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A three-dimensional finite element analysis of hydrostatic bulging of an integral polyhedron into a spherical vessel
Int. J. Pres. Ves. & Piping 60 (1994) 133-138 © 1994 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0308-0161/94]$07.00
ELSEVIER
A three-dimensional finite element analysis of hydrostatic bulging of an integral polyhedron into a spherical vessel J. Hashemi & Q. S. Zheng Department of Mechanical Engineering, Texas Tech University, Lubbock, Texas 79409, USA (Received 15 September 1993; accepted 19 September 1993)
The recently developed integral hydro-bulge forming (IHBF) process has been introduced as a novel method of manufacturing single- and multi-layered spherical pressure vessels. The process of manufacturing spherical vessels using IHBF is rather simple in application but complex in analysis. Thus far, all studies and analyses have been experimental. The interaction of welds and flat plate elements during the hydrostatic bulging process and the ensuing state of stress and large deformation constitute a complex three-dimensional metal forming process that is not understood well. In this study, a three-dimensional finite element model of the IHBF process used in manufacturing of spherical pressure vessels is developed using the ABAQUS finite element code. The results of the forming simulation and its comparison with experimental observations as well as the deformation history in the flat plate elements are presented in this paper.
INTRODUCTION
theories such as that offered by Hill ~° the solutions have become more accurate and forthcoming. However, most investigations to date have been fundamental in nature and have dealt with the hydrostatic bulging of simple axisymmetric geometries. The modeling of the hydrostatic bulging in relation to large deformation metal forming processes has not appeared as frequently. In this paper, the results of the modeling efforts on the integral hydro-bulge forming (IHBF) process, which is in essence an application of the hydrostatic bulging process to the forming of spherical pressure vessels, will be given.
Hydrostatic bulging of metal sheets has been studied extensively in recent years. Its importance is in the determination of strain hardening characteristics of metals at large strains and also as a metal forming operation in its own right. The hydrostatic bulging process was used by Brown and Sachs, t Gleyzal, 2 Brown and Thompson 3 and Mellor 4 to determine the stress-strain curves for materials such as soft copper and mild steel. The advantage of the bulge test over the uniaxial tension test is that in the former much larger strain can be achieved before fracture. Theoretical approaches in the analysis of diaphragms, membranes, and thin sheets of metals under hydrostatic bulging are numerous and date back to the work done by Hill, 5 Weil 6'7 and the more recent work of Ilahi et al. 8'9. The objective of the theoretical and numerical work to date has been to predict the behavior and characteristics of the metal under the hydrostatic bulging process and to determine the relationships among pressure, strain, and geometrical changes as accurately as possible. With the development of new yield
MANUFACTURING PROCEDURE A complete procedural description of the IHBF process and it advantages is given by Hashemi et al. tl However, for clarity, a brief explanation of key steps and results will be given here. The IHBF process is a process of hydrostatically bulging a structure known as the integral 133
J. Hashemi, Q. S. Zheng
134
polyhedron into a spherical vessel. In this process, flat blank sheets of metal are cut in the shape of pentagonal and hexagonal fiat elements. The elements are then assembled according to a specific pattern. The pattern used in this design is one that satisfies Euler's theory of polyhedrons. According to Euler's theory the number of elements, edges, and vertices on a polyhedron must satisfy the following equation:
V+F=E+2
(1)
where V is the number of vertices, E is the number of edges, and F is the number of face elements in the polyhedron. Although there are numerous patterns that can represent the surface, the pattern shown in Fig. l(a) was determined as
the most effective and suitable for the proposed application. The integral polyhedron, shown in Fig. 1, consists of 32 face elements (20 hexagons and 12 pentagons), 60 vertices, and 90 edges. All pentagons and hexagons are cut with equal side dimensions, a. The boundaries of each flat element are then welded to its adjacent flat elements. This structure is referred to as an integral polyhedron shell. To prepare the shell for bulging, two holes are machined at the centers of two diametrically opposite fiat elements. The shell is filled with water through one of the holes. A vacuum pump is used to extract all of the trapped air bubbles through another hole. The vacuum pump is removed and a pressure gauge is positioned at that hole. The pressurization process is performed at a slow rate and in many cycles. Accompanied with every pressurization cycle is a relaxation cycle to allow the weld seams to adapt to the surrounding deformation field. The cyclic process of pressurization-relaxation is repeated until the desired spherical shape is achieved, as shown in Fig. l(b). The bulging pressure (the pressure at which the polyhedron becomes fully spherical) differs according to the t/2r ratio. The bulging pressure's for the 30.48, 60.96, and 91.45 cm diameter vessels of approximately 2 mm thickness are, respectively, 6.21, 4.09, and 2.06 MPa, as shown in Table 1.
