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1 Available online at SCIENCE ELSEVIER Theoretical and Applied Fracture Mechanics 42 (2004) 53-6 theoretical and applied fracture mechanics Evaluation of crack initiation angle under mixed mode loading at diverse strain rates LH. Hernández-Gómez *, I. Sauceda-Meza, G. Urriolagoitia-Calderón, A.S. Balankin, O. Susarrey Instituto Politécnico Nacional. Sección de Estudios de Posgrado e Investigación de la ESIME, Edificio 5, 3er. Piso de la Unidad Profesional Adolfo López.Mateos, Col. Lindavista México, DF, Mexico Abstraet Crack initiation angle, under mixed mode loading at several strain rates, is analysed using an experimental-numerical approach. The physical phenomenon for the problem at hand is influenced by the local and global conditions. Qne of such factors is the strain rafe at the crack tipo For this purpose, PMMA plates with centred angled cracks under mixed mode loading were tested. The strain rafe at the neighbourhood of the crack tip before crack propagation was evaluated. Considering that this material is strain rate sensitive, the numerical models were calibrated with the modulus of eiasticity measured in tension tests at the observed strain Tales. Numerical evaluations were performed with the finite eiement method in conjunction with the volume energy density criterion. An improvement in the evaluation of the crack propagation angle was observed. In order to complete the analysis, the crack initiation angle was algo evaluated with the strain energy density factor S, considering the mechanical properties of PMMA, as evaluated at the observed strain rates, and the stress intensity factors k, and k2. Results are in agreement with those observed 2004 Elsevier Ltd. AII rights reserved.. Introduction ing work in []. lt considered the crack initiation Crack propagation stability is an engineering direction for an angled crack which was predicted with the maximum circumferential stress criterion. problem that has attracted attention for many This work inspired the investigation of several years. Much work has been done since the pioneer- additional factors that are involved in the fracture process. Crack extension stability was studied. Corresponding author. using the eigenfunction series expansion [2,3]. This address: (LH. Hernández- approach wasalso considered in [4] to study the Gómez). influence of transversely applied stress by /$ - see front 2004 Elsevier Ltd. Al! rights reserved. doi: 0.06/j.tafmec

2 54 LB. Hernández-Gómez et al. I Theoretica/ and Applied Fracture Mechanics 42 (2004) 53-6 including the second term of the series obtained in [2,3]. This was extended for the case oftension and bending [5,6]. Regarding crack path extension, the work in [7] investigated effects caused by the applied load direction,the curvature crack radius and the complete stress and/or energy field. Later, crack growth instability was evaluated numerically [8,9] and experimentally [lo]. All this work is based on the single edge Batch specimen under biaxial loading. Recently, crack initiation and propagation in beams with edge crack under mixed loading was analysed []. This was done by using an energetic approach based on the S-theory [2]. The aforementioned works have shown that there are many factors involved. Some of them are local while others are global. These different factors have to be separated in arder to establish their individual contribution to the fracture process. This complicates further when mixed mode loading conditions are developed. In all these cases, quasi-static loading conditions were assumed. However, when the time of load application varíes, strain rate conditions are developed at the crack tipo In materials, which are rate sensitive, mechanical properties change with the loading rateo In other words, fracture energy decreases with increasing strain rateo This has been observed by others [3,4]. As a consequence, the evaluation of the crack initiation angle must take into account these factors. Under these consideraliaos, the question being: how will the crack initiation angle be affected under mixed mode loading? This situation is analysed in this work following an experimental-numerical approach. The results are then compared with those obtained by means of the strain energy density factor S [5]. 2. Experimental analysis The experimental work has been divided in two parís. The first part evaluates the PMMA modulus of elasticity at different strain rates and the second part evaluates the crack initiation angle. The objective is to first evaluate the mechanical properlíes under tensile load at the observed strain rates at the crack tip when crack initiation takes place. Table PMMA modulus of elasticityat dilferentstrain Tales Strain rate Ref. (S-I) Modulus of elasticity (GPa) [6] [6] This work This work This work [7] [7] This was done with an 8502 Instron machine with several loading rates. Appropriate tension specimens were tested. Table swiunarises this data. For the comparison of this data, it has to be borne in mind, atleast, two points, namely () the compliance of the testing machine and (2) the batch variation characteristics of the raw material. These factors were minimized as much as possible by making direct readings of the specimen deformation on the gage length of the specimen with a strain gage. Resides, all the specimens were produced from the same sheet. For the analysis of crack propagation direction, PMMA plates with central angledcracks were loaded under tension, along its majar axis of symmetry. The crack angle with respect to the horizontal plane was set to be 0, 30, 45 and 60. Fig. shows the dimensions of the specimens. In all these cases, the strain field was evaluated with a strain gage located at the crack tip neighbourhood. Specimens were tested with a 8502 Instron machifle with the following loading rates: 300 and 3000mm!min and the strain field variation was recorded with a SYSTEM O0 of Vishay of Measurement Group, Inc. Reproducibility of the results was checked, by testing 0 specimens with the same geometrical conditions. It is important to keep in mind that, it is difficult to establish the strain rate at the very crack tiposo, an average evaluation was obtained with a 3mm strain gage located as clase as possible to the very crack tipo The case of 0 is well known and the results were used to validate the experimental and numerical results. A typical strain rate history is shown in Fig. 2. This corresponds to a specimen tested at 3000mm/min with an horizontal crack. As it can

