Criteria for Mixed-Mode Fracture rediction in Ductile Material N. Recho, S. Ma, X.B. Zhang University of Blaise ascal, IUT of Montluçon. B2235, Av. A. Briand, 03100, Montluçon, France recho@moniut.univ-bclermont.fr ma@moniut.univ-bclermont.fr A. irondi Deartment of Industrial Engineering,University of arma arco Area delle Scienze 181A, I-43100 arma, Italy irondia@ulsar.eng.unir.it C. Dalle Donne German Aerosace Center DLR, Institute of Materials Research, Linder Höhe, D-51170 Cologne, Germany Claudio.Dalle-Donne@dlr.de Abstract In order to assess a crack in a ductile structure subjected to mixed mode I-II loading, the tensile-shear fracture mode cometition must be considered. Then it is ossible to determine the critical load and the crack kinking angle. A mixed-mode fracture criterion is roosed for redicting the onset and the direction of a ductile crack growth under mixed mode loading. The transition between tensile tye fracture and shear tye fracture (TS-transition) will be evaluated using the J-M criterion roosed in (Li, Zhang, Recho, 2003, [1]). In order to validate the criterion, the values of the fracture locus J c as a function of M and of the kink angle will be estimated by finite element simulation of the corresonding mixed-mode I/II fracture exeriments erformed in (irondi, Dalle Donne 2001, [2]), (Dalle Donne, 1999, [3]). The connection of J-M criterion with the criteria resented in [2] and [3] will be finally addressed. Introduction The failure behaviour in ductile material cannot be redicted by elastic fracture models. For a decade, more and more exerimental studies and numerical simulations have been devoted to the investigation of mixed mode ductile fracture. It is well known that both tensile and shear fracture mechanisms can occur at the crack ti in ductile material. The cometition between tensile fracture (T-tye fracture) and shear fracture (S-tye fracture) determines the fracture toughness of a cracked metallic structure. The value of the mixed-mode fracture toughness deends on the actual crack roagation mode and on the fracture arameter used. Ductile steel often dislayed a decrease of fracture toughness with increasing mode II comonents [2]. In order to asses fracture toughness in a ductile cracked structure under mixed mode loading, the Tensile- Shear fracture cometition must be considered at first. Then it is necessary to determine
the critical fracture loads and the crack bifurcation angles. So a series of criteria are needed. The first is the criterion of TS transition, which is used to determine the tye of fracture, Tensile-tye or Shear-tye; secondly, the criterion of critical loading is needed for both tyes of fracture to estimate whether the crack begins to roagate under a mixed mode loading. The third one is the criterion of bifurcation angle. It can be used to redict the crack growth angle whatever the crack tye is. For elastic material, different criteria can be used to calculate the bifurcation angle. For examle, the maximum circumferential stress σ θθmax criterion (Erdogan and Sih, 1963, [6]), the maximum energy release rate criterion (alasniswamy and Knauss, 1978, [7]), the stationary strain energy density criterion (Sih, 1974, [8]), the crack ti oening dislacement (or angle) criterion (Sutton et al. 2000, [5]), and so on. Recently we have develoed the J-M based criteria [1] to assess the roagation of a crack in elastic-lastic material under mixed mode loading. The aim of this aer is to roose an exerimental rocedure to establish the mixed mode fracture criteria for redicting the onset and the direction of a ductile crack growth in structure comonents. The J-integral and the lastic mixity arameter M are used as basic fracture arameters in this investigation. The J- M based criteria is adoted to evaluate the transition between Tensile tye fracture and Shear tye fracture (TS-transition) and the bifurcation angle of a crack. The critical loading angle of crack roagation is redicted by our new roosed method on the basis of the exerimental results. The fracture toughness is studied on the CTS secimen made of steel StE550. Finally, by comaring the numerical calculation with the exerimental data, the validation of the roosed criteria is discussed. roosed criteria Combining exeriment and numerical calculation, we roose a rocedure to redict the crack roagation under mixed mode loading. Three criteria are introduced in the rocedure. They are the TS transition criterion, the bifurcation angle criteria for tensile and shear crack, and the critical load criterion. TS transition criterion When a crack exists in an elastic-lastic material, the angle of crack growth deends on the cometition between cleavage tensile fractures (T-tye fracture), related to the void growth and coalescence near the crack ti, and ductile shear fracture (S-tye fracture), deendent on the lastic strain concentration ahead of the crack ti. Recently, we have develoed the J-M based criteria [1] in order to determine this crack growth angle. In the case of a crack in an elastic-lastic material under mixed mode loading, Shih (1981) [9] showed that the stresses, strains and dislacements fields near the crack ti are dominated by the HRR singularity, and can be characterized by two arameters, the J- integral and the mixity arameter M. The mixity arameter M is defined as follows: 2 1 σ θθ ( θ = 0) M = lim tan (1) r 0 π σ ( θ = 0) rθ The main idea of the J-M based criterion is as follows: the TS-transition occurs at a value of M, M c, that is a material constant. M varies from 0 to 1. When M = 0, it is the
