Detonation driving rules for cylindrical casings under asymmetrical multipoint initiations

Yun Li ,Xiogng Li ,Yuqun Wen ,* ,To Suo

a Institute of Extreme Mechanics and School of Aeronautics, Northwestern Polytechnical University, Xi'an, 710072, China

b State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing,100081, China

Keywords: Multipoint initiation Fragment shape Velocity distribution X-ray photography

ABSTRACT The detonation wave-aiming warhead can effectively enhance the lethality efficiency.In the past,rules for casing rupture and velocity distribution under asymmetrical initiations were not adequately investigated.In this study,X-ray photography and numerical modelling are used to examine the casing expansions under centre point,asymmetrical one-point,and asymmetrical two-point (with central angles of 45° and 90°) initiations.The results indicate that early casing ruptures are caused by local high pressures,induced by the initiation,detonation wave interaction,and Mach wave onset.The fragment shapes are controlled by the impact angle of the detonation wave.The fragment velocity distributions differ under different initiation types,and the end rarefaction waves can affect the velocity distribution.This study can serve as a reference for the design and optimization of high-efficiency warheads.

1.Introduction

The fragments produced from a centre-initiated warhead spread uniformly around the warhead;however,the fragment velocities can be significantly increased by employing an asymmetrical arrangement of the initiation points.Based on this characteristic,researchers have developed an asymmetrically initiated aimable warhead,also called the detonation wave-aiming warhead[1].The asymmetrically initiated aimable warhead is not a new concept[2],and many studies have focused on this topic,especially the velocity enhancement [3-6].However,the radial fragment velocity distributions are also very important for clarifying the mechanism of the detonation driving and evaluating the damage efficiency considering the aiming precision [7].

Ye[3],Zhu[4],and Li[5]investigated the velocity enhancement under different asymmetrical initiations using wire screens,tinfoil papers,and laser screens,respectively.In their studies,only fragment velocities in a few directions were obtained,due to the limits of the test methods.The entire fragment velocity distributions cannot be precisely depicted using the limited data.X-ray photography is an efficient method to capture the entire fragment radial dispersion,provided the X-ray tubes are well protected.Using this method,Marnott [8]measured the dispersion of premade fragments in the radial directions,and obtained the fragment velocity distribution under asymmetrical one-line initiation.Similarly,Feng [9]and Huang [10]conducted X-ray tests on fragment velocity distributions of the natural casings under asymmetrical one-point initiation.The obtained fragment velocity distributions provide a solid basis for numerical modelling verifications and theoretical studies.However,compared to the asymmetrical onepoint initiation,the asymmetrical two-or three-point initiations are more powerful initiation strategies [6,11].The experimental data of fragment velocity distributions under asymmetrical twopoint initiation is urgently demanded.

Using X-ray photography,Zhang [12]and Wang [13]obtained the fragment velocity distributions under asymmetrical two-point initiations;however,their tested charges are with holes.The driving processes of the hollow charges under asymmetrical initiations are different from the solid charge configurations [14,15].There are no experimental reports about the fragment velocity distribution of solid charge under asymmetrical two-point initiation.The relative numerical modelling and theoretical studies lack a solid basis [16],constraining the disclosure of the detonation driving mechanism of asymmetrical two-point initiations.More importantly,although the casing fragmentation under centre initiations has a long research history [17,18],the casing rupture rules under asymmetrical initiation are seldomly studied and no relative reports were found.The rules of casing rupture and velocity distribution under asymmetrical multipoint initiations need to be investigated.

In this study,X-ray tests on the detonation driving of natural casings under asymmetrical multipoint initiations were firstly performed.Then,numerical simulations of the experimental configurations were conducted.The relative casing rupture and fragment velocity distribution were analysed to elucidate the driving mechanism of charges under asymmetrical multipoint initiation conditions.The main findings and outlooks are summarized in the conclusion.

2.Experiments

2.1. Experimental setup

A schematic of the test setup is shown in Fig.1.Two parallel Xray tubes(Scandiflash CVR 450)were used to obtain images of the detonation driving process at different instants.Initially,a static image was captured by one X-ray tube,and after detonation,two sequential dynamic images were captured by two X-ray tubes on the digital film.Two reference balls were aligned vertically under the charge.The balls have two functions.First,based on the actual and captured ball sizes,the amplification ratio of the image was obtained.Second,according to the relative ball positions,the X-ray profiles obtained at different instants were aligned to facilitate the subsequent processing.

