ࡱ>  XZW}z|7 EbjbjUU 7|7|3 %l$0008h<<$:G,s*L/9999999$; =9g2NGg2g29+5:+5+5+5g29+5g29+5+58^8 09 $ 0O388:0:8>C4>8+5$$Some Remarks about 2-D and 3-D Finite Element Modelling of Fracture Toughness Specimens with Material and Geometric Mismatch Franjo Mateji ek, Dra~an Kozak University of Osijek, Mechanical Engineering Faculty in Slavonski Brod, Croatia Abstract  A lot of commercial finite element (FE) package is usually tool today to analyse stress and strain fields at fracture toughness specimens. 2-D approach is then very suitable to use, regarding its simplicity and fastness. According to theory of elasticity, the user must suppose plane stress (PS) or plane strain (PE) state for chosen plane section of the specimen. However, this presumption is valid only for very thin or very thick plates, respectively. Most engineering structures have the thickness between these two extremes. In that case, the state of stress in the section's mid-plane could be imagined as mixed. Mixed state of stress combines PE state in the region near the crack tip and PS state outside of this region. The applicability of such approach could be very restricted if the structure has complex geometry. In this paper some remarks are given to influence of stress state assumptions on CTOD (d5) result for Bx2B fracture toughness specimen with material and geometric mismatch. Stress and strain fields obtained from solid FE model are presented to find out which stress state is closer to reality by 2-D FE modelling. One can conclude that plane strain state prevails in the middle of the specimen, while plane stress state is dominating on the surface. Although the specimen with X-welded overmatch joint cracked through the middle seems to have two planes of symmetry, the results for mid-plane could be accepted just as lower bound. Key words: Fracture toughness specimen, material and geometric mismatch, finite element modelling, state of stress 1 Theoretical background Most of 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). The stress state in cracked plates changes from conditions approaching PS for very thin plates to three-dimensional stress (substituted with PE in 2-D FE analysis) for thick plates (Fig. 1). Fig. 1 State of stress by thin and thick cracked plates  According to theory of elasticity, the component of stress sz in the thickness direction is defined as:  EMBED Equation.3  (1.1) In the same time, the component of strain ez could be determined from:  EMBED Equation.3  (1.2) However, the state of stress in most engineering structures changes along the crack front, where PS condition dominates on the surface and PE is more close to reality with nearing to mid-plane of structure. Although the material in mid-plane has the greatest out-of-plane constraint to yield, there is not pure PE, because the state of stress depends also on the distance from the crack tip. Near the tip PE state prevails, while further away from the tip stress state approaches PS. Very detailed investigations of three-dimensional effects near a crack tip in a ductile three-point bend specimen, Narasimhan and Rosakis ten years ago are performed [1]. Their observations indicate that the plane strain field prevails in the interior of the specimen very near the crack front. On the other hand, for distances from the crack tip exceeding about half a specimen thickness B, plane-stress conditions are approached.  EMBED Word.Picture.8  Newman et al [2] proposed original concept to include three-dimensional stress effects around the crack tip of the CT fracture specimen as example. They assumed plane strain finite elements in the PE core surrounding crack tip and out of this core plane stress elements, respectively (Fig. 2). The main shortcoming of this mixed state-of-stress approach is to correctly locate the PE core and its size. Also, this core should be movable by simulation of crack propagation of ductile materials. The shape of PE core should not be strictly rectangle and it was shown in [3] that the size of PE region depends on loading level.Fig. 2 Plane strain core around crack tip Recent papers of Shan, Kolednik and Fischer [4] described new substitution method for 3-D elastic-plastic FE analyses. They divided a specimen into two parts: one consists of the two surface layers with a total thickness of Bs in a PS state and the other of the centre layer with a thickness Be = B Bs in a PE state. Two 2-D analyses are performed independently for the two parts. The determination of Bs may be complicated, because it demands single 3-D analysis with a coarse mesh. In this paper, regarding all aforementioned circumstances, complex mixed state-of-stress approach by finite element modelling will be avoided. More attention will be given to influence of material and geometric mismatch by pure PS or PE finite element analysis. 