Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 55, 2, pp. 659-666, Warsaw 2017 DOI: 10.15632/jtam-pl.55.2.659 NUMERICALLY PREDICTED J-INTEGRAL AS A MEASURE OF CRACK DRIVING FORCE FOR STEELS 1.7147 AND 1.4762 Goran Vukelic University of Rijeka, Faculty of Maritime Studies Rijeka, Croatia e-mail: gvukelic@pfri.hr Josip Brnic University of Rijeka, Faculty of Engineering, Rijeka, Croatia Fracture behavior of two types of steel (1.4762 and 1.7147) is compared based on their numerically obtained J-integral values. The J-integral are chosen to quantify the crack driving force using the finite element (FE) stress analysis applied to single-edge notched bend (SENB) and compact tensile (CT) type fracture specimens. The resulting J-values are plotted for growing crack length (∆a – crack length extension) at different a/W ratios (a/W – relative crack length; 0.25, 0.5, 0.75). Slightly higher resulting values of the J-integral for 1.4762 than 1.7147 can be noticed. Also, higher a/W ratios correspond to lower J-integral values of the materials and vice versa. J-integral values obtained by using the FEmodel of the CT specimen give somewhat conservative results when compared with those obtained by the FEmodel of the SENB specimen. Keywords: crack, steel 1.7147, steel 1.4762, FE analysis 1. Introduction Material imperfections and failures due to themanufacturing process coupledwith severe service conditions can lead to flawappearance in engineering structures.Consequently, crack occurrence and its growth can seriously affect integrity of such structures leading to catastrophic failure. In order to avoid such a scenario, proper selection of materials is a step of great importance in the process of structural design. Selection of an improper material may affect product profi- tability, reduce its service lifetime and finally result in appearance of flaws and failure. Several requirements have to bemet during the material selection process. These requirements include adequate strength of thematerial, acceptable rigidity level, resistance to elevated temperatures, etc., but also the material must be sufficiently resistant to crack propagation. The resistance of the material to crack propagation in fracture mechanics is usually descri- bed through one or more parameters obtained by experimental research, like crack tip opening displacement (CTOD), J-integral or stress intensity factor K. Of all the above mentioned, the J-integral is suitable for trying to quantify thematerial resistance to crack elongation when ob- serving ductile fracture inmetallic materials (Kossakowski, 2012).When dealingwith a growing crack, the obtained J values can be correlated to appropriate crack length extensions∆a giving the resistance R curve. Standardized experimental procedures are used to obtain the R curve. Extensive experimental procedures can be, in some cases, accompanied or even substitutedusing some of the modern numerical methods, e.g. the finite element (FE) method. Recent research on the topic of numerical fracture mechanics includes accuracy check of J-integral values ob- tained by experiments, planar (2D) FE analysis, space (3D) FE analysis or the EPRI method (Qiao et al., 2014). FE analysis ofMode I fracture in a compact tensile (CT) specimen has been conducted to reveal effects on micro, meso and macroscale (Saxena and Ramakrishnan, 2007), 660 G. Vukelic, J. Brnic while plastic geometry factors were determined numerically in order to calculate the J-integral from the load vs. crack mouth opening displacement or load-line displacement curve in the J-R curve test (Huang et al., 2014). Elastic and plastic constraint parameters for 3D problems were studied on single-edge notched bend (SENB) and CT specimens of non-standard configuration to characterize fracture resistance parameters (Shlyannikov et al., 2014). Research on explaining procedures that guarantee the domain independent propertywhen calculating the 3D J-integral for large deformation problems was carried out by Koshima and Okada (2015). A 3D domain integral method based on the extended FE method for extracting mixed-mode stress intensity factors was described byWu et al. (2012). The work presented in this paper is a comparison of numerically obtained J-integral values taken as ameasure of the crack driving force for steels 1.7147 and 1.4762. Steel 1.7147 is usually used in production of spindles, pistons, bolts, levers, camshafts, gears, shafts, etc. The latter is a heat-resistant steel used in furnace industry, ceramics and cement industry, etc., i.e., in applications with high temperature and relatively low tensile requirements. Carbon, low alloy or high alloy ferritic steels can exhibit ductile fracture at elevated temperatures (Zhu and Joyce, 2015). Structuresmadeof these or similar steels aremore than susceptible to theflawappearance and crack growth (Wagner et al., 2010; Zangeneh et al., 2014;Gojic et al., 2011).Observing these examples, it is easy to understand the need for fracture characterization of such materials. 