Fracture Analysis of Three Dimensional Elliptical Crack for Al7075-T651 Plate with Holes

Indian Journal of Science and Technology, Vol 9(34), DOI: 10.17485/ijst/2016/v9i34/100846, September 2016 ISSN (Print) : 0974-6846 ISSN (Online) : 0974-5645 Fracture Analysis of Three Dimensional Elliptical Crack for Al7075-T651 Plate with Holes A. Sivasubramanian and G. Arunkumar Department of Mechanical Engineering, Saveetha School of Engineering, Saveetha University, No.162, Poonamalle High Road, Velappanchavadi, Chennai - 600077, Tamil Nadu, India; sivnay123@yahoo.co.in, garun55@gmail.com Abstract Objectives: The paper deals with the study on analysis of Elliptical Crack for Al7075-T651 for three Dimensional Elliptical crack emanating from hole of the plate. Methods/Statistical Analysis: The impact of Crack shape and location with respect to various tensile loads on stress intensity factor is analysed. The crack in two holes of the plate is differentiated into different cases and they are analyzed for stress intensity factor. The crack case is formed with respect to the location of the through and part through crack in the model. Findings: The stress intensity factor of the crack geometry is attained by Ansys and the results are compared with Polynomial equation derived by displacement Extrapolation method. The paper also analyses the stress intensity factor along with parametric angle and Beta function of through elliptical crack model for various loads. Application/Improvements: The Stress Intensity factor value obtained can be used in finding the fatigue behaviour of materials under various loading conditions and improvements can be made by further improvement in the geometry of the plate. Keywords: Al 7075-T651, Beta Function, Displacement Extrapolation, Elliptical Crack, Fatigue, Stress Intensity Factor 1. Introduction The growth of Crack in engineering structures is a major concern towards the failure of the structures. The practical application of those structures decides the importance of the study of crack. The type of crack and proximity of the crack also plays a vital role in field of crack analysis 1. Stress intensity factor assessment helps in finding the fracture strength or fatigue behaviour materials. Lot of studies and methods have been employed to study the effect of holes with crack 2. There were also studies been done based on shape of the crack. The need for such study is vital in field of aerospace to overcome history of sudden failure in aircraft causing major loss in terms of life 3. The sudden failure of the components or structure risks the life of many people travelling in the aircraft and billions of dollars poured to make the aircraft. The plate modelled in the study is made of Al7075-T651 which is structured plates specifically used in aircraft 4.Materials in Aerospace are subjected to Continuous loading 5. The plate has hole made of rivets and cracks are modelled on the plate for studying the stress intensity factor under various loading conditions. The analysis of stress intensity factor helps to define the stress field around the crack and its point of failure 6. It is obvious that the shape of crack and proximity of the crack with respect to the hole makes the analysis more vibrant and useful 2. The material used for analysis is Aluminium alloy Al7075-T651 usually known as Aircraft aluminium. It is tempered aluminium alloy which has wide application in field of aerospace engineering 7. The nonlinear property of material is taken to consideration for FE analysis 8. 2. Plate Model 1 The plate is modelled using Ansys with two holes. The plate dimensions are based on the following relationship 1. r/t = 0.075, 0.1, 0.2, 0.333, 0.5, 1.0, 2.0, 3.0, 6.0 L = 1D, 2D, 3D, 4D * Author for correspondence

Fracture Analysis of Three Dimensional Elliptical Crack for Al7075-T651 Plate with Holes D 1 /D 2 = 0.5, 1.0, 2.0 B = 1D, 2D, 3D, 4D The plate has portion to clamp on either side so as to apply tensile load.the plate model front view and section view is shown in figure 1& 2.The dimension of the plate is shown in figure 3 & 4 and values are shown in table 1. Figure 4. Plate Dimension (Section Along Center). 3. Crack Model Three dimensional Elliptical crack model is used for analysis of the plate. The cracks which are elliptical in shape are called as elliptical crack. The elliptical crack taken in this analysis is quarter elliptical type crack shown in figure 5. The quarter elliptical type crack takes the shape of one fourth of the ellipse divided at the centre along the X and Y axis. Figure 1. Plate Model (Front View). Figure 2. Plate Model (Section Along the Center). Table 1. Dimension of the Plate t 6.350 mm r 0.5t 3.175 mm D 1 1 D 2 6.350 mm L 4D 25.40 mm B 4D 25.40 mm Figure 5. Quarter Ellipse with Dimension Representation. There are of two types namely elliptical through crack and elliptical part through crack 9. When the dimension of the crack depth (a) is greater than thickness of the plate (t), then it is called as elliptical through crack. When the dimension of the crack depth (a) is lesser than the thickness of the plate (t), then it is called as elliptical part through crack. Here in our case, the elliptical crack emanates from the centre of the hole parallel to Z axis of the plate. The crack model is shown in figure 6 & 7. Figure 3. Plate Dimension (Front View). 2 Vol 9 (34) September 2016 www.indjst.org Indian Journal of Science and Technology

