Issue 29

S.K. Kudari et alii, Frattura ed Integrità Strutturale, 29 (2014) 419-425; DOI: 10.3221/IGF-ESIS.29.37 423 of K I is in agreement with the result presented by Kwon and Sun [5]. Based on their FE results, Kwon and Sun [5] have proposed a formulation to obtain the highest magnitude of 3D K I (at the centre of the specimen) based on magnitude of K I obtained by 2D analysis as: 2 2 3 1 1    D D K K (4) where  is Poisson’s ratio of the material. The highest magnitude of the K I (at the center of the specimen) extracted by the present 3D FE analysis for various a/W ratios is plotted in Fig. 8 along with the 3D K I values estimated by the formulation, Eq. (4) using 2D results shown in Fig. 3. The 2D computation of K I were made in order to verify the possible use of the formula (Eq. (4)) proposed by Kwon and Sun [5]. The Fig.8 shows that there is an acceptable agreement of 3D K I at the center of the specimen and the computations made by the formulation given by Kwon and Sun [5] using present 2D FE results. As magnitude of K I is found to be maximum at the centre of the specimen, it becomes important to do the fracture analysis of material based on the value of K I estimated at the center of the specimen. This demands 3D FE analysis of fracture specimen, which is difficult and time consuming for any fracture analyst. Hence, in this study an effort is made to propose a simple relation to estimate K I at the center of the specimen, without any numerical computations. Further, the magnitude of K I at the center of specimen having various B ( B =2 to 20mm) and  /  y =0.08 to 0.80 and a/W =0.45 to 0.70 are extracted from 3D FE analysis. A typical plot of K I at center against  /  y and B for a/W =0.50 is plotted in Fig.9. Figure 8 : Variation of K I for various a/W ratios obtained by present 3D FEA and Kwon and Sun [5] Figure 9 : A typical plot of K I at specimen centre against / y   for various B. It is interesting to know from this figure that the variation of K I vs .  /  y is linear for various specimen thicknesses ( B ) and is independent of B . This nature of the variation of K I vs .  /  y allows us to obtain the relation between K I and  /  y for particular a/W . Next, the slope of K I vs .  /  y is evaluated by fitting a straight-line equation to all the values of K I estimated at the centre of the specimen shown in Fig. 9. Now, it is required to find the variation of K I / (  /  y ) with the specimen thickness ( B ) for various a/W . K I / (  /  y ) against normalized thickness ( B/W ) for various a/W are plotted in Fig. 10. The plot indicates that the slopes are almost independent of normalized specimen thicknesses ( B/W ). The average slope values of B =2 to 20mm ( B/W =0.1-1.0) are computed for a respective a/W . The plot between average slope value of K I / (  /  y ) and a/W is shown in Fig 11. This figure is used to obtain the relation between K I / (  /  y ) and a/W , in which K I is the only unknown. As relation between K I / (  /  y ) and a/W is nonlinear, we have tried to fit the data with a suitable polynomial. In such exercise, we found a polynomial equation of third order fits the data with least error (Coefficient of determination, COD=0.99885), which is superimposed in Fig.11. From this third order polynomial fit, the relation between K I , ( a/W ) and (  /  y ) can be expressed as:   2 3 2824 35 23133 54 50651 70 41421 48 . . . . / I Y K a a a W W W                          (5)