Analysis of fatigue crack initiation in cyclic microplasticity regime

International Journal of Fatigue xxx (xxxx) xxxx

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Analysis of fatigue crack initiation in cyclic microplasticity regime A. Ustrzycka, Z. Mróz, Z.L. Kowalewski, S. Kucharski Institute of Fundamental Technological Research, Polish Academy of Sciences, Poland

ARTICLE INFO

ABSTRACT

Keywords: Fatigue crack initiation Micro-plasticity Damage evolution Optical ESPI method Micro indentation

The present work provides description of fatigue crack initiation in metals subjected to cyclic loading within the nominal elastic or initial elastic-plastic regimes next passing to elastic response during cyclic deformation and shake down process. It is assumed that damage growth proceeds due to action of local stress, specified as the sum of mean stress and its fluctuations induced by material inhomogeneities such as grain boundaries, inclusions, cavities, boundary asperities, also due to design notches or holes introduced into the element. The damage growth model is proposed, based on the critical plane concept. The macrocrack initiation then corresponds to a critical value of accumulated damage. The modelling of damage growth is supported by Electronic Speckle Pattern Interferometry (ESPI) apparatus using the coherent laser light. The damage growth effect is analysed by microindentation tests. The fatigue tests are performed for high strength steel specimens with central hole.

1. Introduction Recognizing and understanding fatigue failure processes in materials subjected to cyclic loading in the pre-yield, micro-plastic regime of modern high-strength steels are essential. The initial state of fatigue crack nucleation process begins by dislocation cell formation inside a local micro-plastic zone [1]. The location of this plastic zone strongly depends on a type of applied loading and hardness of individual microstructure grains. During the fatigue process under stress amplitudes below the yield point, the plastic deformation is limited to a small number of grains or inclusions in materials selected for their toughness and durability [2]. The stress amplitude must achieve a threshold value, over that the micro-plastic zones are created and the microscopic cracks are initiated. The yielding process may develop at the low mean stress level in some grains due to local strain concentrations at their boundaries [3]. Such stress fluctuations, developing at a fraction of the macroscopic elastic limit, are the source of initial structural defects and microscopic plastic mechanisms controlling the evolution of defect ensemble, tending toward the state of advanced yielding [4]. Therefore, the main aim of this paper is a description of crack initiation mechanisms in elastic-plastic materials subjected to cyclic loading in the pre-yield, micro-plastic regime. Recently, the observation of micro-plasticity exhibited interesting phenomena, that have been detected and considered by numerous researchers [5–11]. The microplasticity is understood as a local phenomenon in all metallic materials showing the micro-plastic flow in small localized domains while the material globally remains in the

elastic state [7]. As a consequence, the study of micro-plastic regime leads to the fundamental understanding of the microscopic origins of plasticity in metallic materials [10]. The cyclic stress-strain curves exhibiting the hysteresis loops obtained for selected cycles with different stress amplitudes σai below the macroscopic yield stress σy highlighting the regime of micro-plasticity are presented in Fig. 1. First deviations from the linear elastic response can be observed on the closed loadingunloading loop (indicating inelastic response). The dislocation structure does not change during cyclic loading as long as the loop remains closed (true elastic limit). The enclosed loop area, representing plastic dissipation is dependent on the strain amplitude and initial dislocation density. The plastic yielding does not start at the same time in all grains. During the cyclic loading process the material deforms heterogeneously and numerous strain concentration spots are visible, evidencing microplastic activity at small fractions of the macroscopic yield strain [11]. Such deviations from elasticity in the pre-yield regime can be seen during time-dependent loading as distinctly visible displacement fluctuations. The aim of this paper is a description of crack initiation mechanisms in elastic-plastic materials subjected to cyclic loading in the pre-yield, micro-plastic regime. Also it provides experimental and analytical description of stress and strain fluctuations and incorporates them into the fatigue criteria based on the local stress values. The stress and strain fluctuations are induced by material inhomogeneities, dislocation microstructure, grain boundaries, cavities or inclusions introduced in the element and free boundary effects. They are also generated by design

E-mail addresses: [email protected] (A. Ustrzycka), [email protected] (Z. Mróz), [email protected] (Z.L. Kowalewski), [email protected] (S. Kucharski). https://doi.org/10.1016/j.ijfatigue.2019.105342 Received 5 July 2019; Received in revised form 17 October 2019; Accepted 18 October 2019 0142-1123/ © 2019 Published by Elsevier Ltd.

Please cite this article as: A. Ustrzycka, et al., International Journal of Fatigue, https://doi.org/10.1016/j.ijfatigue.2019.105342

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plastic mechanisms controlling the evolution of defect ensemble towards the state of advanced yielding. Assume the continuum approach in description of fluctuation states. The fluctuation stress and strain fields ~ (x ) , ~ (x ) superimposed on the mean stress and strain ¯ , ¯ in the representative element, satisfying uniform boundary conditions, provide the total stress and strain fields, thus

(x ) = ¯ + ~ (x ),

(x ) = ¯ + ~ (x )

(1)

Then the total strain energy can be presented as a sum of the mean strain and fluctuation energy

1 · d 2

=

1 ¯·¯ 2

1~ ~ · d 2

+

(2)

where Ω is the volume of material element. For linear elasticity the local stress-strain relation is = E where E = E(x) is the stiffness tensor and for the homogenized element ¯ = E¯ ¯ where E¯ is the effective stiffness tensor. Eq. (2) can now be written in the form

Fig. 1. Schematic stress-strain curve and the hysteresis loops showing the regime of micro-plasticity.

