Study the pattern of J-profile along the crack front through 3D finite element analysis

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1 Page 16 Study the pattern of J-profile along the crack front through 3D finite element analysis Abstract: Badrun Nahar Hamid a*, Michale J Baker b a Principal Scientific Officer, Atomic Energy Research Establishment, Savar, Dhaka.. b Professor Emeritus, School of Engineering, University of Aberdeen, Scotland, UK. *Corresponding author: nhbadrun@yahoo.com This paper is based on 3D finite element (FE) analysis which aims to illustrate the characteristics of fracture parameter, J-integral, along the crack front with the change of crack sizes and thicknesses of the specimen. Both linear elastic and elastic-plastic behaviour of material have been considered in this investigation. FE analysis software ABAQUS in conjunction with 3-D cracked mesh generation software ZENCRACK has been used for this purpose. Numerical examples have been provided to illustrate the present work. Keywords: J-integral, ABAQUS, ZENCRACK. 1. INTRODUCTION In practice, there is no such thing as defect free material and crack-like defects are always inherent in every component. High temperature, radiation embrittlement, corrosion or fatigue damage are all factors that could lead the crack to leakage and cause severe consequences. For assessing the integrity of the engineering components, stress intensity factor, K, for elastic material and J-integral, for elastic-plastic material play important roles in evaluating the failure of engineering components and systems. Although many papers and a number of handbooks provide solutions for K for different specimens with cracks with different geometries, but solutions for J-integral for elastic-plastic material are not available. Again, most of the solutions of K are found in terms of 2-D plane stress or plane strain approximation of the real 3-D body in which the true 3-D constraint effects or tri-axial effects at the crack tip remain undefined and the prediction of failure both in cleavage and ductile fracture may be subject to error. This is particularly important in case of reliability-based structural integrity assessment where it requires to minimize the number of uncertainties that have to be modelled. Thereby, it becomes necessary to obtain the complete behaviour of K and also J along the crack front through 3D analysis. To tackle these issues, the FE analysis software ABAQUS [1] in conjunction with 3-D cracked mesh generation software ZENCRACK [2] has been used. To know more details about K and J-integral, standard text [3] can be referred to. As ABAQUS provides solutions in terms of J and as there is a relation between K and J-integral (eq.1) as long as the material model is linear elastic, then, rather computing K from J- integral, J has been retained as a linear elastic fracture parameter in the present work. J 2 K E, for plane stress; J 2 2 K (1 ν ), for plane strain (1) E In this study, a non-standard compact tension specimen (CT) machined from different orientations and locations in a 50mm thick steel plate (BS 4360 Grade 50DD) have been taken. Due to their non-standard geometries, stress intensity factor solutions were not available, and hence FE analysis was also necessary to obtain values of the J-integral considering elastic analysis first to validate the CT model. Based on reasonable agreement of FE results with Murakami s solution [4], both linear elastic and elastic plastic analyses have been carried later for different sizes of crack with different thickness of specimens in order to observe the profiles of elastic and elastic-plastic J-values along the crack front. In addition, thickness effects of specimens on mid section and surface J values are also illustrated. The present research, in fact, was undertaken in order to support fatigue research as well as reliability based fracture assessment [5, 6]. It has found the research finally provided some original results relating to the effects of specimen thickness and crack size on J-integral values.

2 Page FE MODELLING AND ANALYSIS Finite element package ABAQUS [1] linked with crack generation software ZENCRACK [2] have been used to model CT specimen with different thickness with different crack lengths. Fig.1 shows the sketch diagram of CT specimen and Table 1 contains the detailed about specimen geometry. The material used in CT specimen was weldable structural steel of BS 4360 Grade 50 DD which has extensive use in offshore structures. The representative material properties for this steel were obtained from a laboratory tensile test [6]. The yield stress was taken to be MPa. The elastic modulus of the material was MPa and Poisson s ratio was The finite element mesh contains reduced integration 3-D 20-noded brick elements. The standard crack block st111x5, supplied by ZENCRACK, was used and replaced the 3-D element of the uncracked mesh. A number of crack blocks were placed through the thickness of the specimens to form the longer section of the crack front. Table-1. Modeling detail for CT specimen Specimen ID Number of crack front elements B (mm) Pl Pl Pl W (mm) H (mm) x (mm) The loading pins were modelled as rigid bodies and the CT specimen was loaded by applying a vertical displacement of same magnitude to both the upper and lower pin thus causing a mode I displacement of the crack. All other motions of the pin were restrained. Each FEA was performed in a single load step with a time period of 1. The initial time increment was used as For elastic-plastic analysis, incremental theory of plasticity was used. ABAQUS uses automatic loading and adjusts this increment according to its success in solving the problem. An additional displacement constraint in the appropriate direction was placed on a set of nodes through the thickness to prevent rigid body motion. Finally, Fig.1: CT specimen of thickness B. Fig.2: FE mesh shows crack opening of CT specimen the cracked mesh was submitted to ABAQUS for finite element analysis. Fig. 2 shows the FE meshes of typical CT specimen with crack opening under displacement of pins. In each case analysed, the J-integral output was requested for four contours and the average J-values of 2 nd - 4 th contours were calculated. Mesh sensitivity tests and the path independence of the FE results have shown that the results are of sufficiently high accuracy for the present investigation. 3. VALIDATING THE CT SPECIMEN MODEL Initially, 3-D linear analysis was conducted for the 5 mm CT specimen for a number of crack lengths, ranges from mm to mm. The results have been compared with the standard results given by Murakami [4] which provide solutions of stress intensity factor K for CT specimens but with somewhat different geometry. The K-values given by Murakami [4] are shown in eq. (2) with eq.(3) which are then converted to J using eq. (1) for idealized plane stress and plane strain conditions,