Fig.1.Schematic of the test setup.

The actual test setup is shown in Fig.2.The test specimen was arranged between the X-ray tubes and film using transparent tapes.A laser pencil was used to align the charge and one X-ray tube.To prevent damage to the film,an aluminium plate was placed before the film,except for specimen C-1,which was protected by a steel plate for trial.The X-ray tubes are protected by Plexiglass plates.Moreover,witness plates are arranged beneath and beside the test specimens not only to provide protection in other directions but also to record the fragment distribution.

Fig.2.Test setup.

2.2. Specimens

Guo [19]demonstrated that axial rarefaction waves affect the radial distribution of fragment velocities;thus,the rarefaction waves should be suppressed in experiments.In the study by Zhang[12]on rod driving of an asymmetrical two-line initiation,a charge longer than the casing was adopted to reduce the rarefaction waves.A similar method was used in this study.The charge length was set as 2.77 times the casing length,as depicted in Fig.3.An 8701 explosive with charge dimensions of φ 40 ?36 mm was used.The outer dimensions of the 1045 steel casing were φ 45 ?13 mm.The casing was assembled at the middle of the charge length.Different initiation strategies,namely end-face centre initiation,asymmetrical one-point initiation,and asymmetrical two-point initiation(with central angles of 45°and 90°),were utilised.To assemble the initiation boosters,holes of 5 mm diameter were drilled on the casing.The different initiation strategies were achieved via detonation cords,as illustrated in Fig.4.

Fig.3.Charge and casing specimens.

Fig.4.Initiations via detonation cords:(a)Asymmetrical one-point;(b)Asymmetrical two-point.

Table 1 lists the test specimen numbers,charge densities,and Xray images capture instants of the different initiation types.The trigger of the X-ray tube was controlled by the initiation signal of the detonator for convenience.The time instants at which the images were obtained,were determined via premodelling.The details of the premodelling are identical to the following numerical modelling and can be found in section 3.The fragment acceleration was completed after 20 μs of the charge detonation.Therefore,the two time instants were set as 20 μs and 44 μs(42 μs for the centre initiation)from the charge detonation.However,as shown in Fig.4,there is a detonation transfer time associated with going from the detonator to the main charge through the booster,detonation cord,adaptor,detonation cord,and booster again.This time should be added to the intended time instants to obtain the actual time of image capture based on the detonator trigger.For the different initiation types,the detonation transfer times differ.Based on the detonation cord length,they were roughly computed as 6,6,9,and 10 μs for the end face centre initiation,asymmetrical one-point initiation,and asymmetrical two-point initiation of 45°and 90°,respectively.Subtracting the transfer time from the actual time yields the charge detonation time.However,owing to factors such as differences in the detonation cord length,transfer error between the detonation cord and booster [20],and operation error,the obtained charge detonation time may differ.For example,for specimen A-2,the first time instant is incorrectly set earlier,resulting in a detonation time of 15 μs.

Table 1 Test parameters.

2.3. Postprocessing method

As depicted in Fig.5,after the dynamic images were obtained,the casing fracture rule and the velocity distribution of the casing were analysed.The dynamic images were contrast stretched to be much clearer for the analysis of the casing fracture,as depicted in Fig.5(b).The amplification ratios of the X-ray images were determined by comparing the actual size of the reference balls and that on the X-ray images.The amplification ratio is defined as the ratio between the image size to the actual size.The image size of the reference ball was measured using the AUTOCAD software,and the outlines of the casing at different time instants were obtained using this software.Based on the relative positions of the balls,the‘a(chǎn)lign’function in AUTOCAD was used to transform the different outlines into the same coordinate system,as illustrated in Fig.5(c).Subsequently,36 equally distributed radial lines were drawn from the centre of the static outline with an interval of 10°.Each of these lines intersects the two dynamic outlines at two points.Measurement of the distances between these two points,and dividing the distances by the amplification ratios and image capture intervals,the fragment velocities at each azimuth angle were obtained.Considering the symmetry,the fragment velocity distributions of the angle ranges of 0-180°and 180-360°should be consistent,and their average values are considered the final fragment velocity distributions.

Fig.5.Test data processing method: (a) Static image;(b) Dynamic image;(c) Processed result.

