銑床縱向工作臺(tái)的設(shè)計(jì),銑床縱向工作臺(tái)的設(shè)計(jì),銑床,縱向,工作臺(tái),設(shè)計(jì) Abstract * Corresponding author. E-mail addresses: seitlipm.cz (S. Seitl), kneslipm.cz (Z. Knesl). Available online at Engineering Fracture Mechanics 75 (2008) 857865 0013-7944/$ - see front matter C211 2007 Elsevier Ltd. All rights reserved. When a crack propagates in a non-homogeneous stress field, its crack path is generally curved. The crack path in brittle isotropic homogeneous material has a tendency to be one for which the local stress field at its tip is of a normal mode I type and for which K II tends 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 criterion 2,3, and the stationary Sih strain energy density factor 1,4. All these criteria have in common that if K II 50, the crack extends with non-zero change, h 0 , in the tangent direction to the crack path in such a way that K II ! 0. According to the above criteria, the direction of crack propagation h 0 depends on the ratio of the stress intensity factors corresponding to mode II and mode I, i.e. h 0 = h 0 (K II /K I ). This approach, based on the assumptions of standard fracture mechanics, does not account for the changing of the constraint level dur- ing crack propagation. The fatigue crack path has been studied on a tensile specimen with holes. The experimental crack path trajectories were compared with those calculated numerically. To incorporate the influence of constraint on the crack curving, we predicted the fatigue crack path by using the two-parameter modification of the maximum tensile stress (MTS) criterion. The values of the mixed-mode stress intensity factors K I , and K II as well as the corresponding constraint level characterized by T-stress were calculated for the obtained curvilinear and reference crack path trajectories. It is shown that in the studied configu- ration the eect of T-stress on the crack path is not significant. On the other hand the eect of constraint on the fatigue crack propagation rate is more pronounced. C211 2007 Elsevier Ltd. All rights reserved. Keywords: Constraint; Mixed-mode loading; Fatigue crack; Crack growth; Crack path 1. Introduction Two parameter fracture mechanics: Fatigue crack behavior under mixed mode conditions Stanislav Seitl * , Zdenek Knesl Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zizkova 22, 616 62 Brno, Czech Republic Received 30 November 2006; received in revised form 12 April 2007 Available online 27 April 2007 doi:10.1016/j.engfracmech.2007.04.011 on the plane The second calculated. direction local gating 858 S. Seitl, Z. Knesl / Engineering Fracture Mechanics 75 (2008) 857865 K I sinh 0 K II 3cosh 0 C010 with K II sinh 0 =2 0 8 : ; 3 where K I and K II are the stress intensity factors for the initial crack corresponding to mode I and mode II loading stress field within a small circle of radius r centered at the crack tip. The direction angle of the propa- crack is computed by solving the following equation: two-parameter modification of the maximum tensile stress (MTS) criterion is applied to determine the h 0 value. The MTS criterion was introduced by Erdogan and Sih 1,9,10 and is one of the most widely used theories for 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 the Thus, the estimation of the h 0 is of paramount importance. Various criteria for the crack growth under mixed-mode loading have been proposed in the literature, see e.g. 8. In the present paper a r ij r;h K I 2pr p f I ij h K II 2pr p f II ij hTd 1i d 1j ; as r ! 0; 1 where r ij denotes the components of the stress tensor, K I and K II are the corresponding stress intensity factors and f ij (h) are known angular functions. The fracture parameters K I , K II and T depend on the crack length a, the geometry, size and external loading of the body and corresponding boundary conditions. The T-stress can be characterized by a non-dimensional biaxiality ratio B given by Leevers and Radon 6: B T pa p K I ; 2 where K I is 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 the T-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 in centre 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-stress is tensile a crack is directionally unstable, and following a small random deviation, it does not return to its initial 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 and consequently 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) and two dimensional approximation corresponding to plane strain conditions according to ASTM Standard E 467. 