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Explosive forming of spherical metal vessels without dies
Journal of Materials Processing Technology, 31 (1992) 135-145
135
Elsevier
Explosive forming of spherical metal vessels without dies Zhang Tiesheng, Li Zhensheng, Guo Changji and Tong Zheng Material Science Department, Inner Mongolia Engineering College, Hohhot 010062, Inner Mongolia Autonomous Region, China
Industrial Summary This paper presents a new method of manufacturing spherical vessels based on the technology of non-die explosive forming, that is highly energy efficient. Comparing spherical pressure vessels with other shapes of vessel, they have a higher load capacity, have less area per unit volume, have better economic benefit and have a nicer appearance: of late, they have been used increasingly.
1. Introduction
The classical technology of manufacturing spherical vessels requires the use of huge presses and special dies, it is costly and it gives less size selection, so that few such pressure vessels of less than 100 m 3 capacity have been made. The Warsaw Institute of Precision Mechanics of Poland has studied the explosive forming of elements of spherical vessels [1 ], but has been restrained by the problem of special dies. Wang Zhangren of Harbin University of Industry of China invented the technology of hydraulic expansion forming of spherically-inscribed polyhedrons to manufacture spherical vessels [2], but the quality of the welding cannot be assured because of the large numbers of welding seams. The technology of the explosive forming of a spherical vessel without dies is first to water-fill a pre-manufactured spherically-inscribed multi-cone container and then to explode it into a spherical shape (Fig. 1 ). Through technological test and theoretical analysis, the principle of the design of the structure of a multi-cone spherical container has been obtained, the equation for the calculation of the charge size has been validated and the technological relationships have been found. As a result, both experimental and theoretical bases 0924-0136/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
136
Fig. 1. Two-cone container before and after explosive forming.
Fig. 2. Sketch of the forming principle and testing equipment: 1 - container shell; 2 spherical impulsive wave; 3 - explosive charge; 4 - water; 5 - circumscribed spherical surface; 6 - buffer.
have been provided for the application of the technology, and this new method for manufacturing small and medium-sized spherical pressure vessels, which is highly energy-efficient, has a bright future. 2. Principle of forming
Explosive forming with dies relies mainly on the dies to control the shape and size of the products. However, in explosive forming without dies, there is relative movement between each mass point of the sheet metal under dynamic load. In the process of deformation, dynamic energy is turned into work of deformation and the velocity of each mass point is rearranged, falling to zero as the product assumes its final shape. The major process of manufacturing a spherical vessel by non-die explosive forming is as follows (see Fig. 2 ) : first, two or more conical frustums are rolled; next, they are welded into a spherically-inscribed container; the latter is then filled with water and a spherically-shaped explosive charge detonated at the centre of the container. The spherical impulsive wave created in the water will firstly push the surfaces that are closest to the centres of the sphere. The energy density of the spherical impulsive wave attenuates approximately in an inverse relationship with distance R, and its peak pressure decreases with increase of distance. Consequently, the velocity difference of the mass points pushes the shell very close in shape to the circumscribed sphere. It is necessary to control the energy of the explosive charge so as to make the container just
137 overlap the circumscribed sphere, whilst at the same time not allowing any of the circumferential welding seams to participate in the deformation. 3. Structure design and analysis
In order to achieve the best effect, comparisons and tests have been made on various spherical structures, two principles being obtained: ( 1 ) the multicone spherical container should be an inscribed one; (2) the length of generatrix of each conical frustum should be equal to the diameter of its base plate. Figure 3 is the structure sketch of a four-cone spherically-inscribed container, with the following geometrical parameters o~=180°/(n+ l)
(1)
di = D cos(ia)
(2)
Di = D cos( ( i - 1)o~)
(3)
Hi = D [sin (ia) - s i n ( ( i - 1 ) a ) ]/2
(4)
~i =
(5)
2 arccos (Hi~a)
a = D sin(o~/2) = d
(6)
Wm.x=D[1-cos(a/2)]/2
(7)
where: a is the central spherical angle faced by each cone and base plate; N is the total number of conical frusta; i is the number of the conical band starting from the equator; di is the smaller diameter of the ith frustum; Di is the larger
I
Fig. 3. Structure sketch of the container.
Fig. 4. Drawing of the ith unfolded conical frustum.
