Fatigue Crack Initiation and Propagation Life Prediction of Materials

Transcription

1 Fatigue Crack Initiation and Propagation Life Prediction of Materials Y. S. Upadhyaya and B. K. Sridhara Abstract Strain controlled fatigue life of EN 19 steel and T6 aluminum alloy have been predicted considering both crack initiation and crack propagation phases. Continuum Damage Mechanics (CDM) approach models the crack initiation life with a damage and damage beyond the crack initiation phase is predicted by Fracture Mechanics in terms of crack size. Fatigue life is predicted for a smooth (un-notched) specimen in the strain amplitude range of 0.3 % to 0.7 %, in room temperature at stress ratio of -1. The inputs required for both the models have been determined by conducting monotonic, cyclic, fracture toughness and fatigue crack growth tests. Predicted life is also compared by conducting strain controlled fatigue tests and predicted life compares well with the experimental life. The results show that by using monotonic, cyclic and fracture parameters of a material available in literature and by conducting Multiple Step Test on one fatigue specimen, it is possible to determine strain controlled crack initiation and crack propagation life of metals and alloys. Keywords Continuum damage mechanics, fatigue, fracture mechanics, life prediction. T I. INTRODUCTION HE components used in aircraft, space and automobile applications are designed based on fatigue consideration. Time varying cyclic loads result in failure of components at stress s below the yield or ultimate strength of the material. This phenomenon is called fatigue and fatigue failure of components takes place by the initiation and propagation of a crack until it becomes unstable and then propagates to sudden failure. The total fatigue life is the sum of crack initiation life and crack propagation life. Fatigue life prediction has become important because of complex nature of fatigue as it is influenced by several factors, statistical nature of fatigue phenomena and time consuming fatigue tests. Though a lot of fatigue models have been developed and used to solve fatigue problems, the range of validity of these models is not well defined. No method would predict the fatigue life with the damage by separating crack initiation and propagation phases. The methods used to predict crack initiation life are mainly empirical [1] and they fail to define the damage caused to the material. Y. S. Upadhyaya is working as Assistant Professor with Department of Mechanical and Manufacturing Engineering, Manipal Institute of Technology, Manipal University, Manipal , India. (phone: ; fax: ; ys.upadhyaya@manipal.edu). Dr. B. K. Sridhara is working as Professor with Department of Mechanical Engineering, The National Institute of Engineering, Mysore , India. ( deanaa@nie.ac.in). Stress or strain based approaches followed do not specify the damage caused to the material, as they are mainly curve fitting methods. The Fracture mechanics approaches are suited for the situations in which it is not necessary to know state of stress or damage in the vicinity of crack tip. Loading configuration issues and use of experimental fracture mechanics data to components may not be appropriate if situations such as complex crack size and shape, pronounced plasticity or multiple cracking, are involved. These limitations of fracture mechanics motivated the development of micromechanics models termed as local approaches based on Continuum Damage Mechanics (CDM). The local approaches are based on application of micromechanics models of fracture in which stress/strain and damage at the crack tip are related to the critical conditions required for fracture. These models are calibrated through material specific parameters. Once these parameters are derived for particular material, they can be assumed to be independent of geometry and loading mode and may be used to the assessment of a component fabricated from the same material. Ductile fracture in metals involves considerable plastic damage via, nucleation, growth and coalescence of microvoids. The phenomenon of initiation and growth of cavities and microcracks by large deformations in metals is called ductile plastic damage. This has been extensively studied by means of micromechanics and continuum damage mechanics models [2, 3, 4]. All models mentioned above are a good representation of physical mechanism at the microscale level, but difficulties arise when these are to be included in large scale structures to predict ductile failures. This difficulty has been overcome by developing model incorporating