2.4. Experimental results

2.4.1.Centre initiation

The obtained X-ray images of the end face centre initiation are shown in Fig.6;the specimen numbers are included in the captions.Under centre initiation,the detonation pressure and products push the casing uniformly around the circumferential direction,resulting in uniformly distributed fractures and fragments.

Fig.6.X-ray images of the centre initiation test specimens: (a) C-1;(b) C-2;(c) C-3.

Evidently,the X-ray image of specimen C-1 is darker.For pretesting,this was the first analysed specimen,which used a steel plate to protect the film and a single reference ball of φ 19 mm.The protection plate did not deform significantly under blast loading.Therefore,to obtain clearer X-ray images,an aluminium alloy protection plate was used,and two balls of φ 8 mm were used as the reference balls.In the case of centre initiation,as no direction was selected,the balls were only used to determine the amplification ratios.To determine the fracture patterns of the casing,the X-ray images of C-2 and C-3 were contrast stretched and enlarged,as depicted in Fig.7.By instant t1,the cracks have propagated through the thickness of the casing,but the fragments were not separated from each other.This suggests that the fracturing process had just completed,and the casing diameter at this moment can be assumed to be the fracture diameter.

Fig.7.Enlarged X-ray images of centre initiation test specimens: (a) C-2;(b) C-3.

As is evident from Fig.7,the casing fracture at instantt2is relatively uniform and dominated by shear failure.The fragment thicknesses in specimen C-3 are smaller than those in specimen C-2.This indicates that the fragment dispersion distance or the time instant corresponding to C-3 is larger than that of C-2.As stated previously,although the nominal time instants are the same,the actual time instants may differ,due to errors in the charge detonation transfer paths.Moreover,many small pieces outside the main fragments exist,but their origin is currently unknown.

By measuring the diameters of the two casing outlines and dividing with the time intervals,the fragment velocities under the centre initiation were obtained (Table 2).The average value of the obtained fragment velocities is 1346.28 m/s,with a standarddeviation of 27.55 m/s.From the actual diameter at instantt1,the fracture ratios (ratio of thet1diameter to the initial diameter 45 mm)were computed as 1.82,1.84,and 1.95 for specimens C-1,C-2,and C-3,respectively.These values are consistent with a previously calculated fracture ratio for 1045 steel casing(approximately 1.84)[21].The results also revealed that the acceleration process of the casing was completed at this moment,and the average fragment velocities betweent1andt2can be considered the initiation fragment velocities.

Table 2 Velocities of centre initiation.

To examine the fragment hits on the witness plates,the three witness plates shown in Fig.2 were unfolded according to the arrangement locations,as depicted in Fig.8-Fig.10.

Fig.8.Witness plates after testing of C-1: (a) Left plate;(b) Bottom plate;(c) Right plate.

Fig.9.Witness plates after testing of C-2: (a) Left plate;(b) Bottom plate;(c) Right plate.

Fig.10.Witness plates after testing of C-3: (a) Left plate;(b) Bottom plate;(c) Right plate.

As shown in Fig.8,the witness plates were not perforated by the fragments but created narrow hit straps or dents.Evidently,there were only one fragment dent along the width of the strap.The sizes of the dents are different,and the fragment size distribution is consistent with that observed in the X-ray images.As small dents exist away from the hit straps,labels are added on the left plate image for identification.These small dents are thought to be caused by small pieces outside the fragments,as observed in the X-ray images.Therefore,these small pieces are not by-products of the casing fragmentation,but are produced from the two ends of the casing.A comparison of Fig.8-Fig.10 reveals that good agreement between the results for the three specimens,all of which produce a clear narrow hit strap.Because of arrangement problems between the plates and charges,the hit straps are not aligned with the edges of the witness plates.In the subsequent tests,it was difficult to analyse the fragment hits because of substantial overlapping,and the witness plates were not replaced by new ones in the following experiments according to the experimental design.Thus,the fragment hits on the witness plates were not presented.

2.4.2.Asymmetrical one-point initiation

The X-ray images obtained from the asymmetrical one-point initiation are depicted in Fig.11,which were also contrast stretched.To accurately determine the initiation location,the static images are superimposed on the dynamic images according to the relative positions of the reference balls.This overlapping rendered the central region of some images blurry.

Fig.11.X-ray images of the one-point initiation: (a) A-1;(b) A-2;(c) A-3.