notches or holes introduced in the element. The process of formation of stress fluctuations occurs at several scales (Fig. 2). Irreversible dislocation glides, at the edges of metallic materials, appearing under cyclic loading usually lead to the development of persistent slip bands (PSB) [12]. The extrusions and intrusions in boundary surface grains are created (Fig. 2a). The nature of slip characteristics often dictates the surface topography near the crack initiation site [13]. Intergranular cracking from grain boundaries is an important fatigue damage mode in materials during cyclic deformation [14]. In general, fatigue cracks initiate at grain boundaries where impinging slip causes stress concentration (Fig. 2c). The stress fluctuations induced by dislocations result from dislocation motion during local plastic deformation and nonuniform distribution of dislocation density (Fig. 2d). Within the bulk of material this generates nonuniformity of deformation Fluctuations developing at a fraction of the macroscopic elastic limit are the source of initial structural defects and microscopic

U ( )d

=

1 ¯ E ¯· ¯ 2

+

1 2

E ~ · ~d

(3)

where U ( ) = 2 · is the specific strain energy density. The stress fluctuation can be expressed as follows 1

~ = E ~ + (E

~ E¯ ) ¯ = E ~ + E ¯ = E ~ + p

(4)

~ ~ where E is the fluctuation of stiffness matrix and p = E ¯ is the polarisation stress tensor inducing the fluctuation field. Assume the homogenized stiffness matrix with the mean stress and strain to be determined. The fluctuation stress field then satisfies the equilibrium equations and uniform boundary condition ~ n = 0 on the loaded boundary where n is the normal vector. Similarly, the strain and ~ (x ) are kinematically admissible displacement fluctuation fields ~ (x ) , u ~ = 0 on the displacement controlled boundary. Thus, for any and u statically admissible stress fluctuation field ~ (x ) and kinematically admissible strain and displacement fluctuation fields, one obtains

Fig. 2. Formation of stress and strain fluctuations at different scales (a) boundary fluctuation, (b) stress fluctuation at the inclusion, (c) grain boundary fluctuation (d) stress fluctuation due to dislocations microstructure. 2

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~ · ~d

(E ~ + p)· ~d

=

=0

promising tool for parameter identification of the constitutive model, and for demonstration of its validity.

(5)

In view of Eq. (5) it can be stated that the fluctuation energy functional

=

1 ~~ E · + p· ~ d 2

2. Experimental testing of high strength steel 2.1. Material, specimen preparation and testing apparatus

(6)

The specimens used in this study are manufactured using X10CrMoVNb9-1 (P91) polycrystalline steel. This is a low carbon, creep-resistant steel, typically used for tubes, plates and structural components in the power plant industry. P91 steel is enriched with vanadium and niobium. Such chemical composition of P91 steel improves its mechanical and thermal properties. Because of its attractive thermal properties, it is being increasingly used for power engineering either conventional or nuclear. The content of alloying elements in P91 steel specimens is given in Table 1. Fig. 3 represents the scanning electron microscopy (SEM) microstructure of P91 steel. The average grain sizes were estimated to be within the range 10–30 μm. Selected mechanical and fatigue tests were carried out with an MTS 858 servohydraulic testing machine working in a feedback loop at a frequency of 20 Hz. The machine enables axial loading of specimens within the force capacity of ± 25 kN. The system is equipped with the MTS controller Flex60 and relevant software. The tests were performed using paddle-shaped flat specimens cut out of the steel block delivered by the manufacturer. The dimensions of specimen are described in Fig. 4. All fatigue tests were stress controlled and the specimens were subjected to the axial cyclic load of 7 000 N (R = 0). The area of local

= 0 in view of Eq. (5). Relations (5) and reaches an extremum since (6) provide the variational framework for determination of stress and strain fluctuations, including approximate methods. The strong minimum of the functional (6) can also be demonstrated as has been shown in a separate paper [15]. The strain and stress fluctuations develop and evolve differing scales. For polycrystalline materials, grain boundaries are natural stress concentration sources, generating nearly periodical fluctuations, cf. [16]. Defects, such as inclusions or cavities generate fluctuations rapidly decaying in the material along the normal distance from the defect. Such type of fluctuation will be discussed in Section 5. Similarly, the boundary inhomogeneity and roughness generate boundary fluctuation, cf. [17]. Finally, the microscopic heterogeneous slip modes induced by long range dislocation interactions generate fluctuations on the microscale, their analysis requiring stochastic approach, cf. [18]. Using the potential offered by the novel experimental techniques, it is possible to identify physical phenomena and to describe the mechanisms of degradation and fatigue damage development [19]. The analysis of stress and strain localization preceding crack initiation is performed for high-strength and fine-grained steel by means of the optical method ESPI (Electronic Speckle Pattern Interferometry), apparatus using the coherent laser light. The hole introduced in the element is regarded here as the softest inclusion, enforcing strong localization of strain, allowing to define the damage zone and the related strain and stress fluctuations (Fig. 2b). In this work the stress fluctuations induced by a hole have been analysed using ESPI device [20]. Our objective is to perform experimental study of the local distribution of stress and strain in damage zone at different levels. However, the limitations resulting from the used lens in ESPI device precluded the stress fluctuation detection inside grains and at grain boundaries. Despite of that, the proposed concept of constitutive modelling of crack initiation mechanisms based on the formation of stress fluctuations is universal, no matter what is the origin of strain fluctuation. Additionally, the micro-indentation method was used to estimate variation of local mechanical properties due to fatigue damage. The results of numerical analyses were correlated with experimental data and provided

Fig. 4. Engineering drawing of fatigue specimen with dimensions in [mm]: width of the minimum cross-section a = 17,8 mm, thickness h = 1,5 mm, hole diameter d = 6 mm.

