ࡱ> UlT[i7 >bjbjUU <7|7|8'l8LfB ^p( ' 3AAAAAAA$%E EGfA-; ;;A#A###;48A#;A##e):&<R IBo;&<\A0B <G' |G&<#4th International Congress of Croatian Society of Mechanics September, 18-20, 2003 Bizovac, Croatia EMBED Word.Picture.8 conservation law of J integral type for multi-material bodyAleksandar Sedmak, Drazan Kozak, Gorgi Adziev, Franjo Matejicek, Nenad GubeljakKeywords: J integral, welded joint, multi-material body, finite element method1. IntroductionThe J integral, as defined by Rice [1], has been used extensively as the fracture mechanics parameter. Its popularity follows from the fact that the original introduction of J integral was well established within the basic laws of continuum mechanics. It was proved by Rice [1] that the J integral is path independent (what has enabled its simple evaluation), that it has a physical meaning, i.e. it can be identified with crack driving force, and that it describes stress and strain fields around crack, making it a valid fracture mechanics parameter. Anyhow, as stated in the Rices original paper, J integral is valid only for two-dimensional plane (non-linear) elasticity in absence of volume and thermal forces, and for the homogeneous material, at least in crack direction. Its application beyond these limitations has been questionable, but still not unsuccessful. As for the example, the J integral was successfully applied in the elastic-plastic fracture mechanics without any modifications of the original expression. On the other hand, the introduction of J integral within the basic laws of continuum mechanics enabled its modifications for some problems out of the scope of original definition, like three-dimensional problems, thermal and inertial forces effects. In this paper, the influence of weldment heterogeneity is of primary interest and will be analyzed both theoretically and numerically. Theoretical analysis is applied in order to show that the J integral is not path independent for a generally shaped weldment. However, its path independence can be recovered if the modified J integral is introduced, comprising the original J integral and line integrals along weldment interfaces. Toward this end the modified J integral for multi-material body, representing welded joint with four different material regions (base metal - BM, weld metal - WM, coarse grain heat affected zone - CGHAZ and fine grain heat affected zone - FGHAZ), is defined following Savovic [2]. It was stated already by Rice [1] that the J integral is valid fracture mechanics parameter for a body homogeneous at least in a crack growth direction. Having weldments in mind one can conclude that the shape of material interface is of crucial importance in this respect. Indeed, as shown in [3], there was no problem with J integral path dependence for the (I( shaped welded joints, but problems arose when (X( shaped joints were analyzed. One possible solution is to evaluate the J integral using only the paths not crossing the material interface. Anyhow, this is not generally acceptable, e.g. in the case of J integral direct measurement, when material interfaces have to be crossed by integration path, [4]. Another intriguing part of this analysis is determination of weldment elastic-plastic characteristics in its different regions. Standard experimental procedure can be applied to the BM and usually to the WM, but the HAZ is too small to provide even micro-specimens. Hardness measurement can be helpful, but here special procedure has been applied, based on fitting the results of numerical simulation of experimental data, [5]. In this way it was also proved that two different regions of HAZ are sufficient to describe weldment behaviour correctly. 2. The modified J integral for multi-material body The basic idea, as given in [6], is considered using the bi-material cracked body with the interface between two different materials, as shown in Fig. 1. The J integral can be evaluated along path (1 encompassing the crack and not crossing the interface:  EMBED Equation.2  (1)where W denotes strain energy density, nj unit normal to (1, (ij stress tensor, ui displacement vector, xi Descartes coordinates (x1 along crack) and G crack driving force. One should notice that the path (1 can be separated into three parts, ((1, (((1 and l1, where the last one is along the material interface, Fig. 1.  