility [3]. Furthermore, properties of rubber can change drastically depending on material composition, manufacturing conditions (mixing and curing), loading condition (strain state and strain rate) and other service conditions (temperature and period of usage, for example) [4]. Availability of an effective technique to predict fatigue life under actual complex loading is very valuable to the design procedure. Such a technique or model relates fatigue properties obtained from constant amplitude fatigue characterization tests to variable amplitude (VA) service loading conditions. Most researchers studying fatigue behavior of elastomers have mainly focused on constant amplitude loading conditions, rather than variable amplitude loading applications. In order to be able to predict life of elastomeric components, several inputs are necessary. These include the component geometry, relevant material properties, and the loading history. For analysis, two basic steps are often required. First, stress and/or strain analysis is used, typically by finite element method, to obtain stress/or strain histories at critical location(s) from the applied load history. Then, fatigue life prediction analysis is performed based on a fatigue damage quantification parameter and damage accumulation model using the critical location stress and/or strain history. Linear elastic and small 2

23 strain assumptions, which are typically used for metals, do not apply to elastomers. As a result, continuum mechanics parameters such as deformation gradient, stretch ratio, and Green-Lagrange strain components are used in rubber mechanics analysis. Numerical FE simulation softwares have been broadly used for design and they have become more reliable in recent years. However, both analysis and testing are still required for fatigue design. The process of producing inexpensive and high quality products requires reducing the amount of testing by more accurate analysis. 1.2 Motivation for the study and objectives The motivation for this study was a practical need to predict fatigue life for industrial components made of elastomeric materials. The main input of the study is the material properties obtained from material testing. Different loading histories from simple to complex are evaluated, including variable amplitude non-proportional multiaxial loading, which is a realistic loading condition in many industrial applications. Crack nucleation life approach and total fatigue life approach based on crack growth properties are used for life predictions, and illustrated by using a passenger vehicle cradle mount. At the critical location(s) of most components and structures, multi-axial states of stress and strain are typically present during each loading cycle. Such components and structures are also usually subjected to variable amplitude or random loading histories. Therefore, the study of multiaxial fatigue and deformation under variable amplitude loading is very important. The extreme values for the strain and/or stress components do not usually coincide with each other under complex loading conditions. This indicates the complex nature of the strain history in applications such as vibration isolators. 3

24 The typical life prediction methodologies for uniaxial loading have been well developed and used for many years. These methodologies for the more complex case of multiaxial variable amplitude loading are not yet well established, particularly when the loads are non-proportional. Very limited papers can be found in the literature related to fatigue life prediction of elastomeric components under variable amplitude multiaxial loading. Therefore, the objective of this study is to develop a robust methodology for fatigue life prediction of such complex loading conditions and validate it by using an automobile cradle mount. More specifically, the following are the objectives of this research: 1) To develop CAE analytical techniques for durability and life prediction of elastomeric components. 2) To validate the CAE analytical techniques with cradle mount component testing. 3) To be able to significantly reduce component bench tests to save time and cost, thereby facilitating optimization. 4) To evaluate the recent promising and other commonly used damage parameters for fatigue life assessment of elastomeric industrial components under multi-axial loads. 5) To investigate the effect of dwell periods and loading rate on behavior of rubbers and its industrial applications. 6) To evaluate the capability of commonly used cycle counting procedures (rainflow cycle counting method) and damage accumulation rules (linear damage rule) for life predictions of elastomers under different loading conditions. 4

25 1.3 Outline In Chapter 2, a brief literature review is presented. This chapter reviews different works on material properties and constitutive behavior models, crack initiation and growth approaches, multi-axial fatigue models, FE simulations, damage accumulation and application to variable amplitude loading. An understanding of stress-strain behavior, particularly under cyclic loading, is necessary for the study of fatigue. Material fatigue behavior characterization for crack initiation and crack growth, and constitutive behavior characterization from simple tension and planar tension tests are presented in Chapter 3. Test procedures, testing apparatus, test methods used, and obtained data are presented and discussed in this chapter. Chapter 4 presents the finite element modeling aspects of the component. FE model details, preprocessor modules description, material properties used in FE analysis and analysis type, step and loading considerations are discussed in this chapter. Mesh sensitivity analysis and the effect of load control versus displacement control input on the results are other issues which are discussed in this chapter. Stiffness comparison between FE simulation and experiment as well as simulation results for uniaxial constant amplitude loading of the component are also presented. Chapter 5 presents the component uniaxial constant and variable amplitude loading experiments and life predictions. FE simulations, fatigue crack initiation as well as total fatigue life prediction methodologies are also discussed. Damage evolution in the component is also described in this chapter. 5

26 Component multiaxial fatigue behavior and life predictions are presented in Chapter 6. First, the life prediction methodology for multiaxial and random loading is presented. This methodology discusses both crack initiation and growth approaches. Then, the capability of these damage quantification parameters in fatigue life prediction for different axial-torsion loadings of both constant and variable amplitude is discussed. This comparison is done with bench component testing and by using the simulation results and the methodology presented. The results of the prediction methods are also discussed in this chapter. Cracking behavior and damage evolution are also presented in Chapter 6. Effects of control mode on fatigue life, hysteresis loops, crack initiation location and crack initiation life are also discussed in this chapter. Finally, a summary of the dissertation along with the important conclusions are presented in Chapter 7. Some suggestions for further studies in multiaxial fatigue of elastomers are also presented in this chapter. These suggestions include further studies to develop or improve a multiaxial fatigue life prediction results by finer meshing at the critical locations of the component and utilizing a Fracture Mechanics FE model of the component for simulations. Based on the contents of this dissertation, five papers have been published. Material deformation and fatigue behavior of the natural rubber used in manufacturing the cradle mount used are characterized in [5] based on what is presented and discussed in Chapters 2 and 3. The component FEA and fatigue life analysis and predictions for uniaxial constant amplitude and variable amplitude loadings are presented in [6] and [7], respectively. The materials in these papers are extracted from Chapters 2, 4 and 5. A robust methodology for fatigue life prediction of elastomeric components under 6

27 complex loading conditions which is developed in Chapter 6 is presented in [8] for general random and multiaxial loading. Then applications to constant amplitude axial-torsion in-phase and out-of-phase loading as well as variable amplitude loading are demonstrated using experimental results from the cradle mount. The information presented in this paper is mainly chosen from the contents of Chapter 6. An overview of the CAE durability analytical techniques for predicting crack initiation life of the automotive cradle mount and their validation under constant and variable amplitude uniaxial and multiaxial loadings is also presented in [9]. 7

28 Chapter 2 Literature Survey 2.1 Introduction In this chapter analytical and numerical approaches that are currently available for predicting fatigue life of elastomers are reviewed. Deformation mechanisms of elastomers are first discussed. Different studies on two main approaches of crack initiation and crack growth are then reviewed. Multiaxial fatigue models and different damage quantification parameters related to complex loading conditions constitute the next part of this literature survey. Numerical methods and fatigue life predictions of complex geometries such as those used in industrial applications are also covered in this review. Recent works on damage accumulation and application in variable amplitude loading of elastomers is the last topic which is reviewed in this chapter. 2.2 Rubber deformation mechanisms and behavior Rubber can have a wide range of mechanical properties by changing the compound formulation and manufacturing process. Strain crystallization has been shown 8

29 to have a beneficial effect on fatigue life at moderate or high strain levels. The addition of carbon black to rubber compound could strengthen the material against fatigue failure and drastically change its mechanical properties. Antidegradants are added to rubber compounds to avoid the deleterious effects of oxygen and ozone. Vulcanization is used in thermosetting elastomers to create covalent bonds or crosslinks between adjacent polymer chains. Crosslink density determines the physical properties of rubber, with higher crosslink density resulting in increased stiffness and reduced hysteresis. Compound stiffness has a direct effect on energy release rate. Filled rubbers show more dissipative mechanical responses at both high strains because of network chain breakage and strain crystallization, and at small or moderate strains under alternating loading. Hysteresis is an important aspect when evaluating fatigue properties in terms of deformation, crack nucleation, or crack growth. In natural rubber material, hysteresis behavior appears at high strains above 200 to 250% and is usually associated with the phase transformation process called crystallization. The initial softening, which is called the Mullin s effect, depends non-linearly on the maximum principal strain reached during a cycle [10]. Mars and Fatemi [11] investigated the monotonic and cyclic behaviors of filled natural rubber. They conducted pure axial, pure torsion as well as proportional and non-proportional axial-torsion experiments by using short thin-walled cylindrical specimens. They observed initial softening within the first 10 cycles of fatigue loading compared to monotonic tests and also the stabilized stress-strain response after a short number of cycles. This is associated with irreversible breakage of different types of bonds in the elastomer-filler composite. They also described the deviations from non-linear 9

30 elasticity under cyclic loading for filled natural rubber. Their results also showed the effect of phase angle on the shape and size of the hysteresis loops with the 90 out of phase loading histories leading to the maximum hysteresis. Initial overload was found to have a major effect on the subsequent evolution of the stress amplitude response. By applying the initial overload, the logarithmic decrease trend of stress amplitude drop with cycles totally disappeared and the stress response remained constant. Mars and Fatemi [12] also described the inability of current hyperelastic models to capture the Mullin s effect and proposed a new model relating flaw growth to the softening of cyclic stressstrain response to explain the behavior of the specimen during late portion of fatigue life. 2.3 Fatigue crack initiation and growth approaches Mars and Fatemi [13] conducted a literature study on different approaches associated with fatigue analysis for rubber. Fatigue failure process is divided into two distinct phases, the crack nucleation phase in which crack nucleates in regions that were initially crack free, and crack growth phase when the nucleated cracks grow to the point of failure or rupture. They identified maximum principal strain (or stretch) and strain energy density as two broadly used damage quantification parameters for the nucleation phase. Although it is commonly observed that cracks initiate on a plane normal to the maximum principal tensile strain direction, the strain energy density criterion applied as a scalar criterion can t account for this preferred orientation. For the fatigue crack growth stage, a fracture mechanics analysis is used based on energy release rate. Mars and Fatemi also discussed the relationship between crack nucleation and crack growth approaches by using an integrated power-law model, which is also proposed in [13]. 10

31 Stevenson [14] conducted a study of fatigue crack growth of rubber in compression with a cylindrical specimen. He concluded that crack growth in compression occurs at an approximately constant rate and is restricted to the outer regions of the test specimen with high local shear strains. Legorju-jago and Bathias [15] showed that a mean stress in tension improves the fatigue behavior by crystallization of the stretched bonds in pure tension cycles. On the other hand, a minimum stress in compression seriously damages the material. Abraham et al. [16] investigated the effect of minimum stress and stress amplitude on fatigue life of non-strain crystallizing elastomers such as ethylene propylene (EPDM) and styrene-butadiene (SBR) rubbers. They used cylindrical dumbbell specimens and concluded that increasing minimum stress with constant stress amplitude can increase the service life by a factor of 10 despite the increase in maximum stresses. To account for R ε ratio (minimum to maximum strain ratio) effect on either crack nucleation or crack growth life, Mars and Fatemi [17] proposed a phenomenological model. The ability of the model to represent data for different R ratios was shown to be reasonable based on their own experimental data, as well as data of Lindley et al. [17]. Harbour et al. [18] also utilized this model to represent their non-zero fatigue crack growth rate data. Wang et al. [19] proposed a continuum damage mechanics model for fatigue damage behavior of elastomers. They used the Ogden model to construct the constitutive relation for carbon-filled natural rubber. They introduced an equation for fatigue life as a function of applied nominal strain amplitude. They concluded their proposed relation 11

32 could describe experimental data very well, though restricted to R ratio of zero and fixed frequency values [19]. A crack initiation life diagram is the Haigh diagram which is a contour plot of strain amplitude versus mean strain in which lines of equal fatigue life are plotted [20]. The Cadwell diagram is another format which also summarizes the dependence of the crack nucleation life on the changing limits of constant amplitude cycle. In this diagram base 10 logarithm of the fatigue life is plotted versus minimum strain with the contours of strain amplitude kept constant [21]. Two main mechanisms, decohision and cavitation, are responsible for fatigue damage nucleation which are both independent of the type of loading and dependent on the nature of the inclusion [22]. The decohision process could be easily seen on the fracture surface, because the surface of the inclusion is free of rubber after decohision. It is found to be predominant at rigid inclusions such as SiO 2 and CaCO 3. The interface between the inclusion and rubber matrix is the weakest location, therefore, fatigue damage initiates at the interface which causes the decohision between the matrix and inclusion. Cavitation is the process of sudden void initiation under stress state [22]. Mars and Fatemi [23] observed a direct relationship between pre-existing flaws in the rubber material and the crack initiation process by assuming that the pre-existing flaws in the materials were small cracks and crack initiation occurs when one of these small cracks becomes large. Several factors including environmental conditions, formulation of the rubber, mechanical load history, and dissipative effects of the stress-strain behavior affect the 12

33 fatigue life of rubber. The synergistic effects of all abovementioned factors define the fatigue life. Mars and Fatemi [24] investigated a detailed survey of these factors. Kim and Jeong [25] investigated natural rubber compound with three different filler types to examine the effect of carbon black on fatigue life. They concluded that for larger carbon black agglomerates, separation of fillers from rubber matrix could relatively easily occur resulting in shorter fatigue life, compared to smaller size fillers. Santangelo and Roland [26] verified that double network of natural rubber (NR) has a higher modulus than single network of equal crosslink density. Fatigue life of double networks NR was also found to be as much as a factor of 10 higher than conventionally crosslinked NR, although tensile strength was slightly lower. Environmental effects can significantly affect fatigue life of elastomers. High temperature has a detrimental effect both on crack nucleation life and crack growth rate. This effect is intensified in amorphous rubber. Ozone could shorten the crack growth life due to reaction with carbon bonds at the crack tip. Presence of oxygen increases the fatigue crack growth rate. Mars and Fatemi discussed the aforementioned effects including temperature, ozone, oxygen, and electrical charges in detail in [24]. The crack growth approach is based on the growth of pre-existing flaws or cracks using fracture mechanics. This approach for rubber was developed in the 1950s and 1960s. The strain energy release rate introduced by Griffith for brittle fracture and extended to rubber by Rivlin and Thomas [27] is generally used as the governing fatigue damage parameter for describing fatigue crack growth rates. 13

34 2.4 Multiaxial fatigue behavior and models Rubber structures in service usually encounter time-varying loads in several directions necessitating a proper multiaxial fatigue criterion for relating stress and strain histories to fatigue life. An important issue that should be considered in multiaxial fatigue is in-phase versus out-of-phase loading. When the different loading channels reach to their peak values at the same time, the loading is called in-phase, while when the time of reaching maximum or minimum value for different load channels are different, the loading is called out-of-phase. Life prediction methodologies for uniaxial loading have been relatively well developed. These methodologies for the more complex case of multiaxial variable amplitude loading are not yet well established, particularly when the loads are non-proportional. Very limited papers can be found in the literature related to fatigue life prediction of elastomeric components under variable amplitude multiaxial loading. Multiaxial fatigue life prediction methodologies for rubber can be classified into four main approaches. These consist of the equivalent strain approaches, energy approaches, equivalent stress approaches, and the more recent critical plane approaches. Critical plane models can be used for both proportional and non-proportional loading conditions and are based on the physical process of the damage process. Critical plane approaches could be strain-based (such as the maximum normal strain), stress-based (such as the maximum principal Cauchy stress), or energy-based (such as cracking energy density). Energy-based critical plane approaches, which use both stress and strain, can reflect the constitutive behavior of the material, while stress or strain-based critical plane approaches do not require the constitutive behavior of the material. 14