138
diameter of the ith frustum; fli is the conical angle of the ith frustum; a is the length of the generatrix of each conical frustum; d is the diameter of the base plate; and Wmax is the maximum deforming degree. Every conical frustum is rolled from sectorial sheet metal. Figure 4 is the drawing of the unfolded ith frustum, its calculating equations being as follows Ri = a D i / (Di - di)
(s)
ri = R i
(9)
-a
OLi = 1 8 0 o D i / R i
(lO)
~i = OLi/Ni
(11)
When Ti-%<180 ° Ni = 2Ri sin (~i/2 )
(12)(13)
Mi =Ri - ri cos (~'i/2)
When yi>~180 ° Ni = 2Ri
(14)(15)
Mi =Ri [ 1 + sin ()'i/2- 90 ° ) ]
where: Ri is the external radius of the sector blank; ri is the internal radius of the sector blank; Ni is the number of sectorial pieces, i.e., the ith conical frustum is welded from Ni sectorial pieces; 7i is the central angle of the sectorial piece; Ni is the maximum chord length of the sectorial piece after division; and Mi is the maximum radial length of the sectorial piece after division. The number of multi-cones N is a major parameter, affecting the deformation of the container and the size of the blank. The effect of different N values on each parameter of deformation is shown in Table 1. To select a proper N value it is necessary to consider: (1) the effect of the cold deforming on the mechanical performance of the material. The degree of deformation in the cold processing of spherical segments recommended by many countries is less than TABLE1 Effect of the N value on the parameters of deformation
N
a
WmaJD
AS/~o
Al/lo
2 4 5 6 7 8
60 36 30 25.7 22.5 20
0.067 0.024 0.017 0.013 0.009 0.008
13.4 4.9 3.4 2.5 2.0 1.5
15.4 5.1 3.5 2.6 2.0 1.5
(°)
(%)
(%)
Note: a is the central spherical angle faced by each cone a n d base plate; Wmax is the m a x i m u m deforming degree; D is the larger diameter of the frustum; A6/5o is t h e m a x i m u m relative shrinkage of thickness; a n d A1/lo is the m a x i m u m relative extension.
139 5%, correspondingly, N>~ 4; (2) increasing the utilization ratio of the material and shortening the length of the welding seams, if the size of the sheet metal is fixed. With the aid of a computer program, the N value and the number of sectorial pieces can be optimized. 4. Calculation of the f o r m i n g e n e r g y
Over the last twenty years, a lot of test results have indicated that strain rate has a pronounced effect on the flow stress of materials [3]. However, in the process of isolated explosive forming in water, it is very difficult to establish the mechanical model with viscoplastic behaviour, under the condition of the strain rate dropping from 10 s - ' to zero. In order to facilitate engineering practice, a plastic mechanical model is still used to calculate the strain energy, with the following assumptions being made: (1) the effect of strain rate can be ignored; (2) the sheet material is an incompressible rigid-plastic strain-hardening material; (3) the equivalent strains along the generatrix of each conical frustum and along the diameter of the base plates are distributed in a parabolic manner; (4) the stress is distributed evenly over the thickness of the shell, ignoring bending effects; (5) the multi-cone container is turned into a sphere after deformation, and the circumferential welding seams do not participate in the deformation. Based on the von Mises yielding criterion, under the condition of proportional loading the increment of strain energy per unit volume can be written
as: (16)
du=,~ d~
where d is the equivalent stress; and C is the equivalent strain, given by
#=,,/(o', -a2)2+ ( a2 -a3)2+ ( a3 - a , )21x/2
(17)
e----x/~x/(E, -- e2)2+ (e2 -- e3)2+ (~3 -~ e,)2/3
(18)
Considering strain hardening, the flow curve is expressed by an exponential equation: (~----k(n
(19)
where k is the strength coefficient; and n is the strain-hardening exponent. Therefore, the strain energy per unit volume U and the total strain energy of the container ut can be determined u=
f (~d~=
k n+---i ¢n+1
i
n+---1 gn+l dv
(20) (21)
140
The total strain energy of the container Ut includes the strain energy of each cone Uc and the strain energy of its upper and lower base plates U~,. The derivation procedure is omitted N/2
Ut = 2 ~ Uci + 2 Ud
(22)
Ut =KND2J
(23)
i=l
N/2
KN = ~'--max b~.n+l {} (2--n) sin ½~ ~ cos(i-- ½) ~ + 1--cos ½~ i=1
+ ( n + l ) [ c o s ½ ~ - ½cos3 ½ ~ - }]/sin 2 ½c~}/(n+l)
(24)
where KN is coefficient related to the material, to the structural form of the container and to the degree of deformation. When the material and the container structure are fixed, then KN is constant, without regard to the spherical diameter and the sheet thickness. 5. Charge-size calculation
