Microsoft Word - numero_60_art_31_3514.docx R.R. Yarullin et alii, Frattura ed Integrità Strutturale, 60 (2022) 451-463; DOI: 10.3221/IGF-ESIS.60.31 451 Fatigue growth rate of inclined surface cracks in aluminum and titanium alloys R.R. Yarullin, M.M. Yakovlev Institute of Power Engineering and Advanced Technologies FRC Kazan Scientific Center of RAS, Russia yarullin_r@mail.ru, yakovlev.mikhail.m@yandex.ru ABSTRACT. In this paper the fatigue crack growth tests were carried out on surface-crack tension (SCT) specimens, made of 7050 and Ti6Al4V alloys, with initial semi-elliptical surface cracks. Pure Mode I conditions were realized on SCT specimens with crack plane located orthogonal to the loading direction, while Mixed-mode conditions were observed on SCT specimens with inclined crack. Optical microscope measurements and the crack mouth opening displacement (CMOD) method were respectively used to monitor crack length and calculate crack depth. Current crack shape during the tests was highlighted by alternation of loading spectrum with baseline load block and a marker load block. The stress strain field along the crack front of semi- elliptical cracks in the SCT specimens was assessed by Finite Element Method (FEM) analysis. The stress intensity factors (SIFs) were calculated along crack fronts and equivalent elastic SIF formulation was used for crack growth rate assessment under mixed mode conditions. As a result, the fracture resistance parameters of aluminum and titanium alloys were obtained for two crack propagation directions under Mode I and Mixed-mode loading. The benefits of using the computational and experimental results of SCT specimen for the assessment of the surface crack growth rate in aluminum and titanium alloys under Mixed-mode loading conditions were stated. KEYWORDS. Inclined surface cracks; Fatigue crack growth; Mixed-mode conditions; Aluminum alloy; Titanium alloy. Citation: Yarullin, R.R., Yakovlev, M.M, Fatigue growth rate of inclined surface cracks in aluminum and titanium alloys, Frattura ed Integrità Strutturale, 60 (2022) 451-463. Received: 28.02.2022 Accepted: 18.03.2022 Online first: 21.03.2022 Published: 01.04.2022 Copyright: © 2022 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. INTRODUCTION urface cracks are typical damages in aircraft structure components, and their growth can lead to catastrophic failures. In service, fatigue surface cracks were found to grow in turbine disks, high-pressure compressor disks, turbine blades, load-carrying components of a frame structures, levers and the airplanes’ landing-gear components [1]. A surface flaws feature is that the crack grows simultaneously in two directions and changes its shape from initial i.e. related to the defect nature (risk, pore, pothole) to final i.e. at fracture. At the same time, a significant crack growth rate reduction in the S https://youtu.be/M5b2195HXYI R.R. Yarullin et alii, Frattura ed Integrità Strutturale, 60 (2022) 451-463; DOI: 10.3221/IGF-ESIS.60.31 452 deepest point of the crack front with respect to the crack front intersection with the free surface of the tested specimens can be observed as a function of loading conditions [2]. Although some surface cracks in structural components are subjected to uniaxial or Mode I loading type, in most cases the loading direction and the operational damage plane are not perpendicular to each other and such defects are considered as inclined cracks [3]. The elastic–plastic stress fields and mode mixity parameters for semi-elliptical surface cracks on biaxial loaded plates have been investigated in [4] using detailed three-dimensional finite element calculations. Different degrees of mode mixity are given by combinations of the far-field stress level, biaxial stress ratio and inclined crack angle. Normalized mixed-mode stress intensity factor equations were presented in the paper [5] for deflected and inclined circular surface and corner cracks in finite-thickness plates under uniform tensile loads. In the resulting equations, different deflection and inclination crack angles were considered, and plate thickness effects were included. However, at present time, there are no fracture resistance parameters for materials under Mixed-mode loading conditions for defects with initial shape close to operational damages. In this paper, the computational and experimental results for inclined surface cracks in SCT specimens of aluminum and titanium alloys are provided for Mixed-mode loading conditions. The fracture resistance parameters for inclined surface cracks and their comparisons with results available in the literature are presented. SUBJECT OF THE STUDY, MATERIAL PROPERTIES, AND EXPERIMENTAL CRACK PATHS he subject of this study is surface-crack tension (SCT) specimens of aluminum and titanium alloys. The plates with surface cracks as well as the modified ASTM E740 [6] SCT specimens with inclined surface