35 In general, a good multiaxial fatigue model should be robust, sensitive to load phasing and mean stress, and applicable to variable amplitude loading. Another important characteristic is the ability to consider crack closure effects. Crack initiation is normally related to the continuum mechanics quantities (e.g. stress or strain) which are macroscopic. The most widely used parameters for crack initiation are the maximum principal strain, maximum principal Cauchy stress and strain energy density (SED). Approaches such as the cracking energy density (CED) or based on configurational mechanics (Eshelby stress tensor) are more complex and recent approaches dealing with multiaxial fatigue analysis of elastomers. The maximum principal strain criterion was introduced by Cadwell et al. [21] for unfilled vulcanized natural rubber in 1940 and still remains one of the most commonly used criteria for rubber. For incompressible materials, this criterion always gives a positive value for maximum principal strain. Octahedral shear strain is also another strain-based predictor which also always gives a positive value for rubber-like materials. In addition, since rubber is incompressible, the hydrostatic stress is independent of strain tensor. Ayoub et al. [28] established a relationship between the fatigue life of styrene-butadiene rubber and the stretch amplitude. They conducted multiaxial constant and variable amplitude tests on cylindrical (axisymmetric) specimens. The specimen curvature radius was large enough to minimize the stress triaxiality effect. Their results showed a good agreement between predicted and experimental fatigue lives. The predictions were based on continuum damage mechanics improved by incorporating CED 15

36 criterion. The developed model is both a fatigue damage criterion and an accumulative damage rule. Rivlin and Thomas first applied strain energy density (SED) criterion to rubber material under static loading. This criterion has some drawbacks including inability to differentiate between simple tension and compression, inability to account for crack closure and the fact that all of the stored energy in the material would not release due to crack growth [29]. Strain energy density usually uses a hyperelastic formulation which is defined in terms of strains. If strain energy density and maximum principal strain would be applied as scalar criteria, the fact that cracks are usually observed to orient in specific orientation could not be predicted. Mars and Fatemi [30] designed a novel specimen for investigating the mechanical behavior of elastomers under multiaxial loading conditions. By utilizing this specimen and based on their experimental observations [23] they concluded that the maximum principal strain as a fatigue damage parameter gave the best prediction of fatigue life, while the traditionally used strain energy density gave the worst correlation of experimental data. Mars and Fatemi [31] discussed the observations of crack initiation and small flaw growth in filled natural rubber under multiaxial loading conditions. They used their designed ring specimen with axial, torsion and both proportional and non-proportional axial-torsion loadings. They suggested that crack nucleation in rubber starts from existing flaws in the virgin material such as voids, surface cavities and non-rubber particles. Fatigue crack initiation and growth were observed to occur on preferred failure planes. For axial, torsion and in-phase axial-torsion loading the cracking plane was transverse to 16

37 the maximum principal strain direction. For more complex loading they still observed preferred nucleation planes but their relation to the principal strain directions was sometimes different. Crack closure affected the nucleation plane and more closure was observed in cyclic torsion under static compression. This also happened in fully reversed axial-torsion experiments due to friction during the torsion-compression part of the loading. Andre et al. [32] performed torsion experiments on axisymmetrical notched sample geometry specimens and concluded that the maximum principal Cauchy stress could be related to multiaxial fatigue damage mechanisms. They proposed that the maximum principal Cauchy stress could be used as a fatigue damage quantification parameter and observed that crack orientation is perpendicular to its direction. Brunac et al. [33] extended Haigh s diagram to arbitrary 3D loadings by considering the smallest sphere containing the closed path of the positive part of the Cauchy stress tensor. They concluded that this approach results in successfully predicting fatigue life. Saintier at al. [22] investigated the micro-mechanisms leading to crack initiation under multiaxial fatigue loading conditions and suggested that cracks initiate from inclusions or large carbon black agglomerates. They observed decohision and cavitation as damage mechanisms and suggest that cyclic loading does not produce a new damage mechanism in rubber, in contrast to metallic materials. They observed that the maximum first principal stress reached during the cycle defined the orientation of the crack in all of the fatigue loading conditions they considered. In another study, Saintier et al. [10] proposed a critical plane approach for fatigue crack initiation based on the micro-mechanisms of crack initiation. In their study, under non-proportional multiaxial 17

38 loading, dependent on the material, fatigue crack growth orientation was found to be on the plane of maximum shear stress amplitude (shear cracking), maximum normal stress plane (tensile cracking), or even mixed-mode shear and tensile cracking. Mars and Fatemi [34] concluded that scalar equivalence criteria were not capable of predicting the fatigue initiation life in natural rubber. They suggested using cracking energy density which is the portion of the strain energy density that is available to cause crack growth on a particular plane. The application of this method involves knowledge of the constitutive behavior of the material. Zine et al. [35] applied the cracking energy density criterion in a FE code and found good agreement between numerical and analytical results for common strain states. Their experiments results also showed the efficiency of this criterion to explain fatigue life of elastomers under multiaxial loading conditions. Harbour et al. [36] used the multiaxial ring specimen in [30] and performed both constant and variable amplitude axial-torsion experiments. They used two rubber materials, one which strain crystallizes (natural rubber) and one which does not (SBR). They concluded that both cracking energy density and normal strain approaches were able to predict the dominant crack orientations for some of the test signals in each material (i.e. axial and multilevel axial tests), but not successful for some other loadings such as for fully reversed torsion experiments. In another study Harbour et al. [37] used the maximum normal strain to find the critical plane and the cracking energy density on that plane to determine fatigue life. Their results showed that this criterion produced similar fatigue life results compared to other approaches such as cracking energy density, strain energy density and maximum principal strain. Because maximum normal strain is 18

39 independent of constitutive behavior, the critical plane can be identified more easily than for CED. They [37] also studied the effect of variable amplitude multiaxial loading and concluded that Miner s linear damage rule gave reasonable predictions of their experimental results. Verron et al. [38] proposed a multiaxial criterion for crack nucleation based on the local properties of the Eshelby second-order tensor [39] and defined in terms of continuum mechanics parameter. Wang et al. [29] evaluated fatigue life prediction approaches by using experimental results from proportional and non-proportional loading paths applied to small axisymmetric diabolo specimens made of vulcanized NR. They conclude maximum principal strain and octahedral shear strain provide good predictions of the fatigue life. They also found that compared to traditional criterion of strain energy density, the cracking energy density model gives better life prediction results. Mars [20] investigated the duty cycle on each material plane along with its corresponding damage to transform the multiaxial loading into the localized flaws. After identifying the damaging events, the original duty cycle is simplified and reconstituted to a new duty cycle including the number of the most damaging events of the original duty cycle. The new shortened duty cycle maintains those features of the original duty cycle corresponding to the original mode of failure and shortens the time scale of the test. For multiaxially loaded rubber parts, Flamm et al. [40] proposed discretizing the continuous signal by a level crossing cycle counting method for each loading channel. Then they constituted the stress amplitude history and performed Rainflow cycle counting on these alternating points. 19

40 2.5 Component fatigue testing and life prediction including FE simulations In order to be able to evaluate or predict fatigue life of the component, it is necessary to obtain the stress and strain history for critical locations of the component. Due to hyper-elastic material behavior and complex component geometry, measurements of mechanical quantities such as stress, strain, and strain energy density often necessitate the use of numerical methods such as finite element (FE). The finite element method has been a very useful tool for simulating the actual service performance of components at their design and development stage. This method can significantly help to reduce trial-and-error cycles, prototyping and testing efforts, and time to market and total cost [41]. Selection of a proper failure criterion is another key issue in analyzing fatigue life of rubber components. Many studies discuss capabilities of different criteria in determining fatigue life and correlation of the predicted lives with experimental lives. Mars and Fatemi [13] provide a literature survey of fatigue analysis approaches for rubber, as well as a review of the many factors that affect fatigue life of rubber [24]. Determination and realistic representation of material deformation and fatigue behavior is another important input to accurate FE analysis and life prediction of rubber components. Rubber mounts are often used as vibration isolators in vehicles. They provide damping for high-amplitude as well as low frequency road induced vibrations, and in the case of engine mount, low-amplitude high frequency engine induced vibrations. Li et al. [42] predicted fatigue life of a rubber mount by obtaining material properties and utilizing the maximum principal strain at the critical region as a fatigue damage 20

41 parameter determined from FEA. To validate their predictions, they performed constant amplitude fatigue tests in load control and in directions perpendicular to the axial direction. In another study of an engine rubber mount by Kim et al. [43] the maximum Green-Lagrange strain and the maximum energy density were used as fatigue damage parameters for life predictions. Load versus Green-Lagrange strain relation was determined by FEA of the engine mount and the material fatigue behavior was derived based on fatigue tests of dumbbell-shaped specimens. Based on their load-controlled constant amplitude fatigue tests of rubber mount they concluded that Green-Lagrange strain is a better fatigue damage parameter than strain energy density for component life predictions. Woo et al. [2] performed fatigue life predictions of natural rubber components by using the Green-Lagrange strain at critical location determined from FEA. Fatigue tests under displacement control and for different values of mean displacement and amplitude were performed. They found their predictions to agree fairly well with the experimental fatigue lives. Fatigue failure analyses of rubber springs, which are widely used as anti-vibration components, were conducted by Luo and Wu [44]. They showed that a nonlinear quasi-static FE simulation capable of modeling geometric and material nonlinearities can effectively be used for product design and failure analysis. Luo et al. [45-47] analyzed fatigue life of anti-vibration rubber springs used in a rail vehicle suspension system with the aid of FEA to obtain the stress contour and using the S-N curve obtained under uniaxial loading for fatigue life predictions. They used the Mooney-Rivlin hyper-elastic 21

42 material model and proposed a three-dimensional effective stress criterion describing an ellipsoidal failure envelope for fatigue failure based on principal stress values. Verification tests of this approach carried out on two types of rubber springs in the longitudinal direction under uniaxial loading indicated good agreement with the predictions both for crack location and for crack initiation orientation [47]. Zhao et al. [48] analyzed a rubber mount using FEA. They determined the main reason for fatigue cracking was the stress concentration at the interfaces between the rubber and metal layers, which was also observed in their constant amplitude mount bench testing. They then modified the geometry of the mount by dig grooving in the rubber layers and changing the shape of the inner bushing. This resulted in a reduction of maximum strain due to lower stress concentration and, therefore, improved fatigue life. Takeuchi at al. [49] devised a procedure for endurance fatigue testing using optimum test piece geometry for fatigue testing. The geometry was designed by FEA with the aim of producing the same maximum tensile strain experienced by the rubber component in bench endurance tests. The experimental fatigue life of the material was shown to have a close correlation with component experimental results. Lee and Kim [50] attempted to minimize both the weight and maximum stress of engine rubber mount to maximize the fatigue life cycle subjected to constraints on the static stiffness of the mount. Sundararaman et al. [51] evaluated fatigue crack growth of V-ribbed belts by using fracture mechanics and FE. They used a power-law relation to characterize the fatigue crack growth rate based on J-integral as well as sub-modeling capability of the ABAQUS software, which eases the evaluation of large models by concentrating on the 22

43 critical location in the local sub-model. In their study the J-integral analysis was performed by using hyper-elastic strain energy density function of Ogden. 2.6 Damage accumulation and cycle counting in variable amplitude loading In most component and structures variable amplitude loading is typical in service. In order to analyze variable amplitude loadings, a cycle counting method is used to disintegrate the loading events to constant amplitude loads. Then by using a cumulative damage rule, the amount of damage caused by each individual cycle is defined and the damage is accumulated to predict failure or to evaluate remaining life. Steinweger et al. [52] developed a test time reduction method for rubber parts by using Rainflow filtering to reduce the length of the load block by removing non-damaging cycles. In multi axial loading it is important to preserve the phasing angle between load channels and their model accounts for this aspect. They suggest conventional methods used for steel parts are not applicable to rubber parts due to their nonlinearity at higher load amplitude and damping properties at higher frequencies. Ayoub et al. [28] mentioned that the damage rules used for rubber are generally based on experimental data which were obtained for specific loading cases and materials. Because of its simplicity, Miner s rule [53] is widely used for variable amplitude loading applications. This rule is applicable to loading blocks where the sequence of loading is not important. Klenke and Beste [54] applied the linear damage rule (LDR) to predict rubber fatigue life of a rubber-metal mount. Their results show that the Miner s linear damage rule is a reliable tool for fatigue life predictions. 23

44 Sun et al. [55] studied the effect of step loading sequence on residual strength for different natural rubber and styrene butadiene rubber (SBR) compounds and reported that Miner s rule is not applicable to multilevel loadings. They used two strain levels to investigate the effect of loading sequence. The step-up sequence showed a higher crack growth rate and larger cracks and lower tearing energy, compared to the step-down sequence. The results also indicated that this effect is larger for longer blocks of load and higher strain levels. Flamm et al. [56] studied the effects of very high loads on fatigue life of elastomeric materials made of natural rubber. By using a spectrum of two different amplitudes, they concluded that Miner s LDR rule is applicable and valid for damage accumulation when the loading block contains few excessively high loads. They used Lagrange strain as the damage parameter and concluded that the application of Miner s rule gives damage ratios of close to one. They also found that the combination of small number of cycles of very high load amplitude and large number of low load amplitude cycles could give reasonable values for damage sum close to optimum value of 1. In another study by Flamm et al. [57], it was shown that in many cases LDR is suitable for use as a cumulative damage rule in rubber parts. They conclude that tests with different sequences, mixed signals and signals resulting in a rotation of maximum principal stress of almost 60 had little influence on the fatigue lifetime. For multiaxially loaded rubber parts, Flamm et al. [40] propose discretizing the continuous signal by a level crossing cycle counting method for each loading channel. Then they constituted the 24

45 stress amplitude history and performed Rainflow cycle counting on these alternating points. Harbour et al. [37] studied the effect of VA multiaxial loading on rubber specimens made of two types of rubber materials, natural rubber and SBR. They concluded that the VA loading prediction results by using Miner s linear damage rule gave reasonable predictions compared to the experiments for NR, while the predicted results did not agree well for SBR, although the predicted values were still within a factor of three of the experimental fatigue lives. In another study, Harbour et al. [18] developed a linear crack growth model equivalent to Miner s linear damage rule. This model equates the crack growth rate for variable amplitude loading to the sum of the constant amplitude crack growth rates for each individual cycle. They studied the effects of R ratio, load level, load sequence and dwell period on crack growth rates of planar tension specimens. They used the Mars-Fatemi model [17] to accurately account for the effect of R ratio. They found that changing R-ratio, load level and load sequence did not significantly affect crack growth rates for repeated block test signals. Adding short dwell periods at the near zero stress level between loading blocks produced faster crack growth rates in both NR and especially SBR. This was explained in terms of the time-dependent recovery in the rubber microstructure at the crack tip which could increase the localized stress state at that point. Roland and Sobieski [58] studied some aspects of variable amplitude behaviour with annealing periods in a strained state for rubber material. They used pre-cycles and then annealed the specimens either with or without annealing strains for different designs of natural rubber and synthetic variant rubber to evaluate the effect on fatigue life. They 25