Table 1 Chemical composition of tested steel P91 Pn 10216–2:2004 (in wt%). Element

C

Mn

Cr

Mo

V

Ni

Cu

Si

Nb

P 91

0.2 ÷ 0.5

0.3 ÷ 0.6

8 ÷ 9.5

0.85 ÷ 1.1

0.18 ÷ 0.25

< 0.4

< 0.3

0.08 ÷ 0.12

0.08

Fig. 3. SEM micrographs of the microstructure taken for two magnifications. 3

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but without the central hole. The stress-strain curve was captured for a first loading–unloading-reloading cycle (R = −1) and for stress amplitude ± 700 MPa, Fig. 5. The corresponding strain amplitude was ± 0.04. It can be seen that after the maximum tensile stress 700 MPa was reached, the reverse plasticity started at a compressive stress equal to −300 MPa approximately. The Wöhler curve was determined for cyclic tension (R = 0) and for range of loads 400–700 MPa, Fig. 6. In the second series of experiments the crack initiation was investigated using specimens with a central hole, Fig. 4. The cyclic tension load, with a frequency of 10 kHz, was applied (R = 0) that corresponds to 260 MPa nominal stress and it was far below the elastic limit. In view of the Wöhler curve, Fig. 6, for this level of stress the sample should exhibit unlimited fatigue life. However, the actual, local level of stress was much higher due to stress concentrator (the hole). At the hole boundary a local plasticity was generated and both compressive and tensile stress occurred locally during the nominally tensile cyclic loading. Therefore the fatigue life of these samples is much shorter that that depicted in Fig. 6. The detailed analysis of the second series of experiments is presented in the subsequent sections of this paper.

Fig. 6. S-N curve (Wöhler diagram) for P 91 steel.

stress and strain concentration was induced in the specimens by cutting out the hole at the middle of specimen gauge length. The hole is of the diameter of 6 mm, so the stress concentration is of macroscopic character. The applied loading amplitude coincides with the stress state at a section away from the hole at the maximum σn,max = 260 MPa and at the cross-section of the hole σn,max = 380 MPa. The fatigue specimens with the central hole were produced using electrodischarge machining. In order to eliminate the residual stress (local heating-cooling) and reduce roughness, the surface of specimens was electro-polished in the vicinity of hole. The ultimate goal was to prepare a specimen with the lowest possible roughness. This parameter plays a crucial role in the initiation of fatigue cracks and in microindentation tests. The experimental and theoretical study of the local stress-strain evolution allows for the analysis of crack initiation and growth under fatigue conditions, also for the assessment of local fatigue damage [21]. Monitoring strain distribution and analysing damage growth proceeds in the micro-strain regime. Basing of these observations, a methodology is proposed to estimate the effect of local stress on fatigue life of the specimen.

2.3. Numerical simulations Due to presence of the stress concentrator, the numerical simulations using finite element method (FEM) were applied to analyse stress state in the specimen. The associated flow rule for Huber-Mises-Hencky yield criterion, and the isotropic work hardening were taken into account according to the stress-strain curve presented in Fig. 5. One end of the sample was fixed and at the second end the uniformly distributed stress σ = 260 MPa was applied. This simplified approach enables to capture the local stress state in a vicinity of the hole boundary at the maximum (σn,max = 260 MPa) and minimum (σn,min = 0) stresses. The stress distribution in the specimen corresponding to the maximum and minimum load is presented in Fig. 7. The maximum local stress was equal 550 MPa approximately, and it was beyond the initial yield stress value σy = 545,5 MPa. A compressive residual stress of about 270 MPa was generated at the minimum load. The local stress-strain relationship calculated at the point of maximum stress concentration for first two cycles is presented in Fig. 8. Additionally, the comparison of numerical results with the experimental data is presented. The local strain measured by ESPI in the maximum stress concentration point versus the stress obtained during loading-unloading-reloading cycle for sample without the central hole

2.2. Mechanical behavior of P91 steel In general, our experiments included two series of tests. The first series was conducted in order to specify properties of P91 steel i.e. stress-strain characteristic and fatigue resistance (Wöhler curve). In these tests the samples were employed of the shape presented in Fig. 4

Fig. 5. Tension-compression curve for P91 steel. 4

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Fig. 7. Stress distribution obtained for the first cycle at (a) minimum and (b) maximum load.

Fig. 8. Local stress-strain relationship for the first two cycles calculated using FEM and compared with experimental stress-strain diagram.

Fig. 9. Evolution of the local strain during cyclic loading at the hole boundary.

the elastic strain occurs. It should be noted that there is a quantitative agreement between strain values measured using ESPI system and those calculated using FEM, Fig. 9.

(see Fig. 5) marked with a red curve is presented in Fig. 8. The numerical simulation with the isotropic hardening model indicates that for the applied level of nominal stress and the corresponding small range of strain, the work hardening of material took place at the first half-cycle (loading), and the subsequent unloading (up to 0 nominal stress) does not generate a reverse plasticity. This can be confirmed by analysis of the tension-compression curve, Fig. 5. Even for much greater strain (0.04) that was applied prior to the beginning of unloading and compression stage, the reverse plasticity was initiated at compressive stress greater than 300 MPa. This value was not exceeded by the stress calculated for the examined sample (with hole) at the minimum load (P = 0), Fig. 8, so the reverse plasticity was not achieved in the sample, and a kinematic hardening model then provides the same result. Therefore it is believed, that the applied simple isotropic hardening model is proper for the range of loads applied in our fatigue test and provides accurate values of stress at the hole boundary [22]. Starting from the second cycle the local stress-strain relationship becomes stabilized and exhibits a linear response in both loading and unloading processes. The evolution of total (elastic + plastic) strain at a point of the maximum stress concentration, for loading -unloading -reloading stages, is presented in Fig. 9. The first loading has a non-linear character as it corresponds to the elastic-plastic deformation, while in the unloading-reloading stage only