2 Welded joint shape and materials dissimilarity The Bx2B SENB fracture toughness specimen with X-shape of welded joint has been chosen for experimental and numerical analysis. The crack was located through the middle of the specimen (Fig. 3). The specimen has two planes of symmetry.  EMBED Word.Picture.8  Fig. 3 Three point bending specimen (Bx2B) and planes of symmetry High strength low alloyed HSLA steel (with initial thickness of 40 mm), corresponding to the grade HT80, was used as the base metal (BM). The overmatched X-grooved multipass welded joint was heterogeneous. The root of weldment has been performed with W.B.800 as fill material. This wire has a little better mechanical properties related to BM (Fig. 4). For the cap of welded joint W.B.800 electrode has been used ensuring globally overmatch of over 20 %.  EMBED Excel.Sheet.8  Fig. 4 True stress-strain curves for base and weld metal 3 2-D and 3-D finite element models of the specimen Regarding the symmetry of the specimen, only 1/4 of the specimen has been considered for finite element modelling (Fig. 5). This figure presents the geometry of the model with appropriate materials [5]. Note from the picture that any section plane of the solid model for 2-D finite element modelling consists from different percent of the base metal.  EMBED Word.Picture.8  Fig. 5 One fourth of Bx2B fracture tuoghness specimen In the mid-thickness plane, the part of the overmatch weld metal is the smallest, but constraints are the greatest (PE state), so we can expect here conservative values of characteristic displacements (CTOD (d5) or CMOD). This approach can be applied just from safety reasons. The finite element mesh of this plane section is presented on the fig. 6. It consists from 360 eight-node isoparametric quadrilateral and triangle elements and 1149 nodes. Crack tip elements have a radius of about 50 mm. The mesh is generated automatically using free mesh technique. Extra key-points have been foreseen for CTOD (d5) and CMOD nodes.  EMBED Word.Picture.8  Fig. 6 Finite element mesh of mid-thickness plane of specimen Two analyses were performed assuming PE state and PS state, respectively. The state of plane stress is not real for the middle of the specimen, but such assumption has a purpose to compare possible PS solution with experiment. The finite element mesh for the surface of the specimen is depicted on the fig. 7 (544 elements and 1695 nodes). Here the part of overmatched weld cap material is the greatest, but this effect on results is reduced assuming PS state. PE state here has no physical sense.  EMBED Word.Picture.8  Fig. 7 Finite element mesh of surface plane of specimen Finally, 3-D finite element model is only competent to analyse which assumption for state of stress is more suitable 2-D analysis. The mesh of one specimen's quarter consisted from 28947 nodes and 6528 20-node elements is given on the fig. 8. Solid model has been obtained by extruding of surface plane mesh. The fusion line is not straight. It is a zigzag shaped through the thickness, where each element has appropriate material properties. HAZ is not modelled as special material.  EMBED Word.Picture.8  4 Comparision of CTOD (d5) results The main aim of this study was to assess how many particular FE models are close to reality. All results in the form of force vs CTOD (d5) displacements are presented on the fig. 9.  EMBED Word.Picture.8  Fig. 9 CTOD (d5) results for different FE models It is obviously that PS state in the middle of the specimen is in some measure real only for low load values. In the other hand, PE state assumption for the mid-thickness plane is much nearer to experimental results [6]. Already the same results, but just from the 'upper side' could be obtained supposing PS state for the surface FE model. 