46 concluded that annealing of polyisoprene-based elastomers can either improve or worsen the deterioration properties based on the deformation type of the rubber during annealing period. These changes were more severe under conventional fatigue loads. Kim et al. [59] studied fatigue life prediction methodology of automotive rubber components under variable amplitude loading. They performed displacement-controlled fatigue testing and used maximum Green-Lagrange strain as a damage parameter. The SAE transmission load history [60] was used in their study. Displacement-controlled testing of dumbbell specimens provided the relationship between displacement and fatigue life. They then used FE analysis results by utilizing Ogden strain energy density function to relate the maximum Green-Lagrange strain at the critical component location to displacement, and subsequently to fatigue life. Racetrack cycle counting method was used to reduce the complex load history. The results showed that predicted fatigue lives based on Miner s rule were within a factor of two, compared to the experimental lives. 26

47 Chapter 3 Material Characterization and Fatigue Behavior 3.1 Introduction This chapter presents the experimental program for characterizing material properties and fatigue behavior using simple geometry specimens under constant amplitude loading. The material used and the method of making specimens as well as testing equipments used for testing are described. Different test procedures for obtaining constitutive behavior response and fatigue initiation life and crack growth properties are explained in detail. Applicability of Miner s cumulative damage rule is also investigated by using variable amplitude loading. The experimental results obtained and prediction methodology used for variable amplitude crack initiation experiments are discussed. 3.2 Material and specimens A filled natural rubber material with 21% carbon black and 9.5% plasticizer was used for specimen experiments. The compound used for specimen tests was identical to that used for the cradle mount manufacture. 27

48 The stress states of simple tension and planar tension are often used to characterize the deformation behavior. Simple tension specimens were cut by using specimen cutting dies according to ASTM standard D [61]. The test specimen used in this study had an hourglass shape specimen with 1 mm thickness and dimensions shown in Figure 3.1(a). Specimen preparation conditions are described in the aforementioned ASTM standard. For planar tension deformation and crack growth tests, a wide rectangular specimen of 1 mm thickness with dimensions shown in Figure 3.1(b) was used. For crack growth specimen, a pre-crack is cut at mid-length, as shown in Figure 3.1(b), but for cyclic deformation test no pre-crack was cut. 3.3 Testing equipment An Instron closed-loop servo-hydraulic axial load frame, as shown in Figure 3.2, in conjunction with a digital controller was used to conduct the experiments. The capacity of the load cell was 5 kn. 3.4 Specimen test methods and procedures Deformation behavior characterization Monotonic and cyclic deformation curves properties are typically needed for FE modeling and stress-strain relations for fatigue analysis. Phenomena associated with elastomers, such as Mullin s effect, can also be evaluated. Extension ratio or stretch ratio ( ) is commonly used for finite strain analysis and is defined as the ratio of the extended length (L) to the original length (L 0 ): 28

49 L (3.1) L 0 The relationship between engineering strain (ε) and stretch ratio is given by: 1 (3.2) As the rubber is an incompressible material, there is no change in volume under deformation, V 0. This results in the determinant of the deformation gradient to be one, which in turn results in the following relation for principal stretch ratios: 1 (3.3) Figure 3.3 shows the stretch and stress states for each specimen condition under uniaxial loading. For simple tension condition, uniaxial state of stress is present, with a multiaxial stretch state with the longitudinal stretch value of and two equal transverse stretch values of -1/2 to satisfy the incompressibility condition. Planar tension specimen is under plane stress condition with longitudinal and transverse stresses. The stretch in the width dimension is 1, indicating there is no strain in this direction. A stretch of in loading direction and 1 exist in the out of plane direction. It should also be mentioned that due to the incompressibility condition, Poisson s ratio is 0.5 and the initial modulus of elasticity (E) is approximately equal to three times the shear modulus (G), E = 3G Constant amplitude crack initiation testing Cracks often nucleate from pre-existing flaws in the compound (i.e. such as undispersed carbon black agglomerates), due to processing defects (such as contaminations, voids, and molding flaws), or at stress concentrations. In the context of mechanical design and fatigue analysis, crack initiation life is referred to the life involved in growing a crack from the pre-existing flaw to a small macro-crack typically on the 29

50 order of 1 mm. Strain is commonly used for crack initiation life analysis as an ideal fatigue damage quantification parameter, due to ease of determining displacements in elastomers. The specimen geometry used in crack initiation tests is the geometry shown in Figure 3.1(a). This type of specimen is designed in such a way that when a crack grows to a length on the order of 1 mm, there would be little remaining life to fracture. Therefore, these tests characterize fatigue life for a crack on the order of 1 mm, although the criterion for defining fatigue nucleation life in the test is the number of cycles in which the specimen ruptures completely according to ASTM standard D 4482 [61]. Multiple specimens can be used for each test. The test setup with five specimens used in this study is shown in Figure 3.4. Tests were performed in displacement control and three strain R ratios of 0.02, 0.1 and 0.2 were chosen for testing. As the specimen cannot support compression load, these R ε ratios represent tension-tension loading condition. For each R ε ratio, three levels of maximum strain were tested. The testing frequency used for all conditions was 1 Hz, except those with peak strain values higher than 2.75 which utilized testing frequencies of 0.5 Hz Crack growth testing Crack growth testing of elastomers typically uses a pre-cracked planar tension specimen, as shown in Figure 3.1(b) with a pre-crack length of 25 mm. While the original application of fracture mechanics approach to rubber was to predict static strength, in the late 1950s, Thomas extended the approach to analyze the growth of cracks under cyclic loads in natural rubber [62]. Since rubber is a nonlinear elastic material, energy release 30

51 rate is often used as the crack driving parameter, rather than the stress intensity factor which is typically used for linear elastic nominal material behavior, such as in metals. This parameter relates the global specimen loading to the local stress and strain fields at the crack tip, and is also applicable to finite strain materials, such as elastomers. Energy release rate is the change in the stored mechanical energy du, per unit change in new crack surface area da, also often called the tearing energy T, and given as: T du (3.4) da Under cyclic loading, the maximum energy release rate achieved during a cycle is related to the crack growth rate. Because of a simple relationship between energy release rate and strain energy density and the fact that energy release rate is independent of crack length in planar tension specimen, this specimen is an ideal choice for fatigue crack growth experiments. In the planar tension specimen, shown in Figure 3.1(b), energy release rate depends only on the strain energy density (W) remote from the crack and specimen edges, and the specimen height (h): T Wh (3.5) This relation makes the planar tension specimen geometry quite practical for use in a fatigue crack growth characterization of elastomers. In this specimen, a state of plane stress under uniaxial tension is present, as shown in Figure 3.3. The strain energy density (W) is the area under the loading stress-strain curve for a stable cycle for each peak strain level. The energy release rate and crack growth rate are independent of crack length in the planar tension specimen. Stress-strain hysteresis loops from an uncracked test specimen can be used to obtain the test signal parameters for the 31

52 fatigue crack growth rate. Numerical integration methods (such as Simpson s rule), as explained in the ASTM standard D [61], can be used to calculate the area under the stress-strain curve from the experimental data. For a constant displacement or strain, strain energy density is constant and so the crack growth rate would be constant with repeated cycles of the same amplitude. For non-fully relaxed conditions (R> 0), the peak strain energy density ( W max ) is the sum of the area under the loading stress-strain curve (WL) plus the strain energy density at the minimum strain level ( W ): min W max. W W (3.6) L min. Control parameters for crack growth test with planar tension specimen are the amplitude and mean values of the applied displacement. Figure 3.5 presents the maximum strain energy density as a function of maximum engineering strain obtained from uncracked test specimens. Tests were performed with both fully relaxing conditions ( R 0), as well as at three R W ratios of 0, 0.05, and 0.10, while changing peak strain W values from 40% to 65%, as listed in Table 3.1. To pre-condition the material to avoid transient Mullin s effect, stable deformation curves were defined at the 128 th cycle of planar tension uncracked specimen, where the peak stress was 10% lower than the first cycle stress (see Figure 3.6(b)). For the cracked specimen 110% of maximum strain was used as pre-conditioning load for 500 cycles at each strain level used to minimize the transient deformation response as well as to produce a natural crack tip. A single test specimen can produce results for multiple crack growth tests as long as the crack length and remaining specimen length are sufficiently longer than the 32

53 specimen height of mm. Due to the transient softening, however, it is necessary to conduct tests on the same specimen in ascending order of maximum strain level to make sure that the higher levels of load would not affect the lower load level test results. The desired range of fatigue crack growth rates were between 6 10 and 3 10 mm/cycle. Using a cycling frequency of 5 Hz, it takes about 5 hours to grow the crack 0.1 mm at a crack growth rate of 6 10 mm/cycle. A traveling microscope which can track the growth of the crack tip along the crack line can be used to measure the crack growth. Crack growth data based on a minimum interval of approximately 0.1 mm of crack growth was used for measurements. After obtaining several data points for each test condition, the R ratio was changed by increasing the minimum displacement. After all R ratio test data were obtained, the value of peak strain was then increased to obtain crack growth rate at the next strain level. If the crack grew irregularly or at an inclined angle, re-cutting was performed by a razor blade, followed by some initial cycles to initiate a natural crack tip Variable amplitude crack initiation testing The test specimen geometry used is the same as that used for constant amplitude loading to characterize the material fatigue behavior with a thickness of 1 mm and dimensions shown in Figure 3.1(a). In order to investigate the effect of cumulative damage from variable amplitude loading on fatigue crack initiation life, a series of tests on the simple tension specimen geometry were performed with three peak load levels and three mean load ratios. This helps to examine the applicability of linear damage rule under simple tension stress state without the complication of multiaxial stress effects due to stress concentration at the 33

54 critical location of the component. A random load history [63] depicted in Figure 3.7(a) was used. This load history consists of 16 individual events, and damage from each event calculated based on Rainflow cycle counting method is tabulated in Figure 3.7(b). The mean and amplitude values listed in the table are calculated using maximum and minimum values of ±100. The desired maximum and minimum strain or stress values in the block are reached by scaling the history. This random loading history was used for both specimen tests in displacement control and component tests in load control. 3.5 Specimen experimental results and observations Monotonic and cyclic deformation behavior results To determine the stress-strain behavior of the material, tests were conducted in displacement control. The applied displacement was then converted into gauge section strain and the measured load from the load cell, which had a maximum capacity of 5 kn, was used for stress calculation. The monotonic deformation curve is shown in Figure 3.8, indicating a markedly non-linear behavior. A cyclic incremental test on simple tension specimen in displacement control with the sinusoidal waveform was performed to obtain the cyclic stress-strain curve. Due to Mullin s effect, stabilized cycle s data is used for each strain level (i.e. after about 20 applied cycles) and all tests are performed in ascending order of displacement amplitude. Figure 3.9(a) shows the resultant hysteresis loops, where the area between loading and unloading curves represents energy dissipation during the cycle. Hyperelastic constitutive models are used to represent the nonlinear elastic deformation curve. A Fictitious curve for simple tension condition is obtained by using a 34

55 least squared fit polynomial to the end points of stress and strain from each loop [64]. This curve is depicted in Figure 3.8 and the mathematical representation of maximum engineering stress ( S max condition is given by: ) versus maximum engineering strain ( e max ) for simple tension S 2 max.25emax e max (3.7) The superimposed plot of monotonic and cyclic simple tension curves illustrate that using monotonic properties in a cyclic loading application can underestimate the level of strain which can be present. It is, therefore, important to obtain and use cyclic material properties in fatigue life analysis and applications. Cyclic incremental step tests were also conducted with the planar tension specimen. Figure 3.9(b) shows the hysteresis loops from the 128 th cycle from each incremental step. The 128 th cycle is used as the stable cycle, subsequent to initial softening due to Mullin s effect. As can be seen, after the initial straining the material does not return to zero strain at zero stress due to some degree of permanent deformation. A curve for planar tension condition is also depicted in Figure 3.8. The nominal stressstrain relationship for cyclic planar tension test is given by: S 2 max.84emax e (3.8) max According to Figure 3.8, cyclic simple tension deformation curve is less stiff than cyclic planar tension at strain levels below 0.8. This is consistent with other deformation behavior characterization studies done by Sharma [65] and Mars-Fatemi [11]. However, at higher strain values the simple tension curve is stiffer than planar tension specimen curve. This could be mainly due to the higher amount of strain crystallization in simple tension, which in turn causes higher stiffness. In simple tension specimen there is no 35

56 lateral constraint to cause specimen thinning and therefore the polymer network chains could become sufficiently aligned. The choice of the curve to use in FE simulations depends on the stress state at the critical location of the component being analyzed. The initial softening phenomenon, also called the pre-stretch or preconditioning effect, is widely referred to as the Mullin s effect [66]. Subsequent loadings of equal or lesser magnitude rapidly tend towards a steady state, nonlinear elastic response. This effect is considered to be a consequence of the breakage of links inside the material and both filler-matrix and chain interaction links are involved in the phenomenon [67]. The Mullin s effect is transient and is exhibited primarily by filled rubbers [68]. Figure 3.6(a) provides an illustration of the Mullin s effect as the stress level drops for each successive loop. The stress-strain loop stabilizes after 3 to 30 cycles of loading for most elastomers. For a higher maximum strain, the initial softening is larger, as can be seen from Figure 3.6(b). This Figure also shows that by increasing peak strain, it takes a longer time for the material to show the stabilized response. Therefore, due to the load history dependence associated with the Mullin s effect, peak loading should be a key consideration in fatigue analysis of rubbers, in addition to the load amplitude and the mean load Constant amplitude crack initiation test results Test conditions and result from these tests are tabulated in Table 3.2. The ratio of maximum fatigue life to minimum fatigue life in each testing condition ( N N ) f, max f, min which is an indicator of data scatter is also shown in this Table. This ratio for all crack initiation tests was in the range of 1.11 and 2.58, which is quite reasonable for elastomeric material fatigue tests. 36

57 A mean value for each strain ratio and maximum strain was calculated and used in curve fitting and data analysis. The reason for using a mean life value for each test condition rather than direct fit of all the data is so that there is equal contribution of each test condition in test data analysis. Figure 3.10 shows raw data of maximum strain versus crack nucleation life for all tests at different R rates. It is observed that the power trend lines shown fit the crack initiation data well. The best-fit lines use the least squares fit method with fatigue life as the dependent variable. The fatigue life equations based on the different strain R ratios (denoted by R ) are given by: for R (3.9) 0.30 max 28.9 ( N f ) for R (3.10) 0.25 max 17.4 ( N f ) for R (3.11) 0.14 max 9.41 ( N f ) From Figure 3.10 it can be seen that the effect of R ε = 0.2 cycles is to increase the nucleation life. The life improvement is very significant at low strain, about an order of magnitude, but less important at high strain, less than a factor of two. This is in contrast with metals where a tensile mean load has detrimental effect on fatigue life. This life improvement in natural rubber is mainly due to strain crystallization. By applying a nonzero minimum strain, rubber does not come back from crystalline state to amorphous rubbery state. This crystalline state in which the polymer chains are aligned highly in the loading direction increase resistance to crack growth, therefore fatigue life would be longer. 37

