Stress intensity factors under combined tension and torsion loadings

Indian Journal of Engineering & Materials Sciences Vol. 19, February 01, pp. 5-16 Stress intensity factors under combined tension and torsion loadings A E Ismail a *, A Ariffin b, S Abdullah b & M J Ghazali b a Department Engineering Mechanics, Faculty of Mechanical & Manufacturing, Engineering, Universiti Tun Hussein Onn Malaysia, 86400 Batu Pahat, Johor, Malaysia b Department Mechanical & Materials Engineering, Faculty of Engineering & Built Environment, Universiti ebangsaan Malaysia, 43600 UM Bangi, Selangor, Malaysia Received 13 December 010; accepted February 01 This paper numerically discusses the stress intensity factor (SIF) calculations for surface cracks in round bars subjected to combined loadings. Different crack aspect ratios, a/b ranged from 0.0 to 1. and the relative crack depth, a/d in the range of 0.1 to 0.6 are considered. Since the loading is non-symmetrical, the whole finite element model is constructed. Then, both tension and torsion loadings are remotely applied to the finite element model and the SIFs are determined along the crack front of various crack geometries. An equivalent SIF method is then explicitly used to combine the individual SIF obtained using different loadings. A comparison is made between the combined SIFs obtained using the equivalent SIF method and finite element analysis (FEA) under similar loadings. It is found that the equivalent SIF method successfully predicted the combined SIF for Mode I. However, discrepancies between the results, which have been determined from the different approaches, occurred when F III is involved. Meanwhile, it is also noted that the predicted SIF using FEA is higher than the predicted through the equivalent SIF method due to the crack face interactions. eywords: Stress intensity factor, Surface crack, Finite element analysis, Combined loadings Cylindrical bars are generally used to transmit power from one point to another. The bars can be subjected to cyclic stresses which can cause mechanical damages and sometime experienced premature failure 1. The initiations of fatigue cracks on the surface are normally due to mechanical defects such as notches and metallurgical defects 3,4. In services, a rotating shaft can generally be subjected to combined loading due to its self-weight, which also induces a tension stress instead of torsion loadings. In fact, any arbitrary shapes of crack initiation may grow and take a semi-elliptical shape 5. Linear elastic fracture mechanics (LEFM) has been used to analyse stress intensity factors (SIFs) along the crack front. The solution of SIFs for a wide range of crack geometries under Mode I loading has been reported elsewhere in the literature 6-16. However, the calculated SIFs, subjected to Mode III and the SIFs under combined loadings such as tension and torsion are rarely studied 1-16. Therefore, the aim of this study is to obtain the SIFs for semi-elliptical surface cracks subjected to tension, torsion and the combination of loadings. According to the literature 1-15, the SIFs *Corresponding author (E-mail: emran@uthm.edu.my) subjected to combined loadings are rarely studied. Since the combined SIF can be obtained directly by combining the different mode of SIFs without considering the influence of crack interaction. This numerical work is carried out to investigate whether the SIFs from different modes can be explicitly combined and compared to the ones using the FEA. Finally, the