2. Material properties Twomaterials are compared: structural steel 1.7147 (AISI 5120, 20MnCr5) and high chromium stainless steel 1.4762 (AISI 446, X10CrAlSi25). Chemical compositions of the mentioned mate- rials are given in Tables 1 and 2. Composition of steel 1.7147 can be compared to the standard EN 10084-2008. Here, the content of carbon equals themaximum standard value (0.22%) while the rest of the alloying elements are within the prescribed values. Comparing steel 1.4762 to the standard EN 10095-1999, all of the alloying elements are in the standard ranges. Table 1.Chemical composition of steel 1.7147 (wt%) (Brnic et al., 2014a) Material C Mn Si S Nb Cu Cr Ni P Ti Rest 1.7147 0.22 1.23 0.29 0.025 0.03 0.06 1.11 0.08 0.021 0.02 96.914 Table 2.Chemical composition of steel 1.4762 (wt%) (Brnic et al., 2014b) Material C Mn Si S Mo Al Cr Ni P V Rest 1.4762 0.102 0.519 1.2 0.01 0.116 1.23 23.05 0.6855 0.0217 0.201 72.8648 Engineering stress-strain (σ-ε) diagrams for both steels are given in Fig. 1, while the yield strength σYS, tensile strength σTS and Young’s modulusE are given in Table 3. Table 3. Yield strength σYS, tensile strength σTS and Young’s modulus E of the considered materials (Brnic et al., 2014a,b) Material σYS [MPa] σTS [MPa] E [GPa] 1.7147 398 562 219 1.4762 487 584 192 Numerically predicted J-integral as a measure of crack... 661 Fig. 1. Steel 1.7147 and steel 1.4762: uniaxial engineering stress-strain diagrams 3. Importance of J-integral Rice (1968) introduced the J-integral as a path-independent integral that can be encircled around the tip of a crack and considered equally as an energy release rate parameter and a stress intensity parameter. In a 2D form and with reference to Fig. 2, it can be written as J = ∫ Γ ( wdy−Ti ∂ui ∂x ds ) (3.1) Fig. 2. J-integral arbitrary contour path enclosing the tip of a crack Equation (3.1) comprises of Ti = σijnj that are components of the traction vector, ui are the components of the displacement vector and ds is an incremental length along the integral contour Γ . The strain energy density w can be written as w= ∫ σij dεij (3.2) where εij is the sum of elastic and plastic strains at a specific point. The J-integral is path independent as long as the stress is a function of the strain alone and provided the crack tip is the only singularity within the contour. The J-integral equation shows that the energy of the integral contour increases for the crack growth per unit length. The JIc parameter, that can be 662 G. Vukelic, J. Brnic derived, describes the fracture resistance of the material, i.e. required energy for crack growth per unit length when the contour Γ must shrink to the crack tip J = lim r→0 ∫ Γ ( wdy−Ti ∂ui ∂x ds ) (3.3) 4. Numerical prediction of J-integral The experimental single specimen testmethod following an elastic unloading compliance techni- que was numerically simulated in order to predict fracture behavior of steels 1.7147 and 1.4762. It is an experimental testmethod that estimates the size of the expanding crack based onmeasu- red values of the crack mouth opening displacement. The resulting J values serve as a fracture toughness parameter and can be correlated to crack extension values. The numerical procedure begins with FE stress analysis. Two-dimensional FEmodels of two types of fracture specimens, single edge notched bend (SENB) and compact tensile (CT), are defined according to theASTM standard (2005), see Fig. 3. Three initial relative crack length a/W (W = 50mm) ratios are taken, 0.25, 0.5 and 0.75. As for thematerial behavior, it is considered to bemultilinear isotropic hardening. Specimens are discretized with 8-node isoparamateric quadrilateral elements. High deformation gradients occur in the yielding regions around the crack tip. That is why the FE mesh is refined there. Quasi-static load is imposed on the specimen in order to simulate the compliance procedure of the single specimen test method. Since the specimen is symmetrical, only half of it needs to bemodelled.To simulate crack propagation, the node releasing technique has been used. Fig. 3. Finite elementmodel of: (a) CT specimen, (b) SENB specimen The second step is to extract stress analysis results from the integration points of finite elements enclosing the crack tip. This results are used to evaluate J values in the integration points by Eq. (4.1) (De Araujo et al., 2008) and sum them along the path Γ that encloses the crack tip giving the total value of J, see Fig. 4 J = np ∑ p=1 GpIp(ξp,ηp) (4.1) In Eq. (4.1), Gp represents Gauss weighting factor, np stands for the number of integration points and Ip is the integrand evaluated at each Gauss point p Ip = { 1 2 [ σxx ∂ux ∂x +σxy (∂ux ∂y + ∂uy ∂x )∂ux ∂x +σyy ∂uy ∂y ]∂y ∂η − [ (σxxn1+σxyn2) ∂ux ∂x +(σxyn1+σyyn2) ∂uy ∂x ] √ (∂x ∂η )2 + (∂y ∂η )2 } g (4.2) Numerically predicted J-integral as a measure of crack... 