The free radial expansion of thin cylindrical brass tubes using explosive gas mixtures

Int. J. Mech. Sci. Pergamon Press. 1968. Vol. 10, pp. 385-401. Printed in Great Britain

THE F R E E RADIAL E X P A N S I O N OF THIN CYLINDRICAL BRASS T U B E S USING EXPLOSIVE GAS M I X T U R E S W. A. PoY~TO~,* F. W. T~AV~% and W. Jom~so~$ (Received 10 .November 1967) Summary--Experiments are described in which the ignition of stoichiometrie mixtures of hydrogen and oxygen at various pressures was used to cause the impulsive expansion of thin.walled brass tubes. The speed of radial expansion and the strain distribution for typical specimens are presented and the contribution of reflected pressure waves is demonstrated. The use of an internal baffle to secure a required substantial local deformation of a tube is explored and the results are discussed, along with the consequences of providing nominally simultaneous initiation from each end of the tube. Explanations for the vase-like appearance of some of the expanded tubes are given, based on a discussion of the burning and detonating characteristics of the explosive gas mixture. INTRODUCTION DURING the last decade m a n y new methods of metal working have been investigated; one such group of methods is t h a t collectively described as "High-rate Forming". Particular interest has centred on high-rate sheet metal forming in which the workpiece is impulsively loaded and t h e r e b y given a high velocity. The resulting kinetic energy of the sheet is largely dissipated plastic° ally in achieving a required formed shape. Underwater chemical explosive1 forming is probably the most well known, but other i m p o r t a n t examples are the electrohydraulics and electromagnetica forming methods. A technique which belongs in the same class as those described above, and which has often been proposed but rarely investigated, is t h a t of "Explosive Gas Forming". Very few experimental results have been published, and it is difficult to assess the practicality of the process and the relative importance of the various parameters encountered. I t thus seemed worthwhile to assess the industrial potentialities of explosive gas forming by carrying out and reporting on a series of small-scale experiments. I n an earlier paper, 4 the authors presented the results of their examination of the feasibility of using mixtures of hydrogen, oxygen and air to secure the plastic deformation of mild steel disks, in an a t t e m p t to form model pressure vessel ends. The disks were clamped peripherally at one end of a steel t ube into which the gases were metered and ignition was provided at the other end b y a glow plug. A distribution of strain within the formed disks was found which was * Rolls-Royce, Ltd., Derby, formerly Department of Mechanical Engineering, Unlver. sity of Manchester Institute of Science and Technology. t Production Engineering Section, University of Strathclyde, formerly Department of Mechanical Engineering, University of Manchester Institute of Science and Technology. Department of Mechanical Engineering, University of Manchester Institute of Science and Technology. 385

386

W.A. POYNTO~, F. W. TRAMS and W. JOHNSON

more uniform than that encountered previously in the forming of similar items using underwater high-explosive charges1; this was attributed principally to the fact that the shock waves developed were of longer duration than the shock waves developed in underwater chemical explosive forming operations.

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FIG. 1. Diagrammatic illustration of sequence of events within tube following ignition of the gas mixture. The results presented below relate to the expansion of thin-walled brass tubes using an explosive hydrogen/oxygen mixture. The tubes were sealed at their ends with the help of thick steel plates, filled with the gas mixture and the latter then ignited by either a spark or a glow plug mounted in one of the plates. During these tests the air was not evacuated from the tubes prior to charging with the gas mixture. The present method appears to be a simple and cheap way of forming tubelike components and one which is free of many of the disadvantages associated with the use of high-explosive materials. (These disadvantages are discussed briefly in ref. 4.) Refs. 5 and 6 report some favourable features in the industrial application of explosive gases to metal forming. The forming agent in the work reported in the latter papers, however, was not a detonation wave; deliberate attempts