result discrepancies between the two methods are also discussed in term of the mesh deformation, which mainly focused on the crack face interaction. Evaluation of Fracture Parameters Stress intensity factors The finite element method is an appropriate approach to calculate the stress intensity factor (SIF) for linear elastic fracture mechanics problems. In order to determine the SIFs, a displacement extrapolation method 17 is used in this study. Several other works have implemented a similar method are also available 18,19. In order to analyse the cracks, it is frequently modelled as a semi-elliptical crack shape. This due to the fact, any arbitrary crack shapes will grow to take semi-elliptical crack geometry 5. Figure 1 shows an arbitrary crack shape where the crack face is parallel to the x-axis and the z-axis is normal to the

6 INDIAN J. ENG. MATER. SCI., FEBRUARY 01 Fig. 1 Arbitrary crack shape Fig. 3 Definition of contour path to evaluate the J-integral the SIFs obtained from the analysis are converted into normalized values in order to ensure the generality of the results. A normalized SIF, F, can be defined as follows 10 F I, a I, a = (4) σ π a a F I, b I, b = (5) σ π a b F II II = (6) xy τ π a x-y plane. Figure shows an arrangement of singular finite elements around a crack tip used in this work. After obtaining the elastic finite element solution of the particular problem, nodal displacements between two crack faces are determined and used to compute the SIFs as follows: I II Fig. Singular element around the crack tip G π vb ν d G π v = = 1+ κ r 1+ κ r G π ub ud G π u = = 1+ κ r 1+ κ r (1) () wb wd w III = G π = G π (3) r r where, I, II and III are the respective mode I, II and III SIFs, v, u and w are the relative nodal displacements between two crack faces in the direction of y-axis, x-axis and z-axis, respectively, and G is the modulus of rigidity. For plain strain condition, κ= 3-4υ, where, υ is the Poisson s ratio. All F III III = (7) xy τ π a where, σ a, σ b and τ xy are the axial, bending and shear stresses, respectively and a is a crack depth. J-integral The concepts of SIFs are successfully used as a driving fracture parameter within the scope of linear elastic analysis. However, the application of this parameter breakdown when large amount of plastic deformation induced during loadings especially for the high strength and low toughness materials. Therefore, J-integral is used instead of SIF as a driving fracture parameter. It is firstly introduced by assuming a crack in two-dimensional plate 0, J-integral is defined as a contour, Γ around the crack tip. It is evaluated counter-clockwise as depicted in Fig. 3 and can be expressed as u J = Wdy T. ds (8) z Γ where, T is a outward traction vector along the contour, Γ is defined as T i = σ ij n i or it is a force per

ISMAIL et al.: STRESS INTENSITY FACTOR FOR SURFACE CRACS 7 unit length, u is a displacement vector and ds is an element on the contour, Γ. While, W is a strain energy density expressed as ε T σij εij { σ} { ε} (9) W = d = d ε 0 0 whereas the axial force, F is directly applied to the direction-x on the cross-sectional area of the round bar. At the other end, the component is appropriately constrained. For combined loadings, two types of loading ratios are used, which are defined as: where, ε ij is a strain tensor and {ε} represents as a strain vector. In elastic-plastic analysis J-integral is composed of two parts, elastic J-integral, J e and plastic J-integral, J p as follows 1 J = J