- Email: [email protected]
Effects of U-notches on the dynamic fracture and fragmentation of explosively driven cylinders
Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
Contents lists available at ScienceDirect
Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Effects of U-notches on the dynamic fracture and fragmentation of explosively driven cylinders Minzu Liang, Xiangyu Li, Fangyun Lu ⇑ College of Science, National University of Defense Technology, 410073 Changsha, PR China
a r t i c l e
i n f o
Article history: Available online 19 March 2015 Keywords: Cylinder U-notch Explosive Dynamic fracture Fragmentation
a b s t r a c t Metal cylinder specimens are explosively expanded to fragmentation and the effect of U-notches in walls are investigated on fragmentation behaviors and failure mechanisms of the cylinders experimentally and numerically. Fragments were recovered by sawdust and sorted into four categories according to fragment morphology and fracture mode. The shear fracture is pronounced in the low notch depth condition, while tensile fracture plays a leading role in higher notch depth. Moreover, mass percentage distribution of the fragments appears more affected by the notch depth than width. In addition, fragmentation energy and fragmentation toughness, considered as material properties, are discussed using Grady’s energy-based theory. A suitable correction factor related to the width and depth of the notch is proposed to depict the effects of the U-notch. The effects of U-notches on the deformation and fracture behavior of cylinders are discussed with numerical simulations, indicating the stress concentration was noticeable at the notch tip, and the dynamic stress concentration factor (SCF) of the U-notch was less than that of the V-notch. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Explosively driven fragmentation is a highly complex phenomenon in which the structure undergoes a rather complicated behavioral process of ‘‘expansion to deformation to fracture to fragmentation’’. The fragmentation process is much more involved than simple quasi-static uniaxial tensile fracture and shear fracture [1]. Numerous experimental methods have been used for fragmentation, including explosive cylinder technology [2] and expanding ring technology [3]. Since the cylindrical structure is widely used in the design of munitions and armaments, the dynamic fracture and fragmentation of this structure has been studied immensely since the Second World War. In the 1940s, Gurney [4], Mott [5], and Taylor [6] presented extensive studies establishing the theoretical foundations for this subject. Gurney derived empirical expressions for prediction of fragment velocities as a function of the ratio between the high explosive mass (C) and the metal cylinder mass (M), the energy of the high explosive available to drive the cylinder, and geometric factors. Mott estimated a statistical model for predicting average fragment sizes and fragment mass distributions. In the model, he proposed that the mechanism proceeded through random spatial and temporal occurrence of fractures resulting in a distribution of fragment lengths. A stress rarefaction wave propagates from the fracture ⇑ Corresponding author. Tel.: +86 073184573276; fax: +86 073184573297. E-mail address: [email protected] (F. Lu). http://dx.doi.org/10.1016/j.tafmec.2015.02.004 0167-8442/Ó 2015 Elsevier Ltd. All rights reserved.
while the fracture strain of the material achieves a mean value. The rarefaction unloads the material surrounding the fracture, preventing the possibility of further fracture in adjacent material. This process completes when the fracture-induced rarefaction waves subsume the entire cylinder. Taylor developed a model of fracture strain based on material tensile strength, internal pressure, and wall thickness of the steel cylinder. Studies on this topic, which has been researched for more than half a century, are extensive, and a large numbers of researchers [7–11] have contributed to this field. These early works laid the foundation for a field of study that continues to this day. The performances of fragmentation warheads are usually described with characteristics of the fragment dispersion. These characteristics include mass and shape, projection angle and direction, velocity and distribution density of the fragments. In general, the natural fragmentation process leads to fracture of the cylinder wall into irregular and predominantly small fragments with low performance. In order to control the distribution of fragment masses and provide the desired terminal effects, many techniques have been utilized. The most famous method, cylinder grooving, has been used productively to control the phenomenon of dynamic deformation and dynamic fragmentation. Contrary to natural fragmentation, fragmentation controlled by weakening the cylinder structure with notches or grooves offers the possibility to adapt fragment parameters, such as size and mass, to the performance requirements in a very flexible way. Early fragmentation tests of notched-cylinder bombs were conducted by Philipchuk [12] in
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
1953. The results indicated that better fragment size control could be achieved with internal notch depths up to at least 50% of the wall thickness. Then, Pearson [13] used the shear method to control fragmentation and developed a fragmentation model based on this method. Subsequently, the work reported by Lamborn [14] used external longitudinal and circumferential notches in cylinder walls to affect the break-up of cylinders upon detonation. These effects of notch geometry, hardness of wall material, pitch of notch, and the C/M ratio were discussed in his report. Then, Saunders [15] investigated the effects of the C/M ratio on the degree of control of fragmentation attainable by external longitudinal grooving, both unfilled and filled with an epoxy resin. Hiroe et al. [16] investigated the effects of wall materials, configuration, explosive energy, and initiated locations of fragmentation behavior of exploded notched cylinders both experimentally and numerically. Recently, an investigation on dynamic fracture trajectories of explosive cylinders with internal and external grooves was carried out by Li et al. [17]. Although the fragmentation behaviors and various effects have been determined over the past many years, however, the effects of U-notches on explosive-filled cylinders have not been fully explored. In the present study, a series of conditions for testing were designed to investigate the effects of U-notches on dynamic fracture and fragmentation of explosively driven cylinders. The tests were performed on AISI 1045 steel with heat treatment. Fragments were sorted and measured, and the fracture was calculated. Moreover, the fragmentation energy was discussed and a derived expression of the factor was suggested using Grady’s energy-based theory. Finally, the observed deformation and fracture behaviors are reproduced by numerical simulation using LS-DYNA software. 2. Experimental procedures 2.1. Experiment assembly The developed experiment assembly for axially uniform or axially phased rapid expansion of cylinders driven by explosives is illustrated in Fig. 1. The casing, a stacked set of cylinders made of steel, is filled with explosive charge and a copper sleeve is placed before and after the array in order to confine the detonation wave passing down the cylinders. It is well known that there is significant difference between the dispersion angles of the fragments located in the center of the charge’s surface and the ones located
51
on its edges. The reason is rarefaction waves influencing the detonation gas near the ends. The phenomenon is called end effect. In order to minimize end effects influence on ejection angles of fragments, the use of a LY-12 aluminum cover, confining the detonation gas, was proposed. The end cover, made of LY-12 aluminum, is placed at the end of the array to protect cylinders from rarefaction waves. A plane wave generator is bonded to the end of the high explosive cylinder and is detonated at its apex with a detonator. The purpose of this plane wave generator is to provide near-planar detonation to the main charge. This caused cylinder walls to expand radially outward under the same conditions, thus resulting in similar strain variation in the longitudinal direction. 2.2. Specimen design Fig. 2 shows a basic test specimen used in the experiments. The cylinder consists of a 100 mm outer diameter with a 10 mm wall thickness, and the height t of the cylinder is 5 mm in order to achieve a plane-stress condition, approximately. An axis-symmetric cylinder, notched longitudinally, is used for the testing. The angle h of adjacent external notches, as notches, is evenly spaced at 18°. The angle between adjacent external and internal notches is h/2. The profile of the notch is U-shaped with width w and depth d. The materials and dimensions of the notches are given in Table 1. The cylinder material, AISI 1045 steel, is a medium carbon steel and the chemical constituents and mechanical properties are summarized in Table 2. 2.3. Test setup The test setup is illustrated in Fig. 3. The recovering unit consisted of sawdust placed in a semi-circle with a 2 m radius, which is designed to collect cylinder fragments softly. The height and thickness of the recovering unit are 1.5 m and 1.2 m, respectively. Three trigger circuits connected to an oscilloscope (OSC) measure the velocities of the fragments. The trigger consists of two copper foils and insulating foam. Electrical signals are recorded by the OSC as a fragment penetrates the trigger circuits. Three signals were obtained in the experiment. Based on the time interval of each two signals and a fixed distance between the triggers, the fragment velocity was calculated by the ratio of the distance to the time interval. 3. Experimental results and analysis 3.1. Fragment categories About 70% of the fragments were successfully recovered and minimized any unintentional damage by the sawdust (Fig. 4). Two main types of fracture have been observed in the dynamic fracture of cylinders, namely, shear fracture at approximately 45° to the circumference of the wall, and tensile fracture along the radial direction of the wall. The fragments recovered from cylinders with different sized notches were sorted into four main categories based on fragment morphology and fracture mode (Fig. 5). These categories are: (1) (2) (3) (4)
Fig. 1. Schematics of test assembly. The specimens, a stacked set of cylinders made of steel, are filled with main explosive charge and a copper sleeve is placed before and after the array.