2.4. Fatigue testing and Electronic speckle pattern Interferometry (ESPI) measurement According to the loading parameters determined on basis of the preliminary tests carried out, the number of cycles to failure was estimated at the level of (65 ÷ 80) · 103 cycles. In order to eliminate vibrations of the testing machine during optical measurements using Electronic Speckle Pattern Interferometry (ESPI) the fatigue tests were interrupted several times after selected numbers of cycles. Such technique was also applied in a single loading cycle in a quasi-static regime. Scheme of loading during fatigue test is shown in Fig. 10. As it is shown, the first cycle was conducted manually, and subsequently, a block of cycles was carried out using the testing machine. The MTS 634.31F-26 longitudinal extensometer of 20 mm base was applied for the axial strain measurements that were also used to calibrate the ESPI testing apparatus. During each test the following parameters were recorded: time (s), force (N), piston displacement of the machine (mm), and strain (mm/mm). The use of ESPI method during fatigue enables the identification of areas of the greatest strain concentration where the stressinduced defects are located. The technique also enables identification of damage initiation spots with high accuracy. 5

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Fig. 10. Scheme of loading during the fatigue test.

2.5. Indentation testing The Anton Paar MHT (micro) and UNHT (nano) testers (Open Platform equipment) were used to perform the indentation tests. The MHT micro-indenter was applied to measure load–penetration depth curves at the micro-scale and the elastic modulus was calculated using Oliver-Pharr method [23,24]. The displacement resolution and load resolution were 0.3 nm and 100 μN, respectively, and the nominal radius of the applied diamond tip was R = 50 μm. The maximum applied load was 3 N.

Fig. 11. Force-displacement curves for selected cycles.

displacements field and to capture strain field evolution during cyclic loading. Introduction of a geometric discontinuity in the form of the hole in the central part of specimen introduces advantageous condition (stress and strain concentrations) for crack initiation. On the basis of strain fields specified at different load levels local hysteresis curves can be generated at selected points of specimen (see Fig. 12 (1–8)). The local stress cannot be measured directly in the experiment, so the hysteresis curve shows the nominal stress versus local strain. The

2.6. Applications of ESPI Electronic Speckle Pattern Interferometry (ESPI) is a non-destructive optical method of stress and strain monitoring for early detection, localization and monitoring of damage in materials under fatigue loading [25]. A typical measuring system consists of a CCD camera localized in the head of the system, four light sources and the specimen. A reference speckle pattern formed by a reference beam is also observed by the camera. The light source used in ESPI measuring apparatus is a 75-mW Nd: YAG laser, emitting a green beam with wavelength λ = 785 nm. This is divided by a beam-splitter into four beams. The results obtained from ESPI method derive from physical surface deformations. This noncontact and noninvasive technique is used in the fields of experimental mechanics to obtain the displacement maps during loading conditions. The use of this method during fatigue enables the location of the greatest concentration of stress-induced defects and allows with high accuracy to predict a damage initiation. 3. Evolution of micro-strain regime during high cycle fatigue crack initiation

10-3

4

3

2

1

5

3.1. Global response during fatigue tests The experimental fatigue tests were executed during the load-controlled cyclic deformation. The samples were subjected to an axial cyclic load of 0–7 103N (R = 0). A series of curves (load vs. displacement of the machine grip) with loading and unloading cycles during the whole fatigue process, from the beginning up to the moment of crack initiation was recorded. The load-displacement curves for selected cycles performed with the test machine are shown in Fig. 11. After the first cycle, the hysteresis loops are not visible during the cyclic deformation process, indicating that the elastic shake-down state has been reached. The elastic-plastic deformation proceeds in the first cycle (red line in Fig. 11), inducing residual back stress growth and material hardening [26,27]. In the subsequent cycles the nominal linear elastic response is observed, probably with micro-plastic deformation localised at large strain fluctuation zones. The microplastic effects and hysteresis loops are observed in the last phase of cyclic process before macro-crack initiation. The damage growth and microcracking process then develop in the local strain concentration zones.

8

3.2. Analysis of damage growth based on ESPI method

7

6

Fig.12. Hysteresis curve showing the first loading-unloading cycle paused at selected stress levels for the strain distribution maps capturing.

The optical method ESPI is used to monitor and measure the 6

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nominal stress ¯ = P /(a d ) (P is the axial force, a is the width of minimum cross-section, d is the hole diameter) and the local strain at the perimeter are used to present hysteresis loops in Figs. 12, 15 and 16. The aim of the experiment was to follow the distribution of local strain fluctuations during loading/unloading cycles. The strain distribution maps, obtained for the first loading-unloading cycle, corresponding to the selected points of the curve are also presented in Fig. 12. Near the hole, the local zone of strain concentration has developed. As shown in Fig. 12 (3, 4, 5), during the specimen loading, the strain concentration develops in smeared shear bands (forming V shape) at the hole boundary. The initial stage of fatigue crack nucleation mechanism begins by formation of shear bands. As a consequence of the inhomogeneous plastic deformation at the root of the hole, the residual strain remains after unloading, Fig. 12 (8). The ESPI results in the form of local strain values are in good agreement with numerical results (Figs. 5, 8). The numerical simulations enable to estimate the maximum local stress concentration at the level of 550 MPa in a vicinity of the hole boundary. The maximal stress corresponds to the strain measured by ESPI, and it is still beyond the elastic limit. It should be noted that the stress values at the hole perimeter can be simply estimated on the basis of the measured strain field. To this goal, for each value of strain measured by ESPI at the maximum stress concentration point, the corresponding stress value can be found using tension-compression curve, Fig. 5, as the uniaxial stress state at the hole boundary can be assumed. The local stress values calculated using this simplified approach are equal to 550 MPa and −300 MPa for the maximum and the minimum loads respectively. These values agree with those calculated using FEM, Fig. 7. It can be seen, that in the first two cycles a new local value of R = −0.545 is reached, while the cyclic loading parameter for the whole specimen equals 0. The purpose of this work is to provide experimental analysis of the evolution of micro-strain regime during high cycle fatigue crack initiation. The analysis is also aimed at development of consistent description of the microplastic state of material. The whole fatigue process was divided into blocks of cycles and carried on the hydraulic testing machine. The process of cyclic loading was interrupted several times after selected numbers of cycles in order to perform displacement measurements by means of the ESPI camera. Strain distribution maps at maximum applied load are presented for increasing number of loading cycles for two specimens, Figs. 13 and 14. The different stages of the fatigue process, from the beginning up to the instant of crack initiation and its propagation are presented. The reference image for ESPI measurement was obtained for each individual