Free radial expansion of thin cylindrical brass tubes

387

were made in the design of the rig and in the selection of the composition and pressure of the charging mixture to inhibit the induction of detonation. The values of the peak pressure were of the order of ten times the initial pressure, i.e. a b o u t h a l f o f w h a t m a y b e o b t a i n e d w i t h a f u l l y d e v e l o p e d d e t o n a t i o n w a v e . A n i m p o r t a n t a d v a n t a g e , h o w e v e r , is t h a t s u c h p r e s s u r e s a r e r e l a t i v e l y sustained, whilst the detonation wave and subsequent shock waves are of short duration. I t is advisable here to distinguish clearly between the two types of reaction referred to above; these are: (i) The ignition of a gas mixture leading to the generation of a slowly moving flame front and subsequently to the formation of a detonation wave. W a v e reflection and action are then very strong a n d very important in the forming process. (ii) Ignition which leads to the generation of a slowly moving flame front which gradually consumes all the reactants. A t a n y given instant in time the pressure is uniform throughout the tube, i.e. wave action is very weak and unimportant. The present equipment was designed to make use of reaction (i) above. The two kinds of wave which p l a y important roles in securing the deformation of the tube when this reaction takes place are: (1) Detonation wave--This is a steep-fronted pressure wave which has reactants (i.e. the mixture of oxygen and hydrogen) on the low-pressure side and products (i.e. the water vapour) on the high-pressure side. There is a steep-fronted pressure change together with a chemical change. (2) Shock wave--This is a stcep-fronted pressure wave which has the same gas on the high.pressure side as it has on the low-pressure side. The sequence of events and the occurrence of the detonation wave and reflected shock waves m a y be clarified b y reference to the diagrammatic illustration presented as Fig. 1.

MATERIALS,

EQUIPMENT

AND PROCEDURE

Tube specimens Throughout the tests 3 in. o.d., ~ in. wall thickness, 70/30 brass tubes were employed, in lengths of up to 12 in. The tubes were sawn slightly oversize and their ends turned square a n d to length, after which t h e y were fully annealed. To determine the mechanical properties of the material a hydrostatic bulge test was carried out on one tube from each of the two batches of material used. Steel caps were silver-soldered onto the ends of the tubes a n d the l a t t e r then pressurized hydraulically. Readings were taken of the pressure a n d of the hoop a n d axial strains a t the mid-length of the tube, from which representative stressrepresentative strain characteristics were determined, as shown in Fig. 2(a).

Tube fixture E a c h tube used was charged with a stoichiometric m i n u t e of hydrogen and oxygen, the mixture being contained within the tube b y means of the end-plates. I n order to hold the l a t t e r firmly in place a fixture was m~nufactured, as shown in Fig. 2(b) ; the fixture also served to provide support for a set of "pin-contactors" used to determine the radial velocity of deformation of the tube, as described below. Before assembly, the ends of each tube were coated lightly with gasket sealant, so t h a t when used in conjunction with a ring of gasket material a pressure-tight joint between the tube and the end-plates was established. F o r some of the tests a cylindrical block or "baffle" was positioned a t a specific height within the tube and used to secure increased local deformation. The length of the prop, see Fig. 2(b), was adjusted to suit the required position of the baflie. A hole, ~ in. dia., a t the centre of the baffle allowed the flame front to proceed beyond the baffle and into the remaining length of the tube. E x c e p t where stated otherwise, initiation was b y a glow plug.

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from the tube. Knowing the sweep speed of the oscilloscope and the radial distance between pin stations, the mean radial velocity of the tube between such stations was easily established. W~here two independent sets of pin-contactors were employed, one at each end of the tube, the signals from each set were displayed separately using a Tektronix type 565 dualbeam oscilloscope and photographed using a Polaroid land camera. I n addition, the "gate-out" voltages from the two sweep systems were used to activate the "start" and "stop" circuits of a microsecond timer. By this means the relative time interval between the commencement of deformation at the two opposite ends of the tube was determined. Strain measurement Circumferential lines were scribed on the outside of the tubes at regular axial intervals, prior to testing. Measurement of the mean diameter and the mean wall thickness of the tube at each axial position before and after testing enabled engineering hoop strains ( D - D o ) / D o and engineering thickness strains (t-to)It o to be determined. (Subscript 0 refers to initial values.) ExperimentaZ procedure The experimental procedure adopted in carrying out these tests is reported in ref. 4 a n d in somewhat greater detail in ref. 9. Briefly, the gases were metered into the system separately from compressed gas bottles (200 lbf]in~ m a x i m u m pressure), use being made of simple push-in connectors. The amount of each gas admitted was determined using a pressure gauge (the mass of each gas was calculated using the Gibbs-Dalton law) and after charging, the pressure gauge and the supply line were isolated by stop valves. (Attention might be drawn at this point to ref. 10, which lists the precautions to be taken in the storage, handling and use of gas cylinders.)