damage variable with the concept of effective stress. This stress, written as the mean density of forces acting on the elementary surface that effectively resists, has been introduced by Kachanov [5] to model creep rupture. The CDM approach has been developed further for dissipation and low cycle fatigue in metals [6], for coupling between creep and damage [7] and for creep fatigue interaction [8]. Later the thermodynamics of irreversible process provided the necessary scientific basis to justify CDM as a theory [9]. The degradation in mechanical properties due to damage is an irreversible phenomenon, which takes into account the effect of damage on the mechanical properties of materials and the influence of external conditions on subsequent development of microvoids. This concept was further developed and applied by Lemaitre [10], Simo and Ju [11] and Chaboche 13

2 [12]. A parallel approach to the micromechanical approach based on CDM is the phenomenological approach. In this approach, damage evolution is defined in terms of a single variable. These models are based on the concepts of Kachanov [5] and they account for different loading conditions. Lemaitre s model [10] is one of the models based on the phenomenological approach. The basic assumption made in the Lemaitre s model as, damage is isotropic and ductile in nature. According to the phenomena of ductile damage, plastic strain at threshold and the critical damage corresponding to the void growth and void coalescence respectively are set as criterion to describe the material degradation. Chandrakanth and Pandey [13] developed an isotropic ductile plastic damage model in the framework of internal variable theory of thermodynamics. Yingchun Xiao [14] developed an isotropic CDM framework which can reasonably show the interaction mechanism of brittle damage, ductile damage and creep damage. Abilio Jesus [15] formulated a fatigue model in the framework of CDM based on an explicit definition of fatigue damage. Xiao et al. [16] developed a thermodynamics based CDM model to predict the high cycle fatigue life. A good agreement between the theoretical log S - log N curve by using this model and experimental one is obtained. Thermodynamics based CDM model developed by Baidurya and Bruce [17] predicts crack initiation life for strain controlled fatigue loading. A nonlinear CDM model developed by Jing Jian Ping et al. [18] to assess the low cycle fatigue life of 200 MW steam turbine rotor under cold start and sliding parameter stop, shows the application of CDM model to a component. Similarly, other research work [19, 20] also shows the application of CDM at component level. From these literatures it is clear that CDM approach has the main advantage of transferability of data from specimen level to component level. Also wide range of CDM models to predict life in case of low cycle fatigue, high cycle fatigue, variable amplitude loading, multiaxial loading, creep-fatigue interaction, specimen/component, have been developed. Therefore, by combining both CDM and fracture mechanics approach, it is possible to predict crack initiation and crack propagation life in fatigue loading. Here, term damage refers to surface density of cracks and cavities in a section (CDM definition) at lower strain amplitudes and crack size (Fracture Mechanics definition) at higher strain amplitudes. II. LIFE PREDICTION MODELS In CDM, the damage, on an elemental cross-sectional plane is quantified by the surface density of cracks and cavities at that section [10]. If damage can be considered the same regardless of the orientation of the cross section on which it is measured, then damage is isotropic, and is quantified by one single scalar variable, D, a dimensionless number between zero and one. D=0 for undamaged material and D=1 for fully broken material in two parts. Damage is considered as isotropic in this study. Using the effective stress concept, the nominal stress is related to effective stress through the damage variable in the material as σ = σ/(1-d) (1) σ - nominal stress σ - effective stress considering the defect in the section D - damage variable If E is the modulus of elasticity of undamaged material, then E is considered as modulus of elasticity of damaged material. Both E and E are related by applying eq. (1) to the elastic strain tensor by assuming that the Poisson s ratio is not affected by damage [21], as mentioned in eq. (2). E = E(1-D) (2) Equation (2) helps to estimate the state of damage in a deformable material by experimentally determining its reduced modulus of elasticity. Considering the experimental difficulties