EMBED PhotoFinish  Figure 1. Basic definitions for bi-material body On the other side, if one chooses closed integration path (2, being completely in the other part of bi-material body, Fig. 1, than analogous expression can be written  EMBED Equation.2   (2)except that J integral in this case equals zero because there is no crack and the energy release rate is zero for the closed path (2. One should notice that the path (2 can be separated into two parts, ((2 and l2, where the last one is also along the material interface, but at the opposite side compared to l1, Fig. 1. Now, by combining eqns (1)-(2) one can get: G = J(1 + J(2 = J( + Jl(3)where it was taken into account that the path ( comprises paths ((1, (((1 and ((2, while the path l comprises paths l1 and l2, Fig. 1. Therefore, it can be concluded that in order to regain path independence of J integral for the bi-material body, the line integral along material interface has to be added. In the special case, when the interface is parallel to the crack direction, additional line integral is zero. The modified J integral for a weldment is introduced as for a multi-material body, represented by four regions of different material properties, Fig. 2: BM, WM and two regions in HAZ - one with fine grain structure (FG) and the other one with coarse grain structure (CG).  Figure 2. Integration paths for multi-material body Applying the same procedure as for the bi-material body, the J integral can be evaluated along path (1 encompassing the crack and not crossing the interface, i.e. eqn (1) is still valid, while eqn (2) has to be written for six closed paths, (2-(7, Fig. 2:  EMBED Equation.2  (4) The J integral along paths (2-(7 reduces to zero because these paths do not encompass any discontinuity. Using equations (1) and (4) one can write:  EMBED Equation.2  (5)where la denote the closed contour around material interface. The expression (5) defines the modified J integral for a weldment, represented by four regions of different material properties. The modified J integral is path independent, as shown by Savovic [2], and has the following physical meaning: the first integral term represents the force acting on both the crack tip and material interfaces (discontinuities of stress and strain), whereas the second one eliminates the force on the interfaces. Thus, the complete integral expression represent only the force acting on the crack tip, and can be identified with the energy release rate due to the unit crack growth. 3. Numerical procedure and results Numerical analysis of elastic-plastic material behaviour is performed using modified version of programme by Fawkes and Owen [7] for elastic-plastic analysis of two-dimensional problems (2D) by the finite element method. Collapsed isoparametric eight-noded element around the crack tip are used, producing r-1 singularity. The finite element mesh around crack tip, Fig. 3, was generated in accordance with the ESIS recommendation, [8].  crack tip Figure 3. The finite element mesh around crack tip In order to check a weldment heterogeneity influence on the J integral value the finite element method was applied to 2D plane strain problem, using a mesh consisting of 297 eight noded elements and 822 nodes, Fig. 4. Both integral terms in eqn (5) were numerically evaluated on different paths (J1-J3), Fig. 4. Data for mechanical properties (yield stress Reh and hardening coefficient H') of weldment regions are given in Tab. 1. Table 1. Values for Reh, H' (MPa) for different weldment regions BMWMCGHAZFGHAZ758, 500 585, 2600760, 300595, 2300 The results are given in Tab. 2, showing the average value of J integral for six inner paths, Jave, close to the crack tip and not intersecting material boundaries (two paths in each of three rings of elements around the crack tip, Fig. 3), the values of first integral term in the modified J integral for the remote paths intersecting the material boundaries (J1, J2 and outer path J3 - Fig. 4), and the values of second integral term in the modified J integral along the boundaries between WM and CG HAZ (J4, Fig. 5), between CG HAZ and FG HAZ (J5, Fig. 5), and between FG HAZ and BM (J6, Fig. 5). Table 2. Results for J1-J6 and Jave J1J2J3J4J5J6JAVE41.306242.278244.6796-0.790050.34121-0.350139.7158100.48195.6042106.957-5.582505.80860-4.167092.5672136.271127.582142.554-8.463729.49186-7.5983125.372202.666187.190206.002-16.322216.4976-14.327188.772247.370225.526247.658-21.401623.0434-20.973232.632  Figure 4. Finite element mesh with some details  Fig. 5. Integration paths 4. Discussion As can be seen from Tab. 2 the finite element results confirm theoretical analysis of material interface effect on the J integral value. Namely, for all load levels, the Rices J integral is path dependent because its values for different paths differ out of the limits of numerical error. The largest difference (J1 and J2) is cca 9%, while the numerical error can be estimated to cca 2.5%, as shown in [9]. On the other hand, if values of the modified J integral (defined by eqn (5) and denoted here as JW), shown in Tab. 3 are analyzed, one