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Petroleum Science 2005 Vo l . 2 N o .3 Applications of the Finite Element Method to PDC Mold Design Zhou Sizhu (School of Mechanical Engineering, Yangtze University, Jingzhou, Hubei 434102, China) Received May 8, 2005 Abstract: This paper describes the structure of the molds for making polycrystalline diamond compact (PDC) drilling bits. It represents the mold shapes by using the Finite Element technique, and compares the analytical results with available experimental data. Based on the results of Finite Element analysis, some areas of stress concentrations are determined, and modifications made, to the PDC (polycrystalline diamond compact) mold. A displacement plot and several stress contour plots are presented. Techniques of the mold design are discussed. Key words: Finite Element Method PDC mold design 1. Introduction Since their introduction in the early 1970s, PDC bits have almost completely replaced three-cone bits for use in relatively soft, nonabrasive formations. They sometimes replace three-cone bits in harder or slower drilling intervals, if the section is uniform. It is fair to say, however, that drillers do not normally consider selecting a PDC bit when drilling goes on in harder formations or even in soft formations with infrequent hard streaks. The problem is that the service life of the bit is too short. A PDC bit consists of a crown, a shank and a pin. The crown contains all the structural components necessary to drill the formation (Gaddy, 1999). The key component necessary to better durability of the bit is polycrystalline diamond compact (PDC) (Fig. 1, from Gaddy, 1999). Fig. 1 Polycrystalline diamond compact The PDC manufacturer must use a procedure similar to a Die-cast or plastic-injection molding process used in toy and automobile industries. The PDC mold design is very important. The need for a higher pressure and temperature in the PDC manufacturing process today has led to very intricate designs of the mold (Fig. 2). Fig. 2 PDC mold 1-mold, 2-steel belts The exotic alloys and complex geometries used require sophisticated analytical methods to optimize mold design. Finite Element Analysis, based on the principle of minimization of total potential energy, is becoming widely accepted as a powerful design tool. This technique is being successfully used to eliminate excess weight and to accurately predict stresses. 2. Theory of three-dimensional Finite Element Analysis A brief summary of the theory follows. For more details refer to the appendix (Zienkiewicz, 1971). The finite element method involves solving k( =R (1) where all variables in brackets are matrices. k represent a n n matrix containing the stiffness constants for each variable in the system of n linear equations. ( is the nodal displacement matrix of all the nodes after dividing the PDC mold into finite Petroleum Science 200512 elements. The matrix R is the 1n column matrix representing the reactions or loads on the system. Solving the Eq.(1), the displacements and stresses can be obtained. The element-stiffness matrix of the three-dimensional element is that = zyxBDBk ddd T (2) where x, y, z i unified coordinates , i local coordinates J i Jacobian matrix D i elastic matrix B i strain matrix 3. Finite element modeling Fig. 2 shows the PDC mold without any modifications. Based on this configuration a three dimensional model is established. A complete grid is established (1/4 ) in Fig. 3, and, for a better perspective, a hidden line plot is shown in Fig. 4. In the Finite Element model, the three dimensional iso-parametric 8 nodal solid element is used (Flugg, 1960; Zhou, 1997; 1998). The total number of elements is 180 corresponding to 308 nodes. Fig. 4 Hidden line plot Fig. 4 Hidden line plot of the mold 4. Stress analysis Finite Element Analysis combined with tests can give the pressure distribution on the internal wall and external surface of the PDC mold. Fig. 5 shows the pressure distribution. Fig. 5 The pressure distribution Figs. 6 and 7 are the tangent stress contours and radial stress contours. The maximum stress value is shown in the Fig 6. Fig. 6 The tangent stress contours Fig. 7 The radial stress contours Fig. 3 Grid generated for the PDC mold (1/4) = 1 1 1 1 1 1 T B ddd JBD Applications of the Finite Element Method to PDC Mold Design Vol .2 No .3 13 Fig. 8 is a displacement plot of the PDC mold, and Fig. 9 shows a hidden initial geometric plot. Fig. 8 The displacement plot Fig. 9 The hidden initial geometry plot Table 1 Encloses the analytical result and experimental data at key points. The location of the measured points is shown in Fig.10. Table 1 The analytical results and experimental data at key points Times 1 2 3 6 1 MPa ( 6 1 )/4 MPa 6 t MPa Error % 1 732 220 75 444.63 2 720 190 132 435.34 3 724 168 154 435.53 4 728 179 188 438.41 438.47 404.63 7.72 Fig. 10 The position of the point measured In the table 6 t is the analytical main stress based on the Finite Element Analysis, 1 the measured