RESULTS

Fig. 4(a) is a photograph of the final shape of tubes in which a mixture has been exploded, and shows the effect of decreasing the length of the tube whilst maintaining the charging pressure constant. The manner in which the distribution of hoop strain in the formed tube varies with the length of tube tested is shown in Fig. 5(a). For typical formed specimens, Fig. 5(b) shows the distribution of measured hoop and thickness strain and also calculated length strain (in the plane of the thickness). The latter value is derived from the former values on the usual plasticity assumption of volume constancy. Fig. 4(b) shows the effects of initiating the gas mixture from both ends of the tube. A spark plug, supplied with its own coil, was mounted in each of the end-plates of the fixture. The low-tension circuits of the two coils were completed using a common switch, so that upon opening the switch the two plugs were sparked simultaneously. Fig. 4(b) also shows the different results obtained when the spark plugs were mounted with their electrodes set some distance into the end-plates (instead of being flush with the inner surface, as in all other tests). The distribution of hoop strain for each of the formed tubes of Fig. 4(b) is shown in Fig. 5(c). Tests were carried out in which a loose-fitting (approximately 0.005-0.010 in. radial clearance) internal baffle was placed at different positions within the tube, all other factors remaining constant, in order to produce a local increase in deformation. Hoop strain-curves for these tests are shown in Fig. 6(a). The results are presented in Fig. 6(b) of further tests in which the baffle was rendered solid in the tube b y casting on top of it a n approximately ¼ in. thickness of tube-bending alloy, whilst retaining the central hole in the baffle at ½in. dia. Using the pin-contactor set-up of Fig. 3, the displacement-time history for both ends of deforming tubes was recorded. Results are presented in Fig. 7 for tubes of 10 in. length, tested at charging pressures of 200 and 150 lbf/in~.

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Fig. 7. T h e a v e r a g e t i m e b e t w e e n t h e c o m m e n c e m e n t of d e f o r m a t i o n a t t h e b o t t o m a n d t h e t o p e n d o f t h e t u b e is 105 psee, c o r r e s p o n d i n g t o a v e l o c i t y o f t h e reflected p r e s s u r e w a v e o v e r t h e 10 in. t u b e l e n g t h of 7930 ft/sec. T h e h y p o t h e s i s o u t l i n e d a b o v e h a s i n t r o d u c e d f a c t o r s t h a t r e q u i r e a closer e x a m i n a tion, and these are now considered individually. I n d u c t i o n length T h e earliest w o r k o n t h e m e a s u r e m e n t of i n d u c t i o n l e n g t h , a c c o r d i n g t o J o s t Is, s e e m s t o h a v e b e e n c a r r i e d o u t in t h e 1920's b y L a f i t t e a n d D u m a n o i s . T h e i r findings o n h y d r o g e n / o x y g e n m i x t u r e s c o m p a r e well w i t h a m o r e r e c e n t r e p o r t b y B o l l i n g e r e$ al. 18. T h e latter show how the induction length varies with composition, initial pressure and initial temperature. I t c a n b e c o n c l u d e d t h a t t h e i n d u c t i o n l e n g t h increases as (i) t h e i n i t i a l p r e s s u r e d e c r e a s e s ; (ii) t h e i n i t i a l t e m p e r a t u r e i n c r e a s e s ; (iii) t h e degree of d e p a r t u r e f r o m t h e s t o i c h i o m e t r i e c o n d i t i o n s ; (iv) t h e c h a m b e r d i a m e t e r increases. (Bollinger et al. Is r e p o r t t h a t t h i s is o n l y s i g n i f i c a n t w i t h fuel-rich m i x t u r e s . ) I n a d d i t i o n t o t h e s e p a r a m e t e r s , Lewis a n d F r i a u f ~4 s t a t e t h a t t h e m e t h o d o f i g n i t i o n a n d t h e c o n d i t i o n of t h e i n t e r i o r surface of t h e t u b e h a v e a m a r k e d effect. F o r a h y d r o g e n / o x y g e n m i x t u r e , t y p i c a l v a l u e s of i n d u c t i o n l e n g t h , w h e n u s i n g a t u b e o f 15 m m dia. a n d a n i n i t i a l t e m p e r a t u r e of 100°F, a r e g i v e n i n T a b l e 1, t a k e n f r o m ref. 13. TABLE