cycle before loading, therefore the strain distribution maps at particular cycles do not demonstrate strain accumulation. In the first cycle the strain concentration spot is visible near the hole in both presented cases, for specimens 1 and 2 (Fig. 13, 14, 1 cycle). In the next cycles the material is strengthened (Fig. 13, 14, 2 cycles), but the local zones of strain concentration in the vicinity of the hole can be still observed. In specimen 1, a high strain resulting from the crack localisation on the left side of the hole is presented (Fig. 13, 78057 cycles). In specimen 2, the high strain concentration in the damage zone on the right side of the hole before micro-crack propagation is visible. Strain distribution maps for this specimen with registered strain localization is presented in Fig. 14 (71145 cycles). The zone of local microplasticity was identified using ESPI technique. Based on that, one can conclude that during cyclic loading the micro-cracks start to nucleate in the zones of large strain accumulation. Afterwards, an evolution of the local hysteresis loops shape was studied using also ESPI technique. The hysteresis curves were plotted for the local strain, at the hole boundary, on the abscissa axis and mean value of stress on the cross-section with the hole on the ordinate axis. They show regime of micro-plasticity on both sides of the hole, Fig. 15 (specimen 1) and 16 (specimen 2). The slope and shape changes of local stress-strain hysteresis loops reflect the changes of local microplasticity zone and the evolution of micro-zone damage. During this analysis, the process of micro-crack initiation can be identified. The fact that the curves representing the first cycle (red curves) have the character of hysteresis loops implies that the local deformation is not entirely elastic. The area within the hysteresis loops represents the energy dissipation during the loading-unloading process. As a consequence, the material is strengthened and can withstand higher stresses. In the subsequent loading cycles, the enclosed area in the loops indicate that the cyclic loading and unloading is linear up to the moment of micro-crack initiation (see Figs. 15a, 16b). The final hysteresis loops (Figs. 15a, 16b, blue curve) show strongly a local effect of damage accumulation during fatigue represented by reduced stiffness. It is worth emphasizing that the hysteresis loops obtained for mean values of strain on the cross-section of the hole practically do not change for subsequent cycles (see Figs. 15c and 16c). A series of stress-strain hysteresis curves were recorded to reveal the role of local plasticity during fatigue failure process in materials subjected to cyclic loading in the pre-yield, micro-plastic regime. The relationship between stiffness degradation and damage evolution is shown. The evolution of the local hysteresis curves indicates the energy

10-3

1 cycle

2 cycles

40001 cycles

60001 cycles

78057 cycles (crack initiation)

Fig.13. Strain distribution maps on the surface of plane specimen using ESPI for different stages of the fatigue process (sample 1). 7

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10-3

1 cycle

2 cycles

40001 cycles

60001 cycles 71145 cycles (strain localization)

Fig. 14. Strain distribution maps on the surface of plane specimen using ESPI for different stages of the fatigue process (sample 2).

dissipation processes by the micro-damage evolution. The damage accumulation during fatigue is represented by the change of the slope due to reduction of stiffness at late fatigue cycles (see Fig. 16a). On the other hand, the reduction of the hysteresis loops width indicates cyclic hardening (Fig. 16a and b). The increase of the hysteresis loops width (Fig. 16b, blue curve) is compatible with the strain concentration obtained on the strain distribution maps for the same cycle (Fig. 14). The proposed concept of crack initiation mechanisms assumes that damage occurs due to action of mean strain and its fluctuations. These strain fluctuations are the source of initial structural defects and

microscopic plastic mechanisms controlling damage evolution. The strain profiles for two specimens 1 and 2 and the mean value in the cross-section of the hole for the second loading cycle (elastic regime) obtained from ESPI method are presented in Fig. 17. Each strain profile produces a corresponding stress profile in the cross-section of the specimen. The amplitude of fluctuation B2 as the difference between the mean and maximum value of strain is shown in Fig. 17. For both specimens the characteristics of strain curves are similar. The strong strain localisations at the vicinity of hole on the right side are observed in both cases.

Fig. 15. Local hysteresis loops (mean stress – local strain) obtained during selected cycles showing the regime of micro-plasticity: (a) left side of the hole; (b) right side of the hole; (c) mean values of strain on the cross-section of the hole (specimen 1). 8

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Fig. 16. Local hysteresis loops (mean stress – local strain) obtained during selected cycles showing the regime of micro-plasticity: (a) left side of the hole; (b) right side of the hole; (c) mean values of strain on the cross-section of the hole (specimen 2).

