Numerical Investigation of the Effect of Residual Stresses on the Behavior of Semi-Elliptical Surface Cracks in Aluminum Plates

Numerical Investigation of the Effect of Residual Stresses on the Behavior of Semi-Elliptical Surface Cracks in Aluminum Plates | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Numerical Investigation of the Effect of Residual Stresses on the Behavior of Semi-Elliptical Surface Cracks in Aluminum Plates Sina Mirzajani Soluosh, Mahmoud Afshari, Hossein Afshari, Ali Bazrafshan Tanha This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8912144/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This study numerically investigates the fracture behavior of semi-elliptical surface cracks in 5000-series aluminum sheets, with emphasis on the role of residual stresses. Since the coexistence of cracks and residual stresses can seriously compromise the integrity and safety of engineering structures, their combined effect must be evaluated for reliable fracture prediction. A three-dimensional finite element model was developed in ABAQUS and the responses of specimens with and without residual stresses were compared. The mechanical properties of the alloy were obtained from tensile tests conducted in accordance with ASTM B557-02a and implemented in the simulations. To improve accuracy near the crack front, a refined, spider-web (focused) mesh was employed to capture the stress–strain fields with higher resolution. Elastoplastic fracture behavior was assessed using the J-integral as the primary fracture mechanics parameter. Residual tensile stresses were generated through a simulated four-point bending procedure; after unloading, the specimens were subjected to tensile loading. The results indicate that tensile residual stresses increase the crack-driving force and reduce the load-carrying capacity in the elastic–plastic regime. At low applied loads, a significant difference in J-integral values was observed between the two conditions, whereas with increasing load and plastic zone development, the influence of residual stresses diminished and the responses converged. Mechanical Engineering Semi-elliptical surface crack residual stress finite element analysis 5000-series aluminum alloy Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Introduction With the continuous growth of various industries, the demand for employing diverse manufacturing methods for producing metallic and non-metallic components has become more pronounced than ever [ 1 ]. Each manufacturing process is associated with specific mechanical properties as well as characteristic defects [ 2 ]. Among these defects, cracks are considered one of the most critical and destructive discontinuities in engineering components and structures. This is mainly due to the sharp geometry of the crack tip, which leads to severe stress concentration in its vicinity and facilitates crack propagation across the component cross-section [ 3 ]. Such defects may originate at different stages of manufacturing, including forming processes [ 4 ], machining operations [ 5 ], or heat treatment procedures. Among the various types of cracks, semi-elliptical surface cracks are of particular importance due to their high prevalence in industrial components; however, their accurate analysis has always been associated with significant challenges [ 6 ]. The geometric complexity of this type of crack, along with the non-uniform stress distribution along the crack front, makes the evaluation of their mechanical behavior and the prediction of their growth a challenging task. Moreover, experience has shown that under cyclic loading conditions, initial defects with irregular shapes gradually evolve into cracks with more regular geometries, such as semi-circular or semi-elliptical cracks [ 7 ]. Another important factor influencing crack behavior is the presence of residual stresses in the material. These stresses, which typically remain in components as a result of mechanical or thermal processes such as welding, rolling, bending, or heat treatment, can significantly alter the stress field around a crack and consequently affect its initiation and growth. The coexistence and interaction of semi-elliptical surface cracks with residual stresses under various loading conditions further increase the need for a detailed and quantitative investigation of this phenomenon [ 8 ]. From this perspective, numerical modeling based on the principles of fracture mechanics becomes an essential tool for evaluating the detrimental effects of these factors on the structural integrity and safety of engineering components. Within the theoretical framework of fracture mechanics, the behavior and stability of cracks under applied loading can be described and analyzed using characteristic parameters such as the elastic stress intensity factor (K) or the elastic–plastic integral parameter (J) [ 9 ]. These parameters make it possible to determine the critical crack size, beyond which unstable fracture of the component is likely to occur. The determination of this critical size depends on several key factors, including the crack location and orientation relative to the stress field, the magnitude and type of applied stresses, and the fracture toughness of the base material. Simultaneous consideration of these factors is essential for accurate prediction of fracture behavior and for improving the reliability of engineering structures. Various methods can be employed to study cracks, one of the most effective being finite element simulation [ 10 ]. Finite element analysis enables the evaluation of the effects of different parameters—such as residual stresses, crack size, component strength, and similar factors—on crack formation and, consequently, on the fracture behavior of components. Surface cracks are typically characterized by two main parameters: crack depth and crack length. The analysis of surface cracks therefore involves accurate prediction of crack growth and the fracture toughness of the material [ 11 ]. In recent years, numerous studies have been conducted on crack behavior, reflecting the continued importance of this topic in engineering research. Jukić et al. [ 12 ] investigated the effect of residual stresses induced by arc welding on crack behavior in two butt-welded plates with a thickness of 20 mm. In their study, finite element (FE) and weight function (WF) methods were employed to evaluate stress intensity factors at the deepest point of semi-elliptical surface cracks with different geometries, orientations, and positions