663 Fig. 4. J-integral path Γ encircled around the crack tip through FE integration points Although the crack tip plastic zone radius can be taken as variable using von Mises yield criterion (Bian, 2009), here it is taken as a constant value. Since a slight variation of J values is possible in the numerical analysis, three different paths around the crack tip are defined in each example. The average value of these three paths is taken as the final value of the J-integral. In order to verify the procedure, the J-integral values are first compared to the available experimental results. Since there is no available experimental results for steels 1.7147 and1.4762, the procedurehas beenfirst validated on steel 1.6310 (Narasaiah et al., 2010), Fig. 5. Fig. 5. Validation of numerically obtained J-integral values on steel 1.6310 Good compatibility of the experimental and numerical results encouraged further use of the numerical procedure for steels 1.7147 and 1.4762. Figures 6 and 7 show the final J values for 1.7147 and 1.4762 taken as a measure of the crack driving force for different initial crack lengths a/W according to the crack propagation ∆a. Fig. 6. J-integral values obtained numerically for steel 1.7147: (a) CT specimen, (b) SENB specimen 664 G. Vukelic, J. Brnic Fig. 7. J-integral values obtained numerically for steel 1.4762: (a) CT specimen, (b) SENB specimen 5. Discussion Fracture behavior of steel 1.7147 and steel 1.4762 can be predicted based on the numerical investigation results presented in Figs. 6 and 7 using J-integral values as ameasure of the crack driving force. Observing the obtained diagrams, it is clear that steel 1.4762 has slightly higher values of the J-integral than 1.7147. This makes steel 1.4762 a bit more adequate to structures that need less susceptibility to fracture. Thepredicted discrepancy in the numerically obtainedJ values and, therefore, the difference in resistance tocrack extension comparing steels 1.7147 and1.4762 canbecontributed todifferent composition and properties (Tables 1-3) of the two steels. Steel 1.4762 has a somewhat higher value of the nickel content which can add to the noted behavior. Nickel, as the alloying element, is usually added to stainless steels to reach a certain level of increased strength and hardness without compromising ductility and toughness levels. Nickel also improves the oxidization and corrosion resistance when added in suitable quantities to stainless steels. Although steel 1.7147 has an elevated chromium content (1.11%) making it suitable for corrosive environment. Steel 1.4762 is a true stainless steel inwhich chromiumexceeds 12%content (here 23.05%) significantly improving corrosion resistance. Benefits of chromium as an alloying element in steel are also improved strength, hardenability, wear resistance and response to heat treatment. Also, observing Figs. 6 and 7, lower a/W ratios corresponding to higher J values exhibit a trend observed by other authors (Cravero and Ruggieri, 2003). Also, the J-integral differs greatly for a/W = 0.75 if matched with a/W = 0.25 and 0.5, then they tend to be close in values. In addition, J-integral values obtained by the FE model of the CT specimen give a bit conservative resultswhen comparing themto those obtained fromtheSENBspecimen.That can be ascribed to the specimen geometry and loading effect. As for the crack geometry, a/W ratios are kept equal for both steels in relative specimens. That way, the influence of geometry on the difference in J values for the two steels is negligible. 6. Conclusion Numerical assessment of the J-integral for steels 1.7147 and 1.4762 can be useful as a prediction of the possible fracture behavior of materials. Although not validated by an experiment, the fine correspondence between numerical and experimental results for steel 1.6310 assures confi- dence in using J-integral values for steels 1.7147 and 1.4762. In the structural design procedure that includes any of the considered material, the obtained results can be useful in the initial assessment of the material susceptibility to crack growth. Numerically predicted J-integral as a measure of crack... 665 The presented work intends to attract attention on the need for fracture behavior characte- rization ofmaterials recommended for use in specific engineering components. Here, the novelty of the research lies in numerically predicted values of the J-integral taken as a measure of the crack driving force for steels 1.7147 and 1.4762 which are, according to the authors’ knowled- ge, unavailable to construction designers in the presented form. Both materials offer improved corrosion resistance and can be considered for engineering applications intended to be used in corrosive environment and susceptible to crack growth and fracture like spiral bevel gears in truck differential systems prone to failure (Sekercioglu andKovan, 2007) or stainless steel tubes found in recuperators and exposed to elevated temperatures that cause failures (Bhattacharyya et al., 2008). The results of the investigation presented in this paper can be used to avoid such failure scenarios. Acknowledgment Research presented in this paper has been supported by Croatian Science Foundation (project 6876) and by University of Rijeka (projects 13.09.1.1.01 and 13.07.2.2.04). References 1. 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