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I N D U C T I O N LEI~GTH M E A S U R E D I N C E N T I M E T R E S

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156 53 45 31

A 118 A A

A : n o d a t a t a k e n ; B : d i d n o t d e t o n a t e ; C: d a t a n o t reliable. B y i n t e r p o l a t i o n with t h e s e values, it is possible to a r r i v e a t a n i n d u c t i o n l e n g t h for a s t o i c h i o m e t r i c m i x t u r e a t a t o t a l p r e s s u r e of 200 l b f / i n 2 (13.3 a r m ) , of a p p r o x i m a t e l y 4 in. T h i s l e n g t h m u s t b e c o n s i d e r e d as a p p r o x i m a t e , since t h e t u b e of ref. 13 w a s s m a l l e r i n d i a m e t e r t h a n t h a t of t h e p r e s e n t tests, t h e i n i t i a l t e m p e r a t u r e w a s h i g h e r a n d t h e inside surface of t h e t u b e was h o n e d t o a m i r r o r finish. T h e c h a n g e i n t h e n a t u r e of t h e s t r a i n d i s t r i b u t i o n , or s h a p e , b e t w e e n s p e c i m e n s 5 a n d 6 o f Fig. 4(a) s u g g e s t s t h a t t h e d e f o r m a t i o n m e c h a n i s m c h a n g e s b e t w e e n t u b e l e n g t h s of 5 a n d 3 in. T h u s , i t is p r o b a b l e t h a t d e t o n a t i o n occurs for t h e f o r m e r case a n d t h a t t h e m e c h a n i s m is as o u t l i n e d a b o v e , w h i l s t in t h e l a t t e r case d e f o r m a t i o n is d u e s i m p l y t o t h e u n i f o r m p r e s s u r e p r o d u c e d b y c o m b u s t i o n of t h e gas, as i n Ref. 5, a n d as discussed earlier. T h e t e s t s p e r f o r m e d w i t h a n i n t e r n a l baffle also s u g g e s t t h a t t h e i n d u c t i o n l e n g t h is less t h a n 5 in., see Fig. 6, since local b u l g i n g does n o t o c c u r w h e n t h e baffle is m o v e d closer t h a n 5 in. f r o m t h e i g n i t i o n end. I n a n earlier p a p e r , 4 t h e a u t b o r s d e m o n s t r a t e d t h a t t h e s t o i c h i o m e t r i c c o n d i t i o n m a y n o t b e t h e o p t i m u m as far as t h e m e t a l - f o r m i n g c a p a b i l i t i e s of t h e d e t o n a t i o n w a v e is c o n c e r n e d ; i n t h e i r earlier work, c i r c u l a r m i l d steel b l a n k s were c l a m p e d a t o n e e n d of a 3 in. i.d., 36 in. long t u b e a n d c o m b u s t i o n was i n i t i a t e d a t t h e o t h e r end. F o r h y d r o g e n / oxygen mixtures, the largest deformations occurred with hydrogen contents in the two regions o f 35 a n d 77 p e r cent. I t was also s h o w n t h a t t h e p r e s e n c e o f a i r i n a n o t h e r w i s e s t o i e h i o m e t r i c m i x t u r e ( e q u i v a l e n t t o a w e a k m i x t u r e plus s o m e n i t r o g e n ) s i m i l a r l y h a d t h e effect of i n c r e a s i n g t h e d e f o r m a t i o n of t h e b l a n k . W i t h t h e s e r e s u l t s i n m i n d , t e s t s were p e r f o r m e d o n 10 in. l e n g t h b r a s s t u b e s w i t h gas m i x t u r e s h a v i n g t h e s e c o m p o s i t i o n s . H o w e v e r , l i t t l e t u b e d e f o r m a t i o n t o o k place a n d p r e s u m a b l y d e t o n a t i o n d i d n o t occur.

397

Free radial expansion of thin cylindrical brass tubes Table 1 indicates miYture, and an even mixtures, detonation Bollinger et a/ . do not there is an increase in

an induction length of about 17 in. for a 77 per cent hydrogen greater length for one of 35 per cent hydrogen. Hence, with such would not occur in the tubes of the present tests. Although tabulate results for hydrogen/oxygen/air mixtures, they show t h a t the induction length compared with the stoichiometric condition.