relative to the weld line. For cracks oriented perpendicular to the weld line, the FE and WF results showed good agreement for small crack sizes; however, as the crack size increased, the discrepancy between the results obtained from the two methods also increased. Smirnov and Vidyushenkov [ 13 ] investigated the estimation of the approximate value of the stress intensity factor for a cylindrical specimen containing a circumferential crack. In their study, Irwin’s concept was employed to extend and extrapolate stress concentration formulas at the notch tip, which had previously been developed by Neuber for a body of revolution with an annular hyperbolic recess under uniaxial tensile loading. A structural fracture criterion was used to estimate the specimen dimensions, including the ligament diameter and the outer diameter. Their results indicated that the ratio of the ligament diameter to the specimen diameter is not a predefined value, but rather is determined through a gradual and step-by-step selection process. Panontin and Hill [ 15 ] employed a micromechanical damage model to predict crack initiation in ductile and brittle materials and demonstrated that, with increasing load, the effect of residual stresses on the fracture toughness of ductile materials can be neglected. Dodds et al. [ 16 ] experimentally investigated the J-integral fracture parameter for a specimen containing a surface crack and compared the results with finite element solutions. In the present study, the finite element method is used to investigate surface cracks in aluminum plates in the presence of residual stresses. To validate the developed numerical model, the extracted results are compared with those reported in previous studies. Methodology In this study, finite element simulations were carried out using the ABAQUS software. To determine the mechanical properties of the aluminum alloy, tensile test specimens were prepared in accordance with the ASTM B557-02a standard, and tensile tests were performed using a SANTAM testing machine. Figure 1 presents the stress–strain curve obtained from the tensile test. To improve computational efficiency and reduce solution time, geometric and loading symmetries of the specimens were exploited, and only one quarter of the full specimen geometry was modeled in the finite element analysis. Accordingly, appropriate symmetric boundary conditions were applied to the corresponding surfaces to ensure that the mechanical behavior of the entire specimen was accurately represented. To characterize the stress and strain fields in the vicinity of the crack tip, the J-integral is employed as a path-independent parameter. The J-integral is a fundamental concept in elastic–plastic fracture mechanics, as it is capable of describing both elastic and plastic deformation behavior around the crack tip. It consists of two components, namely the elastic and plastic parts, and is calculated according to Eq. ( 1 ) [ 17 ]. $$\:J={J}_{el}+{J}_{pl}$$ 1 For surface cracks, the elastic component of the J-integral Jel is evaluated using the stress intensity factor through Equations ( 2 ) and ( 3 ). These expressions incorporate geometric and correction factors to account for the crack shape and loading conditions. The parameters Q and F, which appear in these formulations, are obtained from Equations ( 4 ) and ( 5 ), respectively [ 17 ]. $$\:{\text{K}}_{\text{I}}=\left({\text{S}}_{\text{t}}\right)\sqrt{{\pi\:}\frac{\text{a}}{\text{Q}}}\text{F}(\frac{\text{a}}{\text{t}},\frac{\text{a}}{\text{c}},\frac{\text{c}}{\text{b}},{\upvarphi\:})$$ 2 $$\:{\text{J}}_{\text{e}\text{l}}=\frac{{\text{K}}^{2}(1-{{\upsilon\:}}^{2})}{\text{E}}$$ 3 $$\:\text{Q}=1+1.464\left(\frac{\text{a}}{\text{c}}{)}^{1.65}\right(\frac{\text{a}}{\text{c}}\le\:1)$$ 4 $$\:\text{F}=[{\text{M}}_{1}+{\text{M}}_{2}(\frac{\text{a}}{\text{t}}{)}^{2}+{\text{M}}_{3}\left(\frac{\text{a}}{\text{t}}{)}^{4}\right]{\text{f}}_{{\upvarphi\:}}.\text{g}.{\text{f}}_{\text{w}}$$ 5 In this part of the project, to validate the accuracy of the software-based solution in calculating the J-integral for elastic–plastic material behavior, a model previously reported in the literature is adopted [ 18 ]. Table 1 presents the geometric specifications of the model introduced in the earlier study [ 18 ]. Table 1 Geometric specifications of the model adopted from the previous study [ 18 ] Thickness (mm) Crack length (mm) Specimen width (mm) Specimen length (mm) 8 40 400 400 The material behavior is described by Eq. ( 6 ), and the corresponding constants are listed in Table 2 . $$\:{\epsilon\:}=\left\{\begin{array}{c}\frac{{\sigma\:}}{\text{E}}\:\:\:\:\:for\:\sigma\:\le\:{{\sigma\:}}_{0}\\\:\alpha\:{\left(\frac{{\sigma\:}}{{{\sigma\:}}_{0}}\right)}^{\text{n}-1}\frac{{\sigma\:}}{\text{E}}\:\:\:\:\:for\:\sigma\:>{{\sigma\:}}_{0}\end{array}\right.$$ 6 Table 2 Material constants used in Eq. ( 6 ) n α E σ₀ 5 0.3 1 200 GPa 205 MPa The model was subjected to bending loading as a percentage of the critical load, as listed in Table 3 . In this case, the critical load is obtained from Eq. ( 7 ). $$\:{\text{P}}_{\text{L}}=\frac{1.455\text{B}(\text{W}-\text{a}{)}^{2}{{\sigma\:}}_{\text{Y}}}{2\text{H}}=116.4\left[\text{k}\text{N}\right]\:$$ 7 Table 3 Applied load levels in the model reported in the previous study [ 18 ], expressed as a percentage of the critical load P/P L 0.25 0.5 0.75 1.0 1.25 1.5 Surface traction 53.26 106.52 159.78 213.04 266.30 319.56 It should be noted that, due to quarter-model simulation, the load values in Table 3 are halved when applied to the model. The calculated J-integral values for different loading levels are presented in Fig. 2 and compared with the results of the previous study. In this study, the specimens were modeled in three dimensions, and the mesh was designed to provide sufficient refinement in the vicinity of the crack tip. This type of meshing, commonly used for semi-elliptical and semi-circular cracks, is known as a spider mesh and enables high accuracy in the evaluation of fracture mechanics parameters. In the numerical modeling, linear reduced-integration brick elements (C3D8R) were employed. After defining the meshing strategy and selecting the element type, the mesh size in the region surrounding the crack tip was examined in order to achieve an appropriate and convergent mesh configuration. To assess mesh adequacy, the effect of mesh size on variations of the maximum J-integral along the front of the surface crack tip in the experimental specimen model was investigated. For this purpose, two variables