4. Damage assessment by micro-indentation

cyclic loading is commonly accepted as a measure of fatigue damage. As it was shown in the numerical solution, the work hardening of the material occurs in the examined zone, manifested as the increase of hardness. It takes place in the first loading cycle. So, the indentation tests, that are considered as a reference were performed after first loading cycle, when the material was already strain hardened and one can expect that the subsequent tests are affected only by a change of the elastic modulus due to fatigue damage. In Fig. 18 the representative (mean) load penetration curves for samples 1 and 2, measured in the initial state (after second loading cycle) and after 60,000 loading cycles, in the maximum stress concentration zone, are presented. It can be seen that the residual depth is practically unaffected during fatigue process, so the contact area and consequently hardness (for fixed P) remain unchanged. However, the inclination of the unloading curve, that is proportional to the elastic modulus, decreases due to fatigue damage. The evolution of the Young modulus during increasing number of

The micro-indentation test was used to examine local mechanical properties in the vicinity of hole in the initial state and after fatigue test. The spherical indenter of radius 50 µm was used and maximum penetration depth was approximately 5 μm. For these test parameters, the diameter of residual impression is relatively small, so several indents can be made in small stress concentration zones. On the other hand, the material volume examined in one test consists of several grains. The indentation was performed in quasi-static regime. The advantage of use of spherical indenters is, that in contrast to sharp tips, moderate deformations are generated. The micro-indentation tests were performed at different stages of fatigue process i.e. at the beginning of fatigue test and after 60,000 cycles. In general, the indentation test provides two material parameters, namely hardness and Young modulus. We focused on evaluation of the latter, as the evolution of elastic modulus during

Fig.17. Strain fluctuations B2 in the cross-section of the hole obtained from ESPI method. 9

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Fig. 18. Load penetration curves for specimens 1 and 2 in the initial state (after second loading cycle) and after 60,000 loading cycles: (a) specimen 1, (b) specimen 2.

concentration and microplasticity effects in the form of slip bands and short cracks. The model proposed in this paper is based on the continuum approach with account for local stress fluctuations. The mathematical description of fatigue damage growth and crack initiation is based on the concept of critical plane, proposed by Seweryn and Mróz [30,31]. The model is aimed to predict damage growth on an extremal plane in terms of averaged contact stress or strains for both monotonic and variable load actions [32–34]. Referring to Fig. 20, consider a notched element and the expected critical plane segment inclined at the angle θ to x-axis of the reference system (x,y). Denote the normal and shear stress on the critical plane facet of length d0 by n and n . The failure stress function is then introduced to specify damage and crack initiation condition for a combined stress action on the plane. The local stress condition for static failure corresponds to the critical value of the failure function

Fig. 19. Variation of Young modulus with increasing number of cycles obtained during cyclic loading tests for two stress amplitudes, compared with microindentation tests.

n

Rf = maxR

cycles determined using classical method and micro-indentation technique is presented in Fig. 19. For each curve, the reduction of the Young modulus is presented for different stress amplitudes, therefore the number of cycles to failure is different. The greatest decrease of local elastic modulus is observed for the number of cycles close to the fatigue life. At low amplitude, 500 MPa, the initiation stage may even last for the majority of the lifetime. Then, Young modulus is practically constant during the entire fatigue process, and changes rapidly at the moment of rupture. At a high amplitude, equal to 600 MPa, the initiation is usually accomplished within a small fraction of the fatigue life. The variations of Young modulus, during the cyclic loading process, for specimens 1 and 2 are also measured by micro-indentation. Monitoring Young modulus evolution provides a possibility to estimate the level of material degradation and fatigue life prediction by means of the indentation technique.

( , x 0)

where

0

Rf

n,

c

n

,

n

=1

(7)

c

denote the critical values of normal and shear stress,

1 is a failure factor, R

(

n c

,

n c

) is a stress function expressed in

terms of contact stress components n , n acting on the physical plane and x0, y0 is the origin of a local coordinate system (r, θ) specifying the crack initiation point. For stress concentrations and large stress gradients, the locally averaged stress function value over the size

5. Mathematical modelling of fatigue crack initiation 5.1. Critical plane damage model formulation A major problem in fracture mechanics is associated with the formulation of sufficiently simple and accurate conditions of crack initiation and growth in stress concentration domains [28]. Fatigue crack initiation mechanisms at the microscale have been discussed in detail in numerous papers, cf. recent review by Tu and Zhang [29], Chowdhury and Sehitoglu [16]. Three types of models have been developed to simulate damage growth: discrete dislocation models, continuum dislocation models and continuum models. The emphasis is laid on plastic strain localization zones, slip bands interacting with grain phase boundaries or matrix inclusion interfaces. There is also strong effect of free surface microstructure, generating stress

Fig. 20. Scheme of damage zone initiation. 10

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A. Ustrzycka, et al.

parameter d0 is applied, thus n

Rf = maxR ( , x 0)

n

,

c

is required that the stress resultant on the segment [0, 2 ] vanishes.

1 , x 0) d 0

d0

= max (

c

0

n

R

,

c

n

dr = 1

c

2

(8) (9)

(R ) dR

I( ) =

where

dR =

dR 0

for for

dR dR

0 0

R 1

(R ) = A 0

R0 R0

1 R0

1

n

,R =

c

, R0 =

(10)

c

=

c

(1

0

=

0

0 c

dD (1

D) (n + 1) p = A0

n

0

c

0

d c

dI ( ) = d

3÷5

1

e

x 2 cos(

x ) dx

(21)

(22)

=C

4

(23)

x 2cos(

e

0

x ) dx

=0

=

0

x 2dx

e

=

1 2

(24)

(25)

2

1 2

e

=0

4

(26)

Ultimately, the parameter B1 is determined in the following form

B1 =

2

B2 4

e

4

(27)

Based on the above expression, the inclusion fluctuation function takes the form

y (x ) =

1 B2 2

1 4

e

2

4

+e

x 2cos(

x)

(28)

Since the integration on the segment [0, 2 ] in Eq. (17) was changed to [0, ], a scaling parameter η was introduced in order to distinguish between analytical expression I(β)an and numerical I(β)num calculation of the integral I(β)

I ( )an

(29)

I ( )num =

For = and = 1 it is obtained = 10 11, so the integration error at this level is neglected. Nevertheless, the parameter can be included in the final expression as follows 1 6

Table 2 Material parameters.