Ezra and Schauer from the University of Denver, USA, and Alting from the Technical University of Denmark and their colleagues have done systematic studies on the energy transferred from an underwater explosive charge to the blank [4]. With the aid of their research achievements, the effective energy E t transferred from the explosive charge to the container is known to be Et --~c~l W e
(25)
where, t/c is the efficiency of energy transfer, considering only the geometrical relationship of the container to the explosive charge (for a spherical shell, ~/c= 1 ); Yl is the efficiency of energy transfer when the effect of pressure transfer medium is considered, which includes the energy of the first impulsive wave and the second loading; W is the weight of the explosive charge; and e is the specific energy of the explosive material. Zheng Zhemin from the Mechanics Research Institute of the Academy of Sciences of China and his colleagues have made special studies on the energy utilization rate of an explosive charge when it is exploded inside the sphere [5,6]. Their test results are: ~ ~ 54.5%-56.6%, here ~1 = 55%. Since Et = Ut, the equation for calculation can be obtained, i.e., by using only the amount W of explosive material, the multi-cone container can be just expanded into a sphere, while every circumferential welding seam remains undeformed.
141 W=KND2J/j7 e = CND2J CN = KN/~Ie
(26) (27)
where CN is a coefficient dependent on the material, the structure of the container, the specific energy of the explosive material and the medium of pressure transfer. When the above factors are fixed, CN is a constant and can be determined by calculation or test. 6. Result of p r o o f tests
Experiments into the non-die explosive forming on two-cone and four-cone containers of LF21M aluminium alloy and common carbon structure A3 steel, have been carried out successfully. The experimental equipment is shown in Fig. 2. The appearances of the model aluminium two-cone container before and after expansion are shown in Fig. 1. In order to analyse and prove the shell deformation, the latitudinal true strain el and the longitudinal true strain e2 of each point should be measured and calculated with the method of mesh -coordinates. The thickness of the shell wall can be measured with an ultrasonic meter and the true strain e3 calculated. A practical example of test-model calculation is listed below. The comparison of the charge size calculated theoretically and used practically is shown in Table 2. The conclusions of the test are: (1) The equivalent strain along the generatrix of the cone and the diameter of the base plate is distributed in a shape of parabola, as is the reduction in thickness of the wall. (2) The maximum reduction in thickness, the maximum equivalent strain at the central point of the generatrix of the cone and that at the central point of the diameter of the base plate are approximately equal, and given by emax~ln[1/cos(c~/2) ] ~ (AS/8o)max
(28)
(3) The difference between the tested explosive charge weight and the calculated weight is less than 4%, the circumferential welding seams on the container are virtually undeformed and their degree of non-sphericity is less than 3%. (4) The equation for the calculation of the charge size is sufficiently accurate for material LF21M, but for other materials, because of the different effect of strain hardening on the strain distribution and the difference in strain sensitivity, it may need some modification. (5) The explosive charge size is in direct proportion to the square of the sphere's diameter D, and the thickness of the sheet metal. Under the condition that the material type, the structural form of the container, the specific energy
142 TABLE 2
Comparison of the calculated charge Wt" with the charge used Wpb Container structure
Explosive charge
Wp (g)
Wt (g)
N=2, D = 2 1 0 cm
8 # ed c
1.0
0.96
(1) 8 # e d + 2 g c-4 and (2) 8 # ed 8 # ed
4.6
4.36
1.0
1.03
8 # e d + 2 g c-4
3.6
3.62
of= 1.5 cm N = 2, D = 400 cm 5 = 2 cm N = 4 , D = 4 0 0 cm ~ = 1.5 cm N=4, D = 6 5 0 cm fi=2 cm
material explosive medium coefficient a b c
- LF21M, K = 2 0 k g f / m m 2, n--0.18; - T N T ( e = 3 9 3 k g f m / g ) , c-4 ( e = 5 2 7 k g f m / g ) ;
- water, q=ql=55%; -
N=2 (K2=3.129 kgf/mm, C2= 1.448 101 g), N=4 (K4=0.927 kgf/mm, C4=4.289
101 g). Wt is converted into TNT weight according to the specific energy of the explosive. Wt is converted into TNT weight according to eqn. (26). ed - electric detonator.