case of ure mode II and when M = 1, it is the case of ure mode I. A numerical method has been develoed (Li, 1998, [4]) to determine these two arameters for a crack subjected to mixed mode loading. Exerimental studies showed that, for an elastic-lastic material, there is a transition from T-tye fracture to S-Tye fracture. This transition is controlled by the critical value of the mixity arameter M c which can be determined by means of exeriments according to the critical fracture toughness JIC and J IIC (J IC is obtained from a ure mode I tensile test and J IIC from a ure mode II shear test). If the mixity arameter M for a given loading case is greater than M c, the crack will roagate by T-tye fracture. On the other hand, if M is smaller than, the crack will roagate by S-tye fracture M c Bifurcation angle criterion for a tensile crack When the crack grows in tensile manner, the maximum hoo stress (MHS) criterion is widely used in crack bifurcation angle assessment. It means the crack will roagate in the direction of the maximum circumferential stress σ θθmax. For elastic-lastic cracks, the angles of the near-ti can be reresented as function of M and hardening coefficient n. These relationshis have been given in Fig. 1 [1]. Bifurcation 0 angle (degree) -10-20 -30-40 n = 2,3,5,7,9,13-50 n=13-60 -70-80 -90 n=2 0 0,2 0,4 0,6 0,8 1 M Figure 1 Bifurcation angles of a tensile crack according to the MHS criterion Bifurcation angle criterion for a shear crack When the crack grows in shear manner, the situation is more comlicated. The crack will follow one of the slis lines near the crack ti. According to Li et al.[1], although several sli bands can be observed to develo from the crack ti, only two sli bands are ossible directions of the shear crack growth because the shear stress along these two sli bands are larger comaring to those along others. These two bands are aroximately at 45 with
resect to the initial crack lane, and, nearly along the crack lane. The bifurcation angles versus M and n are lotted in Figure 2. Shear-band ang le ( degree) 55 45 n=3 n=5 n=7 n=2 35 n=9 Sli band 1 25 15 5 Sli band 2 n=2,3,5,7,9-5 0 Figure 2 Bifurcation angles of a shear crack according to the sli band criterion Critical load criterion 0,2 0,4 It is well known that the values of the mixed mode fracture toughness deend on the actual crack roagation mode and on the fracture arameter used. Ductile steels often dislayed a decrease of fracture toughness and tearing resistance curves with increasing mode II comonents, whereas materials with a lower ductility generally showed higher mode II crack resistance values [2]. On the basis of the exeriment erformed by Dalle Donne and irondi [2], a criterion is develoed to assess the initial roagation of a mixed mode crack. The first ste is the mixed mode exeriment. The fracture tests are conducted on CTS secimens, which are mounted on the loading device shown in Fig. 3 so that they can be subjected to different loads with different loading angles. Several loading angle such as 0, 45, 75, and 90 are selected in the exeriments. The crack begins to roagate at a (exerimentally evaluated) value of crack ti oening of 0.19mm. For each loading angle, the crack ti oening, crack length and alied load are recorded at the same time (for details see [2]). Therefore, the critical load is the load corresonding to a crack initiation. The second ste consists in a numerical calculation. For a given loading angle and the corresonding critical load, the J c and M can be calculated numerically by means of FEM [4]. Consequently, the J c versus M fracture locus of the material can be obtained. For any loading angle, one can assess the corresonding fracture toughness J c from this curve. M 0,6 0,8 1
90 27 27 Loading holes 148 a 0 =45 α 54 1 2 3 4 5 6 7 54 crack 7' 6' 5' 2' 4' 3' 1' 15 Figure 3 mixed-mode loading device (dimensions in mm) To sum u, the roosed rocedure for redicting the crack roagation are described as follows: (1) Determination of the fracture locus J c -M (2) Calculation of J and M for a given loading (3) Comarison between M and M c. If M > M c, the crack will roagate by T-tye fracture, if M <M c, the crack will roagate by S-tye fracture (4) Comarison between J and J c. If J > J c, crack will begin to grow (5) Determination of the bifurcation angle (and crack growth ath) Validation of the rocedure In this study, the fracture tests were conducted on 4mm thickness and 90 mm width CTS secimens of the fine grained structural steel StE550, the yield strength is about 580 Ma and ultimate strength is about 650 Ma, the critical mixity arameter M c is about 0.77, which is issued from elastic mixity arameter M e c = 0.68. A fatigue re-crack was introduced u to a/w 0.6. The secimens were tested under 0, 15, 45, and 75 loading with resect to the crack axis. When loading angle α = 0, this is the case of ure mode II loading, on the other hand, when α = 90, it is ure mode I loading. The fracture toughness data are resented in terms of J c and M in this investigation. Table 1 lists the exerimental results. Table 1 exerimental results Loading angle α 75 45 15 0 Critical load (KN) 27 30 37 35 Fracture toughness J c (KN.m) 71 56 39 26 Mixity arameter M (FEM results of Eq.1) 0.98 0.89 0.52 0 The fracture locus J c -M is shown in Fig.4. For a given loading condition, we can determine easily whether the crack begins to roagate or not from Fig.4. For examle,