2. Theoretical background 2.1. The direction of crack growth The growth of a fatigue crack is usually numerically simulated as a number of discrete incremental steps. After each crack length increment the corresponding crack propagation direction h 0 has to be crack path and propagation rate under mixed mode of normal and shear loading conditions. The in- constraint level is described by using T-stress. elastic T-stress represents a constant tensile stress acting parallel to the crack flanks. It is related to the term (the first non-singular term) in the Williams expansion of the stress field 5: In the contribution two-parameter constraint based fracture mechanics is used to assess the constraint level , respectively, and h 0 is the predicted direction angle. K II and (e.g. material intens Erdo pose, Fig. 1 mode formed crack S. Seitl, Z. Knesl / Engineering Fracture Mechanics 75 (2008) 857865 859 that in the vicinity of the holes the fatigue crack path is curved as a consequence of the shear mode of loading caused by holes. In all cases the experimental data (the propagation rate and the position of the crack tips) were I was found K Ith = 6.0 MPa m and the cyclic yield stress is r 0 = 202 MPa. Experiments were per- under load controlled conditions (the load amplitude was kept constant). The experimentally obtained path is shown in Fig. 2. Note that all crack lengths a are measured along the crack path. It can be seen steel) has the following parameters: Youngs modulus, E = 2.1 10 5 MPa and Poissons ratio, m = 0.3. The materials constants in the ParisErdogan region are C = 2.49 10 C09 and m = 2.97. The threshold value for 1/2 the experiments were performed on center cracked plate tension specimens (CCT) with two holes (see ). The crack paths were measured optically with resolution 0.1 mm. The material used (medium carbon To assess the influence of the changing constraint level on the fatigue crack propagation rate, the modified form of the ParisErdogan law for two parameter fracture mechanics was introduced in 16. This makes it possible to account for the eect of constraint on the fatigue propagation rate in the form: da=dN CkT=r 0 K I C138 m ; 6 where C and m are the material constants obtained for the conditions corresponding to T = 0 and r 0 is the cyclic yield stress. The value of the T-stress in Eq. (6) represents the level of the constraint corresponding to the given specimen geometry and kT=r 0 1C00:33 T r 0 C18C19 0:66 T r 0 C18C19 2 C00:445 T r 0 C18C19 3 : 7 The approximation Eq. (7) holds for C00.6 T/r 0 0.4. The results of the theoretical analyses based on Eqs. (6) and (7) correspond well with the experimental data; see 16,17 for details. 3. Experiment a crack approaching a hole This section describes a testing procedure used for experimental verification of the modeling techniques proposed for predicting the curved crack fatigue problem under mixed mode I and II loading. For this pur- ity factor K I = K I (K II ), but to estimate the fatigue crack propagation rate, the standard version of Paris gan law for normal mode loading, da/dN = CK I m , can be used. The curvature of naturally growing cracks is usually slight. In the paper by Knesl 14,15, fatigue crack propagation under slightly changing mixed mode conditions was studied and it was concluded that under con- ditions K II 4 mm, where the value K II 50. 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 direction angle h 0. In this case, the crack propagation direction is given by the corresponding ratio K II /K I . With the intention of assessing the influence of the constraint level on the crack path, the numerical simulation is repeated, but to estimate the crack propagation direction Eq. (4), is applied. In this case, the crack propaga- S. Seitl, Z. Knesl / Engineering Fracture Mechanics 75 (2008) 857865 863 tion direction depends on the ratio K II /K I and corresponding value of T-stress. During the simulation at each crack increment the values of K I , K II and T-stress and corresponding value of h 0 have to be calculated. The results are shown in Figs. 6 and 7. The dependence h 0 = h 0 (K II /K I ,T,d) following from Eq. (4) is shown in Fig. 