of the explosive and the pressure transfer medium all remain unchanged, C N is a constant. In the case of the absence of accurate data of the values of K, n and e, the value of CN can be determined by means of a model test: consequently the charge size for bigger spheres can be determined by scaling, in accordance with the equations for calculating the charge size. 7. E n g i n e e r i n g
applications
The test results indicate that the technology of liquid-filled explosive forming without dies has opened a new way of manufacturing spherical pressure vessels. The method needs no special dies and it is easy to change size and variety; there is no need of forming equipment and the method is highly energy-efficient. In practical production, the energy of the explosive material is sealed in a spherical space, so that it causes no pollution, the tests being conductable under any environment. The authors have applied this technology in the manufacturing of spherical vessels of different material and of different volume, and have had successes each time. Figure 5 shows a photograph of liquid synthetic ammonia vessel manufactured for a chemical factory, which has a diameter of 4 m, a volume of 33 m 3, a wall thickness of 18 mm and is made of 16MnR steel. Checking its sphericity, the clearance on a length of 500 mm is less than 5 mm. One research institute ordered hemispherical high voltage caps, of which the material is
i
143
Fig. 5. Spherical vessel (D=4 m).
Fig. 6. Four-cone container before expansion.
1Cr18Ni9Ti stainless steel, the thickness is ca. 1-2 mm, the radius is 460 + 0.5 mm, and for which the surface finish should be the same as for mirror polishing. Figure 6 is a photograph of the structure of a four-cone container before explosive forming. After the latter process, it has completely achieved the technical requirements. Figure 7 shows small-sized spheres and hemi-spheres of different material and for various purposes: they are all made by explosive forming.
144
Fig. 7. Small spheres and hemi-spheres.
In practical production, usually one container is expanded in several shots, so as to avoid cracks in the welding seams caused by a larger quantity of explosive or a higher peak pressure of the impulsive wave. Before explosive forming, an overall check on the quality of welding seams is required. After explosive forming, the quality and performance of the metal have no evident difference from those in the classical cold forming of spherical segments [7]. As long as the number of cones N is no less than 4, both the reduction in the thickness and the percentage elongation of the material are lower than the international limit of degree of deformation in the cold forming of a spherical segment, which is less than 5%. 8. Prospects of application
This paper presents the research achievements in the explosive forming of spherical pressure vessels in the technical press for the first time, but it has already been applied in engineering practice. Because of its particular advantage, it can be predicted that this technology will be applied widely in the production of small- and medium-sized spherical vessels for various purposes. Further, it has particular advantages in the following respects: (1) Special materials: extra-high strength steel, martensite ageing steel and compound metal, etc. (2) Special size requirements: e.g., a spherical vessel of extra-thin wall, of which D/J > 1,000; a radar-beacon sphere, a high voltage discharging cover etc., which have extremely high requirements as to the precision of size and surface polishing. (3) Special structures such as: double-layered spherical vessels (vessels to store liquid hydrogen, liquid oxygen, etc.); hemispherical domes, large-sized ellipsoidal heads, decorative spheres for architecture, and so on. References 1 H. AndrzewskiandZ. Krurpa, ExplosiveFormingofElementsofSphericalVessels, Institute of Precision Mechanics, Warsaw, Poland.
145
2 Wang Zhangren, New technology of sphericalvesselmanufacturing, Research Report, ScientificResearch Department, Harbin Universityof Industry,China, 1985. 3 T.Z. Blazynski, Explosive Welding, Forming and Compaction, Applied Science Publishers, Barking, 1983. 4 A.A. Ezra, Principlesand Practice of Explosive Metalworking, IndustrialNewspapers Ltd., 1973. 5 Xiang Zensheng, Approach to mechanism of spherical explosive forming, Research Report, Mechanics Research Institute,the Academy of Sciences of China, 1964. 6 Zheng Zhemin, Energy criterionand deformation calculationof spheres, Research Report, Mechanics Research Institute,the Academy of Sciences of China, 1964. 7 M.A. Meyers and L.E. Murr, Shock Waves and High-Strain-Rate Phenomena in Metals: Concepts and Applications, Plenum Press, New York, 1981.