Tensile-mode fragment. Shear-mode fragment. Mix-mode fragment. End fragment.
As Fig. 5 shows, the tensile-mode fragments were the result of cracks, which formed and grew from the root of adjacent external notches or adjacent external and internal notches. The sizes of tensile-mode fragments consisted of two categories, large and small,
52
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
Fig. 2. Specimens of tests. The dE and wE are depth and width of the external notch, and the dI and wI are depth and width of the internal notch.
Table 1 Materials and dimensions of the notches.
a b
Cylinder materials
Explosive
External notch sizes (mm)
Internal notch sizes (mm)
AISI 1045 steela
Composition B
dE = 1.0, 2.0, 3.0, 4.0, 5.0, 6.0b xE = 1.0, 1.5, 2.0, 2.5, 3.0
dI = 2.0 xI = 1.0
Heat treatment: 850 °C quenching, 350 °C tempering. Machining accuracy: ±0.02 mm.
Table 2 Chemical constituents and mechanical properties of AISI 1045 steel (Wt%). Chemical compositions C Mn Si
0.47 0.6 0.21
S P Cr
Mechanical properties 0.025 0.03 0.22
Yield strength (MPa) Tensile strength (MPa) Elongation (%)
420 600 16
with the larger fragments approximately twice the size of the smaller ones. The shear-mode fragments were generally triangular or parallelogram-shaped in cross section, and were formed by shear fractures emanating from the internal notch to the external notch. The size of the parallelogram types were approximately twice the size of the triangular ones. The mix-mode fragments were formed by a shear fracture trace and a radial tensile fracture trace. There were some other fragments, called end fragments, which fractured longitudinally as a result of natural fragmentation mechanisms operating in the wall of the cylinders between adjacent notches. The size was generally random and the mass was smaller than the three main types. The end fragments resulted in mass reduction of the regular fragments from cylinders, thus decreasing the degree of fragmentation control. Fig. 6 expresses the categories of fragments as a percentage by mass of the total
recovery. Total recovery was used here rather than original cylinder mass to allow, to some extent, for anomalous recoveries. It was observed that shear-mode fragments were more pronounced in cylinders of low dE/t ratios, whereas tensile-mode fragments were in the highest flight at high dE/t ratios. This effect was the result of the location of the tensile fracture being lower in the notches in cylinders with deeper notches, together with the virtual elimination of natural fragmentation mechanisms. However, this effect was not so obvious for different notch widths (Fig. 7). 3.2. Fracture calculation After the fragments were retrieved, they were cleaned and measured individually. The fracture strain, calculated by the thicknesses of the measured fragments, could be determined by
ef ¼ ett ¼ lnðtf =ti Þ;
ð1Þ
where ef is critical fracture strain of the cylinder, ett is the logarithmic thickness strain, and ti and tf are the initial and final wall thicknesses of the cylinder, respectively. Assuming the radial expansion of the cylinder is plane strain, an effective plastic strain (eps) at failure can be defined as
Fig. 3. Sketch of the experimental set-up.
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
(a) Notch: ωE =1.0 dE =2.0
(b) Notch: ωE =2.0 dE =3.0
(c) Notch: ωE =1.0 dE =3.0
(d) Notch: ωE =1.0 dE =6.0
53
Fig. 4. Typical fragments recovered from the tests.
2 1=2 ep ¼ pffiffiffi ½e2t þ e2h þ et eh ; 3
ð2Þ
where eh is the logarithmic height strain. Note that when eh = 0, pffiffiffi ep ¼ ð2= 3Þef . The mean eps at fracture is 0.11 with a standard deviation of 0.02. The initial velocity of fragments can be calculated by the exponential equation
v 0 ¼ v x =eax ;
ð3Þ
where vx is fragment velocity at distance x, and was measured in the tests. a is a coefficient related to fragment shape and mass, defined as
1 3
a ¼ cd qa xmf1=3 ;
ð4Þ
where qa is the air density, and x, cd, and mf are the cross-sectional area, drag coefficient, and fragment mass, respectively. Gurney developed an equation for predicting maximum velocity, Vmax, of cylindrical shells subject to internal explosive loading. The Gurney velocity for a cylinder is given by
V max ¼
pffiffiffiffiffiffi 2EðM=C þ 1=2Þ1=2 ;
ð5Þ
where M/C is the ratio of the cylinder mass to the explosive mass. pffiffiffiffiffiffi The empirical Gurney constant 2E, determined by experiments involving a particular type of explosive, is related to the detonation energy of the explosive. The data of initial velocities and Gurney velocities are shown in Fig. 8. As the gas products vent first due
to notching, the velocities of the fragments from experimental data are less than those obtained from the Gurney equation, for obvious reasons. The fracture energy is estimated from the equation proposed by Grady [18] and used as follows
C¼
qf e_ 2f s3f 24
;
ð6Þ
where C is Grady’s fragmentation energy per unit area in forming a crack, qf is the material density, sf is the average distance between cracks corresponding to curvature, and e_ f is circumferential strain rate at the estimated fracture time. The distance between cracks s is obtained by the circumferential width of the fragments. Just prior to break up, the cylinder body is in circumferential tension and undergoing uniform circumferential stretching at the rate e_ , which is given by the ratio
e_ f ¼
de v 0 v0 ¼ ¼ ; dt rf r 0 eett
ð7Þ
where v0 is the initial fragment velocity, rf is the cylinder radius corresponding to the direction of the crack, and r0 is the initial cylinder radius. A material property called fragmentation toughness along with the relation of fragmentation energy and elastic modulus, was defined by Kipp and Grady [19] to characterize the material property of fragmentation. Fragmentation toughness is expressed as
Kf ¼
pffiffiffiffiffiffiffiffiffi 2EC;
ð8Þ
54
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
Fig. 7. Mass percentage of the three categories of fragments: xE = 1.0, 2.0, 3.0 mm, dE = 2.0 mm.