to determine the reduced modulus of elasticity, Marcilio et al. [22] have developed a numerical simulation method. In the context of CDM, failure is not necessarily fracture, but is the point at which the damage causing process becomes localized and leads to the growth of a dominant defect. In slow ductile deformation, damage corresponding to the onset of rupture is termed as critical damage and in fatigue or creep it is associated with initiation of a macroscopic crack. This critical damage termed as D c, generally vary in the range between 0.15 and 0.85 [23]. Therefore in CDM, a general statement is made as, failure occurs when the damage variable equals the critical damage D c 1. D c is an intrinsic material property [24] and it helps to predict the failure in a complex loading situation by using the of D c obtained from a simple static tension test carried out on the same material and at same temperature. A. Isotropic damage model for uniaxial loading Baidurya and Bruce [17], developed an isotropic damage growth model under uniaxial (monotonic) loading by using the constitutive law described by Ramberg-Osgood model, applied to effective stress-actual strain relationship. According to this model, plastic strain is mainly responsible for damage growth. According to the model, damage initiates only after the accumulation of threshold plastic strain. The model establishes a relation among damage, plastic strain and two constants which could be derived using monotonic stressstrain parameters. The plastic strain corresponding to fracture ductility ε f, gives the critical damage for the material as C D = ε n p + C 1 where ε p is plastic strain, n is strain hardening exponent and C 1, C 2 are functions of monotonic stress-strain parameters. B. CDM based fatigue damage model for crack initiation Fatigue failure occurs, after number of cycles, at load levels (3) 14

3 below the static monotonic failure stress, but above the endurance limit. With each cycle, additional damage is introduced in the material, and the damage at the end of one cycle acts as the initial damage for the damage increment in the next cycle. No. of cycles, N i is the cycles to macroscopic crack initiation (localization). Crack initiation occurs when the critical of damage is reached. In any given cycle, unloading portion of a hysteresis loop and compressive stresses are assumed as not to contribute to damage. Consequently, only the reloading section above the endurance limit, S e, in the positive stress region causes damage to increase [25, 26]. Using Ramberg-Osgood model for the hysteresis loop and the principle of strain equivalence, Baidurya and Bruce [17], developed model to predict the number of cycles for crack initiation in case of strain controlled loading. The model demands input in terms of parameters of hysteresis curve, apart from monotonic properties and cyclic properties of the material. An important parameter which decides the crack initiation criteria is critical damage D c and is obtained by CDM based uniaxial loading model. C. Fracture Mechanics based fatigue crack growth model Crack propagation life is estimated using Paris equation [27] and for this initial crack length, a i, is approximately estimated [27] as a i =( K th /2S e ) 2 /π (4) where S e is endurance strength of material and K th is the long crack threshold stress intensity. Final crack length corresponds to fracture toughness of the material. III. MATERIALS AND METHODOLOGY The objective of the present study is to predict strain controlled fatigue life of materials by considering a ferrous and a nonferrous material. The literature shows that, not much work has been done on the strain controlled fatigue life prediction with the damage for EN 19 steel and 6082-T6 aluminum alloy. Therefore, EN 19 steel (a 1% typical chromium molybdenum steel with higher molybdenum) and 6082-T6 (solution heat treated, artificially aged and alloyed with manganese, silicon) aluminum alloy are considered in the present study. EN 19 steel is used for gears and high strength shafts and its chemical composition is shown in Table I is typically used in trusses, bridges and transport applications and its chemical composition is shown in Table II. TABLE I CHEMICAL COMPOSITION OF EN 19 STEEL Constituents C Si Mn S Cr Mo % by weight TABLE II CHEMICAL COMPOSITION OF 6082-T6 ALUMINUM ALLOY Constituents Si Cu Fe Mn Mg Pb % by weight The predicted fatigue life correspond to smooth specimen subjected to strain controlled fatigue loading in room temperature at stress ratio of -1. The literature shows that for the same nominal grade of a