CRACK TUNNELING: EFFECT OF STRESS CONSTRAINT
Proceedings of IMECE042004 ASME International Mechanical Engineering CongressNovember 13-20, 2004, Anaheim, California USAIMECE2004-60700CRACK TUNNELING: EFFECT OF STRESS CONSTRAINTJianzheng ZuoDepartment of Mechanical EngineeringUniversity of South CarolinaColumbia, SC 29208, USAXiaomin Deng*Department of Mechanical EngineeringUniversity of South CarolinaColumbia, SC 29208, USAMichael A. SuttonDepartment of Mechanical EngineeringUniversity of South CarolinaColumbia, SC 29208, USAC.-S. ChengGM R&D and PlanningVehicle Development Research LabWarren, MI 48090, USAABSTRACTCrack tunneling is a crack growth feature often seen instable tearing crack growth tests on specimens made of ductilematerials and containing through-thickness cracks with initiallystraight crack fronts. As a specimen is loaded monotonically,the mid-section of the crack front will advance first, which willbe followed by crack growth along the rest of the crack front,leading to the formation of a thumb-nail shaped crack-frontprofile. From the viewpoint of fracture mechanics, cracktunneling will occur if the operating fracture criterion is metfirst in the mid-section of the crack front, which may be due toa higher fracture driving force and/or a lower fracturetoughness in the mid-section. A proper understanding of thisfracture behavior is important to the development of a threedimensional fracture criterion for general stable tearing crackgrowth in ductile materials.In this paper, the phenomenon of crack tunneling duringstable tearing crack growth in a single-edge crack specimen isinvestigated by considering the effect of stress constraint on thefracture toughness. Crack growth in the specimen undernominally Mode I loading conditions is considered. In this case,crack tunneling occurs while the initially flat crack surface(which is normal to the specimen’s lateral surfaces) evolvesinto a final slanted fracture surface. A mixed-mode CTODfracture criterion and a custom three-dimensional fracturesimulation code, CRACK3D, are used to analyze the tunnelingand slanting process in the specimen. Results of thisinvestigation suggest that the critical CTOD value (which is thefracture toughness) has a clear dependence on the crack frontstress constraint (which is the ratio of the mean stress to the vonMises effective stress). This dependence seems to be linear*within the range of stress constraint values found, with thetoughness decreasing as the constraint increases. It is found thatcrack tunneling in this case is mainly the result of a higherstress constraint (hence a lower fracture toughness) in the midsection of the crack front. Details of the crack growthsimulation and other findings of this study will also bepresented.KEYWORDS: crack tunneling, stress constraint, fracturecriterion, three-dimensional crack growth simulationINTRODUCTIONPrediction of ductile fracture in metallic materials has beenan important subject of fracture mechanics research. A keyfocus of this research effort has been the development offracture criteria for determining the onset and/or direction ofcrack growth. Currently, several approaches have beenproposed to characterize the process of stable crack growth(e.g. J-Integral [1], J-A2 [2], CTOD [3, 4] or CTOA [5] and soon).Experimental results (e.g. [6-8]) show that crack tunnelingis a common crack growth feature in stable tearing fracture inspecimens made of ductile materials. For a specimen containinga through-thickness crack with an initially straight crack front,as the a specimen is loaded monotonically, the mid-section ofthe crack front will advance first and then the rest of the crackfront will grow, leading to the formation of thumb-nail shapedcrack-front profiles. In addition to tunneling, slant crack growthmay also occur in ductile materials. Stable tearing testsperformed on specimens made of AL2024-T3 sheets indicate[6, 7] that slanting occurs when the specimens are in the LTCorresponding author; Email: deng@engr.sc.edu1Copyright 2004 by ASME
orientation whereas it does not occur when the specimens are inthe TL orientation. On the other hand, specimens made of AL2024-T351 display both flat and slant crack growth in LTorientation [8]. However, what is consistent from these tests isthat crack tunneling always occurs regardless of specimenorientation. However, tunneling in flat fracture is usually moresevere than that in slant fracture.From the viewpoint of fracture mechanics, thephenomenon of crack tunneling may be explained using afracture criterion if a non-uniform fracture driving force and/ora non-uniform fracture toughness exists along the crack front.Specifically, tunneling will occur if the operating fracturecriterion is met first in the mid-section of the crack front, eitherdue to a higher fracture driving force and/or a lower fracturetoughness in the mid-section. A proper understanding of thisfracture behavior is important to the development of a threedimensional fracture criterion for