3 L.H. Hernández-Gómez et al I Theoreticaland Applied Fracture Mechanics 42 (2004) 53-6/ 55 I~ 50mm.,~ 50mm. 5mm ~ ~p 8~ 28mm. 5~mm I~ ~ 8mm I-.:! mm 28mm (a) (b) I~ 50mm. I~ 50mm., :Jp 8m~~ 8mm 28mm jp 8 mmjj~ ~~mm 28 min (e) (d) Fig.l. Dimension ofthe cracked specimens and strain gage location.ln al! cases the thickness is 6.35mm. (a) 0, (b) 30, (c) 45 and (d) 60. be seen, the strain cate varies as the loading process takes place. Crack initiation is well defined with a curve peak. 3. Numerical analysis The numerical work has been divided in three parts: () Model development, (2) Model calibration and (3) Evaluation of crack initiation direction.more specifically,franc code [8] was used for this purpose. A finite element model with 4300 nodes was generated. This mesh has rectangular elements of eight nodes and eight quarter point elements were used at the crack tipo The whole plate was simulated because there no sym- metry prevails when angled cracks are introduced. Crack initiation angle was evaluated with the energy density criterion. In this case, K and Ku are required for these calculations. In order to validate FRANC for this purpose, these parameters were evaluated under quasi-static loading with the same geometry of the cracked specimens. They were loaded quasi-statically.. in the same range. The numerical results were compared with those obtained with photoelasticity, using polycarbonate specimens. Convergence between the experimental and numerical results was obtained. This suggests that FRANC is adequate for such calculations. In a second step, the model calibration was done. Its purpose was to obtain the numerical reproduction of the strain cate recorded by the

4 56 LB. Hernández-Gómez et al. I Theoretica/ and Applied Fracture Mechanics 42 (2004) 53-fi E e E Cñ o TIme (sec.) Fig. 2. Strain rate at the crack tip neighbourhood of a specimen with an horizontal crack, and tested with a loading rate of 3000mm/min. strain gages bonded near the crack tipo All the calculations were performed with the same loading rate that was applied by the testing machine. Besides, in order to make a comparison, another finite element analysis was done with the typical modulus of easticity under quasi-static loading (2.9GPa) commonlyreported in the open literature, as in [6]. In Fig. 3, it is shown the typical divergence between these results. This situation was improved when the value of the modulus of elasticity at the observed strain rate, during the loading process, was introduced in the calculations. As shown in Fig. 4, both results are in agreement. In the third step, the fracture analysis of the centred angled crack plate, loaded under tension, '\ E e "(ij Cñ Experimental ---.Numerical o. Time (sec.) 0. Fig. 3. Divergence between the numerical evaluation of the strain rate under quasi-static conditions and the experimental resu!ts.

5 LH. Hernández-GÓmez et al. I Theoretícaland Applied Fracture Medlanícs 42 (2004) 53-6/ E E -; éi Experimental Numerical O (a) Time (sec) E 0.00 Ē c: 'ro éi Experimental Numerical (b) O Time (sec) Fig.4. Converge between numerical and experimental evaluation of the strain rafe at the crack tip neighbourhood, (a) 300mm/min, (b) 3000mm/min. was done. KI and Kn were evaluated. In arder to make a comparison, this evaluation was made with the mechanical properties at the observed strain rates and under quasi-static loading. In the last case, a modulus of elasticity, equals to 2.9GPa was also taken into account. Crack initiation angle was evaluated in accordance with the energy criterían with the following equation: -2KIKII () = arc tan 2 2 [ KI + KII] The results are summarised in Table 2. () 4. Analysis with the strain energy density concept Another way to evaluate the crack initiation angle is with the fracture theory based on the field strength of the local strain energy density pro po sed in [5]. In this case, the energy release rate is not required and mixed moje crack extension problems may be treated. The fundamental parameter of this theory, the strain energy density factor S, is direction sensitive and it is evaluated in two dimensions with the following relation:

6 58 LB. Hernández-Gómez et al. I Tlzeoretical and Applied Fracture Meclzanics 42 (2004) 53~ Table 2 Crack initiation angle evaluated with the energy criterion under quasi-static and dynamic loading Crack Load speed K quasi-static K observed KII quasi-static KII observed Crack Crack initiation angle (O) (mm/min) (MpamI/2) strain raje (MPamI/2) strain raje initiation angle angle at observed (MPamI/2) (MpamI/2) quasi-static (O) strain rate (O) O O s = al!~ + 2al2kk2+ a22~ (2) where k and k2 are the stress intensity factors under loading mode I and, respectively. These parameters are related to the energy release rafe, in generalized plane stress, there results G= (nk )/E and Gil = (n~)/e. The coefficients aij (i,j =,2) are given by al! = l6jl [(3-4v - cos 0)( + cos O)] a2 = ~Jl2 sin O[cos O- ( - 2v)] a22 = l6jl[4(- v)(l - coso) + ( + cos 0)(3cos O-)] (3) where Ois the polar angle, which varies around the crack tip, v is the Poisson's ratio and Jlis the shear modulus of elasticity. Fig. 5 shows the parameters involved in the crack initiation angle evaluation at the crack tip neighbourhood. Once S is established, crack initiation will take place in a radial direction O, from the crack tip, along which the strain energy density is minimum. Hence the crack initiation angle determined from as =0 00 (4) In this work, the crack inciination angle is taken into account in the calculations by means of the values of the SIF k and kz, because their values are function of the orientation of the crack planeo These parameters were caiculated numerically with the finite element method. Besides, the shear mody Fig. 5. Parameters involved in the calculation of the crack initiation angle. ulus is strain rafe dependent, therefore it was calculated from the modulus of elasticity observed during the experimental tests. The results obtained are shown in Table 3 and they are comparedwith those obtained with the maximum circumferential stress criterion. In all the cases reported in Table 3, the load speed was 3000rnm/min. These are the situations, which are more sensitive to the strain rafe. 5. Discussion of the results The tension tests have shown that PMMA becomes more rigid as the strain rafe increases. This is reflected with the increment of the modulus of elasticity as the strain rafe growths. Comparing KI and Kn evaluations at the observed strain rates ay ay 'Ixy ax x

7 Table 3 L.H. Hernández-Gómez et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 53-ól 59 Crack initiation angle evaluated with the strain energy density factor S and the rnaxirnurn circurnferential stress criterion Crack inclination angle Crack initiation angle This work Strain energy density concept Maxirnurn circurnferential stress with those made under quasi-static conditions (Table 2) a divergence of results is found. In fact, the biggest difference of K values is for angles lesser than 45. This is the range in which K is a dominant factor in the fracture process. On the other hand, the upper divergence of Kn líes on the range of angles greater than 45. This is the case in which this parameter plays an important role in crack direction. In the case of the specimen with an horizontal crack, Kn should be zero. Nonetheless, the numerical results show that this value is small. This can be explained by the fact that the whole plate was modelled and some nades may not lay on the crack plane, especially with those that are used in the fracture parameters calculation. Regarding the crack initiation angle, it is neariy O. This was expected. In spite of these differences, it can be considered that the numerical evaluations are in line with reality. With respect to the other cases, the specimens have been made in such a way that the crack runs along the horizontal plane, because Fig. 6. Fracture surface of a 60 cracked specirnen tested at 3rnrn/rnin.

8 60 LH. Hernández-Gómez et al. I Theoretical ami App/ied Fracture Mechanics 42 (2004) 53-{j Fig. 7. Fracture surface of a 60 cracked specimen tested at 3000mm/min. it is perpendicular to the applied tension loado This situation is observed when all the calculations are made with the parameters obtained with the calibrated mojel at the observed strain rateo When quasi-static conditions are applied it can be seen there is some divergence. The crack surfaces were microscopically analysed. Typical results are shown in Figs. 6 and 7, in which the real image is magnified 35 times. In both cases, the initial crack front is on the top of each picture. This is the limit between the mark left by tool that completed the crack front, and the crack surface. In the last afea, different sofí of marks are depicted. In the case of the specimen tested at low strain rate (Fig. 6), the crack surface clase to the crack front has many ribs. On the other hand, the crack loaded with the biggest strain rate (Fig. 7), has a surface which is "clean". This reflects that the latter has used less energy. As the crack propagates, there is a bright regían and at a far distance from the crack front the surface becomes rough. Nonetheless, in the case where a lower strain rate was applied (Fig. 6), there are deeper marks. In other words, as the strain rate increases, the crack surface is smoother. It is important to mentían that this evaluation reflects the fact that less fracture energy is required as the strain rate increases. To complete the discussion, the results with the highest strain rate were compared with those obtained with the strain energy density factor S and the maximum circurnferential stress. As it can be seen, the results of this work and those obtained with the S factor are in agreement. Besides, the case of the horizontal crack (0 ), the crack initiation angle is 0, as it was expected. In accordance with the author's opinion, two points have to be kept in mind when this approach is followed. Namely, () the material mechanical properties at the observed strain rates are easily introduced in the calculations by means of the coefficients all, a2 and a22and (2) the SIF's have to be evaluated as accurate as possible. These should lead to an exact evaluation of the crack angle initiation under