can see an excellent agreement between JW1, JW2 and JW3, as well as a good agreement (within the limits of numerical error) between these values and Jave. The relations between J1-J6 (Tab. 2) and JW1-3 (Tab. 3) is as follows: JW1=J1+J4(6)JW2=J2+J4+J5(7)JW3=J3+J4+J5+J6(8)Table 3. Results for the modified J integral and parameter m JaveJW1JW2JW3m39.715840.516141.829344.58081.5892.567294.899195.8303103.0161.55125.372127.807128.610135.9831.56188.772186.343187.365191.8501.58232.632225.968227.167228.3261.60 Speaking in practical terms the effect of material interfaces is neither significant nor negligible. Having in mind the shape of weldment and differences in properties one can hardly think of more critical situation when similar materials are welded. Anyhow, dissimilar materials (e.g. ferrite and martensite or austenite steels) would produce much larger differences between the J integral for the outer contour and the modified J integral (i.e. crack driving force). Therefore, in such cases J integral direct measurement has to be followed (or preceded) by a numerical analysis in order to correct the experimental values. One should notice that only one practically important case of the cracked weldment is analyzed here (central crack in WM). Some other important cases, like weldment with crack along diffusion zone are left for the future analysis. In this case diffusion zone separates crack faces and different materials produce asymmetrical deformation, making also CTOD difficult to define and evaluate. Finally, in order to verify the modified J integral the crack tip opening displacement (CTOD) was calculated as well, using 45( definition, Fig. 6.  Fig. 6. CTOD calculation procedure The results are given in Tab. 3 in form of non-dimensional parameter, m, used commonly to define the correlation between J integral and CTOD: m=(J/Reh)/CTOD(9) As one can see from the Tab. 3 values for parameter m are constant, verifying both parameters and the relation (9), at the same time. 5. Conclusions From the results and their discussion the following conclusions can be made: Directly measured J integral for weldments is generally not equal to the crack driving force because of path dependence problem caused by material interfaces between BM, HAZ and WM. The effect of material interfaces can be evaluated using the modified J integral, i.e. the additional line integral, obtained by theoretical analysis in order to regain the J integral path independence. References Rice, J.R. "A path independent integral and the approximate analysis of strain concentration by notches and cracks". J. Appl. Mechanics, Trans. ASME E35, 1968, pp. 379-386. Savovic, N. "Theoretical, experimental and numerical analysis of weldment fracture mechanics parameters" (in Serbian). Doctoral thesis, Faculty of Mechanical Engineering, University of Belgrade, 1994 Kozak, D., Sedmak, A., Ad~iev, G. and Gubeljak, N. "Application of J-integral to strength overmatch welded structures", Proceedings of the 6th International Research/Expert Conference 'Trends in the development of machinery and associated technology', TMT 2002, Neum, 2002, pp 353-356 Sedmak, A. "The role of weldment interfaces in fracture mechanics parameters evaluation". Proc. of ICF 9, Sydney, Pergamon Press, 1997, Vol. 5, p. 2345-2356 Savovic, N. and Sedmak, A. "Numerical simulation of weldment heterogeneity". Welding & Welded Structures, Journal of Yugoslav Welding Society, Vol. 39, Num. 4, 1994, pp. 185-188. Smelser, R.E. and Gurtin, M. "On the J integral for bi-material body". Int Journ of Fracture, Vol. 13, 1977, pp. 382-384. Fawkes, A.J. and Owen, D.J.R. "Engineering Fracture Mechanics/Numerical Methods and Applications. Pineridge Press Ltd., Swansea, 1983 ESIS Recommendations for Use of FEM in Fracture Mechanics. ESIS Newsletter No.15, p. 3, 1991 Lacarac, V. and Sedmak, A, "The influence of crack length and material heterogeneity on J integral" (in Serbian), Proc. of the International Symposium Welding 96, Belgrade, 1996 Prof. dr Aleksandar Sedmak Faculty of Mechanical Engineering, University of Belgrade, 27. Marta 80, 11000 Belgrade, Serbia and Montenegro, phone 381-11-3370352, fax 381-11-3370364, asedmak@mas.bg.ac.yu Doc. dr. Drazan Kozak Mechanical Engineering Faculty, University of Osijek, Trg I.B. Mazuranic, HR-35000 Slav. Brod, Croatia phone 385-35-446188, fax 385-35-446446, dkozak@sfsb.hr Dr Gorgi Adziev Faculty of Mechanical Engineering, University of Skopje, Karpos bb, 91000 Skopje, Macedonia, phone 389-2-399290, fax 389-2-363180, gadziev@ereb1.mf.ukim.edu.mk Prof. dr Franjo Matejicek Mechanical Engineering Faculty, University of Osijek, Trg I.B. Mazuranic, HR-35000 Slav. 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