crack are widely used for the Mixed-mode fatigue growth under tension and bending loading [7-10]. The SCT specimens’ thickness B is equal to 10 mm, the specimen’s width W is equal to 40 mm, and the specimen’s length L is equal to 56 mm. The SCT specimens with starter notch a0=2 mm, c0=2 mm and initial inclined plane α=0° are used for pure Mode I loading conditions (Fig. 1a). The SCT specimens with starter notch a0=2 mm, c0=2 mm and initial inclined plane α≈30°-45° are used to provide 3D Mixed-mode problems (Fig. 1b). a) b) Figure 1: SCT specimen geometry for pure Mode I (a) and Mixed Mode loading (b). A typical aerospace material, 7050 aluminum and Ti6Al4V titanium, were chosen for the experiments due to its extended use in industrial applications. 7050 aluminum possesses high specific strength and stiffness, excellent fabricability and low cost. The Ti6Al4V titanium is applied in the aeronautical industry due to high specific mechanical strength, fracture toughness, fatigue strength and high temperature stability. The SCT specimens were taken from rolled aluminum and titanium alloys such that longitudinal (L) grain direction would be under investigations. The tensile properties of considered alloys at room temperature were determined according to the ASTM standard E8 and are listed in Table 1: σ0 is the monotonic tensile yield strength, σs is the nominal ultimate tensile strength, σu is the true T R.R. Yarullin et alii, Frattura ed Integrità Strutturale, 60 (2022) 451-463; DOI: 10.3221/IGF-ESIS.60.31 453 ultimate tensile strength, α is the strain hardening coefficient, n is the strain hardening exponent, E is the Young’s modulus, δ is the elongation and ψ is the reduction of area. Material σ0, MPa σS, MPa σu, MPa α n E, GPa δ, % ψ, % 7050 472 524 701 1.570 10.85 70.57 11 28 Ti6Al4V 886 964 1290 1.225 12.59 118.00 16 29 Table 1: Main mechanical properties of aluminum and titanium alloys at room temperature. Fatigue precracking of SCT specimens was performed under three-point bending mode according to the method described in [6]. SCT specimen with starter notch plane α=30°-45° was turned relative to test tool set in order to obtain Mode I conditions at notch tip (Fig. 2). Precracking was completed when the crack length c reached 4.5-5.0 mm. The nominal stresses magnitude during fatigue precracking was such to not exceed material yield strength and was equal to 157.5 MPa. Figure 2: Fatigue precracking procedure for SCT Mixed-mode loading specimen. Figure 3: Test equipment for SCT specimens fatigue crack growth tests. R.R. Yarullin et alii, Frattura ed Integrità Strutturale, 60 (2022) 451-463; DOI: 10.3221/IGF-ESIS.60.31 454 The SCT specimens fatigue crack growth (FCG) tests are carried out on an MTS Landmark servo-hydraulic test system with a maximum capacity of 100 kN at 10 Hz frequency and with Rs=0.1 stress ratio (Fig. 3). The Mode I and Mixed-mode tests of the 7050 and Ti6Al4V SCT specimens were performed by applying uniaxial forces P=42 kN and P=60 kN, respectively. All tests were carried out with sinusoidal loading. Load control was estimated to be better than ±1%. The crack length on the specimen’s free surface was monitored using an optical microscope. Crack mouth opening displacement (CMOD) measurements was used for crack depth determination. The displacement gage length was equal to 10 mm. For SCT Mixed-mode specimen the CMOD device was placed in the load application direction (Fig. 3). Two different stress ratio values (0.1 and 0.5) were applied several times to each SCT specimen in order to fix the current crack front position. During each test, beach marks were produced for each SCT specimen by increasing the applied stress ratio from 0.1 to 0.5 at a constant value of the maximum cyclic nominal stress, while the surface crack length was increased by approximately 0.1-0.2 mm. As shown in [11-13], the beach mark loading does not induce load history effects or overload retardation. The typical fracture surface marks are shown in Fig. 4a and 4b for SCT Mode I and SCT Mixed-mode specimens, respectively. (a) (b) Figure 4: Fracture surfaces for (a) Mode I and (b) Mixed-mode loading. Factory roof patterns were found on fracture surface of SCT Mixed-mode specimen from titanium alloy. Factory-roofs initiate by elementary Mode I branches at particular sites on the fronts of semi-elliptical surface cracks growing under Mixed- mode II + III [14]. The factory roof patterns’ roughness (or visibility) particularly depends on the applied cyclic shear stress amplitude and also the material microstructure significant influence. These conditions, along with the material yield strength, constitute the main reasons why the factory roof patterns are not observed in the 7050 aluminum alloy. From the crack front shape obtained by beach mark procedure, the crack length measurements c on the free surface (width/height direction) and the crack depth a of