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Two parameter fracture mechanics: Fatigue crackbehavior under mixed mode conditionsStanislav Seitl*, Zdene k Kne slInstitute of Physics of Materials, Academy of Sciences of the Czech Republic, Zizkova 22, 616 62 Brno, Czech RepublicReceived 30 November 2006; received in revised form 12 April 2007Available online 27 April 2007AbstractThe fatigue crack path has been studied on a tensile specimen with holes. The experimental crack path trajectories werecompared with those calculated numerically. To incorporate the influence of constraint on the crack curving, we predictedthe fatigue crack path by using the two-parameter modification of the maximum tensile stress (MTS) criterion. The valuesof the mixed-mode stress intensity factors KI, and KIIas well as the corresponding constraint level characterized by T-stresswere calculated for the obtained curvilinear and reference crack path trajectories. It is shown that in the studied configu-ration the effect of T-stress on the crack path is not significant. On the other hand the effect of constraint on the fatiguecrack propagation rate is more pronounced.? 2007 Elsevier Ltd. All rights reserved.Keywords: Constraint; Mixed-mode loading; Fatigue crack; Crack growth; Crack path1. IntroductionWhen a crack propagates in a non-homogeneous stress field, its crack path is generally curved. The crackpath in brittle isotropic homogeneous material has a tendency to be one for which the local stress field at its tipis of a normal mode I type and for which KIItends to zero. This is consistent with the various proposed mixed-mode criteria such as, e.g. the maximum tensile stress criterion 1, the maximum energy release rate criterion2,3, and the stationary Sih strain energy density factor 1,4. All these criteria have in common that if KII5 0,the crack extends with non-zero change, h0, in the tangent direction to the crack path in such a way thatKII! 0. According to the above criteria, the direction of crack propagation h0depends on the ratio of thestress intensity factors corresponding to mode II and mode I, i.e. h0= h0(KII/KI). This approach, based onthe assumptions of standard fracture mechanics, does not account for the changing of the constraint level dur-ing crack propagation.0013-7944/$ - see front matter ? 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.engfracmech.2007.04.011*Corresponding author.E-mail addresses: seitlipm.cz (S. Seitl), kneslipm.cz (Z. Kne sl).Available online at Engineering Fracture Mechanics 75 (2008) 857865/locate/engfracmechIn the contribution two-parameter constraint based fracture mechanics is used to assess the constraint levelon the crack path and propagation rate under mixed mode of normal and shear loading conditions. The in-plane constraint level is described by using T-stress.The elastic T-stress represents a constant tensile stress acting parallel to the crack flanks. It is related to thesecond term (the first non-singular term) in the Williams expansion of the stress field 5:rijr;h KIffiffiffiffiffiffiffi2prpfIijh KIIffiffiffiffiffiffiffi2prpfIIijh Td1id1j; as r ! 0;1where rijdenotes the components of the stress tensor, KIand KIIare the corresponding stress intensity factorsand fij(h) are known angular functions. The fracture parameters KI, KIIand T depend on the crack length a,the geometry, size and external loading of the body and corresponding boundary conditions. The T-stress canbe characterized by a non-dimensional biaxiality ratio B given by Leevers and Radon 6:B TffiffiffiffiffiffipapKI;2where KIis the stress intensity factor corresponding to the crack length a.The facts in the literature confirm that the directional stability of a growing mode I crack is governed by theT-stress. If the T-stress is compressive and there is a small random crack deviation, perhaps due to microstruc-tural irregularity, then the direction of mode I crack growth is towards the initial crack line. A mode I crack incentre cracked sheet loaded in uniaxial tension is directionally stable in this sense. Repeated random devia-tions mean that the crack follows a zig-zag path about the initial crack line. Theoretically, when the T-stressis tensile a crack is directionally unstable, and following a small random deviation, it does not return to itsinitial line. The straight path is shown to be stable under mode I for T 0 (high constraint), see 7 for details.The aim of the present paper is to show how the T-stress influences the crack propagation direction andconsequently the crack path under mixed mode loading conditions. At the same time the influence of con-straint on the fatigue crack propagation rate under mixed-mode conditions is discussed. The basic assump-tion of the paper are small yielding conditions (corresponding to high cycle fatigue conditions) andtwo dimensional approximation corresponding to plane