ࡱ> <>;~r7 >bjbjUU $e7|7|W1lxxxx&&&&4ZDP}:>Al'D8OOOOOOO$Q S\O_E8}:_E_EO_Gxx%P_G_G_G_ExD(O_G_EO_Gt_GLOD)O @T8 &#F<O)OP0P!OT_GT)O_Gxxxx1. Introduction Structural integrity assessment is unimaginable without knowledge about fracture mechanics. Therefore, fracture mechanics approach to the structures where cracks may appear during production or exploitation (f. i. welded structures) becomes necessary today. To quantify fracture behavior some known parameters have to be used. One of the widely applied in rate-independent quasi-static fracture analysis is J-integral (Rice, 1966). As a parameter characterizing crack tip field it could be used for linear-elastic as well as elastic-plastic material behaviour. Most of all major commercial finite element (FE) packages (f.e. ABAQUS, ADINA, ANSYS etc.) have developed code for 2-D and/or 3-D J-integral calculation. However, 2-D finite element models demand assumption about the state-of-stress that govern in the specimen. This is not questionable by very thin structures where plane stress (PS) state prevails or by very thick structures where plane strain (PE) state is predominant. But, which state of stress is more appropriate for the structures with medium thickness? Three-dimensional effects near a crack tip in a ductile three-point bend specimen were investigated recently (Narasimhan & Rosakis, 1990). Plane stress vs. plain strain was also the topic of a large number of the papers in the past 10 years (Thaulow et al, 1997). Unfortunately, there is no unique and precise conclusion. Additional problem is the fact, that level of the loading plays also the role by state-of-stress assignment. Finite element estimation of J-integral obtained by solid model is complex, because 3-D modeling of cracked structures is always connected to specific problems (e.g. crack faces modeling, boundary conditions setting etc.). CPU time is significant longer than in 2-D calculations, but obtained results are much closer to reality. This paper is devoted to deviation in the J-integral numerical estimation for the structure with medium thickness. In that purpose both two-dimensional (2-D) and three-dimensional (3-D) finite element analyses (FEA) were employed. 2-D FEA supposes plane stress state on the surface of the 36 mm thick specimen and plane strain state in the middle of the specimen, respectively. These plane J-integral solutions were compared with J value calculated from solid model. Experimentally obtained J was considered as the reference value. 2. J-integral definition The J-integral can be defined as a path-independent contour integral that measures the strength of the singular stresses and strains near a crack tip. Its value should be approximate constant far-field as well as near-crack field. However, J-integral constancy may be questionable after crack initiation. Also, dominance of the J-integral becomes more debatable if the structure composition is heterogeneous. Equation (1) shows an expression for J in its 2-D form, where crack lies in the XY plane with x-axis parallel to the crack (Fig. 1):  EMBED Equation.DSMT4   EMBED Equation.3  (1)  EMBED Word.Picture.8   EMBED Equation.3  Fig. 1. Definition of contour for J-integral evaluation In equation (1) G means any path surrounding the crack tip, W is strain energy density, sij is component stress and ui is displacement vector. In this work J-integral calculations were performed by writing of macro for ANSYS code (Swanson Analysis System Inc, 1996) based on virtual crack extension method. 3. Experimental analysis Standard B x B fracture toughness specimen (Kozak, 2001) with thickness of B = 36 mm notched at the surface (a0/W ratio was about 0,5) has been subjected to bending in three point (3PB). The surface notch was in the middle of the specimen. The material of the specimen is high strength low alloyed steel with basic mechanical properties: E = 201351 MPa, Rp0,2=711 MPa, Rm= 838 MPa, work hardening exponent n = 0,095, elongation of 19% and the average toughness values of Charpy tests are 85 and 54 Joules by 10C and -40C, respectively. Before fracture testing all specimens were fatigued. During testing of the specimen some characteristic fracture parameters were recorded as force F, load line displacement LLD and crack mouth opening displacement CMOD (Fig. 2).  EMBED Word.Picture.8  Fig. 2. Three point bending of the fracture toughness specimen J-integral can be calculated from the experiment by:  EMBED Equation.3  (2) where K denotes linear-elastic stress intensity factor, E* = E for plane stress conditions or E* = E/(1-n2) for plane strain state, Up presents plastic part of the energy absorbed by the specimen (area below the F-LLD curve), B(W-a) is cross-sectional area and p is dimensionless constant which depends on a/W ratio (p = 2 for a deeply cracked plate in pure bending). Expression (3) for calculation of p is valid for a wide rang of a/W ratios (Wang and Gordon, 1992):  EMBED Equation.DSMT4  (3) 4. Plane stress (PS) vs. plane strain (PE) Most of the classical solutions in fracture mechanics reduce the problem to two dimensions. That is, at least one of the principal stresses or strains is assumed to equal zero (PS or PE, respectively). In general, the conditions ahead of a crack (Fig. 3) are neither PS nor PE, but are three-dimensional. There are, however, limiting cases where a 2-D assumption is valid, or at least provides a good approximation.  