A. Sivasubramanian and G. Arunkumar Figure 6. Crack Model along the Center. Figure 9. Elliptical Crack Location on the Plate (Sectional Along Center). The crack case is formed with respect to the location of the through crack in the model. The crack I, IV, V and VIII corresponds to the through elliptical crack. 5. Through Elliptical Crack Figure 7. Crack Model with Dimension Representation. The elliptical crack is modelled with respect to the dimension in Ansys. The crack is introduced into the model using sub-model method. The crack is shown in figure 8. Figure 8. Part through Elliptical Crack. 4. Through Elliptical Crack Analysis The elliptical crack is defined by crack length (c) and crack depth (a). The crack length is the along the XY plane of the model and the crack depth is defined in the XZ plane of the model.the elliptical crack location along the centre is shown in figure 9. The crack in the two holes is differentiated into different cases and they are analyzed for the stress intensity factor. When the dimension of the crack depth (a) is greater than thickness of the plate, then it is called as elliptical through crack. The dimension of the crack is determined by the ratio of a/c and a/t. Based on the dimension of the thickness (t), the crack length(c) and crack depth (a) is determined. The dimension and location of crack is shown in Table 2 and Figure 10. a/c = 0.1, 0.125, 0.1667, 0.25, 0.5, 1.0, 2.0, 4.0, 6.0, 8.0, 10.0 a/t = 1.05, 1.5, 2.0, 3.0, 4.0, 5.0, 10.0 5.1 Dimensions 1. Thickness of the plate = 6.35mm. 2. Diameter of Hole1 and Hole2 = 6.35mm. Table 2. Dimension of through Elliptical Crack a/t 1.05 a/c 0.5 Thickness t mm 6.35 Crack Depth a mm 6.6675 Crack Length c mm 13.335 The ANSYS software cannot model the ellipse directly. The Ellipse has to modelled using keypoints and spline to obtain a smooth curve. The ellipse is constructed using AUTOCAD and the quarter elliptical curve is divided in numerous keypoints. The ellipse is constructed in Global co-ordinate system and the XYZ co-ordinates corresponding to the quarter elliptical crack is obtained for individual keypoints. The keypoints are transferred to Vol 9 (34) September 2016 www.indjst.org Indian Journal of Science and Technology 3

Fracture Analysis of Three Dimensional Elliptical Crack for Al7075-T651 Plate with Holes the ANSYS software and using the BSPLINE command, a smooth curve is formed. This curve is later used for the modelling of the three dimensional crack model. The analysis is carried out by forming a model with the crack dimensions. The load is applied to the model into different load step and the results are obtained. The model is analyzed for different tensile load cases and the results are compared with Newman Raju s calculations for finding Stress Intensity Factor.The load cases are shown in table 3. The stress intensity factor for the elliptical crack is analyzed and the beta polynomial equation is obtained through curve fitting 10. Table 3. Load Cases LOAD CASE TENSILE LOAD (MPa) 1.1 25 1.2 50 1.3 75 1.4 100 1.5 125 1.6 150 6. Fracture Analysis for Tensile Loads 25MPA,50MPA,75MPA,100MPA The value of Beta function for various loads are found and the polynomial equations are formed and results of polynomial equation is compared with ansys results. The initial gragh is drwan for parametric angle and beta function and then values of parametric function is substitued in polynomial function to find the stress intensity factor and it is compared with ansys results. K = bs pc (1) The above equation is used for finding the beta function by knowing the values of K,and c.stress Intensity factors are considered as paramount for calculating the material behaviour in crack analysis 11. The analysis for variuos tensile load is found and tabulated and values of Beta function is comapred with parametric angle 2φ/π. The table 4 shows the beta function for load 25 Mpa. After finding the values and graphs are drawn for forming polynomial equation.the polynomial equation is used to find the stress intensity factor and that stress intensity factor is compared with Ansys value for checking its correctness.compartive study of Paramteric angle vs. beta function various cases shown in figure 10, figure 11, Figure 12.The comparision for Beta Function for load 50,75 Mpa is tabulated in Table 5 and Table 6. Table 4. Beta Function for Load 25mpa CASE1.1 K 2φ/π Beta β 122.11 0 1.067 77.434 0.0665 0.677 77.412 0.1309 0.677 72.667 0.1923 0.635 76.699 0.2493 0.670 75.913 0.3036 0.664 79.645 0.3548 0.696 80.426 0.4024 0.703 86.551 0.4479 0.757 88.229 0.4916 0.771 94.718 0.5333 0.828 99.249 0.5735 0.868 108.36 0.6127 0.947 118.53 0.6506 1.036 Figure 10. Parametric Angle vs. Beta Function for Case 1.1. 4 Vol 9 (34) September 2016 www.indjst.org Indian Journal of Science and Technology