growing surface crack (thickness direction) can be obtained for SCT specimens using an optical instrumental microscope. The experimental crack sizes combinations for SCT Mode I and Mixed-mode specimens for aluminum and titanium alloys are presented in Table 2. R.R. Yarullin et alii, Frattura ed Integrità Strutturale, 60 (2022) 451-463; DOI: 10.3221/IGF-ESIS.60.31 455 Materials Mode I Surface crack sizes a/c (mm/mm) Mixed-mode Surface crack sizes a/c (mm/mm) 1st front 2nd front 3rd front 4th front 1st front 2nd front 3rd front 7050 3.19/5.00 4.32/6.27 6.19/8.51 - 2.53/4.55 4.47/7.25 7.70/12.50 Ti6Al4V 2.50/5.08 4.10/6.18 5.90/7.73 7.75/9.73 2.50/4.65 4.00/6.70 7.70/13.00 Table 2: Experimental crack sizes for SCT specimens. For SCT Mixed-mode specimens the crack length c was measured along the curvilinear crack path on the specimen free surface, the crack depth a was measured on a plane orthogonal to the specimen's axis (Fig. 1b). The fracture surface of SCT Mixed-mode specimens, as shown in Fig. 4b, illustrate the tortuous path the crack propagated through the microstructure. The curvilinear shape and orientation of the growing crack need to be modeled based on FEM methodology, which is not simple. This assumption was made because the tortuous crack path did not facilitate direct crack inclination angle measurements from the fracture surface. An alternative way is to do the initial inclination angle measurements (after precracking, Fig. 5a) and final inclination angle on the back side of the specimen (after fracture, Fig. 5b). Based on the crack plane inclination angle α performance along SCT Mixed-mode specimens thickness (Fig. 5c) it is possible to calculate the inclination angle for any experimental crack fronts. It is clear that while the crack, in its initial configuration, starts out as a Mixed-mode crack, after a substantial growth, the crack configuration is in a near pure Mode I state. (a) (b) (c) Figure 5: Crack plane inclination angle α (a) on the front side, (b) on the back side and (c) behavior along thickness of SCT Mixed-mode specimens. R.R. Yarullin et alii, Frattura ed Integrità Strutturale, 60 (2022) 451-463; DOI: 10.3221/IGF-ESIS.60.31 456 As mentioned above, crack growth was monitored using the optical microscope and CMOD for Mode I and Mixed mode loading conditions on SCT specimens’ tests. The relationships between CMOD and crack length on the free surface for both considered alloys under different loading conditions are plotted in Fig. 6. A strong correlation was found between these two parameters, and this information can be very useful for the crack growth rate diagram’s interpretation in thickness direction. Thus, the obtained experimental data of the crack front shape and orientation for various fatigue failure process stages will be used in this study to calculation the fracture parameters distributions along the crack fronts in terms of elastic and equivalent SIFs for all tested SCT specimens. (a) (b) Figure 6: Relationship between CMOD and crack length on the free SCT specimen surface for (a) aluminium and (b) titanium alloys under different loading conditions. MIXED MODE CRACK GROWTH PARAMETERS Equivalent stress intensity factor D Mixed Mode problems are characterized by the fracture superposition Modes I, II and III. While an existing crack under Mode I loading conditions will propagate within the original crack plane, Mode II loading generally leads to a crack kinking, Mode III loading causes the crack front twisting, for 3D Mixed Mode cases depending on the Mode II- and Mode III-portions a more or less intense crack deflection or crack twisting can be observed. This means, that within the linear-elastic fracture mechanics scope the SIFs KI, KII and KIII are of importance for the fracture risk estimation in structures as well as for the stable crack propagation evaluation processes [15] and can be defined by Eq. (1):   I y IK a Y ,   II xy IIK a Y ,   III yz IIIK a Y (1) In general, the SIF depends on the stress (σy, τxy or τyz), the crack length a, and on the boundary correction factors (YI, YII or YIII). Shlyannikov [16] generalized the numerical method to calculate the geometry dependent correction factors YI, YII, and YIII for the SIFs KI, KII, and KIII under mixed mode fracture. The present study explores the direct use of FE solution results for calculating the SIFs KI, KII, KIII, ahead of the crack tip (θ=0º):   2FEM IK r ,   2FEM II rK r ,   2FEM IIIK r (2) where r, θ, and ω are polar coordinates centered at the crack tip, and  FEM i are the stresses obtained from the FE solution. To describe the Mixed-mode crack growth along the curvilinear crack path the equivalent elastic SIF includes Mixed-mode effects such that 3 R.R. Yarullin et alii, Frattura ed Integrità Strutturale, 60 (2022) 451-463; DOI: 10.3221/IGF-ESIS.60.31 457          2 2 2 21 1eqv I II IIIK K K K (3) was obtained from the energy release rate