material, properties reported by researchers are different. Therefore, in this study, all inputs required for the model have been determined by conducting monotonic, cyclic, fracture toughness and fatigue crack growth tests on both the materials. Specimens of EN 19 steel have been prepared by using a single rod of 80 mm diameter and 6082-T6 aluminum alloy specimens are prepared from the single 150 x 30 mm flat bar. Finally, the predicted results have been compared with experimental results by conducting strain controlled fatigue tests. IV. MATERIALS CHARACTERIZATION In order to use CDM and Fracture Mechanics models for fatigue life predictions, various material properties are required. These material properties have been determined for EN 19 steel and 6082-T6 aluminum alloy by conducting monotonic, cyclic, fracture toughness and fatigue crack growth test, on smooth specimen at room temperature. All these different types of tests have been conducted on a ±50 kn axial type srvohydraulic testing machine as per the respective ASTM standards and details of each test is mentioned below in brief. A. Monotonic test Specimen of 6.35 mm of gauge diameter and having threaded ends for gripping, were prepared as per ASTM standard E8-04 [28]. A total of 8 specimens have been tested to get consistency in the monotonic properties. Initially for a known material, a dummy test was conducted and the results obtained were correct. Extensometer of 25 mm is used and a stroke rate of 2 mm/ min. is maintained during the test. The monotonic properties have been determined as mean of results of 8 specimens and are shown in Table III and Table IV for both the materials. TABLE III MONOTONIC PROPERTIES OF EN 19 STEEL Ultimate tensile strength (S u ) MPa ductility (ε f ) % offset yield % elongation on strength (S y ) MP 25 mm gauge Monotonic strength strength (σ f ) MPa coefficient, K (MPa) Modulus of Monotonic strain 185 elasticity (E) GPa hardening exponent, n TABLE IV MONOTONIC PROPERTIES OF 6082-T6 ALUMINUM ALLOY Ultimate tensile strength (S u ) MPa 0.2 % offset yield strength (S y ) MP strength (σ f ) MPa ductility (ε f ) % elongation on 25 mm gauge Monotonic strength coefficient, K (MPa)

4 Modulus of elasticity (E) GPa B. Cyclic test 73.9 Monotonic strain hardening exponent, n The cyclic stress-strain response of a material is characterized by the cyclic stress-strain curve (CSSC). CSSC is defined as the dependence of the saturation stress amplitude on saturation plastic strain amplitude. Cyclic properties of a material presented as cyclic strength coefficient K and cyclic strain hardening exponent n are obtained by determining cyclic stress strain curve. CSSC could be determined from Companion Specimen Test (CST) or Multiple Step test (MST) or Incremental Step Test (IST). These test methods are different because of strain cycling of test specimens and the number of test specimens required. A comparison of results obtained by different tests [29] shows that MST procedure is recommended as it uses only one specimen and also gives good result. In MST procedure single specimen is subjected to multiple steps of fully reversed axial strain cycles, each step with different strain amplitude, being applied one after other, until saturation is achieved at each step. Therefore here, MST procedure is followed to determine CSSC at 0.2 Hz, in the strain amplitude range from 0.1% to 1.2% with step increment of 0.1% using one fatigue specimen. Fatigue specimens required for MST and fatigue test have been designed for 6.35 mm gauge diameter and with threaded ends for gripping as per ASTM standard E [30]. The stress-strain curve have been obtained by MST for both the materials. By plotting stabilized stress vs plastic strain amplitude in a log-log plot, cyclic properties are determined as cyclic strength coefficient K = MPa and cyclic strain hardening exponent n = (for EN 19 steel) and cyclic strength coefficient K = MPa and cyclic strain hardening exponent n = (for 6082-T6 aluminum alloy). C. Fracture toughness (K IC ) test Fracture toughness is a material property and is influenced by material thickness. Here, plane strain fracture toughness of the material is determined as it is independent of material thickness. Fracture toughness test is conducted on Compact Tension C(T) specimens as per ASTM standard E [31]. Assuming K IC = 50 MPa m for steel and K IC = 25 MPa m for aluminum [32], C(T) specimen thickness of 12.5, 20 & 25 mm are considered using eq (5) for both materials. All specimens have been prepared as per ASTM standard E [31], and the notch is obtained by wire cut EDM method. and 25 mm