A. Sivasubramanian and G. Arunkumar Table 5. Beta Function for Load 50mpa CASE1.2 K 2φ/π Beta β 972.42 0 1.406 713.09 0.0665 0.961 655.69 0.1309 0.942 592.05 0.1923 0.873 621.57 0.2493 0.909 617.38 0.3036 0.902 650.72 0.3548 0.940 657.66 0.4024 0.952 715.25 0.4479 1.022 736.37 0.4916 1.044 816.2 0.5333 1.122 878.01 0.5735 1.179 988.76 0.6127 1.286 1155.2 0.6506 1.408 Table 6. Beta Function for Load 75mpa CASE1.3 K 2φ/π Beta β 591.34 0 1.723 411.59 0.0665 1.199 390.33 0.1309 1.137 356.96 0.1923 1.040 371.63 0.2493 1.083 368.91 0.3036 1.075 387.08 0.3548 1.128 390.79 0.4024 1.139 425.3 0.4479 1.239 434.32 0.4916 1.266 477.71 0.5333 1.392 508.9 0.5735 1.483 570.28 0.6127 1.662 647.62 0.6506 1.887 Figure 11. Parametric Angle vs. Beta Function for Case 1.2. Figure 12. Parametric Angle vs. Beta Function for Case 1.2. Vol 9 (34) September 2016 www.indjst.org Indian Journal of Science and Technology 5

Fracture Analysis of Three Dimensional Elliptical Crack for Al7075-T651 Plate with Holes The various load cases are taken into consideration and the results are studied for the behaviour of Stress intensity factor and BETA function to derive polynomial Equations using Displacement Extrapolation method 12 so as to come with a positive agreement of results obtained from ANSYS and SIF results.the parametric angle vs. Beta Function for cases are shown in figure 13 and Table 4 shows beta function for load 25 mpa. The Polynomial Equations for various load conditions are tabulated (Table 7). Table 7. Polynomial Equation for Load Cases Load Polynomial Equation 25Mpa y = 2.262x 3-0.501x 2 0.107x + 0.684 50Mpa y = 2.474x 3 + 0.266x 2 0.613x + 0.998 75Mpa y = 5.352x 3 0.826x 2 0.787x + 1.238 100Mpa y = 7.855x 3 1.176x 2 1.263x + 1.611 6.1 Comparison of Beta Values for Load Cases (Ansys Results) The stress intensity factor values obtained using ANSYS is used to derive the Beta function. The beta values of different load case are plotted against 2φ/π. The curves of all the load cases are similar and variation in the value of the load cases can be seen in the Figure 14. As the load increase the variation in the beta value also increases corresponding to previous load case. The change in the value of beta is not linear for different load case are shown in table 8. Table 8. Beta Values for Load Cases from Ansys 2φ/π CASE1.1 CASE1.2 CASE1.3 CASE 1.4 0 1.067 1.406 1.723 2.125 0.06647 0.677 0.961 1.199 1.558 0.1309 0.677 0.942 1.137 1.433 0.19234 0.635 0.873 1.040 1.294 0.24933 0.670 0.909 1.083 1.358 0.30361 0.664 0.902 1.075 1.349 0.35478 0.696 0.940 1.128 1.422 0.40241 0.703 0.952 1.139 1.437 0.44789 0.757 1.022 1.239 1.563 0.49157 0.771 1.044 1.266 1.609 0.53331 0.828 1.122 1.392 1.784 0.57352 0.868 1.179 1.483 1.919 0.61265 0.947 1.286 1.662 2.161 0.65061 1.036 1.408 1.887 2.524 6.2 Comparison of Beta Values for Load Cases (Polynomial Equation) The 2φ/π values are substituted in the polynomial equation and the beta values are obtained. The polynomial equation corresponding to different load cases are plotted against 2φ/π. The curve shown in Figure 14 is smoother when compared to the Figure 15. The profile of the curve is similar for all the cases and the increase in the load causes increase in the beta value. The change in the beta shown in table 9 value is non linear for different load cases. The beta values at the ends increases rapidly compared to the beta values at the center. Figure 13. Parametric Angle vs. Beta Function for Case 1.4. 6 Vol 9 (34) September 2016 www.indjst.org Indian Journal of Science and Technology