Detonation The phenomenon of detonation was discovered 16 in 1881, and by 1905 a widely applicable theory was developed. 16 This was progressively refined, and by 1930 extremely good correlation between theory and experiment was available, x4 Ref. 14 sets down clearly the operations to be performed for calculating detonation characteristics from a knowledge of the initial conditions of the mixture; the complications introduced when gaseous dissociation arises are also discussed. The most recent calculations for H s / 0 a appear to have been performed by Be]linger and Edse 17, where initial pressures of up to 100 arm are considered, and also elevated temperatures. Table 2, taken from rcf. 18, gives values of detonation characteristics, as calculated in ref. 14, on the assumption t h a t there is gaseous dissociation. Also given, for purposes of comparison, are experimental values of detonation velocity, i.e. the velocity of the detonation wave, and values calculated assuming no dissociation. All mixtures are considered to be at an initial pressure and temperature of 1 arm and 291°K respectively. TABLE 2. DETONATION CHARACTERISTICS

Explosive mixture (2Hz + (2H 2 + (2H2 + (2Hz ÷ (2H 2 + (2H 2 + (2H2 +

03) 03) + 02 03) + 30~ 03) + 5Oz Oz) + 1~2 0~) + 3N 03) + 5Nz (2H~ + Oz) + 2H 2 (2H z + 03) + 4H 2 (2H~ + Oz) + 6H8

P2 arm 18.05 17.4 15.3 14.13 17-37 15-63 14.39 17"25 15-97 14-18

Detonation velocities (ft/sec)

Ta °K 3583 3390 2970 2620 3367 3003 2685 3314 2976 2650

A

B

C

2806 2302 1925 1732 2378 2033 1850 3354 3627 3749

2819 2314 1922 1700 2407 2055 1822 3273 3527 3532

3278 2630 2092 1825 2712 2194 1927 3650 3769 3802

A: calculated assuming dissociation; B: experimental; C: calculated assuming no dissociation. I t appears t h a t contrary to what is the case for induction length, the detonation velocity varies little with initial pressure and temperature ; it is virtually independent of tube diameter (above a certain m i n i m u m diameter) and is independent of the method of ignition. The relationship between the run of the detonation wave and its intensity I m m ed i at el y after the onset of detonation, it is usually found t h a t the detonation characteristics are different from those calculated on the basis of the initial temperature and pressure of the gas mixture. This complication arises as follows, la During the induction period, compression waves are formed which travel ahead of the flame front and cause compression and heating of the unburnt gas. The longer the induction period, the greater is the compression of the unburnt gas. When detonation is induced, the detonation wave is therefore some distance behind the precompression wave, and although it rapidly overtakes the latter, initially it is travelling through unburnt gas which is at a higher pressure and temperature than originally prevailing. Under these conditions the detonation wave will be intensified. Once the detonation wave has reached

398

W. A. poxr~-TO~, F. W. TRAVXS and W. JOHNSON

the precompression wave, it travels through gas at the initial pressure and temperature, so t h a t the detonation characteristics are then as predicted from the initial conditions. Experiments by Br~nl~ley and Lewis l° support the existence of this mechanism. A conclusion to be drawn from the above is t h a t the most effective detonation wave will result if detonation does not occur until almost all the gas has been consumed by the deflagration wave. Precompression is then at a high level and results in a considerable increase in the intensity of the detonation wave. This explains, qualitatively, the observations of the earlier paper by the present authors, 4 which were referred to briefly, above. To be able to choose the most effective detonation wave, one must therefore be able to calculate where detonation occurs, the details of the detonation wave and the details of the precompression wave.