Fig. 5. The formation of three major categories observed: tensile-mode fragment, shear-mode fragment and mix-mode fragment.
Fig. 8. Initial velocities of the tests and Gurney velocities: xE = 2.0 mm, dE = 1.0, 2.0, 3.0, 4.0, 5.0, 6.0.
Table 3 Values of calculated fragmentation energy and fragmentation toughness based on Grady’s energy-based fragmentation model: xE = 2.0 mm.
Fig. 6. Mass percentage of the fragments: xE = 2.0 mm, dE = 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 mm.
where Kf is fragmentation toughness, and E is the elastic modulus. Fragmentation toughness values from testing are shown in Tables 3 and 4. The calculated Kf values are affected by the notches. However, fragmentation toughness is a constant. A modification factor k was suggested by Hiroe et al. [16] for this notch effect which is related as
Notch depth, dE (mm)
1.0
2.0
3.0
4.0
5.0
6.0
Ave. frag. energy (kJ m2) Frag. toughness (MPa m1/2)
84.9 130.3
114.4 151.3
120.5 155.2
133.8 163.6
170.4 184.6
191.4 239.6
Table 4 Values of calculated fragmentation energy and fragmentation toughness based on Grady’s energy-based fragmentation model: dE = 2.0 mm. Notch width, wE (mm)
1.0
1.5
2.0
2.5
3.0
Ave. frag. energy (kJ m2) Frag. toughness (MPa m1/2)
99.6 141.1
108.5 147.4
114.4 151.3
122.1 156.3
129.3 160.8
kðdE ; wE Þ ¼ f ðdE =tÞ gðwE =tÞ:
C¼
kqf e_ 2f s3f 24
:
ð9Þ
Further study is necessary to explore this modification factor. For our experimental results, the factor is related to only two parameters of the U-notch, namely, depth and width. The modification factor can be assumed as
ð10Þ
Based on the fragmentation energy results, the expressions can be obtained by linear curve fitting of
f ðdE =t; wE =t ¼ 0:2Þ ¼ 1 þ 3:1ðdE =tÞ ð0:1 dE =t 0:6Þ
ð11Þ
g ðwE =t; dE =t ¼ 0:2Þ ¼ 1 þ 1:7ðwE =tÞ ð0:1 wE =t 0:3Þ:
ð12Þ
55
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
Fig. 9. Simulation model of the tests and detailed view of the notch tip.
Table 5 J–C model parameters of cylinder material. Material
Tmelt (K)
A (MPa)
B (MPa)
n
c
m
AISI 1045
1792
496
320
0.28
0.064
0.804
However, these two expressions are dependent. The revised expressions are given as
f ðdE =t; wE =t ¼ 0:2Þ ð0:1 dE =t 0:6Þ g ðwE =t ¼ 0:2Þ g ðwE =t; dE =t ¼ 0:2Þ ð0:1 wE =t 0:3Þ: gðwE =tÞ ¼ f ðdE =t ¼ 0:2Þ
f ðdE =tÞ ¼
ð13Þ ð14Þ
f ðdE =t; wE =t ¼ 0:2Þ g ðwE =t; dE =t ¼ 0:2Þ g ðwE =t ¼ 0:2Þ f ðdE =t ¼ 0:2Þ 0:1 dE =t 0:6 ¼ 0:33ð1 þ 3:1dE =tÞð1 þ 1:7wE =tÞ 0:1 wE =t 0:3
kðdE ; wE Þ ¼
ð15Þ Substituting Eq. (15) into Eq. (9), we obtain the fragmentation energy expression for the U-notch cylinder as
ð1 þ 3:1dE =tÞð1 þ 1:7wE =tÞqf e_ 2f s3f
72
m T T room 1 ; T melt T room
e_ r ¼ ½A þ Bens 1 þ c ln _ s er
ð17Þ
where es is the plastic strain of the steel, e_ s is strain rate of the steel, e_ r is the reference strain rate, T is the temperature, A is the quasi-
Eqs. (10), (13) and (14) can be combined to obtain
C¼
The material model and mesh generation are the most important factors that affect the accuracy of the results from the numerical simulation. The average mesh size is 0.5 mm in specimens with a mesh size of 0.2 mm around the notches. The Johnson–Cook (J–C) model was applied to describe the deformation behaviors of the specimen. This material model is specifically used for metals and is appropriate for simulation of problems where strain rate varies greatly or when adiabatic temperature changes lead to material softening. The equation of this material model is given as
0:1 dE =t 0:6 0:1 wE =t 0:3
: ð16Þ
static yield stress, B and n are the strain hardening coefficients, c is the strain rate hardening coefficient, and m is the thermal softening coefficient. For a high explosive model, the Jones–Wilkins–Lee (JWL) method was used to describe the detonation of composition B. This equation defines the pressure field as a function of relative volume and internal energy per initial volume as
xqe R1 qqe0e xqe R2 qqe0e xqe e e p¼A 1 þB 1 þ E ; R1 qe0 R2 qe0 qe0 m0
ð18Þ
where p is the detonation pressure, qe is the explosive density, qe0 is the initial explosive density, and A, B, R1, R2 and x are material constants. The parameters in the Johnson–Cook equation and the explosive parameters are listed in Tables 5 and 6.
4. Numerical simulation and discussion
4.2. Numerical simulation
4.1. Finite element model
4.2.1. Cylinder fracture The dynamic fracture and fragmentation process of cylinders can be broken down into four main steps according to the numerical simulation (Fig. 10). First, the initial stress in the cylinders is caused by explosive shock waves formed upon detonation, causing rapid outward acceleration of the cylinder. In addition, stress concentrations appear at notch tips (Fig. 10a). Then, strain bands between adjacent internal and external notches are turned out along with cylinder expansion (Fig. 10b). Afterwards, cracks
Numerical simulations of the expanding behavior were performed for all the experiments using the Arbitrary-Lagrange–Euler (ALE) method with LS-DYNA software. LS-DYNA [20], a fully integrated analysis program, is well suited for analysis of non-linear dynamics problems. As the cylinder was symmetrical in structure, a quarter of the specimen was simulated. The finite element model configuration and a typical meshed notch are illustrated in Fig. 9. Table 6 JWL model parameters of explosive. Explosive
Density (kg/m3)
Detonation velocity (m/s)
Chapman–Jouget pressure, PCJ (1011 Pa)
A
B
R1
R2
x
Em0 (kJ/m3)
Composition B
1700
8000
0.34
5.81
0.068
4.1
1.0
0.35
0.09
56
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
(a) First step
(b) Second step
(c) Third step
(d) Last step
Fig. 10. Dynamic fracture and fragmentation process of notched cylinders.
form and grow from the roots of notches, and begin to propagate through the cylinder wall after a very short time (Fig. 10c). Finally, gas products begin to vent as cracks propagate completely through the wall due to very high internal pressure that accelerates the wall outward. Stress and strain in the cylinder wall are released, and the cracks are no longer increasing (Fig. 10d). The study by Li et al. [21] on fracture traces of blast loading cylinders with V-notches showed that fracture traces were various. The main fracture types were tensile fractures along the radial direction and shear fractures following the trace between the roots of adjacent notches. This similar fracture pattern was exhibited in U-notched cylinders. As Fig. 11 shows, numerical simulation results coincided with experimental results. 4.2.2. Effect of stress concentration The resulting distributions of effective stress (see Fig. 12) are non-uniform, and stress concentration is noticeable at the notch tip. With regard to the crack case, a relationship between the SCF and the notch geometry should be considered. The static SCF is usually presented as
Ks ¼
rmax r0
ð19Þ
where Ks is the static SCF, rmax is the maximum stress at the discontinuity, and r0 is the average stress of the same section. The
Fig. 11. Fracture trace of U-notched cylinder. Typical fragments in the simulation are in good agreement with experimental results.