As shown in Fig.11,tiny pieces were captured around the detonation cords in the dynamic images at t1.As listed in Table 1,the time t1 corresponding to specimen A-2 is earlier than those of specimens A-1 and A-3;thus,the image of A-2 obtained at t1 shows a small diameter.Casing breakages occurred at the initiation points and other locations,but were not clear.In the images captured at t2,the sizes of fragments around these early casing breakages were relatively larger,which indicate tensile failure.Setting the direction of the initiation point as the azimuth angle of 0°and the opposite direction as 180°,the fragment size from 0 to 180°decreased.In contrast to centre initiation,the fragment shapes were not distinguishable.However,the general shear failure pattern at 60-160°azimuth angles can still be recognised.Around 60°,there is a clear transformation from tensile to shear failure.Small pieces were observed outside the main fragments at the initiation side.The formation mechanism may be the same as that of the centre initiation.

Following the processing method described in Section 2.3,the fragment velocities at different azimuth angles were obtained(see Fig.12),with amplification ratios of 1.348,1.356,and 1.327 for specimens A-1,A-2,and A-3,respectively.The two lines in the figures represent the two velocity distributions,obtained from the two sides of the connection line from the initiation point and charge centre.Ideally,they should coincide with each other because of symmetry.As is evident from Fig.12,the two lines almost coincide for specimens A-1 and A-3 but not for A-2.As shown in Fig.11(b),the static picture of specimen A-2 has an uneven thickness.This implies that the central line of the charge is at an angle with respect to the X-ray tube,which may be the reasons for the inconsistency in the velocity distribution.

Fig.12.Fragment velocities of the one-point initiation: (a) A-1;(b) A-2;(c) A-3.

By averaging the fragment velocity distributions in Fig.12,the final fragment velocities of specimens A-1,A-2,and A-3 were obtained,as shown in Fig.13.The velocity distributions of the different specimens are consistent,and the trend of the velocity distributions approaches the shape of a sine wave.

Fig.13.Velocity distributions of the one-point initiation.

2.4.3.Asymmetrical two-point initiation of 45°

The obtained X-ray images of the two-point initiation at a central angle of 45°are shown in Fig.14,wherein the initiation locations can be clearly observed.The static images are not overlapped,as in the case of the asymmetrical one-point initiation.

Fig.14.X-ray images of the two-point initiation of 45°: (a) A45-1;(b) A45-2;(c) A45-3.

As is evident from the images in Fig.14 which were captured at instant t1,thickness reduction occurs around the initiation points.In the middle of the two initiation points (azimuth angle 0°),the casing thickens and breaks apart.On the opposite side of the initiation points(in the azimuthal 180°direction),the casing forms an outer bulge,with a decreasing curvature.In all results of the three test specimens,the casing breaks at certain locations in addition to the midpoint of the initiation points.In specimen A45-1,the breakage occurs at an azimuth angle of approximately 60°.In the other two test specimens,it occurs at approximately 80°.These casing breakages are also reflected in the t2 images.Large gaps and fragments exist at the breakage locations.The fragments between the middle of the initiation points and the other breakage locations are also large.The thinning casing in images captured at t1 ruptures into large fragments in images captured at t2.From the casing breakage location to the initiation opposite direction (180°),the fragment size decreases gradually.A general shear failure pattern at azimuth angles of 90-160°was observed.The casing portion on the opposite side of the initiation point fractures is relatively complete,and its occupied range appears thicker than the other portions,which indicate spalling.

The X-ray images were manipulated to obtain the fragment velocity distributions of the asymmetrical two-point initiation of 45°,as shown in Fig.15.The amplification ratios for specimens A45-1,A45-2,and A45-3 are 1.322,1.335,and 1.410,respectively.This shows that a good consistency between the velocity distributions obtained from the two sides of the initiation points midline.The final fragment velocity distributions obtained by averaging the results of each test specimen are shown in Fig.16.

Fig.15.Fragment velocities of the two-point initiation of 45°:(a)A45-1;(b)A45-2;(c)A45-3.

Fig.16.Velocity distributions of the two-point initiation of 45°.

As is evident from Fig.16,the results for the different test specimens are in good agreement.The fragment velocity increases rapidly from 0°.It then increases gradually and rapidly again.The curvature changes at an angle of approximately 20°.The entire velocity distribution trend resembles the shape of an exponential curve.