e + J p (10) where J e can be obtained numerically using finite element method or by the following expression J e I = (11) κ where, is the SIF, κ=e for plane stress and κ = E / (1-υ ) for plane strain. Finite Element Modelling The geometry of the crack shown in Fig. 4 can be described by the dimensionless parameters a/d and a/b, the so-called relative crack depth and crack aspect ratio, respectively, where D, a and b are the diameter of the bar, the crack depth and the major diameter of the ellipse. In this work, a/b ranged between 0.0 to 1. 6-1, while, a/d is in the range of 0.1 to 0.6 6-1 which are based on the experimental observations -5. Any arbitrary point, P on the crack front can also be normalised through the ratio of x/h, where h is the crack width, and x is the arbitrary distance of P. The outer diameter of the cylinder is 50 mm and the total length is 00 mm. A finite element model is developed using ANSYS 6, and a special attention is paid to the crack tip by employing 0-node iso-parametric quadratic brick elements. The square-root singularity of stresses and strains is modelled by shifting the mid-point nodes to the quarter-point locations around the cracktip region. A quarter finite element model is shown in Fig. 5. In order to remotely apply loadings on the bar, a rigid element or multi-point constraint (MPC) is used to connect the nodes at a circumferential line at the end of the component to an independent node. Figure 6 shows a technique of constructing the independent node connected to the model using a rigid beam element. The bending moment, M y and the torsion moment, T x are directly applied to this node, Fig. 4 Nomenclature of semi-elliptical surface crack Fig. 5 Symmetrical finite element model and the associated singular element around the crack tip Fig. 6 Remotely applied moments using MPC184 element

8 INDIAN J. ENG. MATER. SCI., FEBRUARY 01 ϑ σ σ b = (1) a τ xy λ = (13) σ x where ϑ is the loading ratio between the bending stress, σ b and the axial stress, σ a, and λ is the ratio between the shear stress, τ xy and the bending or axial stresses, σ x. The ratios for both Eqs (1) and (13) are 0.5, 1.0 and.0. In order to obtain a suitable finite element model, it is needed to compare the proposed model with available results in the literature 11,5,8. Figure 7 shows a comparison of the dimensionless SIFs under tension loading. Two crack aspect ratio, a/b are used for the validation purposes which are 0.0 and 1.0. It has been found that the findings of this study are in agreement with those determined by the previous models where the curves have coincident to each others. The solution of Mode III SIFs is difficult to obtain 10,1,14-16,9 and consequently compared with the present results. Therefore, it can be assumed that the present model is also suitable to analyse Mode III condition in a satisfactory way. Results and Discussion Stress intensity factors under loading Figures 8-10 respectively show F I,a, F II and F III along the crack fronts under pure tension stress and torsion moment for the selected crack conditions. The SIFs are calculated for six points along the crack front. However, the SIF at the intersection, between the crack and the surface, is not determined due to the square-root singularity problem. The uses of a quarter point finite element in that area do not generally produce reliable results. Many works have discussed such problems 6-16 where the nearest point is approximated 83% from the deepest crack point (x/h = 