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
(a) Internal notch
57
(b) External notch Fig. 12. Effective stress distributions around the notch tip.
the shear fracture was pronounced in the low notch depth condition, while tensile fracture played a leading role in higher notch depth. Moreover, mass percentage distribution of the fragments appeared more affected by the notch depth than width. The mass and size of recovered fragments were measured, and the initial velocities were obtained from experiments. Then, the fragmentation energy and fragmentation toughness based on Grady’s energy-based theory were discussed. The notch effect was obvious based on the calculated results, and a revised fragmentation model was suggested to account for the notch effect. Numerical simulations were performed using the LS-DYNA software. The fracture trace of the simulation results were in good agreement with experimental results. In particular, the stress concentration was noticeable at the notch tip, and the dynamic SCF of the U-notch was less than that of the V-notch. Acknowledgements Fig. 13. The evolution history of the stress concentration: final average internal I E dynamic SCF kd = 2.6, and final average external dynamic SCF kd = 2.3.
average stress is easily determined from the applied load for static situations; however, the average load is much more difficult to obtain for blast loading conditions. Therefore, the average stress is based on the nominal section without a discontinuity, and the dynamic SCF is determined by the following expression [22]
K d ðtÞ ¼
rmax ðtÞ rnom ðtÞ
ð20Þ
where Kd is the dynamic SCF, rmax is the maximum stress as a function of time, and rnom is the nominal stress as a function of time. The evolutionary history of the stress concentration is illustrated in Fig. 13. Comparing to literature data [23,24], the dynamic SCF of the U-notch is less than that of the V-notch. 5. Conclusions Notched cylinder specimens were expanded explosively to fragmentation by varying U-notch dimensions. The fragments were collected to determine dynamic fracture and fragmentation characteristics. Fragments were sorted into four categories according to fragment morphology and fracture mode. It was found that
The authors wish to acknowledge financial supported by the China National Natural Science Funding (11202237, 11132012), and the State Key Laboratory of Explosion Science and Technology Opening Fund (KFJJ12-16MM). References [1] M. Singh, H. Suneja, M. Bola, S. Prakash, Dynamic tensile deformation and fracture of metal cylinders at high strain rates, Int. J. Impact Eng 27 (2002) 939–954. [2] D. Goto, R. Becker, T. Orzechowski, H. Springer, A. Sunwoo, C. Syn, Investigation of the fracture and fragmentation of explosively driven rings and cylinders, Int. J. Impact Eng 35 (2008) 1547–1556. [3] M.Z. Liang, X.Y. Li, J.G. Qin, F.Y. Lu, Improved expanding ring technique for determining dynamic material properties, Rev. Sci. Instrum. 84 (2013). [4] R.W. Gurney, The initial velocities of fragments from bombs shell and grenades, BRL Report No. 405, Army Ballistic Research Laboratory, MD, USA, 1943. [5] N.F. Mott, Fragmentation of shell cases, NATO ASI Ser., Ser. C 189 (1947) 300– 308. [6] G.I. Taylor, The fragmentation of tubular bombs, in: Scientific Papers of G.I. Taylor, Cambridge University Press, London, 1963, pp. 387–390. [7] A. Ivanov, Explosive deformation and destruction of tubes, Strength Mater. 8 (1976) 1303–1306. [8] C.R. Crowe, W.H. Holt, W. Mock Jr, O.H. Griffin, Dynamic fracture and fragmentation of cylinders, TR Report No. 3449, Naval Surface Weapons Center Dahlgren Laboratory, MD, USA, 1976. [9] C.R. Hoggatt, R.F. Recht, Fracture behavior of tubular bombs, J. Appl. Phys. 39 (1968) 1856–1862. [10] D. Grady, D. Benson, Fragmentation of metal rings by electromagnetic loading, Exp. Mech. 23 (1983) 393–400.
58
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
[11] P. Elek, S. Jaramaz, Modeling of fragmentation of rapidly expanding cylinders, Theoret. Appl. Mech. 32 (2005) 113–130. [12] V. Philipchuk, Fragmentation tests of notched-ring bombs, Report No. 1138, Aberdeen Proving Ground, MD, USA, 1953. [13] J. Pearson, The shear control method of warhead fragmentation, in: 4th International Symposium on Ballistics Monterey, CA, USA, 1978, pp. 17–19. [14] I.R. Lamborn, The fracture of small internally detonated cylinders partly prefragmented by notching, MRL Report No. 545, 1973. [15] D.S. Saunders, The effects of C/M ratio on the controlled fragmentation of cylinders using external notching, MRL Report No. 800, Defence Science and Technology Organisation Materials Research Laboratories, Australia, 1980. [16] T. Hiroe, K. Fujiwara, H. Hata, H. Takahashi, Deformation and fragmentation behaviour of exploded metal cylinders and the effects of wall materials, configuration, explosive energy and initiated locations, Int. J. Impact Eng 35 (2008) 1578–1586. [17] X. Li, M. Liang, M. Wang, G. Lu, F. Lu, Experimental and numerical investigations on the dynamic fracture of a cylindrical shell with grooves subjected to internal explosive loading, Propellants, Explos., Pyrotech. 35 (2014) 1–10.
[18] D. Grady, Fragmentation of Rings and Shells: The Legacy of NF Mott, Springer, 2007. [19] M.E. Kipp, D.E. Grady, Dynamic fracture growth and interaction in one dimension, J. Mech. Phys. Solids 33 (1985) 399–415. [20] J.O. Hallquist, LS-DYNA Theoretical Manual, Livermore Software Technology Corporation, California, 1998. [21] X.Y. Li, M.Z. Liang, M.F. Wang, G.X. Lu, F.Y. Lu, Experimental and numerical investigations on the dynamic fracture of a cylindrical shell with grooves subjected to internal explosive loading, Propellants, Explos., Pyrotech. 35 (2014) 1–10. [22] W. Altenhof, N. Zamani, W. North, B. Arnold, Dynamic stress concentrations for an axially loaded strut at discontinuities due to an elliptical hole or double circular notches, Int. J. Impact Eng 30 (2004) 255–274. [23] R. Afshar, F. Berto, Stress concentration factors of periodic notches determined from the strain energy density, Theoret. Appl. Fract. Mech. 56 (2011) 127–139. [24] P. Lazzarin, R. Afshar, F. Berto, Notch stress intensity factors of flat plates with periodic sharp notches by using the strain energy density, Theoret. Appl. Fract. Mech. 60 (2012) 38–50.