EMBED Word.Picture.8  Fig. 3. Characteristic plane stress and plane strain state for a thick and thin plate Material near the crack front is loaded to higher stress than the surrounding material. In the interior of the structure near the crack front, contraction is constrained by surrounded material. This constraint causes a triaxial state of stress near the crack tip:  (4) so in the midplane (z = 0), PE conditions exist and ez = 0. Material on the structure surface is in a state of plane stress (PS), however, because there are no stresses normal to the free surface (sz = 0) and contraction can be expressed as:  (5) The transition from PE (at mid thickness) to PS (at the surface) depends on the plastic zone size relative to the structure thickness. Plane strain conditions exist at the boundary if the plastic zone is small compared to the thickness, but the stress state is predominantly plane stress if the plastic zone is of the same order as the thickness (Anderson, 1995). Although the stress state at the elastic-plastic boundary is predominantly PS when the plastic zone is of the order of half the structure thickness (or larger), a 3-D stress state may exist deep inside the plastic zone. With further plastic deformation, however, the level of stress triaxility at the crack tip relaxes. Fracture toughness specimens with medium thickness have significant transition region where state-of-stress is mixed (Fig. 4). Three-dimensional effects could not be neglected. Pure PS conditions prevail just for small among of loading, but increasing the loading causes more hardened area around the crack, where PE state is closer to reality. In this case PE solution becomes nearer to 3-D solution.  EMBED Word.Picture.8  Fig. 4. Variation of the PE region depending on the structure thickness 5. Finite element models of the fracture toughness specimen The aim of this paper was to determine J-integral value by FE through 2-D and 3-D analysis and to compare obtained results with experimental one. Before J-integral calculation it is necessary to check validity of the finite element model. Numerically obtained characteristic displacements for certain values of forces should not significant deviate from the measured values, because J-integral value depends directly on the stresses and strains values mapped on the path around the crack. 2-D J-integral FE calculations were performed assuming PS or PE conditions using eight-noded isoparametric elements. Symmetry of the specimen has been taken into account. Free meshing technique was applied with singular elements in the first row of the sharp crack. The mesh around the crack tip is refined, the element length being 50 mm. The calculations have been performed using small and large deformations (parameter NLGEOM), respectively. The choice of the deformation theory influences the stresses and strains in the vicinity of the crack tip, but not the global behaviour (Brocks & Scheider, 2001). Known Ramberg Osgood equation was used to describe material law of yielding. Calculating a contour integral like eq. (1) is quite unfavourable in finite element codes as coordinates and displacements refer to nodal points and stresses and strains to Gaussian integration points. Hence, a domain integral method is commonly used to evaluate contour integrals. This method is quite robust in the sense that accurate values are obtained even with coarse meshes, because the integral is taken over a domain of elements. Here the J-integral is defined in terms of the energy release rate, eq. (6) associated with a fictitious small crack advance a:  EMBED Equation.3  (6) B multiplied by da corresponding to increase in crack area Ac and the integration domain is presented as the grey area in Fig. 5.  EMBED Word.Picture.8  Fig. 5. Virtual crack extension Solid finite element model consisting of 1116 twenty-noded elements and 5145 nodes used for J-integral calculation is presented on the Fig. 6. It has been extruded from two-dimensional FE model. One quarter of the force was divided on 9 layers, where the edge nodes are loaded with F/216 and mid-nodes with F/54 and F/108, respectively. The same path for all 2-D J-integral estimations cross over 4 nodes and it does not touch the boundary of the structure. The domain was chosen as large as possible and J-values were not averaged from various domains: the highest value is the best approximation to the real far-field J.   Fig. 6. Solid FE model of one quarter of fracture toughness specimen 6. Results with discussion It was no large difference in J-integral results up to the point of crack initiation (Fig. 7). After initiation PS state becomes less dominant in the specimen, so even for maximum load it exists no converged solution. This is in according with general statement that J-integral is derived for plane stress and displacement fields, i.e. no dependence on z-axis. Comparing J-integral values obtained from solid model and 2-D model with PE assumption, it is obviously more coincidence. One can conclude that PE state is more suitable assumption for the specimens with medium thickness, especially after crack initiation. However, numerical investigations of the J-integral for a growing crack have shown that the near field contour integral loses its path independence immediately after the onset of crack growth. Therefore, abovementioned conclusion has to be accepted with some restriction. Namely, validity of J-dominance is lost under plane strain condition (crack tip stress triaxility).  EMBED Excel.Sheet.8 Fig. 7. J-integral solutions by different procedures 6. References Anderson, T.L. (1995). Fracture Mechanics: Fundamentals and Applications, CRC Press, ISBN 0-8493-4260-0, Boca Raton, Florida Ansys (1999). Users Manual Rev. 5.6, Swanson Analysis System Inc. Brocks, W. & Scheider, I. (2001). Numerical Aspects of the Path-Dependence of the J-Integral in Incremental Plasticity, Technical Note GKSS/WMS/01/08 Kozak, D. (2001). The contribution to numerical and experimental analysis of fracture behaviour of heterogeneous structures (in Croatian), Dissertation, University of Zagreb Narasimhan, R. & Rosakis, A. J. (1990). Three-dimensional effects near a crack tip in a ductile three-point bend specimen: Part I - A numerical investigation, Journal of applied mechanics, Vol. 57, pp. 607-617 Rice, J.R. (1968). A path independent integral and the approximate analysis of strain concentrations by notches and cracks, Journal of applied mechanics, Transactions of ASME E35, pp. 379-386. Thaulow, C.; Ranestad, .; Hauge, M.; Zhang, Z.; Toyoda, M. & Minami, F. (1997). FE calculations of stress fields from cracks located at the fusion line of weldments, Engineering Fracture Mechanics, Vol. 57, No. 6, pp. 637-651 Wang, Y.-Y.; Gordon, J.R. (1995). 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ZB  s *D`B @ 0ZDZB  s *DTB   c $D TB ! c $D NB % S DjJ NB & S DjJ  ( |BUC DEFX::. MTlv4Ocj7e6E:/|wrv~R .    IS70gUF7!   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