2.4.4.Asymmetrical two-point initiation of 90°

The X-ray images of the asymmetrical two-point initiation of 90°are shown in Fig.17.Notably,the X-ray tubes during testing of specimen A90-2 did not get triggered for unknown reasons;thus,only the static image was obtained.

Fig.17.X-ray images of the two-point initiation of 90°: (a) A90-1;(b) A90-2;(c) A90-3.

Similar to the case of the two-point initiation for 45°,the casing thickness near the initiation points decreased,as shown in the t1 images.A few breakages were observed in the envelope between the two initiation points.The breakage at the middle of the two initiation points is readily apparent.The casing near this breakage becomes thicker,which suggests the possibility of squeeze between each other.The casing on the opposite side of the initiation point(180°) bulged and broke.As shown in the t2 images,the casing ruptured completely.Some large fragments were observed around the initiation points.The size of the fragments decreased gradually from the initiation point to the opposite point (180°).A general shear failure pattern at azimuth angles of 120-160°can be observed.At 180°,the casing underwent a more complete rupture.

The X-ray images were processed to obtain the amplification ratios and fragment velocities.The amplification ratios for A90-1 and A90-3 were 1.360 and 1.266,respectively.The obtained fragment velocities are shown in Fig.18.The fragment velocity distributions were averaged to obtain the final velocity distributions for each test specimen,as shown in Fig.19.The velocity distributions of the different test specimens exhibit a similar trend,and the discrepancy in the velocity values is negligible,with an error of approximately 5%.The curvature inflection point of the velocity distribution appears at an angle of approximately 70°.

Fig.18.Fragment velocities for the two-point initiation of 90°: (a) A90-1;(b) A90-3.

Fig.19.Velocity distributions for the two-point initiation of 90°.

3.Numerical modelling

3.1. Modelling approach

To obtain more information,numerical modelling was also performed.The 3D problem could be simplified as a 2D plane strain problem,because longer charges were used to suppress the influence of the end rarefaction waves.A plane strain model,shown in Fig.20,was established to reduce the computation time.The dimensions of the charge and casing were the same as those of the experimental configuration.Euler meshes were used for the charge and air domains,and Lagrange meshes were employed for the casing.The explosive material was filled in the air domain using the‘initial volume fraction’ option in the LS-DYNA software.The fluid-structure interaction between the Euler and Lagrange parts was defined.The‘initiation detonation’option was used to achieve centre initiation,asymmetrical one-point initiation,and asymmetrical two-point initiation of 45°and 90°.

Fig.20.Two-dimensional element model.

To obtain more precise fragment information,relatively small mesh sizes were used for the air domain and casing.Through a mesh convergence study,the mesh sizes of the air and casing were determined to be 0.15 and 0.2 mm,respectively.

3.2. Material parameters

A high explosive burn model and the Jones-Wilkins-Lee equation of state (EOS) were used to model 8701 explosive;the corresponding material parameters were adopted from Ref.[22]and listed in Table 3.

Table 3 Explosive parameters [22].

The Johnson-Cook constitutive model and Grüneisen EOS [18]were used to describe 1045 steel with the corresponding parameters listed in Table 4.A plastic failure strain of 0.65[23,24]was used to model the casing rupture.Under high-velocity driving,the influence of small random defects on the casing rupture pattern is negligible [25,26];thus,the random imperfections were not considered for the casing.The Null material model and the linear polynormal EOS were used for the air domain,with a density of 0.00129 g/cm3and EOS parameters of-1.0E-6,0.4,0.4,and 2.5E-6 forC0,C4,C5,andE0,respectively.

Table 4 Parameters of 1045 steel [18].

3.3. Modelling results

3.3.1.Casing expansion

3.3.1.1.Centre initiation.The casing expansion process of the centre initiation is shown in Fig.21,in which presents some typical instants.The pressure contours are shown in addition to the casing fracture pattern.

Fig.21.Casing expansion for the centre initiation.

As shown in Fig.21,at 2.4 μs,the detonation wave arrived at the casing and caused an increase in pressure.In the subsequent detonation driving,the pressure in the casing reflects back and forth (6.6 μs).At 9.4 μs,fractures appear at the inner part of the casing,and subsequently,the fractures expand increasingly,as observed at 11.8 μs.The fractures finally propagate through the casing thickness (13.0 μs),which produced fragments that dispersed outwards continually (16.0 μs).As observed from the fracture pattern and fragment shape,the fractures are mainly caused by shear failure.The fragments have shapes of parallelograms,trapezoids,and triangles,which are consistent with the experimental results shown in Fig.7.