Contents lists available at ScienceDirect
Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Effects of U-notches on the dynamic fracture and fragmentation of explosively driven cylinders Minzu Liang, Xiangyu Li, Fangyun Lu ⇑ College of Science, National University of Defense Technology, 410073 Changsha, PR China
a r t i c l e
i n f o
Article history: Available online 19 March 2015 Keywords: Cylinder U-notch Explosive Dynamic fracture Fragmentation
a b s t r a c t Metal cylinder specimens are explosively expanded to fragmentation and the effect of U-notches in walls are investigated on fragmentation behaviors and failure mechanisms of the cylinders experimentally and numerically. Fragments were recovered by sawdust and sorted into four categories according to fragment morphology and fracture mode. The shear fracture is pronounced in the low notch depth condition, while tensile fracture plays a leading role in higher notch depth. Moreover, mass percentage distribution of the fragments appears more affected by the notch depth than width. In addition, fragmentation energy and fragmentation toughness, considered as material properties, are discussed using Grady’s energy-based theory. A suitable correction factor related to the width and depth of the notch is proposed to depict the effects of the U-notch. The effects of U-notches on the deformation and fracture behavior of cylinders are discussed with numerical simulations, indicating the stress concentration was noticeable at the notch tip, and the dynamic stress concentration factor (SCF) of the U-notch was less than that of the V-notch. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Explosively driven fragmentation is a highly complex phenomenon in which the structure undergoes a rather complicated behavioral process of ‘‘expansion to deformation to fracture to fragmentation’’. The fragmentation process is much more involved than simple quasi-static uniaxial tensile fracture and shear fracture [1]. Numerous experimental methods have been used for fragmentation, including explosive cylinder technology [2] and expanding ring technology [3]. Since the cylindrical structure is widely used in the design of munitions and armaments, the dynamic fracture and fragmentation of this structure has been studied immensely since the Second World War. In the 1940s, Gurney [4], Mott [5], and Taylor [6] presented extensive studies establishing the theoretical foundations for this subject. Gurney derived empirical expressions for prediction of fragment velocities as a function of the ratio between the high explosive mass (C) and the metal cylinder mass (M), the energy of the high explosive available to drive the cylinder, and geometric factors. Mott estimated a statistical model for predicting average fragment sizes and fragment mass distributions. In the model, he proposed that the mechanism proceeded through random spatial and temporal occurrence of fractures resulting in a distribution of fragment lengths. A stress rarefaction wave propagates from the fracture ⇑ Corresponding author. Tel.: +86 073184573276; fax: +86 073184573297. E-mail address: [email protected] (F. Lu). http://dx.doi.org/10.1016/j.tafmec.2015.02.004 0167-8442/Ó 2015 Elsevier Ltd. All rights reserved.
while the fracture strain of the material achieves a mean value. The rarefaction unloads the material surrounding the fracture, preventing the possibility of further fracture in adjacent material. This process completes when the fracture-induced rarefaction waves subsume the entire cylinder. Taylor developed a model of fracture strain based on material tensile strength, internal pressure, and wall thickness of the steel cylinder. Studies on this topic, which has been researched for more than half a century, are extensive, and a large numbers of researchers [7–11] have contributed to this field. These early works laid the foundation for a field of study that continues to this day. The performances of fragmentation warheads are usually described with characteristics of the fragment dispersion. These characteristics include mass and shape, projection angle and direction, velocity and distribution density of the fragments. In general, the natural fragmentation process leads to fracture of the cylinder wall into irregular and predominantly small fragments with low performance. In order to control the distribution of fragment masses and provide the desired terminal effects, many techniques have been utilized. The most famous method, cylinder grooving, has been used productively to control the phenomenon of dynamic deformation and dynamic fragmentation. Contrary to natural fragmentation, fragmentation controlled by weakening the cylinder structure with notches or grooves offers the possibility to adapt fragment parameters, such as size and mass, to the performance requirements in a very flexible way. Early fragmentation tests of notched-cylinder bombs were conducted by Philipchuk [12] in
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
1953. The results indicated that better fragment size control could be achieved with internal notch depths up to at least 50% of the wall thickness. Then, Pearson [13] used the shear method to control fragmentation and developed a fragmentation model based on this method. Subsequently, the work reported by Lamborn [14] used external longitudinal and circumferential notches in cylinder walls to affect the break-up of cylinders upon detonation. These effects of notch geometry, hardness of wall material, pitch of notch, and the C/M ratio were discussed in his report. Then, Saunders [15] investigated the effects of the C/M ratio on the degree of control of fragmentation attainable by external longitudinal grooving, both unfilled and filled with an epoxy resin. Hiroe et al. [16] investigated the effects of wall materials, configuration, explosive energy, and initiated locations of fragmentation behavior of exploded notched cylinders both experimentally and numerically. Recently, an investigation on dynamic fracture trajectories of explosive cylinders with internal and external grooves was carried out by Li et al. [17]. Although the fragmentation behaviors and various effects have been determined over the past many years, however, the effects of U-notches on explosive-filled cylinders have not been fully explored. In the present study, a series of conditions for testing were designed to investigate the effects of U-notches on dynamic fracture and fragmentation of explosively driven cylinders. The tests were performed on AISI 1045 steel with heat treatment. Fragments were sorted and measured, and the fracture was calculated. Moreover, the fragmentation energy was discussed and a derived expression of the factor was suggested using Grady’s energy-based theory. Finally, the observed deformation and fracture behaviors are reproduced by numerical simulation using LS-DYNA software. 2. Experimental procedures 2.1. Experiment assembly The developed experiment assembly for axially uniform or axially phased rapid expansion of cylinders driven by explosives is illustrated in Fig. 1. The casing, a stacked set of cylinders made of steel, is filled with explosive charge and a copper sleeve is placed before and after the array in order to confine the detonation wave passing down the cylinders. It is well known that there is significant difference between the dispersion angles of the fragments located in the center of the charge’s surface and the ones located
51
on its edges. The reason is rarefaction waves influencing the detonation gas near the ends. The phenomenon is called end effect. In order to minimize end effects influence on ejection angles of fragments, the use of a LY-12 aluminum cover, confining the detonation gas, was proposed. The end cover, made of LY-12 aluminum, is placed at the end of the array to protect cylinders from rarefaction waves. A plane wave generator is bonded to the end of the high explosive cylinder and is detonated at its apex with a detonator. The purpose of this plane wave generator is to provide near-planar detonation to the main charge. This caused cylinder walls to expand radially outward under the same conditions, thus resulting in similar strain variation in the longitudinal direction. 2.2. Specimen design Fig. 2 shows a basic test specimen used in the experiments. The cylinder consists of a 100 mm outer diameter with a 10 mm wall thickness, and the height t of the cylinder is 5 mm in order to achieve a plane-stress condition, approximately. An axis-symmetric cylinder, notched longitudinally, is used for the testing. The angle h of adjacent external notches, as notches, is evenly spaced at 18°. The angle between adjacent external and internal notches is h/2. The profile of the notch is U-shaped with width w and depth d. The materials and dimensions of the notches are given in Table 1. The cylinder material, AISI 1045 steel, is a medium carbon steel and the chemical constituents and mechanical properties are summarized in Table 2. 2.3. Test setup The test setup is illustrated in Fig. 3. The recovering unit consisted of sawdust placed in a semi-circle with a 2 m radius, which is designed to collect cylinder fragments softly. The height and thickness of the recovering unit are 1.5 m and 1.2 m, respectively. Three trigger circuits connected to an oscilloscope (OSC) measure the velocities of the fragments. The trigger consists of two copper foils and insulating foam. Electrical signals are recorded by the OSC as a fragment penetrates the trigger circuits. Three signals were obtained in the experiment. Based on the time interval of each two signals and a fixed distance between the triggers, the fragment velocity was calculated by the ratio of the distance to the time interval. 3. Experimental results and analysis 3.1. Fragment categories About 70% of the fragments were successfully recovered and minimized any unintentional damage by the sawdust (Fig. 4). Two main types of fracture have been observed in the dynamic fracture of cylinders, namely, shear fracture at approximately 45° to the circumference of the wall, and tensile fracture along the radial direction of the wall. The fragments recovered from cylinders with different sized notches were sorted into four main categories based on fragment morphology and fracture mode (Fig. 5). These categories are: (1) (2) (3) (4)
Fig. 1. Schematics of test assembly. The specimens, a stacked set of cylinders made of steel, are filled with main explosive charge and a copper sleeve is placed before and after the array.