3 Page 18 P K I f I α, 1/2 BW a α (2) W f I (2 α)( α 13.32α 14.72α 5.6α ) (3) 1 α 3/2 Fig. 3 shows the plane stress (MPS) and plane strain (MPN) values of J as a function of crack size, computed from the Murakami solutions compared with the newly calculated 3-D mid-section J - values (BM). Similar plots were produced for 10 mm and 20 mm CT specimens which have not shown in here due to space limitations. But, these all plots show that the mid-section J-integral values in the present analysis are in reasonably good agreement with the Murakami s plane strain solutions. Based on this reasonable agreement between calculated results and Murakami s solution, the model later has been used for further calculations to be carried on as discussed in section 4. Fig.3: Comparison of present 3-D results with MPS and MPN for 5mm thick specimen 4. ELASTIC AND ELASTIC PLASTIC J-INTEGRAL ALONG THE CRACK FRONT In this investigation, CT specimens with mentioned three thicknesses have been considered to have common crack lengths of mm, mm and mm. Fig. 4 (a), Fig. 5(a) and Fig. 6 (a) show the distribution of elastic J-integral along the crack front, together with the plane stress and plane strain solutions of Murakami for a CT specimen of thickness 5 mm, for three different crack lengths. For comparison with these, Fig. 4(b), Fig. 5 (b) and Fig. 6 (b) show the distribution of elastic-plastic J-integral along the crack front for the same geometries. Similarly, Fig. 7 to Fig. 9 and Fig. 10 to Fig. 12 show thee same information for CT specimens of 10 mm and 20 mm thickness, respectively. For the crack length of mm, with B/a = 0.349, as in Figure 4(a), the J-integral profile is in fairly close agreement with the Murakami plane stress solution, but exceeds this by about 8% at the mid-section of the plate. For the mm crack, with B/a = 0.248, as in Figure 5 (a), the maximum J-integral value is somewhat lower and the mid-section value equals the Murakami plane strain solution, and for the mm long crack, with B/a = 0.163, Figure 6 (a) shows the maximum J-integral value lies between the theoretical plane stress and plane strain values. Very similar results are found for the 10 mm and 20 mm thick plates. Although these results for 10 mm thick plate are illustrated through Figures 7(a), 8(a) and 9(a) but for space limitations, only linear elastic analysis for 14.33mm length crack 20 mm specimen is provided in here. From these results, it is noticed that the profiles for the three thicknesses of plate are all very similar with roughly uniform J-integral values over most of the central part of the plate but with lower values towards the plate surfaces. If now focus given on the elastic-plastic J-integral profiles as shown in Fig. 4 (b), Fig. 5 (b) and Fig. 6 (b) for the 5 mm thick specimen for the three different crack lengths at the load corresponding to a relative pin displacement of 0.25 mm, the profiles are radically different from the elastic J-integral profiles, being much more rounded in shape, with high values at the mid-section of the plate which then progressively reduce towards the plate surface. Very similar J-integral profiles have been found for the two other plate thicknesses, as shown