were considered to characterize the mesh density around the crack tip: the mesh size in the radial direction ( \(\:{n}_{1}\) ​) and the number of elements in the circumferential direction ( \(\:{n}_{2}\) ​), as illustrated in Fig. 3 . To investigate the effect of element size, the model without residual stresses was loaded in the same manner as the experimental model, and the average value of the maximum J-integral from the third to the eighth contours along the surface crack front was calculated. These values were then compared for different mesh configurations in order to evaluate the influence of mesh size. The results of this investigation are presented in Table 4 Given the small differences observed in the J-integral values at this stage, n 1 = 26 was selected as the appropriate mesh size for the number of elements in the radial direction. In the next step, while keeping n 1 = 26 constant, the number of elements in the circumferential direction was varied by doubling and halving the initial value to determine the suitable mesh density. Based on this procedure, the minimum adequate number of elements was determined to be 18168._ Table 4 Effect of the number of elements in the radial direction with a constant number of elements in the circumferential direction Number of elements in the model Percentage difference from previous mesh J [N/mm] (n 2 ) (n 1 ) 18,168 – 3.70 4 26 16,920 0.5 3.68 4 13 20,664 0 3.70 4 52 In this section, residual stresses were introduced into the finite element model by applying a four-point bending load with a force magnitude of 4800 N. Figure 4 illustrates the finite element model subjected to the four-point bending loading used to generate the residual stress field. Results and Discussion Figure 5 shows the deflection generated at the center of the specimen after unloading and following the crack release stage. The developed finite element model was subjected to tensile loading once in the presence of residual stresses and once without residual stresses. The load versus crack mouth opening displacement (CMOD) curves were extracted for both cases. Figure 6 presents these results for the specimens with and without residual stresses. As shown in Fig. 6 , the elastic load-carrying capacity of the specimen without residual stresses obtained from the finite element analysis is approximately 60 kN, whereas for the specimen containing residual stresses it is about 35 kN. This indicates an increase in the crack driving force due to the tensile residual stresses introduced into the model. Furthermore, the finite element results show that the load required to produce the same level of plastic deformation in the specimen with residual stresses is lower than that required for the specimen without residual stresses. In other words, under the same applied load, the maximum allowable surface crack size is smaller for the specimen containing residual stresses compared to the specimen without residual stresses. According to Fig. 6 , at higher load levels the load–CMOD curves for specimens with and without residual stresses tend to converge. This behavior can be attributed to the development of extensive plasticity at the crack tip under high loads, which reduces the influence of residual stresses on the fracture behavior of the specimens. In addition, the difference between the loads required to produce the same plastic deformation in specimens with and without residual stresses becomes negligible at small deformation levels. Figure 7 J-integral values obtained from finite element analysis for specimens with and without residual stresses As shown in Fig. 7 , at low load levels a noticeable difference exists between the J-integral values obtained from the finite element analysis for specimens with and without residual stresses. This difference is attributed to the influence of the residual stress field generated at the crack tip as a result of the four-point bending loading. At higher load levels, however, the two curves tend to coincide due to the development of significant plastic deformation at the crack tip. Consequently, under loads exceeding approximately 100 kN, the effect of residual stresses on the crack-tip field can be considered negligible. Conclusion The results of this study indicate that the presence of tensile residual stresses increases the driving force for crack growth and reduces the load-carrying capacity of the specimen in the elastic–plastic regime, such that at low load levels a noticeable difference is observed between the responses of specimens with and without residual stresses. However, at higher load levels and with the expansion of the plastic zone at the crack tip, the influence of residual stresses diminishes and the fracture responses of the two cases become closer. It was also found that the maximum allowable crack size in the presence of residual stresses is smaller than that in the absence of residual stresses. 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J Strain Anal Eng Des, p. 03093247241310346 Govindan P, Joshi SS (2012) Analysis of micro-cracks on machined surfaces in dry electrical discharge machining. J Manuf Process 14(3):277–288 Marazani T, Madyira DM, Akinlabi ET (2017) Repair of cracks in metals: A review, Procedia Manufacturing, vol. 8, pp. 673–679 Jones R (1952) A method of studying the formation of cracks in a material subjected to stress. Br J Appl Phys 3(7):229 Riemer A, Richard HA (2016) Crack propagation in additive manufactured materials and structures, Procedia Structural Integrity, vol. 2, pp. 1229–1236 Mohan Kumar S, Rajesh Kannan A, Pramod R, Siva Shanmugam N, Dhinakaran V, Krishnaveni A (2022) A study on evaluation of stress intensity factor (KI) and J-integral for 40Ni2Cr1Mo28 alloy (structural steel): analytical and finite element analysis approach. Materialwiss Werkstofftech 53(12):1504–1517 Afshari H, Ashrafi A (2023) Experimental and numerical study of the hydroforming process of copper-aluminum double layered tube. Iran J Manuf Eng 10(3):18–33 Song P, Shieh Y (2004) Crack growth and closure behaviour of surface cracks. Int J Fatigue 26(4):429–436 Jukić K, Perić M, Tonković Z, Skozrit I, Jarak T (2021) Numerical calculation of stress intensity factors for semi-elliptical surface cracks in buried-arc welded thick plates, Metals. 