Tensile-mode fragment. Shear-mode fragment. Mix-mode fragment. End fragment.
As Fig. 5 shows, the tensile-mode fragments were the result of cracks, which formed and grew from the root of adjacent external notches or adjacent external and internal notches. The sizes of tensile-mode fragments consisted of two categories, large and small,
52
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
Fig. 2. Specimens of tests. The dE and wE are depth and width of the external notch, and the dI and wI are depth and width of the internal notch.
Table 1 Materials and dimensions of the notches.
a b
Cylinder materials
Explosive
External notch sizes (mm)
Internal notch sizes (mm)
AISI 1045 steela
Composition B
dE = 1.0, 2.0, 3.0, 4.0, 5.0, 6.0b xE = 1.0, 1.5, 2.0, 2.5, 3.0
dI = 2.0 xI = 1.0
Heat treatment: 850 °C quenching, 350 °C tempering. Machining accuracy: ±0.02 mm.
Table 2 Chemical constituents and mechanical properties of AISI 1045 steel (Wt%). Chemical compositions C Mn Si
0.47 0.6 0.21
S P Cr
Mechanical properties 0.025 0.03 0.22
Yield strength (MPa) Tensile strength (MPa) Elongation (%)
420 600 16
with the larger fragments approximately twice the size of the smaller ones. The shear-mode fragments were generally triangular or parallelogram-shaped in cross section, and were formed by shear fractures emanating from the internal notch to the external notch. The size of the parallelogram types were approximately twice the size of the triangular ones. The mix-mode fragments were formed by a shear fracture trace and a radial tensile fracture trace. There were some other fragments, called end fragments, which fractured longitudinally as a result of natural fragmentation mechanisms operating in the wall of the cylinders between adjacent notches. The size was generally random and the mass was smaller than the three main types. The end fragments resulted in mass reduction of the regular fragments from cylinders, thus decreasing the degree of fragmentation control. Fig. 6 expresses the categories of fragments as a percentage by mass of the total
recovery. Total recovery was used here rather than original cylinder mass to allow, to some extent, for anomalous recoveries. It was observed that shear-mode fragments were more pronounced in cylinders of low dE/t ratios, whereas tensile-mode fragments were in the highest flight at high dE/t ratios. This effect was the result of the location of the tensile fracture being lower in the notches in cylinders with deeper notches, together with the virtual elimination of natural fragmentation mechanisms. However, this effect was not so obvious for different notch widths (Fig. 7). 3.2. Fracture calculation After the fragments were retrieved, they were cleaned and measured individually. The fracture strain, calculated by the thicknesses of the measured fragments, could be determined by
ef ¼ ett ¼ lnðtf =ti Þ;
ð1Þ
where ef is critical fracture strain of the cylinder, ett is the logarithmic thickness strain, and ti and tf are the initial and final wall thicknesses of the cylinder, respectively. Assuming the radial expansion of the cylinder is plane strain, an effective plastic strain (eps) at failure can be defined as
Fig. 3. Sketch of the experimental set-up.
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
(a) Notch: ωE =1.0 dE =2.0
(b) Notch: ωE =2.0 dE =3.0
(c) Notch: ωE =1.0 dE =3.0
(d) Notch: ωE =1.0 dE =6.0
53
Fig. 4. Typical fragments recovered from the tests.
2 1=2 ep ¼ pffiffiffi ½e2t þ e2h þ et eh ; 3
ð2Þ
where eh is the logarithmic height strain. Note that when eh = 0, pffiffiffi ep ¼ ð2= 3Þef . The mean eps at fracture is 0.11 with a standard deviation of 0.02. The initial velocity of fragments can be calculated by the exponential equation
v 0 ¼ v x =eax ;
ð3Þ
where vx is fragment velocity at distance x, and was measured in the tests. a is a coefficient related to fragment shape and mass, defined as
1 3
a ¼ cd qa xmf1=3 ;
ð4Þ
where qa is the air density, and x, cd, and mf are the cross-sectional area, drag coefficient, and fragment mass, respectively. Gurney developed an equation for predicting maximum velocity, Vmax, of cylindrical shells subject to internal explosive loading. The Gurney velocity for a cylinder is given by
V max ¼
pffiffiffiffiffiffi 2EðM=C þ 1=2Þ1=2 ;
ð5Þ
where M/C is the ratio of the cylinder mass to the explosive mass. pffiffiffiffiffiffi The empirical Gurney constant 2E, determined by experiments involving a particular type of explosive, is related to the detonation energy of the explosive. The data of initial velocities and Gurney velocities are shown in Fig. 8. As the gas products vent first due
to notching, the velocities of the fragments from experimental data are less than those obtained from the Gurney equation, for obvious reasons. The fracture energy is estimated from the equation proposed by Grady [18] and used as follows
C¼
qf e_ 2f s3f 24
;
ð6Þ
where C is Grady’s fragmentation energy per unit area in forming a crack, qf is the material density, sf is the average distance between cracks corresponding to curvature, and e_ f is circumferential strain rate at the estimated fracture time. The distance between cracks s is obtained by the circumferential width of the fragments. Just prior to break up, the cylinder body is in circumferential tension and undergoing uniform circumferential stretching at the rate e_ , which is given by the ratio
e_ f ¼
de v 0 v0 ¼ ¼ ; dt rf r 0 eett
ð7Þ
where v0 is the initial fragment velocity, rf is the cylinder radius corresponding to the direction of the crack, and r0 is the initial cylinder radius. A material property called fragmentation toughness along with the relation of fragmentation energy and elastic modulus, was defined by Kipp and Grady [19] to characterize the material property of fragmentation. Fragmentation toughness is expressed as
Kf ¼
pffiffiffiffiffiffiffiffiffi 2EC;
ð8Þ
54
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
Fig. 7. Mass percentage of the three categories of fragments: xE = 1.0, 2.0, 3.0 mm, dE = 2.0 mm.