135
Elsevier
Explosive forming of spherical metal vessels without dies Zhang Tiesheng, Li Zhensheng, Guo Changji and Tong Zheng Material Science Department, Inner Mongolia Engineering College, Hohhot 010062, Inner Mongolia Autonomous Region, China
Industrial Summary This paper presents a new method of manufacturing spherical vessels based on the technology of non-die explosive forming, that is highly energy efficient. Comparing spherical pressure vessels with other shapes of vessel, they have a higher load capacity, have less area per unit volume, have better economic benefit and have a nicer appearance: of late, they have been used increasingly.
1. Introduction
The classical technology of manufacturing spherical vessels requires the use of huge presses and special dies, it is costly and it gives less size selection, so that few such pressure vessels of less than 100 m 3 capacity have been made. The Warsaw Institute of Precision Mechanics of Poland has studied the explosive forming of elements of spherical vessels [1 ], but has been restrained by the problem of special dies. Wang Zhangren of Harbin University of Industry of China invented the technology of hydraulic expansion forming of spherically-inscribed polyhedrons to manufacture spherical vessels [2], but the quality of the welding cannot be assured because of the large numbers of welding seams. The technology of the explosive forming of a spherical vessel without dies is first to water-fill a pre-manufactured spherically-inscribed multi-cone container and then to explode it into a spherical shape (Fig. 1 ). Through technological test and theoretical analysis, the principle of the design of the structure of a multi-cone spherical container has been obtained, the equation for the calculation of the charge size has been validated and the technological relationships have been found. As a result, both experimental and theoretical bases 0924-0136/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
136
Fig. 1. Two-cone container before and after explosive forming.
Fig. 2. Sketch of the forming principle and testing equipment: 1 - container shell; 2 spherical impulsive wave; 3 - explosive charge; 4 - water; 5 - circumscribed spherical surface; 6 - buffer.
have been provided for the application of the technology, and this new method for manufacturing small and medium-sized spherical pressure vessels, which is highly energy-efficient, has a bright future. 2. Principle of forming
Explosive forming with dies relies mainly on the dies to control the shape and size of the products. However, in explosive forming without dies, there is relative movement between each mass point of the sheet metal under dynamic load. In the process of deformation, dynamic energy is turned into work of deformation and the velocity of each mass point is rearranged, falling to zero as the product assumes its final shape. The major process of manufacturing a spherical vessel by non-die explosive forming is as follows (see Fig. 2 ) : first, two or more conical frustums are rolled; next, they are welded into a spherically-inscribed container; the latter is then filled with water and a spherically-shaped explosive charge detonated at the centre of the container. The spherical impulsive wave created in the water will firstly push the surfaces that are closest to the centres of the sphere. The energy density of the spherical impulsive wave attenuates approximately in an inverse relationship with distance R, and its peak pressure decreases with increase of distance. Consequently, the velocity difference of the mass points pushes the shell very close in shape to the circumscribed sphere. It is necessary to control the energy of the explosive charge so as to make the container just
137 overlap the circumscribed sphere, whilst at the same time not allowing any of the circumferential welding seams to participate in the deformation. 3. Structure design and analysis
In order to achieve the best effect, comparisons and tests have been made on various spherical structures, two principles being obtained: ( 1 ) the multicone spherical container should be an inscribed one; (2) the length of generatrix of each conical frustum should be equal to the diameter of its base plate. Figure 3 is the structure sketch of a four-cone spherically-inscribed container, with the following geometrical parameters o~=180°/(n+ l)
(1)
di = D cos(ia)
(2)
Di = D cos( ( i - 1)o~)
(3)
Hi = D [sin (ia) - s i n ( ( i - 1 ) a ) ]/2
(4)
~i =
(5)
2 arccos (Hi~a)
a = D sin(o~/2) = d
(6)
Wm.x=D[1-cos(a/2)]/2
(7)
where: a is the central spherical angle faced by each cone and base plate; N is the total number of conical frusta; i is the number of the conical band starting from the equator; di is the smaller diameter of the ith frustum; Di is the larger
I
Fig. 3. Structure sketch of the container.
Fig. 4. Drawing of the ith unfolded conical frustum.