4 Page 19 in Fig. 7 (b), Fig. 8 (b) and Fig. 9 (b) for 10 mm thick plate, and in Fig. 10 (b), Fig. 11 and Fig. 12 for 20 mm thick plate. A clear finding from these graphs is that the profiles become less uniform and more rounded with increase in crack length, i.e. increase in the ratio B/a. The possible cause for such behavior is expected to arise from stress condition at crack tip as a result of material ductility, crack length as well as specimen thickness, etc. It is clear from the above that these findings are of importance both in the assessment of fracture and fatigue behavior. A conservative approach would be to use the maximum value of the J-integral for each profile in any assessment, but further research is clearly required to determine how conservative this is. Fig. 4(a): J-integral profile for Linear Elastic Analysis with B = 5mm and a = 14.33mm. Fig. 4(b): J-integral profile for Elastic Plastic Analysis with B = 5mm and a = 14.33mm. Fig. 5(a): J-integral profile for Linear Elastic Analysis with B = 5mm and a = mm. Fig. 5(b): J-integral profile for Elastic Plastic Analysis with B = 5mm and a = mm. Fig. 6(a): J-integral profile for Linear Elastic Analysis with B = 5mm and a = mm. Fig. 6(b): J-integral profile for Elastic Plastic Analysis with B = 5mm and a = mm. Fig. 7(a) J-integral profile for Linear Elastic Analysis with B = 10 mm and a = mm. Fig. 7(b) J-integral profile for Elastic Plastic Analysis with B = 10 mm and a = mm.

5 Page 20 Fig. 8(a) J-integral profile for Linear Elastic Analysis with B = 10 mm and a = mm. Fig. 8(b) J-integral profile for Elastic Plastic Analysis with B = 10 mm and a = mm. Fig. 9(a) J-integral profile for Linear Elastic Analysis with B = 10 mm and a = mm. Fig. 9(b) J-integral profile for Elastic Plastic Analysis with B = 10 mm and a = mm. Fig. 10(a) J-integral profile for Linear Elastic Analysis with B = 20 mm and a = mm. Fig. 10 (b) J-integral profile for Elastic Plastic Analysis with B = 20 mm and a = mm. Fig. 11: J-integral profile for Elastic Plastic Analysis with B = 20 mm and a = mm. Fig. 12: J-integral profile for Elastic Plastic Analysis with B = 20 mm and a = mm.

6 Page THICKNESS EFFECTS ON THE MID-SECTION AND SURFACE J-VALUES Based on the above discussion, Fig. 13 shows the variation of mid-section J-integral with plate thickness, B, for the three crack lengths of 14.33mm, 20.15mm and 30.63mm for the case of elastic-plastic material behavior. It is found that the mid-section J-integral value can be seen to decrease with plate thickness, corresponding to a change in stress state closer to that of plane strain. In comparison with Fig.13, Fig. 14 shows the computed values of the J-integral on the two surfaces of the plate, for the same three crack lengths, for elastic-plastic material behavior. These also reduce with increasing plate thickness, but these surface values cannot be considered to be reliable because of mesh sensitivity problems, as discussed in the work of Badrun [6]. Fig. 13: Mid-section J-values for specimens of different thickness having different lengths of cracks 6. CONCLUDING REMARKS The present analysis provides new results of J-integral for somewhat non-standard CT specimens from consideration of both elastic and elastic plastic behavior of material. Also, the distribution of J-integral how varies along the crack front for number of crack lengths with the change of specimen thickness are also investigated. The important conclusions from the present research are that The distribution of elastic-plastic J-values along the crack front for CT specimens is less uniform compared to that of elastic analysis for the same size of crack in the same thickness specimen. The distribution of elastic-plastic J-values along the crack front is strongly influenced by the ratio of specimen thickness to crack length (B/a). Mid section J-values at same crack length decreases with the increase of specimen thickness. It is clear from the present research that the variation of J-values along the crack front both for elastic and elastic plastic analysis should be encountered for the accurate estimation of component failure probability rather than following the conservative approach where only mid (maximum) value of J-values is considered. The present research is important in the sense that as it has provided some new information in fracture mechanics. ACKNOWLEDGEMENT The authors are grateful to the Safety Engineering Unit, University of Aberdeen, UK and also to the Chairman, Bangladesh Atomic Energy Commission for allowing to conduct such research as a part of PhD program on reliability based fracture assessment. REFERENCES Fig. 14: Surface J-values for specimens of different thickness having different lengths of cracks 1. Anderson, T.L., Fracture Mechanics-Fundamentals and Applications, CRC Press, London, ABAQUS/Standard User s Manual, Version 6.4, Hibbit, Karlsson & Sorensen, USA, ZENCRACK, Automatic 3-D Fracture Mechanics Mesh Generation Software, Version Murakami, Y., Stress Intensity Factors Hand Book, Volume-1, Pergamon Press, Stanley, I; PhD Thesis, University of Aberdeen, Scotland, UK, Badrun Nahar Hamid; PhD Thesis, University of Aberdeen, Aberdeen, UK, 2006.