G definition for plane strain [17]:             22 2 2 1 2 3 1 1 III I II K G G G G K K E (4) where E is the Young's modulus,  is the Poisson's ratio. NUMERICAL STUDY Elastic–plastic stress–strain fields along the crack front he numerical calculations in this study are connected with the stress-strain state (SSS) SCT specimens’ analysis with Mode I and inclined surface cracks. The ANSYS FE code [18] is used in the mechanical analysis. Twenty nodal solid brick elements with quadratic interpolation were used to mesh the 3D FE model configurations, which is a quarter of the SCT Mode I specimen and full geometry of the SCT Mixed-mode specimen. The FE models of SCT specimens were loaded with forces which coincides with the maximum experimental loads value and are P=42 kN and P=60 kN for 7050 and Ti6Al4V alloys, respectively. In order to perform numerical calculations, the main mechanical properties listed in Table 1 were used. The crack tip shapes obtained by beach mark procedure has been considered in the numerical part of the study. The crack sizes for both alloys are presented in Table 2 and the crack inclination angles for SCT Mixed-mode specimens have been defined consistently from the Fig. 5c. As a result, for each considered material type, from 6 to 7 3D FE models with different crack front positions were analyzed under experimental loading conditions. The SIFs were calculated using the topology building principles of FE meshes, the elements sizes, and their distribution density in the radial and circumferential directions, as applied to surface defects in real structures, components and specimens, which are described in [2, 4, 9, 19-21]. Thus, in order to accurately characterizing the influence of the strain gradient, a very refined mesh is used near the crack tip, where the elements' size is in the one micrometer order. The nodes number in the 3D FE models were varied from 1 000 000 to 2 500 000. Typical FE meshes for the SCT Mode I and SCT Mixed-mode specimens with surface crack are illustrated in Fig. 7a,b and 8a,b, respectively. The typical equivalent stress distributions for the SCT Mode I and SCT Mixed-mode specimens with surface crack are illustrated in Fig. 7c and 8c, respectively. (a) (b) (c) Figure 7: Typical (a, b) FE meshes and (c) equivalent stress distributions for SCT Mode I specimen with surface crack. T R.R. Yarullin et alii, Frattura ed Integrità Strutturale, 60 (2022) 451-463; DOI: 10.3221/IGF-ESIS.60.31 458 (a) (b) (c) Figure 8: Typical (a, b) FE meshes and (c) equivalent stress distributions for SCT Mixed Mode specimen with surface inclined crack. Stress intensity factors distributions One of the main objectives of the present study is the elastic fracture mechanical parameters calculation for real shape and surface cracks sizes, which were obtained by uniaxial tension tests on two SCT specimens types. It should be recalled that the SCT specimens with crack plane located orthogonal to the loading direction was used to realize pure Mode I loading conditions, and the SCT specimens with inclined crack was proposed in order to reproduce the Mixed-mode loading conditions around the crack tip. The elastic SIFs KI, KII and KIII distributions and elastic equivalent Mixed-mode SIFs Keqv were obtained by Eq. (2) and Eq. (3), respectively. All parameters are determined at the crack tip distance range r/a=0.0025- 0.01, where the numerical solution provides a stabilized result. To compare the parameter distributions along surface crack tip, it is convenient to introduce dimensionless coordinates in the following form:                       0 0 0 0cos , sin , cos , sin , cos , sin ,c c c c i i i ix y x y x y          0 0, ,i c c (5)       0 0 0 0 , ,i i i i c c x x y y X Y x x y y    2 21 , 0,1 2 i i iR RX Y (6) where 0 is the angle determining the crack tip initial point position, and c is the angle corresponding to the crack tip last point. (a) (b) Figure 9: Distributions of the elastic SIFs along the crack fronts for (a) aluminum and (b) titanium alloys under model I. R.R. Yarullin et alii, Frattura ed Integrità Strutturale, 60 (2022) 451-463; DOI: 10.3221/IGF-ESIS.60.31 459 In the following numerical results representation, the authors will use the variable R in the 0 to 1. R=0 indicates the SCT specimens exterior free surface (c direction, Fig. 1), while R=1 indicates the crack tip deepest point (a direction, Fig. 1). The SIFs KI distributions along several crack fronts for both alloys at pure uniaxial tension loading conditions are plotted in Fig. 9. KII and KIII are nearly zero during all crack fronts (consequently, Keqv and KI values nearly coincide), therefore were not reported. Looking at this figure, it can be noted that the elastic SIFs KI for the SCT Model I specimens are changed by a moderate amount in the range of 0> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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