138
diameter of the ith frustum; fli is the conical angle of the ith frustum; a is the length of the generatrix of each conical frustum; d is the diameter of the base plate; and Wmax is the maximum deforming degree. Every conical frustum is rolled from sectorial sheet metal. Figure 4 is the drawing of the unfolded ith frustum, its calculating equations being as follows Ri = a D i / (Di - di)
(s)
ri = R i
(9)
-a
OLi = 1 8 0 o D i / R i
(lO)
~i = OLi/Ni
(11)
When Ti-%<180 ° Ni = 2Ri sin (~i/2 )
(12)(13)
Mi =Ri - ri cos (~'i/2)
When yi>~180 ° Ni = 2Ri
(14)(15)
Mi =Ri [ 1 + sin ()'i/2- 90 ° ) ]
where: Ri is the external radius of the sector blank; ri is the internal radius of the sector blank; Ni is the number of sectorial pieces, i.e., the ith conical frustum is welded from Ni sectorial pieces; 7i is the central angle of the sectorial piece; Ni is the maximum chord length of the sectorial piece after division; and Mi is the maximum radial length of the sectorial piece after division. The number of multi-cones N is a major parameter, affecting the deformation of the container and the size of the blank. The effect of different N values on each parameter of deformation is shown in Table 1. To select a proper N value it is necessary to consider: (1) the effect of the cold deforming on the mechanical performance of the material. The degree of deformation in the cold processing of spherical segments recommended by many countries is less than TABLE1 Effect of the N value on the parameters of deformation
N
a
WmaJD
AS/~o
Al/lo
2 4 5 6 7 8
60 36 30 25.7 22.5 20
0.067 0.024 0.017 0.013 0.009 0.008
13.4 4.9 3.4 2.5 2.0 1.5
15.4 5.1 3.5 2.6 2.0 1.5
(°)
(%)
(%)
Note: a is the central spherical angle faced by each cone a n d base plate; Wmax is the m a x i m u m deforming degree; D is the larger diameter of the frustum; A6/5o is t h e m a x i m u m relative shrinkage of thickness; a n d A1/lo is the m a x i m u m relative extension.
139 5%, correspondingly, N>~ 4; (2) increasing the utilization ratio of the material and shortening the length of the welding seams, if the size of the sheet metal is fixed. With the aid of a computer program, the N value and the number of sectorial pieces can be optimized. 4. Calculation of the f o r m i n g e n e r g y
Over the last twenty years, a lot of test results have indicated that strain rate has a pronounced effect on the flow stress of materials [3]. However, in the process of isolated explosive forming in water, it is very difficult to establish the mechanical model with viscoplastic behaviour, under the condition of the strain rate dropping from 10 s - ' to zero. In order to facilitate engineering practice, a plastic mechanical model is still used to calculate the strain energy, with the following assumptions being made: (1) the effect of strain rate can be ignored; (2) the sheet material is an incompressible rigid-plastic strain-hardening material; (3) the equivalent strains along the generatrix of each conical frustum and along the diameter of the base plates are distributed in a parabolic manner; (4) the stress is distributed evenly over the thickness of the shell, ignoring bending effects; (5) the multi-cone container is turned into a sphere after deformation, and the circumferential welding seams do not participate in the deformation. Based on the von Mises yielding criterion, under the condition of proportional loading the increment of strain energy per unit volume can be written
as: (16)
du=,~ d~
where d is the equivalent stress; and C is the equivalent strain, given by
#=,,/(o', -a2)2+ ( a2 -a3)2+ ( a3 - a , )21x/2
(17)
e----x/~x/(E, -- e2)2+ (e2 -- e3)2+ (~3 -~ e,)2/3
(18)
Considering strain hardening, the flow curve is expressed by an exponential equation: (~----k(n
(19)
where k is the strength coefficient; and n is the strain-hardening exponent. Therefore, the strain energy per unit volume U and the total strain energy of the container ut can be determined u=
f (~d~=
k n+---i ¢n+1
i
n+---1 gn+l dv
(20) (21)
140
The total strain energy of the container Ut includes the strain energy of each cone Uc and the strain energy of its upper and lower base plates U~,. The derivation procedure is omitted N/2
Ut = 2 ~ Uci + 2 Ud
(22)
Ut =KND2J
(23)
i=l
N/2
KN = ~'--max b~.n+l {} (2--n) sin ½~ ~ cos(i-- ½) ~ + 1--cos ½~ i=1
+ ( n + l ) [ c o s ½ ~ - ½cos3 ½ ~ - }]/sin 2 ½c~}/(n+l)
(24)
where KN is coefficient related to the material, to the structural form of the container and to the degree of deformation. When the material and the container structure are fixed, then KN is constant, without regard to the spherical diameter and the sheet thickness. 5. Charge-size calculation