11(11):1809 Smirnov V, Vidyushenkov S (2021) Stress intensity factor for cylindrical specimen with external circular crack under tension, in International Scientific Siberian Transport Forum, : Springer, pp. 408–417 Chen Y, Lambert S (2005) Numerical modeling of ductile tearing for semi-elliptical surface cracks in wide plates. Int J Press Vessels Pip 82(5):417–426 Panontin T, Hill M (1996) The effect of residual stresses on brittle and ductile fracture initiation predicted by micromechanical models. Int J Fract 82(4):317–333 Dodds RH Jr, Read DT (1990) Experimental and numerical studies of the J-integral for a surface flaw. Int J Fract 43(1):47–67 Newman J Jr, Raju I (1979) Analyses of Surface Cracks in Finite Farahani M, Sattari-Far I, Akbari D, Alderliesten R (2013) Effect of residual stresses on crack behaviour in single edge bending specimens. Fatigue Fract Eng Mater Struct 36(2):115–126 Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8912144","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":593525608,"identity":"3de29795-161d-44c4-979e-a4fa27e91bb8","order_by":0,"name":"Sina Mirzajani Soluosh","email":"","orcid":"","institution":"1*Department of Mechanical Engineering, Amirkabir University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Sina","middleName":"Mirzajani","lastName":"Soluosh","suffix":""},{"id":593525609,"identity":"91f8a0b7-901a-482b-8f46-ec7db0d129a2","order_by":1,"name":"Mahmoud 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alloy\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8912144/v1/608d8b28fd9d91e7690c6165.jpg"},{"id":103045352,"identity":"9ddecf90-bc99-4192-849b-5c08b75c5289","added_by":"auto","created_at":"2026-02-20 06:04:50","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":37564,"visible":true,"origin":"","legend":"\u003cp\u003eComparison between the results obtained from the software-based analysis in this project and the results reported in the previous study [18]\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8912144/v1/c3534e1726256bc21e7ff20e.jpg"},{"id":103045356,"identity":"d4fc230b-e137-4128-80da-06ffcef7db59","added_by":"auto","created_at":"2026-02-20 06:04:50","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":40455,"visible":true,"origin":"","legend":"\u003cp\u003eElement size variables in the vicinity of the crack tip\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8912144/v1/fb0ed96174a309380fb06af8.jpg"},{"id":103045358,"identity":"fb27f56d-be52-4a58-a883-b06acdff380c","added_by":"auto","created_at":"2026-02-20 06:04:50","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":82849,"visible":true,"origin":"","legend":"\u003cp\u003eFinite element model loaded to generate residual stresses using the four-point bending method\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8912144/v1/e42b58e857eacece73ba9f2f.jpg"},{"id":103045354,"identity":"7b217997-c6e6-4e73-ac0d-aaf5f0c60c93","added_by":"auto","created_at":"2026-02-20 06:04:50","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":92592,"visible":true,"origin":"","legend":"\u003cp\u003eDeflection at the mid-span of the finite element model induced by four-point bending loading\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8912144/v1/9ca4a5ba99c23eac4326c21a.jpg"},{"id":103045355,"identity":"96d96264-685e-4461-8428-f48413dec3ef","added_by":"auto","created_at":"2026-02-20 06:04:50","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":38984,"visible":true,"origin":"","legend":"\u003cp\u003eLoad versus crack mouth opening displacement (CMOD) for specimens with and without residual stresses obtained from finite element analysis\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8912144/v1/3d04e4e06fbcfbea7417c11e.jpg"},{"id":103050590,"identity":"4feec08d-b5e7-4aa1-afd0-bdcf7f46c304","added_by":"auto","created_at":"2026-02-20 07:50:41","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":32936,"visible":true,"origin":"","legend":"\u003cp\u003eJ-integral values obtained from finite element analysis for specimens with and without residual stresses\u003c/p\u003e","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8912144/v1/bb56ad1fc62ba807e7a72772.jpg"},{"id":103051097,"identity":"b8eadbbf-b3c4-4059-ad78-d4480086224d","added_by":"auto","created_at":"2026-02-20 07:58:15","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":898441,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8912144/v1/558fc4dc-4793-4431-b34d-1ebae25274a7.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eNumerical Investigation of the Effect of Residual Stresses on the Behavior of Semi-Elliptical Surface Cracks in Aluminum Plates\u003c/p\u003e","fulltext":[{"header":"Introduction","content":"\u003cp\u003eWith the continuous growth of various industries, the demand for employing diverse manufacturing methods for producing metallic and non-metallic components has become more pronounced than ever [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. Each manufacturing process is associated with specific mechanical properties as well as characteristic defects [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eAmong these defects, cracks are considered one of the most critical and destructive discontinuities in engineering components and structures. This is mainly due to the sharp geometry of the crack tip, which leads to severe stress concentration in its vicinity and facilitates crack propagation across the component cross-section [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. Such defects may originate at different stages of manufacturing, including forming processes [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], machining operations [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e], or heat treatment procedures.\u003c/p\u003e \u003cp\u003eAmong the various types of cracks, semi-elliptical surface cracks are of particular importance due to their high prevalence in industrial components; however, their accurate analysis has always been associated with significant challenges [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. The geometric complexity of this type of crack, along with the non-uniform stress distribution along the crack front, makes the evaluation of their mechanical behavior and the prediction of their growth a challenging task. Moreover, experience has shown that under cyclic loading conditions, initial defects with irregular shapes gradually evolve into cracks with more regular geometries, such as semi-circular or semi-elliptical cracks [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eAnother important factor influencing crack behavior is the presence of residual stresses in the material. These stresses, which typically remain in components as a result of mechanical or thermal processes such as welding, rolling, bending, or heat treatment, can significantly alter the stress field around a crack and consequently affect its initiation and growth. The coexistence and interaction of semi-elliptical surface cracks with residual stresses under various loading conditions further increase the need for a detailed and quantitative investigation of this phenomenon [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. From this perspective, numerical modeling based on the principles of fracture mechanics becomes an essential tool for evaluating the detrimental effects of these factors on the structural integrity and safety of engineering components.