58 The influence of strain ratio can be estimated by an empirical relationship by the Mars-Fatemi model [17] that relates fatigue failure at a given life for R 0 conditions to fatigue failure at the same life for R 0 condition by the following relation: max,0 G( R) G(0) max, R G( R) [1 ] G(0) c (3.12) G( R) R (3.13) where max, R is the maximum engineering strain at a given R ratio, max, 0 is the equivalent maximum engineering strain at R 0, G (R) is the power law exponent, and Figure 3.11 shows correlation of all crack nucleation data on a single plot for all strain ratios. The equivalent R = 0 maximum strain ( max, 0 ) is then given by: c max,0 3 (1 74R ) max, R 3 ( 74R ) 6.25 (3.14) Once max, 0 is determined for a given R ε loading condition, it is then used in Equations (3.9)-(3.11) to calculate the fatigue crack initiation life Crack growth tests results Fatigue crack growth rate (da/dn) is obtained by fitting a linear relationship to the crack length versus cycles data and determining the slope of the linear fit. Crack growth data fits at different energy release rate ratios and peak strain levels are shown in Figure The test conditions used for the constant amplitude fatigue crack growth experiments produces crack growth rates in the region of crack growth that can be characterized by a power-law relation. The coefficient and exponent of this relation are calculated from R 0 data as: T 38

59 da dn ( Tmax ) (3.15) Note that for planar tension specimen, due to the relation given by Equation (3.5), R T = R W. In strain crystallized rubbers, such as the natural rubber used in this study, increasing the minimum strain or energy release rate ratio has a significant beneficial effect on fatigue life compared to R 0 condition, as fatigue crack growth rate T decreases. This behavior is depicted in Figure 3.13 where the fits to data show that increasing R ratio causes slower crack growth rate and, therefore, being beneficial to fatigue life. The power law equations for the two energy release rate ratios higher than zero are obtained as: da dn da dn ( Tmax ) for T ( Tmax ) for T R (3.16) R (3.17) By using Mars-Fatemi model [17], it is possible to reduce fatigue data taken at various levels of R ratio to a single equivalent R = 0 curve. This is achieved by plotting the crack growth rates against the equivalent R = 0 maximum energy release rate, T max, 0. The Mars-Fatemi model is given by: T da max F ( R) rc ( ) (3.18) dn Tc wheret 10 kj/m 2 is based on Lindley s estimate for fatigue crack growth of natural c rubber [17]. The critical crack growth rate for this condition is defined as r mm/cycle based on R = 0 crack growth Equation. The equivalent R = 0 c maximum energy release rate ( T max, 0 ) is given by: 39

60 T max,0 F ( R) F (0) max, R T ( R) (1 ) T (0) c T T (3.19) where: F4 R F R) F e T ( (3.20) 0 This form of F(R) is used when there are limited data points and it adds just one variable to be calculated to the material properties obtained from R = 0 analysis. Based on Lindley s model and using energy release rate ratios of 0 and 0.10 constants, the values of F 0, obtained from R = 0 power law fit data for this material, and F4 are calculated as 2 and 8.27, respectively. Figure 3.14 shows correlation of all fatigue crack growth (FCG) data on a single plot for different R T ratios by the use of Mars-Fatemi model Relationship between crack growth and crack initiation approaches In a single edge cut simple tension specimen geometry the energy release rate depends on the gauge section strain energy density W, crack length a, and a stretch dependent parameter k [69]: T = 2kWa (3.21) By combining the power-law fatigue crack growth rate relation (Equation (3.15)) and the above equation and then integrating the resulting equation, the following relationship is obtained [13]: N [ ] (2 ) f (3.22) F F F F B kw a a 0 f where B and F are the fatigue power-law coefficient and exponent for R = 0 condition, respectively. If the initial flaw size a0 is much smaller than the critical flaw size, a f, the life becomes nearly independent of the critical flaw size: 40

61 1 1 1 N f (3.23) F F 1 B(2kW) a F 1 0 Effective flaw sizes in the range of 0.02 mm to 0.06 mm were observed in a study by Lake and Lindley, which covered different polymer types, and various fillers, curatives and other compounding variables [70]. In order to compare the relationship between crack initiation and crack growth approaches, Equation (3.23) was used to obtain fatigue life based on the crack growth approach and to compare with the fatigue life based on the crack initiation approach. By assuming a mm, utilizing B 4 10 and F = 2 as crack growth rate coefficient and exponent, respectively, in Equation (3.23), and by assuming k 2 (in the stretch range studied), relatively good agreement (i.e. about a factor of two in fatigue life) between the R = 0 crack initiation life and crack growth life is obtained, as shown in Figure Therefore, the crack growth approach could be used as a total life approach, based on growth of pre-existing flaws to failure. Correlation between crack nucleation life obtained from simple tension specimen and crack growth life obtained from planar tension specimen was studied by Mars and Fatemi [68] and they also found good agreement between the results. It should be mentioned that a change of initial assumed crack length from 0.02 mm to 0.04 mm resulted in very small change in the obtained results. 5 If N f, represents the fatigue life associated with a long final crack size (i.e. a crack size much longer than the initial crack size of 0.02 mm), the following relation based on Equation (3.22) is obtained [71]: N N f f, [1 a F a 1 a F 1 0 F 1 f ] 1 a a 0 f F 1 (3.24) 41

62 The failure crack nucleation size used for this study is a 1mm. Therefore, for the initial flaw size in the range observed by Lake and Lindley, it is estimated that the observed initiation life is in the range of 94% to 98% of the life if crack grew in an infinitely wide specimen. Thus, the crack initiation results obtained are nearly geometry independent. f Variable amplitude crack initiation tests analysis methodology In order to be able to predict the fatigue damage, there are several general prerequisites. First, it is necessary to identify the fatigue critical location and map the load history for this location by using a proper cycle counting method. A fatigue damage parameter is then needed to compute damage for the loading cycles identified by the cycle counting method. Finally, by using a proper cumulative damage criterion the damage is integrated and the fatigue life is calculated for the critical location of the specimen or component. For complex geometries, such as industrial components, a numerical method such as FEA is also necessary. The component life prediction procedure can utilize crack initiation and/or crack growth approaches. Constant amplitude specimen test data are used as the basis of fatigue life curve for crack nucleation, while specimen crack growth data and the initial natural flaw size in the material are used for crack growth or total fatigue life of the component. The objective of all cycle counting methods is to be able to compare the effect of variable amplitude loading to fatigue data and properties obtained from constant amplitude load cycles. Many cycle counting methods have been proposed, with the most popular method being the Rainflow method proposed by Matsuishi and Endo [72]. Miner s LDR [53] is the simplest form of cumulative damage rule. However, load 42

63 sequence effect or interaction between cycles is not accounted for in this rule [37].The mathematical representation of Miner s linear damage rule is as follows: N N i f i N N 1 f 1 N N 2 f 2 Ni 1 (3.25) N f i where N i is the number of applied cycles at a given load level and N fi is the constant amplitude fatigue initiation life at the same load level. Continuum mechanics parameters such as different definitions of stretch or strain, stress, and energy are used as damage quantification parameters. Cauchy stress, maximum Green-Lagrange strain and strain energy density are some examples. Due to the ease of measuring strain, maximum principal strain is used for life predictions in this work. This agrees with the experimental observation of fatigue cracks usually initiating and growing perpendicular to the maximum principal strain direction [31]. The strain R ratio (R ε ) effect during each loading cycle at the fatigue critical location can be accounted for by calculating a maximum equivalent R = 0 engineering strain (ε 1,max,0 ) based on the Mars-Fatemi R ratio material model [17] as discussed by Equation (3.14). By using the material strain-life, as shown in Figure 3.11, represented by the equation below, crack initiation life is then predicted at each equivalent strain level: ,max, 0) N ( (3.26) f Crack initiation life based on blocks of loading is then calculated based on the linear damage rule, as expressed by Equation (3.25) Variable amplitude crack initiation tests results and analysis Table 3.3 tabulates the variable amplitude fatigue crack initiation experimental results for the simple tension specimens. While the first crack to traverse the specimen 43

64 cross section determines the experimental fatigue life, many cracks were visible on the specimen surface, similar to that observed in constant amplitude loading in [68]. As can be seen from this table, scatter of fatigue lives for each test condition is within a factor of about two. Table 3.3 also shows the life prediction results. Rainflow cycle counting method in conjunction with the linear damage rule was used for life predictions. Each cycle s damage was calculated based on constant amplitude specimen behavior by using Equation (3.14) to account for strain R-ratio and Equation (3.26) for fatigue life. Experimental fatigue life versus predicted fatigue life is depicted in Figure Based on the comparison of predicted and experimental fatigue crack initiation lives, it is observed that most of the predictions (73%) fall within a scatter band of two and nearly all the remaining experimental data fall within a predicted factor of five. Considering a scatter factor of two in duplicate experimental data, this can be considered reasonable prediction of the crack initiation life based on LDR. The effects associated with a rest period between blocks of cyclic loading on the material fatigue behavior was investigated by 33 seconds hold periods (dwell period) between load blocks in a R ~ 0 strain ratio test with strain range of 3.14, with the results included in Table 3.3. As can be seen from comparison of fatigue lives between tests with or without rest period, no detrimental effect on fatigue life is observed. In variable amplitude crack growth behavior of another filled natural rubber in [18], dwell periods effect, although not significant, increased the fatigue crack growth rate by a factor of two. The effect of test frequency or loading rate on the variable amplitude fatigue crack initiation life was investigated by using a loading rate three times faster than the original baseline test. The results are also included in Table 3.3. As can be observed, 44

65 fatigue life results at higher and at lower rates are nearly identical, indicating no significant effect of test frequency on the fatigue life for the range of frequency investigated. 3.6 Conclusions A key ingredient of fatigue analysis and life prediction of elastomeric components is relevant material properties. These properties include deformation behavior under different stress states, crack initiation life, and crack growth rate properties. Specific experimental procedures and data analysis techniques were presented and discussed to obtain each of these material characterizations. Two simple specimen geometries can be used for deformation and fatigue behavior characterizations of elastomers: simple tension and planar tension specimens. In the simple tension specimen a uniaxial state of stress with a multiaxial stretch state is present, while the planar tension specimen is under plane stress condition with longitudinal and transverse stresses. As monotonic and cyclic deformation behaviors can be vastly different, it is important to obtain and use cyclic deformation properties in fatigue life analysis and applications. Cyclic incremental tests on simple tension and/or planar tension specimen can be performed to obtain the stabilized cyclic stress-strain curve. The choice of the curve to use in FE simulations depends on the stress state at the critical location of the component being analyzed. Due to the load history dependence associated with the Mullin s effect, peak stress (or strain) should be a key consideration in fatigue analysis of elastomers, in addition to the stress (or strain) amplitude and the mean stress. Fatigue crack initiation tests can be performed with the simple tension specimen geometry in displacement 45

66 control and with different strain ratios (R ε ratios) representing tension-tension loading conditions. Best fit lines in log-log scale, where the fatigue life is treated as the dependent variable, represent maximum strain versus crack nucleation life for each strain ratio. The effect of R ε > 0 is to increase the nucleation life and is significant at low strains. This is in contrast to metals and this effect in natural rubber can be attributed to strain crystallization. Energy release rate is often used as the crack driving parameter in characterizing elastomers fatigue crack growth behavior. A pre-cracked planar tension specimen is typically used for fatigue crack growth tests since energy release rate is independent of crack length for this specimen geometry. Therefore, a single specimen can produce results for multiple crack growth tests with different loading conditions and R T ratios. A power-law relation can be used to describe crack growth rates in terms of the maximum energy release rate. The crack growth approach could be used as a total life approach, based on growth of pre-existing flaws to failure. The Mars-Fatemi model can be used to correlate test results from different R ratio conditions. This model can be used to obtain an equivalent maximum strain in crack initiation tests, or an equivalent maximum energy release rate in crack growth tests, to account for the effect of R ratio. Most of the life predictions for specimen random loading history tests based on Rainflow cycle counting method in conjunction with the linear damage rule were within the factor of two scatter bands. The R-ratio equation for fatigue life based on constant amplitude material data was found to be applicable to random loading. The effect associated with a rest period (dwell period) between blocks of cyclic loading on the 46

67 material fatigue behavior was found to be insignificant. Also, loading rate in the range investigated did not have a significant effect on fatigue life in specimen fatigue crack initiation tests. 47

68 Table 3.1: Constant amplitude fatigue crack growth test conditions min max R R W da/dn T max Freq. (mm/cycle) (kj/m 2 ) (Hz) E E E E E E E E E E E E E E E E E E E E E Table 3.2: Crack nucleation test conditions and results Freq Number of Mean N min max R f Standard a N (Hz) Tests (cycles) Deviation f,max / N f,min ,528 12, , , ,940 6, ,321 3, ,966 30, ,641 4,

69 Table 3.3: Simple tension specimen fatigue test conditions and results with the random loading history used. ε min ε max ε R ε Load block time duration (sec) Number of blocks to failure, B f Test #1 Test #2 Test #3 Test #4 Test #5 Mean Std. Dev. B fmax /B fmin Predicted blocks to failure * ,941 1,945 2,023 2,064 2,385 2, ,279 1,364 1,412 1,767 1,863 1, This load block time consists of 33 seconds of loading time block and 33 seconds of dwell period between each block at near zero strain. * This load block time is one third of normal block duration for high rate effect evaluation. 49

70 (a) (b) Figure 3.1: Specimen geometry and dimensions for (a) simple tension, and (b) planar tension. Specimen thickness is 1 mm for both geometries. 50

71 Figure 3.2: Axial servo-hydraulic Instron frame used for specimen testing. 51

72 Figure 3.3: Simple tension and planar tension specimens with the corresponding stretch and stress states [36]. 52

73 Maximum SED (MPa) Figure 3.4: Crack initiation test setup showing five specimens and grips y = 0.439x x R 2 = Maximum Engineering Strain Figure 3.5: Maximum strain energy density versus maximum engineering strain obtained from uncracked planar tension specimen. 53

74 (a) (b) Figure 3.6: (a) Mullins effect showing initial transient softening in planar tension specimen at 133% maximum strain, and (b) stabilization of stress with applied cycles in displacement-controlled incremental step cyclic deformation tests of planar tension specimen at different maximum strain levels. 54

75 (a) cycle max min range mean % Damage 2-3-2' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' (b) Figure 3.7: (a) Loading history used for random loading tests of simple tension specimens in displacement control and of cradle mounts in load control, and (b) Rainflow cycle count for max/min values of ±100 with relative damage distribution from each cycle [63]. 55

76 Figure 3.8: Superimposed plot of monotonic and stable cyclic curves for simple tension and planar tension specimens. 56

77 Engineering Stress (MPa) Engineering Stress (MPa) Engineering Strain (a) Engineering Strain (b) Figure 3.9: Stable cyclic stress-strain loops from incremental step tests at different peak strain levels from (a) simple tension test, and (b) planar tension test. 57