9 L.H. Hernández-Gómez et al. I Theoretical and Applied Fracture Mechanics 42 (2004) mixed loading mode, as the material is strain rafe dependent. Finally, the work in [5]has shown that the predictions based on the maximum stress criterion do deviate from those obtained by the strain energy density criterion. The results reported in Table 3 are in agreement with ibis statement. 6. Concusions The accurate evaluation. of the stress intensity factors of a crack under mixed mode loading is relevant in arder to establish the crack initiation angle. However, when the strain rafe increases at the crack tip, a quasi-static evaluation may no be valido Under ibis scheme, the crack direction angle of propagation calculated varíes in relation with the one observed in quasi-static conditions. This is confirmed with the values of fracture parameters, such as K and KIl' The calculations are improved when the changes of mechanical properties with strain rafe are taken into account. This situation is also valid when the strain energy density factor S is applied. The crack initiation angle was calculated along the radial direction on which the strain energy density is a mínimum. The obtained results are in agreement with those observed experimentally. AIso, ibis confirms the fact that the stationary value of Smin can be used as an intrinsic material parameter, and from ibis a mixed mode fracture criterion can be stated. Resides, the expected divergence with maximum circumferential stress was also observed. Acknowledgments The grant U awarded by Consejo Nacional de Ciencia y Tecnología and the support given to ibis project by COF AA and CGEPI of Instituto Politécnico Nacional are gratefui acknowledged. AIso, the authors thank Mr. Cándido Zamora for the final numerical cal- - culations. References [] F. Erdogan, O.e. Sih, On crack extension in plates under plane loading and transverse shear, J. Basic Eng. Trans. ASME 85(D) (963) [2] B. Cotterell, On brittle fracture paths, In!. J. Fract. Mech. (965) [3] B. Cotterell, Notes on the paths and stability of cracks, Int. J. Fract. TechnoI. 2 (966) [4] J.O. Williams, P.D. Ewing, Fracture under complex stress: The angled crack problem, Int. J. Frac!. Mech. 8 (972) [5] L Finnie, A. Saith, A note on the angled crack problem and directional stability of a crack, In!. J. Frac!. 9 (973) [6] P.D. Ewing, J.L Swedlow, Further results on the angled crack problem, In!. J. Fract. 2 (976) [7] M.E. Kipp, O.C. Sih, The strain energy density failure criterion applied to notched elastic solids, Int. J. Solids Struc!. (975) [8] O. Urriolagoitia-Calderón, LH. Hernández-Oómez, Evaluation of crack propagation stability with the Williams stress function. Part, Int. J. Compu!. Struct. 6 (4) (996) [9] O. Urriolagoitia-Calderón, L.H. Hernández-Oómez, Evaluation of crack propagation stability with the Williams stress function. Part II, Int. J. Computo Struct. 63 (5) (997) [0] O. Urriolagoitia-Calderón, LH. Hernández Oómez, Experimental analysis of crack propagation stability in single edge notch specimens, Theor. AppI. Fract. Mech. 28 (997) [] L Nobile, Mixed mode crack initiation and direction in beams with edge crack, Theor. AppI. Fract. Mech. 33 (2000) [2] O.C. Sih, Strain density factor applied to mixed mode crack problems, Int. J. Frac!. 0 (974) [3] O.C. Sih, D.Y. Tzou, Dynamic fracture tale of Charpy V- notch specimen, Theor. AppI. Fract. Mech. 5 (986) [4] L.H. Hernández-Oómez, C. Ruiz, Assessment of data for dynamic crack initiation under shock pressure loading: Part I-Experiment, Theor. AppI. Fract. Mech. 9 (993) [5] O.C. Sih, Mechanics of Fracture Initiation and Propagation, Kluwer Academic Publishers, Dordrecht, 99, pp [6] LCJ. The properties of "Pespex" acrylic materials, Plastics Division, Welwyn. [7] LH. Hernández-Oómez, C. Ruiz, Assessment of data for dypamic crack initiation under shock pressure loading: Part II-Analysis, Theor. AppI. Fract. Mech. 9 (993) [8] D. Swenson, J. Mark, FRANC2D/L: A crack propagation simulator for plane layered structures, version.3 User's guide, Kansas State University.

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