Shock-wave reflection I t is well known that when a shock wave is reflected from a rigid surface its amplitude is increased. The method of calculating this increase is discussed by Woods ~°, and from graphs given it is possible to obtain both the pressure ratio and the speed of the reflected shock wave. Unfortunately, however, the magnitude of the shock waves experienced in the present experiments is usually greater than the magnitude of those investigated by Woods. I n a paper by Gealer and Churchill sl, results are presented of detonation characteristics measured at initial pressures of up to 1000 lbf/in ~. The detonation wave is allowed to reflect from a rigid end-plate and it is shown t h a t the degree of intensification experienced is approximately 2.5 for a wide range of initial pressures. I f the end-plates of the present tests were flexible, motion of the plates would relieve the incident pressure. However, the end-plates are rigid and the intensified pressure wave is relieved only by the yielding of the tube walls. Ref. 4 shows t h a t the time constant of the pressure wave (i.e. the time for the peak pressure to fall to 1/e of its initial value) in explosive-gas forming operations is of the order of 200 ~sec. This suggests t h a t the deformation occurs over a time interval of the order of (or less than) the latter value. Fig. 7 shows t h a t the greater part of the deformation of the tube ends occurs over a period of about 70/~sec.

Operational ej~ciencies The Appendix details the operations t h a t are performed in calculating the plastic work done on the tube, the calorific energy of the gaseous m i x t u r e and hence the efficiency of the process; at the most, this value is of the order of 0.5 per cent. W i t h the deformation mechanism advanced above, a high efficiency is not to be expected because only partial expansion of the gas takes place, and at the end of the deformation process the gas pressure will still be high (or alternatively, as is envisaged, there will be a general loss of gas from between the tube ends and the end-plates on account of the axial contraction of the tube during straining). As high working temperatures are involved there will also be heat losses from the gas to the tube walls and the end-plates. I n the expansion of water-filled brass tubes of 8 ft length, 11 in. o.d. and i in. wall thickness, using axial high-explosive charges, 2~ however, efficiencies of the order of 50 per cent were realized. This is attributable in part to the high velocity of detonation of the explosive material (compared with the velocity of propagation of underwater shock waves), allowing the shock waves arriving at the inner surface of the tube to be of close-to-normal incidence. (A photograph of the shock wave developed by a detonated underwater line explosive charge is presented in ref. 23, and clearly shows t h a t the front of the shock wave progressing into the surrounding water is at an angle of about 20 ° to the axis of the charge.) I t thus appears t h a t if explosive gas forming is to be applied to tube expansion, a method of inducing detonation in a radial direction would be advantageous.

Attempta to localize bulging (a) Simultaneous ignition. Once the mechanism of deformation of the tube ends is clear, the idea suggests itself of providing simultaneous initiation from each end of the tube so t h a t conditions equivalent to normal incidence on a perfectly rigid surface will result

Free radial expansion of thin cylindrical brass tubes

399

when the two detonation waves meet in the vicinity of the tube mid-length. I t was anticipated that a greatly increased local deformation would then result at this point. Results of such tests are presented in Figs. 4(b) and 5(c), from which it may be deduced that (i) detonation may not have been completely induced because of the different method of ignition (spark plug instead of glow plug) and also because the effective tube length is halved, re~ulting in only a slight deformation near the tube mid-length; (ii) this technique is unsatisfactory for locating with any accuracy, the region of increased expansion; it is well known from work on spark-ignition petrol engines that the generation of the flame front depends critically on the nature of the ~rnall sample of gas in the vicinity of the spark, and even if the two sparks are fired at the same time the generation of the two flame fronts may occur at different times; (iii) even less deformation appears to be secured using buried electrodes, which is probably attributable to the effect on the induction length of the initiation process. (b) Use of an internaZ ba~e. An even simpler technique in providing increased local deformation of the tubes was then considered; an internal ba~e, as shown in Fig. 2(b), was used to diffract incident pressure waves. Results of these tests are shown in Figs. 6(a) and (b). The ba~e is seen to be effective, provided that it is positioned a sufficient dis. tance from the point of initiation; this is in agreement with previous induction length considerations. Where the time interval between the commencement of the local deformation and the deformation at the initiating end of the tube was determined, as described earlier, similar results were obtained; thus it can be concluded that the deformation mechanism is unchanged in principle in these tests; however, it will be noted that wave action occurs separately in each of the two parts of the tube separated by the baffle. If the baffle is positioned too close to the initiation end of the tube, greater bulging is experienced at its rear face, presumably due to incidence of the pressure wave reflected from the bottom end-plate. I t will be noted that whilst the use of a baffle can provide local straining at a selected point, the maximum hoop strains are less than those occurring at the ends of the tube in its absence. During these tests it was found that a certain amount of asymmetry was developed in the local bulges. This was initially attributed to an angular variation in the radial clearance between the baffle and the tube, due to a slight ovality of the latter. In the tests of Fig. 6(b) a layer of Fry's tube bending alloy (which has the advantage of expanding upon solidification) was poured on top of the baffle, in an attempt to e]imlnate the effect of the tube ovality. It was found that the previously encountered asymmetry was only slightly reduced by the presence of the cast layer. A peculiarity in the strain curves of Figs. 6(a) and (b) may be noted, in that for the test where the baffle had been set at 5 in. from the ignition end of the tube, deformation was almost entirely confined to that part of the tube beyond the baffle.