FINITE E L E M E N T M O D E L
(a)
I
The general purpose finite element code A B A Q U S was used for the modeling of the I H B F process. A B A Q U S uses the incremental plasticity theory and has demonstrated its potential as a code suitable for modeling of large strain processes such as those encountered in metal forming operations. Figure l(b) shows the fully assembled integral polyhedron. Even though the hydrostatic bulging Table 1. Summary of bulging data for three different size spheres
(b) Fig. L Experimental observation of the hydrostatic bulging of an integral polyhedron into a sphere: (a) before bulging; (b) after bulging.
Average initial shell diameter (cm)
Final shell diameter (cm)
Bulging pressure (mPa)
% Increase in volume
30"48 60.96 91"45
31 '65 63.41 95-32
6.34 4" 19 2"14
11.42 12-54 13"24
Hydrostatic bulging of an integral polyhedron into a sperical shell
135
where: a = arcsin(cos 36/cos 30) R = a cos 30/cos a
(3)
0 = arcsin(2a cos 36/R)
e~
A
I I
) /
Fig. 2. A schematic representation of the polyhedron pattern and the corresponding element of symmetry, AABC. process of the above polyhedron is essentially a three-dimensional phenomenon, advantage can be taken of the inherent geometric symmetry of the problem. The polyhedron pattern shows that every pentagonal fiat element is surrounded by five hexagonal fiat elements, as shown in Fig. 2. Furthermore, if the center point of the pentagon (marked as A on Fig. 2) and the centers of the two adjacent, neighboring, hexagons (marked as B and C on Fig. 2) are connected, the simplest geometry representing the whole polyhedron is produced. By providing the proper boundary conditions and loading on this segment of the geometry a proper three-dimensional solution can be developed. The isolated region presented in Fig. 2 was first used to simulate the bulging process. The coordinates of all critical points are calculated based on the length a of the pentagons and hexagons. For instance the coordinates of points Po through />5 are calculated using the following equations:
and a is the length of the sides. Any other coordinates may be determined by simply adjusting to proper angles and rotations. The coordinates of vertices on the other faces can be determined by varying the x, y, and z axes in a cyclic order. It should be mentioned that in this simulation the weld material was not modeled. It was decided that the inherent geometrical stiffness at the juncture between elements is suitable for a first try. Displacement boundary conditions were assigned to the nodes on the edges AB, BC, and CA of the region such that the nodes on those boundaries were constrained in a direction normal to the edge on which they were located. In addition to displacement boundary conditions, rotational boundary conditions at the edges had to be imposed. The rotation boundary conditions, however, were not implementable since the edges were in a state of constant change and deformation. Subsequently, the local normal and tangential vectors, about which rotations must be defined, change as a function of location of the point on the edge as well as time as represented in Fig. 3. Therefore, the simulation could not be performed using the above simplified geometry due to the complexity of the boundary conditions. It was therefore decided that the whole polyhedron be modeled. Modeling of the whole structure would make the simulation more accurate and there would be no need for boundary condition implementation.
T= t2 A
~tlx"
~"
e"
Po = (a/2 + a cos 36, a/2 + a cos 36, a/2 + a cos 36) P, = (O, O, R) P2 = (0, 2a cos 36, R cos 0) P3 e4
=
=
(a/2, 2a cos 36, R cos 0)
(a cos 36, R - a cos ot cos 30, a) />5 = (0, R, a/2)
(2)
D Fig. 3. A schematic representation of time and position rate of change of the local coordinate system on edge A D of the AABC.