0.0). Figures 8a and 8b show the variations of F I,a along the crack front under the tension stress for two crack aspect ratios, a/b = 0.4 and 1.. As for a/b = 0.4, the SIF is uniformly distributed along the crack front. When x/h approached the outer surface of the bar, F I,a is found to be slightly higher than the others. It is clearly shown that the maximum F I,a always occurred at the intersection point area. Under tension stress, the crack growth started at the intersection point and the semi-elliptical crack front might be flattened as the cracks grew 5. For the case of a/b = 1., F I,a is not affected by the a/d, particularly when a/d 0.5. When F I,a reached x/h 0.6, F I,a is diverged to its individual values. Fig. 7 Validation of finite element model under bending loading Fig. 8 Behaviour of F I,a along the crack front (a) a/b = 0.4 and (b) a/b = 1.

ISMAIL et al.: STRESS INTENSITY FACTOR FOR SURFACE CRACS 9 By referring to F II in Fig. 9, it is found that at the deepest point (x/h = 0.0), F II = 0.0 and it is observed to steeply increase when x/h approached the outer surface. Figure 10 shows F III along the crack front subjected to torsion loading. The role of F III is strongly related to the relative crack depth, a/d. For a/d < 0.3, the maximum F III occurred at the deepest point of the crack front, and when a/d > 0.3, the maximum F III is shifted to the outer surface area. The movement of x/h is observed to move to the outer edge of the bar when a/b is increased implying different crack evolutions can be obtained during the crack growth and before the final failure. Tables 1 and show the F II and F III results obtained under pure torsion moments. Stress intensity factors under combined loadings In previous studies 14-16, it is hard to find the SIFs under combined loadings. This is because it is assumed that the combined SIFs can be obtained using a superposition technique 6. This assumption is established for a similar type of loading mode, for example mode I 16. The normalized SIFs under combined loading, F * EQ, under Mode I loadings are here obtained by the superposition method defined as F = F + ϑf (14) EQ I, a I, b where ϑ is the stress ratio defined in Eq. (1). Then, these combined SIFs are compared with the combined SIFs, F * FEA, obtained numerically using finite element analysis, with an excellent agreement as shown in Fig. 11. Further enhancement of the superposition technique is required to include F II and F III. Therefore, the equivalent SIF method 30 is used instead of a superposition method. The equivalent SIF is defined as the following expression III EQ = I + II + (15) 1 ν Fig. 9 Behaviour of F II along the crack front (a) a/b = 0.6 and (b) a/b = 1.0 Fig. 10 Behaviour of F III along the crack front (a) a/b = 0.4 and (b) a/b = 1.0

10 INDIAN J. ENG. MATER. SCI., FEBRUARY 01 x/h a/d Table 1 List of mode II normalized SIF, F II a/b 0.0 0. 0.4 0.6 0.8 1.0 1. 0.1 0.00 0.0016 0.0010 0.0014 0.001 0.001 0.0005 0. 0.0014 0.0013 0.0013 0.001 0.0011 0.0011 0.0010 0.00 0.3 0.0004 0.0054 0.0000 0.0001 0.0000 0.0008 0.0005 0.4 0.0018 0.0015 0.0015 0.0014 0.0013 0.001 0.0011 0.5 0.0031 0.007 0.007 0.007 0.006 0.004 0.001 0.6 0.0031 0.007 0.008 0.009 0.008 0.007 0.004 0.1 0.3580 0.3464 0.353 0.586 0.048 0.1675 0.141 0. 0.507 0.456 0.380 0.13 0.1798 0.159 0.131 0.17 0.3 0.175 0.053 0.051 0.1884 0.1663 0.1455 0.180 0.4 0.091 0.057 0.1967 0.1813 0.168 0.1448 0.188 0.5 0.179 0.140 0.041 0.1877 0.1693 0.1517 0.1353 0.6 0.431 0.415 0.89 0.317 0.1891 0.1684 0.1495 0.1 0.6450 0.677 0.583 0.483 0.3964 0.3340 0.874 0. 