3-D FE model may be accepted as good enough in view of neglecting HAZ as special material or irregularity of crack front shape through the thickness. So, we use the stress and strain results for further analyses. The intention was to investigate sz stress field and ez strain field according to half-thickness B/2. From those fields is it possible to conclude which state of stress assumption is more valid for 2-D FE analyses. 5 Stress and strain field through the thickness of the specimen The fig. 10 presents stress and strain distribution in the z-direction:  EMBED Word.Picture.8  EMBED Word.Picture.8  Fig. 10 Stress and strain fields through the half-thickness of the specimen The stress triaxility can be neglected in the region close to surface of the specimen (Fig. 10a). Therefore, PS state assumption for surface FE models is valid. In the same region, the contraction of the specimen is free, which is reflected in negative strain ez (Fig. 10b). In the other hand, the yielding of crack front surrounding material in the middle of the specimen is constrained. This causes stress in the thickness direction sz appearance. Triaxial stress field in the middle of the specimen could be idealised as PE state by 2-D FE modelling. This can be proved by calculating of the factor of stress triaxility k (the ratio between hydrostatic and equivalent stress):  EMBED Equation.3  (5.1) If its value greater then 2,5 may PE state be supposed with great probability. The routine for estimation of stress triaxility factor k that runs under ANSYS is written for this paper. The changing of k through the ligament of the specimen for the surface plane and mid-plane is presented on the fig. 11. It is obviously that triaxility of the stress is more in presence for the mid-plane, but the value of k is less than 1. This leads to the PS state assumption for any plane of the specimen. By nearing to the end of the specimen, the value of stress triaxility increases because of loading acting effect.  EMBED Word.Picture.8  Fig. 11 Factor of stress triaxility change for the mid-plane and surface plane 6 Conclusions The effect of specimens contraction on the free surface is more present by thin-walled structures. Here the PS state assumption by 2-D FE modelling is not in question. For the mid-plane of thick structures may be supposed PE state with good accuracy. It is difficult to find a quantitative measure to assess a validity of assumed state of stress for the structures with medium thickness. Materials dissimilarity or geometric mismatch may decision about proper state of stress make more difficult. One possibility to express present out-of-plane constraint is to calculate the factor of stress triaxility. If its measure less than 2,5 may PS state be assumed with great probability. Although, for the specimen analysed in this paper (B=36mm) may be concluded that PS state assumption gives good results for surface FE model, but PE state is still dominant in some region around crack front for the mid-plane of the specimen. 7 Literature: [1] Narasimhan, R., Rosakis, A. J., 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, 1990, pp. 607-617 [2] Newman, J.C.Jr., Booth, B.C. and Shivakumar, K.N.: An Elastic-Plastic Finite-Element Analysis of the J-Resistance Curve Using a CTOD Criterion, Fracture Mechanics: Eighteenth Symposium, ASTM STP 945, ASTM, 1988, pp. 665-685 (3( Kozak, D., Gubeljak, N., Rak, I., Modificirano numeri no modeliranje lomnega vedenja zvarnega spoja v prehodnem podro ju iz duktilnega h krhkemu lomu (in Slovenian), KOVINE, ZLITINE, TEHNOLOGIJE, 32 (5), 1998, pp. 337-342 (4( Shan, G.X.; Kolednik, O; Stwe, H.P. and Fischer, F.D.: A Substitution Method for 3D Elastic-Plastic FE Analyses of Fracture Mechanics Specimens, Engineering Fracture Mechanics, Vol. 41, No. 5, 1992, pp. 625-633 (5( Kozak, D., The contribution to numerical and experimental analysis of fracture behaviour of heterogeneous structures (in Croatian), PhD dissertation, Faculty of Mechanical Engineering and Naval Architecture, Zagreb, 2001 (6( Gubeljak, N., Kozak, D., Mateji ek, F., Prediction of Strength Mis-match Welded Joints Failure by Using of Crack Driving Force Curve and Crack Tip Opening Displacement (CTOD) Resistance Curve, 42 (2000) 3,4, pp.85-100  PAGE 1  EMBED Word.Picture.8  :<   # , @pqr\^.0VXZ\p$%PQhսղսղՁvjUmHnHuCJ jEHUjPsJ? 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