J. Hashemi, Q. S. Zheng
136
However, the large n u m b e r of elements that were required to provide a reasonably accurate simulation would m a k e the p r o b l e m computationally intensive. N u m e r o u s elemental patterns were analyzed and finally an optimized mesh pattern that was least expensive (computationally) and most accurate was selected. The initial g e o m e t r y of the simulation and the utilized elemental pattern is p r e s e n t e d in Fig. 4(a). A combination of eight-node quadrilateral shell elements (360 elements) and three node triangular shell elements (720 elements) were used to discretize the pentagons and hexagons in the polyhedron. The quadrilateral shell elements
were used a r o u n d the three-way edge regions to increase the accuracy of the simulation. In this particular simulation, the length a was selected as 10 cm and the thickness of the p o l y h e d r o n was chosen as 2 mm. A strain hardening material model was used. The value of the strain hardening coefficient, C, was 250 MPa and that of strain hardening exponent, n, was 0.22.
(a)
(b)
O"= f e n ( 4 ) To simulate the experiment, an inner uniform pressure was applied on the inside surface of the
(c) (d) l~g. 4. Different stages of the hydrostatic bulging processas simulated by ABAQUS. (a) Initial geometry, P = O; (b) small amount of bulging observed at center points, P = 0.98 MPa; (c) significant bulging observed specifically on hexagonal plates, P = 2.4 MPa; (d) final stage of the simulation showing a completely bulged sphere, P = 4.1 MPa.
Hydrostatic bulging of an integral polyhedron into a sperical shell
polyhedron. The pressurization path was selected as a linear function of time starting from zero to a prescribed final value.
RESULTS Figure 4 shows the hydrostatic bulging simulation at different stages of the pressurization. Figure 4(a) shows the initial geometry where the elemental pattern used is shown. Each pentagon and hexagon is modeled by higher order shell elements at the edges and by constant stress shell element at the center. This was done because the deformation at the boundaries was experimentally observed to be very complex and dynamic. Accurate calculations were needed to simulate the deformation of the boundaries realistically. Four stages are selected to show the gradual transition from a polyhedron shell (Fig. 4(a)) to a full sphere (Fig. 4(d)). Minor discontinuity of deformation may be observed across each pentagon or hexagon but this is clearly due to connection of higher order elements to constant
1.1
2.2
137
stress shell elements and could be solved by a more accurate mesh. Comparison of Fig. 4 with Fig. 1 shows a qualitative agreement between the experiment and the numerical simulation. In general, the evolution of the deformation and forming as a function of pressure agrees well with the experimental observations presented earlier. The bulging pressure was also determined numerically. This was done by following the radial deformation history of the critical points, described in Fig. 2, as a function of pressure (or time). The bulging pressure corresponded to that pressure at which all those points had almost the same radial coordinates, implying that a full sphere was formed. The final stage of the bulging, Fig. 4(d), shows the deformed sphere and the corresponding pressure of 4.1 MPa. This pressure is consistent with the range of pressures observed in the experiment. Figure 5 shows the radial displacement changes versus time (or pressure) for a series of points lying on a line from the center of a pentagon (point A) to the middle point of an edge (point D). The displacement-time curve for the center point (point A) shows the largest positive slope
Pressure In MPa 3.3
4.4
0.2490 0.2475 0.2460 0.2445 0.2430 0.2415 0.2400 0.2385 '5
.c_ In "D
o
0.237O 0.2355 0.2340
A
0.2325
A
0.2310 0.2295 0.2280 0.2265 0.2250 0.0
5.0
I 10.0
I 1~.0
i 20.0
I ~.0
I 30.0
I ,,~).0
I 40.0
I 4~'.0
I 50.0
I ~.0
60.0
Time in Second
Fig. 5. Radial displacement versus time (pressure) history of selected points on a pentagonal element.