0.4943 0.4854 0.4674 0.4185 0.358 0.309 0.705 0.33 0.3 0.4389 0.4145 0.4161 0.3816 0.337 0.967 0.67 0.4 0.451 0.5001 0.4035 0.4890 0.3333 0.967 0.645 0.5 0.449 0.4365 0.417 0.3846 0.347 0.3110 0.773 0.6 0.494 0.4906 0.466 0.480 0.3868 0.3446 0.3057 0.1 0.8540 0.8369 0.7706 0.6707 0.5740 0.5018 0.4445 0. 0.733 0.7 0.6913 0.638 0.541 0.4760 0.443 0.50 0.3 0.6758 0.6388 0.6470 0.597 0.553 0.4656 0.4161 0.4 0.6646 0.6585 0.6400 0.5876 0.589 0.4716 0.418 0.5 0.6940 0.6860 0.6589 0.613 0.5546 0.497 0.4437 0.6 0.771 0.7684 0.7343 0.6778 0.6149 0.5494 0.4880 0.1 1.0193 1.0076 0.978 0.853 0.7568 0.6834 0.630 0. 0.9851 0.975 0.8993 0.8610 0.7555 0.6745 0.6094 0.67 0.3 0.9574 0.91 0.9361 0.9760 0.7676 0.687 0.6145 0.4 0.9637 0.9560 0.9533 0.8788 0.796 0.7104 0.6356 0.5 1.0139 1.0013 0.9780 0.953 0.8481 0.763 0.6787 0.6 1.150 1.1174 1.0857 1.016 0.9384 0.8466 0.755 0.1 1.440 1.54 1.1670 1.1407 1.044 0.9415 0.8760 0. 1.3674 1.3563 1.3344 1.754 1.144 1.0054 0.9178 0.83 0.3 1.4197 1.3801 1.4477 1.3648 1.63 1.0865 1.1306 0.4 1.468 1.461 1.5366 1.4366 1.331 1.1906 1.0540 0.5 1.5488 1.540 1.5385 1.517 1.4476 1.31 1.167 0.6 1.7060 1.6959 1.6778 1.6438 1.5781 1.4808 1.35 where, EQ is the equivalent SIF and ν is the Poisson s ratio. It is assumed that EQ = * where * is a combined SIF. Substituting Eqs (4) or (5), (6) and (7) into Eq. (15) yields the following expression FIIIτ xy π a = ( I, xσ x π ) + ( IIτ xy π ) + 1 ν * F a F a (16) where σ x can be represented as axial or bending stresses and F I,x can also be represented as the normalized SIF under bending or tension stress, respectively. Substituting Eq. (13) into Eq. (16), we obtain the following expression λfiii = ( σ x π ) ( I, x ) + ( λ II ) + 1 ν * a F F (17)

ISMAIL et al.: STRESS INTENSITY FACTOR FOR SURFACE CRACS 11 x/h a/d Table List of mode III normalized SIF, F III a/b 0.0 0. 0.4 0.6 0.8 1.0 1. 0.1 0.8093 0.7760 0.7088 0.6377 0.5338 0.4615 0.405 0. 0.7196 0.6964 0.6380 0.5557 0.4700 0.3964 0.3379 0.00 0.3 0.667 0.591 0.5918 0.5098 0.419 0.3397 0.779 0.4 0.6435 0.643 0.5679 0.480 0.384 0.90 0.34 0.5 0.6533 0.636 0.5711 0.4768 0.369 0.556 0.1745 0.6 0.7118 0.6884 0.6177 0.5076 0.3707 0.354 0.130 0.1 0.7641 0.7331 0.6671 0.599 0.5180 0.456 0.3987 0. 0.7046 0.6819 0.630 0.543 0.4610 0.3910 0.3361 0.17 0.3 0.663 0.5870 0.5873 0.5053 0.4154 0.3373 0.779 0.4 0.6447 0.655 0.5698 0.4831 0.3830 0.96 0.53 0.5 0.6597 0.6389 0.5784 0.4831 0.368 0.598 0.1789 0.6 0.745 0.709 0.6315 0.503 0.3818 0.449 0.140 0.1 0.6554 0.6307 0.5696 0.581 0.4760 0.473 0.3861 0. 0.6648 0.6449 0.5850 0.5115 0.4377 0.3767 0.3307 0.33 0.3 0.6499 0.575 0.5779 0.4955 0.4068 0.334 0.787 0.4 0.6499 0.6314 0.578 0.4890 0.387 0.966 0.3 0.5 0.680 0.6596 0.5998 0.5037 0.3856 0.740 0.197 0.6 0.7635 0.7415 0.6705 0.5578 0.415 0.78 0.1638 0.1 0.574 0.5114 0.4605 0.4484 0.4179 0.3881 0.368 0. 0.609 0.5936 0.5360 0.4714 0.4063 0.3557 0.310 0.50 0.3 0.6350 0.569 0.5699 0.4871 0.3988 0.377 0.807 0.4 0.6635 0.6463 0.599 0.5061 0.4005 0.3077 0.45 0.5 0.703 0.7001 0.640 0.545 0.417 0.306 0.180 0.6 0.8357 0.819 0.748 0.677 0.4777 0.343 0.054 0.1 0.4070 0.407 0.3699 0.383 0.3653 0.3484 0.3358 0. 0.5495 0.5400 0.4945 0.4458 0.3869 0.344 0.3150 0.67 0.3 0.683 0.5665 0.5810 0.5681 0.418 0.3399 0.944 0.4 0.7009 0.6863 0.6563 0.5593 0.448 0.3460 0.784 0.5 0.8014 0.7850 0.7330 0.6378 0.5063 0.3703 0.78 0.6 0.971 0.9540 0.8858 0.767 0.606 0.497 0.873 0.1 0.3148 0.336 0.394 0.3549 0.3304 0.3153 0.3076 0. 