A. Sivasubramanian and G. Arunkumar Figure 14. Comparison of Beta Function for Load Case (Ansys Results). The graph is drawn to find the percentage of deviation in results.the comparision of the values helps in studying the dependability of results obtaioned in various forms. Table 9. Beta Values for Load Cases From Polynomial Equation 2φ/π CASE1.1 CASE1.2 CASE1.3 CASE1.4 0 0.684 0.998 1.239 1.611 0.0665 0.675 0.959 1.184 1.525 0.1309 0.667 0.928 1.133 1.444 0.1923 0.661 0.907 1.095 1.381 0.2493 0.661 0.900 1.074 1.345 0.3036 0.669 0.906 1.073 1.339 0.3548 0.684 0.924 1.094 1.366 0.4024 0.707 0.956 1.137 1.424 0.4479 0.739 0.999 1.201 1.515 0.4916 0.779 1.055 1.288 1.639 0.5333 0.827 1.122 1.396 1.795 0.5735 0.884 1.201 1.525 1.982 0.6127 0.950 1.291 1.677 2.202 0.6506 1.025 1.393 1.851 2.455 7. Conclusion The values of stress intensity factor vs 2φ/π for different load cases shows the stress intensity factor values are similar for all the load cases.the change in values of SIF with respect to BETA function for various load cases shows that the deviation in result is around 1% of the values obtained through ANSYS results which clearly indicates that the results are dependable anddeviation of 1% is negligible.similarly the values of BETA with 2φ/π also shows a limited deviation of results.the crack propagation is highly defined by the load applied and Makes clear scenerio that shape and size of crack also considered for Fracture behaviour.the Elliptical crack has higher impact on material crack propagtion along with the position of crack from the hole.the higher degree of non linearity is seen existing in the results of stress intensity factor along the crack tip of the elliptical curve studied. The crack geometry influence the fracture initiation and proximity of the crack with respect to the hole plays a Figure 15. Comparison of Beta Function for Load Case (Polynomial Equation). Vol 9 (34) September 2016 www.indjst.org Indian Journal of Science and Technology 7

Fracture Analysis of Three Dimensional Elliptical Crack for Al7075-T651 Plate with Holes major role in increase in Stress intensity factor.the study clearly showws that the closer the crack higher the Stress Intensity factor which makes the propagation crack faster even at a small increase in load applied. 8. Refrences 1. Arunkumar SG. Analysis of Stress Intensity Factor of plate with crack holes-a constant Load Method. Australian Journal of Basic and Applied Sciences. 2013; 7(14):281-91. 2. Sivasubramanian A, Arunkumar G. Analysis of Stress Intensity factor of Al7075-T651 Plate with Cracked hole-a constant crack Length Method. ARPN Journal of Engineering and Applied Sciences. 2015; 10:10627-33. 3. Arunkumara PC, Dinesh P. Study of Fatigue crack growth in heat Treated Aluminium 2024. International Journal of Innovative Research in Science. Engineering and Technology. 2014; 3(2):9782-89. 4. Pedersen KO. Fracture mechanisms of aluminium alloy AA7075-T651 under various loading conditions. Materials & Design. 2011; 32:97-107. 5. Schijve J. Comparison between empirical and calculated stress intensity factors of hole edge cracks. Engineering Fracture Mechanics. 1985; 22:49-58. 6. Lin XB, Smith RA. Stress intensity factors for corner cracks emanating from fastener holes under tension. Engineering Fracture Mechanics. 1999; 62:535-53. 7. Lin XB, Smith RA. Fatigue shape analysis for corner cracks at fastener holes. Engineering Fracture Mechanics. 1998; 59:73-87. 8. Arivukkarasan S, Dhanalakshmi V, Babu AS, Aruna M. Performance Study on Fatigue Behaviour in Aluminium Alloy and Alumina Silicate Particulate Composites. Journal of Applied Science and Engineering.2013; 16:127134. 9. Chen LS, Kuang JH. A Displacement Extrapolation Method for Determining the Stress Intensity Factors Along Flaw Border. International Journal of Fracture. 1992; 57:R51-R58. 10. Fawaz SA. Stress intensity factor solutions for part-elliptical through cracks. Engineering Fracture Mechanics. 1999; 63:209-26. 11. Fawaz SA, Andersson B, Newman JC. Experimental Verification of Stress Intensity Factor Solutions for Corner Cracks at A Hole Subject to General Loading. Fatigue of Aeronautical Structures as an Engineering Challenge, ICAF. 2003; p.1-31. 12. Anand Babu K, Kumar GV, Venkataramaiah P. Prediction of Surface Roughness in Drilling of Al 7075/10% - SiCp Com posite under MQL Condition using Fuzzy Logic. Indian Journal of Science and Technology. 2015 June; 8(12):1-5. 8 Vol 9 (34) September 2016 www.indjst.org Indian Journal of Science and Technology