138
J. Hashemi, Q. S. Zheng
implying that the deformation starts at the center point and there is an initial m a x i m u m radial deformation rate at that point. The deformation curve for points 1 and 2 show the same positive slope but a smaller magnitude as the point is located closer to the edge. The point on the edge (point D), however, shows a negative slope until the pressurization level is close to the bulging pressure at which point it experiences a sudden change in its displacement-time slope. This implies that the point is actually moving downward, slightly, for a large part of the loading process and is pushed out suddenly at a critical pressure value. The same p h e n o m e n o n is observed experimentally but it is difficult to capture. The only evidence of that p h e n o m e n o n was provided by the strain gauge readings during the early stages of the loading. The strain data for points close to the welded edges of plates showed negative values. The compressive strain measurements showed that the element was concaving at that region. The strain values at points far from the edge were positive throughout the process. Average radial position of all nodes is presented by the dashed line and the average radial position of all points on line A D is given by the dotted line in Fig. 5. This implies that the points close to the edge (points 2 and D) expand less than the points close to the center (points 1 and A). This is probably due to the geometric stiffness at the edges or the relatively coarse mesh used. All points under consideration show a sharp increase in the slope of their displacement-time curve at around 32 s or 3.6 MPa which is consistent with the experimental observations where the expansion of the polyhedron becomes more significant and more visible.
CONCLUSION A finite element model was developed to simulate the complex three-dimensional process of hydrostatic bulging of a polyhedron into a full sphere. The simulation results helped explain some of the questions about the mechanics of the I H B F process. It was verified that the three-way junction points move inward during the initial stages of the process and at later stages move rapidly outward. This is the stage of the process during which leakage may occur. The model also predicted the bulging pressure with reasonable
accuracy and produced accurate patterns of stress distribution during the bulging process. In general, the comparison of the experimental and numerical results was satisfactory. For future research the model could certainly be used to determine the effect of pattern design changes or in relation to fluid-filled multi-layered structures or prestress vessel design using the IHBF process. A factor that was not addressed in the simulation was the simulation of the weld material which could be modeled; however, the authors do not believe that the weld material addition will change the results significantly.
ACKNOWLEDGEMENTS This work was supported by the State of Texas' Line Item Research, under the Research E n h a n c e m e n t program. All the computations were performed on C R A Y - Y M P supercomputer. The supercomputing time was provided by a grant from Pittsburgh Supercomputing Center through National Science Foundation. The authors wish to thank Dr J. Rasty for his help in setting up the initial A B A Q U S model.
REFERENCES 1. Brown, W.F. & Sachs, G., Strength and characteristics of thin circular membranes. Trans. ASME, 70 (1948) 241. 2. Gleyzat, A., Plastic deformation of a circular diaphragm under pressure. J. Appl. Mech., 22 (1948) 533. 3. Brown, W.F. & Thompson, F.C., Strength and failure characteristics of metal membranes in circular bulging. Trans. ASME, 71 (1949) 575. 4. Mellor, P.B., Stretch forming under fluid pressure. J. Mech. Phys. Solids, 4 (1956) 41. 5. Hill, R., A theory of plastic bulging of a metal diaphragm by lateral pressure. Phil. Mag., 41, ser. 7 (1950) 1133. 6. Weil, N.A., Rupture characteristics of safety diaphragms. J. Appl. Mech., 26 (1959) 621. 7. Weil, N.A., Approximation to plastic behavior of membranes. J. Engng Mech. Div. ASCE, 83, 412. 8. Ilahi, M.F., Parmar, A. & Mellor, P.B., Hydrostatic bulging of a circular aluminum diaphragm. Int. J. Mech. Sci., 23 (1981) 221. 9. Ilahi, M.F. & Paul, T.K., Hydrostatic bulging of a circular soft brass diaphragm. Int. J. Mech. Sci., 27 (1985) 275. 10. Hill, R., Theoretical plasticity of textured aggregates. Math. Proc. Camb. Phil Sci., 85 (1979) 179. 11. Hashemi, J., Rasty, J., Li, S.D. & Tseng, A.A., Integral hydro-bulge forming of single and multi-layered spherical pressure vessels. A S M E J. Pres. Ves. Technol., 115 (1993) 249.