0.50 0.514 0.5100 0.4866 0.4180 0.3651 0.3347 0.83 0.3 0.6779 0.6464 0.6780 0.6145 0.516 0.4184 0.3586 0.4 0.839 0.8178 0.8396 0.747 0.601 0.4851 0.3883 0.5 0.9976 0.9938 0.968 0.8903 0.7541 0.5777 0.4335 0.6 1.601 1.537 1.1959 1.095 0.9333 0.7181 0.5130 Rearranging Eq. (17) in terms of combined dimensionless SIF, F * is given by λf III ( I, x ) ( λ II ) F = = F + F + σ ( 1 x π a ν ) (18) Eq. (18) can be divided into two separate equations given as F FE * = = F (19) I, x III, FE σ π a x λf F = ( F ) + ( λf ) + = F ( 1 ν ) III * I, x II I, x III, EQ (0)

1 INDIAN J. ENG. MATER. SCI., FEBRUARY 01 Fig. 11 Comparisons of combined mode I SIF, F * I (a) a/b = 0. and (b) a/b = 0.6 where F * I,x-III,FE is the normalised SIF obtained directly from finite element analysis under combined loadings, and F * I,x-III,EQ is the normalised SIF obtained explicitly by combining the individual SIFs F I,b, F I,a, F II and F III. In order to simplify the analysis, the work focused on this location at x/h = 0.0 where F II = 0.0. Therefore, Eq. (0) can be then reduced to the following expression * λf III I, x III, EQ =, + F ( FI x ) ( 1 ν ) (1) Figure 1 shows the SIFs obtained under single loading conditions at x/h = 0.0. It is indicated that, all the SIFs have decreased when a/b is increased and no significant effect on the SIFs for the relatively straight-fronted cracks (a/b 0.). For mode I conditions as shown in Fig. 11, the SIFs showed the increasing trends as a/d increased. Figure 1b on the other hand showed the SIFs under torsion moment obtained at x/h = 0.0. It is shown that lower values of F III occurred when a/b is increased. This occurrence Fig. 1 Behaviour of normalized SIF at different a/d (a) F I and (b) F III may be caused by the influence of crack geometries where the cracks become deeper are obtained using a/b = 1.. These type of cracks attained fully constraint mechanisms around the crack front and producing lower F III at x/h = 0.0. It is hard to obtain a single value of SIF directly from ANSYS. Therefore, an elastic J-integral is used by assuming that a single value of J-integral under combined loading represented the unified SIFs consisting of I, II and III. This is because in ANSYS, if J-integral is used in the elastic or plastic regions, it calculates only a single value of J-integral even under combined loadings. The elastic J-integral, J e, is as in Eq. (11) and it is rearranged in the terms of and assuming that = * FE, where * FE is a SIF under combined loadings obtained using finite element analysis, yields the following expression * FE E 1 ν = Je ()

ISMAIL et al.: STRESS INTENSITY FACTOR FOR SURFACE CRACS 13 Fig. 14 Deformed meshes under torsion moment (a) whole model, (b) enlarged area around the crack tip and (c) arial view of surface crack. Fig. 13 Comparisons of normalized SIF, F * I,a-III under combined loadings using two approaches, (a) a/b = 0., (b) a/b = 0.6 and (c) a/b = 1.0 Eq. () is used to convert the J-integral into the combined SIF for plain strain condition and it is substituted into Eq. (19). Then, F I,a are combined explicitly with F III through Eq. (1) using different stress ratio values, λ. The calculated F * EQ and F * FE are compared and the results are presented in Fig. 13. Figure 13 shows the plot of F * I,a-III against a/d for different a/b subjected to combined tension and torsion loadings. It is indicated that the loading ratio, γ played an important role in determining the discrepancies among the results. Reducing these ratios from.0 to 1.0, the Fig. 15 Deformed