when the crack is subjected to 35 KN, 30 loading, we get M = 0.72 and J = 55. From Fig4, when M = 0.72, J c is about 47, therefore the crack will grow. Jc 80 70 60 50 40 30 20 10 Jc_M 0 0 0,2 0,4 0,6 0,8 1 M Figure 4 fracture curve as a function of J c and M In order to verify the J-M criteria, we calculate the bifurcation angle for each loading condition and comare them with the exerimental data. Fig. 5 shows the bifurcation angle of 45 loading. Table 2 lists the results of kink angles. θ FEM θex figure 5 bifurcation angle of 45 loading Table 2 results of bifurcation angles θ Loading angle α 75 45 15 0 M (FEM result of Eq.1 ) 0.98 0.89 0.52 0 Fracture tye T-tye T-tye S-tye S-tye Numerical results -12-23 -1.7 0 Exerimental results -13.5-27.6 5.7-11.1
According to J-M criteria, when loading angles are 75 and 45, M are greater than M c (0.77), they are T-tye fracture and the crack grows in the direction of -12 and -23 with resect to the crack axis. When loading angle are 15 and 0, M are smaller than M c, they are S-tye fracture and the crack grows along one of sli bands, the crack growth angle are -1.5 and 0.The numerical results of T-tye crack growth are close to the exerimental data. On the other hand, the S-tye crack growth angles calculated numerically follow a different trend with resect to the exerimental values. In [2] it was ostulated that the S-tye crack growth is romoted by a slight ositive triaxiality, which is not accounted for by the sli band criterion. Finally, in order to verify the accuracy of our method for evaluating J-integral, we evaluate J values by another exerimental based method [2], in which J (J _ exeriments in Fig. 6) is defined by J = J e + J (2) where, K 2 2 I + K II J e = (3) E U J =η (4) B( W a) and U is calculated from the crack ti oening and shearing dislacements δ I, and δ II resectively, monitored during exeriments and η is the mixed-mode lastic η factor [9]. Fig.6 shows the J results of 30, 45 and 60 loading. It can be noted that our J values are in good agreement with the results of equation 2. 30 45 60 160 J_exeriment 140 120 100 80 60 30 loading 45 loading 60 loading 40 20 0 0 20 40 60 80 100 120 140 160 J_FEM Figure 6 J results of 30, 45, and 60 loading
4. CONCLUSION An exerimental rocedure was roosed to redict the roagation of a mixed mode crack in ductile material. This rocedure includes three criteria to determine fracture tye, to evaluate crack bifurcation angles, and to estimate the critical loads. This method is validated numerically and exerimentally with CTS secimens of ferritic steel. The material constant M c and critical loads of several loading angles are given by mixed mode exeriments, and then the criteria can be determined by FEM calculation. So it is easy to realize. Further exeriments are needed to verify the fracture criteria so that they can be used in different material. Reference 1. Li J., Zhang X.B. and Recho N., J-M based criteria for bifurcation assessment of a crack in elastic-lastic materials under mixed mode I-II loading. Engineering Fracture Mechanics 71 (2004) 329-343 2. A. irondi, C. Dalle Donne, Characterisation of ductile mixed-mode fracture with the crack-ti dislacement vector. Engineering Fracture Mechanics 68, 1385-1402, 2001 3. C. Dalle Donne, The crack ti dislacement vector aroach to mixed-mode fracture. ASTM ST 1359, ASTM, West Conshohocken, A, USA, 21-39, 1999 4. Li J, estimation of the mixity arameter of a lane strain elastic-lastic crack by using the associated J_integral, Engineering Fracture Mechanics, No. 61,.355-368, 1998 5. Sutton M.C., Deng X., Ma F., Newman Jr. J.C. and James M., Develoment and alication of a crack ti oening dislacement-based mixed mode fracture criterion. Int. J. Struct. Solids, 37, 3591-3618, 2000 6. Erdogan F. and Sih, G.C., On the crack extension in lates under lane loading and transverse shear. Transaction of the ASME. J. Basic Eng, 85, 519-527, 1963 7. alaniswamy K. and Knauss, W.G., On the roblem of crack extension in brittle solids under general loading. In: Memat-Nasser S. (Ed.), Mechanics Today, vol. 4, ergamon ress, 87-148, 1978 8. Sih, G.C., Strain energy density factor alied to mixed mode crack roblem. Int. J. Fractures, 10, 305-321, 1974 9. Dalle Donne C., irondi A., J-integral evaluation of single edge notched secimens under mixed-mode loading, Journal of Testing and Evaluation, Vol. 29, No. 3, 239-245,2001