Ezra and Schauer from the University of Denver, USA, and Alting from the Technical University of Denmark and their colleagues have done systematic studies on the energy transferred from an underwater explosive charge to the blank [4]. With the aid of their research achievements, the effective energy E t transferred from the explosive charge to the container is known to be Et --~c~l W e
(25)
where, t/c is the efficiency of energy transfer, considering only the geometrical relationship of the container to the explosive charge (for a spherical shell, ~/c= 1 ); Yl is the efficiency of energy transfer when the effect of pressure transfer medium is considered, which includes the energy of the first impulsive wave and the second loading; W is the weight of the explosive charge; and e is the specific energy of the explosive material. Zheng Zhemin from the Mechanics Research Institute of the Academy of Sciences of China and his colleagues have made special studies on the energy utilization rate of an explosive charge when it is exploded inside the sphere [5,6]. Their test results are: ~ ~ 54.5%-56.6%, here ~1 = 55%. Since Et = Ut, the equation for calculation can be obtained, i.e., by using only the amount W of explosive material, the multi-cone container can be just expanded into a sphere, while every circumferential welding seam remains undeformed.
141 W=KND2J/j7 e = CND2J CN = KN/~Ie
(26) (27)
where CN is a coefficient dependent on the material, the structure of the container, the specific energy of the explosive material and the medium of pressure transfer. When the above factors are fixed, CN is a constant and can be determined by calculation or test. 6. Result of p r o o f tests
Experiments into the non-die explosive forming on two-cone and four-cone containers of LF21M aluminium alloy and common carbon structure A3 steel, have been carried out successfully. The experimental equipment is shown in Fig. 2. The appearances of the model aluminium two-cone container before and after expansion are shown in Fig. 1. In order to analyse and prove the shell deformation, the latitudinal true strain el and the longitudinal true strain e2 of each point should be measured and calculated with the method of mesh -coordinates. The thickness of the shell wall can be measured with an ultrasonic meter and the true strain e3 calculated. A practical example of test-model calculation is listed below. The comparison of the charge size calculated theoretically and used practically is shown in Table 2. The conclusions of the test are: (1) The equivalent strain along the generatrix of the cone and the diameter of the base plate is distributed in a shape of parabola, as is the reduction in thickness of the wall. (2) The maximum reduction in thickness, the maximum equivalent strain at the central point of the generatrix of the cone and that at the central point of the diameter of the base plate are approximately equal, and given by emax~ln[1/cos(c~/2) ] ~ (AS/8o)max
(28)
(3) The difference between the tested explosive charge weight and the calculated weight is less than 4%, the circumferential welding seams on the container are virtually undeformed and their degree of non-sphericity is less than 3%. (4) The equation for the calculation of the charge size is sufficiently accurate for material LF21M, but for other materials, because of the different effect of strain hardening on the strain distribution and the difference in strain sensitivity, it may need some modification. (5) The explosive charge size is in direct proportion to the square of the sphere's diameter D, and the thickness of the sheet metal. Under the condition that the material type, the structural form of the container, the specific energy
142 TABLE 2
Comparison of the calculated charge Wt" with the charge used Wpb Container structure
Explosive charge
Wp (g)
Wt (g)
N=2, D = 2 1 0 cm
8 # ed c
1.0
0.96
(1) 8 # e d + 2 g c-4 and (2) 8 # ed 8 # ed
4.6
4.36
1.0
1.03
8 # e d + 2 g c-4
3.6
3.62
of= 1.5 cm N = 2, D = 400 cm 5 = 2 cm N = 4 , D = 4 0 0 cm ~ = 1.5 cm N=4, D = 6 5 0 cm fi=2 cm
material explosive medium coefficient a b c
- LF21M, K = 2 0 k g f / m m 2, n--0.18; - T N T ( e = 3 9 3 k g f m / g ) , c-4 ( e = 5 2 7 k g f m / g ) ;
- water, q=ql=55%; -
N=2 (K2=3.129 kgf/mm, C2= 1.448 101 g), N=4 (K4=0.927 kgf/mm, C4=4.289
101 g). Wt is converted into TNT weight according to the specific energy of the explosive. Wt is converted into TNT weight according to eqn. (26). ed - electric detonator.