3.3.1.2.Asymmetrical one-point initiation.The expansion process of the asymmetrical one-point initiation is depicted in Fig.22.The casing near the initiation point was subjected to the detonation pressure at first,as observed at 0.6 μs.Along with the propagation of the detonation wave,the pressure in the casing acted on the two sides and begun to affect the opposite position at 7.2 μs.At this moment,the expansion distance of the initiation side is larger than that of the opposite side.At 9.0 μs,fractures occur inside the casing,and by 12.4 μs,the fractures have completely propagated through the casing at some locations,such as around the initiation point.By 15.0 μs,the casing has ruptured completely,and fragments fly outwards continually,as observed at 19.0 μs.From the fragment morphology,fragments in a range of 130°around the initiation point are large.For the azimuth angles in the range of 70-160°,the casing was sheared in one direction,which produced parallelogram-shaped fragments.This is consistent with the related experimental results,as shown in Fig.11.At positions around 180°,a two-direction shear failure occurred,which produced triangular and trapezoidal fragments.

Fig.22.Casing expansion of the one-point initiation.

3.3.1.3.Asymmetrical two-point initiation of 45°.The casing expansion of the two-point initiation of 45°is shown in Fig.23.Similarly,the casing near the initiation points suffered the detonation pressure at first,and then the pressures propagated towards the two directions.They collide and finally reflected at the opposite positions,as observed at 5.2 μs? At 7.6 μs,the casing near the initiation points became thin,and the casing thickness at the middle portion increased slightly.Fractures appear inside the casing at the midline and the opposite side of the initiation points.As the expansion continued,the case ruptured at some locations at 11.6 μs,such as the initiation points and middle of the initiation points.In particular,the casing rupture at the azimuth angle of 80°is in good agreement with the experimental results (Fig.14).Ruptures at the initiation points and their middle positions cause pressure or stress release,which produced relatively large fragments.By 14.6 μs,the casing had ruptured completely and the fragments flew outwards.The fragment shapes in the range of 90-160°indicate shear failure,as in the experiments,whereas the fragments at other positions underwent composite failures in different shear directions.

Fig.23.Casing expansion of the two-point initiation of 45°.

3.3.1.4.Asymmetrical two-point initiation of 90°.The expansion process of the two-point initiation of 90°is illustrated in Fig.24.Along with the detonation wave propagation,the pressure in the casing is transmitted from the initiation points to other directions and affects the opposite side.At 7 μs,casing fracture occurs at the middle of the initiation points (0°).When the expansion reached 11.4 μs,the casing broke apart at the locations of the initiation points,middle point (0°),and opposite position (180°).These locations coincide with the rupture positions of the experimental results at instant t1,as shown in Fig.17.By 14.0 μs,the casing has ruptured completely.Large fragments were observed,as in the experimental results,at the middle of the two initiation points.Single-direction shear failure occurred in the range of 120-160°,as in the experiments,whereas shear failure occurred in different directions at other positions.

Fig.24.Casing expansion of the two-point initiation of 90°.

3.3.2.Fragment dispersions

Fig.25 shows the fragment dispersion patterns at the same instants as in the experimental images.Comparing these with the Xray images (Fig.7,Fig.11,Fig.14 and Fig.17) revealed that the obtained morphologies of the fragment dispersions are consistent with those in the experimental results.

Fig.25.Fragment dispersions of different initiations: (a) Centre (20 and 42 μs);(b)One-point(20 and 44 μs);(c)Two-point of 45° (20 and 44 μs);(d)Two-point of 90° (20 and 44 μs).

The maximum dimensions of the fragment dispersion patterns at the two time instants were measured and compared with that of the experimental results,as presented in Table 5.The values of the experiments took the average results of the three specimens.The maximum dimensions of the modelling are all bigger than the experiments,indicating larger fragment velocities of the modelling.There may exist rarefaction waves in the experiments,which reduced the fragment velocities.

Table 5 Maximum dimensions of the fragment dispersion patterns.