strain conditions according to ASTM Standard E467.2. Theoretical background2.1. The direction of crack growthThe growth of a fatigue crack is usually numerically simulated as a number of discrete incrementalsteps. After each crack length increment the corresponding crack propagation direction h0has to becalculated. Thus, the estimation of the h0is of paramount importance. Various criteria for the crack growthdirection under mixed-mode loading have been proposed in the literature, see e.g. 8. In the present paper atwo-parameter modification of the maximum tensile stress (MTS) criterion is applied to determine the h0value.The MTS criterion was introduced by Erdogan and Sih 1,9,10 and is one of the most widely used theoriesfor mixed mode I and II crack growth. It states that a crack propagates in the direction for which the tangen-tial stress is maximum. It is a local approach since the direction of crack growth is directly determined by thelocal stress field within a small circle of radius r centered at the crack tip. The direction angle of the propa-gating crack is computed by solving the following equation:KIsinh0 KII3cosh0 ? 1 0 withKIIsinh0=2 08:;3where KIand KIIare the stress intensity factors for the initial crack corresponding to mode I and mode IIloading, respectively, and h0is the predicted direction angle.858S. Seitl, Z. Kne sl / Engineering Fracture Mechanics 75 (2008) 857865The two parameter modification of the MTS criterion takes into account the existence of the second term inthe Williams expansion of the stress field, see Eq. (1), and has the following form (see e.g. 11 for details):KIsinh0 KII3cosh0 ? 1 163T sinh02?cosh0ffiffiffiffiffiffiffi2prp 0:4Here T is the corresponding value of the T-stress. The crack growth direction h0can now be expressed in thefollowing form:h0 h0KII=KI;T;d;5where d is an additional length scale representing the fracture process zone size, see e.g. the article by Kimet al. 12. Based on Eq. (4) it can be found that the negative T-stress decreases the crack initiation angleh0, but the positive T-stress increases the angle h0.2.2. Fatigue crack propagation rateUnder mixed-mode conditions the fatigue crack propagation rate generally depends on the values KI, andKIIand a modified ParisErdogan law for mixed mode fatigue crack growth rate estimation is usually used(e.g. see the work by Henn et al. 13). The disadvantage of the approach is the fact that the correspondingmaterial constants are not known in most cases.The curvature of naturally growing cracks is usually slight. In the paper by Kne sl 14,15, fatigue crackpropagation under slightly changing mixed mode conditions was studied and it was concluded that under con-ditions KII 0.25 KI, the shear mode of loading affects the crack path and consequently the values of the stressintensity factor KI= KI(KII), but to estimate the fatigue crack propagation rate, the standard version of ParisErdogan law for normal mode loading, da/dN = CKIm, can be used.To assess the influence of the changing constraint level on the fatigue crack propagation rate, the modifiedform of the ParisErdogan law for two parameter fracture mechanics was introduced in 16. This makes itpossible to account for the effect of constraint on the fatigue propagation rate in the form:da=dN CkT=r0KI?m;6where C and m are the material constants obtained for the conditions corresponding to T = 0 and r0is thecyclic yield stress. The value of the T-stress in Eq. (6) represents the level of the constraint correspondingto the given specimen geometry andkT=r0 1 ? 0:33Tr0? 0:66Tr0?2? 0:445Tr0?3:7The approximation Eq. (7) holds for ?0.6 T/r0 4 mm, where the value KII5 0.Then the corresponding fatigue crack is numerically simulated. In the case of standard fracture mechanics, i.e.without consideration of the constraint influence, Eq. (3) is used to estimate the crack propagation directionangle h0.In this case, the crack propagation direction is given by the corresponding ratio KII/KI. With theintention of assessing the influence of the constraint level on the crack path, the numerical simulation isrepeated, but to estimate the crack propagation direction Eq. (4), is applied. In this case, the crack propaga-tion direction depends on the ratio KII/KIand corresponding value of T-stress. During the simulation at eachcrack increment the values of KI, KIIand T-stress and corresponding value of h0have to be calculated. Theresults are shown in Figs. 6 and 7. The dependence h0= h0(KII/KI,T,d) following from Eq. (4) is shown inFig. 