CONCLUSIONS The use of d e t o n a t i n g gas m i x t u r e s for t h e free (or dieless) expansion of thin-walled t u b e s appears m o s t suited to t h e p r o d u c t i o n of localized bulging, r a t h e r t h a n to securing a general e x p a n s i o n ; for t h e l a t t e r operation preference m a y advisedly be given to t h e use of line high-explosive charges. I f necessary, split dies could be used to f o r m such local bulges t o a specific contour. H o w ever, t h e n u m b e r of bulges t h a t can be f o r m e d a n d their positioning, using t h e present t y p e o f set-up, is severely limited b y i n d u c t i o n length a n d shock-wave reflection considerations. Explosive gas f o r m i n g is a t an a d v a n t a g e in t h e circumstances outlined in ref. 4, where t h e aim is t o p r o d u c e a dished c o m p o n e n t f r o m a fiat workpiece, a n d f u r t h e r w o r k is in h a n d ~4 to investigate t h e forming o f large c o m p o n e n t s using this m e t h o d .

400

W . A . PoY-ff~oN, F. W. TRoves and W. JOHNSON

Acknow~nta~The authors wish to acknowledge the financial support of the Science Research Council in this work. During the course of these investigations, the first author was the recipient of the Sir Charles Renold Fellowship, for which he would like to express his gratitude to Renolds Ltd. Thanks are duo to W. Heywood and G. Blackburn for their practical assistance throughout the tests, and to Dr. W. A. Woods for his helpful discussion of this work. REFERENCES I. W. JOHNSON, W. A. Port,toN, I-I. SINGH and F. W. TRAY/S, Int. J. Mech. Sci. 8, 237 (1966). 2. J. L. DUNCA-~ and W. JoHNson, Prec. Inst. mech. Engrs 179, Part 1, No. 7, p. 234 (1964-65). 3. S. T. S. AL-HASSA_~I,J. L. DUNC~'~ and W. JOHHSO~, Prec. 8th Int. M . T . D . R . Conf. (Manchester), Pergamon Press, September (1967). 4. W. JOHnSOn, W. A. POY~TON and F. W. TRAWS, Int. Conf. Manufact. Tech., C I R P ASTME, pp. 801-815. A n n Arbor, Michigan, September (1967). 5. High Velocity Working of Me2xds. ASTME. Prentice Hall, New Jersey (1964). 6. J. M~.T.~.Rand P. Y~USE, Metal Prog. August (1961). 7. S. M_r~Snu,.T.,J. Appl. Phys. 26, No. 4 (1955). 8. W. JOHNSO~¢, K. K O R ~ and F. W. T~AvIs, Int. J. Mech. Sc/. 6, 287 (1964). 9. W. A. POYN'rON, Ph.D. Thesis, University of Manchester (December 1967). 10. Fire Prevention Association, London, Technical Information Sheet 3017 March (1966). 11. Underwat. Explos. Res. 3, (1950). 12. W. JesT, Explosion and Combustion Processes in Gazes. McGraw-Hill, New York (1946). 13. L. E. BOT.T.r~G~.R,M. C. FONG and R. EDS~., J. Am. Rocket Soc. May (1961). 14. B. L~.wIs and J. B. F ~ L ¢ ~ , J. Am. Chem. Soc. 52, 3905 (1930). 15. M. B~'~'maLOT and P. VIELLE, C. r. hebd. Sdanc. Acad. Sci. Paris 93, 18 (1881). 16. E. JOUaUET~ J. Math. 347 (1905). 17. L. E. BoI~rNGER and R. EDS~., J. Am. Rocket Soc. F e b r u a r y (1961). 18. B. LEwis and G. y o n ELB~, Combustion, Flames and Explosions of Gazes. Academic Press, New York (1961). 19. S. R. B ~ n ~ Y and B. LEWIS, 7th Syrup. on Combustion, p. 807. Butterworths, London (1959). 20. W. A, WooDs, Prec. Inst. Mech. Engrs 186, Part 3J (1965-66). 21. R. L. G~.ALE~and S. W. CHtrRCmLT.,J. Am. Chem. Engrs 6, September (1960). 22. W. JOHNSON, E. DOEGE and F. W. TRAWS, Prec. Inst. Mech. Engrs 179, Part 1, (1964-65). 23. W. S. H o ~ s , Conf. Engng Mater. Design, Paper 11 (London), November (1963). 24. W. JOHHSO~, W. A. POY~TON and F. W. TRAWS, Prec. 9th. Int. M . T . D . R . Conf. (Birmingham), September, (1968). 25. D. G. DALRYM'PL~.and W. JohNson, Int. J. Mech. Sci. 8, 353 (1966). APPENDIX