stress according to the experimental data . The equation of the measured stress can be expressed as where E and 0 are the elastic coefficients of the steel used for the PDC Mold. E=597.8 GPa, 0 =0.215 5. Design of the mold According to the elasticity theory, if the ratio (K) of the external radius to the internal radius for a single layer cylinder is more than 4 (K4), an increase of the outside radius does not improve the performance of the cylinder. The value of the k for synthetic diamond manufacturing molds in use today is equal to 3.27, close to this critical value . The mold for synthetic diamond manufacture is made of tungsten carbide. The pressure for manufacturing the synthetic diamond is in excess of 60,000 atmospheric pressures. The limits of the compression strength and tensile strength for tungsten carbide are different. The value of the compression strength limit is 6,200MPa. In a working condition the value cannot be reached. The value of the tensile strength limit is about 1,000 to 1,200 MPa. which is a critical value for designing the mold. The main items of designing the mold are to determine the relationships between pressures (internal and external pressure) and tangent stresses in the inner wall of the mold. The finite element model is developed according to the mold structure (Zienkiewicz, 1971; Flugg , 1960). Various values of the pressure act on the mold surface (internal surface pressure (IP): 4,000, 4,500, 5,000, 5,500, 6,000, 6,500MPa, external surface pressure (EP): 1,700, 1,750, 1,800, 1,850, 1,900, 1,950, 2,000, 2,050, 2,100, 2200MPa). The stresses in the mold for each group of internal and external surface pressure are calculated. Table 2 shows the results of the () () () ()() 2 3 2 2 213 1 1 1212 + + + = Petroleum Science 200514 calculation. The relationships between the external surface pressure and the internal tangent stresses for the mold are showed in Fig. 11. The lines s1, s2, s3, s4 coincide with internal surface pressures 4,000, 4,500, 5,000, 5,500MPa. Table 2 Tangent stresses on internal wall of the mold (different internal and external surface pressure) EP, MPa IP, MPa 1700 1750 1800 1850 1900 1950 2000 2050 2100 2200 4000(s1) -612.1 -731.7 -851.4 971.11 -1090.7 -1210.4 -1330.1 -1449.7 -1569.4 1808.76 4500(s2) -180.0 -299.7 -419.3 539.05 658.71 778.38 898.04 -1017.7 -1137.3 1376.70 5000(s3) 252.01 132.35 12.68 106.98 226.65 346.31 465.98 585.64 705.31 944.64 5500(s4) 684.08 564.41 444.75 325.08 205.42 85.75 33.915 153.58 273.25 512.56 6000 1116.1 996.47 876.81 757.14 637.48 517.81 398.15 278.48 158.817 80.51 6500 1548.2 1428.5 1308.8 1189.20 1069.54 949.88 830.21 710.55 590.88 351.55 Fig. 11 Relationships between external surface pressure and tangent stress on internal wall From Table 2 and Fig.11, we can see that for a special internal pressure, there is an approximately linear relationship between the external surface pressure and the internal tangent stress. When a mold is designed, the pressure value acting on the out side of the mold can be determined according to the requirement of a lower tangent stress (600MPa or 1,000MPa). The tangent stress on the inner wall directly affects the lifetime for the mold. Tables 3 and 4 show the relationships between the internal pressure and the external pressure under different internal tangent stress requirements Table 3 Tangent stress on internal wall is 600MPa Internal pressure MPa 4000 4500 5000 5500 6000 6500 External pressure MPa 1193.54 1374.07 1554.60 1735.13 1915.66 2096.19 Table 4 Tangent stress on internal wall is 1,000MPa Internal pressure MPa 4000 4500 5000 5500 6000 6500 External pressure MPa 1026.40 1206.94 1387.47 1568.00 1748.53 1929.06 There is a linear relationship between the internal pressure and the external pressure under a special tangent stress on the inner wall of the mold, as Figs. 12 and 13 show. When the internal working pressure is 6,000MPa, the external pressure acting on the mold should be 1,915.66MPa, corresponding to the inner wall tangent pressure 600MPa, and 1,748.53MPa to 1,000MPa. The steel belts shown in Fig.2 provide these outside pressure values. 6. Conclusions A mechanical calculation model for the design of molds for polycrystaline diamond compact (PDC) drilling bits has been developed. Since the model is Fig. 12 Relationships between internal pressure and external pressure (Tangent stress on the internal wall is 600MPa) Applications of the Finite Element Method to PDC Mold Design Vol .2 No .3 15 Fig. 13 Relationships between internal and external pressures (Tangent stress of internal wall is 1,000MPa) generated interactively, it is easy to make modifications and study the effect of the changes. Once calibration is achieved, the Finite Element technique proves very economical by saving a lot of prototype testing. For the stress analysis, Finite Element provides a comprehensive picture of stress contours and displacements. The stress contours identifies the areas of concentrations and thus appropriate modifications can be mad