meshes under tension moment (a) whole model, (b) enlarged area around the crack tip and (c) arial view of surface crack results are almost agreed to each other. The detail of F * I,a-III,EQ and F * I,a-III,FE are given in Tables 3 and 4. The discrepancies of combined SIFs, F * between the two approaches remarkably depend on the values of a/b, a/d and λ. For all cases of a/b, F * EQ is considerably in agreement with F * FE for all values of loading ratios except for λ.0. When a/b increases, the discrepancies between the results obtained from the two different approaches are tremendously reduced, and all the F * values then converge when deeper cracks are used. Crack Deformation Mechanisms Figures 14-16 present the stress distribution around the tip that is situated at the outer surface. Meanwhile,

14 INDIAN J. ENG. MATER. SCI., FEBRUARY 01 Table 3 List of F * I,a-III,EQ using an equivalent SIF method χ a/d a/b 0.0 0. 0.4 0.6 0.8 1.0 1. 0.1 1.0830 1.0436 0.9686 0.8718 0.7545 0.6704 0.6006 0. 1.0793 1.1548 1.0791 0.9678 0.8487 0.7301 0.6451 0.5 0.3 1.0579 1.948 1.888 1.145 0.9788 0.860 0.7110 0.4 1.036 1.7893 1.6606 1.453 1.100 0.985 0.8133 0.5 1.00.5116.3188.010 1.659 1.563 0.9741 0.6 1.0866 3.8779 3.5808 3.0743.4180 1.7558 1.408 0.1 1.81 1.18 1.0934 0.9840 0.8457 0.7473 0.6661 0. 1.099 1.577 1.1717 1.0463 0.919 0.7833 0.6889 1.0 0.3 1.1573 1.3739 1.3566 1.1993 1.038 0.8611 0.7383 0.4 1.0994 1.844 1.7096 1.4936 1.406 1.0071 0.889 0.5 1.067.551.3546.0389 1.6465 1.695 0.980 0.6 1.1097 3.9091 3.6080 3.0957.435 1.7639 1.444 0.1 1.6883 1.6 1.4916 1.34 1.1398 0.9975 0.8807 0. 1.6311 1.6049 1.4856 1.3144 1.1341 0.967 0.8418.0 0.3 1.4898 1.6530 1.5994 1.4039 1.1867 0.9888 0.8386 0.4 1.3336.0493 1.8931 1.645 1.3560 1.0904 0.8884 0.5 1.119.7080.495.1501 1.765 1.31 1.0133 0.6 1.1977 4.0314 3.7147 3.1798.4897 1.7958 1.586 Table 4 List of F * I,a-III,FE using FEA χ a/d a/b 0.0 0. 0.4 0.6 0.8 1.0 1. 0.1 1.089 1.0495 0.9797 0.8776 0.8417 0.7505 0.6745 0. 1.1881 1.1569 1.0810 1.0188 0.8911 0.7773 0.6900 0.5 0.3 1.407 1.3850 1.793 1.1798 1.0091 0.8558 0.739 0.4 1.8433 1.7904 1.6609 1.4531 1.111 0.9886 0.819 0.5.5776.5106.370.019 1.6336 1.658 0.985 0.6 3.9949 3.8905 3.5969 3.0944.4471 1.7796 1.644 0.1 1.1965 1.1487 1.0789 0.969 1.0701 0.9446 0.8477 0. 1.619 1.80 1.1616 1.364 1.073 0.973 0.8316 1.0 0.3 1.4764 1.4408 1.3449 1.350 1.1495 0.9691 0.839 0.4 1.8837 1.890 1.7158 1.4998 1.51 1.0193 0.846 0.5.6113.5410.3736.0547 1.6681 1.917 1.0041 0.6 4.0195 3.9134 3.641 3.1355.481 1.8071 1.83 0.1 1.5308 1.4834 1.378 1.613 1.6854 1.4841 1.3187 0. 1.580 1.4867 1.3995 1.6406 1.5857 1.3591 1.1999.0 0.3 1.6859 1.6387 1.594 1.6534 1.5615 1.3081 1.1166 0.4.0441 1.9857 1.8557 1.6178 1.5404 1.90 0.9967 0.5.730.6588.4799.146 1.736 1.3409 1.0397 0.6 4.115 4.0055 3.741 3.00.5307 1.8349 1.3056

ISMAIL et al.: STRESS INTENSITY FACTOR FOR SURFACE CRACS 15 Fig. 16 Deformed meshes under combined loadings (a) whole model, (b) enlarged area around the crack tip and (c) arial view of surface crack Figs 14 and 15 included the deformed meshes of the cracks subjected to pure torsion moment and tension loading, respectively. The F I,a and F III as seen in Figs 1a and 1b are explicitly combined together and then compared with the F *, FE that is directly retrieved from the FEA. The situation in Fig. 16 is then produced using the FEA subjected to a combined loading. It is clearly shown that under the pure torsion loading, the crack faces are completely closed due to the absence of Mode I loading. Under the