Fig. 5. The formation of three major categories observed: tensile-mode fragment, shear-mode fragment and mix-mode fragment.
Fig. 8. Initial velocities of the tests and Gurney velocities: xE = 2.0 mm, dE = 1.0, 2.0, 3.0, 4.0, 5.0, 6.0.
Table 3 Values of calculated fragmentation energy and fragmentation toughness based on Grady’s energy-based fragmentation model: xE = 2.0 mm.
Fig. 6. Mass percentage of the fragments: xE = 2.0 mm, dE = 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 mm.
where Kf is fragmentation toughness, and E is the elastic modulus. Fragmentation toughness values from testing are shown in Tables 3 and 4. The calculated Kf values are affected by the notches. However, fragmentation toughness is a constant. A modification factor k was suggested by Hiroe et al. [16] for this notch effect which is related as
Notch depth, dE (mm)
1.0
2.0
3.0
4.0
5.0
6.0
Ave. frag. energy (kJ m2) Frag. toughness (MPa m1/2)
84.9 130.3
114.4 151.3
120.5 155.2
133.8 163.6
170.4 184.6
191.4 239.6
Table 4 Values of calculated fragmentation energy and fragmentation toughness based on Grady’s energy-based fragmentation model: dE = 2.0 mm. Notch width, wE (mm)
1.0
1.5
2.0
2.5
3.0
Ave. frag. energy (kJ m2) Frag. toughness (MPa m1/2)
99.6 141.1
108.5 147.4
114.4 151.3
122.1 156.3
129.3 160.8
kðdE ; wE Þ ¼ f ðdE =tÞ gðwE =tÞ:
C¼
kqf e_ 2f s3f 24
:
ð9Þ
Further study is necessary to explore this modification factor. For our experimental results, the factor is related to only two parameters of the U-notch, namely, depth and width. The modification factor can be assumed as
ð10Þ
Based on the fragmentation energy results, the expressions can be obtained by linear curve fitting of
f ðdE =t; wE =t ¼ 0:2Þ ¼ 1 þ 3:1ðdE =tÞ ð0:1 dE =t 0:6Þ
ð11Þ
g ðwE =t; dE =t ¼ 0:2Þ ¼ 1 þ 1:7ðwE =tÞ ð0:1 wE =t 0:3Þ:
ð12Þ
55
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
Fig. 9. Simulation model of the tests and detailed view of the notch tip.
Table 5 J–C model parameters of cylinder material. Material
Tmelt (K)
A (MPa)
B (MPa)
n
c
m
AISI 1045
1792
496
320
0.28
0.064
0.804
However, these two expressions are dependent. The revised expressions are given as
f ðdE =t; wE =t ¼ 0:2Þ ð0:1 dE =t 0:6Þ g ðwE =t ¼ 0:2Þ g ðwE =t; dE =t ¼ 0:2Þ ð0:1 wE =t 0:3Þ: gðwE =tÞ ¼ f ðdE =t ¼ 0:2Þ
f ðdE =tÞ ¼
ð13Þ ð14Þ
f ðdE =t; wE =t ¼ 0:2Þ g ðwE =t; dE =t ¼ 0:2Þ g ðwE =t ¼ 0:2Þ f ðdE =t ¼ 0:2Þ 0:1 dE =t 0:6 ¼ 0:33ð1 þ 3:1dE =tÞð1 þ 1:7wE =tÞ 0:1 wE =t 0:3
kðdE ; wE Þ ¼
ð15Þ Substituting Eq. (15) into Eq. (9), we obtain the fragmentation energy expression for the U-notch cylinder as
ð1 þ 3:1dE =tÞð1 þ 1:7wE =tÞqf e_ 2f s3f
72
m T T room 1 ; T melt T room
e_ r ¼ ½A þ Bens 1 þ c ln _ s er
ð17Þ
where es is the plastic strain of the steel, e_ s is strain rate of the steel, e_ r is the reference strain rate, T is the temperature, A is the quasi-
Eqs. (10), (13) and (14) can be combined to obtain
C¼
The material model and mesh generation are the most important factors that affect the accuracy of the results from the numerical simulation. The average mesh size is 0.5 mm in specimens with a mesh size of 0.2 mm around the notches. The Johnson–Cook (J–C) model was applied to describe the deformation behaviors of the specimen. This material model is specifically used for metals and is appropriate for simulation of problems where strain rate varies greatly or when adiabatic temperature changes lead to material softening. The equation of this material model is given as
0:1 dE =t 0:6 0:1 wE =t 0:3
: ð16Þ
static yield stress, B and n are the strain hardening coefficients, c is the strain rate hardening coefficient, and m is the thermal softening coefficient. For a high explosive model, the Jones–Wilkins–Lee (JWL) method was used to describe the detonation of composition B. This equation defines the pressure field as a function of relative volume and internal energy per initial volume as
xqe R1 qqe0e xqe R2 qqe0e xqe e e p¼A 1 þB 1 þ E ; R1 qe0 R2 qe0 qe0 m0
ð18Þ
where p is the detonation pressure, qe is the explosive density, qe0 is the initial explosive density, and A, B, R1, R2 and x are material constants. The parameters in the Johnson–Cook equation and the explosive parameters are listed in Tables 5 and 6.
4. Numerical simulation and discussion
4.2. Numerical simulation
4.1. Finite element model
4.2.1. Cylinder fracture The dynamic fracture and fragmentation process of cylinders can be broken down into four main steps according to the numerical simulation (Fig. 10). First, the initial stress in the cylinders is caused by explosive shock waves formed upon detonation, causing rapid outward acceleration of the cylinder. In addition, stress concentrations appear at notch tips (Fig. 10a). Then, strain bands between adjacent internal and external notches are turned out along with cylinder expansion (Fig. 10b). Afterwards, cracks
Numerical simulations of the expanding behavior were performed for all the experiments using the Arbitrary-Lagrange–Euler (ALE) method with LS-DYNA software. LS-DYNA [20], a fully integrated analysis program, is well suited for analysis of non-linear dynamics problems. As the cylinder was symmetrical in structure, a quarter of the specimen was simulated. The finite element model configuration and a typical meshed notch are illustrated in Fig. 9. Table 6 JWL model parameters of explosive. Explosive
Density (kg/m3)
Detonation velocity (m/s)
Chapman–Jouget pressure, PCJ (1011 Pa)
A
B
R1
R2
x
Em0 (kJ/m3)
Composition B
1700
8000
0.34
5.81
0.068
4.1
1.0
0.35
0.09
56
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
(a) First step
(b) Second step
(c) Third step
(d) Last step
Fig. 10. Dynamic fracture and fragmentation process of notched cylinders.
form and grow from the roots of notches, and begin to propagate through the cylinder wall after a very short time (Fig. 10c). Finally, gas products begin to vent as cracks propagate completely through the wall due to very high internal pressure that accelerates the wall outward. Stress and strain in the cylinder wall are released, and the cracks are no longer increasing (Fig. 10d). The study by Li et al. [21] on fracture traces of blast loading cylinders with V-notches showed that fracture traces were various. The main fracture types were tensile fractures along the radial direction and shear fractures following the trace between the roots of adjacent notches. This similar fracture pattern was exhibited in U-notched cylinders. As Fig. 11 shows, numerical simulation results coincided with experimental results. 4.2.2. Effect of stress concentration The resulting distributions of effective stress (see Fig. 12) are non-uniform, and stress concentration is noticeable at the notch tip. With regard to the crack case, a relationship between the SCF and the notch geometry should be considered. The static SCF is usually presented as
Ks ¼
rmax r0
ð19Þ
where Ks is the static SCF, rmax is the maximum stress at the discontinuity, and r0 is the average stress of the same section. The
Fig. 11. Fracture trace of U-notched cylinder. Typical fragments in the simulation are in good agreement with experimental results.