specimens. Averaging these s, recommended K IC = 44.8 MPa m for EN 19 steel and K IC = 21.1 MPa m for 6082-T6 aluminum alloy. D. Fatigue crack growth test Fatigue crack growth (FCG) test is required to determine the constants C, m, used in the Paris equation. FCG test is conducted using C(T) specimen having a thickness of 6.25 mm, as per ASTM standard E [33]. As all fatigue tests are carried out at R = -1, to model the crack growth, FCG test be also conducted at R = -1. The literature shows that majority of researchers, conduct FCG test using C(T) specimens, as it best models crack growth in an opening mode. The C(T) specimen is not recommended for tension compression testing because of uncertainties introduced into the loading experienced at the crack tip [33]. Therefore, though the FCG test is conducted using C(T) specimen at R=0.3, the constants C and m have been modified for R = -1, using Walker equation [27]. The da/dn vs ΔK curve obtained from FCG test on EN 19 steel and 6082-T6 aluminum alloy at R=0.3. Using experimental curve of da/dn vs ΔK and Walker equation, calculated for C= 3.37x10-10, m=3.37, ΔK th = MPa m (for EN 19 steel) and C= 4.84 x10-10, m=3.37, ΔK th = 9.54 MPa m (for 6082-T6 aluminum alloy) at R= -1. V. FATIGUE LIFE PREDICTION AND VALIDATION Using isotropic damage model of uniaxial loading as mentioned in eq. (4) and monotonic properties, critical damage D c is obtained at ε p = ε f and its is D c =0.54 (for EN 19 steel) and D c =0.44 (for 6082-T6 aluminum alloy). The crack in fatigue loading initiates at D = D c. Crack initiation life is obtained using CDM based fatigue model and input to the model have been provided by cyclic test result and the parameters of stabilized hysteresis curve. It is also observed that, crack initiation life is sensitive to each parameter of the stabilized hysteresis curve. Crack initiation life corresponds to an approximate crack size obtained using eq. (4) and this size is the initial crack size for crack growth model. Crack propagation life is determined using Paris equation, by integrating between the initial and final crack size. Crack propagation life is sensitive to initial crack size. The predicted crack initiation and crack propagation life are as shown in Fig.1 and Fig. 2. B > 2.5(K IC /S y ) 2 (5) Initial precracking is done by fatigue loading and fracture toughness is obtained by tensile loading. Fracture toughness s have been obtained for all the specimens. As the experimental and assumed of K IC are close, the design of C(T) specimen in terms of thickness is valid. Using the experimental of K IC and eq. (5), the plane strain fracture toughness, correspond to thickness of 12.5, 20 16

5 ACKNOWLEDGMENT The authors would like to thank Bangalore Integrated System Solutions (P) Ltd. (BiSS), Bangalore, India, for providing the facility to conduct all types of tests. Fig. 1 Fatigue life of EN 19 steel Fig. 2 Fatigue life of 6082-T6 aluminum alloy Validation of predicted fatigue life is carried out by conducting a few fatigue tests. By observing the hysteresis curve obtained by MST procedure, strain amplitudes in the range 0.3% to 0.7% are considered, for strain controlled fatigue tests. Two specimens have been tested at each strain amplitude, to get repeatability and four strain amplitudes are considered for fatigue testing. All tests are axial type, conducted in room temperature at 0.2 Hz frequency and at stress ratio of -1. The experimental fatigue lives obtained by these tests are also shown in Fig. 1 and Fig. 2. The predicted total life agrees with the test s within experimental error. The results trend presented are in line with literature [17]. VI. CONCLUSION One of the critical parameter used to predict total fatigue life is D c, which decides the crack initiation life. As it is difficult to experimentally determine the crack initiation life, total predicted life is used for comparision. Predicted total fatigue life agreeing with experimental work shows that D c obtained by monotonic CDM model, could be used to predict crack initiation life under fatigue condition. Generally monotonic, cyclic and fracture parameters of a material are available in literature for a material, but hysteresis curve with its parameters are rarely reported. Using one specimen and by conducting MST it is possible to get stabilized hysteresis curve and its parameters at any strain amplitude. Therefore, using literature data and by conducting MST, strain controlled fatigue life could be predicted for metals and alloys. REFERENCES [1] Q. W. Wang, Y. Berard, S. Rathery, and C. 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