\u003c/p\u003e \u003cp\u003eWithin the theoretical framework of fracture mechanics, the behavior and stability of cracks under applied loading can be described and analyzed using characteristic parameters such as the elastic stress intensity factor (K) or the elastic\u0026ndash;plastic integral parameter (J) [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. These parameters make it possible to determine the critical crack size, beyond which unstable fracture of the component is likely to occur. The determination of this critical size depends on several key factors, including the crack location and orientation relative to the stress field, the magnitude and type of applied stresses, and the fracture toughness of the base material. Simultaneous consideration of these factors is essential for accurate prediction of fracture behavior and for improving the reliability of engineering structures.\u003c/p\u003e \u003cp\u003eVarious methods can be employed to study cracks, one of the most effective being finite element simulation [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. Finite element analysis enables the evaluation of the effects of different parameters\u0026mdash;such as residual stresses, crack size, component strength, and similar factors\u0026mdash;on crack formation and, consequently, on the fracture behavior of components. Surface cracks are typically characterized by two main parameters: crack depth and crack length. The analysis of surface cracks therefore involves accurate prediction of crack growth and the fracture toughness of the material [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. In recent years, numerous studies have been conducted on crack behavior, reflecting the continued importance of this topic in engineering research.\u003c/p\u003e \u003cp\u003eJukić et al. [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e] investigated the effect of residual stresses induced by arc welding on crack behavior in two butt-welded plates with a thickness of 20 mm. In their study, finite element (FE) and weight function (WF) methods were employed to evaluate stress intensity factors at the deepest point of semi-elliptical surface cracks with different geometries, orientations, and positions relative to the weld line. For cracks oriented perpendicular to the weld line, the FE and WF results showed good agreement for small crack sizes; however, as the crack size increased, the discrepancy between the results obtained from the two methods also increased. Smirnov and Vidyushenkov [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] investigated the estimation of the approximate value of the stress intensity factor for a cylindrical specimen containing a circumferential crack. In their study, Irwin\u0026rsquo;s concept was employed to extend and extrapolate stress concentration formulas at the notch tip, which had previously been developed by Neuber for a body of revolution with an annular hyperbolic recess under uniaxial tensile loading. A structural fracture criterion was used to estimate the specimen dimensions, including the ligament diameter and the outer diameter. Their results indicated that the ratio of the ligament diameter to the specimen diameter is not a predefined value, but rather is determined through a gradual and step-by-step selection process.\u003c/p\u003e \u003cp\u003ePanontin and Hill [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] employed a micromechanical damage model to predict crack initiation in ductile and brittle materials and demonstrated that, with increasing load, the effect of residual stresses on the fracture toughness of ductile materials can be neglected. Dodds et al. [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] experimentally investigated the J-integral fracture parameter for a specimen containing a surface crack and compared the results with finite element solutions.\u003c/p\u003e \u003cp\u003eIn the present study, the finite element method is used to investigate surface cracks in aluminum plates in the presence of residual stresses. To validate the developed numerical model, the extracted results are compared with those reported in previous studies.\u003c/p\u003e"},{"header":"Methodology","content":"\u003cp\u003eIn this study, finite element simulations were carried out using the ABAQUS software. To determine the mechanical properties of the aluminum alloy, tensile test specimens were prepared in accordance with the ASTM B557-02a standard, and tensile tests were performed using a SANTAM testing machine. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e presents the stress\u0026ndash;strain curve obtained from the tensile test.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTo improve computational efficiency and reduce solution time, geometric and loading symmetries of the specimens were exploited, and only one quarter of the full specimen geometry was modeled in the finite element analysis. Accordingly, appropriate symmetric boundary conditions were applied to the corresponding surfaces to ensure that the mechanical behavior of the entire specimen was accurately represented.\u003c/p\u003e \u003cp\u003eTo characterize the stress and strain fields in the vicinity of the crack tip, the J-integral is employed as a path-independent parameter. The J-integral is a fundamental concept in elastic\u0026ndash;plastic fracture mechanics, as it is capable of describing both elastic and plastic deformation behavior around the crack tip. It consists of two components, namely the elastic and plastic parts, and is calculated according to Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e].