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
(a) Internal notch
57
(b) External notch Fig. 12. Effective stress distributions around the notch tip.
the shear fracture was pronounced in the low notch depth condition, while tensile fracture played a leading role in higher notch depth. Moreover, mass percentage distribution of the fragments appeared more affected by the notch depth than width. The mass and size of recovered fragments were measured, and the initial velocities were obtained from experiments. Then, the fragmentation energy and fragmentation toughness based on Grady’s energy-based theory were discussed. The notch effect was obvious based on the calculated results, and a revised fragmentation model was suggested to account for the notch effect. Numerical simulations were performed using the LS-DYNA software. The fracture trace of the simulation results were in good agreement with experimental results. In particular, the stress concentration was noticeable at the notch tip, and the dynamic SCF of the U-notch was less than that of the V-notch. Acknowledgements Fig. 13. The evolution history of the stress concentration: final average internal I E dynamic SCF kd = 2.6, and final average external dynamic SCF kd = 2.3.
average stress is easily determined from the applied load for static situations; however, the average load is much more difficult to obtain for blast loading conditions. Therefore, the average stress is based on the nominal section without a discontinuity, and the dynamic SCF is determined by the following expression [22]
K d ðtÞ ¼
rmax ðtÞ rnom ðtÞ
ð20Þ
where Kd is the dynamic SCF, rmax is the maximum stress as a function of time, and rnom is the nominal stress as a function of time. The evolutionary history of the stress concentration is illustrated in Fig. 13. Comparing to literature data [23,24], the dynamic SCF of the U-notch is less than that of the V-notch. 5. Conclusions Notched cylinder specimens were expanded explosively to fragmentation by varying U-notch dimensions. The fragments were collected to determine dynamic fracture and fragmentation characteristics. Fragments were sorted into four categories according to fragment morphology and fracture mode. It was found that
The authors wish to acknowledge financial supported by the China National Natural Science Funding (11202237, 11132012), and the State Key Laboratory of Explosion Science and Technology Opening Fund (KFJJ12-16MM). References [1] M. Singh, H. Suneja, M. Bola, S. Prakash, Dynamic tensile deformation and fracture of metal cylinders at high strain rates, Int. J. Impact Eng 27 (2002) 939–954. [2] D. Goto, R. Becker, T. Orzechowski, H. Springer, A. Sunwoo, C. Syn, Investigation of the fracture and fragmentation of explosively driven rings and cylinders, Int. J. Impact Eng 35 (2008) 1547–1556. [3] M.Z. Liang, X.Y. Li, J.G. Qin, F.Y. Lu, Improved expanding ring technique for determining dynamic material properties, Rev. Sci. Instrum. 84 (2013). [4] R.W. Gurney, The initial velocities of fragments from bombs shell and grenades, BRL Report No. 405, Army Ballistic Research Laboratory, MD, USA, 1943. [5] N.F. Mott, Fragmentation of shell cases, NATO ASI Ser., Ser. C 189 (1947) 300– 308. [6] G.I. Taylor, The fragmentation of tubular bombs, in: Scientific Papers of G.I. Taylor, Cambridge University Press, London, 1963, pp. 387–390. [7] A. Ivanov, Explosive deformation and destruction of tubes, Strength Mater. 8 (1976) 1303–1306. [8] C.R. Crowe, W.H. Holt, W. Mock Jr, O.H. Griffin, Dynamic fracture and fragmentation of cylinders, TR Report No. 3449, Naval Surface Weapons Center Dahlgren Laboratory, MD, USA, 1976. [9] C.R. Hoggatt, R.F. Recht, Fracture behavior of tubular bombs, J. Appl. Phys. 39 (1968) 1856–1862. [10] D. Grady, D. Benson, Fragmentation of metal rings by electromagnetic loading, Exp. Mech. 23 (1983) 393–400.
58
M. Liang et al. / Theoretical and Applied Fracture Mechanics 77 (2015) 50–58
[11] P. Elek, S. Jaramaz, Modeling of fragmentation of rapidly expanding cylinders, Theoret. Appl. Mech. 32 (2005) 113–130. [12] V. Philipchuk, Fragmentation tests of notched-ring bombs, Report No. 1138, Aberdeen Proving Ground, MD, USA, 1953. [13] J. Pearson, The shear control method of warhead fragmentation, in: 4th International Symposium on Ballistics Monterey, CA, USA, 1978, pp. 17–19. [14] I.R. Lamborn, The fracture of small internally detonated cylinders partly prefragmented by notching, MRL Report No. 545, 1973. [15] D.S. Saunders, The effects of C/M ratio on the controlled fragmentation of cylinders using external notching, MRL Report No. 800, Defence Science and Technology Organisation Materials Research Laboratories, Australia, 1980. [16] T. Hiroe, K. Fujiwara, H. Hata, H. Takahashi, Deformation and fragmentation behaviour of exploded metal cylinders and the effects of wall materials, configuration, explosive energy and initiated locations, Int. J. Impact Eng 35 (2008) 1578–1586. [17] X. Li, M. Liang, M. Wang, G. Lu, F. Lu, Experimental and numerical investigations on the dynamic fracture of a cylindrical shell with grooves subjected to internal explosive loading, Propellants, Explos., Pyrotech. 35 (2014) 1–10.
[18] D. Grady, Fragmentation of Rings and Shells: The Legacy of NF Mott, Springer, 2007. [19] M.E. Kipp, D.E. Grady, Dynamic fracture growth and interaction in one dimension, J. Mech. Phys. Solids 33 (1985) 399–415. [20] J.O. Hallquist, LS-DYNA Theoretical Manual, Livermore Software Technology Corporation, California, 1998. [21] X.Y. Li, M.Z. Liang, M.F. Wang, G.X. Lu, F.Y. Lu, Experimental and numerical investigations on the dynamic fracture of a cylindrical shell with grooves subjected to internal explosive loading, Propellants, Explos., Pyrotech. 35 (2014) 1–10. [22] W. Altenhof, N. Zamani, W. North, B. Arnold, Dynamic stress concentrations for an axially loaded strut at discontinuities due to an elliptical hole or double circular notches, Int. J. Impact Eng 30 (2004) 255–274. [23] R. Afshar, F. Berto, Stress concentration factors of periodic notches determined from the strain energy density, Theoret. Appl. Fract. Mech. 56 (2011) 127–139. [24] P. Lazzarin, R. Afshar, F. Berto, Notch stress intensity factors of flat plates with periodic sharp notches by using the strain energy density, Theoret. Appl. Fract. Mech. 60 (2012) 38–50.