3.3.3.Fragment velocities

In addition,fragment velocities were extracted from the modelling results,and the velocity distributions are shown in Fig.26.Assuming the densities of 8701 explosive and 1045 steel to be 1.67 g/cm3and 7.85 g/cm3,respectively.The charge ratio(ratio of the charge mass to that of the casing)is computed nominally as 0.8.This ignores the charge mass of the longer portions.It is known that the Gurney constant of 8701 explosive is 2794 m/s,and the fragment velocity can be calculated using the Gurney formula as 2113.4 m/s.By contrast,the average fragment velocity obtained via modelling of centre initiation is 2010.2 m/s.This is 0.95 times as high as that of the theoretical calculation,which shows that the numerical modelling yields reasonably reliable results [27].As observed in Fig.26,different initiation strategies produce different fragment velocity distributions.A detailed comparison with that of the experiments will be presented in the discussion section.

Fig.26.Modelling results of velocity distributions.

4.Discussion

4.1. Casing rupture

For casing rupture,consistency was found between the modelling and experimental results.The modelling results reproduced the typical break positions and fragment dispersion patterns of the experiments.Combining the modelling and experimental results,the casing near the initiation points reduces its thickness and ruptures,producing large fragments.This is caused by the high local pressure at the onset of initiation.As the propagation of the detonation wave progresses,its curvature continuously increases,making it more difficult for local ruptures to occur.In the case of the asymmetrical two-point initiations,the two detonation waves collide at the midline of the initiation points,which produced high local pressures on the casing at relative positions.These local high pressures from collision also lead to early breakage of the casing,for example,at the middle point(0°)of the two-point initiations of 45°and 90°.Similar to the early breakage just at the middle point of the two-point initiation of 45°,breakage also occurred at the opposite position(180°) for the two-point initiation of 90°.This is because different initiation central angles produced different detonation pressures at opposite positions [6],that is,the produced local pressure of 90°is higher than that of 45°.In addition to the local pressures at the locations of the initiation points and detonation wave interaction,interaction between the detonation wave and casing occurs in addition to the propagation of the detonation wave.Mach reflection may occur when the incident angle of the detonation wave exceeds a certain value.At the onset of the Mach wave,the local pressure at the relative position is the highest.This is believed to be the reason for the early casing breakage at the azimuth angle of 80°for the asymmetrical two-point initiation of 45°.To validate this hypothesis,the detonation pressure propagation around the azimuth angle of 80°is plotted in Fig.27.There is a clear onset of the triple point from the azimuth angle of 80°.Therefore,the high local pressures during initiation,detonation wave collision,and onset of Mach reflection caused early breakages of the casing at the relative positions.These early breakages release the internal pressure of the casing and produced relatively large fragments.

Fig.27.Detonation pressure for the two-point initiation of 45°.

From the casing rupture status of the centre initiation(as shown in Fig.7 and Fig.21),different shear directions of the casing were observed under normal impact of the detonation wave,which resulted in fragment shapes of parallelograms,trapezoids,and triangles.Under the asymmetrical initiation of one point and two points,the impact angle of the detonation wave near the initiation points is approximately zero (normal impact),as shown in Fig.28(a).Thus,the fragments in these regions had different morphologies,such as parallelograms,trapezoids,and triangles.Once the distance was covered,the detonation wave slid against the casing,as illustrated in Fig.28(b).This sliding impact induced onedirectional shear failure of the casing,which produced only parallelogram-shaped fragments,for example,in the ranges of 70-160°for the one-point initiation,90-160°for the two-point initiation of 45°,and 120-160°for the two-point initiation of 90°.In the opposite position of the one-point initiation,the detonation wave interacted normally with the casing;consequently,the fragment shapes in this region are mostly triangular.The opposite casing of the two-point initiation of 45°did not rupture early,and the detonation impact is almost normal;thus,multiple fragment shapes exist.Therefore,the casing fracture pattern is controlled by the impact angle of the detonation wave.The normal impact produced shear failure in different directions,which resulted in different fragment shapes.By contrast,the sliding or oblique impact of the detonation wave only caused one-direction shear,producing parallelogram-shaped fragments.

Fig.28.Impact angle of detonation wave: (a) Normal impact;(b) Sliding impact.