\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:J={J}_{el}+{J}_{pl}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eFor surface cracks, the elastic component of the J-integral Jel is evaluated using the stress intensity factor through Equations (\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) and (\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). These expressions incorporate geometric and correction factors to account for the crack shape and loading conditions. The parameters Q and F, which appear in these formulations, are obtained from Equations (\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) and (\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e5\u003c/span\u003e), respectively [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e].\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:{\\text{K}}_{\\text{I}}=\\left({\\text{S}}_{\\text{t}}\\right)\\sqrt{{\\pi\\:}\\frac{\\text{a}}{\\text{Q}}}\\text{F}(\\frac{\\text{a}}{\\text{t}},\\frac{\\text{a}}{\\text{c}},\\frac{\\text{c}}{\\text{b}},{\\upvarphi\\:})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:{\\text{J}}_{\\text{e}\\text{l}}=\\frac{{\\text{K}}^{2}(1-{{\\upsilon\\:}}^{2})}{\\text{E}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:\\text{Q}=1+1.464\\left(\\frac{\\text{a}}{\\text{c}}{)}^{1.65}\\right(\\frac{\\text{a}}{\\text{c}}\\le\\:1)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:\\text{F}=[{\\text{M}}_{1}+{\\text{M}}_{2}(\\frac{\\text{a}}{\\text{t}}{)}^{2}+{\\text{M}}_{3}\\left(\\frac{\\text{a}}{\\text{t}}{)}^{4}\\right]{\\text{f}}_{{\\upvarphi\\:}}.\\text{g}.{\\text{f}}_{\\text{w}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn this part of the project, to validate the accuracy of the software-based solution in calculating the J-integral for elastic\u0026ndash;plastic material behavior, a model previously reported in the literature is adopted [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e presents the geometric specifications of the model introduced in the earlier study [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e].\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eGeometric specifications of the model adopted from the previous study [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eThickness (mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCrack length (mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSpecimen width (mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eSpecimen length (mm)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e400\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e400\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe material behavior is described by Eq.\u0026nbsp;(\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e), and the corresponding constants are listed in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:{\\epsilon\\:}=\\left\\{\\begin{array}{c}\\frac{{\\sigma\\:}}{\\text{E}}\\:\\:\\:\\:\\:for\\:\\sigma\\:\\le\\:{{\\sigma\\:}}_{0}\\\\\\:\\alpha\\:{\\left(\\frac{{\\sigma\\:}}{{{\\sigma\\:}}_{0}}\\right)}^{\\text{n}-1}\\frac{{\\sigma\\:}}{\\text{E}}\\:\\:\\:\\:\\:for\\:\\sigma\\:\u0026gt;{{\\sigma\\:}}_{0}\\end{array}\\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMaterial constants used in Eq.\u0026nbsp;(\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e)\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003en\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eα\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eσ₀\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e200 GPa\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e205 MPa\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe model was subjected to bending loading as a percentage of the critical load, as listed in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. In this case, the critical load is obtained from Eq.\u0026nbsp;(\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e).\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\:{\\text{P}}_{\\text{L}}=\\frac{1.455\\text{B}(\\text{W}-\\text{a}{)}^{2}{{\\sigma\\:}}_{\\text{Y}}}{2\\text{H}}=116.4\\left[\\text{k}\\text{N}\\right]\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eApplied load levels in the model reported in the previous study [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e], expressed as a percentage of the critical load\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eP/P\u003csub\u003eL\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.25\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.5\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.75\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.25\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.5\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSurface traction\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e53.26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e106.52\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e159.78\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e213.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e266.30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e319.56\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eIt should be noted that, due to quarter-model simulation, the load values in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e are halved when applied to the model.\u003c/p\u003e \u003cp\u003eThe calculated J-integral values for different loading levels are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e and compared with the results of the previous study.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn this study, the specimens were modeled in three dimensions, and the mesh was designed to provide sufficient refinement in the vicinity of the crack tip. This type of meshing, commonly used for semi-elliptical and semi-circular cracks, is known as a spider mesh and enables high accuracy in the evaluation of fracture mechanics parameters. In the numerical modeling, linear reduced-integration brick elements (C3D8R) were employed. After defining the meshing strategy and selecting the element type, the mesh size in the region surrounding the crack tip was examined in order to achieve an appropriate and convergent mesh configuration.