8. It follows from the above results that for the configuration studied the influence of T-stress on the fati-gue crack propagation path is very small. The maximum difference between h0values obtained from Eqs. (3)and (4) is about a few degrees. Consequently, for the CCT specimen with holes which was studied the influenceof constraint on the crack path can be neglected. This is documented in Fig. 2, where the results of both sim-ulations (with and without T-stress) are practically identical.Due to the existence of holes the values of the stress intensity factor KIdecrease in comparison with stan-dard CCT specimen; see Figs. 3, 4 and 6. The presence of the holes changes the constraint level as well. The T-values in CCT standard specimen are negative and indicate the loss of the constraint, see Fig. 3. In the case ofthe CCT specimen with holes the loss of the constraint is not so high, see Fig. 5 (for the reference curve), andFig. 7 (for the simulated crack path). The increase of the constraint level generally leads to the decrease of thefatigue crack propagation rate 17. This is demonstrated in Fig. 9, where the calculated values of the fatiguecrack propagation rate based on one and two parameters fracture mechanics approach are presented. Theexperimentally determined values of the crack fatigue propagation rate are in good agreement with the twoparameter approach, see Fig. 9.0481216crack length a mm10-710-610-510-4da/dN mm/cycleda/dN(K)da/dN(K,T)Experiment Fig. 9. Calculated of fatigue crack propagation rate for CCT specimen with holes, one parameter (full line), two-parameter (dash line)fracture mechanics and experimentally determined values (points) are compared.S. Seitl, Z. Kne sl / Engineering Fracture Mechanics 75 (2008) 8578658636. ConclusionsTwo-parameter constraint based on fracture mechanics was applied to assess the influence of the constrainton the fatigue crack propagation behavior. Two aspects of fatigue crack propagation, namely the fatigue crackpropagation direction and the fatigue crack propagation rate, were analyzed. Generally under mixed modeconditions, the fatigue crack path and the propagation rate are influenced by the corresponding constraintlevel.The behavior of the fatigue crack near a hole in a tensile CCT with holes specimen was experimentally stud-ied and the results obtained were compared with those of the corresponding numerical simulation. The fatiguecrack path was predicted by using the modified maximum tensile stress criterion. The following results wereobtained:(1) In the present case of a fatigue crack propagation in the tensile specimen with two holes the influence ofconstraint on the fatigue crack path is non-significant. Both simulated crack paths (with and without T-stress) are practically identical and correspond to the experimentally determined curve well.(2) The constraint level influences the fatigue crack propagation rate. For the configuration studied the exis-tence of the holes leads to the increase of constraint values and contributes to the decrease of the fatiguecrack propagation rate.(3) For naturally growing cracks where the crack curvature is slight the fatigue crack propagation rate canbe calculated by using the ParisErdogan law corresponding to the normal load of loading only, but thecorresponding values of KIand T have to be calculated for the curved crack path. The resulting fatiguerack propagation rate is influenced by the constraint level and to estimate it Eqs. (6) and (7) have to beapplied.AcknowledgementsThis investigation reported in this paper was supported by Grants No. 101/04/P001 of the Grant Agency ofthe Czech Republic and by Institutional Research Plan AV OZ 204 105 07.References1 Sih GS. A special theory of crack propagation. Mechanics of fracture, vol. I. Leiden: Noordhoff; 1972.2 Sih GC, Macdonald B. Fracture mechanics applied to engineering problem-strain energy density fracture criterion. Engng DraftMech 1974:36186.3 Wu CH. Fracture under combined loads by maximum energy release rate criterion. J Appl Mech 1978;45:5538.4 Sladek J, Sladek V, Fedelinski P. Contour integrals for mixed-mode crack analysis: effect of nonsingular terms. Theo Appl FractMech 1999;27:11527.5 Williams JG, Ewing PD. Fracture under complex stress the angled crack problem. Int J Fract 1972:41641.6 Leevers PS, Radon JC. Inherent stress biaxiality in various fracture specimen geometries. Int J Fract 1982;19:31125.7 Pook LP. Linear elastic fracture mechanics for engineers: theory and applications. WIT Press; 2000.8 Qian J, Fatemi A. Mixed mode fatigue crack growth: a literature survey. Engng Fract Mech 1996;55(6):96990.9 Anderson TL. Fracture mechanics: fundamentals and applications. Boca Raton: CRC Press LLC; 1995.10 Broek D. Elementary engineering fracture mechanics. NordhoffInternational Publishing; 1974.11 Bouchard PO, Bay F, Chastel Y. Numerical modeling of crack propagating automatic remeshing and comparison of differentcriteria. Comput Methods Appl Mech Engng 2003;192:3887908.12 Kim JH, Paulino GH. T-stress, mixed-mode stress intensity factors, and crack initiation angles in functionally graded materials: aunified approach using the interaction integral method. Comput Meth
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