general stable tearing crackgrowth in ductile materials.The present work investigates crack tunneling duringstable tearing crack growth in a single-edge crack specimen [6,7], with a focus on the understanding of the effect of stressconstraint on fracture toughness [9, 10], based on a CTODfracture criterion [3, 4]. It is organized as follows. For referencepurposes, Section 2 gives a brief description of theexperimental results considered in this study. Section 3introduces the finite element model and computationalapproach for simulating crack front evolution observed in thetests. Section 4 presents the distributions of CTOD and stressconstraint along different crack fronts based on the simulationresults and establishes a correlation between CTOD toughnessvalues and stress constraint values in terms of a linear equation.As an application/verification, this equation is then used inSection 5 as part of the CTOD criterion to enable the simulationof the stable tearing tests to predict crack tunneling profilesduring crack growth. The predicted results are compared withexperimental measurements. Section 6 presents the conclusionsfrom this study.CRACK TUNNELING MEASUREMENTFigure 1 provides a schematic of the single-edge crackspecimen geometry [6]. The specimens were machined fromrolled 2.3mm-thick sheets made of aluminum alloy 2024-T3.The specimens were then fatigue pre-cracked in the LTorientation so that the initial crack/width ratio, a/w, is 0.0833.To assess the level of crack tunneling, a procedure described in[11] was used to obtain crack front profiles on crack surfaces,which are shown in Figure 2 for stable tearing crack growth ina specimen under remote Mode I loading conditions. Thecorresponding load for each crack front profile is availablefrom [6].FINITE ELEMENT MODEL AND COMPUTATIONALAPPROACHThe finite element method is used to analyze the stabletearing crack growth tests described above in order tounderstand the crack tunneling phenomenon. To carry out suchanalyses, a finite element code capable of simulating generalthree-dimensional (3D) crack growth in elastic-plastic solids isrequired. In this study, the custom code CRACK3D developedat the University of South Carolina is used. Preprocessing (forgenerating the initial finite element mesh) and postprocessing(for analyzing stress and deformation state at a particular stageof crack growth) are performed using the commercial codeANSYS, through interface options in CRACK3D.Two simulation options are available in CRACK3D. In thefirst option (the nodal release option), the crack front is made toadvance along a prescribed path, which is accomplishedthrough the release of nodal pairs (the two nodes in a pair areinitially tied together by rigid springs) along the crack pathwhen a certain condition (e.g. when a critical load from a test isreached) is satisfied. This option is useful for analyzing fracturetests. In the second option (the local remeshing option), thecrack front position is not prescribed but is predicted by afracture criterion (the CTOD criterion in this study) and a userspecified region around the new crack front is remeshed as thecrack front grows.Figure 3 shows frontal planar views of a 3D finite elementmesh used in analyzing the stable tearing tests based on thenodal release option, where Fig. 3(a) is for the entire problemdomain and Fig. (b) for a local view of the mesh around thecrack front (note that the fatigue crack front has extended fromthe initial notch into the region on the right). The mesh consistsof 8,917 ten-node tetrahedral elements with 14,315 nodes. Theminimal element size around the crack front is 0.2 mm. Aremote Mode I load was applied in terms of a monotonicallyincreasing displacement, so that nodes at the bottom edge of thespecimen were constrained with zero displacements in alldirections, and nodes at the specimen’s top edge were made tomove in the y-direction (vertical direction) (whiledisplacements in the x and z directions were held to zero).The crack front profiles in the finite element model, whichare created based on experimental measurements ([6]), areshown in Fig. 2. Due to slant crack growth, the fracture surfaceis not flat, hence Fig. 2 is only a projected section view of theactually slant fracture surface (viewed from the positive ydirection; see Fig. 1). The profiles numbered 1, 2, 4, and 6 arefor crack fronts measured based on fatigue striation marksduring interrupted crack growth tests, and the profilesnumbered 3 and 5 are interpolated from profiles 2 and 4, and 4and 6, respectively. Since the x-y plane coincides with thespecimen’s mid-plane (where z is zero), the back surface of thespecimen corresponds to negative z coordinates and the frontsurface corresponds to positive z coordinates. A typical 3Dcrack front profile is shown in Fig. 4.It is noted that crack front #1 is the initial fatigue crackfront and has slight tunneling in the middle. As the specimen isloaded monotonically to the critical load for the onset of crackgrowth, the middle region of the specimen grows first while theregions near specimen’s front and back surfaces remain still. Asthe load is further increased, the crack front evolves from crackfront #1 to #6, with increasing tunneling. The maximum loadrequired for continued crack growth occurred at crack front #6,after which the required load began to decrease gradually.2Copyright 2004 by ASME