ELSEVIER
A three-dimensional finite element analysis of hydrostatic bulging of an integral polyhedron into a spherical vessel J. Hashemi & Q. S. Zheng Department of Mechanical Engineering, Texas Tech University, Lubbock, Texas 79409, USA (Received 15 September 1993; accepted 19 September 1993)
The recently developed integral hydro-bulge forming (IHBF) process has been introduced as a novel method of manufacturing single- and multi-layered spherical pressure vessels. The process of manufacturing spherical vessels using IHBF is rather simple in application but complex in analysis. Thus far, all studies and analyses have been experimental. The interaction of welds and flat plate elements during the hydrostatic bulging process and the ensuing state of stress and large deformation constitute a complex three-dimensional metal forming process that is not understood well. In this study, a three-dimensional finite element model of the IHBF process used in manufacturing of spherical pressure vessels is developed using the ABAQUS finite element code. The results of the forming simulation and its comparison with experimental observations as well as the deformation history in the flat plate elements are presented in this paper.
INTRODUCTION
theories such as that offered by Hill ~° the solutions have become more accurate and forthcoming. However, most investigations to date have been fundamental in nature and have dealt with the hydrostatic bulging of simple axisymmetric geometries. The modeling of the hydrostatic bulging in relation to large deformation metal forming processes has not appeared as frequently. In this paper, the results of the modeling efforts on the integral hydro-bulge forming (IHBF) process, which is in essence an application of the hydrostatic bulging process to the forming of spherical pressure vessels, will be given.
Hydrostatic bulging of metal sheets has been studied extensively in recent years. Its importance is in the determination of strain hardening characteristics of metals at large strains and also as a metal forming operation in its own right. The hydrostatic bulging process was used by Brown and Sachs, t Gleyzal, 2 Brown and Thompson 3 and Mellor 4 to determine the stress-strain curves for materials such as soft copper and mild steel. The advantage of the bulge test over the uniaxial tension test is that in the former much larger strain can be achieved before fracture. Theoretical approaches in the analysis of diaphragms, membranes, and thin sheets of metals under hydrostatic bulging are numerous and date back to the work done by Hill, 5 Weil 6'7 and the more recent work of Ilahi et al. 8'9. The objective of the theoretical and numerical work to date has been to predict the behavior and characteristics of the metal under the hydrostatic bulging process and to determine the relationships among pressure, strain, and geometrical changes as accurately as possible. With the development of new yield
MANUFACTURING PROCEDURE A complete procedural description of the IHBF process and it advantages is given by Hashemi et al. tl However, for clarity, a brief explanation of key steps and results will be given here. The IHBF process is a process of hydrostatically bulging a structure known as the integral 133
J. Hashemi, Q. S. Zheng
134
polyhedron into a spherical vessel. In this process, flat blank sheets of metal are cut in the shape of pentagonal and hexagonal fiat elements. The elements are then assembled according to a specific pattern. The pattern used in this design is one that satisfies Euler's theory of polyhedrons. According to Euler's theory the number of elements, edges, and vertices on a polyhedron must satisfy the following equation:
V+F=E+2
(1)
where V is the number of vertices, E is the number of edges, and F is the number of face elements in the polyhedron. Although there are numerous patterns that can represent the surface, the pattern shown in Fig. l(a) was determined as
the most effective and suitable for the proposed application. The integral polyhedron, shown in Fig. 1, consists of 32 face elements (20 hexagons and 12 pentagons), 60 vertices, and 90 edges. All pentagons and hexagons are cut with equal side dimensions, a. The boundaries of each flat element are then welded to its adjacent flat elements. This structure is referred to as an integral polyhedron shell. To prepare the shell for bulging, two holes are machined at the centers of two diametrically opposite fiat elements. The shell is filled with water through one of the holes. A vacuum pump is used to extract all of the trapped air bubbles through another hole. The vacuum pump is removed and a pressure gauge is positioned at that hole. The pressurization process is performed at a slow rate and in many cycles. Accompanied with every pressurization cycle is a relaxation cycle to allow the weld seams to adapt to the surrounding deformation field. The cyclic process of pressurization-relaxation is repeated until the desired spherical shape is achieved, as shown in Fig. l(b). The bulging pressure (the pressure at which the polyhedron becomes fully spherical) differs according to the t/2r ratio. The bulging pressure's for the 30.48, 60.96, and 91.45 cm diameter vessels of approximately 2 mm thickness are, respectively, 6.21, 4.09, and 2.06 MPa, as shown in Table 1.