\u003c/p\u003e \u003cp\u003eTo assess mesh adequacy, the effect of mesh size on variations of the maximum J-integral along the front of the surface crack tip in the experimental specimen model was investigated. For this purpose, two variables were considered to characterize the mesh density around the crack tip: the mesh size in the radial direction (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{n}_{1}\\)\u003c/span\u003e\u003c/span\u003e​) and the number of elements in the circumferential direction (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{n}_{2}\\)\u003c/span\u003e\u003c/span\u003e​), as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTo investigate the effect of element size, the model without residual stresses was loaded in the same manner as the experimental model, and the average value of the maximum J-integral from the third to the eighth contours along the surface crack front was calculated. These values were then compared for different mesh configurations in order to evaluate the influence of mesh size. The results of this investigation are presented in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e Given the small differences observed in the J-integral values at this stage, n\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;26 was selected as the appropriate mesh size for the number of elements in the radial direction. In the next step, while keeping n\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;26 constant, the number of elements in the circumferential direction was varied by doubling and halving the initial value to determine the suitable mesh density. Based on this procedure, the minimum adequate number of elements was determined to be 18168._\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eEffect of the number of elements in the radial direction with a constant number of elements in the circumferential direction\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNumber of elements in the model\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePercentage difference from previous mesh\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eJ [N/mm]\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e(n\u003csub\u003e2\u003c/sub\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e(n\u003csub\u003e1\u003c/sub\u003e)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e18,168\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026ndash;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e26\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e16,920\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.68\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e13\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e20,664\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e52\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eIn this section, residual stresses were introduced into the finite element model by applying a four-point bending load with a force magnitude of 4800 N. Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e illustrates the finite element model subjected to the four-point bending loading used to generate the residual stress field.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"Results and Discussion","content":"\u003cp\u003eFigure \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e shows the deflection generated at the center of the specimen after unloading and following the crack release stage.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe developed finite element model was subjected to tensile loading once in the presence of residual stresses and once without residual stresses. The load versus crack mouth opening displacement (CMOD) curves were extracted for both cases. Figure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e presents these results for the specimens with and without residual stresses.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, the elastic load-carrying capacity of the specimen without residual stresses obtained from the finite element analysis is approximately 60 kN, whereas for the specimen containing residual stresses it is about 35 kN. This indicates an increase in the crack driving force due to the tensile residual stresses introduced into the model. Furthermore, the finite element results show that the load required to produce the same level of plastic deformation in the specimen with residual stresses is lower than that required for the specimen without residual stresses. In other words, under the same applied load, the maximum allowable surface crack size is smaller for the specimen containing residual stresses compared to the specimen without residual stresses.\u003c/p\u003e \u003cp\u003eAccording to Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, at higher load levels the load\u0026ndash;CMOD curves for specimens with and without residual stresses tend to converge. This behavior can be attributed to the development of extensive plasticity at the crack tip under high loads, which reduces the influence of residual stresses on the fracture behavior of the specimens. In addition, the difference between the loads required to produce the same plastic deformation in specimens with and without residual stresses becomes negligible at small deformation levels. Figure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003eJ-integral values obtained from finite element analysis for specimens with and without residual stresses\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, at low load levels a noticeable difference exists between the J-integral values obtained from the finite element analysis for specimens with and without residual stresses. This difference is attributed to the influence of the residual stress field generated at the crack tip as a result of the four-point bending loading. At higher load levels, however, the two curves tend to coincide due to the development of significant plastic deformation at the crack tip. Consequently, under loads exceeding approximately 100 kN, the effect of residual stresses on the crack-tip field can be considered negligible.\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eThe results of this study indicate that the presence of tensile residual stresses increases the driving force for crack growth and reduces the load-carrying capacity of the specimen in the elastic\u0026ndash;plastic regime, such that at low load levels a noticeable difference is observed between the responses of specimens with and without residual stresses. However, at higher load levels and with the expansion of the plastic zone at the crack tip, the influence of residual stresses diminishes and the fracture responses of the two cases become closer. It was also found that the maximum allowable crack size in the presence of residual stresses is smaller than that in the absence of residual stresses.