78 Maximum Engineering Strain Re=0.02 R ε = 0.02 Re=0.10R ε = 0.10 Re=0.20R ε = Crack Nucleation Life (Nf), cycles Figure 3.10: Crack nucleation life as a function of peak strain in simple tension tests. 58

79 Figure 3.11: Equivalent fatigue life as a function of equivalent maximum strain by using Mars-Fatemi R-ratio model. 59

80 (a) (b) (c) (d) Figure 3.12: Crack length versus cycle linear fits from crack growth test at (a) R T 0, (b) R 0, (c) R 0. 05, and (d) R T T T 60

81 Crack Growth Rate (mm/cycle) 1.00E-02 R T = 0 RT = 0 RRT T = 0.05 RRT T = 0.10 RRT T = 0 Power (RT = 0) 1.00E E E Maximum Energy Release Rate, T max,(kj/m 2 ) Figure 3.13: Fatigue crack growth rate data comparisons at different R-ratios. 61

82 Crack Growth Rate (mm/cycle) 1.00E E-03 R T = 0.05 RT=0.05 R T = 0.10 RT=0.10 R T = 0 RT=0 R T = 0 RT=0, specimen1 da/dn = (T max,0 ) E E Maximum Equivalent Energy Release Rate, T max,0,(kj/m 2 ) Figure 3.14: Fatigue crack growth rate data correlations at different R-ratios based on Mars-Fatemi R-ratio model. 62

83 Figure 3.15: Comparison of fatigue lives obtained from crack initiation and crack growth approaches. 63

84 Figure 3.16: Experimental versus predicted fatigue lives in blocks to failure for simple tension specimen tests subjected to variable load history. 64

85 Chapter 4 Component Finite Element Model and Results 4.1 Introduction Because the measurement or analytical calculation of stress and strain for the complex mount geometry is difficult, a numerical method such as finite element method must be used to compute them. FE methods are commonly used tools for simulating the actual service performance of the components at their design and development stage. The nonlinear FE program ABAQUS was used to simulate the nonlinear and large deformation behavior of the mount. This section discusses the relevant details and aspects of the FE simulations including material deformation characterization, FE model construction, and the obtained results from the FE model. Numerical results which are obtained from FE simulations will then be used in fatigue life predictions. 4.2 Hyperelastic material deformation characterization There are some common approaches in FE software to deal with hyper-elastic materials. Rubber can be considered as a hyper-elastic material, showing highly nonlinear 65

86 elastic isotropic behavior with incompressibility. The relationship between stress and strain in a hyper-elastic material is generally characterized by the strain energy potential. Strain energy potential defines the strain energy stored at a point as a function of the strain at that point assuming the material is isotropic and homogeneous. There are two general approaches for isothermal rubber mechanical behavior, the kinetic theory which is based on statistical thermodynamics considerations, and the phenomenological approach which treats the material as a continuum regardless of its micro-structural and molecular nature [3]. The kinetic energy approach is based on statistical distribution of rubber molecular chains and assumes rubber elastic deformation is originated from the decrease in entropy resulting from the increase in applied deformation. For example, Arruda and Boyce proposed a three-dimensional model of eight chain network [73]. The phenomenological approach assumes that the elastic properties can be described in terms of a strain energy function W which is either defined as a polynomial function of strain invariants, or in terms of principal stretch ratios. Rivlin [74] proposed that the strain energy density function can be approximated by a power series in terms of strain invariants, which for an incompressible material it reduces to: cij i j 1 i j W ( I1 3) ( I 2 3) (4.1) where c ij are material constants and I 1 and I 2 are the first and second strain invariants given by: I , I 2 ( ) (4.2) 66

87 and 1, and 2 are principal stretches or extension ratios. Note that I 3 = ( ) 2 is the square of the ratio between the volumes of a material element in the deformed and undeformed states and for an incompressible material = 1 and, therefore, I 3 1. By using the first term of Equation 4.1, the well-known Neo-Hookean model is generated and by using both terms the Mooney-Rivlin form is generated [75]. Ogden [76] presented the strain energy density function based on a polynomial of principal stretches, which for incompressible materials is expressed as: W n i 1 2 i i i ( i i ) 3 (4.3) where μ i and α i are constants to be determined from experimental data. It should be noted that the aforementioned strain energy density functions assume the material behaves as a perfectly elastic material. This is an idealization of filled rubber behavior since such rubbers exhibit some degree of hysteresis, stress softening, and plastic deformation upon unloading. Of the several strain energy potential forms available in ABAQUS to model incompressible isotropic elastomers, the Marlow form was used in this study. In this case strain energy potential is constructed in a way to reproduce the test data exactly for a given deformation state, for example simple tension, and have reasonable behavior in other deformation states such as planar tension. In order to minimize undesirable nonlinearity, sufficient data points should be specified in the low, intermediate, and high strain ranges. The form of the Marlow strain energy potential which is based on deviatoric part of strain energy for an incompressible material is given by [75]: U U I ) U ( J ) (4.4) dev ( 1 vol el 67

88 where U is the strain energy per unit of reference volume with and U dev as its deviatoric part U vol as its volumetric part and J el is elastic volume ratio. The deviatoric part of the potential is defined by providing test data, while the volumetric part is not considered due to incompressibility [75]. The Marlow model does not assume any explicit form [77], as this function is calculated based on integrating the stress-strain curve up to a certain strain ε 1 which is a function of the first strain invariant, ε 1 = 1 (I 1 ) 1. It should also be noted that there is no curve-fit involved. If simple tension, planar tension and equi-biaxial tension data are all available, then the Ogden or Van-der-Waals [78] forms are more accurate in fitting the experimental results [75]. Strain in this study is defined as the nominal strain. Nominal strain can be expressed in terms of the principal stretch ratios as [75]: 3 i 1 i i T i V I ( 1) n n (4.5) where V T F F is the left stretch tensor, F is the deformation gradient, and n i s are the principal stretch directions in the deformed shape. Principal values of nominal strain show direct values of deformation and could be used in life predictions due to the fact that these values are the ratios of change in length to length in the initial shape. These strain values show the importance of deformation gradient in analyzing rubber-like material continuum mechanics terms. 68

89 4.3 Uniaxial FE model definition and specifications The FE model consists of half of the actual mount due to the symmetry of the mount geometry and uniaxial loading applied in a symmetry plane. The mount and the FE mesh are shown in Figure 4.1. To adequately model the half-mount geometry, a fine mesh with 14,355 elements was used. Load and/or displacement are applied to the reference point which is coupled with inner FE nodes of the mount at the interface with metallic part by utilizing the rigid body constraint capability of the software. In load control simulations half of the testing load was applied to the model due to the model symmetry. To prevent rigid body motion the outer portion of the mount was fixed. Proper element type, integration scheme and reasonable meshing strategy are all important factors to consider for appropriate modeling and simulation of large deformation rubber components. The fully hybrid formulation elements are recommended for use with hyper-elastic materials [75]. Wang et al. [41] showed that hybrid elements with full integration and lower-order interpolation show less distortion than higher order reduced integration elements and are suitable for large deformation simulation computations. ABAQUS/Standard has a special family of hybrid elements that could be used to model the fully incompressible behavior. The element type used in this work was the eight node linear brick (hexahedral) element. Hybrid element locking is another key issue in the large deformation simulation computations of 3-D models, which is caused by large element distortion. Element locking can be lessened by proper meshing and element selections. Too fine a mesh exhibits more sensitivity to element volumetric locking, especially in large strain areas. The mesh verification showed no error in element meshing. This tool verifies elements 69

90 aspect ratios, and larger or smaller face corner angles to assure there would be no inappropriate element distortion. Therefore, the mesh quality was quite good for performing FE analysis. For judging the quality of fit to experimental data a Drucker stability check is performed in the software [75] to assure the tangential material stiffness is positive definite. According to this criterion, the deformation of the body with time-independent properties under isothermal conditions is stable provided that the work done by infinitesimal increments of generalized forces associated with infinitesimal increments of displacements is positive [79]. Quasi-static simulation was performed for constant amplitude axial loading. For cyclic loading, the amplitude tool of the FE software was used, so that a loading cycle could be exactly simulated. The stress strain relationship of the natural rubber showed significant softening during initial straining cycles but stabilized after almost 20 cycles. This softening is associated with the Mullin s effect. Therefore, a stabilized cyclic stress strain response was used for determining the cyclic material properties. The stable loops at different strain levels are shown in Figure 3.9(a). Due to the application of stable deformation curve, initial softening (Mullin s effect) is inherently considered. Primary Hyper-elastic curve is constructed through fictitious curve points. This curve is obtained from peak strain and peak stress points of the different stabilized hysteresis loops [64], as shown in Figure 3.8. To evaluate mesh sensitivity, a refined mesh with 114,840 elements was utilized. Although the results changed by 19% for strain at the critical location for 550 N load, critical locations were the same for both analyses. The run time, memory usage and 70

91 output database file size for the refined mesh, however, increased 34, 11, and 8 times, respectively. In addition, the possibility of element locking at large strain area and under high load increases with the refined mesh. Therefore, it was decided to use the original mesh from a practical application viewpoint. In order to evaluate the displacement versus load input effect on simulation results two sets of simulation are performed, under load control and under displacement control. The displacement-controlled simulation used the displacements obtained from the load-controlled simulation. As expected, load, displacement, maximum principal stress, and strain values between the two analyses were identical. FE simulations assume non-linear perfectly elastic constitutive behavior, therefore, the same deformation curve for loading and unloading. This is not unreasonable with regards to actual component behavior obtained from actual component fatigue testing at midlife. The hysteresis loops shown in Figure 4.2 for two R ratios (i.e. minimum to maximum load ratios) and several load levels all indicate small amount of hysteresis during a stabilized cycle. Brief description of the model and the parameters used are summarized in Table Stiffness test results comparison with predictions A displacement-controlled monotonic compression test of the mount was conducted to compare with the FEA results. For FE simulation of this test condition monotonic stress-strain curve was used. Also, 30 pre-cycles were applied at the intermediate displacement level to account for initial softening before the monotonic test. Figure 4.3(a) shows stiffness comparison between the FEA and the test results indicating reasonable agreement between results of less than 20%. 71

92 Incremental step displacement-controlled cyclic tests at 1 Hz and 3 Hz with R ~ 0 were also performed on the cradle mount to compare with FEA results. In order to avoid Mullin s effects, 50 pre-cycles at intermediate displacement levels were applied prior to each displacement amplitude increment. At each displacement level 260 cycles were applied. The data from the last cycle are assumed to represent the stabilized behavior for that test level. FE simulations were based on the stabilized cyclic material behavior from specimen simple tension tests and SED function of Marlow, as described earlier. Figure 4.3(b) shows the comparison between the simulation and test results. As can be seen, the experimental cyclic curves agree well with FEA predictions. 4.5 Uniaxial constant amplitude FE simulation results The numerical simulation results show that multiaxial states of stress and strain, although proportional, are present at the critical location under axial load. Figure 4.4 shows the strain and stress history for R = 0.2 and load amplitude of 1,500 N during a sinusoidal load cycle. As can be seen, one normal and one shear component of strain and stress have the largest values compared to the other components. By examining the principal strain and stress histories of the critical element during a cycle, the observed states of stress and strain result in the mid and minimum principal stress and strain values to be small. Maximum principal strain distributions of an engine mount under R = 0.2 and load amplitude of 1,500 N is shown in Figure 4.5. The maximum principal strain occurs at the interface between the inner bushing and the bulk rubber material at a location of abrupt geometry change. This region is the critical region where fatigue cracks are 72

93 expected to nucleate. The maximum principal strain at the critical location determined from FEA will be used for evaluating the fatigue damage parameter of the material. 4.6 Conclusions This chapter presented a generalized fatigue FE simulation analysis for rubber components. Application of the approach was illustrated by analysis of a vehicle cradle mount and verified by means of component stiffness tests. As it was discussed, quasistatic simulations can be performed for cyclic loading where by using the stabilized cyclic stress strain curve the initial softening associated with the Mullin s effect is inherently accounted for. Due to large deformations, strain may be considered a natural choice to characterize the fatigue behavior of rubber. Among the strain quantities, maximum principal strain can be used for predictions since fatigue cracks in rubber typically initiate and grow perpendicular to the maximum principal strain directions. The R ratio can vary at different critical locations of the component even under constant amplitude loading. As the R ratio can significantly influence fatigue life, its effect can be taken into account by using a model such as the Mars-Fatemi R ratio model. Simulation of component deformation and load-deflection curves under monotonic as well as cyclic loadings agreed well with the component test results under the corresponding loading condition. 73

94 Table 4.1: Summary of parameters used for numerical model. Parameter Conditions Material Behavior Elastic, Isotropic Poisson s Ratio ~0.5, Incompressible Material Property Used Stabilized Cyclic Deformation Curve Strain Energy Potential Model Marlow Element Type Hybrid Continuum 8-Node Brick Simulation Type Quasi-Static Implicit Simulation Control Mode Uniaxial Load 74

95 (a) (b) Figure 4.1: Vehicle cradle mount used as illustrative example (a), and rubber mount FE model where due to symmetry half of the model is shown (b). 75

96 (a) (b) Figure 4.2: Component mid-life hysteresis loops under axial loading condition for (a) R ~ 0 and, (b) R =

97 (a) (b) Figure 4.3: Stiffness comparison between FEA simulation and monotonic test of the mount (a) and cyclic tests of the mount (b). 77

98 (a) (b) Figure 4.4: Strain (a) and stress (b) histories at the critical element location obtained from FE simulation during one sinusoidal loading cycle between 750 N and 3750 N. 78

99 Figure 4.5: Maximum principal strain distribution under R =0.2 and load amplitude of 1,500N. 79

100 Chapter 5 Component Uniaxial Fatigue Behavior 5.1 Introduction The main objective of this chapter is developing and validating CAE durability analytical techniques under constant and variable amplitude uniaxial loading. Application of the approach is illustrated by analysis of a vehicle cradle mount. The approach is then verified by component test results. The capabilities of Rainflow cycle counting procedure and Miner s linear damage rule are also evaluated with component tests. A filled natural rubber material with 21% carbon black and 9.5% plasticizer was used for component experiments. In this chapter, first the component life prediction methodology and predictions involving cycle counting, cumulative damage, and damage parameter are presented and discussed, where both crack initiation and crack growth approaches are considered. Finally, verification of the approach used is presented based on the conducted fatigue tests of both constant amplitude and variable amplitude loadings. Validation of life prediction methodology used is discussed with respect to both failure location and fatigue life. 80

101 5.2 Component life predictions In order to be able to predict the fatigue damage, there are several general prerequisites. First, it is necessary to identify the fatigue critical location and map the load history for this location by using a proper cycle counting method. A fatigue damage parameter is then needed to compute damage for the loading cycles identified by the cycle counting method. Finally, by using a proper cumulative damage criterion the damage is integrated and the fatigue life is calculated for the critical location of the specimen or component. For complex geometries, such as industrial components, a numerical method such as FEA is also necessary. The component life prediction procedure developed is based on both crack initiation and crack growth approaches. Constant amplitude specimen test data are used as the basis of fatigue life curve for crack nucleation, while specimen crack growth data and the initial natural flaw size in the material are used for crack growth or total fatigue life of the component. It should be noted, however, that the load levels used for component testing as listed in Table 5.1 are higher than those applied to the component in actual service condition. This was necessary in order to induce fatigue failure and in a reasonable time duration (i.e. hours and days, rather than months or years) Crack initiation life predictions There are generally two methods to characterize the fatigue behavior of the material. One approach is based on nominal stress amplitude (S N equation and curve). Another approach is based on local strain amplitude (ε N equation and curve). Because strain is readily calculated from deformation and is relatively easy to measure due to 81