FINITE E L E M E N T M O D E L
(a)
I
The general purpose finite element code A B A Q U S was used for the modeling of the I H B F process. A B A Q U S uses the incremental plasticity theory and has demonstrated its potential as a code suitable for modeling of large strain processes such as those encountered in metal forming operations. Figure l(b) shows the fully assembled integral polyhedron. Even though the hydrostatic bulging Table 1. Summary of bulging data for three different size spheres
(b) Fig. L Experimental observation of the hydrostatic bulging of an integral polyhedron into a sphere: (a) before bulging; (b) after bulging.
Average initial shell diameter (cm)
Final shell diameter (cm)
Bulging pressure (mPa)
% Increase in volume
30"48 60.96 91"45
31 '65 63.41 95-32
6.34 4" 19 2"14
11.42 12-54 13"24
Hydrostatic bulging of an integral polyhedron into a sperical shell
135
where: a = arcsin(cos 36/cos 30) R = a cos 30/cos a
(3)
0 = arcsin(2a cos 36/R)
e~
A
I I
) /
Fig. 2. A schematic representation of the polyhedron pattern and the corresponding element of symmetry, AABC. process of the above polyhedron is essentially a three-dimensional phenomenon, advantage can be taken of the inherent geometric symmetry of the problem. The polyhedron pattern shows that every pentagonal fiat element is surrounded by five hexagonal fiat elements, as shown in Fig. 2. Furthermore, if the center point of the pentagon (marked as A on Fig. 2) and the centers of the two adjacent, neighboring, hexagons (marked as B and C on Fig. 2) are connected, the simplest geometry representing the whole polyhedron is produced. By providing the proper boundary conditions and loading on this segment of the geometry a proper three-dimensional solution can be developed. The isolated region presented in Fig. 2 was first used to simulate the bulging process. The coordinates of all critical points are calculated based on the length a of the pentagons and hexagons. For instance the coordinates of points Po through />5 are calculated using the following equations:
and a is the length of the sides. Any other coordinates may be determined by simply adjusting to proper angles and rotations. The coordinates of vertices on the other faces can be determined by varying the x, y, and z axes in a cyclic order. It should be mentioned that in this simulation the weld material was not modeled. It was decided that the inherent geometrical stiffness at the juncture between elements is suitable for a first try. Displacement boundary conditions were assigned to the nodes on the edges AB, BC, and CA of the region such that the nodes on those boundaries were constrained in a direction normal to the edge on which they were located. In addition to displacement boundary conditions, rotational boundary conditions at the edges had to be imposed. The rotation boundary conditions, however, were not implementable since the edges were in a state of constant change and deformation. Subsequently, the local normal and tangential vectors, about which rotations must be defined, change as a function of location of the point on the edge as well as time as represented in Fig. 3. Therefore, the simulation could not be performed using the above simplified geometry due to the complexity of the boundary conditions. It was therefore decided that the whole polyhedron be modeled. Modeling of the whole structure would make the simulation more accurate and there would be no need for boundary condition implementation.
T= t2 A
~tlx"
~"
e"
Po = (a/2 + a cos 36, a/2 + a cos 36, a/2 + a cos 36) P, = (O, O, R) P2 = (0, 2a cos 36, R cos 0) P3 e4
=
=
(a/2, 2a cos 36, R cos 0)
(a cos 36, R - a cos ot cos 30, a) />5 = (0, R, a/2)
(2)
D Fig. 3. A schematic representation of time and position rate of change of the local coordinate system on edge A D of the AABC.