810

0

I( )

1 2

2 B1 + B2

where α, β, B1 and B2 are the material parameters dependent on size and properties of inhomogeneities. Here, B2 is assumed to correspond to the inclusion fluctuation measurement [9]. The present symmetric form (18) corresponds to the case of small inhomogeneity size relative to the tested element size. In view of equilibrium condition satisfied by the stress fluctuation it

430

2

It means that the solution of Eq. (17) is

(16)

p

x ) dx =

So the constant C is determined

where y (x ) denotes the fluctuation function. The function describing the inclusion fluctuation is proposed in the form, Fig. 21,

n

x 2 cos(

4

I ( = 0) =

(15)

[MPa]

e

Alternatively, by expressing Eq. (18) with the same condition β = 0

(14)

c

0

=0

The local stress fluctuation ~ (x ) develops at inhomogeneities, cavities or inclusions and grain boundaries of tested elements. It is related to the mean stress and can be expressed as follows

[MPa]

2

= 0: I ( = 0) = Ce

C=

cos ( x ))

x )|xx = =0

2

(13)

~ (x ) = ¯ y (x )

(19)

2

n

(x ) = ¯ + ~ (x )

x 2 sin(

2

I ( ) = Ce

For 0

x ) dx

The solution of Eq. (21) takes the form

The normal stress profile σ(x) on the critical plane is expressed as the sum of mean stress ¯ and fluctuations stress ~ (x ) , thus

0

(18)

The differential equation can be solved by using the method of separation of variables

5.2. Stress fluctuation on the critical plane

x2

>0

(20)

(11)

where D denotes the scalar measure of damage (0 D 1) and p is the material parameter. The values of parameters 0 and c are designated from the Wöhler curve and are presented in Table 2. An alternative high cycle fatigue model proposed by Ottosen et al. [21] does not refer to the critical plane concept but to stress dependent damage surface.

~ (x ) = ¯ (B + B e 1 2

x ) dx ,

x 2sin(

xe

0

dI ( ) 1 = e d 2

In view of (11) and (12), the damage accumulation rule (9) can be presented in the form n

(17)

Integration by parts gives

(12)

D) p

(1

x 2cos(

e

0

dI ( ) = d

where A0 and n denote material parameters, 0 is the varying threshold value for the damaged material, c denotes the damage dependent failure stress in tension for the damaged material, 0 and c are the respective values for the undamaged material. Both 0 and c depend on the damage state according to the formula

D) p ,

x ) ) dx = 0

Differentiating (18) with respect to the parameter , leads to

and R > R0 or R R0

Here 0 R 0 1 denotes the threshold value of the failure function for the damage growth in a cyclic loading state, and (R ) is the damage growth function. For the case of uniaxial tension, the following form of the damage function is assumed, cf. [31] n

x 2cos(

In order to obtain analytical solution, first the integration on the segment [0, ] is considered

The damage growth is expressed by the following rule

dD =

(B1 + B2 e

0

B1 =

B2 4

2

e

4

+

Finally, the inclusion function takes the form 11

(30)

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A. Ustrzycka, et al.

Fig. 21. Function of inclusion fluctuation.

~ (x ) = ¯ y (x ) = ¯ B 2

1 4

2

e

4

+

+e

x 2 cos(

x)

allows to identify the initial plastic deformation field and the subsequent microplastic cyclic response. It has been assumed in the previous paper [17], that the damage growth occurs due to action of mean stress and its fluctuations generated by defects, grain boundaries and free boundaries of the tested structural element [35]. Then for the specified stress amplitude, determined by the ESPI method, and knowing the threshold and failure stress values σ0 and σc for the damaged material, the macrocrack initiation state can be specified. The crack propagation stage can be treated in a similar manner. The damage identification can be provided from the microindentation tests [36]. It is based on the measurement of local variation of Young modulus in the course of cyclic deformation process. In the framework of Continuum Damage Mechanics (CDM), the scalar damage variable D represents microcrack surface density affecting the material compliance, with the stress-strain relations taking the following forms for undamaged and damaged states

(31)

where the parameters α and β depend on the material grain microstructure. In order to perform a numerical calculations of damage growth the material parameters are presented in Table 2. Following the mathematical description, the numerical analysis of damage evolution under mechanical loads in elastic-plastic solids was made. Damage evolution related to number of cycles for different values of parameter n is shown in Fig. 22. The macro-crack initiation is assumed to occur at the critical value of damage Dc = 0.3. For the sake of numerical calculations, a dimensionless quantity is introduced. The point distance from the specimen axis is denoted as x, a denotes the width of the minimum cross-section and d is the hole diameter. Values of percentage correlation of the numerical predictions and the experimental results are 95% for n = 2.5 and 96% for n = 3.

0

= E 0 and

= E 0 (1

D ) p,

= E^

(32)

0

where E and E are the values of Young modulus in two states. Then, for the same strain value one obtains

5.3. Damage related to stiffness modulus variation The experimental characterisation of fatigue crack initiation in cyclic microplasicity regime has been presented with the proposed damage growth rule based on the critical plane concept. The imperfection introduced in the form of circular hole induces strong strain fluctuation field, which can be measured by the ESPI method. This

D=1

E E0

1 p

(33)

where p is the constant exponent. Fig. 23 presents the damage evolution predicted by the constitutive

Fig. 22. Damage evolution related to number of cycles for different values of parameter n. 12

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A. Ustrzycka, et al.

free boundary effects, the local strain concentrations develop. The initial loading then induces plastic deformation and subsequent residual stress field generating the elastic shake-down state. Both the fixed residual stress field and the cyclically varying fluctuation fields can be determined by applying the Electronic Speckle Pattern Interferometer (ESPI), used for displacement and strain measurement on the specimen surface. Additionally, the microindentation method has been applied to determine the damage variable from the data of the stiffness modulus. It is interesting to note that the initial plastic deformation and the shear band generation near the hole perimeter did not affect the damage growth and macrocrack orientation. In this paper, the new concept of constitutive modelling of fatigue crack initiation mechanisms is proposed. The new model is based on the continuum approach with account for local stress fluctuations, usually neglected in formulation of the damage models. Depending on the accuracy of description of stress and strain fluctuations, such type of modelling may become close to microstructural models, usually requiring numerous material parameters.