102 large deformations, the ε N equation and curve may be considered a natural choice to characterize the fatigue behavior of rubber. Among the strain quantities, maximum principal strain is used for predictions. This is because fatigue cracks in rubber typically initiate and grow perpendicular to the maximum principal strain directions [31]. Maximum principal strain at the critical location is obtained from FEA; thereby local strain R ratio during the loading cycle can be obtained at this location. Maximum equivalent R = 0 engineering strain is then calculated by using the Mars-Fatemi R ratio material model [17] as discussed in chapter 3 and given by Equation (3.14). Predictions for failure at critical location based on crack initiation approach are then made based on the material strain-life curve shown in Figure 3.11 and represented by Equation (3.26). Crack initiation lives of the rubber mount at different loads were then obtained by substituting the maximum equivalent principal strains at the critical location into this equation. Values of R ε, ε 1,max, ε 1,max,0, and predicted life N f for different component tests conducted are tabulated in Table Crack growth-based life predictions Maximum principal strain is also used for crack growth-based fatigue life predictions, where crack growth life is used as fatigue life. Maximum principal strain and R ratio at 10 elements along the crack path adjacent to the critical location element was obtained from FEA. Maximum equivalent R = 0 engineering strain could then be calculated for each element by using the Mars-Fatemi R ratio material model based on Equation (3.14). Mars and Fatemi [80] used the energy release rate estimate for small edge crack under simple tension condition which is given by [27]: 82

103 T w 2kWa (5.1) where W is the strain energy density, k is a factor dependent on maximum principal strain [69], and a is the crack length. Strain energy release rate (pseudo-energy release rate equation) is then obtained by observing the fact that energy density varies approximately with the square of the maximum principal strain, given by [80]: 2 T, max,0 C ( 1,max,0 ) a (5.2) where C is the material s initial Young s modulus obtained from monotonic stress-strain curve and is 2.34 MPa for the material in this investigation and a is crack length. It should be mentioned that the constant C defined as the initial Young s modulus value is not strictly consistent with the classical form of the strain energy release rate for simple tension frequently used for rubber strength and fatigue analyses. However, Equation (5.2) still represents the important characteristic of energy release rate varying linearly with crack length. Recent work by Ait-Bachir et al. [81] validates the form of Equation (5.2) by showing energy release rate to depend on crack length and one far-field property, irrespective of the state of loading. Crack growth rate is then obtained based on specimen crack growth rate data which is shown in Figure 3.14 and represented by: da dn 5 max, ,max, ( T ) 4 10 C a (5.3) By using the maximum equivalent R = 0 engineering strain as a damage parameter along the crack, which is in the depth direction for the component considered, the relationship between strain and crack length (depth) was obtained. Integrating Equation (5.3), fatigue life can then be obtained. Based on Lake and Lindley [82], the natural flaw size for 8 different polymer types and with various fillers, curatives and 83

104 other compounding variables were reported to be around 0.02 to 0.05 mm. Mars and Fatemi [68] obtained an initial crack size of 0.01 mm for natural rubber based on agreement between crack initiation and crack growth properties. Based on this approach, for the natural rubber material of the component used in this study the effective initial flaw size was found to be 0.02 mm. This value was, therefore, used as an initial crack length at the critical location in this study. Predictions based on integration results along the critical dominating crack path are finally obtained from: N f 10 i i,1,max, 0 1 ai 1 a i 1 (5.4) The variation of maximum equivalent R = 0 principal strain versus crack length (depth) is shown in Figure 5.1. This figure shows the importance of initial flaw size in fatigue life prediction, due to the drastic decrease of the value of 1,max, 0 away from this critical element. Therefore, the critical element has the highest effect in total life prediction. The total fatigue lives of the rubber mounts were predicted using this procedure and the predicted lives for the four load levels and two R ratios considered are tabulated in Table Variable amplitude loading life predictions As discussed in section 3.5.5, many cycle counting methods have been proposed. Miner s LDR [53] is the simplest form of cumulative damage rule. Continuum mechanics parameters are used as damage quantification parameters. Due to the ease of measuring strain, maximum principal strain is used for life predictions in this work. This agrees with the experimental observation of fatigue cracks usually initiating and growing perpendicular to the maximum principal strain direction [31]. 84

105 The strain R ratio (R ε ) effect during each loading cycle at the fatigue critical location can be accounted for by calculating a maximum equivalent R = 0 engineering strain (ε 1,max,0 ) based on the Mars-Fatemi R ratio material model (Equation (3.14)). By using the material strain-life, as given in Equation (3.26) crack initiation life is then predicted at each equivalent strain level. Crack initiation life based on blocks of loading is then calculated based on the linear damage rule, as expressed by Equation (3.25). By utilizing maximum equivalent R = 0 engineering strain as a damage parameter along the crack (depth direction for this component), the relationship between strain and crack length (depth) was obtained. Predictions based on the damage parameter along the critical dominating crack element for each individual event is obtained from: N f ,max, 0 1 a0 1 a1 (5.5) where a 0 and a 1 are natural flaw size and critical element s depth, respectively. It should be noted that the unit of crack length in Equations (5.2) and (5.5) is mm. It is found that the critical element has the highest effect in the total life prediction. Total fatigue lives of the rubber mounts were predicted by using the linear damage rule based on blocks of loading shown in Figure

106 5.3 Constant amplitude fatigue experimental program and validation of life predictions Component fatigue tests Component level fatigue tests were conducted in order to validate the accuracy of the fatigue life predictions discussed. Fatigue tests were conducted at ambient temperature under constant amplitude load-controlled conditions with a sinusoidal waveform of 0.5 to 1 Hz using a servo-hydraulic fatigue testing load frame with the load capacity of 50 kn. Displacement response of each test specimen was periodically recorded. In order to define the fatigue crack nucleation life of the engine mount in a consistent manner, cracking of several engine mounts was also monitored. The load amplitudes chosen resulted in fatigue lives between about 19,000 cycles and 161,500 cycles. With initial cycling, the maximum displacement increased due to the Mullin s effect. The load-displacement response of the mount then stabilized after several hundred cycles. When the fatigue crack initiated and then grew to a critical size, the displacement amplitude increased rapidly due to reduced stiffness and failure occurred. Failure was defined as the number of cycles at which the displacement amplitude increased drastically from its gradual linear change, as illustrated for a typical test in Figure 5.2. The experimental fatigue lives at different load levels are tabulated in Table 5.1. As can be observed from Table 5.1, experimental life scatter is within a factor of two between duplicate tests. This table also shows the first crack observations for the tests which were stopped at regular intervals for monitoring. As can be observed, the crack growth life is a significant portion of the total life in all tests. The experimental 86

107 results show that by increasing the mean load or R ratio, the total component life increases. This is the same as what was observed for specimen testing of this material. Displacement amplitude versus applied cycles for all of the fatigue experiments is depicted in semi-logarithmic scale in Figure 5.3. As can be seen, near failure there is a steep change in displacement amplitude due to reduced stiffness resulting from presence of macro-crack(s). In order to preliminary investigation of the effect of higher frequency on the component life, one test with the loading rate of 3 Hz was performed to compare with the test at 1 Hz. The fatigue life was similar to the lives at 1 Hz. Frequency effect has been observed to be small for rubber compounds which strain crystallize under loading (i.e. filled natural rubber) under isothermal conditions for the frequency range of 10-3 to 50 Hz [83]. Young [84] also reported that while the strain rate could affect the crack growth rate for natural rubber compounds at high energy release rates close to fracture, for smaller values of energy release rate changing the strain rate by factor of 20 does not significantly change the crack growth rate. To evaluate the difference between load control (LC) and displacement control (DC) on component fatigue life, one axial test in DC was also performed based on midlife response displacement amplitude and R ratio of the corresponding LC test. The crack grew more rapidly under load control than under displacement control, as also observed in [85]. DC test total life was 61,450 cycles, compared to 69,683 and 107,411 cycles in LC tests (see Table 5.1) 87

108 5.3.2 Damage development Crack growth life was a significant portion of total fatigue life (between 70% and 93%). For R ~ 0 tests at the two load levels the ratio of crack initiation life to total life decreases compared to R = 0.2, but still exceeds 70% of total component life. For R = 0.2 ratio and load amplitudes of 1,100N and 1,500 N, this ratio increases to about 78% and 93%, respectively. Crack length as well as its depth and their changes were measured at critical (failure) location by periodic test interruptions and visual inspection. Crack length did not have a significant effect on fatigue life, as even at a length of about 4 cm, the component is still at only about half of its total life. Crack depth had a more dominant effect on life than crack length. When crack depth was on the order of 3 cm, the component was near failure. Stiffness drop also correlated better with crack depth than with crack length Comparison of predicted and experimental fatigue lives Finite element contours at the maximum load are shown in Figure 4.5. The value of highest maximum principal strain defines the crack initiation location. The same critical element was observed in all of the FE simulations. Experiments failure locations and crack growth directions are schematically depicted in the Figure 5.4. The experiments show the same location as predicted by FE simulations. Experimental and predicted initiation lives are tabulated in the Table 5.2. The results show life predictions based on crack initiation are within about a factor of two of experimental lives. The crack size used for both experimental and predicted nucleation lives was 1 mm, as the specimen fatigue initiation life data equation was generated for a crack length of about 1 mm. 88

109 Figure 5.5(a) shows crack initiation life prediction versus experimental life. Initiation life for the tests not monitored for crack growth is calculated based on crack initiation to total life ratio for each loading condition. It can be seen that most of the predictions are within a factor of two and the predictions for R ~ 0 loading were closer to the experimental results. This may be mainly due to the fact that the strain-life equation was developed for R ~ 0 condition and an R ratio model was used to find the equivalent R ~ 0 strain for the R = 0.2 condition. Figure 5.5(b) shows predictions based on total life (crack growth approach) as compared to experimental life. This figure illustrates that for R ~ 0 loading condition, similar to the crack initiation approach, predictions are better than for R = 0.2 loading. The predictions for total component life are within a factor of two and on the conservative side for most of the loading conditions. 5.4 Variable amplitude fatigue experimental program and validation of life predictions Component fatigue tests To evaluate variable amplitude loading damage accumulation and life prediction methodology, the rubber component shown in Figure 4.1(a) was used. Quasi-static FE simulation was performed for the variable amplitude loading shown in Figure 3.7, using ABAQUS software, so that a loading block was exactly simulated. Because the critical element location is the site for crack initiation, maximum principal strain and R ratio at the integration point of this element (therefore, the averaged value in the element) were obtained from FEA for each individual cycle. 89

110 Four component fatigue tests were conducted in order to investigate the applicability of fatigue life prediction methodology discussed. Load R ratio of near zero with two load ranges of 3,000 N and 3,600 N were used and each test was duplicated to evaluate variability. The fatigue test results are presented in Table 5.3. A sinusoidal waveform with 16.5 second duration per block was used and tests were conducted using a servo-hydraulic fatigue testing machine with load capacity of 50 kn. Displacement response of each test specimen was recorded based on the logarithmic intervals of 2. In order to define the crack initiation life of the mount in a consistent manner, damage evolution of two of the mounts was monitored by periodic visual inspection. For the tests in which crack initiation life was not monitored, nucleation life was estimated based on the ratio of the average initiation life to total life of the tests for which crack initiation was monitored. After application of the initial few blocks of load, the maximum displacement increased because of Mullin s effect. The response then stabilized after several blocks. After fatigue crack initiation occurred, the displacement amplitude gradually increased with macro-crack growth because of reduced stiffness. Once the crack grew to a critical size, the displacement amplitude increased rapidly leading to failure. Failure was defined as the number of blocks at which the displacement amplitude increased abruptly from its gradual linear change, as shown in Figure 5.6 for a typical test. 90

111 5.4.2 Damage development As can be seen in Table 5.3, experimental life scatter was within a factor of two between duplicate tests. This table also tabulates the first crack observations for the tests which were stopped at regular intervals for inspection. By increasing the load range, the total life decreases, as expected. By increasing the load range, the ratio of crack initiation life to total life decreases. This means that more life is spent in crack growth. Crack growth life is a significant portion of the total fatigue life (about 80% to 90%). Crack length and its growth were measured at critical (failure) location by periodic test interruptions and visual inspection. Crack length did not have a significant effect on fatigue life, while crack depth had a more dominant effect and this fact is illustrated in Figure Comparison of predicted and experimental fatigue lives Maximum principal strain history at the critical element was generated from FEA. By using Rainflow cycle counting and Miner s linear damage rule, life predictions were performed for both crack initiation and total life approaches using the methodology discussed in section 5.2. The predicted results are also tabulated in Table 5.3. Figure 5.8 shows maximum principal strain contour for R ~ 0 and loading range of 3,000 N at the highest peak load during a block of loading. The location of critical element which is the indicator of crack initiation site is shown in red color and it matches observed crack initiation location in experiments. Figure 5.9 shows experimental versus predicted fatigue initiation as well as total lives. For the lower load range initiation predictions are within a factor of 3, whereas for higher load range initiation predictions are about a factor of 5 with the predictions being on the non-conservative side. 91

112 Total life predictions are within a factor of 2, thus total life gives more reasonable predictions than crack initiation. The total life predictions are more meaningful since the component fatigue life is dominated by macro-crack growth. 5.5 Conclusions This chapter presented a generalized fatigue analysis and life prediction approach for rubber components. Application of the approach was illustrated by analysis of a vehicle cradle mount and verified by means of component tests. Based on the analysis and test results presented, the following conclusions can be made: Quasi-static simulations can be performed for cyclic loading where by using the stabilized cyclic stress strain curve the initial softening associated with the Mullin s effect is inherently accounted for. Due to large deformations, strain may be considered a natural choice to characterize the fatigue behavior of rubber. Among the strain quantities, maximum principal strain can be used for predictions since fatigue cracks in rubber typically initiate and grow perpendicular to the maximum principal strain directions. The R ratio can vary at different critical locations of the component even under constant amplitude loading. As the R ratio can significantly influence fatigue life, its effect can be taken into account by using a model such as the Mars-Fatemi R ratio model. The location of critical element as indicator of the crack initiation site based on the life prediction methodology used was identical to the observed crack initiation location in the cradle mount component experiments for both CA and VA loadings. The predicted fatigue lives based on the crack initiation approach used were within about a factor of two of the experimental lives for CA loadings. Comparison of predicted versus 92

113 experimental cradle mount fatigue lives show satisfactory life predictions based on the linear cumulative damage rule and the maximum principal strain as a damage criterion. Crack growth constitutes a significant portion of the component total fatigue life. Life predictions for total component life based on a crack growth analysis approach were within a factor of three of the experimental lives, with most of the predictions being on the conservative side. The total life approach used resulted in more accurate fatigue life predictions (within a factor of 2), as compared to the crack initiation approach. 93