J. Hashemi, Q. S. Zheng
136
However, the large n u m b e r of elements that were required to provide a reasonably accurate simulation would m a k e the p r o b l e m computationally intensive. N u m e r o u s elemental patterns were analyzed and finally an optimized mesh pattern that was least expensive (computationally) and most accurate was selected. The initial g e o m e t r y of the simulation and the utilized elemental pattern is p r e s e n t e d in Fig. 4(a). A combination of eight-node quadrilateral shell elements (360 elements) and three node triangular shell elements (720 elements) were used to discretize the pentagons and hexagons in the polyhedron. The quadrilateral shell elements
were used a r o u n d the three-way edge regions to increase the accuracy of the simulation. In this particular simulation, the length a was selected as 10 cm and the thickness of the p o l y h e d r o n was chosen as 2 mm. A strain hardening material model was used. The value of the strain hardening coefficient, C, was 250 MPa and that of strain hardening exponent, n, was 0.22.
(a)
(b)
O"= f e n ( 4 ) To simulate the experiment, an inner uniform pressure was applied on the inside surface of the
(c) (d) l~g. 4. Different stages of the hydrostatic bulging processas simulated by ABAQUS. (a) Initial geometry, P = O; (b) small amount of bulging observed at center points, P = 0.98 MPa; (c) significant bulging observed specifically on hexagonal plates, P = 2.4 MPa; (d) final stage of the simulation showing a completely bulged sphere, P = 4.1 MPa.
Hydrostatic bulging of an integral polyhedron into a sperical shell
polyhedron. The pressurization path was selected as a linear function of time starting from zero to a prescribed final value.
RESULTS Figure 4 shows the hydrostatic bulging simulation at different stages of the pressurization. Figure 4(a) shows the initial geometry where the elemental pattern used is shown. Each pentagon and hexagon is modeled by higher order shell elements at the edges and by constant stress shell element at the center. This was done because the deformation at the boundaries was experimentally observed to be very complex and dynamic. Accurate calculations were needed to simulate the deformation of the boundaries realistically. Four stages are selected to show the gradual transition from a polyhedron shell (Fig. 4(a)) to a full sphere (Fig. 4(d)). Minor discontinuity of deformation may be observed across each pentagon or hexagon but this is clearly due to connection of higher order elements to constant
1.1
2.2
137
stress shell elements and could be solved by a more accurate mesh. Comparison of Fig. 4 with Fig. 1 shows a qualitative agreement between the experiment and the numerical simulation. In general, the evolution of the deformation and forming as a function of pressure agrees well with the experimental observations presented earlier. The bulging pressure was also determined numerically. This was done by following the radial deformation history of the critical points, described in Fig. 2, as a function of pressure (or time). The bulging pressure corresponded to that pressure at which all those points had almost the same radial coordinates, implying that a full sphere was formed. The final stage of the bulging, Fig. 4(d), shows the deformed sphere and the corresponding pressure of 4.1 MPa. This pressure is consistent with the range of pressures observed in the experiment. Figure 5 shows the radial displacement changes versus time (or pressure) for a series of points lying on a line from the center of a pentagon (point A) to the middle point of an edge (point D). The displacement-time curve for the center point (point A) shows the largest positive slope
Pressure In MPa 3.3
4.4
0.2490 0.2475 0.2460 0.2445 0.2430 0.2415 0.2400 0.2385 '5
.c_ In "D
o
0.237O 0.2355 0.2340
A
0.2325
A
0.2310 0.2295 0.2280 0.2265 0.2250 0.0
5.0
I 10.0
I 1~.0
i 20.0
I ~.0
I 30.0
I ,,~).0
I 40.0
I 4~'.0
I 50.0
I ~.0
60.0
Time in Second
Fig. 5. Radial displacement versus time (pressure) history of selected points on a pentagonal element.