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eTaher F, Afshari M, Houmani A, Samadi MR, Bakhshi S, Afshari H (2024) Simultaneous enhancement of the impact strength and tensile modulus of PP/EPDM/TiO2 nanocomposite fabricated by fused filament fabrication. Colloid Polym Sci 302(3):393\u0026ndash;407\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKhademian A et al (2025) Investigating the mechanical and electrical properties of hybrid joint of aluminum to copper obtained by friction stir welding and brazing. J Mater Sci: Mater Electron 36(23):1440\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKahlin M, Ansell H, Moverare J (2022) Fatigue crack growth for through and part-through cracks in additively manufactured Ti6Al4V. Int J Fatigue 155:106608\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eAlavi SA, Afshari M, Afshari H, Samadi MR, Abbaas H, Abualigah L (2025) Investigating the effect of hot preforming combined with superplastic forming on thickness distribution and forming time of Ti-6Al-4V alloy using finite element analysis. J Strain Anal Eng Des, p. 03093247241310346\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eGovindan P, Joshi SS (2012) Analysis of micro-cracks on machined surfaces in dry electrical discharge machining. J Manuf Process 14(3):277\u0026ndash;288\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMarazani T, Madyira DM, Akinlabi ET (2017) Repair of cracks in metals: A review, Procedia Manufacturing, vol. 8, pp. 673\u0026ndash;679\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJones R (1952) A method of studying the formation of cracks in a material subjected to stress. Br J Appl Phys 3(7):229\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eRiemer A, Richard HA (2016) Crack propagation in additive manufactured materials and structures, Procedia Structural Integrity, vol. 2, pp. 1229\u0026ndash;1236\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMohan Kumar S, Rajesh Kannan A, Pramod R, Siva Shanmugam N, Dhinakaran V, Krishnaveni A (2022) A study on evaluation of stress intensity factor (KI) and J-integral for 40Ni2Cr1Mo28 alloy (structural steel): analytical and finite element analysis approach. Materialwiss Werkstofftech 53(12):1504\u0026ndash;1517\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eAfshari H, Ashrafi A (2023) Experimental and numerical study of the hydroforming process of copper-aluminum double layered tube. Iran J Manuf Eng 10(3):18\u0026ndash;33\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSong P, Shieh Y (2004) Crack growth and closure behaviour of surface cracks. Int J Fatigue 26(4):429\u0026ndash;436\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJukić K, Perić M, Tonković Z, Skozrit I, Jarak T (2021) Numerical calculation of stress intensity factors for semi-elliptical surface cracks in buried-arc welded thick plates, Metals. 11(11):1809\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSmirnov V, Vidyushenkov S (2021) Stress intensity factor for cylindrical specimen with external circular crack under tension, in International Scientific Siberian Transport Forum, : Springer, pp. 408\u0026ndash;417\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eChen Y, Lambert S (2005) Numerical modeling of ductile tearing for semi-elliptical surface cracks in wide plates. Int J Press Vessels Pip 82(5):417\u0026ndash;426\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePanontin T, Hill M (1996) The effect of residual stresses on brittle and ductile fracture initiation predicted by micromechanical models. Int J Fract 82(4):317\u0026ndash;333\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eDodds RH Jr, Read DT (1990) Experimental and numerical studies of the J-integral for a surface flaw. Int J Fract 43(1):47\u0026ndash;67\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eNewman J Jr, Raju I (1979) Analyses of Surface Cracks in Finite\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eFarahani M, Sattari-Far I, Akbari D, Alderliesten R (2013) Effect of residual stresses on crack behaviour in single edge bending specimens. Fatigue Fract Eng Mater Struct 36(2):115\u0026ndash;126\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Semi-elliptical surface crack, residual stress, finite element analysis, 5000-series aluminum alloy","lastPublishedDoi":"10.21203/rs.3.rs-8912144/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8912144/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study numerically investigates the fracture behavior of semi-elliptical surface cracks in 5000-series aluminum sheets, with emphasis on the role of residual stresses. Since the coexistence of cracks and residual stresses can seriously compromise the integrity and safety of engineering structures, their combined effect must be evaluated for reliable fracture prediction. A three-dimensional finite element model was developed in ABAQUS and the responses of specimens with and without residual stresses were compared. The mechanical properties of the alloy were obtained from tensile tests conducted in accordance with ASTM B557-02a and implemented in the simulations. To improve accuracy near the crack front, a refined, spider-web (focused) mesh was employed to capture the stress\u0026ndash;strain fields with higher resolution. Elastoplastic fracture behavior was assessed using the J-integral as the primary fracture mechanics parameter. Residual tensile stresses were generated through a simulated four-point bending procedure; after unloading, the specimens were subjected to tensile loading. The results indicate that tensile residual stresses increase the crack-driving force and reduce the load-carrying capacity in the elastic\u0026ndash;plastic regime. At low applied loads, a significant difference in J-integral values was observed between the two conditions, whereas with increasing load and plastic zone development, the influence of residual stresses diminished and the responses converged.\u003c/p\u003e","manuscriptTitle":"Numerical Investigation of the Effect of Residual Stresses on the Behavior of Semi-Elliptical Surface Cracks in Aluminum Plates","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-20 06:04:45","doi":"10.21203/rs.3.rs-8912144/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"0d4bfe2f-585c-4a6c-8ff2-4cbc9b1b8e05","owner":[],"postedDate":"February 20th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":63158455,"name":"Mechanical Engineering"}],"tags":[],"updatedAt":"2026-02-20T06:04:45+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-20 06:04:45","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8912144","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8912144","identity":"rs-8912144","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}