114 Table 5.1: Summary of component fatigue test results. Control Mode R ratio Control Amplitude Control Mean Frequency (Hz) Midlife Response Amplitude Life to Failure (Cycles) Initiation Total Load ~ N N mm - >148,000 Load ~ N N mm 20,455 * 69,683 Load ~ N N mm 28,000<N<35, ,411 Load ~ N N mm - 36,359 Load ~ N N mm 5,685 * 19,069 Load ~ N N mm 5,667 * 19,100 Load ~ N N mm 4,000<N<8,000 20,180 Load N N mm 16,000<N<32, ,129 Load N N mm 35,189 * 161,447 Load N N mm N<3,000 20,414 Load N N mm 1,915 * 26,034 Displacement mm mm N N<7,000 61,450 1 st observed crack (with the length of 1 mm) by periodic test interruptions and visual examination * Estimated initiation life calculated based on the ratio of initiation life to total life of the test with the same loading which was stopped in regular intervals for crack growth monitoring 94

115 Table 5.2: Predicted nucleation and total fatigue lives of the rubber mount component and comparison with experimental fatigue lives. R P Load Amplitude (N) Mean Load (N) FEA results at critical Location ε 1, max ε 1, min R ε ε 1,max,0 Predicted Nucleation Life (Cycles) Predicted Total Life (Cycles) Experimental Nucleation Life (Cycles) Experimental Total Life (Cycles) ( N ( N nuc ) ) pred nuc exp ~0 1,025-1, , ,752 - >148, ~0 1,100-1, , ,516 20,455 69, , , ~0 1,300-1, ,511 37,496-36, ,685 19, ~0 1,500-1, ,045 18,493 5,667 19, ,000 20, ,100-1, ,178 53,687 24, , , , ,500-2, ,642 9,195 1,500 20, ,915 26, ( N ( N ) total pred total) exp 95

116 Table 5.3: Component fatigue test results and predictions under uniaxial variable amplitude loading with R~0. R P Control Range (N) Predicted Nucleation Life (Blocks) Predicted Total Life (Blocks) Experimental Nucleation 1 Life (Blocks) Experimental Total Life (Blocks) ( B ( B nuc ) ) pred nuc exp ( B ( B ) total pred total) exp ~0 3,000 1,861 4,268 ~0 3,600 8,38 1,631 0<B<1,500 3, <B<2, , <B<300 2, <B< , First observed surface crack (with the length of 1 mm) by periodic test interruptions and visual examination 2 Estimated initiation life ranges calculated based on the ratio of initiation life range to total life of the test with the same loading which was stopped in regular intervals for crack growth monitoring 96

117 Figure 5.1: Maximum R = 0 equivalent engineering strain versus crack depth for load amplitude of 1500 N and R = 0.2 from FEA simulation. 97

118 Figure 5.2: tests. Component failure definition based on stiffness degradation in fatigue 98

119 Figure 5.3: Displacement amplitude versus cycles in fatigue tests. 99

120 Figure 5.4: Experimental failure locations and crack growth direction in component tests. 100

121 (a) (b) Figure 5.5: Experimental versus predicted fatigue life based on crack initiation approach (a) and based on crack growth approach (b). 101

122 Figure 5.6: Component displacement amplitude response as a function of applied blocks of loading under variable amplitude fatigue loading. Figure 5.7: Crack length versus applied load blocks for a test at R ~ 0 and with a load range of 3,600 N. 102

123 Figure 5.8: Maximum principal strain contour results for R~0 and loading range of 3,000 N obtained from FE simulation. 103

124 Figure 5.9: Component fatigue testing and predictions for variable amplitude random loading at R~0, both for crack initiation and total component lives. 104

125 Chapter 6 Component Multiaxial Fatigue Behavior 6.1 Introduction At the critical location(s) of most elastomeric components and structures, multi-axial states of stress and strain typically exist. Such components and structures are also usually subjected to variable amplitude or random loading histories. The extreme values for the strain and/or stress components are not usually coincide with each other under complex loading conditions. This indicates the complex nature of strain history in applications like vibration isolators. Therefore, the study of multiaxial fatigue and deformation of elastomers under variable amplitude loading is an important issue for their proper design and life prediction analysis. In this chapter, first the life prediction methodology for general random and multiaxial loading is discussed. Both crack initiation and crack growth approaches are included and commonly used and more recently developed fatigue damage quantification parameters are discussed. Then applications to constant amplitude axial-torsion in-phase and out-of-phase loading as well as variable amplitude loading are demonstrated using experimental results from a vehicle cradle mount made of natural rubber. Finally, predictions for both approaches for all loadings including uniaxial CA and VA are 105

126 tabulated and the correlation results will be described. Material deformation and fatigue behavior of the mount are characterized in Chapter 3. The component FEA and fatigue life analysis and predictions for uniaxial constant and variable amplitude loadings are presented in Chapters 4 and Life prediction methodology for general random and multiaxial loading of components Component fatigue damage is typically localized at points of high strains or stresses. For complex component geometry, finite element analysis is often performed to identify the critical location and obtain the stress and strain states for that location. A cycle counting method and a cumulative damage rule are then used for damage calculations. The cycle counting procedure relates the damage effect of variable amplitude loading to constant amplitude material fatigue data and fits. The most popular method is the Rainflow method proposed by Matsuishi and Endo [72], which is also used here. Linear damage rule is the simplest form of cumulative damage rule, which is used in this work. However, load sequence effect or interaction between cycles is not accounted for in this rule. A damage quantification parameter is also needed to relate the component multiaxial stresses and strains to uniaxial specimen test and data. Harbour et al. [37] evaluated maximum normal strain as a critical plane approach. Normal strain can be defined from: e ( N ˆ ) Nˆ. C. Nˆ 1 (6.1) 106

127 where C = F T F is the Green s deformation tensor, F is the deformation gradient tensor and Nˆ is the unit vector normal to the plane in space. By using a MATLAB script, the maximum value of normal strain and its direction could be defined on all planes in space in spherical coordinates. Then, the normal strain history on maximum normal strain (MNS) plane is calculated. The next step is using the strain-life equation obtained from uniaxial fatigue tests, as shown in Figure 3.11 and represented by Equation (3.26). The strain R ratio effect during each loading cycle at the critical location can be accounted for by using maximum equivalent R = 0 engineering strain ( 1,max, 0 ) based on Mars-Fatemi R ratio material model [17] discussed in Chapter 3 and given by Equation (3.14). For variable amplitude loading, cycle counting is performed on the critical plane (i.e. MNS) and damage is calculated for each cycle using the linear damage rule (LDR). Another successful damage quantification parameter mentioned previously is cracking energy density (CED). The increment in cracking energy density dw c in the spatial description is defined in terms of the traction vector T and the unit displacement vector d on a given plane in the instantaneous deformed configuration and is given by: dwc T. d (6.2) The normal to the plane is defined by the unit vector r. By converting from the spatial description to the material description, the final expression for CED increment is given by [71]: dw c T ~ R C S de R T R C R T ~ R (2E I) S de R T R (2E I) R 0 0 (6.3) 107

128 where / 0 is the ratio of the deformed mass density (volume) to the undeformed mass density (volume), F is the deformation gradient, S ~ is the 2 nd Piola-Kirchhoff stress tensor, and E is the Green-Lagrange strain tensor. The relationship between unit vector in the current configuration r and the unit vector in the undeformed configuration R is then given by: F R r (6.4) F R Also C = 2E + I, where C is the Green deformation tensor. Equation (6.3) gives the cracking energy density in terms of the stress and strain measures of the material description and in terms of a unit vector in the undeformed configuration. Two series of calculations were made, one for calculating cracking energy density (CED) on maximum normal strain (MNS) plane, and another for calculating CED on CED critical plane with the increment of 5 in spherical coordinates. Crack closure effect is considered in each approach since the normal traction on each plane is used in the calculation. Calculation of CED history on MNS plane is only on a single plane, while calculation of CED history on the critical CED plane is performed on 1296 (= 36 36) planes to find the critical plane with the highest damage value. Therefore, significant difference in computation time exists between the two approaches. SED (W) and CED values are the same for uniaxial loading. The SED or CED life Equation is given by: 0.47 W 148( N ) (6.5) f This Equation is obtained by uniaxial crack initiation experiments. 108

129 Fatigue crack initiation approach is typically related to the fatigue life to grow a crack from a natural flaw to a length on the order of about a millimeter. Crack growth approach then considers fatigue life from this crack length to failure. Crack initiation is often used for applications where micro-cracks dominate the total fatigue life. Crack growth approach is usually considered when a large portion of the total life is involved with macro-crack growth. In this study both approaches were considered. To calculate total component fatigue life, the same methodology discussed in section 5.2 is used. The cradle mount was initially made with a snubber part which stops application of the high loads to the component. Moreover, the amount of load or displacement applied to this component was much less than needed to produce fatigue failure. Therefore, the stop or snubber parts of the mounts are removed prior to testing and in simulations for analyzing this component in order to produce fatigue failure sooner. This elastomeric cradle mount was not designed for applying torsion loads. Therefore it is not appropriate to evaluate criteria like SED efficiently with this geometry. The load levels chosen for testing were much higher than actual loading of the component and experimental fatigue lives were much shorter than fatigue life of the cradle mount in service. The predictions are highly conservative, since they are based on initiation life of a crack in the order of a millimeter. Almost all of the experiments showed that significant life was left from this point to final failure of the component. Therefore, macro-crack growth involves most of the cradle mount fatigue life. 109

130 6.3 Applications to multiaxial constant amplitude loading Experimental program and results Component level fatigue tests were conducted in order to validate the fatigue life prediction methodology discussed above. Figure 4.1(a) shows the component used. The finite element model of the component is shown in Figure 6.1. Axial loading with amplitude of 1100 N and R ~ 0 and torsion loading with amplitude of 30 N.m. and R = -1 were used as the experimental loading conditions. Both in-phase and 90 out-of-phase tests were conducted with two tests for each condition. One test of each condition was stopped in regular intervals to monitor crack initiation and growth. Fatigue tests were conducted with a sinusoidal waveform of 1 Hz using a servo-hydraulic axial-torsion load frame and axial as well as rotational displacement responses of each test specimen were periodically recorded. The loading conditions chosen resulted in fatigue lives between about 21,500 cycles and 67,500 cycles. The maximum axial and rotational displacements increased initially due to the Mullin s effect and then stabilized after several hundred cycles. When a fatigue crack initiated and then grew to a critical size, the displacement and rotation amplitudes increased rapidly due to reduced stiffness. Failure was defined as the lower number of cycles at which the axial or rotational displacement amplitudes increased drastically from its gradual linear change, as illustrated for a typical test in Figure 6.2. The experimental fatigue lives at different load levels are tabulated in Table 6.1. Test results from uniaxial fatigue tests reported in Chapter 5 are also included in this table. Experimental response for constant amplitude axial-torsion loading of components is shown in Figure 6.3(a) for in-phase and 6.3(b) for 90 out-of-phase tests. The life 110

131 comparisons show shorter life for 90 out-of-phase loading, as compared to in-phase loading. Experimental life scatter between duplicate tests was within a factor of three for in-phase tests and within a factor of two for out-of-phase tests. This table also shows the first crack observations for the tests which were stopped at regular intervals for monitoring. The crack nucleation life for a duplicate test for which initiation was not monitored was calculated based on the ratio of crack nucleation life to total life of the monitored test of the same loading condition. Crack growth life was a significant portion of the total fatigue life (about 75%) for in-phase test, while for out-of-phase test, this ratio was about 30%. In these load and torque controlled tests both displacement and rotation amplitudes increased with increasing cycles. Displacement versus rotation at different cycles during the test is shown for both In-phase and 90 out-of-phase loading in Figure 6.4. By increasing the number of cycles more distinct change in maximum displacement (in compression) and negative rotation value is observed. This is due to the evolution of micro-cracks and subsequent coalescence of those cracks. This affects the stiffness behavior of the component. Displacement versus rotation at midlife is shown in Figure 6.5. Crack length as well as its depth and their changes were measured at critical (failure) location by periodic test interruptions and visual inspection. For in-phase loading, crack length did not have a significant effect on fatigue life, as even at a length of about 4 cm, the component was still at only about half of its total life. Crack depth had a more dominant effect on life than crack length. When crack depth was on the order of 3 cm, the component was near failure. Stiffness drop also correlated better with crack 111

132 depth than with crack length. Crack length and depth versus cycles are shown in Figure 6.6(a) for in-phase loading. For out-of-phase loading, crack length was a key factor, rather than crack depth and this is shown in Figure 6.6(b). In out-of-phase loading, the ratio of crack initiation life to total life increased about 3 times of the same loading level for in-phase condition, therefore, less life is spent in crack growth for OP loading. Since this ratio was much higher in OP tests compared to what was observed for uniaxial load cases, phasing had much more effect on the ratio of initiation to total life. This is shown in Figure 6.7. To evaluate any difference between load control (LC) and displacement control (DC) mode, IP and OP displacement-controlled tests were also performed based on midlife displacements of the LC tests. In DC tests failure was defined based on 40% load or torque drop, whichever occurred sooner. The midlife load-displacement and torque-rotation was similar in both LC and DC tests. Cracks initiated more rapidly under LC than under DC. This is the same as what was observed for CA uniaxial tests in Chapter 5. The results are tabulated in Table 6.1. Crack initiation location was the same for all cases of LC and DC mode tests, however Life predictions and validation of methodology The nonlinear FE program ABAQUS was used to simulate the nonlinear and large deformation hyper-elastic behavior of the mount. The Marlow strain energy density form was used to model deformation of incompressible isotropic elastomers. The mount FE mesh is shown in Figure 6.1. To adequately model the geometry, a relatively fine mesh with about 30,000 elements was used. The element type used was the eight node brick (hexahedral) element. For cyclic loading, the amplitude tool of the FE software was used, 112

133 so that a loading cycle could be exactly simulated. Stable deformation curve was applied as an input to the model, therefore initial softening (Mullin s effect) was inherently considered. More details of FE model can be found in Chapter 4. Maximum principal strain distributions of the mount under IP loading is shown in Figure 6.8, which is similar to that for OP loading. The maximum principal strain occurs at the interface between the inner bushing and the bulk rubber material at a location of abrupt geometry change. Locations, at which the maximum principal strain occurred, as indicated in Figure 6.8 for in-phase loading, were also observed to be the site(s) of fatigue cracking during all fatigue tests of the component. The maximum principal strain at the critical location determined from FEA was, therefore, one of the parameters used for evaluating the fatigue damage. By using the data from critical element (defined as the location with the highest maximum principal strain) of the mount in each loading cycle, life predictions were performed. Values of the maximum normal strain (MNS), critical cracking energy density (CED), and CED on MNS plane are listed in Table 6.2. Through a tensor transformation program developed in MATLAB, the maximum normal strain value and direction was defined. Equations (3.14) and (3.26) based on uniaxial fatigue properties were then used to perform life predictions. The correlation of the predicted and experimental results is shown in Figure 6.9(a). The predictions are within a factor of two for all experiments by using the maximum normal strain parameter. Since only strain is involved in the calculations, this approach does not require the constitutive behavior of the material under multiaxial loading. Therefore, numerically it is an efficient parameter for determining the critical or failure plane(s) through a 3-D search process. 113