4.2. Fragment velocity distributions

By averaging the fragment velocities obtained from the different test specimens under the same initiation type,the velocity distributions of the different initiation strategies were obtained,as shown in Fig.29.Evidently,different velocity distribution trends are produced by different types of initiation.The shape of the velocity distribution of the asymmetrical one-point initiation is somewhat similar to a sine wave,which is consistent with previous findings[9,10].In contrast to the asymmetrical one-point initiation,the shapes of the velocity distributions under the asymmetrical two-point initiation resembles an exponential curve.The change in the curvature of the velocity distribution for the two-point initiation of 90°is readily apparent.The obtained velocity distributions result from the combined effects of the detonation wave interaction and driving of the detonation products.

Fig.29.Experimental velocity distributions.

Fig.30.3D modelling of two-point initiation 90°: (a) SPH model;(b) Fragment dispersion at 45 μs.

Comparisons of the experiment and modelling results are shown in Fig.30.The modelling results have larger magnitudes than the experimental results;however,the velocity distribution trends coincide with each other,except for that of the one-point initiation.Compared with those of the modelling,the low results of the experiments indicate that the rarefaction waves are not completely suppressed by the elongated charge portions.The rarefaction waves reduced the entre fragment velocity values,which is also observed from the fragment dispersions (section 3.3.2).To check this,the 3D modelling was constructed according to the experiment configurations.Constrained by the high computing demand of the 3D ALE algorithm,SPH models of the AUTODYNsoftware were established (which is similar to the work of Ref.[23]).Fig.30 presents the SPH model and fragment dispersion about the two-point initiation of central angle 90°.The material model and parameters were identical to that of the 2D modelling.The fragment velocity distribution obtained from 3D modelling are presented in Fig.31.It is observed that the 3D results coincide well with the experimental results,demonstrating that the experimental configuration indeed not able to suppress the rarefaction waves.Other experimental configurations may be designed to better deal with the rarefaction waves.It needs to say that although the 3D modelling could produce much consistent fragment velocity distributions,the experimental casing fracture pattern can only be achieved via the 2D modelling,rather than the 3D modelling.

Fig.31.Velocity distribution comparisons: (a) Centre initiation;(b) One-point initiation;(c) Two-point initiation of 45°;(d) Two-point initiation of 90°.

As illustrated in Fig.31,the experimental fragment velocities near the initiation point of the asymmetrical one-point initiation were much lower than those of the modelling.However,under the asymmetrical two-point initiation,the fragment velocities near the initiation points do not exhibit a similar trend.This demonstrates that the velocity distribution differences in the asymmetrical onepoint initiation were not caused by the detonation holes used to place the boosters[28].Instead,these may have been caused by the influence of rarefaction waves on the fragment velocity distributions [19].However,the velocity distributions of the experiments and modelling are consistent for the other two asymmetrical twopoint initiations,which indicate that the rarefaction waves do not significantly affect the velocity distributions of the asymmetrical two-point initiation.However,this proposition needs further research.

5.Conclusions

In this study,X-ray imaging and numerical modelling were used to investigate the casing expansions under asymmetrical multipoint initiations.The casing ruptures and fragment velocity distributions were analysed,and the following conclusions were drawn:

(1) Local high pressures are induced during initiation,detonation wave interaction,and onset of Mach reflection.These local pressures cause early rupture of the casing,release the internal pressure,and finally produce large fragments.

(2) The casing rupture pattern is influenced by the interaction angle between the detonation wave and casing.When the detonation wave impacts the casing normally,multipledirection shear failure occurs,producing triangular-,trapezoidal-,and parallelogram-shaped fragments.By contrast,the oblique impact causes single-direction shear failure,resulting in only parallelogram-shaped fragments.

(3) Different initiation types result in different velocity distributions.The velocity distribution trend for the asymmetrical one-point initiation resembles the shape of a sine function,whereas that of the asymmetrical two-point initiation resembles the shape of an exponential curve.

(4) The technique of elongating the charge length cannot completely suppress rarefaction waves.Rarefaction waves decrease the total fragment velocities and may affect the velocity distribution of the asymmetrical one-point initiation.However,understanding the influence of rarefaction waves on the velocity distribution of two-point asymmetrical initiations requires further research.

The findings of this study can guide the design and optimization of high-efficiency warheads.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was funded by the National Natural Science Foundation of China [Grant No.12002178],opening project of the State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology) [Grant No.KFJJ22-17M],and the Fundamental Research Funds for Central Universities.