CRACK3D is used to analyze the tunneling and slantingprocess in the specimen. With respect to the criterion usedduring the simulation, the measured critical load correspondingto each crack front is used to control the crack growth atdifferent crack length. The total CTOD (defined as the vectormagnitude of its opening, shearing and tearing components)and stress constraint (defined as the ratio of the mean stress tothe von Mises effective stress) with regard to the normal planeat each node on crack front are evaluated at the critical instantjust before the crack propagation.In the CTOD criterion [3, 4], the driving force is the totalCTOD, which is measured at a fixed distance behind the crackfront. In this study, CTOD was strictly computed at a distanceof 0.5 mm behind the crack front (along a line normal to thecrack front). Since it is impractical to use an extremely refinedmesh and an extremely small distance behind the crack frontfor computing CTOD, and since severe crack tunneling createsa length scale that may not sufficient large compared to 0.5mm, the computed CTOD is expected to be less accurate in themiddle region of a crack front than away from that region. Thereason is that, due to severe crack tunneling in the middleregion, a point at 0.5 mm (or another small distance) behind thecrack front along a line normal to the crack front may be tooclose to other parts of the crack front, making this pointinappropriate for calculating CTOD. To alleviate this problem,it is chosen in the calculation to compute a nominal CTODvalue, when the situation noted above occurs, by using thecrack-front CTOA value to extrapolate to a distance of 0.5 mmbehind the crack front.It is important to note that, in this analysis phase of thestudy, crack growth simulations were performed using thenodal release option, with the experimentally measured crackfront profile positions and the corresponding loads. The CTODcriterion was not applied even though CTOD variations alongthe crack fronts were evaluated.EFFECT OF STRESS CONSTRAINT ON CRITICALCTODThe main results of interests from the finite elementsimulations described in the preceding section are the variationsof CTOD and stress constraints along the measured crackfronts, which will be used to establish a correlation betweenCTOD fracture toughness and stress constraint. Since theCTOD values along a crack front correspond to the critical loadthat causes growth of the crack front, they are critical CTODvalues and equal to the corresponding CTOD fracturetoughness values along the crack front.Figures 5, 6, 7 provide the variations of the total(combined) CTOD and CTOD components for Mode I(opening), Mode II (shearing) and Mode III (tearing) alongcrack fronts #4, #5, and #6, respectively, at a distance of 0.5mm behind the crack front (along a line normal to the crackfront). In the figures, the through-thickness value refers to the zcoordinate value along the crack front. Results for crack frontsin the early stage of crack growth are not shown because thesecrack fronts have growth only in the middle section and do notprovide reliable critical CTOD values (note that CTOD valuesalong parts of the crack front that are not at the impendingmoment of growth are not critical values and do not equal toCTOD fracture toughness values there).Two important observations can be made from thesefigures. First, due to crack tunneling and slanting, a perfectsymmetry about the specimen’s mid-plane, a feature expectedin Model I crack growth, is seen to disappear as crack grows,which lead to a truly three-dimensional mixed-mode CTODdistribution along the crack front, especially in the middleregion of the crack fronts. The crack front is Mode I dominantonly in the middle region. Second, the total CTOD value(which represents the CTOD-based fracture toughness duringcrack growth) is not a constant along the crack front—it islower in the middle region than near the specimen surfaces. Asdiscussed later, this variation in fact reflects the dependence ofCTOD toughness on the stress constraint Am.Alternatively, variations of the total CTOD along the crackfronts can be plotted in one figure, as shown in Fig. 8. It is seenthat the trend shown in the variations along different crackfronts is basically the same. This observation will be utilizedsubsequently to relate CTOD variation with stress constraintvariation along the crack fronts. Before this is done, it must bepointed out that CTOD variations shown so far are strictlycomputed at a distance of 0.5 mm behind the crack front.Because of severe tunneling (as discussed earlier), thecomputed CTOD values are