Fig. 23. Damage identification using microindentation tests and the numerical prediction of damage evolution with increasing number of cycles.

model using ESPI measurement data and the microindentation tests discussed in Section 3. There is the fair correlation of predictions by two methods. The microindentation tests could be executed only at several selected stages for differing numbers of cycles, requiring to stop the cyclic loading process and to perform the state indentation tests.

Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgments

6. Conclusions

This work has been partially supported by the National Science Centre through the Grant No 2014/15/B/ST8/04368. The authors also acknowledge Mirosław Wyszkowski and Leszek Urbański for contribution to the conducted experiment.

The present paper is devoted to simulation of fatigue crack initiation in materials subjected to cyclic loading within the nominal elastic regimes. Due to imperfections (inclusions, cavities), grain boundaries,

Appendix A. The working principle of ESPI (Electronic speckle pattern Interferometer) ESPI method based on the application of the elementary wave phenomenon – interference [37,38]. The main purpose of this method is to obtain fringe patterns which represent wave interference from a coherent source light sources projected on a surface. When the object is displaced in the direction normal to the viewing direction, distance travelled by the object beam changes, the phase of one beam changes and therefore the amplitude of the combined beams changes. The speckle effect is a result of the interference of waves having different amplitudes. Superposition of a pair of simple coherent waves lead to the interference intensity

U1 (x , y ) = A1 e

i( t

1 (x,y)) ,

U2 (x , y ) = A2 e

i( t

(A1)

2 (x,y))

where (x,y) represent the coordinates of the image plane, A is the magnitude of the displacement, (x , y ) = 1 (x , y ) 2 (x , y ) represents the phase difference between the two beams and ω denotes the angular frequency. The intensity of the light wave can be obtained from the relation

I (x , y ) =

(U1 (x , y ) + U2 (x , y ))(U1 (x , y ) + U2 (x , y )) dt

(A2)

where U is a complex conjugate of U . Consequently, using the Eqs (A1) and (A2), the intensity of the light wave can be determined

I (x , y ) = A12 + A22 + 2A1 A2 cos (

1 (x,

y)

2 (x,

y))=A12 + A22 + 2A1 A2 cos( (x , y ))

(A3)

Thus, the amplitude of the light at any point in the image is the sum of the light from the object and the second reference beam. As a result of scattering of beam light on the specimen surface, the interference of secondary waves occurs, which leads to the formation of the characteristic speckle pattern. When the specimen is deformed, the speckle pattern obtained before loading (reference image) is subtracted from the speckle pattern obtained after loading (measurement image), fringes are obtained which represent contours of displacement. Based on the fringe image, it is not possible to obtain the information on the direction of displacement. Consequently, the intensity distributions I1 (x , y ) and I2 (x , y ) recorded before and after the object displacement can be respectively written as (A4)

I1 (x , y ) = A12 + A22 + 2A1 A2 cos( (x , y )) I2 (x ,

y )=A12

where signals

I (x , y )after

+

A22

+ 2A1 A2 cos( (x , y ) +

(A5)

(x , y ))

(x , y ) is the additional phase change introduced due to the object deformed. A new interference fringes is obtained by subtracting the I (x , y )before = 2A1 A2 [cos ( (x , y ) +

(x , y ))

(A6)

cos ( (x , y ))]

The intensity of the interference pattern can be written as

I j (x , y ) = A12 + A22 + 2A1 A2 cos( (x , y ) +

(A7)

j)

where j is the amount of phase shifting and j = 1,2,3,…N integer number depending on the number of phase shifts introduced. Applying the fourphase shift method (−3π/4, −π/4. π/4, 3π/4), the phase of the wavefront computed from the four interferograms is 13

International Journal of Fatigue xxx (xxxx) xxxx

A. Ustrzycka, et al.

Fig. A1. (a) Phase map; (b) displacement map; (c) Gradient vector and the partial derivatives with respect to the coordinate axes.

(x , y ) = arctg

I4 (x , y ) I1 (x , y )

I2 (x , y ) I3 (x , y )

(A8)

There are three unknown values: A1, A2, φ(x,y) and αj is a known optical phase shift. With the intensity maps the optical phase at each pixel can be calculated. Minimum three images with different relative optical path length are acquired. The difference between the phase maps calculated after and before the deformation is given by

(x , y )after

(x , y )

(A9)

(x , y )before

(x , y ) of the wavefront from the object surface before and after deformation is directly associated with the On the other hand, the phase change component of displacement u(x) and is expressed as

(x , y ) =

4

sin u (x , y )

(A10)

where λ is the laser wavelength, and θ is the incident angle between two light beams illuminating the object, and u (x ) is the in-plan displacement component. Finally, one can obtained the optical phase for each deformation stage (Fig. A1a). The information about displacement is encoded in a carrier as levels of grey that represent the intensity at a given pixel I(x,y), a scalar field [39]. Finally, the displacement map is shown in Fig. A1b. The next step is to calculate the strain. Determining the gradient of the fringe pattern is shown in Fig. A1c. The following relationships are valid

tg

u

u / y

=

u x

(A11)

Appling relationship between the derivative of displacements and the fringe orientation of the object before and after deformation, the designation of the strain components is possible according to the following equations

u = x

xx ,

u = y

xy,

v = x

yx ,

v = y

yy

(A12)

Similar relationships can be derived for the field of the v-displacements. ESPI is implemented by producing interference between an optical wave front scattered from the object and a fixed reference wave, yielding the displacement for each point in the image of the object.

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