of the explosive and the pressure transfer medium all remain unchanged, C N is a constant. In the case of the absence of accurate data of the values of K, n and e, the value of CN can be determined by means of a model test: consequently the charge size for bigger spheres can be determined by scaling, in accordance with the equations for calculating the charge size. 7. E n g i n e e r i n g
applications
The test results indicate that the technology of liquid-filled explosive forming without dies has opened a new way of manufacturing spherical pressure vessels. The method needs no special dies and it is easy to change size and variety; there is no need of forming equipment and the method is highly energy-efficient. In practical production, the energy of the explosive material is sealed in a spherical space, so that it causes no pollution, the tests being conductable under any environment. The authors have applied this technology in the manufacturing of spherical vessels of different material and of different volume, and have had successes each time. Figure 5 shows a photograph of liquid synthetic ammonia vessel manufactured for a chemical factory, which has a diameter of 4 m, a volume of 33 m 3, a wall thickness of 18 mm and is made of 16MnR steel. Checking its sphericity, the clearance on a length of 500 mm is less than 5 mm. One research institute ordered hemispherical high voltage caps, of which the material is
i
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Fig. 5. Spherical vessel (D=4 m).
Fig. 6. Four-cone container before expansion.
1Cr18Ni9Ti stainless steel, the thickness is ca. 1-2 mm, the radius is 460 + 0.5 mm, and for which the surface finish should be the same as for mirror polishing. Figure 6 is a photograph of the structure of a four-cone container before explosive forming. After the latter process, it has completely achieved the technical requirements. Figure 7 shows small-sized spheres and hemi-spheres of different material and for various purposes: they are all made by explosive forming.
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Fig. 7. Small spheres and hemi-spheres.
In practical production, usually one container is expanded in several shots, so as to avoid cracks in the welding seams caused by a larger quantity of explosive or a higher peak pressure of the impulsive wave. Before explosive forming, an overall check on the quality of welding seams is required. After explosive forming, the quality and performance of the metal have no evident difference from those in the classical cold forming of spherical segments [7]. As long as the number of cones N is no less than 4, both the reduction in the thickness and the percentage elongation of the material are lower than the international limit of degree of deformation in the cold forming of a spherical segment, which is less than 5%. 8. Prospects of application
This paper presents the research achievements in the explosive forming of spherical pressure vessels in the technical press for the first time, but it has already been applied in engineering practice. Because of its particular advantage, it can be predicted that this technology will be applied widely in the production of small- and medium-sized spherical vessels for various purposes. Further, it has particular advantages in the following respects: (1) Special materials: extra-high strength steel, martensite ageing steel and compound metal, etc. (2) Special size requirements: e.g., a spherical vessel of extra-thin wall, of which D/J > 1,000; a radar-beacon sphere, a high voltage discharging cover etc., which have extremely high requirements as to the precision of size and surface polishing. (3) Special structures such as: double-layered spherical vessels (vessels to store liquid hydrogen, liquid oxygen, etc.); hemispherical domes, large-sized ellipsoidal heads, decorative spheres for architecture, and so on. References 1 H. AndrzewskiandZ. Krurpa, ExplosiveFormingofElementsofSphericalVessels, Institute of Precision Mechanics, Warsaw, Poland.
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2 Wang Zhangren, New technology of sphericalvesselmanufacturing, Research Report, ScientificResearch Department, Harbin Universityof Industry,China, 1985. 3 T.Z. Blazynski, Explosive Welding, Forming and Compaction, Applied Science Publishers, Barking, 1983. 4 A.A. Ezra, Principlesand Practice of Explosive Metalworking, IndustrialNewspapers Ltd., 1973. 5 Xiang Zensheng, Approach to mechanism of spherical explosive forming, Research Report, Mechanics Research Institute,the Academy of Sciences of China, 1964. 6 Zheng Zhemin, Energy criterionand deformation calculationof spheres, Research Report, Mechanics Research Institute,the Academy of Sciences of China, 1964. 7 M.A. Meyers and L.E. Murr, Shock Waves and High-Strain-Rate Phenomena in Metals: Concepts and Applications, Plenum Press, New York, 1981.