less accurate in the middle regionof a crack front than away from that region. It is suggestedearlier that improved CTOD values in the middle region can beobtained by using the crack-front CTOA value to extrapolate toa distance of 0.5 mm behind the crack front. This seems to be aviable approach for crack fronts after a certain amount of crackgrowth, so that crack blunting usually seen at the early stage ofcrack growth can be avoided. Based on this approach, the totalCTOD variations in Fig. 8 are updated and are shown in Fig. 9.Shown in Fig. 10 are the variations of Am along crackfronts #3, #4, #5, and #6. The constraint value at each crackfront point is computed based on an integrated average from thecrack front to 0.3 mm ahead of the crack front, in the directionnormal to the crack front, within the plane of the crack surface.It must be pointed out that, at and very near the specimen’sfront and back surfaces, a distance of 0.3mm ahead of a crackfront may go outside the specimen domain. In this case, we usethe constraint value of the next interior crack front point toapproximate the constraint value for the current crack frontpoint. Similar to the total CTOD variations in Fig. 9, a commontrend in the stress constraint variations can also be seen.To reduce scatter in the subsequent data reduction forpossible correlation between CTOD toughness and stressconstraint Am, the variations in Figs. 9-10 need to be smoothed.To this end, an averaged total CTOD variation is obtained fromFig. 9 and an averaged stress constraint variation is obtainedfrom Fig. 10. Now, a correlation between CTOD and Am can beobtained by plotting all (CTOD, Am) pairs from the same crackfront locations, and this correlation is shown in Fig. 11. Clearly,there is a linear correlation between CTOD and Am, which canbe described by the fitted linear equation below (within therange of values for Am)3Copyright 2004 by ASME
CTOD 0.0932-0.0312 Am(1)Since the CTOD and Am values are extracted fromsimulation results of actual fracture tests, with experimentallyrecorded crack fronts and load values during stable crackgrowth, there is strong reason to believe that Eq. (1) mayrepresent the dependence of the CTOD based fracturetoughness on stress constraint Am for the material in concern,aluminum alloy 2024-T3. Eq. (1) suggests that the CTODfracture toughness value is lower when the stress constrain ishigher.If validated by further studies, Eq. (1) can be treated as amaterial property and will be extremely useful in threedimensional fracture mechanics applications where constrainteffects on fracture toughness are important.PREDICTION OF CRACK TUNNELINGThe relationship between the CTOD toughness and stressconstrain can be used to predict crack tunneling based on theCTOD fracture criterion. To demonstrate this application, theMode I stable tearing fracture test described in previoussections is now simulated. Two types of simulations usingCRACK3D are performed.In the first case, the nodal release option is used with acoarse mesh (to save computation time) with design similar tothat in Fig. 3. The mesh consists of 3,308 ten-node tetrahedralelements with 6,169 nodes. The minimal element size aroundthe crack front is 0.4mm. The difference between thesimulation here and that in section 4 is that, there are nospecified crack front positions on the crack surface except forthe initial fatigue crack front (see Fig. 12). The crack frontshape is predicted by the CTOD criterion and Eq. (1) eventhough the crack front is constrained to grow on the measuredslant fracture surface. Since the nodal release option is used, theelements are fixed. As a result, some mesh dependence in thepredicted crack front positions is expected (see Fig. 13). Thedependence can be reduced if finer meshes are used (a coarsemesh is used here). Alternatively, the crack front profile can besmoothed based on predicted crack growth amounts atdifference points along the crack front.In the second case, instead of using a fixed mesh and thenodal release option, the local remeshing option is used so thatthe mesh in a user-specified region around the current crackfront is remeshed each time the crack front is predicted to grow.Again, the CTOD criterion and the relationship between theCTOD toughness and stress constraint are employed to predictthe crack front profile although the crack front is stillconstrained to grow on the measured slant fracture surface.All simulations are conducted using CRACK3D. In thesimulations, the CTOD criterion is evaluated node by nodealong the current crack front. A crack-front point will advancealong the measured slant fracture surface when the CTODvalue at that point reaches the critical value defined by thestress constraint Am ahead of the crack front according to Eq.(1).The predicted crack front profiles (without smoothing)during stable crack growth are shown in Fig. 13 (with nodalrelease) and Fig. 14 (with local re-meshing). In order to
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