138
J. Hashemi, Q. S. Zheng
implying that the deformation starts at the center point and there is an initial m a x i m u m radial deformation rate at that point. The deformation curve for points 1 and 2 show the same positive slope but a smaller magnitude as the point is located closer to the edge. The point on the edge (point D), however, shows a negative slope until the pressurization level is close to the bulging pressure at which point it experiences a sudden change in its displacement-time slope. This implies that the point is actually moving downward, slightly, for a large part of the loading process and is pushed out suddenly at a critical pressure value. The same p h e n o m e n o n is observed experimentally but it is difficult to capture. The only evidence of that p h e n o m e n o n was provided by the strain gauge readings during the early stages of the loading. The strain data for points close to the welded edges of plates showed negative values. The compressive strain measurements showed that the element was concaving at that region. The strain values at points far from the edge were positive throughout the process. Average radial position of all nodes is presented by the dashed line and the average radial position of all points on line A D is given by the dotted line in Fig. 5. This implies that the points close to the edge (points 2 and D) expand less than the points close to the center (points 1 and A). This is probably due to the geometric stiffness at the edges or the relatively coarse mesh used. All points under consideration show a sharp increase in the slope of their displacement-time curve at around 32 s or 3.6 MPa which is consistent with the experimental observations where the expansion of the polyhedron becomes more significant and more visible.
CONCLUSION A finite element model was developed to simulate the complex three-dimensional process of hydrostatic bulging of a polyhedron into a full sphere. The simulation results helped explain some of the questions about the mechanics of the I H B F process. It was verified that the three-way junction points move inward during the initial stages of the process and at later stages move rapidly outward. This is the stage of the process during which leakage may occur. The model also predicted the bulging pressure with reasonable
accuracy and produced accurate patterns of stress distribution during the bulging process. In general, the comparison of the experimental and numerical results was satisfactory. For future research the model could certainly be used to determine the effect of pattern design changes or in relation to fluid-filled multi-layered structures or prestress vessel design using the IHBF process. A factor that was not addressed in the simulation was the simulation of the weld material which could be modeled; however, the authors do not believe that the weld material addition will change the results significantly.
ACKNOWLEDGEMENTS This work was supported by the State of Texas' Line Item Research, under the Research E n h a n c e m e n t program. All the computations were performed on C R A Y - Y M P supercomputer. The supercomputing time was provided by a grant from Pittsburgh Supercomputing Center through National Science Foundation. The authors wish to thank Dr J. Rasty for his help in setting up the initial A B A Q U S model.
REFERENCES 1. Brown, W.F. & Sachs, G., Strength and characteristics of thin circular membranes. Trans. ASME, 70 (1948) 241. 2. Gleyzat, A., Plastic deformation of a circular diaphragm under pressure. J. Appl. Mech., 22 (1948) 533. 3. Brown, W.F. & Thompson, F.C., Strength and failure characteristics of metal membranes in circular bulging. Trans. ASME, 71 (1949) 575. 4. Mellor, P.B., Stretch forming under fluid pressure. J. Mech. Phys. Solids, 4 (1956) 41. 5. Hill, R., A theory of plastic bulging of a metal diaphragm by lateral pressure. Phil. Mag., 41, ser. 7 (1950) 1133. 6. Weil, N.A., Rupture characteristics of safety diaphragms. J. Appl. Mech., 26 (1959) 621. 7. Weil, N.A., Approximation to plastic behavior of membranes. J. Engng Mech. Div. ASCE, 83, 412. 8. Ilahi, M.F., Parmar, A. & Mellor, P.B., Hydrostatic bulging of a circular aluminum diaphragm. Int. J. Mech. Sci., 23 (1981) 221. 9. Ilahi, M.F. & Paul, T.K., Hydrostatic bulging of a circular soft brass diaphragm. Int. J. Mech. Sci., 27 (1985) 275. 10. Hill, R., Theoretical plasticity of textured aggregates. Math. Proc. Camb. Phil Sci., 85 (1979) 179. 11. Hashemi, J., Rasty, J., Li, S.D. & Tseng, A.A., Integral hydro-bulge forming of single and multi-layered spherical pressure vessels. A S M E J. Pres. Ves. Technol., 115 (1993) 249.