(a) Calculation of plastic work done The method of calculation follows that in ref. 25. The work of plastic deformation per unit of undeformed length is _ f ~av W = 21rrt| 5de

(1)

jo

where gay is the average value of the representative strain over unit length of the tube, r is the initial radius of the tube and t is the thickness of the tube. The representative strain g can, with reasonable accuracy, be taken to be equal to the natural hoop strain ea at a point. Thus, if the representative stress-representative strain relationship for a work-hardening material is written in the form A+Bg n (2) =

F r e e r a d i a l e x p a n s i o n of t h i n c y l i n d r i c a l b r a s s t u b e s

401

t h e n e q u a t i o n (1) b e c o m e s

w F o r t h e r e s u l t s o f Fig. l(a), e q u a t i o n (2) b e c o m e s 5 =

2 5 , 6 0 0 + 87,500g °'.8 l b f / i n ~

I f t h e v a r i a t i o n of W w i t h eh h a s b e e n e s t a b l i s h e d , v a l u e s o f p l a s t i c w o r k d o n e c a n b e t a b u l a t e d for all t h e t u b e s u s e d i n t h e p r e s e n t tests. T h e T a b l e b e l o w s h o w s t y p i c a l v a l u e s for t u b e 1 of Fig. 4(a).

Element of tube

Average natural h o o p s t r a i n ea

Plastic work on element in lbf

0 - ½ ½- 1 1 - 2 2 - 3 3 - 4 4 - 5 5 - 6 6 - 7 7 - 8 8 - 9 9 -10 10 - 1 1 11 -11½ 11~-12

0.331 0.270 0.180 0.115 0-060 0.037 0.024 0.018 0.017 0-018 0.028 0.078 0.241 0.382

56O 425 495 285 135 80 50 40 35 40 60 135 370 673

T o t a l p l a s t i c w o r k d o n e is 3433 in lbf. (b) Calculation of calorific energy and process efficiency C o n s i d e r i n g t u b e 1 o f Fig. 4(a), t h e v o l u m e o c c u p i e d b y t h e gaseous m i x t u r e (2H s + Oz) is 12rr 1.52 i n s. T h e p a r t i a l p r e s s u r e o f h y d r o g e n i n t h i s m i x t u r e a t a t o t a l p r e s s u r e o f 150 l b f / i n 2 is 100 l b f / i n ~, t h u s t h e v o l u m e o c c u p i e d b y t h e h y d r o g e n a t S.T.P. is g i v e n b y

v=

273 100 121r 1.52fta,

~'1-~.7"

or

0.317 f t a.

12 3

A s s u m i n g t h e lower calorific v a l u e of h y d r o g e n t o b e 288 B . t . u . / s t a n d a r d f t a, t h e calorific e n e r g y o f t h e m i x t u r e is g i v e n b y E = 288.0.317 B . t . u .

or

8"58.105 in l b f

I f t h e process efficiency is defined as t h e r a t i o o f t h e p l a s t i c w o r k d o n e i n d e f o r m i n g t h e t u b e t o t h e t o t a l calorific e n e r g y s u p p l i e d , t h e n Efficiency -

3-433.103 8-58.105 x 100 -- 0.4 p e r c e n t .