134 Maximum normal strain plane direction and Cauchy stress history were used in another MATLAB program to obtain cracking energy density on the MNS plane. Equation (6.5) for CED based on uniaxial fatigue properties was used to perform life predictions. The correlation of predicted results with experimental results for CED is shown in Figure 6.9(b) and the same plot for CED on MNS plane is shown in Figure 6.9(c). The results show that using CED on MNS gives similar predictions to the critical CED criterion. As discussed earlier, this is a computationally efficient approach (by a factor of about 45), to predict crack initiation life of the component, since CED history is calculated on a single plane. Total life of the component was also predicted based on Equation (5.5). The predicted and experimental results are tabulated in Table 6.2. The correlation of predicted versus experimental total lives are shown in Figure The predictions are all within a factor of three of the experiments. 6.4 Applications to variable amplitude multiaxial loading For evaluating more complex loading conditions, variable amplitude proportional and non-proportional axial-torsion tests were also conducted on the elastomeric component. Two in-phase and two out-of-phase tests were conducted. For in-phase loading the peaks and valleys in both axial and torsion load signals are reached at the same time, while for out-of-phase loading they are not (see Figure 6.11). Experiments were conducted with the load range of 3,680 N in comparison with minimum load of near zero and torque range of 50 N.m. with mean value of near zero. For one duplicate test of each condition, the test was stopped at regular intervals (every 10% of the expected life) to monitor crack initiation and growth. 114

135 Test results for variable amplitude tests are given in Table 6.1. Experimental life scatter between duplicate tests was within a factor of two for both in-phase and out-of-phase loading. Crack initiation range is also tabulated in Table 6.1. Crack growth life constitutes a significant portion of the total life in all tests. In out-of-phase tests the ratio of crack initiation life to total life increased about 1.5 times, as compared to in-phase loading. The same observation was made in constant amplitude loading. This is shown in Figure 6.7. Crack length and depth measurement results showed that crack depth was key factor in component fatigue life for both in-phase and out-of-phase tests. This was also observed in constant amplitude uniaxial tests. The maximum principal strain occurs at the interface between the inner bushing and the bulk rubber material at a location of abrupt geometry change, similar to all other loading cases and corresponding to that observed during the fatigue tests. This crack initiation site remained the same for all of the simulations, either IP or OP. The same methodology used for constant amplitude axial-torsion loading condition was used to define crack initiation life as well as total fatigue life of the component. The only difference for variable amplitude loading is the use of Rainflow cycle counting method for event identification and then using Miner linear rule to accumulate the fatigue damage. Figures 6.12(a) and 6.12(b) show normal strain history on maximum normal strain plane for in-phase and out-of-phase loadings, respectively. As can be seen from this figure, the maximum normal strain history for both IP and OP conditions at critical location is different from the nominal loading history applied to the component (see Figure 6.11). Cycle counting was performed on the shown MNS history. 115

136 After event identification, by utilizing Miner rule, the fatigue life prediction was performed. Life prediction results, shown in Figure 6.9 for different damage parameters, indicate that MNS criterion is not accurate for complex variable amplitude multiaxial loading. Both critical CED and CED on MNS criteria give better predictions. Most of the total fatigue life predictions are within a factor of three of experiments in this case according to Figure Discussion of Results Predicted crack nucleation locations based on the developed methodology matched observed failure locations for all loading conditions. Crack initiation lives were predicted well (mostly within a factor of two) based on the maximum normal strain and the developed methodology for most of the loading conditions. Because only strain is involved in this approach and no constitutive behavior is used, maximum normal strain can be a very efficient parameter to determine critical or failure plane. Most of the predictions were on non-conservative side (i.e. predicted life longer than experimental life). This is, at least partly, because of the fact that in the FE model the strain and stress at critical location were lower than in the component since a finer mesh could not be used due to practical considerations. Of the two CED criteria, the CED on MNS plane criterion gives quite acceptable predictions with much less calculation time, compared to the critical CED plane criterion. This is because CED history is calculated on just one MNS plane and not on many planes in space. Calculation of CED requires knowledge of the constitutive behavior of the 116

137 material, however. Both of these approaches resulted in better life predictions than MNS for multiaxial variable amplitude loading conditions. Two criteria of Smith-Watson-Topper (SWT) commonly used for metallic materials and strain energy density (SED) traditionally used for elastomers were also evaluated. All of the predictions based on SWT criterion were overly conservative predictions by up to a factor of 23. Predictions based on SED resulted mostly in life within a factor of two for this particular component and the loading conditions considered. SED criterion gives close predictions compared to CED criterion because of the particular component geometry where based on the FE results the maximum principal strain at the critical location is much higher than the other two principal strains. This indicates essentially uniaxial strain state, although the loading of the component was multiaxial. In addition, the principal strain at the critical location is tensile, while for compression strain there would be a large deviation between SED and CED criteria predictions. The difference between SED and CED criteria is also large when torsion loading is dominant compared to axial loading, which was not the case for the loading considered in this study. Differences between CED and SED criteria based on different loading conditions are also discussed in [38]. Overall, of all fatigue crack initiation criteria studied here, those which use critical plane approaches work better than the scalar damage parameters or those damage parameters which do not take constitutive behavior into account. Crack growth constitutes a significant portion of the component total fatigue life. The ratio of initiation to total fatigue life varied between 6 and 69 % for different loading cases. For constant amplitude or variable amplitude uniaxial tests, by increasing the load 117

138 amplitude, the ratio of crack initiation life to total life decreased (i.e. more life spent in crack growth). This is expected, since higher load level initiates a crack quicker and fatigue life is dominated by crack growth. Multiaxial load phasing had an important effect on the ratio of crack initiation to total life. For constant amplitude out-of-phase loading, this ratio increased about 3 times compared to in-phase loading at the same level. This means more life was involved in crack initiation for out-of-phase loading, compared to in-phase loading. Total life comparison showed shorter life for out-of-phase loading compared to in-phase loading for both constant and variable amplitude loadings, indicating much faster crack growth rate once a crack has initiated in out-of-phase loading. The fracture mechanics approach was used for total fatigue life prediction for each loading condition based on specimen crack growth data and FE simulation results. The methodology used for component total fatigue life resulted in reasonable predictions for nearly all the loading conditions. The prediction results are sensitive to initial crack length used for analysis, but not very sensitive to the final crack length used. This was also observed in uniaxial behavior of the component, as discussed in Chapter 5, where the critical element of the component had the highest effect in the total life prediction. 6.6 Conclusions 1. Shorter total life was observed for out-of-phase loading compared to in-phase loading at the same level for both constant and variable amplitude loadings. In addition, for out-of-phase loading the ratio of crack initiation life to total life was about 3 times higher than for in-phase loading. This indicates more life was involved 118

139 in crack initiation but much faster crack growth rate once a crack initiated in out-ofphase loading, as compared to in-phase loading. 2. Crack initiation location was observed to remain the same for all of the loading conditions used for fatigue tests of the component. This location corresponded to the point at which the maximum principal strain occurred. Therefore, maximum normal strain (MNS) is an effective parameter to determine the critical or failure plane. 3. Although the maximum normal strain (MNS) parameter correlated the constant amplitude fatigue life data within mostly a factor of about two, it could not correlate the data satisfactorily for the more complex case of variable amplitude loading. 4. Amongst the different approached evaluated, a computationally efficient and relatively accurate fatigue crack initiation life prediction approach for complex loading was found to be a critical plane approach based on MNS and quantifying damage on this plane using the cracking energy density (CED) parameter. 5. For variable amplitude loading the rainflow cycle counting of the maximum normal strain (MNS) history was found to be an efficient method. Miner linear damage rule was then used to accumulate damage on the MNS plane based on constant amplitude fatigue data, resulting in satisfactory life predictions for variable amplitude loading. 6. Crack growth constituted a significant portion of the component total fatigue life. The fracture mechanics approach was used for total fatigue life prediction based on specimen crack growth data and FE simulation results. The methodology used for component total fatigue life resulted in reasonable predictions with nearly all the life predictions being within a factor of three of the experimental lives. 119

140 Table 6.1: Component experimental conditions and test results. Test Type Load Type Load Path Control Mode Control Amplitude (Range) R P /R M or R /R Frequency (Hz) a / a or P a / M a Fatigue Life (Cycles or Blocks) Initiation 1 Total 2 A 3 CA N.A. LC 1025 N ~ mm >148,000 A CA N.A. LC 1100 N ~ mm 18,165<N<22, ,683 A CA N.A. LC 1100 N ~ mm 28,000<N<35, ,411 A CA N.A. LC 1300 N ~ mm 36,359 A CA N.A. LC 1500 N ~ mm 3,780<N<7,560 19,069 A CA N.A. LC 1500 N ~ mm 4,000<N<8,000 20,180 A CA N.A. LC 1500 N ~ mm 3,786<N<7,572 19,100 A CA N.A. LC 1100 N 0.2 1,0.5, mm 23,456<N<46, ,447 A CA N.A. LC 1100 N mm 16,000<N<32, ,129 A CA N.A. LC 1500 N mm N<3,826 26,034 A CA N.A. LC 1500 N mm N<3,000 20,414 A VA N.A. LC (3000 N) ~ S/B mm B<2,598 6,327 A VA N.A. LC (3000 N) ~ S/B 8.77 mm B<1,500 3,653 A VA N.A. LC (3600 N) ~ S/B 9.94 mm B<320 2,615 A VA N.A. LC (3600 N) ~ S/B 9.87 mm B<300 2,

141 Table 6.1 (Continued): Component experimental conditions and test results. Test Type Load Type Load Path Control Mode Control Amplitude (Range) R P /R M or R /R Frequency (Hz) a / a or P a / M a Fatigue Life (Cycles or Blocks) Initiation 1 Total 2 A-T CA IP LC 1100 N/30 N.m ~0/ mm/ ,510 67,406 A-T CA IP LC 1100 N/30 N.m ~0/ mm/ ,200 31,290 A-T CA IP DC 6.95 mm/ / N/28.2 N.m 0<N<3,600 74,230 A-T CA OP LC 1100 N/30 N.m ~0/ mm/ ,806<N<18,967 25,119 A-T CA OP LC 1100 N/30 N.m ~0/ mm/ ,500<N<16,200 21,454 A-T CA OP LC 1100 N/30 N.m ~0/ mm/21.97 >21,250 A-T CA OP DC 7.53 mm/20.68 ~0/ N/30 N.m 43,750 A-T VA IP LC (3680 N/50 N.m) ~0/ S/B mm/ <B<244 1,393 A-T VA IP LC (3680 N/50 N.m) ~0/ S/B 9.81 mm/ <B<300 1,710 A-T VA OP LC (3680 N/50 N.m) ~0/ S/B mm/ <B<638 1,501 A-T VA OP LC (3680 N/50 N.m) ~0/ S/B mm/ <B< A-T VA OP LC (3680 N/50 N.m) ~0/ S/B 7.63 mm/ <B<510 2, st observed crack (with the length of 1 mm) by periodic test interruptions and visual examination 2 In LC defined by sudden increase of displacement amplitude (log scale), in DC defined by 40% load drop 3 A- Axial, T- Torsion, CA- Constant Amplitude, VA- Variable Amplitude, LC- Load-Controlled, DC- Displacement-Controlled, IP- In-Phase, OP- Out-of-Phase 4 Bold font indicates the estimated initiation life which is calculated based on the ratio of initiation to total life of the test with the same loading which is stopped in regular intervals for crack growth monitoring 5 S/B is seconds per block 121

142 Table 6.2: Summary of component crack initiation and total life experiments and predictions. Test Type Load Type Load Path ε 1,max,0 Control Amplitude (Range) R P /R M Experimental 1.Measured 2.Estimated Initiation Life (Cycles or Blocks) Prediction (MNS) Prediction (CED on MNS Plane) Prediction (CED Critical Plane) Total Life (Cycles or Blocks) Experimental Prediction A CA N.A N ~0 1: >148, ,752 A CA N.A N ~0 1: 28,000<N<35,000 2: 18,165<N<22,706 20,008 20,983 20,809 1: 69,683 2: 107, ,516 A CA N.A N ~0 1: 36,359 37,496 A CA N.A N ~0 1: 4,000<N<8,000 2: 3,780<N<7,560 2: 3,786<N<7,572 6,339 6,420 6,383 1: 19,069 2: 20,180 3: 19,100 18,493 A CA N.A N 0.2 1: 16,000<N<32,000 2: 23,456<N<46,912 12,779 10,632 10,543 1: 110,129 2: 161,447 53,687 A CA N.A N 0.2 1: N<3,000 2: N<3,826 3,665 2,932 2,883 1: 20,414 2: 26,034 9,195 A VA N.A (3000 N) ~0 1: B<1500 2: B<2,598 1,951 1, : 6,327 2: 3,653 4,268 A VA N.A (3600 N) ~0 1: B<300 2: B< : 2,615 2: 2,454 1,

143 Table 6.2 (continued): Summary of component crack initiation and total life experiments and predictions. Test Type Load Type Load Path ε 1,max,0 Control Amplitude (Range) R P /R M Experimental 1.Measured 2.Estimated Initiation Life (Cycles or Blocks) Prediction (MNS) Prediction (CED on MNS Plane) Prediction (CED Critical Plane) Total Life (Cycles or Blocks) Experimental Prediction A-T CA IP N, 30 N.m ~0/-1 1: 7,200 2: 15,510 11,348 10,209 10,119 1: 67,406 2: 31,290 25,646 A-T CA OP N, 30 N.m ~0/-1 1: 13,500<N<16,200 2: 15,806<N<18,967 21,585 25,389 18,677 1: 25,119 2: 21,454 55,765 A-T VA IP 2.29 (3680 N, 50 N.m) ~0/-1 1: 150<B<300 2: 122<B< : 1,393 2: 1,710 3,680 A-T VA OP 2.30 (3680 N, 50 N.m) ~0/-1 1: 170<B<340 1: 340<B<510 2: 319<B<638 1, : 1,501 2: 800 3: 2,000 4,

144 Figure 6.1: Vehicle cradle mount FE model. Figure 6.2 : Definition of failure illustrated by variation of axial and rotational displacements in constant amplitude in-phase axial-torsion tests of the cradle mount. 124

145 (a) (b) Figure 6.3: Experimental response for constant amplitude axial-torsion loading of components for (a) IP and (b) OP tests. 125

146 (a) (b) Figure 6.4: Axial displacement versus rotation angle of the component for different cycles throughout the CA and A-T tests of (a) IP and (b) 90 OP. 126

147 Figure 6.5: Axial displacement versus rotation angle of the component for mid-life cycles for CA and A-T IP and 90 OP tests. 127

148 (a) (b) Figure 6.6: Evolution of crack length and depth for constant amplitude axial-torsion (a) in-phase test with N f = 31,290, and (b) out-of-phase test with N f = 21,

149 Figure 6.7: Crack initiation to total fatigue life ratio for all types of loadings. 129

150 Figure 6.8: Maximum principal strain location for constant amplitude in-phase loading simulation. 130