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 dierence between h 0 values obtained from Eqs. (3) and (4) is about a few degrees. Consequently, for the CCT specimen with holes which was studied the influence of 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 K I decrease 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 of the CCT specimen with holes the loss of the constraint is not so high, see Fig. 5 (for the reference curve), and Fig. 7 (for the simulated crack path). The increase of the constraint level generally leads to the decrease of the fatigue crack propagation rate 17. This is demonstrated in Fig. 9, where the calculated values of the fatigue crack propagation rate based on one and two parameters fracture mechanics approach are presented. The experimentally determined values of the crack fatigue propagation rate are in good agreement with the two parameter approach, see Fig. 9. 0481216 crack length a mm 10 -7 10 -6 10 -5 10 -4 d a /d N mm/cycle da/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. corresponding values of K I and T have to be calculated for the curved crack path. The resulting fatigue rack propagation rate is influenced by the constraint level and to estimate it Eqs. (6) and (7) have to be 864 S. Seitl, Z. Knesl / Engineering Fracture Mechanics 75 (2008) 857865 applied. Acknowledgements This investigation reported in this paper was supported by Grants No. 101/04/P001 of the Grant Agency of the Czech Republic and by Institutional Research Plan AV OZ 204 105 07. References 1 Sih GS. A special theory of crack propagation. Mechanics of fracture, vol. I. Leiden: Noordho; 1972. 2 Sih GC, Macdonald B. Fracture mechanics applied to engineering problem-strain energy density fracture criterion. Engng Draft Mech 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: eect of nonsingular terms. Theo Appl Fract Mech 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. Nordho International Publishing; 1974. 11 Bouchard PO, Bay F, Chastel Y. Numerical modeling of crack propagating automatic remeshing and comparison of dierent criteria. 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: a unified approach using the interaction integral method. Comput Methods Appl Mech Engng 2003:146394. 13 Henn K, Richard HA, Linning W. Fatigue crack growth under mixed mode and mode II cyclic loading. In: Proc. of the 7th european conference on fracture; 1998. p. 110413. 14 Knesl Z. Numerical simulation of crack behaviour under mixed mode conditions. Part I: linear-elastic fracture mechanics, ACTA Technica C SAV 1978;5:60320. 15 Knesl Z. A fracture mechanics approach to the optimum design of cracked structures under cyclic loading. Handbook Fatigue Crack Propagation Metallic Struct 1994:55177. 6. Conclusions Two-parameter constraint based on fracture mechanics was applied to assess the influence of the constraint on the fatigue crack propagation behavior. Two aspects of fatigue crack propagation, namely the fatigue crack propagation direction and the fatigue crack propagation rate, were analyzed. Generally under mixed mode conditions, the fatigue crack path and the propagation rate are influenced by the corresponding constraint level. 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 fatigue crack path was predicted by using the modified maximum tensile stress criterion. The following results were obtained: (1) In the present case of a fatigue crack propagation in the tensile specimen with two holes the influence of constraint 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 fatigue crack propagation rate. (3) For naturally growing cracks where the crack curvature is slight the fatigue crack propagation rate can be calculated by using the ParisErdogan law corresponding to the normal load of loading only, but the 16 Knesl Z, Bednar K, Radon JC. Influence of T-stress on the rate of propagation of fatigue crack. Phys Mesomech 2000:59. 17 Hutar P, Seitl S, Knesl Z. Quantification of the eect of specimen geometry on the fatigue crack growth response by two-parameter fracture mechanics. Mater Sci Engng 2004;A387389:4914. 18 Knesl Z, Hutar P, Seitl S. Calculation of stress intensity factor by finite element method, analysis of constructions by finite element method. Prague 2002:6980 in Czech. 19 Seitl S, Hutar P, Knesl Z. Determination of T-stress by finite element analysis. Analysis of constructions by finite element method. Brno 2003:11322 in Czech. 20 Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook. New York (NY): The American Society of Mechanical Engineers; 2000. 10016. 21 Rasid MM. The arbitrary local mesh replacement method: An alternative to remeshing for crack propagation analysis. Comput Methods Appl Mech Engrg 1998;154:15099. S. Seitl, Z. Knesl / Engineering Fracture Mechanics 75 (2008) 857865 865