combined loadings, opening crack faces are observed even the bar is subjected to torsion loading, implicating that the mechanism was responsible to produce the discrepancies among the obtained results using the two distinct methods. In FEA, the SIFs are calculated by referring to the relative distance between the two nodes situated on the crack faces. It is important to note that under the combined loadings, F * FE is found to be greater than F * EQ, due to the fact that longer relative node distances are produced when tension stress involved. Therefore, ANSYS has calculated greater F III compared to the F III obtained under completely closed crack faces under pure torsion loading. Conclusions Finite element analyses (FEA) are performed for semi-elliptical surface cracks in round bars under combined tension and torsion loadings. No available solutions are carried out to calculate the normalised SIFs, particularly the SIF under combined loadings. It is assumed that the SIFs can be explicitly combined together without considering the crack face interactions. Based on the findings of this study, the direct SIF combinations are rather questionable and inappropriate when different failure modes were involved. Meanwhile, the discrepancies of the results between the explicitly combined SIFs and the SIFs obtained using FEA are due to the different mechanisms of crack face interactions shown by the deformed meshes, in which the crack faces are closed under the pure torsion, and vice versa under the combined loadings. The opening crack faces under the combined loadings increased the relative node distances. ANSYS then used these relative distances to calculate the SIFs. As a result, higher F * FE is obtained, relative to the F * EQ due to the different mechanisms of crack deformations. References 1 Li Y D, Zhang H C & Lee Y, Indian J Eng Mater Sci, 16 (009) 95-300. Ismail A E, Ariffin A, Abdullah S & Ghazali M J, Int J Automot Tech, 1() (011) 5-3. 3 Mahmoud M, Theo Appl Frac Mech, 48 (007) 15-160. 4 Gray G T, Thompson A W & William J C, Metall Trans, 16A (1985) 753-760. 5 Lin X B & Smith R A, Int J Fatigue, 19 (1997) 461-469. 6 Raju I S & Newman J C, Fract Mech, 17 (1986) 89-805. 7 Shiratori M, T Miyoshi, Sakay Y & Zhang G R, Analysis and application of influence coefficients for round bar with a semi-elliptical surface crack (Oxford Pergamon Press), 1987. 8 Murakami Y & Tsuru H, Stress intensity factor equations for semi-elliptical surface crack in a shaft under bending (Oxford Pergamon Press), 1987. 9 Carpinteri A, Fatigue Fract Eng Mater Struct, 15 (199) 1141-1153. 10 Fonte M D & Freitas M D, Int J Fatigue, 1 (1999) 457-463. 11 Carpinteri A, Brighenti R & S, Vantadori, Int J Fatigue, 8 (006) 51-60. 1 Shahani A R & Habibi S E, Int J Fatigue, 9(1) (007) 8 140. 13 Toribio J, Matos J C, Gonzalez B & Escuadra J, Eng Failure Anal, 16(6) (009) 18-630. 14 Ismail A E, Ariffin A, Abdullah S & Ghazali M J, Int Rev Mech Eng, 4(7) (010) 87-83. 15 Ismail A E, Ariffin A, Abdullah S & Ghazali M J, Meccanica (in-press) 16 Ismail A E, Ariffin A, Abdullah S, Ghazali M J Daud R & AbdulRazzaq M, J Zhejiang Univ Sci A, 13(1) (01) 1-8. 17 Guinea GV, Planas J & Elices M, Eng Fract Mech, 66(3) (000) 43-55. 18 Aslantas A, Int J Solids Struct, 40(6) (003) 7475-7481. 19 Aslantas, Ergun E & Tasgetiren S, J Mech Mater Des, 3() (006) 01-08. 0 Rice J R, Appl Mech, 35 (1968) 379-386. 1 im Y J, im J S, Shim D J & im Y J, J Strain Anal, 39 (004) 45-60.

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